Long-time dynamics and semi-wave of a delayed nonlocal epidemic model with free boundaries

Abstract

This paper is concerned with a nonlocal reaction–diffusion system with double free boundaries and two time delays. The free boundary problem describes the evolution of faecally–orally transmitted diseases. We first show the well-posedness of global solution, and then establish the monotonicity and asymptotic property of basic reproduction number for the epidemic model without delays, which is defined by spectral radius of the next infection operator. By introducing the generalized principal eigenvalue defined in general domain, we obtain an upper bound of the limit value of basic reproduction number. We discuss the spreading and vanishing phenomena in terms of the basic production number. By employing the perturbed approximation method and monotone iteration method, we establish the existence, uniqueness and monotonicity of solution to semi-wave problem. When spreading occurs, we determine the asymptotic spreading speeds of free boundaries by constructing suitable upper and lower solutions from the semi-wave solutions. Moreover, spreading speeds for partially degenerate diffusion case are provided in a similar way.

1. Introduction

In this paper, to study the evolution of faecally–orally transmitted diseases, such as hand, foot and mouth diseases, cholera and so on, we consider the following nonlocal reaction–diffusion system with double free boundaries and two delays:

(1.1)\begin{equation} \left\{\begin{array}{@{}l} u_{t}=d_{1}u_{xx}-a_{1}u+h\Bigg(\displaystyle\int_{-\infty}^{+\infty}J_1(x-y)v(t-\tau_1,y){\rm d}y\Bigg),\\ \quad\quad t>0, s_1(t)< x< s_2(t),\\ v_{t}=d_{2}v_{xx}-a_{2}v+g\Bigg(\int_{-\infty}^{+\infty}J_2(x-y)u(t-\tau_2,y){\rm d}y\Bigg),\\ \quad\quad t>0, s_1(t)< x< s_2(t),\\ u(t,x)=v(t,x)=0,\quad t>0,\ x\leq s_{1}(t)~\mbox{or}~x\geq s_{2}(t),\\ s_{1}^\prime(t)={-}\mu [u_x(t,s_1(t))+\rho v_x(t,s_1(t))],\quad t>0,\\ s_{2}^\prime(t)={-}\mu [u_x(t,s_2(t))+\rho v_x(t,s_2(t))],\quad t>0,\\ s_{1}(0)={-}s_0,~s_{2}(0)=s_0,\\ u(\theta,x)=u_0(\theta,x),\quad-\,\tau_2\leq\theta\leq0,~s_1(\theta)\leq x\leq s_2(\theta),\\ v(\theta,x)=v_0(\theta,x),\quad-\,\tau_1\leq\theta\leq0,~s_1(\theta)\leq x\leq s_2(\theta), \end{array}\right. \end{equation}

where $u(t,\,x)$ and $v(t,\,x)$ represent the density of bacteria in the environment and infective human population, respectively; $d_1$ and $d_2$ are the diffusion coefficients; $a_1$ and $a_2$ are the natural death rate of the bacteria and the fatality rate of the infective human population, respectively; the nonlocal term $h(\int _{-\infty }^{+\infty }J_1(x-y)v(t-\tau _1,\,y){\rm d}y)$ is the contribution of the infective human population in a neighbourhood of $x$ to the density of bacteria, $g(\int _{-\infty }^{+\infty }J_2(x-y)u(t-\tau _2,\,y){\rm d}y)$ gives the ‘force of infection’ on human due to the concentration of bacteria, $J_{1}$ and $J_{2}$ are transfer kernels, $\tau _1$ and $\tau _2$ describe the delays-in-time of positive feedback interaction between the bacteria and infective human; $(s_1(t),\,s_2(t))$ is the infected area at time $t$, and its boundary fronts $s_1(t)$ and $s_2(t)$, depending on time $t$, are called free boundaries. We assume that the expanding rate of the infected area is proportional to a linear combination of the spatial gradients of bacteria and infective human population at the fronts, i.e., $s_1(t)$ and $s_2(t)$ satisfy the Stefan conditions. All the parameters are positive constants. Since the infected area may vary over time during the evolution of faecally–orally transmitted diseases, the fixed boundary problem is not suitable to be applied to understand how the bacteria spread spatially to larger area from the initial infected area, which motivates us to consider the free boundary problem (1.1).

The epidemic model in problem (1.1)

(1.2)\begin{equation} \left\{\begin{array}{@{}l} u_{t}=d_{1}u_{xx}-a_{1}u+h\Bigg(\int_{-\infty}^{+\infty}J_1(x-y)v(t-\tau_1,y){\rm d}y\Bigg),\\ v_{t}=d_{2}v_{xx}-a_{2}v+g\Bigg(\int_{-\infty}^{+\infty}J_2(x-y)u(t-\tau_2,y){\rm d}y\Bigg) \end{array}\right. \end{equation}

was studied in [38]. The authors investigated the global attractivity of the equilibria, the spreading speed of a general system without quasi-monotone conditions, and travelling wave solutions for (1.2) in whole space. System (1.2) is a generalization of the epidemic models proposed by Capasso-Maddalena [8, 9] (without delay), Thieme-Zhao [27] (with a time delay) and Wu-Hsu [37] (with two time delays). A basic feature in these models is the positive feedback interaction between the infective human and the bacteria in the environment.

Some simplified forms of (1.1) without time delays, including the partially degenerate diffusion case ($d_{2}=0$) [1, 19, 40] and non-degenerate diffusion case ($d_{2}>0$) [30], have been recently studied. Moreover, the corresponding nonlocal diffusion models were considered in [17, 31, 32, 41]. The authors established the spreading–vanishing dichotomy, discussed the influence of different parameters on the spreading and vanishing, and determined the asymptotic spreading speeds of the free boundaries. These results are extensions of the work of Du and Lin [15], in which they proposed a free boundary problem for homogeneous logistic equation to model the species invasion. Except the above-mentioned works, the results in [15] have also been extended to other population models and epidemic models, for example, time-periodic case [13, 28, 29], nonlocal case [6, 7, 14, 18] and general nonlinearities case [16].

We mention in particular that, based on [15], free boundary problems for time-delayed biological models have also been studied in very recent years, but still quite few. To model the biological invasion of an age-structured species, Sun and Fang [24] first derived a local free boundary problem for Fisher-KPP equation with time delay. Tang et al. [25] subsequently extended some results of [24] to a two-species weak competition model with time delays. By considering the diffusion rate of the immature population, Du et al. [12] further derived a nonlocal free boundary problem with time delay. For the epidemic model (1.2) with $J_{1}=J_{2}=\delta$ (Dirac delta function), the corresponding free boundary problems with a time delay ($\tau _1=d_{2}=0, \tau _{2}>0$) and two time delays ($\tau _{1}, \tau _{2}, d_{2}>0$) were also considered in [10, 11], respectively.

The purpose of this paper is to establish the long-time dynamical behaviours of (1.1), and determine the asymptotic spreading speeds when spreading happens. Throughout this paper, we define

\[ [a, b]\times[s_1, s_2]:= \Bigg\{(t, x):~t\in[a, b],~x\in [s_1(t), s_2(t)]\Bigg\}. \]

The sets $(a,\,b]\times [s_1,\,s_2]$, $(a,\,b)\times (s_1,\,s_2)$, etc., are defined similarly. We always assume that the initial functions in (1.1) satisfy

(1.3)\begin{equation} \left\{\begin{array}{@{}l} u_0(\theta,x)\in C^{1,2}([-\tau_{2}, 0]\times[s_1, s_2]),~ v_0(\theta,x) \in C^{1,2}([-\tau_{1}, 0]\times[s_1, s_2]),\\ u_0(\theta,x) \left\{\begin{array}{@{}l} >0\quad\mbox{for}~\theta\in[-\tau_{2}, 0],~x\in (s_1(\theta), s_2(\theta)),\\ \equiv0\quad\mbox{for}~\theta\in[-\tau_{2}, 0],~x\not\in (s_1(\theta), s_2(\theta)), \end{array}\right.\\ v_0(\theta,x) \left\{\begin{array}{@{}l} >0\quad\mbox{for}~\theta\in[-\tau_{1}, 0],~x\in (s_1(\theta), s_2(\theta)),\\ \equiv0\quad\mbox{for}~\theta\in[-\tau_{1}, 0],~x\not\in (s_1(\theta), s_2(\theta)), \end{array}\right. \end{array}\right. \end{equation}

as well as the compatible condition

(1.4)\begin{equation} [s_1(\theta), s_2(\theta)]\subset[{-}s_0, s_0]\quad\mbox{for}~\theta\in[-\max\{\tau_{1},\tau_{2}\}, 0]. \end{equation}

The kernel functions $J_i(\cdot )$ $(i=1,\,2)$ and nonlinearities $g(\cdot )$, $h(\cdot )$ satisfy the following assumptions:

$(\textbf {J})$: $J_i\in C(\mathbb {R})$, $J_{i}(0)>0$, $J_i(-x)=J_i(x)\geq 0$ for $x\in \mathbb {R}$, $\int _{-\infty }^{+\infty }J_i(y){\rm d}y=1$, and $\int _{-\infty }^{+\infty }J_i(y)e^{-\lambda y}{\rm d}y<+\infty$ for any $\lambda >0$;

$(\textbf {H})$: $h\in C^2([0,\,+\infty ))$, $g\in (C^2\cap L^\infty )([0,\,+\infty ))$, $h(0)=g(0)=0$, and $h^\prime (z)$, $g^\prime (z)>0$ for any $z\in [0,\,+\infty )$, $h^{\prime \prime }(z)\leq 0$, $g^{\prime \prime }(z)<0$ for all $z>0$.

A typical example is $J_{i}(x)=\frac {1}{\sqrt {4\pi \varrho }}e^{-\frac {x^2}{4\varrho }}$, $h(x)=ax$ and $g(x)=\frac {px}{1+qx}$ with some $\varrho, a, p, q>0$.

For (1.1), the nonlocal reaction terms and time delays cause several difficulties which require different treatment from earlier works. Firstly, for our nonlocal epidemic model without delays, the basic reproduction number has no explicit expression as in [22], and its monotonicity and asymptotic property with respect to the domain are not easy to obtain. We define the basic reproduction number by spectral radius of the next infection operator, and pay much effort to establish its monotonicity and asymptotic property, especially provide an upper bound of the limit value by introducing the generalized principal eigenvalue defined in general domain. Secondly, to overcome the effects of nonlocal terms on spreading and vanishing, we need to construct the upper and lower solutions from the principal eigenfunctions of perturbed nonlocal eigenvalue problems, instead of the unperturbed ones as in [11, 30]. Thirdly, the delayed nonlocal semi-wave problem we considered is different from the previous works. It is difficult to get the critical value $c_{\tau }^*$ of speed $c$ for semi-wave by discussing the distribution of real roots of transcendental equation as in [10, 11, 24]. Motivated by the works [14, 17] for nonlocal diffusion models, we first study the corresponding perturbed semi-wave problem, and apply the iteration monotone method to cope with the existence and monotonicity of perturbed semi-wave solution. Then we build a dichotomy between monotone travelling wave and monotone semi-wave, which ensures that the critical values of their speeds are equal. Finally, we determine the spreading speeds for partially degenerate diffusion case without delays, which was considered in [19]. The upper bounds of spreading speeds were provided in [19], but their precise values are still unknown due to the effect of nonlocal term. We give a complete answer to the problem in this paper.

Let us now describe the results of this paper more precisely.

For the following epidemic model without delays

(1.5)\begin{equation} \left\{\begin{array}{@{}l} \phi_{t}=d_1\phi_{xx}-a_1\phi +h^\prime(0)\int_{{-}l}^{l}J_{1}(x-y)\varphi(t,y){\rm d}y,~t>0,~x\in({-}l,l),\\ \varphi_{t}=d_2\varphi_{xx}-a_2\varphi +g^\prime(0)\int_{{-}l}^{l}J_{2}(x-y)\phi(t,y){\rm d}y,~t>0,~x\in({-}l,l),\\ (\phi(t,\pm l),\varphi(t,\pm l))=(0,0),~t>0,\\ (\phi(0,x),\varphi(0,x))=(\phi_{0}(x),\varphi_{0}(x)),~x\in[{-}l,l], \end{array}\right. \end{equation}

we define the basic reproduction number $\mathcal {R}_{0}^{l}$ by spectral radius of the next infection operator.

Theorem 1.1 Basic reproduction number

$(i)$ $\mathcal {R}_{0}^{l}-1$ has the same sign as $\lambda _{1}$, where $\lambda _{1}$ is the principle eigenvalue of the following eigenvalue problem

(1.6)\begin{equation} \left\{\begin{array}{@{}l} d_1\phi_{xx}-a_1\phi +h^\prime(0)\int_{{-}l}^{l}J_{1}(x-y)\varphi(y){\rm d}y =\lambda\phi,~x\in({-}l,l),\\ d_2\varphi_{xx}-a_2\varphi +g^\prime(0)\int_{{-}l}^{l}J_{2}(x-y)\phi(y){\rm d}y =\lambda\varphi,~x\in({-}l,l),\\ (\phi({\pm} l),\varphi({\pm} l))=(0,0). \end{array}\right. \end{equation}

$(ii)$ $\mathcal {R}_{0}^{l}=\frac {1}{\mu _{0}^{l}}$, where $\mu _{0}^{l}$ is the unique positive principle eigenvalue of the following eigenvalue problem

(1.7)\begin{equation} \left\{\begin{array}{@{}l} -d_1\phi_{xx}+a_1\phi =\mu h^\prime(0)\int_{{-}l}^{l}J_{1}(x-y)\varphi(y){\rm d}y,~x\in({-}l,l),\\ -d_2\varphi_{xx}+a_2\varphi =\mu g^\prime(0)\int_{{-}l}^{l}J_{2}(x-y)\phi(y){\rm d}y,~x\in({-}l,l),\\ (\phi({\pm} l),\varphi({\pm} l))=(0,0) \end{array}\right. \end{equation}

with a positive eigenfunction $\Phi _{\mu _{0}}^{l}(x)=(\phi _{\mu _{0}}^{l}(x),\,\varphi _{\mu _{0}}^{l}(x))$.

$(iii)$ $\mathcal {R}_{0}^{l}$ is increasing in $l>0$, and

\[ \begin{array}{l} \mathcal{R}_{0}^{l}\rightarrow \mathcal{R}^{*} \leq\mathcal{R}_0:=\sqrt{\dfrac{h^\prime(0)g^\prime(0)}{a_1a_2}} \end{array} \]

as $l\rightarrow +\infty$, where $\mathcal {R}_0$ is the basic reproduction number of the corresponding ordinary differential equations.

Denote $s_{i,\infty }=\lim _{t\rightarrow +\infty }s_i(t)$ for $i=1,\,2$. We call that the bacteria are spreading if $s_{2,\infty }-s_{1,\infty }=+\infty$ and $\limsup _{t\rightarrow +\infty }(\|u(t,\,\cdot )\|_{C([s_1(t), s_2(t)])}+\|v(t,\,\cdot )\|_{C([s_1(t), s_2(t)])})>0$, and the bacteria are vanishing if $s_{2,\infty }-s_{1,\infty }<+\infty$ and $\lim _{t\rightarrow +\infty }(\|u(t,\,\cdot )\|_{C([s_1(t), s_2(t)])}+\|v(t,\,\cdot )\|_{C([s_1(t), s_2(t)])})=0$. In terms of the basic reproduction number, we can discuss the spreading and vanishing phenomena.

Theorem 1.2 Spreading and vanishing

$(i)$ If $0<\mathcal {R}_0\leq 1$, then the solution of (1.1) satisfies $\lim _{t\rightarrow +\infty }(\|u(t,\,\cdot )\|_{C([s_1(t),s_2(t)])} +\|v(t,\,\cdot )\|_{C([s_1(t),s_2(t)])})=0$.

$(ii)$ Assume that $\mathcal {R}^{*}>1$. Then there exists $\mu ^*\in [0,\,+\infty )$ such that $s_{2,\infty }-s_{1,\infty }=+\infty$ for $\mu >\mu ^*$, and $s_{2,\infty }-s_{1,\infty }<+\infty$ for $0<\mu \leq \mu ^*$.

Assume that $\mathcal {R}^{*}>1$, we consider the delayed nonlocal semi-wave problem

(1.8)\begin{equation} \left\{\begin{array}{@{}l} c\phi^\prime(\xi)=d_1\phi^{\prime\prime}(\xi)-a_1\phi(\xi)+h\Bigg(\int_{-\infty}^{+\infty} J_1(y)\varphi(\xi-y-c\tau_1){\rm d}y\Bigg),~\xi>0,\\ c\varphi^\prime(\xi)=d_2\varphi^{\prime\prime}(\xi)-a_2\varphi(\xi)+g\Bigg(\int_{-\infty}^{+\infty} J_2(y)\phi(\xi-y-c\tau_2){\rm d}y\Bigg),~\xi>0,\\ (\phi(\xi), \varphi(\xi))=(0,0),~\xi\leq0,\\ (\phi(+\infty),\varphi(+\infty))=(u^*,v^*), \end{array}\right. \end{equation}

where $(u^*, v^*)$ is the unique positive equilibrium of the equations, which is guaranteed by the condition $\mathcal {R}_{0}\geq \mathcal {R}^{*}>1$. By employing the perturbed approximation method and monotone iteration method, we can establish the (non)existence of semi-wave solution to (1.8). The critical value of speed $c$ for semi-wave is $c^*_{\tau }$.

Theorem 1.3 Semi-wave solution

The semi-wave problem (1.8) admits an increasing solution for $0< c< c^*_{\tau }$, but has no increasing solution for $c\geq c^*_{\tau }$.

For any fixed $\mu,\,\rho >0$, it is shown that there exists a unique $c_{\tau }=c^{\mu,\rho }_{\tau }\in (0,\,c^*_{\tau })$ such that $\mu [(\phi _{\tau }^{c_{\tau }})^\prime _+(0)+\rho (\varphi _{\tau }^{c_{\tau }})^\prime _+(0)]=c_{\tau }$, where $(\phi _{\tau }^{c_{\tau }}, \varphi _{\tau }^{c_{\tau }})$ is the semi-wave solution of (1.8) with $c=c_{\tau }$. By constructing suitable upper and lower solutions from the semi-wave, we can determine the asymptotic spreading speeds of free boundaries when spreading happens.

Theorem 1.4 Spreading speed

Assume that $\mathcal {R}^{*}>1$. If spreading happens to (1.1), then $-\lim _{t\rightarrow +\infty }\frac {s_1(t)}{t}=\lim _{t\rightarrow +\infty }\frac {s_2(t)}{t}=c_{\tau }$.

In a similar way, we can provide the spreading speeds for partially degenerate diffusion case in [19]. More details are provided in § 6.

The rest of this paper is organized as follows. In § 2, we first establish the well-posedness of the solutions to (1.1) and two comparison principles, and then give the proof of theorem 1.1 related to the basic reproduction number. In § 3, we discuss the spreading and vanishing. The existence and monotonicity of solutions to a delayed nonlocal semi-wave problem are investigated in § 4. The spreading speeds of free boundaries in (1.1) and partially degenerate diffusion case are determined in §5 and 6, respectively. The results of generalized principal eigenvalue are presented in Appendix.

2. Preliminaries

In this section, we first exhibit the well-posedness and comparison principles for the free boundary problem (1.1), and then discuss the basic reproduction number of (1.5).

2.1 Well-posedness

Theorem 2.1 $(i)$ For any $\gamma \in (0,\,1)$, there exists a $T>0$ such that problem (1.1) with the initial date $(u_0(\theta,\,x),\,v_0(\theta,\,x);s_1(\theta ),\,s_2(\theta ))$ satisfying (1.3) and (1.4), admits a unique solution $(u(t,\,x),\,v(t,\,x);s_1(t),\,s_2(t))$ with $u,\,v\in C^{\frac {1+\gamma }{2},\,1+\gamma }(D_T)$, $s_1,\,s_2\in C^{1+\frac {\gamma }{2}}([0,\,T])$, where $D_T=[0,\,T]\times [s_1,\,s_2]$.

$(ii)$ For the local solution $(u,\,v;s_1,\,s_2)$ obtained in $(i)$, there exist positive constants $M_{1}, M_{2}$ and $M_{3}$ independent of $T$, such that $0< u(t,\,x)\leq M_{1}$, $0< v(t,\,x)\leq M_{2}$ and $0<-s_1^{\prime }(t), s_2^{\prime }(t)\leq M_{3}$ for any $0< t\leq T$ and $s_1(t)< x< s_2(t)$.

$(iii)$ The solution $(u,\,v;s_1,\,s_2)$ of (1.1) exists and is unique for all $t\in (0,\,+\infty )$.

Proof. We only prove that $u(t,\,x)\leq M_{1}$ and $v(t,\,x)\leq M_{2}$ in $(ii)$, since the remaining part can be obtained by similar arguments as in the proof of Theorems 2.4–2.5 in [12] and Theorem 2.1 in [30].

For any $z>0$, by Taylor's formula and the concavity of $h$, we have $h(z)=h(z)-h(0)=h^{\prime }(0)z+\frac {1}{2}h^{\prime \prime }(\xi )z^{2}\leq h^{\prime }(0)z$ for some $\xi \in (0,\,z)$, which implies $\frac {h(z)}{z}\leq h^{\prime }(0)$. Since $g$ is bounded, we can choose $M_{i}$ $(i=1,\,2)$ sufficiently large such that

\[ M_{2}\geq\frac{\|g\|_{L^{\infty}}}{a_{2}},\quad M_{1}\geq h^{\prime}(0)\frac{M_{2}}{a_{1}}. \]

It follows that $\frac {g(M_{1})}{M_{2}}\leq \frac {\|g\|_{L^{\infty }}}{M_{2}} \leq a_{2}$, $\frac {h(M_{2})}{M_{1}}=\frac {h(M_{2})}{M_{2}}\cdot \frac {M_{2}}{M_{1}}\leq h^{\prime }(0)\frac {M_{2}}{M_{1}}\leq a_{1}$. We may assume that

\[ \begin{array}{l} u_{0}(\theta,x)\leq M_{1} \quad \mbox{for}~(\theta,x)\in[-\tau_{2},0]\times[{-}s_{0},s_{0}],\\ v_{0}(\theta,x)\leq M_{2} \quad \mbox{for}~(\theta,x)\in[-\tau_{1},0]\times[{-}s_{0},s_{0}]. \end{array} \]

Define $(U(t,\,x),\,V(t,\,x)):=e^{-kt}(M_{1}-u(t,\,x),\,M_{2}-v(t,\,x))$, where $k>0$ is a constant to be determined. Then, for $t>0$ and $s_1(t)< x< s_2(t)$, $(U,\,V)$ satisfies

(2.1)\begin{align} U_{t}& =d_{1}U_{xx} -(a_{1}+k)U+e^{{-}kt}\Bigg[a_{1}M_{1}-h\Bigg(\int_{-\infty}^{+\infty}J_{1}(x-y)v(t-\tau_{1},y){\rm d}y\Bigg)\Bigg]\nonumber\\ & =d_{1}U_{xx}-(a_{1}+k)U+e^{{-}kt}(a_{1}M_{1}-h(M_{2}))\nonumber\\ & \quad +e^{{-}k\tau_{1}}h^{\prime}(\xi)\int_{-\infty}^{+\infty}J_{1}(x-y)V(t-\tau_{1},y){\rm d}y\nonumber\\ & \geq d_{1}U_{xx}-(a_{1}+k)U +e^{{-}k\tau_{1}}h^{\prime}(\xi)\int_{-\infty}^{+\infty}J_{1}(x-y)V(t-\tau_{1},y){\rm d}y,\nonumber\\ V_{t} & =d_{2}V_{xx} -(a_{2}+k)V+e^{{-}kt}\Bigg[a_{2}M_{2}-g\Bigg(\int_{-\infty}^{+\infty}J_{2}(x-y)u(t-\tau_{2},y){\rm d}y\Bigg)\Bigg]\nonumber\\ & =d_{2}V_{xx}-(a_{2}+k)V+e^{{-}kt}(a_{2}M_{2}-g(M_{1}))\nonumber\\ & \quad +e^{{-}k\tau_{2}}g^{\prime}(\eta)\int_{-\infty}^{+\infty}J_{2}(x-y)U(t-\tau_{2},y){\rm d}y\nonumber\\ & \geq d_{2}V_{xx}-(a_{2}+k)V+e^{{-}k\tau_{2}}g^{\prime}(\eta)\int_{-\infty}^{+\infty}J_{2}(x-y)U(t-\tau_{2},y){\rm d}y, \end{align}

where $\xi$ lies between $M_{2}$ and $\int _{-\infty }^{+\infty }J_{1}(x-y)v(t-\tau _{1},\,y){\rm d}y$, $\eta$ lies between $M_{1}$ and $\int _{-\infty }^{+\infty }J_{2}(x-y)u(t-\tau _{2},\,y){\rm d}y$.

We claim that $U(t,\,x),\,V(t,\,x)\geq 0$ in $(0,\,+\infty )\times (s_{1},\,s_{2})$. Assume by contraction that there exist some $T_0>0$ and $(t_{0},\,x_{0})\in (0,\,T_0]\times (s_1,\,s_2)$ such that

\[ \min\{U(t_0,x_0),V(t_0,x_0)\} =\min_{(t,x)\in[0,T_0]\times[s_{1},s_{2}]}\min\{U(t,x),V(t,x)\}<0. \]

If $U(t_0,\,x_0)=\min \{U(t_0,\,x_0),\,V(t_0,\,x_0)\}<0$, then $U_{t}(t_0,\,x_0)\leq 0$ and $U_{xx}(t_0,\,x_0)\geq 0$. On the other hand, if $t_{0}\leq \tau _{1}$, then $V(t_{0}-\tau _{1},\,x)\geq 0>U(t_0,\,x_0)$ for any $x\in \mathbb {R}$; if $t_{0}>\tau _1$, then $V(t_{0}-\tau _{1},\,x)\geq U(t_0,\,x_0)$ for $x\in [s_{1}(t_{0}-\tau _{1}), s_{2}(t_{0}-\tau _{1})]$ and $V(t_{0}-\tau _{1},\,x)\geq 0>U(t_0,\,x_0)$ for $x\in \mathbb {R}\setminus [s_{1}(t_{0}-\tau _{1}), s_{2}(t_{0}-\tau _{1})]$. Thus, $V(t_{0}-\tau _{1},\,x)\geq U(t_0,\,x_0)$ holds for all $x\in \mathbb {R}$. It follows that

\begin{align*} & -(a_{1}+k)U(t_0,x_0)+e^{{-}k\tau_{1}}h^{\prime}(\xi)\int_{-\infty}^{+\infty}J_{1}(x_{0}-y)V(t_{0}-\tau_{1},y){\rm d}y\\ & \quad\geq{-}(a_{1}+k)U(t_0,x_0)+e^{{-}k\tau_{1}}h^{\prime}(\xi)\int_{-\infty}^{+\infty}J_{1}(x_{0}-y)U(t_0,x_0){\rm d}y\\ & \quad=({-}a_{1}-k+e^{{-}k\tau_{1}}h^{\prime}(\xi))U(t_0,x_0)\geq ({-}a_{1}-k+h^{\prime}(\xi))U(t_0,x_0). \end{align*}

Choose

\[ k=\max\Bigg\{\|h^{\prime}\|_{L^{\infty}([0,\max\{M_{2},K_{2}\}])}, \|g^{\prime}\|_{L^{\infty}([0,\max\{M_{1},K_{1}\}])}\Bigg\} \]

with $K_1=\|u\|_{L^{\infty }([-\tau _{2},\,T_{0}-\tau _{2}]\times [s_{1},\,s_{2}])}$ and $K_2=\|v\|_{L^{\infty }([-\tau _{1},\,T_{0}-\tau _{1}]\times [s_{1},\,s_{2}])}$. Thus,

\[ -(a_{1}+k)U(t_0,x_0)+e^{{-}k\tau_{1}}h^{\prime}(\xi)\int_{-\infty}^{+\infty}J_{1}(x_{0}-y)V(t_{0}-\tau_{1},y){\rm d}y>0, \]

which contradicts with the first equation in (2.1). If $V(t_0,\,x_0)=\min \{U(t_0,\,x_0), V(t_0,\,x_0)\} <0$, we can prove the claim in a similar way. This completes the proof.

We introduce two comparison principles for the free boundary problem (1.1), which can be proved similarly as the proof of Lemma 2.5 in [1].

Lemma 2.2 Let $T\in (0,\,+\infty )$, $\bar {s}_1,\,\bar {s}_2\in C([-\max \{\tau _1,\,\tau _2\},\,T])\cap C^{1}((0,\,T])$, $\bar {u}_0 (\theta,\,x)\in C^{1,2}([-\tau _2,\,0]\times [\bar {s}_1,\,\bar {s}_2])$, $\bar {v}_0(\theta,\,x)\in C^{1,2}([-\tau _1,\,0]\times [\bar {s}_1,\,\bar {s}_2])$, $\bar {u}(t,\,x), \bar {v}(t,\,x) \in C^{1,2}((0,\,T]\times [\bar {s}_1,\,\bar {s}_2])$, and

\[ \left\{\begin{array}{@{}l} \bar{u}_t\geq d_1\bar{u}_{xx}-a_1\bar{u} +h\Bigg(\int_{-\infty}^{+\infty}J_1(x-y)\bar{v}(t-\tau_1,y){\rm d}y\Bigg),\\ \quad\quad 0< t\leq T, \bar{s}_1(t)< x<\bar{s}_2(t),\\ \bar{v}_t\geq d_2\bar{v}_{xx}-a_2\bar{v} +g\Bigg(\int_{-\infty}^{+\infty}J_2(x-y)\bar{u}(t-\tau_2,y){\rm d}y\Bigg),\\ \quad\quad 0< t\leq T, \bar{s}_1(t)< x<\bar{s}_2(t),\\ \bar{u}(t,x)=\bar{v}(t,x)=0,~0< t\leq T,~x\leq\bar{s}_1(t)~\mbox{or}~x\geq\bar{s}_2(t),\\ \bar{s}_1^\prime(t)\leq{-}\mu[\bar{u}_x(t,\bar{s}_1(t))+\rho\bar{v}_x(t,\bar{s}_1(t))],~0< t\leq T,\\ \bar{s}_2^\prime(t)\geq{-}\mu[\bar{u}_x(t,\bar{s}_2(t))+\rho\bar{v}_x(t,\bar{s}_2(t))],~0< t\leq T,\\ \bar{u}(\theta,x)=\bar{u}_0(\theta,x),~-\tau_2\leq\theta\leq0,~\bar{s}_1(\theta)\leq x\leq\bar{s}_2(\theta),\\ \bar{v}(\theta,x)=\bar{v}_0(\theta,x),~-\tau_1\leq\theta\leq0,~\bar{s}_1(\theta)\leq x\leq\bar{s}_2(\theta). \end{array}\right. \]

If $(u,\,v;s_1,\,s_2)$ is a solution of (1.1) with $[s_1(\theta ),\,s_2(\theta )]\subset [\bar {s}_1(\theta ),\,\bar {s}_2(\theta )]$ for $\theta \in [-\max \{\tau _1,\,\tau _2\},\,0]$, $u_0(\theta,\,x)\leq \bar {u}_{0}(\theta,\,x)$ for $(\theta,\,x)\in [-\tau _{2},\,0]\times [s_1, s_2]$ and $v_0(\theta,\,x)\leq \bar {v}_{0}(\theta,\,x)$ for $(\theta,\,x)\in [-\tau _{1},\,0]\times [s_1, s_2]$, then $[s_1(t),\,s_2(t)]\subset [\bar {s}_1(t),\,\bar {s}_2(t)]$ and $(u(t,\,x),\,v(t,\,x))\leq (\bar {u}(t,\,x),\,\bar {v}(t,\,x))$ for $t\in (0,\,T],\,~x\in (s_1(t),\,s_2(t))$.

Lemma 2.3 Let $T\in (0,\,+\infty )$, $\bar {s}_2\in C([-\max \{\tau _1,\,\tau _2\},\,T])\cap C^{1}((0,\,T])$, $\bar {u}_0(\theta,\,x)\in C([-\tau _2,\,0]\times (-\infty,\,\bar {s}_2])\cap C^{1,2}([-\tau _2,\,0]\!\times\! (0,\,\bar {s}_2])$, $\bar {v}_0(\theta,\,x)\in C([-\tau _1,\,0]\!\times\! (-\infty,\,\bar {s}_2]) \cap C^{1,2}([-\tau _1,\,0]\times (0,\,\bar {s}_2])$, $\bar {u}(t,\,x),\,~\bar {v}(t,\,x)\in C([0,\,T]\times (-\infty,\bar {s}_2])\cap C^{1,2}((0,\,T]\times (0,\,\bar {s}_2])$, and

\[ \left\{\begin{array}{@{}l} \bar{u}_t\geq d_1\bar{u}_{xx}-a_1\bar{u} +h\Bigg(\int_{-\infty}^{+\infty}J_1(x-y)\bar{v}(t-\tau_1,y){\rm d}y\Bigg),~0< t\leq T,~0< x<\bar{s}_2(t),\\ \bar{v}_t\geq d_2\bar{v}_{xx}-a_2\bar{v} +g\Bigg(\int_{-\infty}^{+\infty}J_2(x-y)\bar{u}(t-\tau_2,y){\rm d}y\Bigg),~0< t\leq T,~0< x<\bar{s}_2(t),\\ \bar{u}(t,x)=\bar{v}(t,x)=0,~0< t\leq T,~x\geq\bar{s}_2(t),\\ \bar{u}(t,x)\geq u(t,x),~\bar{v}(t,x)\geq v(t,x),~0< t\leq T,~x\leq 0,\\ \bar{s}_2^\prime(t)\geq{-}\mu[\bar{u}_x(t,\bar{s}_2(t))+\rho\bar{v}_x(t,\bar{s}_2(t))],~0< t\leq T,\\ \bar{u}(\theta,x)=\bar{u}_0(\theta,x),~-\tau_2\leq\theta\leq0,~-\infty< x\leq\bar{s}_2(\theta),\\ \bar{v}(\theta,x)=\bar{v}_0(\theta,x),~-\tau_1\leq\theta\leq0,~-\infty< x\leq\bar{s}_2(\theta). \end{array}\right. \]

If the solution $(u,\,v;s_1,\,s_2)$ of (1.1) satisfies that $s_2(\theta )\leq \bar {s}_2(\theta )$ for $\theta \in [-\max \{\tau _1,\,\tau _2\},\,0]$, $u_0(\theta,\,x)\leq \bar {u}_{0}(\theta,\,x)$ for $(\theta,\,x)\in [-\tau _{2},\,0]\times (-\infty, s_2]$ and $v_0(\theta,\,x)\leq \bar {v}_{0}(\theta,\,x)$ for $(\theta,\,x)\in [-\tau _{1},\,0]\times (-\infty, s_2]$, then $s_2(t)\leq \bar {s}_2(t)$ and $(u(t,\,x),\,v(t,\,x))\leq (\bar {u}(t,\,x),\,\bar {v}(t,\,x))$ for $t\in (0,\,T],\,~x\in (0,\,s_2(t))$.

Remark 2.4 $(\bar {u},\,\bar {v};\bar {s}_1,\,\bar {s}_2)$ in lemma 2.2 is called an upper solution of (1.1), and $(\bar {u},\,\bar {v}; \bar {s}_2)$ in lemma 2.3 is that of one-side case. Lower solutions of (1.1) can be defined analogously by reversing all the inequalities.

2.2 Basic reproduction number

In epidemiology, the basic reproduction number is an important index of measuring the transmission potential of a disease. In order to discuss the spreading and vanishing phenomenon for the free boundary problem (1.1), we need to study the basic reproduction number of the epidemic model without delays (1.5). In theorem 1.1, we establish the relationship between the basic reproduction number and principle eigenvalues, and provide the monotonicity and asymptotic properties of the basic reproduction number with respect to the domain.

In this section, we first introduce the definition of the basic reproduction number of (1.5), and then give the proof of theorem 1.1.

For any $l>0$, let $T_{i}(t)$ be the solution semigroup on $C_{0}([-l,\,l],\,\mathbb {R})$ associated with the following linear parabolic equation

\[ \left\{\begin{array}{@{}l} w_{t}=d_{i}w_{xx}-a_{i}w,\quad t>0,-l< x< l,\\ w(t, \pm l)=0,\quad t>0. \end{array}\right. \]

Let

\[ T(t)\Phi:= \Bigg(T_{1}(t)\phi, T_{2}(t)\varphi\Bigg),\quad \forall\Phi=(\phi,\varphi)\in X:=C_{0}([{-}l,l], \mathbb{R}^{2}),~t\geq 0. \]

It is clear that $T(t)$ is a positive $C_{0}$-semigroup on $X$. We further define a positive linear operator $\mathcal {G}$ from $X$ to $Y:=C([-l,\,l],\,\mathbb {R}^{2})$ by

\[ \mathcal{G}(\Phi)(x):=(\mathcal{G}_{1}(\Phi)(x),\mathcal{G}_{2}(\Phi)(x)),\quad \forall \Phi=(\phi,\varphi)\in X, \]

where

\[ \begin{array}{l} \mathcal{G}_{1}(\Phi)(x) =h^\prime(0)\int_{{-}l}^{l}J_{1}(x-y)\varphi(y){\rm d}y,\\ \mathcal{G}_{2}(\Phi)(x) =g^\prime(0)\int_{{-}l}^{l}J_{2}(x-y)\phi(y){\rm d}y. \end{array} \]

Then the distribution of total new infection of human is

\[ \int_{0}^{+\infty}h^\prime(0)\int_{{-}l}^{l}J_{1}(x-y)(T_{2}(t)\varphi)(y){\rm d}y{\rm d}t =\int_{0}^{+\infty}\mathcal{G}_{1}(T_{1}(t)\phi, T_{2}(t)\varphi)(x){\rm d}t, \]

and the distribution of total new infection of bacteria is

\[ \int_{0}^{+\infty}g^\prime(0)\int_{{-}l}^{l}J_{2}(x-y)(T_{1}(t)\phi)(y){\rm d}y{\rm d}t =\int_{0}^{+\infty}\mathcal{G}_{2}(T_{1}(t)\phi, T_{2}(t)\varphi)(x){\rm d}t. \]

It follows that

\[ \mathcal{L}(\Phi) :=\int_{0}^{+\infty}\mathcal{G}(T(t)\Phi){\rm d}t =\mathcal{G}\int_{0}^{+\infty}(T(t)\Phi){\rm d}t \]

is the next infection operator, which maps the initial distribution $\Phi$ of infectious bacteria and humans to the distribution of the total infective bacteria and humans produced during the infection period.

We define the basic reproduction number of the epidemic model (1.5)

\[ \mathcal{R}_{0}^{l}:=r(\mathcal{L}), \]

where $r(\mathcal {L})$ is the spectral radius of $\mathcal {L}$. Here, we use the notation $\mathcal {R}_{0}^{l}$ to emphasis the dependence of the basic reproduction number on the domain $(-l,\,l)$.

Proof of theorem 1.1 $(i)$ The corresponding linear evolution system (1.5) generates a compact, strongly positive semigroup on $Z_+:=Z\cap X_+$, where $Z:=C_{0}^{1}([-l,\,l], \mathbb {R}^{2})$, and $X_+:=C_{0}([-l,\,l], \mathbb {R}_+^{2})$ is the cone of nonnegative functions in $X$. Therefore, by standard arguments in Theorem 6.1 of [23], we deduce that the elliptic problem (1.6) has a principal eigenvalue $\lambda _1$ with a strictly positive eigenvector.

Similar as the proof of Lemma 2.2 in [33], we can prove that $\mathcal {R}_{0}^{l}-1$ has the same sign as $\lambda _{1}$. We also refer the readers to [20] (Theorem 3.7).

$(ii)$ We first prove that (1.7) admits a unique positive principle eigenvalue with a positive eigenvector.

Define $L_{i}=-d_{i}\partial _{xx}+a_{i}$, $i=1,\,2$, and let $L_{i}$ also denote the realization of $L_{i}$ in $C([-l,\,l], \mathbb {R})$ subject to Dirichlet boundary condition. Let

\[ L\Phi=(L_{1}\phi,L_{2}\varphi)=({-}d_1\partial_{xx}\phi+a_1\phi,-d_2\partial_{xx}\varphi+a_2\varphi), ~\forall \Phi=(\phi,\varphi)\in{\rm dom}(L)\subset Z. \]

The operator $L: Z\supset {\rm dom}(L)\rightarrow Y:=C([-l,\,l], \mathbb {R}^{2})$ is invertible, with compact inverse. It follows that the problem (1.7) is equivalent to the equation

\[ \Phi=\mu L^{{-}1}\mathcal{G}(\Phi). \]

Define $\mathcal {A}_{\mu }:=\mu L^{-1}\mathcal {G}$, and let $r(\mathcal {A}_{\mu })$ be its spectral radius. Note that $\mathcal {G}$ is a bounded linear operator from $Z$ to $Y$, and $L^{-1}$ is a compact linear operator from $Y$ to $Z$. Then for any fixed $\mu >0$, $\mathcal {A}_{\mu }: Z\rightarrow Z$ is a compact linear operator, and strongly positive with respect to the solid cone $Z_+$, i.e., $\mathcal {A}_{\mu }(Z_+\setminus \{\textbf {0}\})\subset {\rm Int}~Z_+\neq \emptyset$. By the Krein–Rutman theorem (strong form), $r(\mathcal {A}_{\mu })>0$ and there exists $\Phi _{\mu }\in {\rm Int}~Z_+$ such that $\mathcal {A}_{\mu }\Phi _{\mu }=r(\mathcal {A}_{\mu })\Phi _{\mu }$. Moreover, $r(\mathcal {A}_{\mu })$ is a geometrically simple eigenvalue.

By the Gelfand's formula, $r(\mathcal {A}_{\mu })=\lim _{n\rightarrow \infty }\|\mathcal {A}_{\mu }^{n}\|_{Z}^{\frac {1}{n}} =\mu \lim _{n\rightarrow \infty }\|(L^{-1}\mathcal {G})^{n}\|_{Z}^{\frac {1}{n}} =\mu r(\mathcal {A}_{1})$, $\forall \mu >0$. Since $r(\mathcal {A}_{1})>0$, there exists a unique $\mu _0^{l}>0$ such that $r(\mathcal {A}_{\mu _0^{l}})=1$. In fact, $\mu _0^{l}=\frac {1}{r(\mathcal {A}_{1})}$. Then, we have $\Phi _{\mu _0^{l}}=\mu _0^{l} L^{-1}\mathcal {G}(\Phi _{\mu _0^{l}})$, which implies that (1.7) admits a unique positive principle eigenvalue $\mu _0^{l}$ with a positive eigenvector $\Phi _{\mu _0^{l}}$.

The equality $\mathcal {R}_{0}^{l}=\frac {1}{\mu _{0}^{l}}$ can be proved by similar arguments as in the proof of Theorem 3.2 in [34]. We also refer the readers to [20] (Theorem 3.8).

$(iii)$ To stress the dependence of $\mathcal {A}_{1}$, $L^{-1}$ and $\mathcal {G}$ on $l$, here we use the notations $\mathcal {A}_{1}^{l}$, $(L^{l})^{-1}$ and $\mathcal {G}^{l}$. Obviously, for any $l_{2}>l_{1}>0$ and $\Phi =(\phi,\,\varphi )\in Z_+$,

\begin{align*} \mathcal{G}^{l_{2}}(\Phi)(x)& =(\mathcal{G}_{1}^{l_{2}}(\Phi)(x),\mathcal{G}_{2}^{l_{2}}(\Phi)(x))\\ & =\Bigg(h^\prime(0)\int_{{-}l_{2}}^{l_{2}}J_{1}(x-y)\varphi(y){\rm d}y, g^\prime(0)\int_{{-}l_{2}}^{l_{2}}J_{2}(x-y)\phi(y){\rm d}y\Bigg)\\ & \geq\Bigg(h^\prime(0)\int_{{-}l_{1}}^{l_{1}}J_{1}(x-y)\varphi(y){\rm d}y, g^\prime(0)\int_{{-}l_{1}}^{l_{1}}J_{2}(x-y)\phi(y){\rm d}y\Bigg)\\ & =(\mathcal{G}_{1}^{l_{1}}(\Phi)(x),\mathcal{G}_{2}^{l_{1}}(\Phi)(x)) =\mathcal{G}^{l_{1}}(\Phi)(x). \end{align*}

Moreover, by the maximum principle for elliptic equations, we know that

\[ (L^{l})^{{-}1}(\Phi_{2})\geq(L^{l})^{{-}1}(\Phi_{1}) \quad \text{for any }l>0~\mbox{and}~\Phi_{1}, \Phi_{2}\in Y_+~\mbox{with}~\Phi_{2}\geq\Phi_{1}, \]

and

\[ (L^{l_{2}})^{{-}1}(\Phi)\geq(L^{l_{1}})^{{-}1}(\Phi)\quad \text{for any}~l_{2}>l_{1}>0~\text{ and }~\Phi\in \tilde{Y}_+, \]

where $Y_+:=C([-l,\,l], \mathbb {R}_+^{2})$ and $\tilde {Y}_+:=C([-l_{2},\,l_{2}], \mathbb {R}_+^{2})$. Thus, for any $l_{2}>l_{1}$ and $\Phi \in Z_+$,

\[ (L^{l_{2}})^{{-}1}\mathcal{G}^{l_{2}}(\Phi) \geq(L^{l_{2}})^{{-}1}\mathcal{G}^{l_{1}}(\Phi) \geq(L^{l_{1}})^{{-}1}\mathcal{G}^{l_{1}}(\Phi). \]

Since each $(L^{l})^{-1}\mathcal {G}^{l}$ is a positive and bounded linear operator on $Z$, by Theorem 1.1 in [5] we know that $r(\mathcal {A}_{1}^{l})=r((L^{l})^{-1}\mathcal {G}^{l})$ is an increasing function of $l$. It follows that $\mu _{0}^{l}=\frac {1}{r(\mathcal {A}_{1}^{l})}$ is decreasing in $l$, and $\mathcal {R}_{0}^{l}=\frac {1}{\mu _{0}^{l}}$ (by $(ii)$) is increasing in $l$.

Note that $(L_{1}^{l})^{-1}(f)(x)=\int _{-l}^{l}G(x,\,\xi )f(\xi ){\rm d}\xi$, where $G(x,\,\xi )$ is the Green's function defined as

\[ G(x,\xi)= \left\{\begin{array}{@{}l} \dfrac{(e^{\lambda (l-x)}-e^{-\lambda (l-x)})(e^{\lambda (l+\xi)}-e^{-\lambda( l+\xi)})}{2d_{1}\lambda(e^{2\lambda l}-e^{{-}2\lambda l})},\quad -l\leq\xi\leq x,\\ \dfrac{(e^{\lambda (l+x)}-e^{-\lambda (l+x)})(e^{\lambda (l-\xi)}-e^{-\lambda (l-\xi)})}{2d_{1}\lambda(e^{2\lambda l}-e^{{-}2\lambda l})},\quad x\leq\xi\leq l \end{array}\right. \]

with $\lambda =\sqrt {\frac {a_{1}}{d_{1}}}$. It is easy to check that $G(x, \xi )>0$ and

\begin{align*} 0& <\int_{{-}l}^{l}G(x,\xi){\rm d}\xi =\frac{1}{d_{1}\lambda^{2}} -\frac{(e^{\lambda (l-x)}-e^{-\lambda (l-x)})+(e^{\lambda (l+x)}-e^{-\lambda (l+x)})} {d_{1}\lambda^{2}(e^{2\lambda l}-e^{{-}2\lambda l})} \\ & \leq \frac{1}{d_{1}\lambda^{2}}=\frac{1}{a_{1}}. \end{align*}

Then, we have

(2.2)\begin{equation} \|(L_{1}^{l})^{{-}1}(f)\|_{\infty} =\Bigg\|\int_{{-}l}^{l}G({\cdot},\xi)f(\xi){\rm d}\xi\Bigg\|_{\infty} \leq \Bigg\|\int_{{-}l}^{l}G({\cdot},\xi){\rm d}\xi\Bigg\|_{\infty}\|f\|_{\infty} \leq \frac{1}{a_{1}}\|f\|_{\infty}. \end{equation}

Moreover,

\[ \partial_{x}G(x,\xi)= \left\{\begin{array}{@{}l} \dfrac{-\lambda(e^{\lambda (l-x)}+e^{-\lambda (l-x)})(e^{\lambda (l+\xi)}-e^{-\lambda( l+\xi)})}{2d_{1}\lambda(e^{2\lambda l}-e^{{-}2\lambda l})},\quad -l\leq\xi< x,\\ \dfrac{\lambda(e^{\lambda (l+x)}+e^{-\lambda (l+x)})(e^{\lambda (l-\xi)}-e^{-\lambda (l-\xi)})}{2d_{1}\lambda(e^{2\lambda l}-e^{{-}2\lambda l})},\quad x<\xi\leq l. \end{array}\right. \]

Direct calculations yield

\begin{align*} 0<\int_{{-}l}^{l}|\partial_{x}G(x,\xi)|{\rm d}\xi & =\frac{1}{d_{1}\lambda} -\frac{e^{-\lambda x}(e^{\lambda l}-e^{-\lambda x}) +e^{\lambda x}(e^{\lambda l}-e^{\lambda x})} {d_{1}\lambda(e^{2\lambda l}-e^{{-}2\lambda l})}\\ & \qquad -\frac{e^{-\lambda l}(e^{\lambda x}\!-\!e^{-\lambda l}) +e^{-\lambda l}(e^{-\lambda x}\!-\!e^{-\lambda l})} {d_{1}\lambda(e^{2\lambda l}-e^{{-}2\lambda l})}\!\leq\! \frac{1}{d_{1}\lambda}\!=\!\frac{1}{\sqrt{a_{1}d_{1}}}. \end{align*}

Then, we have

\begin{align*} \|\nabla (L_{1}^{l})^{{-}1}(f)\|_{\infty} & =\Bigg\|\int_{{-}l}^{l}\partial_{x}G({\cdot},\xi)f(\xi){\rm d}\xi\Bigg\|_{\infty}\\ & \leq \Bigg\|\int_{{-}l}^{l}|\partial_{x}G({\cdot},\xi)|{\rm d}\xi\Bigg\|_{\infty}\|f\|_{\infty} \leq \frac{1}{\sqrt{a_{1}d_{1}}}\|f\|_{\infty}, \end{align*}

which together with (2.2) imply

\[ \|(L_{1}^{l})^{{-}1}(f)\|_{\infty}+\|\nabla (L_{1}^{l})^{{-}1}(f)\|_{\infty} \leq \left(\frac{1}{a_{1}}+\frac{1}{\sqrt{a_{1}d_{1}}}\right)\|f\|_{\infty}. \]

In a similar way, we can prove $\|(L_{2}^{l})^{-1}(f)\|_{\infty }+\|\nabla (L_{2}^{l})^{-1}(f)\|_{\infty } \leq (\tfrac {1}{a_{2}}+\tfrac {1}{\sqrt {a_{2}d_{2}}})\|f\|_{\infty }$. Thus,

(2.3)\begin{equation} \|(L^{l})^{{-}1}\|_{Y\rightarrow Z}\leq \frac{1}{a_{1}}+\frac{1}{a_{2}} +\frac{1}{\sqrt{a_{1}d_{1}}}+\frac{1}{\sqrt{a_{2}d_{2}}}. \end{equation}

On the other hand,

\begin{align*} \|\mathcal{G}_{1}^{l}(\Phi)\|_{\infty}& =\Bigg\|h^\prime(0)\int_{{-}l}^{l}J_{1}({\cdot}{-}y)\varphi(y){\rm d}y\Bigg\|_{\infty}\\ & \leq h^\prime(0)\Bigg\|\int_{{-}l}^{l}J_{1}({\cdot}{-}y){\rm d}y\Bigg\|_{\infty}\|\varphi\|_{\infty} \leq h^\prime(0)\|\Phi\|_{\infty}. \end{align*}

Similarly, $\|\mathcal {G}_{2}^{l}(\Phi )\|_{\infty }\leq g^\prime (0)\|\Phi \|_{\infty }$. Thus,

(2.4)\begin{equation} \|\mathcal{G}^{l}\|_{Z\rightarrow Y} \leq h^\prime(0)+g^\prime(0). \end{equation}

By (2.3) and (2.4), we have

\begin{align*} r(\mathcal{A}_{1}^{l})& \leq\|\mathcal{A}_{1}^{l}\|_{Z} =\|(L^{l})^{{-}1}\mathcal{G}^{l}\|_{Z} \leq \|(L^{l})^{{-}1}\|_{Y\rightarrow Z}\|\mathcal{G}^{l}\|_{Z\rightarrow Y}\\ & \leq \left(\frac{1}{a_{1}}+\frac{1}{a_{2}} +\frac{1}{\sqrt{a_{1}d_{1}}}+\frac{1}{\sqrt{a_{2}d_{2}}}\right) (h^\prime(0)+g^\prime(0))=:M. \end{align*}

It follows that $\mu _{0}^{l}$ has a positive lower bound independent of $l$, i.e.,

\[ \mu_{0}^{l} =\frac{1}{r(\mathcal{A}_{1}^{l})} \geq \frac{1}{M}, \quad \forall l>0, \]

which together with the fact that $\mu _{0}^{l}=\frac {1}{r(\mathcal {A}_{1}^{l})}$ is decreasing in $l$ imply that $\mu ^*:=\lim _{l\rightarrow +\infty }\mu _{0}^{l}$ exists and satisfies $\mu ^*\geq \frac {1}{M}>0$. Then $\mathcal {R}^{*}:=\lim _{l\rightarrow +\infty }\frac {1}{\mu _{0}^{l}}=\frac {1}{\mu ^{*}}$ is well-defined and satisfies $0<\mathcal {R}^{*}\leq M$.

Now, we provide a more accurate upper bound of $\mathcal {R}^{*}$, i.e., $\mathcal {R}^{*}\leq \mathcal {R}_0:=\sqrt {\frac {h^\prime (0)g^\prime (0)}{a_1a_2}}$. It is sufficient to show that $\mu ^*\geq \sqrt {\frac {a_1a_2}{h^\prime (0)g^\prime (0)}}$.

Recall that $r(\mathcal {A}_{\mu }^{l})$ is a geometrically simple eigenvalue of $\mathcal {A}_{\mu }^{l}$ by the Krein–Rutman theorem. We may assume that the corresponding positive eigenvector $\Phi _{\mu _{0}^{l}}^{l}=(\phi _{\mu _{0}^{l}}^{l},\,\varphi _{\mu _{0}^{l}}^{l})$ of (1.7) satisfies $\|\Phi ^{l}_{\mu _{0}^{l}}\|_{\infty }=1$. Thus, there exist a sequence $\{l_{n}\}$ and positive function $\Phi ^{*}$ satisfying $\|\Phi ^{*}\|_{\infty }=1$, such that $\Phi _{\mu _{0}^{l_{n}}}^{l_{n}}\rightarrow \Phi ^{*}$ in $C^{2}_{loc}(\mathbb {R})$ as $n\rightarrow \infty$. Then, $(\mu ^*, \Phi ^{*})$ solves

(2.5)\begin{equation} \left\{\begin{array}{@{}l} -d_1\phi^{*}_{xx}+a_1\phi^{*} =\mu^{*} h^\prime(0)\int_{-\infty}^{+\infty}J_{1}(x-y)\varphi^{*}(y){\rm d}y,~x\in(-\infty,+\infty),\\ -d_2\varphi^{*}_{xx}+a_2\varphi^{*} =\mu^{*} g^\prime(0)\int_{-\infty}^{+\infty}J_{2}(x-y)\phi^{*}(y){\rm d}y,~x\in(-\infty,+\infty). \end{array}\right. \end{equation}

As in [3, 4], we define the generalized principal eigenvalue in a (possibly unbounded) domain $\Omega \subset \mathbb {R}$ as follows

(2.6)\begin{equation} \begin{array}{l} \mu_{1}(\Omega) :=\sup E^{\Omega}\\ \qquad~~ :=\sup\Bigg\{\mu\in \mathbb{R}: ~\exists (\phi,\varphi)\in C^{2}(\Omega,\mathbb{R}^{2})\cap C_{loc}^{1}(\overline{\Omega},\mathbb{R}^{2}), ~(\phi,\varphi)>\textbf{0}~\mbox{in}~\Omega,\\ \qquad\qquad\qquad\qquad\qquad\mbox{and} -d_1\phi_{xx}+a_1\phi \geq\mu h^\prime(0)\int_{\Omega}J_{1}(x-y)\varphi(y){\rm d}y,\\ \qquad\qquad\qquad\qquad\qquad -d_2\varphi_{xx}+a_2\varphi \geq\mu g^\prime(0)\int_{\Omega}J_{2}(x-y)\phi(y){\rm d}y\quad\mbox{for}~x\in\Omega \Bigg\}. \end{array} \end{equation}

Here, $C_{loc}^{1}(\overline {\Omega },\,\mathbb {R}^{2})$ denotes the set of functions $(\phi,\,\varphi )\in C^{1}(\Omega,\,\mathbb {R}^{2})$ for which $(\phi,\,\varphi )$ and $(\phi _{x},\,\varphi _{x})$ can be extended by continuity on $\partial \Omega$, but which are not necessarily bounded. From $(ii)$ and (2.5), we know $E^{(-l,l)}, E^{\mathbb {R}}\neq \emptyset$ for any $l>0$. We claim that $(i)$ $\mu _{1}((-l,\,l))=\mu _{0}^{l}$ for any $l>0$, where $\mu _{0}^{l}$ is the principal eigenvalue of (1.7); $(ii)$ $\mu _{1}((-l,\,l))\rightarrow \mu _{1}(\mathbb {R})$ as $l\rightarrow +\infty$, and then $\mu ^*=\mu _{1}(\mathbb {R})$. The claim can be proved by similar arguments as in the proofs of Proposition 4.4 and Theorem 2.2 in [2]. For the convenience of the reader, we provide the details in proposition A of Appendix.

Assume that $(\tilde {\mu },\,\tilde {\Phi }(x))=(\tilde {\mu }, c_{1}, c_{2})$ is a solution of (2.5) with $\|\tilde {\Phi }\|_{\infty }=1$, where $c_{1}, c_{2}$ are positive constants. Due to $\int _{-\infty }^{+\infty }J_{i}(x){\rm d}x=1$, $i=1,\,2$, we have

\[ \left\{\begin{array}{@{}l} a_1 c_1=\tilde{\mu} h^\prime(0)c_{2},\\ a_2 c_{2}=\tilde{\mu} g^\prime(0)c_{1},\\ c_{1}+c_{2}=1. \end{array}\right. \]

By simple calculations,

\[ (\tilde{\mu},c_{1},c_{1}) =\Bigg(\sqrt{\frac{a_1a_2}{h^\prime(0)g^\prime(0)}}, 1-\frac{a_{1}}{a_{1}+h^{\prime}(0)\sqrt{\frac{a_1a_2}{h^\prime(0)g^\prime(0)}}}, \frac{a_{1}}{a_{1}+h^{\prime}(0)\sqrt{\frac{a_1a_2}{h^\prime(0)g^\prime(0)}}}\Bigg). \]

Then $\tilde {\mu }=\sqrt {\frac {a_1a_2}{h^\prime (0)g^\prime (0)}}\in E^{\mathbb {R}}$, which implies that $\mu _{1}(\mathbb {R})\geq \tilde {\mu }=\sqrt {\frac {a_1a_2}{h^\prime (0)g^\prime (0)}}$. It follows that $\mathcal {R}^{*}=\frac {1}{\mu ^*}=\frac {1}{\mu _{1}(\mathbb {R})}\leq \mathcal {R}_{0}:=\sqrt {\frac {h^\prime (0)g^\prime (0)}{a_1a_2}}$, which completes the proof.

Remark 2.5 (1) We remark that $\mathcal {R}^*$ may be not equal to $\mathcal {R}_0$. Here, we give two cases to illustrate the result $\mathcal {R}^*\leq \mathcal {R}_0$.

Case I. If $J_{1}=J_{2}=\delta$ (Dirac delta function), then

\[ \mathcal{R}_{0}^{l} =\sqrt{\frac{h^{\prime}(0)g^{\prime}(0)}{[d_1(\frac{\pi}{2l})^2+a_1][d_2(\frac{\pi}{2l})^2+a_2]}}. \]

As $l\rightarrow \infty$, we have $\mathcal {R}_{0}^{l}\rightarrow \sqrt {\frac {h^{\prime }(0)g^{\prime }(0)}{a_{1}a_{2}}}$. Therefore $\mathcal {R}^*=\mathcal {R}_0$. More details can be seen in [22].

Case II. If $d_{i}=d$, $a_{i}=a$, $J_{i}=J$ $(i=1,\,2)$ and $h=g$, then, by taking $\psi =\phi +\varphi$, (1.7) reduces to the following single equation

\[ \left\{\begin{array}{@{}l} -d\psi_{xx}+a\psi =\mu h^\prime(0)\int_{{-}l}^{l}J(x-y)\psi(y){\rm d}y,~x\in({-}l,l),\\ \psi({\pm} l)=0. \end{array}\right. \]

From the variational characterization of the principal eigenvalue, we have

\[ \mu_{0}^{l}=\inf_{\psi\in H_{0}^{1}(({-}l,l))\atop \|\psi\|_{L^{2}=1}} \Bigg\{\frac{\int_{{-}l}^{l}d|\nabla \psi|^{2}{\rm d}x+a} {h^{\prime}(0)\int_{{-}l}^{l}\int_{{-}l}^{l}J(x-y)\psi(y)\psi(x){\rm d}x{\rm d}y}\Bigg\}, \]

and then

\[ \mathcal{R}_{0}^{l} =\sup_{\psi\in H_{0}^{1}(({-}l,l)) \atop \|\psi\|_{L^{2}}=1} \Bigg\{\frac{h^{\prime}(0)\int_{{-}l}^{l}\int_{{-}l}^{l}J(x-y)\psi(y)\psi(x){\rm d}x{\rm d}y} {\int_{{-}l}^{l}d|\nabla \psi|^{2}{\rm d}x+a}\Bigg\}. \]

Thus,

\[ \mathcal{R}^* \leq \frac{h^{\prime}(0)}{a}\sup_{\psi\in H^{1}(\mathbb{R}) \atop \|\psi\|_{L^{2}}=1} \Bigg\{\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}J(x-y)\psi(y)\psi(x){\rm d}x{\rm d}y\Bigg\}. \]

Note that

\[ \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}J(x-y)\psi(y)\psi(x){\rm d}x{\rm d}y \leq \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}J(x-y)\frac{\psi(y)^{2}+\psi(x)^{2}}{2}{\rm d}x{\rm d}y = 1, \]

and the first equality holds if and only if $\psi$ is a constant function, which contradicts with $\psi \in H^{1}(\mathbb {R})$. Therefore $\mathcal {R}^*<\frac {h^{\prime }(0)}{a}=\mathcal {R}_0$.

1. (2) If the interval $(-l,\,l)$ is replaced by $(a,\,b)$ with $-\infty < a<0< b<+\infty$, then the conclusions in theorem 1.1 are still valid.

3. Spreading and vanishing

In this section, we discuss the spreading and vanishing phenomenon of the bacteria in terms of the basic reproduction number, and then provide the sharp criteria for spreading and vanishing.

Denote $s_{1,\infty }=\lim _{t\rightarrow +\infty }s_1(t)$ and $s_{2,\infty }=\lim _{t\rightarrow +\infty }s_2(t)$. Then, we have the following results.

Lemma 3.1 $(i)$ If $s_{2,\infty }-s_{1,\infty }=+\infty$, then $s_{2,\infty }=-s_{1,\infty }=+\infty$.

$(ii)$ If $s_{2,\infty }-s_{1,\infty }<+\infty$, then $s_0<-s_{1,\infty }, s_{2,\infty }<+\infty$, and

\[ \lim_{t\rightarrow+\infty} \Bigg(\|u(t,\cdot)\|_{C([s_1(t),s_2(t)])} +\|v(t,\cdot)\|_{C([s_1(t),s_2(t)])}\Bigg)=0. \]

$(iii)$ If $\mathcal {R}^*>1$ and $s_{2,\infty }-s_{1,\infty }=+\infty$, then $\lim _{t\rightarrow +\infty }(u(t,\,x),\,v(t,\,x))=(u^*,\,v^*)$ locally uniformly for $x\in \mathbb {R}$, where $(u^*,\,v^*)$ is the unique equilibrium of (1.1).

Proof. All the conclusions can be proved by similar arguments as Lemmas 3.1–3.2 and Theorem 4.5 in [1] with minor modifications (see also Theorem 2.3 in [11]), here we omit the details. We remark that the condition $\mathcal {R}^*>1$ in $(iii)$ is assumed for applying theorem 1.1 in the proof.

Theorem 3.2 If $0<\mathcal {R}_0\leq 1$, then the solution $(u,\,v;s_1,\,s_2)$ of (1.1) satisfies

\[ \lim_{t\rightarrow+\infty}\Bigg(\|u(t,\cdot)\|_{C([s_1(t),s_2(t)])} +\|v(t,\cdot)\|_{C([s_1(t),s_2(t)])}\Bigg)=0. \]

Proof. Let $(w_1(t), w_2(t))$ be the unique solution of

(3.1)\begin{equation} \left\{\begin{array}{@{}l} w_1^{\prime}={-}a_{1}w_1+h(w_2(t-\tau_1)),~ t>0,\\ w_2^{\prime}={-}a_{2}w_2+g(w_1(t-\tau_2)),~ t>0,\\ w_1(\theta)=\|u_0\|_{C([-\tau_{2},0]\times[s_1, s_2])},~\theta\in[-\tau_2,0],\\ w_2(\theta)=\|v_0\|_{C([-\tau_{1},0]\times[s_1, s_2])},~\theta\in[-\tau_1,0]. \end{array}\right. \end{equation}

From the comparison principle, we know that $(u(t,\,x), v(t,\,x))\leq (w_1(t), w_2(t))$ in $[0, +\infty )\times [s_1, s_2]$.

Since $h^{\prime \prime }(z)\leq 0$, $g^{\prime \prime }(z)<0$ for all $z>0$, we have

\begin{align*} \frac{h(\lambda z)}{\lambda z}& =\frac{h(\lambda z)-h(0)}{\lambda z}\geq\frac{h(z)-h(0)}{z} =\frac{h(z)}{z},\\ \frac{g(\lambda z)}{\lambda z}& =\frac{g(\lambda z)-g(0)}{\lambda z}>\frac{g(z)-g(0)}{z}=\frac{g(z)}{z} \end{align*}

for $z>0$ and $\lambda \in (0,\,1)$. That is, $h,\,g$ are subhomogeneous. From Theorem 3.2 in [42], we know that $(0,\,0)$ is globally asymptotically stable for (3.1). That is, $(w_1,\,w_2)$ satisfies $\lim _{t\rightarrow +\infty }(w_{1}(t), w_{2}(t))=(0,\,0)$, which implies $\lim _{t\rightarrow +\infty }(\|u(t,\,\cdot )\|_{C([s_1(t), s_2(t)])} +\|v(t,\,\cdot )\|_{C([s_1(t), s_2(t)])})=0$. More details can be seen in [10, 11].

Remark 3.3 Due to the effects of delays, we can not show $s_{2,\infty }-s_{1,\infty }<+\infty$ as in [1], even for the local, partially degenerate case with one delay considered in [10]. We leave it for further consideration.

Next, we discuss the spreading and vanishing phenomenon of (1.1) for $\mathcal {R}^*>1$ in terms of $\mathcal {R}_0^{s_0}$. The selected forms of upper and lower solutions in the proof of next two theorems have been used in many related works. However, to overcome the effects of nonlocal terms, in this paper we construct upper and lower solutions from the principle eigenfunctions of perturbed eigenvalue problems. This idea is inspired by the work of Huang and Wang [18].

Theorem 3.4 If $\mathcal {R}_{0}^{s_0}\geq 1$, then $s_{2,\infty }-s_{1,\infty }=+\infty$.

Proof. By theorem 1.1 $(iii)$, we know that the basic reproduction number is strictly increasing with respect to the domain. Note that $(-s_0, s_0)\subset (s_{1}(t_0), s_{2}(t_0))$ for any fixed $t_0>0$. If $\mathcal {R}_{0}^{s_0}=1$, then the basic reproduction number of (1.5) with $(-l,\,l)$ replaced by $(s_{1}(t_0), s_{2}(t_0))$ is larger than 1. In such a case, we can choose some $t_0>0$ as initial time. Therefore it suffices to consider the case $\mathcal {R}_{0}^{s_0}>1$.

From theorem 1.1 $(i)$, we have $ {\rm sign}~(\mathcal {R}_{0}^{s_0}-1)= {\rm sign}~\lambda _1>0$. Then there exists a constant $0<\delta ^*\ll 1$ such that $\lambda _1^{\delta }>0$ for all $0<\delta <\delta ^*$, where $\lambda _1^{\delta }$ is the principle eigenvalue of the following perturbed eigenvalue problem

(3.2)\begin{equation} \left\{\begin{array}{@{}l} d_1\phi_{xx}-a_1\phi +(h^\prime(0)-\delta)\int_{{-}s_0}^{s_0}J_{1}(x-y)\varphi(y){\rm d}y =\lambda\phi,~x\in({-}s_0,s_0),\\ d_2\varphi_{xx}-a_2\varphi +(g^\prime(0)-\delta)\int_{{-}s_0}^{s_0}J_{2}(x-y)\phi(y){\rm d}y =\lambda\varphi,~x\in({-}s_0,s_0),\\ (\phi({\pm} s_0),\varphi({\pm} s_0))=(0,0). \end{array}\right. \end{equation}

Let $\Phi ^{\delta }(x)=(\phi ^{\delta }(x),\,\varphi ^{\delta }(x))$ be the positive eigenfunction of (3.2) associated with $\lambda _1^{\delta }$.

Define $\underline {u}(t,\,x)=\varepsilon \phi ^{\delta }(x)$, $\underline {v}(t,\,x)=\varepsilon \varphi ^{\delta }(x)$ for $(t,\,x)\in [\max \{\tau _1,\,\tau _2\},\,+\infty )\times [-s_0,\,s_0]$, where $\varepsilon$ is a small positive constant to be determined. Direct calculations yield

\begin{align*} & \underline{u}_t-d_1\underline{u}_{xx}+a_1\underline{u} -h\Bigg(\int_{-\infty}^{+\infty}J_1(x-y)\underline{v}(t-\tau_1,y){\rm d}y\Bigg)\\ & \quad={-}\varepsilon \lambda_{1}^{\delta}\phi^{\delta} +\varepsilon (h^{\prime}(0)-\delta)\int_{{-}s_0}^{s_0}J_{1}(x-y)\varphi^{\delta}(y){\rm d}y -h\Bigg(\int_{{-}s_0}^{s_0}\varepsilon J_{1}(x-y)\varphi^{\delta}(y){\rm d}y\Bigg)\\ & \quad={-}\varepsilon \Bigg[\lambda_{1}^{\delta}\phi^{\delta} +\Bigg(\delta+h^{\prime}(\eta_{1}^{\varepsilon}(x))-h^{\prime}(0)\Bigg)\int_{{-}s_0}^{s_0}J_{1}(x-y)\varphi^{\delta}(y){\rm d}y\Bigg] \end{align*}

and

\begin{align*} & \underline{v}_t-d_2\underline{v}_{xx}+a_2\underline{v} -g\Bigg(\int_{-\infty}^{+\infty}J_2(x-y)\underline{u}(t-\tau_2,y){\rm d}y\Bigg)\\ & \quad={-}\varepsilon \Bigg[\lambda_{1}^{\delta}\varphi^{\delta} +\Bigg(\delta+g^{\prime}(\eta_{2}^{\varepsilon}(x))-g^{\prime}(0)\Bigg)\int_{{-}s_0}^{s_0}J_{2}(x-y)\phi^{\delta}(y){\rm d}y\Bigg], \end{align*}

where $\eta _{1}^{\varepsilon }(x)\in (0, \varepsilon \int _{-s_0}^{s_0}J_{1}(x-y)\varphi ^{\delta }(y){\rm d}y)$ and $\eta _{2}^{\varepsilon }(x)\in (0, \varepsilon \int _{-s_0}^{s_0}J_{2}(x-y)\phi ^{\delta }(y){\rm d}y)$.

Note that $\eta _{1}^{\varepsilon }(x), \eta _{2}^{\varepsilon }(x)\rightarrow 0$ as $\varepsilon \rightarrow 0$. We can choose $\varepsilon >0$ sufficiently small such that $\delta +h^{\prime }(\eta _{1}^{\varepsilon }(x))-h^{\prime }(0)>0$ and $\delta +g^{\prime }(\eta _{2}^{\varepsilon }(x))-g^{\prime }(0)>0$, which imply that

\[ \left\{\begin{array}{l} \underline{u}_t-d_1\underline{u}_{xx}+a_1\underline{u} -h\Bigg(\int_{-\infty}^{+\infty}J_1(x-y)\underline{v}(t-\tau_1,y){\rm d}y\Bigg)\leq 0,\\ \underline{v}_t-d_2\underline{v}_{xx}+a_2\underline{v} -g\Bigg(\int_{-\infty}^{+\infty}J_2(x-y)\underline{u}(t-\tau_2,y){\rm d}y\Bigg)\leq 0, \end{array}\right. \]

for any $t>\max \{\tau _1,\,\tau _2\}$ and $-s_0< x< s_0$. We can also assume $\varepsilon >0$ small such that

\begin{align*} \begin{array}{l} \underline{u}(\theta,x)=\varepsilon \phi^{\delta}(x)\leq u(\theta,x), ~\max\{\tau_{1},\tau_{2}\}-\tau_{2}\leq\theta\leq\max\{\tau_{1},\tau_{2}\}, -s_0\leq x\leq s_0,\\ \underline{v}(\theta,x)=\varepsilon \varphi^{\delta}(x)\leq v(\theta,x), ~\max\{\tau_{1},\tau_{2}\}-\tau_{1}\leq\theta\leq\max\{\tau_{1},\tau_{2}\}, -s_0\leq x\leq s_0. \end{array} \end{align*}

Moreover, it is easy to deduce that

\[ \begin{array}{l} \underline{u}(t,x)=\underline{v}(t,x)=0, ~t\geq \max\{\tau_1,\tau_2\},~x\leq{-}s_0~\mbox{or}~x\geq s_0,\\ 0=s_0^\prime\leq{-}\mu [\underline{u}_x(t,s_0)+\rho \underline{v}_x(t,s_0)],~t>\max\{\tau_1,\tau_2\},\\ 0={-}s_0^\prime\geq{-}\mu [\underline{u}_x(t,-s_0)+\rho \underline{v}_x(t,-s_0)],~t>\max\{\tau_1,\tau_2\},\\ {}[{-}s_0,s_0]\subseteq[s_1(\theta),s_2(\theta)],~t>\max\{\tau_1,\tau_2\}. \end{array} \]

Therefore, $(\underline {u},\,\underline {v};-s_0,\,s_0)$ is a lower solution of (1.1). By the comparison principle,

\[ \liminf_{t\rightarrow+\infty}\Bigg(\|u(t,\cdot)\|_{C([s_1(t),s_2(t)])} +\|v(t,\cdot)\|_{C([s_1(t),s_2(t)])}\Bigg)\geq\varepsilon(\phi^{\delta}(0)+\varphi^{\delta}(0))>0, \]

which implies $s_{2,\infty }-s_{1,\infty }=+\infty$.

Theorem 3.5 If $\mathcal {R}_0^{s_0}<1$, then $s_{2,\infty }-s_{1,\infty }<+\infty$ provided that $\mu$ is sufficiently small.

Proof. From theorem 1.1 $(i)$, we have $ {\rm sign}~(\mathcal {R}_{0}^{s_0}-1)= {\rm sign}~\lambda _1<0$. Then there exists a constant $0<\delta ^*\ll 1$ such that $\lambda _1^{\delta }<0$ for all $0<\delta <\delta ^*$, where $\lambda _1^{\delta }$ is the principle eigenvalue of the following perturbed eigenvalue problem

(3.3)\begin{equation} \left\{\begin{array}{@{}l} d_1\phi_{xx}-a_1\phi +(h^\prime(0)+\delta)\int_{{-}s_0}^{s_0}J_{1}(x-y)\varphi(y){\rm d}y =\lambda\phi,~x\in({-}s_0,s_0),\\ d_2\varphi_{xx}-a_2\varphi +(g^\prime(0)+\delta)\int_{{-}s_0}^{s_0}J_{2}(x-y)\phi(y){\rm d}y =\lambda\varphi,~x\in({-}s_0,s_0),\\ (\phi({\pm} s_0),\varphi({\pm} s_0))=(0,0). \end{array}\right. \end{equation}

Let $\Phi ^{\delta }(x)=(\phi ^{\delta }(x),\,\varphi ^{\delta }(x))$ be the positive eigenfunction of (3.3) associated with $\lambda _1^{\delta }$.

We define

\begin{align*} k(t)& =s_0(1+\sigma-\frac{\sigma}{2}e^{-\sigma t}),\quad t>0,\\ \bar{u}(t,x) & =\alpha e^{-\sigma t}\phi^{\delta} \Bigg(\frac{s_{0} x}{k(t)}\Bigg),\quad t>0,\ x\in[{-}k(t),k(t)],\\ \bar{v}(t,x) & =\alpha e^{-\sigma t}\varphi^{\delta} \Bigg(\frac{s_{0} x}{k(t)}\Bigg),\quad t>0,\ x\in[{-}k(t),k(t)],\\ k(\theta)& \equiv k(0)=s_0(1+\frac{\sigma}{2}),\quad \theta\in[-\max\{\tau_{1},\tau_{2}\},0],\\ \bar{u}(\theta,x)& =\bar{u}(0,x) =\alpha \phi^{\delta}\Bigg(\frac{2x}{2+\sigma}\Bigg), \quad\theta\in[-\tau_2,0],~x\in[{-}k(\theta),k(\theta)],\\ \bar{v}(\theta,x)& =\bar{v}(0,x) =\alpha \varphi^{\delta}\Bigg(\frac{2x}{2+\sigma}\Bigg),\quad\theta\in[-\tau_1,0],~x\in[{-}k(\theta),k(\theta)] \end{align*}

and extend $\bar {u}(t,\,x)$ (resp. $\bar {v}(t,\,x)$) by $0$ for $t\in [-\tau _2,\,+\infty ),\,~x\in (-\infty,\,-k(t))\cup (k(t),\,+\infty )$ (resp. $t\in [-\tau _1,\,+\infty )$, $x\in (-\infty,\,-k(t))\cup (k(t),\,+\infty )$).

Direct calculations show that

\begin{align*} & \bar{u}_t-d_1\bar{u}_{xx}+a_1\bar{u} -h\Bigg(\int_{-\infty}^{+\infty}J_1(x-y)\bar{v}(t-\tau_{1},y){\rm d}y\Bigg)\\ & \quad={-}\sigma \bar{u}-\frac{s_{0}xk^{\prime}(t)}{k^{2}(t)} \alpha e^{-\sigma t}(\phi^{\delta})^{\prime}\Bigg(\frac{s_{0} x}{k(t)}\Bigg) -d_{1}\alpha e^{-\sigma t}\Bigg(\frac{s_{0}}{k(t)}\Bigg)^{2} (\phi^{\delta})^{\prime\prime}\Bigg(\frac{s_{0} x}{k(t)}\Bigg)\\ & \qquad+a_{1}\bar{u}-h\Bigg(\int_{-\infty}^{+\infty}\ J_1(x-y)\bar{v}(t-\tau_{1},y){\rm d}y\Bigg)\\ & \quad\geq{-}\sigma \bar{u}-\frac{s_{0}xk^{\prime}(t)}{k^{2}(t)} \alpha e^{-\sigma t}(\phi^{\delta})^{\prime}\Bigg(\frac{s_{0} x}{k(t)}\Bigg) \\ & \qquad-\alpha e^{-\sigma t}\Bigg(\frac{s_{0}}{k(t)}\Bigg)^{2} \Bigg[(a_{1}+\lambda_{1}^{\delta})\phi^{\delta}\Bigg(\frac{s_{0} x}{k(t)}\Bigg)\\ & \qquad -(h^{\prime}(0)+\delta)\int_{{-}s_{0}}^{s_{0}}J_{1}\left(\frac{s_{0}x}{k(t)}-y\right)\varphi^{\delta}(y){\rm d}y\Bigg]\\ & \qquad+a_{1}\bar{u}-h^{\prime}(0)\int_{-\infty}^{+\infty}J_1(x-y)\bar{v}(t-\tau_{1},y){\rm d}y\\ & \quad={-}\frac{\alpha\sigma^{2}e^{{-}2\sigma t}s_{0}}{2k(t)}z(\phi^{\delta})^{\prime}(z) +\Bigg[a_{1}-\sigma-(a_{1}+\lambda_{1}^{\delta})\Bigg(\frac{s_{0}}{k(t)}\Bigg)^{2}\Bigg] \alpha e^{-\sigma t}\phi^{\delta}(z)\\ & \qquad+\Bigg[\alpha (h^{\prime}(0)+\delta)e^{-\sigma t}\Bigg(\frac{s_{0}}{k(t)}\Bigg)^{2} \int_{{-}s_{0}}^{s_{0}}J_{1}(z-y)\varphi^{\delta}(y){\rm d}y \\ & \qquad -h^{\prime}(0)\int_{-\infty}^{+\infty}J_1\Bigg(\frac{k(t)z}{s_0}-y\Bigg)\bar{v}(t-\tau_{1},y)\Bigg){\rm d}y\Bigg]\\ & \quad=:I+II+III, \end{align*}

where $z:=\tfrac {s_{0}x}{k(t)}\in (-s_0,\,s_0)$.

Since $-z(\phi ^{\delta })^{\prime }(z)|_{z=\pm s_0}>0$ by the Hopf boundary lemma, we have $(I+II)|_{z=\pm s_0}>0$. By the continuity, we know that $I+II>0$ in some neighbourhood $\mathcal {O}\subseteq [-s_0,\,s_0]$ of $z=\pm s_0$. For $z\in [-s_0,\,s_0]\setminus \mathcal {O}$, $\phi ^{\delta }(z)\geq c$ with some positive constant $c$, and then $II\rightarrow -\alpha \lambda _{1}^{\delta }\phi ^{\delta }(z) \geq -\alpha \lambda _{1}^{\delta }c>0$ as $\sigma \rightarrow 0$. Note that $\lim _{\sigma \rightarrow 0}I=0$. We can choose $\sigma$ sufficiently small such that $I+II>0$ on $[-s_0,\,s_0]\setminus \mathcal {O}$. Therefore, $I+II$ is positive on $[-s_0,\,s_0]$ for small $\sigma$.

Now we consider the third term. Since $J_{1}$ is a nonnegative, continuous function satisfying $J_{1}(0)>0$, we have $\int _{-s_0}^{s_0}J_{1}(z-y)\varphi ^{\delta }(y){\rm d}y>0$ for any $z\in [-s_0,\,s_0]$. As $\sigma \rightarrow 0$,

\begin{align*} \begin{array}{@{}ll} III {\kern-9pt}& \rightarrow \alpha \delta\int_{{-}s_0}^{s_0}J_{1}(z-y)\varphi^{\delta}(y){\rm d}y\\ & \geq \alpha \delta\min_{z\in [{-}s_0,s_0]}\int_{{-}s_0}^{s_0}J_{1}(z-y)\varphi^{\delta}(y){\rm d}y =:\alpha \delta c_1>0. \end{array} \end{align*}

By choosing $\sigma$ sufficiently small, we can get $III>0$.

In summary, for $(t,\,x)\in (0,\,+\infty )\times (-k,\,k)$,

\[ \bar{u}_t-d_1\bar{u}_{xx}+a_1\bar{u} -h\Bigg(\int_{-\infty}^{+\infty}J_1(x-y)\bar{v}(t-\tau_{1},y){\rm d}y\Bigg) \geq 0. \]

In a similar way, we can prove that

\[ \bar{v}_t-d_2\bar{v}_{xx}+a_2\bar{v} -g\Bigg(\int_{-\infty}^{+\infty}J_2(x-y)\bar{u}(t-\tau_{2},y){\rm d}y\Bigg) \geq 0. \]

Moreover, choose $\alpha$ large enough such that

\[ \bar{u}_{0}(\theta,x)=\alpha\phi^{\delta}\Bigg(\frac{2x}{2+\sigma}\Bigg) \geq\|u_0\|_{L^{\infty}([-\tau_{2},0]\times[s_1, s_2])} \geq u_0(\theta,x) \]

for $(\theta,\,x)\in [-\tau _{2},\,0]\times [s_1, s_2]$, and

\[ \bar{v}_{0}(\theta,x)=\alpha\varphi^{\delta}\Bigg(\frac{2x}{2+\sigma}\Bigg) \geq\|v_0\|_{L^{\infty}([-\tau_{1},0]\times[s_1, s_2])} \geq v_0(\theta,x) \]

for $(\theta,\,x)\in [-\tau _{1},\,0]\times [s_1, s_2]$. Then take $\mu >0$ sufficiently small such that, for $t>0$,

\begin{align*} k^{\prime}(t)& =s_0\frac{\sigma^2}{2}e^{-\sigma t}\geq{-}\mu[\bar{u}_x(t, k(t))+\rho \bar{v}_x(t, k(t))],\\ -k^{\prime}(t)& ={-}s_0\frac{\sigma^2}{2}e^{-\sigma t}\leq{-}\mu[\bar{u}_x(t, -k(t))+\rho \bar{v}_x(t, -k(t))]. \end{align*}

Besides, it is easy to check that

\[ [{-}k(\theta), k(\theta)] =\Bigg[{-}s_0\left(1+\frac{\sigma}{2}\right),s_0\left(1+\frac{\sigma}{2}\right)\Bigg] \supset[{-}s_0,s_0]\supset [s_1(\theta), s_2(\theta)] \]

for $\theta \in [-\max \{\tau _{1},\,\tau _{2}\},\,0]$.

Therefore, $(\bar {u},\,\bar {v};-k(t),\,k(t))$ is an upper solution of (1.1) and we have

\[{-}s_{1,\infty}, s_{2,\infty}\leq\lim_{t\rightarrow+\infty}k(t)=s_0(1+\sigma), \]

which completes the proof.

By using similar arguments as the proof of Theorem 4.4 in [26], we can obtain the following result for the case $\mathcal {R}_0^{s_0}<1<\mathcal {R}^{*}$ with large $\mu$.

Theorem 3.6 If $\mathcal {R}_0^{s_0}<1<\mathcal {R}^{*}$, then $s_{2,\infty }-s_{1,\infty }=+\infty$ provided that $\mu$ is sufficiently large.

In what follows, we exhibit the sharp criteria of (1.1). The proof relies on the conclusions of theorems 3.4–3.6. More details can be seen in the proofs of Theorem 3.9 in [15] and Theorem 4.5 in [26], here we omit the details.

Theorem 3.7 Assume that $\mathcal {R}^{*}>1$. Then there exists $\mu ^*\in [0,\,+\infty )$ such that $s_{2,\infty }-s_{1,\infty }=+\infty$ for $\mu >\mu ^*$, and $s_{2,\infty }-s_{1,\infty }<+\infty$ for $0<\mu \leq \mu ^*$.

Remark 3.8 In theorems 3.2 and 3.7, we discuss the long-time behaviour of solution for $\mathcal {R}_0\leq 1$ and $\mathcal {R}^{*}>1$, respectively. However, the case $\mathcal {R}^{*}\leq 1<\mathcal {R}_0$ is still unknown.

4. Nonlocal semi-wave problem with delays

In this section, we consider the delayed nonlocal semi-wave problem (1.8). The semi-wave solution of (1.8) will play an important role in determining the precise asymptotic spreading speed of (1.1) when spreading occurs. We always assume $\mathcal {R}^*>1$ in this section.

4.1 Perturbed semi-wave problem

To establish the existence of semi-wave solutions to (1.8), we first consider the corresponding perturbed problem:

(4.1)\begin{equation} \left\{\begin{array}{@{}l} c\phi^\prime(\xi)=d_1\phi^{\prime\prime}(\xi)-a_1\phi(\xi) +h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi(\xi-y-c\tau_1){\rm d}y\Bigg),\quad \xi>0,\\ c\varphi^\prime(\xi)=d_2\varphi^{\prime\prime}(\xi)-a_2\varphi(\xi) +g\Bigg(\int_{-\infty}^{+\infty}J_2(y)\phi(\xi-y-c\tau_2){\rm d}y\Bigg),\quad \xi>0,\\ (\phi(\xi), \varphi(\xi))=(\delta u^*,\delta v^*),\quad \xi\leq0,\\ (\phi(+\infty),\varphi(+\infty))=(u^*,v^*), \end{array}\right. \end{equation}

where $\delta \in (0,\,\frac {1}{2})$ is a small parameter. Then the desired semi-wave solution $(\phi (\xi ), \varphi (\xi ))$ of (1.8) can be obtained from the solutions $(\phi ^{\delta }(\xi ), \varphi ^{\delta }(\xi ))$ of (4.1) by taking $\delta \rightarrow 0$.

For convenient, we denote $\beta _{i1}=\frac {c-\sqrt {c^2+4a_id_i}}{2d_i}$, $\beta _{i2}=\frac {c+\sqrt {c^2+4a_id_i}}{2d_i}$, $i=1,\,2$. Let $\Phi =(\phi,\,\varphi )$, we define the operators $(\mathcal {F}_1, \mathcal {F}_2): C(\mathbb {R},\,\mathbb {R}^2)\rightarrow C(\mathbb {R}, \mathbb {R}^{2})$ by

\begin{align*} & \mathcal{F}_1(\Phi)(\xi)\\ & \quad=\left\{\begin{array}{@{}l} \delta u^*e^{\beta_{11} \xi}+\dfrac{1}{d_1(\beta_{12}-\beta_{11})} \Bigg[\int_{0}^{\xi} (e^{\beta_{11} (\xi-s)}-e^{\beta_{11} \xi-\beta_{12} s})\\ \qquad \times h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi(s-y-c\tau_1){\rm d}y\Bigg){\rm d}s \\ \qquad +\int_{\xi}^{+\infty}(e^{\beta_{12} (\xi-s)}-e^{\beta_{11} \xi-\beta_{12} s}) \\ \qquad \times h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi(s-y-c\tau_1){\rm d}y\Bigg){\rm d}s\Bigg],~\xi>0,\\ \delta u^*,~\xi\leq0, \end{array}\right. \end{align*}

and

\begin{align*} & \mathcal{F}_2(\Phi)(\xi)\\ & \quad=\left\{\begin{array}{l} \delta v^*e^{\beta_{21}\xi}+\dfrac{1}{d_2(\beta_{22}-\beta_{21})}\Bigg[\int_{0}^{\xi} (e^{\beta_{21}(\xi-s)}-e^{\beta_{21}\xi-\beta_{22}s}) \\ \qquad \times g\Bigg(\int_{-\infty}^{+\infty}J_2(y)\phi(s-y-c\tau_2){\rm d}y\Bigg){\rm d}s\\ \qquad +\int_{\xi}^{+\infty}(e^{\beta_{22}(\xi-s)}-e^{\beta_{21}\xi-\beta_{22}s})\\ \qquad \times g\Bigg(\int_{-\infty}^{+\infty}J_2(y)\phi(s-y-c\tau_2){\rm d}y\Bigg){\rm d}s\Bigg],~\xi>0,\\ \delta v^*,~\xi\leq0. \end{array}\right. \end{align*}

It is easy to show that the operators $\mathcal {F}_i$ $(i=1,\,2)$ are well-defined and satisfy

\[ \left\{\begin{array}{@{}l} c(\mathcal{F}_1(\Phi))^\prime(\xi)=d_1(\mathcal{F}_1(\Phi))^{\prime\prime}(\xi)-a_1\mathcal{F}_1(\Phi)(\xi) \\ \qquad +\,h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi(\xi-y-c\tau_1){\rm d}y\Bigg),~\xi>0,\\ c(\mathcal{F}_2(\Phi))^\prime(\xi)=d_2(\mathcal{F}_2(\Phi)^{\prime\prime}(\xi)-a_2\mathcal{F}_2(\Phi)(\xi) \\ \qquad +\,g\Bigg(\int_{-\infty}^{+\infty}J_2(y)\phi(\xi-y-c\tau_2){\rm d}y\Bigg),~\xi>0,\\ (\mathcal{F}_1(\Phi)(\xi),\mathcal{F}_2(\Phi)(\xi))=(\delta u^*,\delta v^*),~\xi\leq0. \end{array}\right. \]

Thus, $(\phi,\,\varphi )$ is a fixed point of $(\mathcal {F}_1, \mathcal {F}_2)$ in $C(\mathbb {R}, \mathbb {R}^2)$ if and only if it is a solution of (4.1) in $C(\mathbb {R}, \mathbb {R}^2)$.

We define the set $\Gamma$ as follows:

\begin{align*} \Gamma & =\Bigg\{(\phi,\varphi)\in C(\mathbb{R}, \mathbb{R}^{2}):~ (i)~\phi(\xi), \varphi(\xi)~\mbox{are~increasing~in}~\xi\in \mathbb{R}_+,\\ & \quad (ii)~(\phi(\xi),\varphi(\xi))=(\delta u^*, \delta v^*)~\mbox{for}~\xi\leq0, ~ (iii)~(\phi(+\infty),\varphi(+\infty))=(u^*,v^*)\Bigg\}. \end{align*}

Lemma 4.1 For any $\Phi =(\phi,\,\varphi )\in \Gamma,$ we have

1. (i) $(\mathcal {F}_1(\Phi )(\xi ), \mathcal {F}_2(\Phi )(\xi ))\geq (0,\,0)$ for any $\xi \in \mathbb {R}$;

2. (ii) $(\mathcal {F}_1(\Phi )(\xi ), \mathcal {F}_2(\Phi )(\xi ))$ are increasing in $\xi \in \mathbb {R}$;

3. (iii) if $\Phi _i=(\phi _i,\,\varphi _i)\in \Gamma$ $(i=1,\,2)$ satisfy $\Phi _1\leq \Phi _2$, then $\mathcal {F}_{i}(\Phi _1)(\xi )\leq \mathcal {F}_{i}(\Phi _2)(\xi )$ for any $\xi \in \mathbb {R}$.

Proof. Since $\beta _{i2}>\beta _{i1}$ $(i=1,\,2)$, we can easily check that $(i)$ and $(iii)$ hold. Now we prove $(ii)$.

By the definitions of $\mathcal {F}_i$, it is sufficient to consider the case $\xi >0$. Note that $\beta _{11}<0$ and $\varphi$ is a positive increasing function. For $\xi >0$, we have

\begin{align*} & (\mathcal{F}_1(\Phi))^{\prime}(\xi) =\delta u^*\beta_{11}e^{\beta_{11}\xi} +\frac{1}{d_1(\beta_{12}-\beta_{11})}\\ & \qquad\times \Bigg[\beta_{11}\int_{0}^{\xi} (e^{\beta_{11}(\xi-s)}-e^{\beta_{11}\xi-\beta_{12}s})h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi(s-y-c\tau_1){\rm d}y\Bigg){\rm d}s \\ & \qquad+\int_{\xi}^{+\infty}(\beta_{12}e^{\beta_{12}(\xi-s)}-\beta_{11}e^{\beta_{11}\xi-\beta_{12}s})h\Bigg(\int_{-\infty}^{+\infty} J_1(y)\varphi(s-y-c\tau_1){\rm d}y\Bigg){\rm d}s\Bigg] \\ & \quad\geq\delta u^*\beta_{11}e^{\beta_{11}\xi}+\frac{1}{d_1(\beta_{12}-\beta_{11})}h\Bigg(\int_{-\infty}^{+\infty} J_1(y)\varphi(\xi-y-c\tau_1){\rm d}y\Bigg)\\ & \qquad \times\Bigg[\beta_{11}\int_{0}^{\xi} (e^{\beta_{11}(\xi-s)}-e^{\beta_{11}\xi-\beta_{12}s}){\rm d}s +\int_{\xi}^{+\infty}(\beta_{12}e^{\beta_{12}(\xi-s)}-\beta_{11}e^{\beta_{11}\xi-\beta_{12}s}){\rm d}s\Bigg]\\ & \quad =\delta u^*\beta_{11}e^{\beta_{11}\xi}+\frac{e^{\beta_{11}\xi}}{d_1\beta_{12}} h\Bigg(\int_{-\infty}^{+\infty} J_1(y)\varphi(\xi-y-c\tau_1){\rm d}y\Bigg)\\ & \quad \geq\delta u^*\beta_{11}e^{\beta_{11}\xi}+\frac{e^{\beta_{11}\xi} }{d_1\beta_{12}}h\Bigg(\int_{-\infty}^{+\infty}\delta v^*J_1(y){\rm d}y\Bigg). \end{align*}

In view of $\int _{-\infty }^{+\infty }J_1(y){\rm d}y=1$ and $h$ is subhomogeneous (see the proof of theorem 3.2),

\begin{align*} (\mathcal{F}_1(\Phi))^{\prime}(\xi) & =\delta u^*\beta_{11}e^{\beta_{11}\xi} +\frac{h(\delta v^*)e^{\beta_{11}\xi}}{d_1\beta_{12}} \geq\delta u^*\beta_{11}e^{\beta_{11}\xi}+\frac{\delta h(v^*)e^{\beta_{11}\xi}}{d_1\beta_{12}}\\ & =\delta u^*\beta_{11}e^{\beta_{11}\xi}+\frac{\delta a_1 u^*e^{\beta_{11}\xi}}{d_1\beta_{12}} \geq 0. \end{align*}

Similarly, we can deduce that $(\mathcal {F}_2(\Phi ))^{\prime }(\xi )\geq 0$ for $\xi >0$.

Next, we give the definitions of upper and lower solutions for (4.1).

Definition 4.2 Assume that $(\bar {\phi },\,\bar {\varphi }), (\underline {\phi },\,\underline {\varphi })$ are continuous function pairs from $\mathbb {R}$ into $[\delta u^*, u^*]\times [\delta v^*, v^*]$. We call that $(\bar {\phi },\,\bar {\varphi }), (\underline {\phi },\,\underline {\varphi })$ are respectively an upper solution and a lower solution of (4.1), if $\bar {\phi }, \underline {\phi }$ are twice continuously differentiable on $\mathbb {R}_+\setminus \{\xi _{i}\}_{i=1}^{m}$, $\bar {\varphi },\,\underline {\varphi }$ are twice continuously differentiable on $\mathbb {R}_+\setminus \{\eta _{j}\}_{j=1}^{k}$, and they satisfy

\[ \left\{\begin{array}{@{}l} c\bar{\phi}^\prime(\xi)\!\geq\! d_1\bar{\phi}^{\prime\prime}(\xi)\!-\!a_1\bar{\phi}(\xi) \!+\!h\Bigg(\int_{-\infty}^{+\infty} J_1(y)\bar{\varphi}(\xi-y-c\tau_1){\rm d}y\Bigg),~\xi>0,~ \xi\notin\{\xi_{i}\}_{i=1}^{m},\\ c\bar{\varphi}^\prime(\xi)\!\geq \! d_2\bar{\varphi}^{\prime\prime}(\xi)\!-\!a_2\bar{\varphi}(\xi) \!+\!g\Bigg(\int_{-\infty}^{+\infty} J_2(y)\bar{\phi}(\xi-y-c\tau_2){\rm d}y\Bigg),~\xi>0,~ \xi\notin\{\eta_{j}\}_{j=1}^{k},\\ \bar{\phi}_+^\prime(\xi_{i})\leq \bar{\phi}_{-}^\prime(\xi_{i}), \quad i=1,\cdots, m,\\ \bar{\varphi}_+^\prime(\eta_{j})\leq \bar{\varphi}_{-}^\prime(\eta_{j}), \quad j=1,\cdots, k,\\ (\bar{\phi}(\xi), \bar{\varphi}(\xi))=(\delta u^*, \delta v^*),\quad \xi\leq 0,\\ (\bar{\phi}(+\infty), \bar{\varphi}(+\infty))=(u^*, v^*) \end{array}\right. \]

and

\[ \left\{\begin{array}{@{}l} c\underline{\phi}^\prime(\xi)\!\leq\! d_1\underline{\phi}^{\prime\prime}(\xi)\!-\!a_1\underline{\phi}(\xi) \!+\!h\Bigg(\int_{-\infty}^{+\infty} J_1(y)\underline{\varphi}(\xi-y-c\tau_1){\rm d}y\Bigg), ~ \xi>0,~ \xi\notin\{\xi_{i}\}_{i=1}^{m},\\ c\underline{\varphi}^\prime(\xi)\!\leq \! d_2\underline{\varphi}^{\prime\prime}(\xi)\!-\!a_2\underline{\varphi}(\xi) \!+\!g\Bigg(\int_{-\infty}^{+\infty} J_2(y)\underline{\phi}(\xi-y-c\tau_2){\rm d}y\Bigg), ~ \xi>0,~ \xi\notin\{\eta_{j}\}_{j=1}^{k},\\ \underline{\phi}_+^\prime(\xi_{i})\geq \underline{\phi}_{-}^\prime(\xi_{i}), \quad i=1,\cdots, m,\\ \underline{\varphi}_+^\prime(\eta_{j})\geq \underline{\varphi}_{-}^\prime(\eta_{j}), \quad j=1,\cdots, k,\\ (\underline{\phi}(\xi), \underline{\varphi}(\xi))=(\delta u^*, \delta v^*),\quad \xi\leq 0,\\ (\underline{\phi}(+\infty), \underline{\varphi}(+\infty))\leq(u^*, v^*). \end{array}\right. \]

Now we establish the existence of solution to the perturbed semi-wave problem (4.1) by applying monotone iteration method, which is an efficient method for travelling wave solutions, see [36].

Theorem 4.3 If there exist an upper solution $(\bar {\phi },\,\bar {\varphi })\in \Gamma$ and a lower solution $(\underline {\phi },\,\underline {\varphi })$ (which is not necessary in $\Gamma$) of (4.1), satisfying $(\delta u^*,\,\delta v^*)\leq (\underline {\phi }(\xi ),\,\underline {\varphi }(\xi ))\leq (\bar {\phi }(\xi ),\,\bar {\varphi }(\xi ))\leq (u^*,\,v^*)$ for $\xi \in \mathbb {R}_+$, then the perturbed problem (4.1) admits an increasing solution.

Proof. The proof is divided into the following three steps.

Step 1: For $n=1,\,2,\,\cdots$, we consider the following iteration scheme

\[ \left\{\begin{array}{@{}l} c\phi_n^\prime(\xi)=d_1\phi_n^{\prime\prime}(\xi)-a_1\phi_n(\xi) +h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi_{n-1}(\xi-y-c\tau_1){\rm d}y\Bigg),~\xi>0,\\ c\varphi_n^\prime(\xi)=d_2\varphi_n^{\prime\prime}(\xi)-a_2\varphi_n(\xi) +g\Bigg(\int_{-\infty}^{+\infty}J_2(y)\phi_{n-1}(\xi-y-c\tau_2){\rm d}y\Bigg),~\xi>0,\\ (\phi_n(\xi),\varphi_n(\xi))=(\delta u^*,\delta v^*),~\xi\leq0,\\ (\phi_0,\varphi_0)=(\bar{\phi},\bar{\varphi}). \end{array}\right. \]

Let $\Phi _{n}(\xi )=(\phi _n(\xi ), \varphi _n(\xi ))$, we have

(4.2)\begin{equation} \phi_n(\xi)=\mathcal{F}_1(\Phi_{n-1})(\xi),~\varphi_n(\xi)=\mathcal{F}_2(\Phi_{n-1})(\xi). \end{equation}

Step 2: We claim that, for each $n=1,\,2,\,\cdots$, $(i)$ $(\phi _n,\,\varphi _n)\in \Gamma$; $(ii)$ $(\underline {\phi }(\xi ),\,\underline {\varphi }(\xi ))\leq (\phi _n(\xi ),\,\varphi _n(\xi ))\leq (\phi _{n-1}(\xi ),\,\varphi _{n-1}(\xi )) \leq (\bar {\phi }(\xi ),\,\bar {\varphi }(\xi ))$ on $\mathbb {R}$.

$(i)$ Since $(\bar {\phi }, \bar {\varphi })\in \Gamma$, $(\bar {\phi }, \bar {\varphi })$ is increasing in $\xi \in \mathbb {R}$. From lemma 4.1 $(ii)$, $(\phi _1, \varphi _1)=(\mathcal {F}_1(\bar {\phi })(\xi ), \mathcal {F}_2(\bar {\varphi })(\xi ))$ is also increasing in $\xi$. By repeating this process, we know that $(\phi _n, \varphi _n)$ is increasing in $\xi$ for each $n\geq 1$.

Next, we prove $(\phi _n(+\infty ), \varphi _n(+\infty ))=(u^*, v^*)$. Note that $\beta _{11}<0$ and $\beta _{12}>0$. By the L'Hôpital's rule,

\begin{align*} & \lim_{\xi\rightarrow+\infty}\phi_1(\xi)\\ & \quad=\lim_{\xi\rightarrow+\infty}\delta u^*e^{\beta_{11}\xi}+ \frac{1}{d_1(\beta_{12}-\beta_{11})}\\& \qquad \times\lim_{\xi\rightarrow+\infty} \Bigg[\frac{\int_{0}^{\xi}e^{-\beta_{11}s} h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\bar{\varphi}(s-y-c\tau_1){\rm d}y\Bigg){\rm d}s} {e^{-\beta_{11}\xi}}\\ & \qquad +\frac{\int_{\xi}^{+\infty}e^{-\beta_{12}s} h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\bar{\varphi}(s-y-c\tau_1){\rm d}y\Bigg){\rm d}s} {e^{-\beta_{12}\xi}}\\ & \qquad -e^{\beta_{11}\xi}\int_{0}^{+\infty}e^{-\beta_{12}s} h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\bar{\varphi}(s-y-c\tau_1){\rm d}y\Bigg){\rm d}s\Bigg]\\ & \quad={-}\frac{1}{d_1(\beta_{12}-\beta_{11})} \Bigg(\frac{1}{\beta_{11}} -\frac{1}{\beta_{12}}\Bigg) \lim_{\xi\rightarrow+\infty}h\Bigg(\int_{-\infty}^{+\infty}\ J_1(y)\bar{\varphi}(\xi-y-c\tau_1){\rm d}y\Bigg)\\ & \quad={-}\frac{1}{d_1\beta_{11}\beta_{12}} h\Bigg(v^*\int_{-\infty}^{+\infty}J_1(y){\rm d}y\Bigg)\\ & \quad={-}\frac{h(v^*)}{d_1\beta_{11}\beta_{12}} =\frac{h(v^*)}{a_{1}}=u^*. \end{align*}

Similarly, we can show $\lim _{\xi \rightarrow +\infty }\varphi _1(\xi )=v^*$.

By repeating the above process, we can obtain $\lim _{\xi \rightarrow +\infty }(\phi _n(\xi ), \varphi _n(\xi )) =(u^*, v^*)$ for each $n=2,\,3,\,\cdots$. Thus, $(\phi _n,\,\varphi _n)\in \Gamma$.

$(ii)$ We first prove $(\phi _1(\xi ),\,\varphi _1(\xi ))\leq (\bar {\phi }(\xi ),\,\bar {\varphi }(\xi ))$.

Let $\xi _0=0$ and $\xi _{m+1}=+\infty$. Assume that $\xi \in (\xi _{i}, \xi _{i+1})$ for some $i\in \{0,\,1,\,\cdots,\,m\}$, we have

\begin{align*} & \phi_{1}(\xi)\\ & =\delta u^*e^{\beta_{11}\xi}+\frac{1}{d_1(\beta_{12}-\beta_{11})} \Bigg[\int_{0}^{\xi} (e^{\beta_{11}(\xi-s)}-e^{\beta_{11}\xi-\beta_{12}s})\\ & \qquad \times h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\bar{\varphi}(s-y-c\tau_1){\rm d}y\Bigg){\rm d}s \\ & \qquad +\int_{\xi}^{+\infty}(e^{\beta_{12}(\xi-s)}-e^{\beta_{11}\xi-\beta_{12}s}) h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\bar{\varphi}(s-y-c\tau_1){\rm d}y\Bigg){\rm d}s\Bigg] \\ & \leq\delta u^*e^{\beta_{11}\xi}+\frac{1}{d_1(\beta_{12}-\beta_{11})} \Bigg[\int_{0}^{\xi} (e^{\beta_{11}(\xi-s)}-e^{\beta_{11}\xi-\beta_{12}s})\\ & \qquad \times\Bigg(c\bar{\phi}^\prime(s)-d_1\bar{\phi}^{\prime\prime}(s)+a_1\bar{\phi}(s)\Bigg){\rm d}s \\ & \qquad +\int_{\xi}^{+\infty}(e^{\beta_{12}(\xi-s)}-e^{\beta_{11}\xi-\beta_{12}s}) \Bigg(c\bar{\phi}^\prime(s)-d_1\bar{\phi}^{\prime\prime}(s)+a_1\bar{\phi}(s)\Bigg){\rm d}s\Bigg]\\ & =\delta u^{*}e^{\beta_{11}\xi}-e^{\beta_{11}\xi}\bar{\phi}(0) +\bar{\phi}(\xi)\\ & \qquad +\frac{1}{\beta_{12}-\beta_{11}} \Bigg[\sum_{k=1}^{i}(e^{\beta_{11}(\xi-\xi_{k})}- e^{\beta_{11}\xi-\beta_{12}\xi_{k}})(\bar{\phi}_+^{\prime}(\xi_{k}) -\bar{\phi}_{-}^{\prime}(\xi_{k}))\\ & \qquad +\sum_{k=i+1}^{m}(e^{\beta_{12}(\xi-\xi_{k})} -e^{\beta_{11}\xi-\beta_{12}\xi_{k}})(\bar{\phi}_+^{\prime}(\xi_{k}) -\bar{\phi}_{-}^{\prime}(\xi_{k}))\Bigg]\\ & \leq \bar{\phi}(\xi). \end{align*}

By the continuity, we can get the same result for the endpoints $\xi _{i}$ $(i=1,\,\cdots,\,m)$. In a similar way, we can prove $\varphi _{1}(\xi )\leq \bar {\varphi }(\xi )$.

By lemma 4.1 $(iii)$, we can deduce that $(\phi _n(\xi ),\,\varphi _n(\xi ))$ is decreasing with respect to $n$. It follows that

\[ (\phi_n(\xi),\varphi_n(\xi))\leq(\phi_{n-1}(\xi),\varphi_{n-1}(\xi))\leq\cdots \leq (\phi_1(\xi),\varphi_1(\xi))\leq(\bar{\phi}(\xi),\bar{\varphi}(\xi)) \]

for each $n\geq 2$. Moreover, it is easy to check that $(\phi _n(\xi ),\,\varphi _n(\xi ))\geq (\underline {\phi }(\xi ),\,\underline {\varphi }(\xi ))$ for each $n\geq 1$.

Step 3: We claim that (4.1) has an increasing solution $(\phi,\,\varphi )$.

According to Step 2 $(ii)$, $(\phi (\xi ),\,\varphi (\xi ))=\lim _{n\rightarrow +\infty }(\phi _n(\xi ),\,\varphi _n(\xi ))$ exists for $\xi \in \mathbb {R}$, and satisfies $(\underline {\phi }(\xi ),\,\underline {\varphi }(\xi ))\leq (\phi (\xi ),\,\varphi (\xi )) \leq (\bar {\phi }(\xi ),\,\bar {\varphi }(\xi ))$. Moreover, $(\phi (\xi ),\,\varphi (\xi ))$ is increasing in $\xi \in \mathbb {R}$, $(\phi (\xi ),\,\varphi (\xi ))=(\delta u^*,\,\delta v^*)$ for $\xi \leq 0$, and $(\phi (+\infty ),\,\varphi (+\infty ))=(u^*, v^*)$.

Direct calculations yield that $(\phi (\xi ),\,\varphi (\xi ))=\lim _{n\rightarrow +\infty }(\phi _n(\xi ),\,\varphi _n(\xi ))$ satisfies the equations in (4.1), which completes the proof.

Next, we construct a pair of upper and lower solutions of (4.1).

For any fixed $c>0$, we choose $m>0$ sufficiently large such that

\[ 0<\frac{1}{m}<\min\Bigg\{\frac{c}{a_{1}},\frac{c}{a_{2}},c\tau_1,c\tau_2\Bigg\}. \]

Define

(4.3)\begin{align} \bar{\phi}(\xi)& = \left\{\begin{array}{@{}l} \delta u^*,~\xi\leq0,\\ u^*+u^*(\delta-1)(m\xi-1)^{2},~0<\xi\leq\dfrac{1}{m},\\ u^*,~\xi>\dfrac{1}{m}, \end{array}\right. \end{align}

(4.4)\begin{align} \bar{\varphi}(\xi)& = \left\{\begin{array}{@{}l} \delta v^*,~\xi\leq0,\\ v^*+v^*(\delta-1)(m\xi-1)^{2},~0<\xi\leq\dfrac{1}{m},\\ v^*,~\xi>\dfrac{1}{m}, \end{array}\right. \end{align}

and

\[ (\underline{\phi}(\xi),\underline{\varphi}(\xi))=(\delta u^*,\delta v^*),\quad\xi\in \mathbb{R}. \]

Lemma 4.4 $(\bar {\phi }(\xi ),\,\bar {\varphi }(\xi ))$ and $(\underline {\phi }(\xi ),\,\underline {\varphi }(\xi ))$ as defined above are respectively an upper solution and a lower solution of (4.1). Moreover, $(\bar {\phi },\,\bar {\varphi })\in \Gamma$.

Proof. It is easy to check that $(\bar {\phi },\,\bar {\varphi })\in \Gamma$.

$(i)$ For $0<\xi <\frac {1}{m}$, we have $\delta v^*\leq \bar {\varphi }(\xi )\leq v^*$, $\xi \in \mathbb {R}$. By simple calculations,

\begin{align*} & c\bar{\phi}^\prime(\xi)-d_1\bar{\phi}^{\prime\prime}(\xi)+a_1\bar{\phi}(\xi) -h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\bar{\varphi}(\xi-y-c\tau_1){\rm d}y\Bigg)\\ & \quad=u^*(1-\delta)\Bigg[{-}2mc(m\xi-1)+2d_{1}m^{2}-a_1(m\xi-1)^2\Bigg]+a_1u^*\\ & \qquad-h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\bar{\varphi}(\xi-y-c\tau_1){\rm d}y\Bigg)\\ & \quad\geq u^*(1-\delta)\Bigg[{-}2mc(m\xi-1)+2d_{1}m^{2}-a_1(m\xi-1)^2\Bigg]\\ & \qquad +a_1u^*-h\Bigg(\int_{-\infty}^{+\infty}J_1(y)v^*{\rm d}y\Bigg)\\ & \quad=u^*(1-\delta)\Bigg[{-}2mc(m\xi-1)+2d_{1}m^{2}-a_1(m\xi-1)^2\Bigg] +a_1u^*-h(v^*)\\ & \quad=u^*m^2(1-\delta)\Bigg[{-}2c\left(\xi-\frac{1}{m}\right)+2d_1-a_1(\xi-\frac{1}{m})^2\Bigg]. \end{align*}

Due to $-\frac {c}{a_1}\leq -\frac {1}{m}\leq s-\frac {1}{m}<0$, we have

\[{-}2c\left(\xi-\frac{1}{m}\right)+2d_1-a_1\left(\xi-\frac{1}{m}\right)^2>0. \]

It follows that

\[ c\bar{\phi}^\prime(\xi)-d_1\bar{\phi}^{\prime\prime}(\xi)+a_1\bar{\phi}(\xi) -h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\bar{\varphi}(\xi-y-c\tau_1){\rm d}y\Bigg)\geq0. \]

$(ii)$ For $\xi >\frac {1}{m}$, we have $\bar {\phi }(\xi )=u^*$ and $\delta v^*\leq \bar {\varphi }(\xi )\leq v^*$ for $\xi \in \mathbb {R}_+$. Then

\begin{align*} & c\bar{\phi}^\prime(\xi)-d_1\bar{\phi}^{\prime\prime}(\xi)+a_1\bar{\phi}(\xi) -h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\bar{\varphi}(\xi-y-c\tau_1){\rm d}y\Bigg)\\ & \quad\geq a_1u^*-h\Bigg(\int_{-\infty}^{+\infty}J_1(y)v^*{\rm d}y\Bigg) =a_1u^*-h(v^*)=0. \end{align*}

In summary,

\begin{align*} & c\bar{\phi}^\prime(\xi)-d_1\bar{\phi}^{\prime\prime}(\xi)+a_1\bar{\phi}(\xi) -h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\bar{\varphi}(\xi-y-c\tau_1){\rm d}y\Bigg)\geq0, \\ & \qquad\forall\xi\in \mathbb{R}_+{\setminus} \left\{\frac{1}{m}\right\}. \end{align*}

Similarly, we can prove that

\begin{align*} & c\bar{\varphi}^\prime(\xi)-d_2\bar{\varphi}^{\prime\prime}(\xi)+a_2\bar{\varphi}(\xi) -g\Bigg(\int_{-\infty}^{+\infty}J_2(y)\bar{\phi}(\xi-y-c\tau_2){\rm d}y\Bigg)\geq0,\\ & \qquad\forall\xi\in \mathbb{R}_+{\setminus} \left\{\frac{1}{m}\right\}. \end{align*}

Moreover, $\bar {\phi }^\prime _+(\frac {1}{m})=\bar {\phi }^\prime _-(\frac {1}{m})=0$ and $\bar {\varphi }^\prime _+(\frac {1}{m})=\bar {\varphi }^\prime _-(\frac {1}{m})=0$. Thus, $(\bar {\phi },\,\bar {\varphi })\in \Gamma$ is an upper solution of (4.1).

Next, we prove that $(\underline {\phi }(\xi ),\,\underline {\varphi }(\xi ))=(\delta u^*,\,\delta v^*)$, $\xi \in \mathbb {R}$ is a lower solution of (4.1).

Obviously, for $\xi >0$,

\begin{align*} & c\underline{\phi}^\prime(\xi)-d_1\underline{\phi}^{\prime\prime}(\xi)+a_1\underline{\phi}(\xi) -h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\underline{\varphi}(\xi-y-c\tau_1){\rm d}y\Bigg)\\ & \quad=a_1\delta u^* -h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\underline{\varphi}(\xi-y-c\tau_1){\rm d}y\Bigg)\\ & \quad=a_1\delta u^*-h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\delta v^*{\rm d}y\Bigg) \leq a_1\delta u^*-\delta h(v^*)=0. \end{align*}

Similarly, we can obtain

\[ c\underline{\varphi}^\prime(\xi)-d_2\underline{\varphi}^{\prime\prime}(\xi)+a_2\underline{\varphi}(\xi) -g\Bigg(\int_{-\infty}^{+\infty}J_2(y)\underline{\phi}(\xi-y-c\tau_2){\rm d}y\Bigg)\leq0\quad \mbox{for}~\xi>0, \]

which means that $(\underline {\phi },\,\underline {\varphi })$ is a lower solution of (4.1).

Theorem 4.5 For all $\delta \in (0,\,\frac {1}{2})$, the perturbed semi-wave problem (4.1) admits an increasing solution $(\phi ^\delta (\xi ),\,\varphi ^\delta (\xi ))$. Moreover, $(\phi ^{\delta }(\xi ),\,\varphi ^{\delta }(\xi ))$ obtained in this way is increasing with respect to $\delta \in (0,\,\frac {1}{2})$.

Proof. From theorem 4.3 and lemma 4.4, we can establish the existence of increasing solution to (4.1).

Assume that $0<\delta _{1}<\delta _{2}<\frac {1}{2}$. In view of the definitions of $(\bar {\phi }, \bar {\varphi })$ in (4.3)–(4.4), we have $(\bar {\phi }^{\delta _{2}}, \bar {\varphi }^{\delta _{2}}) >(\bar {\phi }^{\delta _{1}}, \bar {\varphi }^{\delta _{1}})$. Applying the iteration scheme (4.2) and lemma 4.1 $(iii)$, we get

\begin{align*} (\phi_{1}^{\delta_{2}}, \varphi_{1}^{\delta_{2}}) & =(\mathcal{F}_1(\bar{\phi}^{\delta_{2}}, \bar{\varphi}^{\delta_{2}})(\xi), \mathcal{F}_2(\bar{\phi}^{\delta_{2}}, \bar{\varphi}^{\delta_{2}})(\xi))\\ & \geq (\mathcal{F}_1(\bar{\phi}^{\delta_{1}}, \bar{\varphi}^{\delta_{1}})(\xi), \mathcal{F}_2(\bar{\phi}^{\delta_{1}}, \bar{\varphi}^{\delta_{1}})(\xi)) =(\phi_{1}^{\delta_{1}}, \varphi_{1}^{\delta_{1}}). \end{align*}

By repeating the above process, we can obtain $(\phi _{n}^{\delta _{2}}, \varphi _{n}^{\delta _{2}})\geq (\phi _{n}^{\delta _{1}}, \varphi _{n}^{\delta _{1}})$ for each $n\geq 1$. It follows that the two limit solutions satisfy $(\phi ^{\delta _{2}}, \varphi ^{\delta _{2}})\geq (\phi ^{\delta _{1}}, \varphi ^{\delta _{1}})$, which completes the proof.

We remark that, for the perturbed semi-wave problem (4.1), the iteration monotone method is more efficient than the Schauder's fixed point method applied in [10, 11, 35], especially in proving the monotonicity of semi-wave solution with respect to the parameter $\delta$.

4.2 Original semi-wave problem

Theorem 4.6 For any fixed $c>0$, either the semi-wave problem (1.8) or the following problem

(4.5)\begin{equation} \left\{\begin{array}{@{}l} c\phi^\prime(\xi)=d_1\phi^{\prime\prime}(\xi)-a_1\phi(\xi) +h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi(\xi-y-c\tau_1){\rm d}y\Bigg),~\xi\in \mathbb{R},\\ c\varphi^\prime(\xi)=d_2\varphi^{\prime\prime}(\xi)-a_2\varphi(\xi) +g\Bigg(\int_{-\infty}^{+\infty}J_2(y)\phi(\xi-y-c\tau_2){\rm d}y\Bigg),~\xi\in \mathbb{R},\\ (\phi(-\infty),\varphi(-\infty))=(0,0),\quad (\phi(+\infty),\varphi(+\infty))=(u^*,v^*) \end{array}\right. \end{equation}

has an increasing solution $(\phi,\,\varphi )$, but they can not occur simultaneously.

Proof. $(i)$ Assume that $\{\delta _{n}\}_{n=1}^{\infty }$ is a sequence satisfying $\delta _{n}\in (0,\,\frac {1}{2})$ and $\delta _{n}\searrow 0$ as $n\rightarrow \infty$. By theorem 4.5, the perturbed problem (4.1) with $\delta =\delta _{n}$ has an increasing solution $(\phi ^{\delta _{n}}, \varphi ^{\delta _{n}})$. Define $\xi _{n}:=\max \{\xi : \phi ^{\delta _{n}}(\xi )=\frac {1}{2}u^{*}\}$. From theorem 4.5, we can deduce that $\xi _{n}$ is increasing with respect to $n$, and then $\xi _{0}:=\lim _{n\rightarrow \infty }\xi _{n}\in (0,\,+\infty ]$ is well-defined.

Define $(\tilde {\phi }_{n}(\xi ), \tilde {\varphi }_{n}(\xi )):=(\phi ^{\delta _{n}}(\xi +\xi _{n}), \varphi ^{\delta _{n}}(\xi +\xi _{n}))$, $\forall \xi \in \mathbb {R}$. Then $\tilde {\phi }_{n}(0)=\frac {1}{2}u^{*}$ and $(\tilde {\phi }_{n}(\xi ), \tilde {\varphi }_{n}(\xi ))$ satisfies

\[ \left\{\begin{array}{@{}l} c\tilde{\phi}_{n}^\prime(\xi)=d_1\tilde{\phi}_{n}^{\prime\prime}(\xi)-a_1\tilde{\phi}_{n}(\xi) +h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\tilde{\varphi}_{n}(\xi-y-c\tau_1){\rm d}y\Bigg),~ \xi>{-}\xi_{n},\\ c\tilde{\varphi}_{n}^\prime(\xi)=d_2\tilde{\varphi}_{n}^{\prime\prime}(\xi)-a_2\tilde{\varphi}_{n}(\xi) +g\Bigg(\int_{-\infty}^{+\infty}J_2(y)\tilde{\phi}_{n}(\xi-y-c\tau_2){\rm d}y\Bigg),~ \xi>{-}\xi_{n},\\ (\tilde{\phi}_{n}(\xi), \tilde{\varphi}_{n}(\xi))=(\delta_{n} u^*, \delta_{n} v^*),~ \xi\leq{-}\xi_{n},\\ (\tilde{\phi}_{n}(+\infty), \tilde{\varphi}_{n}(+\infty))=(u^*, v^*). \end{array}\right. \]

Note that $\textbf {0}\leq (\phi ^{\delta _{n}}, \varphi ^{\delta _{n}})\leq (u^*,\,v^*)$, i.e., $(\phi ^{\delta _{n}}, \varphi ^{\delta _{n}})$ are uniformly bounded with respect to $n$. From the integration presentations of solution $(\phi ^{\delta _{n}}, \varphi ^{\delta _{n}})=(\mathcal {F}_{1}(\phi ^{\delta _{n}}, \varphi ^{\delta _{n}}), \mathcal {F}_{2}(\phi ^{\delta _{n}}, \varphi ^{\delta _{n}}))$, we can easily deduce that $(\phi ^{\delta _{n}}, \varphi ^{\delta _{n}})$ are uniformly bounded in $C^{2}(\mathbb {R}_+)$ with respect to $n$. By the Arzela–Ascoli theorem, there is a subsequence of $(\tilde {\phi }_{n}, \tilde {\varphi }_{n})$, which converges to $(\tilde {\phi },\,\tilde {\varphi })$ in $C_{loc}^{2}(\mathbb {R})$. Obviously, $(\tilde {\phi }(\xi ),\,\tilde {\varphi }(\xi ))$ is increasing in $\xi$ and satisfies $\tilde {\phi }(0)=\frac {1}{2}u^{*}$.

Case $I$: $\xi _{0}=+\infty$. In such a case, $(\tilde {\phi }(\xi ),\,\tilde {\varphi }(\xi ))$ satisfies

\[ \left\{\begin{array}{@{}l} c\tilde{\phi}^\prime(\xi)=d_1\tilde{\phi}^{\prime\prime}(\xi)-a_1\tilde{\phi}(\xi) +h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\tilde{\varphi}(\xi-y-c\tau_1){\rm d}y\Bigg),~\xi\in \mathbb{R},\\ c\tilde{\varphi}^\prime(\xi)=d_2\tilde{\varphi}^{\prime\prime}(\xi)-a_2\tilde{\varphi}(\xi) +g\Bigg(\int_{-\infty}^{+\infty}J_2(y)\tilde{\phi}(\xi-y-c\tau_2){\rm d}y\Bigg),~\xi\in \mathbb{R}. \end{array}\right. \]

Since $(\tilde {\phi }(\xi ),\,\tilde {\varphi }(\xi ))$ is increasing and uniformly continuous on $\mathbb {R}_+$, by lemma 2.3 in [36] we can deduce that $\lim _{\xi \rightarrow \infty }\tilde {\phi }^{\prime }(\xi ) =\lim _{\xi \rightarrow \infty }\tilde {\phi }^{\prime \prime }(\xi )=0$ and $\lim _{\xi \rightarrow \infty }\tilde {\varphi }^{\prime }(\xi ) =\lim _{\xi \rightarrow \infty }\tilde {\varphi }^{\prime \prime }(\xi )=0$, which imply that $(\tilde {\phi }(\pm \infty ), \tilde {\varphi }(\pm \infty ))=(0, 0)$ or $(u^*, v^*)$. In view of $\tilde {\phi }(0)=\frac {1}{2}u^{*}$, we know that $(\tilde {\phi }(-\infty ), \tilde {\varphi }(-\infty ))=(0, 0)$ and $(\tilde {\phi }(+\infty ), \tilde {\varphi }(+\infty ))=(u^*, v^*)$.

Case $II$: $\xi _{0}\in (0,\,+\infty )$. In such a case, $(\tilde {\phi }(\xi ),\,\tilde {\varphi }(\xi ))$ satisfies

\[ \left\{\begin{array}{@{}l} c\tilde{\phi}^\prime(\xi)=d_1\tilde{\phi}^{\prime\prime}(\xi)-a_1\tilde{\phi}(\xi) +h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\tilde{\varphi}(\xi-y-c\tau_1){\rm d}y\Bigg),~ \xi>{-}\xi_{0},\\ c\tilde{\varphi}^\prime(\xi)=d_2\tilde{\varphi}^{\prime\prime}(\xi)-a_2\tilde{\varphi}(\xi) +g\Bigg(\int_{-\infty}^{+\infty}J_2(y)\tilde{\phi}(\xi-y-c\tau_2){\rm d}y\Bigg),~ \xi>{-}\xi_{0},\\ (\tilde{\phi}(\xi), \tilde{\varphi}(\xi))=(0, 0),~ \xi\leq{-}\xi_{0}. \end{array}\right. \]

Let $(\phi (\xi ),\,\varphi (\xi ))=(\tilde {\phi }(\xi -\xi _{0}),\,\tilde {\varphi }(\xi -\xi _{0}))$, we can also prove $(\phi (+\infty ), \varphi (+\infty ))=(u^*, v^*)$. Obviously, $(\phi (\xi ), \varphi (\xi ))=(0, 0)$ for $\xi \leq 0$. This completes the proof of the first part.

$(ii)$ We prove that the two cases can not happen simultaneously for any fixed $c>0$.

Suppose, to the contrary, that there exists some $c_{0}>0$ such that (4.5) and (1.8) have two increasing solutions $\Phi _{1}(\xi )=(\phi _{1}(\xi ),\,\varphi _{1}(\xi ))$ and $\Phi _{2}(\xi )=(\phi _{2}(\xi ),\,\varphi _{2}(\xi ))$, respectively. Similar as in the proof of lemma 2.10 in [17], for any $\theta \in \mathbb {R}$ and some fixed $k\in (0,\,1)$, we define

\begin{align*} \Phi_{1}^{\theta}(\xi)& =(\phi_{1}^{\theta}(\xi),\varphi_{1}^{\theta}(\xi))=(\phi_{1}(\xi+\theta),\varphi_{1}(\xi+\theta)),\\ \tilde{\Phi}_{2}(\xi)& =(\tilde{\phi}_{2}(\xi),\tilde{\varphi}_{2}(\xi))=(k\phi_{2}(\xi),k\varphi_{2}(\xi))=k\Phi_{2}(\xi),\\ \hat{\Phi}^{\theta}(\xi)& =(\hat{\phi}^{\theta}(\xi), \hat{\varphi}^{\theta}(\xi)) =(\phi_{1}^{\theta}(\xi)-\tilde{\phi}_{2}(\xi),\varphi_{1}^{\theta}(\xi)-\tilde{\varphi}_{2}(\xi)) =\Phi_{1}^{\theta}(\xi)-\tilde{\Phi}_{2}(\xi). \end{align*}

Obviously, $\hat {\Phi }^{\theta }$ is increasing in $\theta$, and then $(\sigma _1(\theta ), \sigma _2(\theta )):=(\inf _{\xi \geq 0}\hat {\phi }^{\theta }(\xi ), \inf _{\xi \geq 0}\hat {\varphi }^{\theta }(\xi ))$ is increasing, continuous in $\theta$.

Note that $\hat {\Phi }^{\theta }(\xi )=\Phi _{1}^{\theta }(\xi )-\tilde {\Phi }_{2}(\xi ) \geq \Phi _{1}^{\theta }(0)-\tilde {\Phi }_{2}(+\infty ) = \Phi _{1}(\theta )-k(u^*, v^*)$ for any $\xi \geq 0$. Since $\lim _{\theta \rightarrow +\infty }\Phi _{1}(\theta )=(u^*, v^*)$, we have $\lim _{\theta \rightarrow +\infty }\hat {\Phi }^{\theta }(\xi )\geq (1-k)(u^*, v^*)$ uniformly on $[0,\,+\infty )$. Then there exists sufficiently large $\bar {\theta }\gg 1$ (independent of $\xi$) such that

(4.6)\begin{equation} \hat{\Phi}^{\theta}(\xi)>\frac{1}{2}(1-k)(u^*, v^*)>\textbf{0} \end{equation}

on $[0,\,+\infty )$ for all $\theta >\bar {\theta }$. Moreover, as $\theta \rightarrow -\infty$,

(4.7)\begin{equation} \hat{\Phi}^{\theta}(1)=\Phi_{1}^{\theta}(1)-\tilde{\Phi}_{2}(1) =\Phi_{1}(1+\theta)-k\Phi_{2}(1)\rightarrow -k\Phi_{2}(1)<\textbf{0}. \end{equation}

Since $(\sigma _1(\theta ), \sigma _2(\theta ))$ is increasing, continuous in $\theta$, by (4.6) and (4.7), there exist $\theta _{1},\,\theta _{2}\in \mathbb {R}$ such that $\sigma _1(\theta )>0$ for $\theta >\theta _{1}$, $\sigma _1(\theta _1)=0$, and $\sigma _2(\theta )>0$ for $\theta >\theta _{2}$, $\sigma _2(\theta _2)=0$.

We may assume that $\theta _{1}\geq \theta _{2}$. It follows that $\hat {\Phi }^{\theta _1}\geq \textbf {0}$ for $\xi \geq 0$. It is easy to check that $(\hat {\phi }^{\theta _1}(+\infty ),\,\hat {\varphi }^{\theta _2}(+\infty ))=(1-k)(u^*,\,v^*)>\textbf {0}$, and $(\hat {\phi }^{\theta _1}(0),\,\hat {\varphi }^{\theta _2}(0))=(\phi _{1}^{\theta _1}(0), \varphi _{1}^{\theta _2}(0))>\textbf {0}$. Then $\sigma _1(\theta _1)=\inf _{\xi \geq 0}\hat {\phi }^{\theta _1}(\xi )=0$ is attainable at some $\xi _1\in (0,\,+\infty )$, i.e., $\hat {\phi }^{\theta _1}(\xi _1)=0$.

Since $k\in (0,\,1)$ and $h,\,g$ are subhomogeneous, we have

\begin{align*} & h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi_{1}(\xi+\theta_{1}-y-c_{0}\tau_1){\rm d}y\Bigg) -kh\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi_{2}(\xi-y-c_{0}\tau_1){\rm d}y\Bigg)\\ & \quad\geq h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi_{1}(\xi+\theta_{1}-y-c_{0}\tau_1){\rm d}y\Bigg) \\ & \qquad -h\Bigg(\int_{-\infty}^{+\infty}J_1(y)k\varphi_{2}(\xi-y-c_{0}\tau_1){\rm d}y\Bigg)\\ & \quad=h^{\prime}(\vartheta)\int_{-\infty}^{+\infty}J_1(y)\hat{\varphi}^{\theta_1}(\xi-y-c_{0}\tau_1){\rm d}y \geq0. \end{align*}

Similarly, we can get

\[ g\Bigg(\int_{-\infty}^{+\infty}J_2(y)\phi_{1}(\xi+\theta_{1}\!-\!y-c_{0}\tau_2){\rm d}y\Bigg) \!-\!kg\Bigg(\int_{-\infty}^{+\infty}J_2(y)\phi_{2}(\xi\!-\!y\!-\!c_{0}\tau_2){\rm d}y\Bigg) \!\geq\!0. \]

Thus, $\hat {\Phi }^{\theta _1}$ satisfies

\[ \left\{\begin{array}{@{}l} c_{0}(\hat{\phi}^{\theta_1})^\prime(\xi)-d_1(\hat{\phi}^{\theta_1})^{\prime\prime}(\xi)+a_1\hat{\phi}^{\theta_1}(\xi) =h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi_{1}(\xi+\theta_{1}-y-c_{0}\tau_1){\rm d}y\Bigg)\\ \qquad -kh\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi_{2}(\xi-y-c_{0}\tau_1){\rm d}y\Bigg) \geq0,~\xi> 0,\\ c_{0}(\hat{\varphi}^{\theta_1})^\prime(\xi)-d_2(\hat{\varphi}^{\theta_1})^{\prime\prime}(\xi)+a_2\hat{\varphi}^{\theta_1}(\xi) =g\Bigg(\int_{-\infty}^{+\infty}J_2(y)\phi_{1}(\xi+\theta_{1}-y-c_{0}\tau_2){\rm d}y\Bigg)\\ \qquad-kg\Bigg(\int_{-\infty}^{+\infty}J_2(y)\phi_{2}(\xi-y-c_{0}\tau_2){\rm d}y\Bigg) \geq0,~\xi> 0,\\ (\hat{\phi}^{\theta_1}(+\infty),\hat{\varphi}^{\theta_2}(+\infty))=(1-k)(u^*,v^*)>\textbf{0},\\ (\hat{\phi}^{\theta_1}(\xi),\hat{\varphi}^{\theta_2}(\xi))\geq \textbf{0},~ \xi\geq 0. \end{array}\right. \]

By the maximum principle for single equation, we have $(\hat {\phi }^{\theta _1}(\xi ),\,\hat {\varphi }^{\theta _2}(\xi ))> \textbf {0}$ on $(0,\,+\infty )$, which contradicts with $\hat {\phi }^{\theta _1}(\xi _1)=0$. This completes the proof.

Remark 4.7 Problem (4.5) with fixed $c>0$ has an increasing solution $(\phi,\,\varphi )$ is equivalent to the evolution system

(4.8)\begin{equation} \left\{\begin{array}{@{}l} \partial_t u=d_1\partial_{xx}u-a_1u+h\Bigg(\int_{-\infty}^{+\infty}J_1(x-y)v(t-\tau_1,y) {\rm d}y\Bigg),~t>0,~x\in \mathbb{R},\\ \partial_t v=d_2\partial_{xx}v-a_2v+g\Bigg(\int_{-\infty}^{+\infty}J_2(x-y)u(t-\tau_2,y) {\rm d}y\Bigg), ~t>0,~x\in \mathbb{R} \end{array}\right. \end{equation}

admits an increasing travelling wave solution $(u(t,\,x),\,v(t,\,x)):=(\phi (x+ct),\,\varphi (x+ct))$.

Define

\[ c_{\tau}^{*}=\inf_{\lambda>0}\frac{\chi_{\tau}(\lambda)}{\lambda}, \]

where $\chi _{\tau }(\lambda )$ is a real root of

(4.9)\begin{equation} \begin{aligned} P(\chi) & := \chi^{2}-[(d_{1}\lambda^{2}-a_{1})+(d_{2}\lambda^{2}-a_{2})]\chi +(d_{1}\lambda^{2}-a_{1})(d_{2}\lambda^{2}-a_{2})\\ & \qquad -h^{\prime}(0)g^{\prime}(0)e^{-\chi(\tau_{1}+\tau_{2})}\int_{-\infty}^{+\infty}J_{1}(y)e^{-\lambda y}{\rm d}y\int_{-\infty}^{+\infty}J_{2}(y)e^{-\lambda y}{\rm d}y\\ & =0, \end{aligned} \end{equation}

and $\chi _{\tau }(\lambda )$ is greater than the real parts of all other roots.

Note that $\lim _{\chi \rightarrow +\infty }P(\chi )=+\infty$ and $P(d_{1}\lambda ^{2}-a_{1}), P(d_{2}\lambda ^{2}-a_{2})<0$. Then

\[ \chi_{\tau}(\lambda)>\max\{d_{1}\lambda^{2}-a_{1}, d_{2}\lambda^{2}-a_{2}\}, \]

which implies that $\lim _{\lambda \rightarrow +\infty }\frac {\chi _{\tau }(\lambda )}{\lambda }=+\infty$. Similarly, since $\mathcal {R}_{0}\geq \mathcal {R}^*>1$, we can deduce $\chi _{\tau }(0)>0$, and then $\lim _{\lambda \rightarrow 0+}\frac {\chi _{\tau }(\lambda )}{\lambda }=+\infty$. Thus, $\inf _{\lambda >0}\frac {\chi _{\tau }(\lambda )}{\lambda }$ is attainable at some $\lambda ^{*}\in (0,\,+\infty )$, i.e.,

\[ c^{*}_{\tau}=\inf_{\lambda>0}\frac{\chi_{\tau}(\lambda)}{\lambda}=\frac{\chi_{\tau}(\lambda^{*})}{\lambda^{*}}. \]

Let $c=\frac {\chi (\lambda )}{\lambda }$, we have $\frac {dc}{d \lambda }|_{\lambda =\lambda ^{*}}$=0. Define

\begin{align*} \Delta(\lambda,c) & :=(d_{1}\lambda^{2}-c\lambda-a_{1})(d_{2}\lambda^{2}-c\lambda-a_{2})\\ & \quad~ -h^{\prime}(0)g^{\prime}(0)e^{{-}c\lambda(\tau_{1}+\tau_{2})}\int_{-\infty}^{+\infty}J_{1}(y)e^{-\lambda y}{\rm d}y\int_{-\infty}^{+\infty}J_{2}(y)e^{-\lambda y}{\rm d}y. \end{align*}

Then $(c^{*}_{\tau },\,\lambda ^{*})$ can be determined as the positive solution to the system

\[ \Delta(\lambda,c)=0,\quad \frac{\partial \Delta(\lambda,c)}{\partial \lambda}=0. \]

Theorem 4.8 The semi-wave problem (1.8) admits an increasing solution for $0< c< c^*_{\tau }$, but has no increasing solution for $c\geq c^*_{\tau }$.

Proof. By the theory of monotone semiflows developed in [21], there exists $c_{0}>0$ such that $c_{0}$ is the asymptotic spreading speed. Moreover, the asymptotic spreading speed $c_{0}$ coincides with the minimal wave speed, that is, (4.8) has an increasing travelling wave solution for $c\geq c_0$, but no such a solution for $0< c< c_0$. If $c_{0}=c^*_{\tau }$, then we can get the desired result by applying remark 4.7 and theorem 4.6.

Now it is sufficient to prove $c_{0}=c^*_{\tau }$. Set $\mathcal {C}=C([-\tau _{2},\,0]\times \mathbb {R}, \mathbb {R})\times C([-\tau _{1},\,0]\times \mathbb {R}, \mathbb {R})$, $\bar {\mathcal {C}}=C([-\tau _{2},\,0], \mathbb {R})\times C([-\tau _{1},\,0], \mathbb {R})$. Let $M_t=(M_{t}^{u}, M_{t}^{v}): \mathcal {C}\rightarrow \mathcal {C}$ be the solution map at time $t$ of the following linear equations

\[ \left\{\begin{array}{@{}l} \partial_{t}u =d_{1}\partial_{xx}u -a_{1}u+h^{\prime}(0)J_{1}*v_{t},\\ \partial_{t}v =d_{2}\partial_{xx}v -a_{2}v+g^{\prime}(0)J_{2}*u_{t}. \end{array}\right. \]

For $\lambda \geq 0$, we define the linear map $B_{t}=(B^{u}_{t},\,B^{v}_{t}): \bar {\mathcal {C}}\rightarrow \bar {\mathcal {C}}$ by

\begin{align*} B^{u}_{t}[(\varphi_{1},\varphi_{2})](\theta) & =M_{t}^{u}[(\varphi_{1},\varphi_{2})e^{-\lambda x}](\theta,0), ~\forall \theta\in [-\tau_{2},0],\\ B^{v}_{t}[(\varphi_{1},\varphi_{2})](\theta) & =M_{t}^{v}[(\varphi_{1},\varphi_{2})e^{-\lambda x}](\theta,0), ~\forall \theta\in [-\tau_{1},0]. \end{align*}

Then $B_{t}=(B^{u}_{t},\,B^{v}_{t}): \bar {\mathcal {C}}\rightarrow \bar {\mathcal {C}}$ is the solution map of the following equations

(4.10)\begin{equation} \left\{\begin{array}{@{}l} u^{\prime}(t) =d_{1}\lambda^{2}u(t) -a_{1}u(t)+h^{\prime}(0)\Bigg(\int_{-\infty}^{+\infty}J_{1}(y)e^{-\lambda y}{\rm d}y\Bigg) v_{t},\\ v^{\prime}(t) =d_{2}\lambda^{2}v(t) -a_{2}v(t)+g^{\prime}(0)\Bigg(\int_{-\infty}^{+\infty}J_{2}(y)e^{-\lambda y}{\rm d}y\Bigg) u_{t}. \end{array}\right. \end{equation}

Let

\[ A(\chi) =\left( \begin{array}{cc} d_{1}\lambda^{2}-a_{1} & h^{\prime}(0)e^{-\chi \tau_{1}}\int_{-\infty}^{+\infty}J_{1}(y)e^{-\lambda y}{\rm d}y \\ g^{\prime}(0)e^{-\chi \tau_{2}}\int_{-\infty}^{+\infty}J_{2}(y)e^{-\lambda y}{\rm d}y & d_{2}\lambda^{2}-a_{2} \\ \end{array} \right). \]

Since (4.10) is a cooperative and irreducible delay equations, it follows that

\[ \det (\chi I-A(\chi))=0, \]

i.e.,

\[ \begin{array}{l} P(\chi)= \chi^{2}-[(d_{1}\lambda^{2}-a_{1})+(d_{2}\lambda^{2}-a_{2})]\chi +(d_{1}\lambda^{2}-a_{1})(d_{2}\lambda^{2}-a_{2})\\ \qquad\qquad -h^{\prime}(0)g^{\prime}(0)e^{-\chi(\tau_{1}+\tau_{2})}\int_{-\infty}^{+\infty}J_{1}(y)e^{-\lambda y}{\rm d}y\int_{-\infty}^{+\infty}J_{2}(y)e^{-\lambda y}{\rm d}y =0, \end{array} \]

admits a real root $\chi _{\tau }(\lambda )$ which is greater than the real parts of all other ones (see Theorem 5.5.1 in [23]).

By Theorem 3.10 in [21], we know that the spreading speed $c_{0}=\inf _{\lambda >0}\frac {\chi _{\tau }(\lambda )}{\lambda }$. Thus, $c_{0}=c^*_{\tau }$, which completes the proof.

Theorem 4.9 For any $c\in (0,\,c^*_{\tau })$, the solution of (1.8) obtained in theorem 4.8 is unique and strictly increasing on $\mathbb {R}_+$.

Proof. $(i)$ (Strict monotonicity) For any $\theta >0$, we have

\begin{align*} & \phi(\xi+\theta)\\ & \quad=\frac{1}{d_1(\beta_{12}-\beta_{11})} \Bigg[\int_{0}^{\xi+\theta}(e^{\beta_{11}(\xi+\theta-s)}-e^{\beta_{11}(\xi+\theta)-\beta_{12}s})\\ & \qquad\quad \times h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi(s-y-c\tau_1){\rm d}y\Bigg){\rm d}s\\ & \qquad+\int_{\xi+\theta}^{+\infty} (e^{\beta_{12}(\xi+\theta-s)}-e^{\beta_{11}(\xi+\theta)-\beta_{12}s}) h\Bigg(\int_{-\infty}^{+\infty} J_1(y)\varphi(s-y-c\tau_1){\rm d}y\Bigg){\rm d}s\Bigg]\\ & \quad=\frac{1}{d_1(\beta_{12}-\beta_{11})} \Bigg[\int_{-\theta}^{\xi} (e^{\beta_{11}(\xi-\tilde{s})}-e^{\beta_{11}(\xi+\theta)-\beta_{12}(\tilde{s}+\theta)})\times\\ & \qquad h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi(\tilde{s}+\theta-y-c\tau_1){\rm d}y\Bigg){\rm d}\tilde{s} +\int_{\xi}^{+\infty}(e^{\beta_{12}(\xi-\tilde{s})}-e^{\beta_{11}(\xi+\theta) -\beta_{12}(\tilde{s}+\theta)}) \\ & \qquad \quad\times h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi(\tilde{s}+\theta-y-c\tau_1){\rm d}y\Bigg){\rm d}\tilde{s}\Bigg]\\ & \quad>\frac{1}{d_1(\beta_{12}-\beta_{11})}\Bigg[\int_{0}^{\xi}(e^{\beta_{11}(\xi-\tilde{s})}- e^{\beta_{11}\xi-\beta_{12}\tilde{s}}) h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi(\tilde{s}-y-c\tau_1){\rm d}y\Bigg){\rm d}\tilde{s} \\ & \qquad +\int_{\xi}^{+\infty}(e^{\beta_{12}(\xi-\tilde{s})}-e^{\beta_{11}\xi-\beta_{12}\tilde{s}}) h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi(\tilde{s}-y-c\tau_1){\rm d}y\Bigg){\rm d}\tilde{s}\Bigg]\ = \phi(\xi). \end{align*}

In a similar way, we can obtain $\varphi (\xi +\theta )>\varphi (\xi )$ for $\xi \in \mathbb {R}_+$.

$(ii)$ (uniqueness) Fix $c\in (0,\,c^*_{\tau })$, suppose that $(\phi _1,\,\varphi _1)$ and $(\phi _2,\,\varphi _2)$ are two strictly increasing solutions of (1.8). Then, for $i=1,\,2$, $(0,\,0)<(\phi _i(\xi ),\,\varphi _i(\xi ))<(u^*,\,v^*)$ for $\xi >0$ and $(\phi _i(+\infty ),\,\varphi _i(+\infty ))=(u^*,\,v^*)$. Moreover, by the Hopf's boundary lemma, we have $(\phi _i)^\prime _+(0)>0$, $(\varphi _i)^\prime _+(0)>0$ for $i=1,\,2$.

We define

\begin{align*} \rho_1& :=\inf\{\rho\geq1:\rho\phi_1(\xi)>\phi_2(\xi),\forall\xi>0\},\\ \rho_2& :=\inf\{\rho\geq1:\rho\varphi_1(\xi)>\varphi_2(\xi),\forall\xi>0\}, \end{align*}

and $\rho ^*:=\max \{\rho _1,\,\rho _2\}$.

We will show that $\rho ^*=1$. Otherwise, $\rho ^*>1$. Denote $\tilde {\phi }=\rho ^*\phi _1-\phi _2$ and $\tilde {\varphi }=\rho ^*\varphi _1-\varphi _2$. Obviously, $\tilde {\phi }(\xi )\geq 0$, $\tilde {\varphi }(\xi )\geq 0$ for $\xi \geq 0$, $\tilde {\phi }(0)=\tilde {\varphi }(0)=0$, $\tilde {\phi }(+\infty )=(\rho ^*-1)u^*$ and $\tilde {\varphi }(+\infty )=(\rho ^*-1)v^*$. Since $h$ is subhomogeneous, we obtain, for $\xi >0$,

\begin{align*} & c\tilde{\phi}^\prime(\xi)-d_1\tilde{\phi}^{\prime\prime}(\xi)+a_1\tilde{\phi}(\xi)\\ & \quad=\rho^{*}h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi_1(\xi\!-\!y\!-\!c\tau_1){\rm d}y\Bigg) -h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi_2(\xi-y-c\tau_1){\rm d}y\Bigg)\\ & \quad=\rho^{*}h\Bigg(\frac{1}{\rho^{*}}\int_{-\infty}^{+\infty}J_1(y)\rho^{*}\varphi_1(\xi-y-c\tau_1){\rm d}y\Bigg)\\ & \qquad -h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi_2(\xi-y-c\tau_1){\rm d}y\Bigg)\\ & \quad\geq h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\rho^{*}\varphi_1(\xi\!-\!y\!-\!c\tau_1){\rm d}y\Bigg) -h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi_2(\xi-y-c\tau_1){\rm d}y\Bigg)\\ & \quad\geq0. \end{align*}

Similarly, we can deduce that $c\tilde {\varphi }^\prime (\xi )-d_2\tilde {\varphi }^{\prime \prime }(\xi )+a_2\tilde {\varphi }(\xi )\geq 0$ for $\xi >0$. By the Hopf's boundary lemma, we have $\tilde {\phi }_+^\prime (0)>0$ and $\tilde {\varphi }_+^\prime (0)>0$.

In view of the L'Hôpital's rule, $\lim _{\xi \rightarrow 0+}\frac {\tilde {\phi }(\xi )}{\phi _{2}(\xi )} =\frac {\tilde {\phi }_+^\prime (0)}{(\phi _2)_+^\prime (0)}>0$ and $\lim _{\xi \rightarrow 0+}\frac {\tilde {\varphi }(\xi )}{\varphi _{2}(\xi )} =\frac {\tilde {\varphi }_+^\prime (0)}{(\varphi _2)_+^\prime (0)}>0$. Note that $\lim _{\xi \rightarrow +\infty }\frac {\tilde {\phi }(\xi )}{\phi _{2}(\xi )} =\frac {(\rho ^*-1)u^*}{u^*}>0$, $\lim _{\xi \rightarrow +\infty }\frac {\tilde {\varphi }(\xi )}{\varphi _{2}(\xi )} =\frac {(\rho ^*-1)v^*}{v^*}>0$. Thus, there exist constants $\varepsilon _1, \varepsilon _2>0$ such that $\frac {\tilde {\phi }}{\phi _2}>\varepsilon _1$ and $\frac {\tilde {\varphi }}{\varphi _2}>\varepsilon _2$ for $\xi >0$. It follows that

\[ \frac{\rho^*}{1+\varepsilon_1}\phi_1(\xi)\geq\phi_2(\xi),\quad \frac{\rho^*}{1+\varepsilon_2}\varphi_1(\xi)\geq\varphi_2(\xi) \quad \mbox{for}~\xi>0, \]

which contradicts the definition of $\rho ^*$. Thus, $\rho ^*=1$, which implies that $(\phi _1(\xi ),\,\varphi _1(\xi ))\geq (\phi _2(\xi ),\,\varphi _2(\xi ))$ for $\xi \geq 0$. Clearly, the same method can be used to prove $(\phi _1(\xi ),\,\varphi _1(\xi ))\leq (\phi _2(\xi ),\,\varphi _2(\xi ))$ for $\xi \geq 0$. Hence, we get the uniqueness of solution.

In what follows, we exhibit some properties of the strictly increasing solution of (1.8).

Lemma 4.10 For any fixed $c\in (0, c^*_{\tau })$, let $(\phi _{\tau }^{c},\,\varphi _{\tau }^{c})$ be the unique strictly increasing solution of (1.8).

1. (i) For $0< c_1< c_2< c^*_{\tau }$, then $((\phi _{\tau }^{c_1})^\prime _+(0),\,(\varphi _{\tau }^{c_1})^\prime _+(0))>((\phi _{\tau }^{c_2})^\prime _+(0), (\varphi _{\tau }^{c_2})^\prime _+(0))$, and $(\phi _{\tau }^{c_1}(\xi ),\,\varphi _{\tau }^{c_1}(\xi ))>(\phi _{\tau }^{c_2}(\xi ),\,\varphi _{\tau }^{c_2}(\xi ))$ for $\xi >0$.

2. (ii) For any fixed $\mu,\,\rho >0$, there exists a unique $c_{\tau }=c^{\mu,\rho }_{\tau }\in (0,\,c^*_{\tau })$ such that

   \[ \mu[(\phi_{\tau}^{c_{\tau}})^\prime_+(0)+\rho(\varphi_{\tau}^{c_{\tau}})^\prime_+(0)]=c_{\tau}. \]

3. (iii) If $(\tau _1,\,\tau _2)\leq (\tilde {\tau }_1,\,\tilde {\tau }_2)$, then $c^{*}_{\tilde {\tau }}\leq c^{*}_{\tau }$, $c_{\tilde {\tau }}\leq c_{\tau }$ with $\tilde {\tau }=\tilde {\tau }_1+\tilde {\tau }_2$ and $\tau =\tau _1+\tau _2$.

Proof. Similarly as the proof of Theorem 4.6 in [30], we can prove $(i)$ and $(ii)$, here we omit the details.

Next we prove $(iii)$. Recall that in (4.9), $\chi _{\tau }(\lambda )$ can be seen as an intersection of two curves:

\begin{align*} f_{1}(\chi)& =\chi^{2}-[(d_{1}\lambda^{2}-a_{1})+(d_{2}\lambda^{2}-a_{2})]\chi +(d_{1}\lambda^{2}-a_{1})(d_{2}\lambda^{2}-a_{2}),\\ f_{2}(\chi)& =h^{\prime}(0)g^{\prime}(0)e^{-\chi\tau} \int_{-\infty}^{+\infty}J_{1}(y)e^{-\lambda y}{\rm d}y\int_{-\infty}^{+\infty}J_{2}(y)e^{-\lambda y}{\rm d}y. \end{align*}

The function $f_{1}$ is independent of $\tau$, and $f_{2}$ is decreasing in $\tau$. If $\tau \leq \tilde {\tau }$, then the two intersections satisfy $\chi _{\tau }(\lambda )\geq \chi _{\tilde {\tau }}(\lambda )$, which implies

\[ c^{*}_{\tau}=\inf_{\lambda>0}\frac{\chi_{\tau}(\lambda)}{\lambda} \geq \inf_{\lambda>0}\frac{\chi_{\tilde{\tau}}(\lambda)}{\lambda}=c^{*}_{\tilde{\tau}}. \]

Now we prove $c_{\tau }\geq c_{\tilde {\tau }}$. Note that $c_{\tilde {\tau }}\in (0, c^{*}_{\tilde {\tau }})$ and $c_{\tau }\in (0, c^{*}_{\tau })$. If $c_{\tau }\geq c^{*}_{\tilde {\tau }}$, then we have $c_{\tau }\geq c^{*}_{\tilde {\tau }}>c_{\tilde {\tau }}$, which completes the proof.

Next, we assume $c_{\tau }< c^{*}_{\tilde {\tau }}$. In such a case, $c_{\tau }, c_{\tilde {\tau }}\in (0, c^{*}_{\tilde {\tau }})$. In view of $(ii)$, to get the desired result, we only need to prove that $((\phi _{\tau }^{c})^\prime _+(0),\,(\varphi _{\tau }^{c})^\prime _+(0))\geq ((\phi _{\tilde {\tau }}^{c})^\prime _+(0), (\varphi _{\tilde {\tau }}^{c})^\prime _+(0))$ for any $c\in (0, c^{*}_{\tilde {\tau }})$.

Since $(\phi ^{c}_{\tilde {\tau }}(\xi ), \varphi ^{c}_{\tilde {\tau }}(\xi ))$ is increasing on $\mathbb {R}$, we have

\[ \left\{\begin{array}{@{}l} c(\phi^{c}_{\tilde{\tau}})^{\prime}(\xi)-d_1(\phi^{c}_{\tilde{\tau}})^{\prime\prime}(\xi) +a_{1}\phi^{c}_{\tilde{\tau}}(\xi) =h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi^{c}_{\tilde{\tau}}(\xi-y-c\tilde{\tau}_1){\rm d}y\Bigg)\\ \leq h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi^{c}_{\tilde{\tau}}(\xi-y-c\tau_1){\rm d}y\Bigg),\quad \xi>0,\\ c(\varphi^{c}_{\tilde{\tau}})^{\prime}(\xi)-d_2(\varphi^{c}_{\tilde{\tau}})^{\prime\prime}(\xi) +a_{2}\varphi^{c}_{\tilde{\tau}}(\xi) =g\Bigg(\int_{-\infty}^{+\infty}J_2(y)\phi^{c}_{\tilde{\tau}}(\xi-y-c\tilde{\tau}_2){\rm d}y\Bigg)\\ \leq g\Bigg(\int_{-\infty}^{+\infty}J_2(y)\phi^{c}_{\tilde{\tau}}(\xi-y-c\tau_2){\rm d}y\Bigg),\quad\xi>0,\\ \phi^{c}_{\tilde{\tau}}(\xi)=\varphi^{c}_{\tilde{\tau}}(\xi)=0,~ \xi\leq0, \end{array}\right. \]

which implies that $(\phi ^{c}_{\tilde {\tau }}(\xi ), \varphi ^{c}_{\tilde {\tau }}(\xi ))$ is a lower solution of the following problem

(4.11)\begin{equation} \left\{\begin{array}{@{}l} \phi_t=d_1\phi_{\xi\xi}-c\phi_{\xi}-a_{1}\phi+h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi(\xi-y-c\tau_1){\rm d}y\Bigg),~ t>0,~\xi>0,\\ \varphi_t=d_1\varphi_{\xi\xi}-c\varphi_{\xi}-a_{1}\varphi+h\Bigg(\int_{-\infty}^{+\infty}J_2(y)\phi(\xi-y-c\tau_2){\rm d}y\Bigg),~ t>0,~\xi>0,\\ \phi(t,\xi)=\varphi(t,\xi)=0,~ t>0,~\xi\leq0,\\ (\phi(0,\xi), \varphi(0,\xi))=(\phi^{c}_{\tilde{\tau}}(\xi), \varphi^{c}_{\tilde{\tau}}(\xi)). \end{array}\right. \end{equation}

By the maximum principle, the solution $(\phi (t,\,\xi ), \varphi (t,\,\xi ))$ of (4.11) is increasing in $t\geq 0$ and satisfies $\lim _{t\rightarrow +\infty }(\phi (t,\,\xi ), \varphi (t,\,\xi ))=(\phi ^*(\xi ), \varphi ^*(\xi ))$, where $(\phi ^*(\xi ), \varphi ^*(\xi ))$ is a solution of (1.8). Clearly, the uniqueness of the solution to (1.8) ensures that $(\phi ^*(\xi ), \varphi ^*(\xi ))=(\phi ^{c}_{\tau }(\xi ), \varphi ^{c}_{\tau }(\xi ))$. Thus, for all $\xi >0$, we have

\begin{align*} (\phi^{c}_{\tilde{\tau}}(\xi), \varphi^{c}_{\tilde{\tau}}(\xi))& =(\phi(0,\xi), \varphi(0,\xi)) \leq(\phi(t,\xi), \varphi(t,\xi))\\ & \leq(\phi(+\infty,\xi), \varphi(+\infty,\xi)) =(\phi^{c}_{\tau}(\xi), \varphi^{c}_{\tau}(\xi)). \end{align*}

Let $\hat {\phi }(\xi )=\phi ^{c}_{\tau }(\xi )-\phi ^{c}_{\tilde {\tau }}(\xi )$ and $\hat {\varphi }(\xi )=\varphi ^{c}_{\tau }(\xi )-\varphi ^{c}_{\tilde {\tau }}(\xi )$, then $(\hat {\phi },\,\hat {\varphi })$ satisfies

\[ \left\{\begin{array}{@{}l} c\hat{\phi}^{\prime}(\xi)-d_1\hat{\phi}^{\prime\prime}(\xi)+a_{1}\hat{\phi}(\xi) =h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi^{c}_{\tau}(\xi-y-c\tau_1){\rm d}y\Bigg)\\ \qquad\qquad\qquad\qquad\qquad\qquad -h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi^{c}_{\tilde{\tau}}(\xi-y-c\tilde{\tau}_1){\rm d}y\Bigg) \geq 0,~\xi>0,\\ c\hat{\varphi}^{\prime}(\xi)-d_2\hat{\varphi}^{\prime\prime}(\xi)+a_{2}\hat{\varphi}(\xi) =g\Bigg(\int_{-\infty}^{+\infty}J_2(y)\phi^{c}_{\tau}(\xi-y-c\tau_2){\rm d}y\Bigg)\\ \qquad\qquad\qquad\qquad\qquad\qquad -g\Bigg(\int_{-\infty}^{+\infty}J_2(y)\phi^{c}_{\tilde{\tau}}(\xi-y-c\tilde{\tau}_2){\rm d}y\Bigg) \geq 0,~\xi>0,\\ \hat{\phi}(0)=0,~\hat{\varphi}(0)=0. \end{array}\right. \]

The Hopf boundary lemma yields $\hat {\phi }^{\prime }(0)>0$ and $\hat {\varphi }^{\prime }(0)>0$, that is, $(\phi ^{c}_{\tau })^{\prime }_+(0)>(\phi ^{c}_{\tilde {\tau }})^{\prime }_+(0)$ and $(\varphi ^{c}_{\tau })^{\prime }_+(0)>(\varphi ^{c}_{\tilde {\tau }})^{\prime }_+(0)$. This completes the proof.

5. Asymptotic spreading speed

In this section, by employing the semi-wave solutions, we determine the asymptotic spreading speeds of free boundaries when spreading occurs.

Proof of theorem 1.4 We divide the proof into the following two steps.

Step 1. We prove $\liminf _{t\rightarrow +\infty }\frac {s_1(t)}{t}\geq -c_{\tau }$ and $\limsup _{t\rightarrow +\infty }\frac {s_2(t)}{t}\leq c_{\tau }$.

Consider the following auxiliary semi-wave problem

(5.1)\begin{equation} \left\{\begin{array}{@{}l} c\phi^\prime(\xi)=d_1\phi^{\prime\prime}(\xi)-(a_1-2\varepsilon)\phi(\xi) +h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi(\xi-y-c\tau_1){\rm d}y\Bigg),~\xi>0,\\ c\varphi^\prime(\xi)=d_2\varphi^{\prime\prime}(\xi)-(a_2-2\varepsilon) \varphi(\xi) +g\Bigg(\int_{-\infty}^{+\infty}J_2(y)\phi(\xi-y-c\tau_2){\rm d}y\Bigg),~\xi>0,\\ (\phi(\xi),\varphi(\xi))=(0,0),~\xi\leq 0,\\ (\phi(+\infty),\varphi(+\infty))=(u_{2\varepsilon}^*,v_{2\varepsilon}^*), \end{array}\right. \end{equation}

where $\varepsilon >0$ is a small constant, and $(u_{2\varepsilon }^*,\,v_{2\varepsilon }^*)$ is the unique positive equilibrium for the first two equations of (5.1). By theorem 4.10 $(ii)$, there exists a unique $c_{\tau,2\varepsilon }>0$ such that

\[ \mu[(\phi_{2\varepsilon}^{c_{\tau,2\varepsilon}})^\prime_+(0)+\rho(\varphi_{2\varepsilon}^{c_{\tau,2\varepsilon}})^\prime_+(0)] =c_{\tau,2\varepsilon}, ~\lim_{\varepsilon\rightarrow0^+}c_{\tau,2\varepsilon}=c_{\tau}, \]

where $(\phi _{2\varepsilon }^{c_{\tau,2\varepsilon }},\,\varphi _{2\varepsilon }^{c_{\tau,2\varepsilon }})$ is the unique strictly increasing solution of (5.1) with $c=c_{\tau,2\varepsilon }$.

Let $(\bar {u}(t),\,\bar {v}(t))$ be the solution of (3.1). Since $\mathcal {R}_{0}\geq \mathcal {R}^{*}>1$, from Theorem 3.2 in [42], we can show that $\lim _{t\rightarrow \infty }(\bar {u}(t),\,\bar {v}(t))=(u^*,\,v^*)$. The comparison principle implies $(u(t,\,x),\,v(t,\,x))\leq (\bar {u}(t),\,\bar {v}(t))$ for $t>0,\,~x\in (s_1(t),\,s_2(t))$. Note that $(u^*,\,v^*)<(u_{\varepsilon }^*,\,v_{\varepsilon }^*)$. Thus, there exists sufficiently large $T_0>0$ such that

\begin{align*} & u(t,x)\leq u_\varepsilon^*,~\forall t\geq T_0,~x\in[s_1(t),s_2(t)],\\ & v(t,x)\leq v_\varepsilon^*,~\forall t\geq T_0,~x\in[s_1(t),s_2(t)]. \end{align*}

Since $(\phi _{2\varepsilon }^{c_{\tau,2\varepsilon }}(+\infty ), \varphi _{2\varepsilon }^{c_{\tau,2\varepsilon }}(+\infty )) =(u_{2\varepsilon }^*,\,v_{2\varepsilon }^*)>(u_{\varepsilon }^*,\,v_{\varepsilon }^*)$, there exists $\xi _{0}>s_{2}(T_{0}+\max \{\tau _1,\,\tau _2\})$ such that

\[ \Bigg(\phi_{2\varepsilon}^{c_{\tau,2\varepsilon}}(\xi_{0}\!-\!s_{2}(T_{0}\!+\!\max\{\tau_1,\tau_2\})),\quad \varphi_{2\varepsilon}^{c_{\tau,2\varepsilon}}(\xi_{0}-s_{2}(T_{0}+\max\{\tau_1,\tau_2\}))\Bigg) >(u_{\varepsilon}^*,v_{\varepsilon}^*). \]

Define

\begin{align*} \bar{s}_2(t)& = c_{\tau,2\varepsilon}(t-T_0)+\xi_0,\quad t\geq T_0,\\ \bar{u}(t,x)& = \left\{\begin{array}{@{}l} \phi_{2\varepsilon}^{c_{\tau,2\varepsilon}}(\bar{s}_2(t)+x),~t\geq T_0,~x\in[-\bar{s}_2(t),0],\\ \phi_{2\varepsilon}^{c_{\tau,2\varepsilon}}(\bar{s}_2(t)-x),~t\geq T_0,~x\in[0,\bar{s}_2(t)], \end{array}\right. \end{align*}

and

\[ \bar{v}(t,x)= \left\{\begin{array}{@{}l} \varphi_{2\varepsilon}^{c_{\tau,2\varepsilon}}(\bar{s}_2(t)+x),~t\geq T_0,~x\in[-\bar{s}_2(t),0],\\ \varphi_{2\varepsilon}^{c_{\tau,2\varepsilon}}(\bar{s}_2(t)-x),~t\geq T_0,~x\in[0,\bar{s}_2(t)]. \end{array}\right. \]

For $t\geq T_{0}+\max \{\tau _1,\,\tau _2\}$ and $x\in [0, \bar {s}_{2}(t))$, by the symmetry of $J_{1}$ and the monotonicity of $\varphi _{2\varepsilon }^{c_{\tau,2\varepsilon }}$ and $h$, we have

\begin{align*} & h\Bigg(\int_{-\infty}^{+\infty}J_{1}(x-y)\bar{v}(t-\tau_{1},y){\rm d}y\Bigg) =h\Bigg(\int_{-\infty}^{+\infty}J_{1}(y)\bar{v}(t-\tau_{1},x+y){\rm d}y\Bigg)\\ & \quad=h\Bigg(\int_{-\infty}^{{-}x}J_{1}(y)\varphi_{2\varepsilon}^{c_{\tau,2\varepsilon}} (\bar{s}_{2}(t-\tau_1)+x+y){\rm d}y \\ & \qquad +\int_{{-}x}^{+\infty}J_{1}(y)\varphi_{2\varepsilon}^{c_{\tau,2\varepsilon}} (\bar{s}_{2}(t-\tau_1)-x-y){\rm d}y\Bigg)\\ & \quad\leq h\Bigg(\int_{-\infty}^{{-}x}J_{1}(y)\varphi_{2\varepsilon}^{c_{\tau,2\varepsilon}} (\bar{s}_{2}(t-\tau_1)-x-y){\rm d}y \\ & \qquad +\int_{{-}x}^{+\infty}J_{1}(y)\varphi_{2\varepsilon}^{c_{\tau,2\varepsilon}} (\bar{s}_{2}(t-\tau_1)-x-y){\rm d}y\Bigg)\\ & \quad=h\Bigg(\int_{-\infty}^{+\infty}J_{1}(y)\varphi_{2\varepsilon}^{c_{\tau,2\varepsilon}} (\bar{s}_{2}(t-\tau_1)-x-y){\rm d}y\Bigg), \end{align*}

and then

\begin{align*} & \bar{u}_{t}-d_{1}\bar{u}_{xx}+a_{1}\bar{u} -h\Bigg(\int_{-\infty}^{+\infty}J_{1}(x-y)\bar{v}(t-\tau_{1},y){\rm d}y\Bigg)\\ & \quad=c_{\tau,2\varepsilon}(\phi_{2\varepsilon}^{c_{\tau,2\varepsilon}})^{\prime}(\bar{s}_2(t)-x) -d_{1}(\phi_{2\varepsilon}^{c_{\tau,2\varepsilon}})^{\prime\prime}(\bar{s}_2(t)-x) +a_{1}\phi_{2\varepsilon}^{c_{\tau,2\varepsilon}}(\bar{s}_2(t)-x)\\ & \qquad -h\Bigg(\int_{-\infty}^{+\infty}J_{1}(x-y)\bar{v}(t-\tau_{1},y){\rm d}y\Bigg)\\ & \quad=2\varepsilon\phi_{2\varepsilon}^{c_{\tau,2\varepsilon}}(\bar{s}_2(t)-x) +h\Bigg(\int_{-\infty}^{+\infty}J_{1}(y)\varphi_{2\varepsilon}^{c_{\tau,2\varepsilon}} (\bar{s}_{2}(t)-x-y-c_{\tau,2\varepsilon}\tau_{1}){\rm d}y\Bigg)\\ & \qquad\quad -h\Bigg(\int_{-\infty}^{+\infty}J_{1}(x-y)\bar{v}(t-\tau_{1},y){\rm d}y\Bigg)\\ & \quad=2\varepsilon\phi_{2\varepsilon}^{c_{\tau,2\varepsilon}}(\bar{s}_2(t)-x) +h\Bigg(\int_{-\infty}^{+\infty}J_{1}(y)\varphi_{2\varepsilon}^{c_{\tau,2\varepsilon}} (\bar{s}_{2}(t-\tau_1)-x-y){\rm d}y\Bigg)\\ & \qquad -h\Bigg(\int_{-\infty}^{+\infty}J_{1}(x-y)\bar{v}(t-\tau_{1},y){\rm d}y\Bigg)\\ & \quad\geq 0. \end{align*}

For $t\geq T_{0}+\max \{\tau _1,\,\tau _2\}$ and $x\in (-\bar {s}_{2}(t), 0)$, we also have

\begin{align*} & \bar{u}_{t}-d_{1}\bar{u}_{xx}+a_{1}\bar{u} -h\Bigg(\int_{-\infty}^{+\infty}J_{1}(x-y)\bar{v}(t-\tau_{1},y){\rm d}y\Bigg)\\ & \quad=c_{\tau,2\varepsilon}(\phi_{2\varepsilon}^{c_{\tau,2\varepsilon}})^{\prime}(\bar{s}_2(t)+x) -d_{1}(\phi_{2\varepsilon}^{c_{\tau,2\varepsilon}})^{\prime\prime}(\bar{s}_2(t)+x) +a_{1}\phi_{2\varepsilon}^{c_{\tau,2\varepsilon}}(\bar{s}_2(t)+x)\\ & \qquad -h\Bigg(\int_{-\infty}^{+\infty}J_{1}(x-y)\bar{v}(t-\tau_{1},y){\rm d}y\Bigg)\\ & \quad=2\varepsilon\phi_{2\varepsilon}^{c_{\tau,2\varepsilon}}(\bar{s}_2(t)+x) +h\Bigg(\int_{-\infty}^{+\infty}J_{1}(y)\varphi_{2\varepsilon}^{c_{\tau,2\varepsilon}} (\bar{s}_{2}(t)+x-y-c_{\tau,2\varepsilon}\tau_{1}){\rm d}y\Bigg)\\ & \qquad -h\Bigg(\int_{-\infty}^{+\infty}J_{1}(y)\bar{v}(t-\tau_{1},x-y){\rm d}y\Bigg)\\ & \quad=2\varepsilon\phi_{2\varepsilon}^{c_{\tau,2\varepsilon}}(\bar{s}_2(t)+x) +h\Bigg(\int_{-\infty}^{+\infty}J_{1}(y)\varphi_{2\varepsilon}^{c_{\tau,2\varepsilon}} (\bar{s}_{2}(t-\tau_1)+x-y){\rm d}y\Bigg)\\ & \qquad -h\Bigg(\int_{-\infty}^{+\infty}J_{1}(y)\bar{v}(t-\tau_{1},x-y){\rm d}y\Bigg)\\ & \quad\geq 0. \end{align*}

The inequality satisfied by $\bar {v}$ can be proved similarly. In terms of the choices of $T_0$ and $\xi _0$, we can check that $(\bar {u}(t,\,x),\,\bar {v}(t,\,x);-\bar {s}_2(t),\,\bar {s}_2(t))$ is an upper solution of (1.1) with $t>0$ in lemma 2.2 replaced by $t\geq T_0+\max \{\tau _1,\,\tau _2\}$. Applying the comparison principle, we have $s_1(t)\geq -\bar {s}_2(t)$ and $s_2(t)\leq \bar {s}_2(t)$ for $t\geq T_0+\max \{\tau _1,\,\tau _2\}$, and then

\begin{align*} & \liminf_{t\rightarrow+\infty}\frac{s_1(t)}{t} \geq\liminf_{t\rightarrow+\infty}\frac{-\bar{s}_2(t)}{t} \geq{-}c_{\tau,2\varepsilon},\\ & \limsup_{t\rightarrow+\infty}\frac{s_2(t)}{t} \leq\limsup_{t\rightarrow+\infty}\frac{\bar{s}_2(t)}{t} \leq c_{\tau,2\varepsilon}. \end{align*}

Taking $\varepsilon \rightarrow 0^+$, we can get the desired result.

Step 2. We show $\limsup _{t\rightarrow +\infty }\frac {s_1(t)}{t}\leq -c_{\tau }$ and $\liminf _{t\rightarrow +\infty }\frac {s_2(t)}{t}\geq c_{\tau }$.

We consider another auxiliary semi-wave problem

(5.2)\begin{equation} \left\{\begin{array}{@{}l} c\phi^\prime(\xi)=d_1\phi^{\prime\prime}(\xi)-a_1\phi(\xi) +h\Bigg(\int_{-\infty}^{+\infty}J_1(y)\varphi(\xi-y-c\tau_1){\rm d}y-\zeta\Bigg),~\xi>0,\\ c\varphi^\prime(\xi)=d_2\varphi^{\prime\prime}(\xi)-a_2\varphi(\xi) +g\Bigg(\int_{-\infty}^{+\infty}J_2(y)\phi(\xi-y-c\tau_2){\rm d}y-\zeta\Bigg),~\xi>0,\\ (\phi(\xi),\varphi(\xi))=(0,0),~\xi\leq 0,\\ (\phi(+\infty),\varphi(+\infty))=(u_{\zeta}^*,v_{\zeta}^*), \end{array}\right. \end{equation}

where $\zeta >0$ is a small constant, and $(u_{\zeta }^*,\,v_{\zeta }^*)$ is the unique positive equilibrium for the first two equations of (5.2). By theorem 4.10 $(ii)$, there exists a unique $c_{\tau,\zeta }>0$ such that

\[ \mu[(\phi_{\zeta}^{c_{\tau,\zeta}})^\prime_+(0)+\rho(\varphi_{\zeta}^{c_{\tau,\zeta}})^\prime_+(0)] =c_{\tau,\zeta}, \quad\lim_{\zeta\rightarrow0^+}c_{\tau,\zeta}=c_{\tau}, \]

where $(\phi _{\zeta }^{c_{\tau,\zeta }},\,\varphi _{\zeta }^{c_{\tau,\zeta }})$ is the unique strictly increasing solution of (5.2) with $c=c_{\tau,\zeta }$.

From lemma 3.1, we know that $\lim _{t\rightarrow +\infty }(u(t,\,x), v(t,\,x))=(u^*, v^*)$ locally uniformly for $x\in \mathbb {R}$. Note that $(u^*, v^*)>(u_{\zeta }^*, v_{\zeta }^*)$. Then for any $L_{0}>0$, there exists sufficiently large $\mathcal {T}_0>0$ such that $s_2(\mathcal {T}_0)>L_0$ and $(u(t,\,x),\,v(t,\,x))\geq (u_{\zeta }^*,\,v_{\zeta }^*)$ for any $(t,\,x)\in [\mathcal {T}_0,\,+\infty )\times [-3L_0,\,L_0]$.

We define

\begin{align*} \underline{s}_2(t)& = c_{\tau,\zeta}(t-\mathcal{T}_0)+L_0,~t\geq \mathcal{T}_0,\\ \underline{u}(t,x)& = \phi_{\zeta}^{c_{\tau,\zeta}}(\underline{s}_2(t)-x),~t\geq \mathcal{T}_0,~x\in[{-}L_0,\underline{s}_2(t)],\\ \underline{v}(t,x)& = \varphi_{\zeta}^{c_{\tau,\zeta}}(\underline{s}_2(t)-x),~t\geq \mathcal{T}_0,~x\in[{-}L_0,\underline{s}_2(t)], \end{align*}

and continuously extend $\underline {u}(t,\,x), \underline {v}(t,\,x)$ to be functions defined on $[\mathcal {T}_0, +\infty )\times (-\infty, \underline {s}_2(t)]$ such that $\underline {u}(t,\,x)\equiv 0$, $\underline {v}(t,\,x)\equiv 0$ on $[\mathcal {T}_0, +\infty )\times (-\infty, -3L_0]$, and

\[ (\phi_{\zeta}^{c_{\tau,\zeta}}(\underline{s}_2(t)-x), \varphi_{\zeta}^{c_{\tau,\zeta}}(\underline{s}_2(t)-x))\leq(\underline{u}(t,x), \underline{v}(t,x))\leq (u_{\zeta}^*,v_{\zeta}^*) \]

on $[\mathcal {T}_0, +\infty )\times [-2L_0,\,-L_0]$. The graph of $\underline {v}(t,\,x)$ is plotted in Fig. 1. It follows that

(5.3)\begin{equation} \begin{aligned} & \int_{{-}L_0}^{+\infty}J_{1}(x-y)\underline{v}(t-\tau_{1},y){\rm d}y =\int_{{-}L_0}^{+\infty}J_{1}(x-y)\varphi_{\zeta}^{c_{\tau,\zeta}} (\underline{s}_{2}(t-\tau_1)-y){\rm d}y,\\ & \int_{{-}2L_0}^{{-}L_0}J_{1}(x-y)\underline{v}(t-\tau_{1},y){\rm d}y \geq\int_{{-}2L_0}^{{-}L_0}J_{1}(x-y)\varphi_{\zeta}^{c_{\tau,\zeta}} (\underline{s}_{2}(t-\tau_1)-y){\rm d}y \end{aligned} \end{equation}

for any $t\geq \mathcal {T}_0+\max \{\tau _1,\,\tau _2\}$ and $x\in [-L_0,\,\underline {s}_2(t)]$.

Figure 1. Lower solution $\underline {v}(t,\,x)$.

Choose $L_0$ sufficiently large such that

\[ v^{*}\int_{-\infty}^{{-}L_0}J_{1}(z){\rm d}z<\zeta,\quad u^{*}\int_{-\infty}^{{-}L_0}J_{2}(z){\rm d}z<\zeta, \]

which imply

(5.4)\begin{equation} \begin{aligned} & \int_{-\infty}^{{-}2L_0}J_{1}(x-y)\varphi_{\zeta}^{c_{\tau,\zeta}}(\underline{s}_2(t-\tau_1)-y){\rm d}y \leq v^*\int_{-\infty}^{{-}2L_0}J_{1}(x-y){\rm d}y\\ & \quad=v^*\int_{-\infty}^{{-}2L_0-x}J_{1}(z){\rm d}z \leq v^*\int_{-\infty}^{{-}L_0}J_{1}(z){\rm d}z<\zeta \end{aligned} \end{equation}

and

\[ \int_{-\infty}^{{-}2L_0}J_{2}(x-y)\phi_{\zeta}^{c_{\tau,\zeta}}(\underline{s}_2(t-\tau_2)-y){\rm d}y \leq u^*\int_{-\infty}^{{-}L_0}J_{2}(z){\rm d}z<\zeta \]

for $t\geq \mathcal {T}_0+\max \{\tau _1,\,\tau _2\}$ and $x\in [-L_0,\,\underline {s}_2(t)]$.

For $t\geq \mathcal {T}_0+\max \{\tau _1,\,\tau _2\}$ and $x\in [-L_0,\,\underline {s}_2(t)]$, we can deduce

\begin{align*} & \underline{u}_{t}-d_{1}\underline{u}_{xx}+a_{1}\underline{u} -h\Bigg(\int_{-\infty}^{+\infty}J_{1}(x-y)\underline{v}(t-\tau_{1},y){\rm d}y\Bigg)\\ & \quad=c_{\tau,\zeta}(\phi_{\zeta}^{c_{\tau,\zeta}})^{\prime}(\underline{s}_2(t)-x) -d_{1}(\phi_{\zeta}^{c_{\tau,\zeta}})^{\prime\prime}(\underline{s}_2(t)-x) +a_{1}\phi_{\zeta}^{c_{\tau,\zeta}}(\underline{s}_2(t)-x)\\ & \qquad -h\Bigg(\int_{-\infty}^{+\infty}J_{1}(x-y)\underline{v}(t-\tau_{1},y){\rm d}y\Bigg)\\ & \quad=h\Bigg(\int_{-\infty}^{+\infty}J_{1}(y)\varphi_{\zeta}^{c_{\tau,\zeta}} (\underline{s}_{2}(t)-x-y-c_{\tau,\zeta}\tau_{1}){\rm d}y-\zeta\Bigg)\\ & \qquad -h\Bigg(\int_{-\infty}^{+\infty}J_{1}(x-y)\underline{v}(t-\tau_{1},y){\rm d}y\Bigg)\\ & \quad=h^{\prime}(\eta)\Bigg(\int_{-\infty}^{+\infty}J_{1}(x-y)\varphi_{\zeta}^{c_{\tau,\zeta}} (\underline{s}_{2}(t-\tau_1)-y){\rm d}y\\ & \qquad -\zeta -\int_{-\infty}^{+\infty}J_{1}(x-y)\underline{v}(t-\tau_{1},y){\rm d}y\Bigg)\\ & \quad\leq 0 \quad \text{with some }\eta>0, \end{align*}

where the last inequality uses (5.3) and (5.4). The inequality satisfied by $\underline {v}$ can be proved similarly. In terms of the choices of $\mathcal {T}_0$ and $L_0$, we can check that $(\underline {u}(t,\,x),\,\underline {v}(t,\,x);\underline {s}_2(t))$ is a lower solution of one-side case with $t>0, 0< x<\bar {s}_{2}(t)$ in lemma 2.3 replaced by $t>\mathcal {T}_0+\max \{\tau _1,\,\tau _2\}, -L_0< x<\underline {s}_{2}(t)$. Therefore, by the comparison principle we have $s_2(t)\geq \underline {s}_2(t)$ for $t\geq \mathcal {T}_0+\max \{\tau _1,\,\tau _2\}$, which implies

\[ \liminf_{t\rightarrow+\infty}\frac{s_2(t)}{t}\geq \liminf_{t\rightarrow+\infty}\frac{\underline{s}_2(t)}{t}\geq c_{\tau,\zeta}. \]

By taking $\zeta \rightarrow 0^+$, we get the desired result. The limit superior of $\frac {s_1(t)}{t}$ can be proved similarly. This completes the proof.

6. Partially degenerate diffusion case without delays

In this section, we aim to determine the asymptotic spreading speeds of free boundaries for the partially degenerate diffusion case considered in [19]. The upper bounds of spreading speeds were provided in [19], but their precise values are still unknown. Here we give a complete answer to the problem. More precisely, we consider the following free boundary model introduced in [19]:

(6.1)\begin{equation} \left\{\begin{array}{@{}l} u_t(t,x)=d_1u_{xx}-a_1u(t,x)+\int_{-\infty}^{+\infty}J_1(x-y)v(t,y){\rm d}y,~t>0, s_1(t)< x< s_2(t),\\ v_t(t,x)={-}a_2v(t,x)+g(u(t,x)),~t>0,s_1(t)< x< s_2(t),\\ u(t,s_1(t))=u(t,s_2(t))=0,~v(t,s_1(t))=v(t,s_2(t))=0,~t>0,\\ s_1^\prime(t)={-}\mu u_x(t,s_1(t)),~s_2^\prime(t)={-}\mu u_x(t,s_2(t)),~t>0,\\ s_1(0)={-}s_0,~s_2(0)=s_0,~ t>0,\\ u(0,x)=u_0(x),~v(0,x)=v_0(x),~-s_0< x< s_0, \end{array}\right. \end{equation}

which is a special case of (1.1).

As in § 4, we consider the corresponding perturbed semi-wave problem

(6.2)\begin{equation} \left\{\begin{array}{@{}l} c\phi^\prime(\xi)=d_1\phi^{\prime\prime}(\xi)-a_1\phi(\xi) +\int_{-\infty}^{+\infty}J_1(y)\varphi(\xi-y){\rm d}y,~\xi>0,\\ c\varphi^\prime(\xi)={-}a_2\varphi(\xi) +g(\phi(\xi)),~\xi>0,\\ (\phi(\xi), \varphi(\xi))=(\delta u^*,\delta v^*),~\xi\leq0,\\ (\phi(+\infty),\varphi(+\infty))=(u^*,v^*). \end{array}\right. \end{equation}

Define $\mathcal {F}_{1}(\Phi )(\xi )$ similarly as in § 4.1, and

\[ \mathcal{F}_2(\Phi)(\xi)= \left\{\begin{array}{@{}l} \delta v^{*}e^{-\frac{a_{2}}{c}\xi} +\frac{1}{c}\int_{0}^{\xi}e^{\frac{a_{2}}{c}(s-\xi)} g(\phi(s)){\rm d}s,~\xi>0,\\ \delta v^*,~\xi\leq0. \end{array}\right. \]

By applying the monotone iteration method, we can also establish the existence of solutions to the perturbed semi-wave problem (6.2).

Similar as theorem 4.6, there is a dichotomy between increasing semi-wave solution and increasing travelling wave solution $(u(t,\,x),\,v(t,\,x))=(\phi (x+ct),\,\varphi (x+ct))$ of

(6.3)\begin{equation} \left\{\begin{array}{@{}l} \partial_t u=d_1\partial_{xx}u-a_1u+\int_{-\infty}^{+\infty}J_1(x-y)v(t,y){\rm d}y,\quad t>0,~x\in \mathbb{R},\\ \partial_t v={-}a_2v+g(u(t,x)), \quad t>0,~x\in \mathbb{R}. \end{array}\right. \end{equation}

In [39], Xu and Zhao proved that there exists $c^{*}>0$ such that (6.3) has an increasing travelling wave solution for $c\geq c^*$, but no such a solution for $0< c< c^*$. Therefore, we can establish the (non-)existence of semi-wave solution. The critical value of speed $c$ for semi-wave is also $c^*$.

Similarly as the proof of Lemma 2.13 in [40] and Lemma 2.10 in [10], we can prove that there exists a unique $c_{\mu }^{*}\in (0, c^*)$ such that $\mu (\phi ^{c_{\mu }^{*}})_+^{\prime }(0)=c_{\mu }^{*}$ for any given $\mu >0$, where $(\phi ^{c_{\mu }^{*}},\,\varphi ^{c_{\mu }^{*}})$ is the semi-wave solution with $c=c_{\mu }^{*}$. Moreover, $\lim _{\mu \rightarrow +\infty }c_{\mu }^{*}=c^*$.

As in § 5, by constructing a pair of upper and lower solutions from semi-wave solutions, we can get the asymptotic spreading speeds for (6.1) as follows

\[ -\lim_{t\rightarrow+\infty}\frac{s_1(t)}{t}=\lim_{t\rightarrow+\infty}\frac{s_2(t)}{t}=c_{\mu}^*. \]

Remark 6.1 We remark that the method in this paper can also be applied to determine the asymptotic speeds for the partially degenerate diffusion case with time delays, i.e., $d_{2}=\rho =0, J_{2}=\delta$ (Dirac delta function).