Each species is subject to various biotic and abiotic factors during growth. This paper formulates a deterministic model with the consideration of various factors regulating population growth such as age-dependent birth and death rates, spatial movements, seasonal variations, intra-specific competition and time-varying maturation simultaneously. The model takes the form of two coupled reaction–diffusion equations with time-dependent delays, which bring novel challenges to the theoretical analysis. Then, the model is analysed when competition among immatures is neglected, in which situation one equation for the adult population density is decoupled. The basic reproduction number $\mathcal{R}_0$ is defined and shown to determine the global attractivity of either the zero equilibrium (when $\mathcal{R}_0\leq 1$ ) or a positive periodic solution ( $\mathcal{R}_0\gt1$ ) by using the dynamical system approach on an appropriate phase space. When the immature intra-specific competition is included and the immature diffusion rate is neglected, the model is neither cooperative nor reducible to a single equation. In this case, the threshold dynamics about the population extinction and uniform persistence are established by using the newly defined basic reproduction number $\widetilde{\mathcal{R}}_0$ as a threshold index. Furthermore, numerical simulations are implemented on the population growth of two different species for two different cases to validate the analytic results.

Steady incompressible inviscid flow past a three-dimensional multiconnected (toroidal) aerofoil with a sharp trailing edge TE is considered, adopting for simplicity a linearized analysis of the vortex sheets that collect the released vorticity and form the trailing wake. The main purpose of the paper is to discuss the uniqueness of the bounded flow solution and the role of the eigenfunction. A generic admissible flow velocity u has an unbounded singularity at TE; and the physical flow solution requires the removal of the divergent part of u (the Kutta condition). This process yields a linear functional equation along the trailing edge involving both the normal vorticity ω released into the wake, and the multiplicative factor of the eigenfunction, a1. Uniqueness is then shown to depend upon the topology of the trailing edge. If δTE=[empty ], as, for example, in an annular-aerofoil configuration, both ω and a1 are uniquely determined by the Kutta condition, and the bounded flow u is unique. If δTE≠[empty ], as, for example, in a connected-wing configuration, there is an infinity of bounded flows, parametrized by a1. Numerical results of relevance for these typical configurations are presented to show the different role of the eigenfunction in the two cases.

Similar to how Hopf–Lax–Oleinik-type formula yield variational solutions for Hamilton–Jacobi equations on Euclidean space, optimal mass transportations can sometimes provide variational formulations for solutions of certain mean-field games. We investigate here the particular case of transports that maximize and minimize the following ‘ballistic’ cost functional on phase space TM, which propagates Brenier’s transport along a Lagrangian L,

$$b_T(v, x):=\inf\left\{\langle v, \gamma (0)\rangle +\int_0^TL(t, \gamma (t), {\dot \gamma}(t))\, dt; \gamma \in C^1([0, T], M); \gamma(T)=x\right\}\!,$$

$M = \mathbb{R}^d$

\begin{align*} \underline{B}_T^s(\mu,\nu):=\inf\left\{\mathbb{E}\left[\langle V,X_0\rangle +\int_0^T L(t, X,\beta(t,X))\,dt\right]\!; X\in \mathcal{A}, V\sim\mu,X_T\sim \nu\right\}\!, \end{align*}

$\mathcal{A}$

Steel corrosion plays a central role in different technological fields. In this article, we consider a simple case of a corrosion phenomenon which describes a pure iron dissolution in sodium chloride. This article is devoted to prove rigorously that under rather general hypotheses on the initial data, the solution of this iron dissolution model converges to a self-similar profile as $t\rightarrow +\infty$. We will do so for an equivalent formulation as presented in the book of Avner Friedman about parabolic equations (Friedman (1964) Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, NJ.). In order to prove the convergence result, we apply a comparison principle together with suitable upper and lower solutions.

We consider the bifurcating solutions for the Ginzburg–Landau equations when the superconductor is a film of thickness 2d submitted to an external magnetic field. We refine some results obtained earlier [1] on the stability of bifurcating solutions starting from normal solutions. We prove, in particular, the existence of curves d [map ] κ0(d), defined for large d and tending to 2−1/2 when d [map ] +∞ and κ [map ] d1(κ), defined for small κ and tending to √5/2 when κ [map ] 0, which separate the sets of pairs (κ, d) corresponding to different behaviour of the symmetric bifurcating solutions. In this way, we give in particular a complete answer to the question of stability of symmetric bifurcating solutions in the asymptotics ‘κ fixed-d large’ or ‘d fixed-κ small’.

The basic constitutive relations for elastoplasticity and elastoviscoplasticity are shown to form a typical boundary layer-type stiff system of ordinary differential equations. Three numerical algorithms are discussed: (i) The singular perturbation method (O'Malley, 1971a, b; Hoppensteadt, 1971; Miranker, 1981; Smith, 1985), which yields accurate results for both the rate-independent and rate-dependent cases, where in the former case, the algorithm is explicit, whereas in the latter case, it is implicit and requires the solution of a nonlinear equation; therefore it is impractical as a constitutive algorithm for large-scale finite-element applications, where the constitutive algorithm is used a great number of times at each finite-element node. (ii) The new constitutive algorithm (Nemat-Nasser, 1991; Nemat-Nasser & Chung, 1989, 1992) which is explicit and accurate for both the rate-independent and rate-dependent cases; the underlying mathematical feature of this new method is investigated, and it is shown that it can be classified as a simplified perturbation method; computable error bounds for this algorithm are obtained, and when the flow rule is given by the commonly used power law, it is shown that the errors are very small, (iii) A modified outer-solution method, which combines the above two techniques, and is simple, explicit, and accurate.

In this article, we consider diffusive transport of a reactive substance in a saturated porous medium including variable porosity. Thereby, the evolution of the microstructure is caused by precipitation of the transported substance. We are particularly interested in analysing the model when the equations degenerate due to clogging. Introducing an appropriate weighted function space, we are able to handle the degeneracy and obtain analytical results for the transport equation. Also the decay behaviour of this solution with respect to the porosity is investigated. There a restriction on the decay order is assumed, that is, besides low initial concentration also dense precipitation leads to possible high decay. We obtain nonnegativity and boundedness for the weak solution to the transport equation. Moreover, we study an ordinary differential equation (ODE) describing the change of porosity. Thereby, the control of an appropriate weighted norm of the gradient of the porosity is crucial for the analysis of the transport equation. In order to obtain global in time solutions to the overall coupled system, we apply a fixed point argument. The problem is solved for substantially degenerating hydrodynamic parameters.

Physical experiments indicate that when an expanding fluid flows through a porous rock then the boundary between the wet and the dry region can be very irregular (e.g. see [OMBAFJ] and the references therein). In fact, it has been conjectured that this boundary is a fractal with Hausdorff dimension about 2.5. The (one-phase) fluid flow in a porous medium can be modelled mathematically by a system of partial differential equations, which, under some simplifying assumptions, can be reduced to a family of semi-elliptic boundary value problems involving the (unknown) pressure p(x) of the fluid (at the point x and at t) and the (unknown) wet region Ut at time t. (See equations (1.5)–(1.7) below). This set of equations, called the moving boundary problem involves the permeability matrix K(x) of the medium at x. A question which has been debated is whether this relatively simple mathematical model can explain such a complicated fractal nature of ∂Ut. More precisely, does there exist a symmetric non-negative definite matrix K(x) such that the solution Ut of the corresponding (expanding) moving boundary problem has a fractal boundary? The purpose of this paper is to prove that this is indeed the case. More precisely, we show that a porous medium which produce fractal wet boundaries can be obtained by distorting a completely homogeneous medium by means of a quasiconformal map.

Suspended fibres significantly alter fluid rheology, as exhibited in for example solutions of DNA, RNA and synthetic biological nanofibres. It is of interest to determine how this altered rheology affects flow stability. Motivated by the fact thermal gradients may occur in biomolecular analytic devices, and recent stability results, we examine the problem of Rayleigh–Bénard convection of the transversely isotropic fluid of Ericksen. A transversely isotropic fluid treats these suspensions as a continuum with an evolving preferred direction, through a modified stress tensor incorporating four viscosity-like parameters. We consider the linear stability of a stationary, passive, transversely isotropic fluid contained between two parallel boundaries, with the lower boundary at a higher temperature than the upper. To determine the marginal stability curves the Chebyshev collocation method is applied, and we consider a range of initially uniform preferred directions, from horizontal to vertical, and three orders of magnitude in the viscosity-like anisotropic parameters. Determining the critical wave and Rayleigh numbers, we find that transversely isotropic effects delay the onset of instability; this effect is felt most strongly through the incorporation of the anisotropic shear viscosity, although the anisotropic extensional viscosity also contributes. Our analysis confirms the importance of anisotropic rheology in the setting of convection.

‘Parametric dependence of exponents and eigenvalues in focusing porous media flows’ by D. G. Aronson, J. B. Van den Berg and J. Hulshof, European Journal of Applied Mathematics, Vol. 14 No. 4 August 2003, pp. 485–512

The superheating that usually occurs when a solid is melted by volumetric heating can produce irregular solid–liquid interfaces. Such interfaces can be visualised in ice, where they are sometimes known as Tyndall stars. This paper describes some of the experimental observations of Tyndall stars and a mathematical model for the early stages of their evolution. The modelling is complicated by the strong crystalline anisotropy, which results in an anisotropic kinetic undercooling at the interface; it leads to an interesting class of free boundary problems that treat the melt region as infinitesimally thin.

The one-dimensional transient solidification of a binary alloy undergoing shrinkage is well-known as an invaluable benchmark for the testing of numerical codes that model macrosegregation. Here, recent work that considered the small-time behaviour of this problem is extended until complete solidification, thereby determining the solute profile across the entire solidified domain. The small-time solution is used as the initial condition for the numerical integration of a problem having three moving boundaries. Of particular significance is the so-called inverse segregation that is observed at the start of solidification, and the extreme segregation that is observed at the end; in the case of the example presented, which is for the often-cited Al–Cu system, the macrosegregation is found to be positive or negative, depending on whether Scheil’s equation or the lever rule is assumed at the microscale, respectively. The relevance of these results for the modelling of steady-state continuous casting processes – in particular, the phenomenon of centreline segregation – is also discussed.

Flowering plants depend on some animals for pollination and contribute to nourish the animals in natural environments. We call these animals pollinators and build a plants-pollinators cooperative model with impulsive effect on a periodically evolving domain. Next, we define the ecological reproduction index for single plant model and plants-pollinators system, respectively, whose threshold dynamics, including the extinction, persistence and coexistence, is established by the method of upper and lower solutions. Theoretical analysis shows that a large domain evolution rate has a positive influence on the survival of pollinators whether or not the impulsive effect occurs, and the pulse eliminates the pollinators even when the evolution rate is high. Moreover, some selective numerical simulations are still performed to explain our theoretical results.

We study a free boundary problem of the form: ut = uxx + f(t, u) (g(t) < x < h(t)) with free boundary conditions h′(t) = −ux(t, h(t)) – α(t) and g′(t) = −ux(t, g(t)) + β(t), where β(t) and α(t) are positive T-periodic functions, f(t, u) is a Fisher–KPP type of nonlinearity and T-periodic in t. This problem can be used to describe the spreading of a biological or chemical species in time-periodic environment, where free boundaries represent the spreading fronts of the species. We study the asymptotic behaviour of bounded solutions. There are two T-periodic functions α0(t) and α*(t; β) with 0 < α0 < α* which play key roles in the dynamics. More precisely, (i) in case 0 < β< α0 and 0 < α < α*, we obtain a trichotomy result: (i-1) spreading, that is, h(t) – g(t) → +∞ and u(t, ⋅ + ct) → 1 with $c\in (-\overline{l},\overline{r})$, where $ \overline{l}:=\frac{1}{T}\int_{0}^{T}l(s)ds$, $\overline{r}:=\frac{1}{T}\int_{0}^{T}r(s)ds$, the T-periodic functions −l(t) and r(t) are the asymptotic spreading speeds of g(t) and h(t) respectively (furthermore, r(t) > 0 > −l(t) when 0 < β < α < α0; r(t) = 0 > −l(t) when 0 < β < α = α0; $0 \gt \overline{r} \gt -\overline{l}$ when 0 < β < α0 < α < α*); (i-2) vanishing, that is, $\lim\limits_{t \to \mathcal {T}}h(t) = \lim\limits_{t \to \mathcal {T}}g(t)$ and $\lim\limits_{t \to \mathcal {T}}\max\limits_{g(t)\leq x\leq h(t)} u(t,x)=0$, where $\mathcal {T}$ is some positive constant; (i-3) transition, that is, g(t) → −∞, h(t) → −∞, $0<\lim\limits_{t \to \infty}[h(t)-g(t)] \lt +\infty$ and u(t, ⋅) → V(t, ⋅), where V is a T-periodic solution with compact support. (ii) in case β ≥ α0 or α ≥ α*, vanishing happens for any solution.

One of the most remarkable aspects of human homoeostasis is bone remodelling. This term denotes the continuous renewal of bone that takes place at a microscopic scale and ensures that our skeleton preserves its full mechanical compliance during our lives. We propose here that a renewal process of this type can be represented at an algorithmic level as the interplay of two different but related mechanisms. The first of them is a preliminary screening process, by means of which the whole skeleton is thoroughly and continuously explored. This is followed by a renovation process, whereby regions previously marked for renewal are first destroyed and then rebuilt, in such a way that global mechanical compliance is never compromised. In this work, we pay attention to the first of these two stages. In particular, we show that an efficient screening mechanism may arise out of simple local rules, which at the biological level are inspired by the possibility that individual bone cells compute signals from their nearest local neighbours. This is shown to be enough to put in place a process which thoroughly explores the region where such mechanism operates.

We consider self-propelled rigid bodies interacting through local body-attitude alignment modelled by stochastic differential equations. We derive a hydrodynamic model of this system at large spatio-temporal scales and particle numbers in any dimension $n \geq 3$. This goal was already achieved in dimension $n=3$ or in any dimension $n \geq 3$ for a different system involving jump processes. However, the present work corresponds to huge conceptual and technical gaps compared with earlier ones. The key difficulty is to determine an auxiliary but essential object, the generalised collision invariant. We achieve this aim by using the geometrical structure of the rotation group, namely its maximal torus, Cartan subalgebra and Weyl group as well as other concepts of representation theory and Weyl’s integration formula. The resulting hydrodynamic model appears as a hyperbolic system whose coefficients depend on the generalised collision invariant.

Ion flow in charged nanopores is strongly influenced by the ratio of the Debye length to the pore radius. We investigate the asymptotic behaviour of solutions to the Poisson–Nernst–Planck (PNP) system in narrow pore like geometries and study the influence of the pore geometry and surface charge on ion transport. The physical properties of real pores motivate the investigation of distinguished asymptotic limits, in which either the Debye length and pore radius are comparable or the pore length is very much greater than its radius This results in a quasi-one-dimensional (1D) PNP model, which can be further simplified, in the physically relevant limit of strong pore wall surface charge, to a fully 1D model. Favourable comparison is made to the two-dimensional (2D) PNP equations in typical pore geometries. It is also shown that, for physically realistic parameters, the standard 1D area averaged PNP model for ion flow through a pore is a very poor approximation to the (real) 2D solution to the PNP equations. This leads us to propose that the quasi-1D PNP model derived here, whose computational cost is significantly less than 2D solution of the PNP equations, should replace the use of the 1D area averaged PNP equations as a tool to investigate ion and current flows in ion pores.

Entropy maximising spatial interaction models have been widely exploited in a range of disciplines and applications: from trade and migration flows to the spread of riots and the understanding of spatial patterns in archaeological sites of interest. When embedded into a dynamic system and framed in the context of a retail model, the dynamics of centre growth poses an interesting mathematical problem, with bifurcations and phase changes, which may be addressed analytically. In this paper, we present some analysis of the continuous retail model and the corresponding discrete version, which yields insights into the effect of space on the evolving system, and an understanding of why certain retail centres are more successful than others. The slowly developing growths and the fast explosive growths that are of particular concern are explained in detail.

Conventional preytaxis systems assume that prey-tactic velocity is proportional to the prey density gradient. However, many experiments exploring the predator–prey interactions show that it is the predator’s acceleration instead of velocity that is proportional to the prey density gradient in the prey-tactic movement, which we call preytaxis with prey-induced acceleration. Mathematical models of preytaxis with prey-induced acceleration were proposed by Arditi et al. ((2001) Theor. Popul. Biol. 59(3), 207–221) and Sapoukhina et al. ((2003) Am. Nat. 162(1), 61–76) to interpret the spatial heterogeneity of predators and prey observed in experiments. This paper is devoted to exploring the qualitative behaviour of such preytaxis systems with prey-induced acceleration and establishing the global existence of classical solutions with uniform-in-time bounds in all spatial dimensions. Moreover, we prove the global stability of spatially homogeneous prey-only and coexistence steady states with decay rates under certain conditions on system parameters. For the parameters outside the stability regime, we perform linear stability analysis to find the possible patterning regimes and use numerical simulations to demonstrate that spatially inhomogeneous time-periodic patterns will typically arise from the preytaxis system with prey-induced acceleration. Noticing that conventional preytaxis systems are unable to produce spatial patterns, our results imply that the preytaxis with prey-induced acceleration is indeed more appropriate than conventional preytaxis to interpret the spatial heterogeneity resulting from predator–prey interactions.