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Asymptotic propagations of a nonlocal dispersal population model with shifting habitats

Part of: Parabolic equations and systems Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  16 June 2021

SHAO-XIA QIAO ,
WAN-TONG LI  and
JIA-BING WANG
SHAO-XIA QIAO
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, P.R. China emails: [email protected]; [email protected]
WAN-TONG LI
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, P.R. China emails: [email protected]; [email protected]
JIA-BING WANG
Affiliation:
School of Mathematics and Physics, Center for Mathematical Sciences, China University of Geosciences, Wuhan 430074, P.R. China email: [email protected]
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Abstract

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This paper is concerned with the asymptotic propagations for a nonlocal dispersal population model with shifting habitats. In particular, we verify that the invading speed of the species is determined by the speed c of the shifting habitat edge and the behaviours near infinity of the species’ growth rate which is nondecreasing along the positive spatial direction. In the case where the species declines near the negative infinity, we conclude that extinction occurs if c > c*(∞), while c < c*(∞), spreading happens with a leftward speed min{−c, c*(∞)} and a rightward speed c*(∞), where c*(∞) is the minimum KPP travelling wave speed associated with the species’ growth rate at the positive infinity. The same scenario will play out for the case where the species’ growth rate is zero at negative infinity. In the case where the species still grows near negative infinity, we show that the species always survives ‘by moving’ with the rightward spreading speed being either c*(∞) or c*(−∞) and the leftward spreading speed being one of c*(∞), c*(−∞) and −c, where c*(−∞) is the minimum KPP travelling wave speed corresponding to the growth rate at the negative infinity. Finally, we give some numeric simulations and discussions to present and explain the theoretical results. Our results indicate that there may exists a solution like a two-layer wave with the propagation speeds analytically determined for such type of nonlocal dispersal equations.

Keywords

Nonlocal dispersal modelasymptotic propagationshifting habitat

MSC classification

Primary: 35K57: Reaction-diffusion equations 35R20: Partial operator-differential equations (i.e., PDE on finite-dimensional spaces for abstract space valued functions)
Secondary: 92D25: Population dynamics (general)
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 4 , August 2022 , pp. 701 - 728
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

Alfaro, M., Berestycki, H. & Raoul, G. (2017) The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition. SIAM J. Math. Anal. 49, 562–596.CrossRefGoogle Scholar
Berestycki, H., Diekmann, O., Nagelkerke, C. J. & Zegeling, P. A. (2009) Can a species keep pace with a shifting climate?. Bull. Math. Biol. 71, 399–429.CrossRefGoogle ScholarPubMed
Berestycki, H., Desvillettes, L. & Diekmann, O. (2014) Can climate change lead to gap formation?. Ecol. Complex. 20, 264–270.CrossRefGoogle Scholar
Berestycki, H. & Fang, J. (2018) Forced waves of the Fisher-KPP equation in a shifting environment. J. Differ. Equations 264, 2157–2183.CrossRefGoogle Scholar
Berestycki, H. & Rossi, L. (2008) Reaction-diffusion equations for population dynamics with forced speed. I. The case of the whole space. Discrete Contin. Dyn. Syst. 21, 41–67.CrossRefGoogle Scholar
Bouhoours, J. & Giletti, T. (2016) Extinction and spreading of a species under the joint influence of climate change and a weak Allee effect: a two-patch model, arXiv:1601.06589v1.Google Scholar
Bouhours, J. & Giletti, T. (2018) Spreading and vanishing for a monostable reaction diffusion equation with forced speed. J. Dyn. Differ. Equations 31, 247–286.CrossRefGoogle Scholar
Coville, J., Dávila, J. & Martnez, S. (2008) Nonlocal anisotropic dispersal with monostable nonlinearity. J. Differ. Equations 244, 3080–3118.CrossRefGoogle Scholar
Du, Y., Wei, L. & Zhou, L. (2018) Spreading in a shifting environment modeled by the diffusive logistic equation with a free boundary. J. Dyn. Differ. Equations 30 1389–1426.CrossRefGoogle Scholar
Fang, J., Lou, Y. & Wu, J. (2016) Can pathogen spread keep pace with its host invasion? SIAM J. Appl. Math. 76, 1633–1657.CrossRefGoogle Scholar
Fang, J., Peng, R. & Zhao, X. Q. (2021) Propagation dynamics of a reaction-diffusion equation in a time-periodic shifting environment. J. Math. Pures Appl. 147, 1–28.CrossRefGoogle Scholar
Garnier, J. (2011) Accelerating solutions in integro-differential equations. SIAM J. Math. Anal. 43, 1955–1974.CrossRefGoogle Scholar
Gonzalez, P., Neilson, R. P., Lenihan, J. M. & Drapek, R. J. (2010) Global patterns in the vulnerability of ecosystems to vegetation shifts due to climate change. Glob. Ecol. Biogeogr. 19, 755–768.CrossRefGoogle Scholar
Hu, C. & Li, B. (2015) Spatial dynamics for lattice differential equations with a shifting habitat. J. Differ. Equations 259, 1967–1989.CrossRefGoogle Scholar
Hu, C., Shang, J. & Li, B. (2020) Spreading speeds for reaction-diffusion equations with a shifting habitat. J. Dyn. Differ. Equations 32, 1941–1964.CrossRefGoogle Scholar
Hu, H., Deng, L. & Huang, J. (2021) Traveling wave of a nonlocal dispersal Lotka-Volterra cooperation model under shifting habitat. J. Math. Anal. Appl. 500, 125100.CrossRefGoogle Scholar
Hu, H., Yi, T. & Zou, X. (2020) On spatial-temporal dynamics of a Fisher-KPP equation with a shifting envirronment. Proc. Am. Math. Soc. 148, 213–221.CrossRefGoogle Scholar
Hu, H. & Zou, X. (2017) Existence of an extinction wave in the Fisher equation with a shifting habitat. Proc. Am. Math. Soc. 145, 4763–4771.CrossRefGoogle Scholar
Lam, K. Y. & Yu, X. Asymptotic spreading of KPP reactive fronts in heterogeneous shifting environments, arXiv:2101.06698v2.Google Scholar
Lewis, M. A., Marculis, N. G. & Shen, Z. (2018) Integrodifference equations in the presence of climate change: persistence criterion, travelling waves and inside dynamics. J. Math. Biol. 77, 1649–1687.CrossRefGoogle ScholarPubMed
Li, B., Bewick, S., Barnard, M. R. & Fagan, W. F. (2016) Persistence and spreading speeds of integro-difference equations with an expanding or contracting habitat. Bull. Math. Biol. 78, 1337–1379.CrossRefGoogle ScholarPubMed
Li, B., Bewick, S., Shang, J. & Fagan, W. F. (2014) Persistence and spread of s species with a shifting habitat edge. SIAM J. Appl. Math. 5, 1397–1417; Erratum: 75, 2379–2380 (2015).CrossRefGoogle Scholar
Li, W. T., Wang, J. B. & Zhao, X. Q. (2018) Spatial dynamics of a nonlocal dispersal population model in a shifting environment. J. Nonlinear Sci. 28, 1189–1219.CrossRefGoogle Scholar
Li, W. T., Wang, J. B. & Zhao, X. Q. (2020) Propagation dynamics in a time periodic nonlocal dispersal model with stage structure. J. Dyn. Differ. Equations 32, 1027–1064.CrossRefGoogle Scholar
Liang, X. & Zhao, X. Q. (2007) Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Commun. Pure Appl. Math. 60, 1–40.CrossRefGoogle Scholar
Parr, C. L., Gray, E. F. & Bond, W. J. (2012) Cascading biodiversity and functional consequences of a global change-induced biome switch. Divers. Distrib. 18, 493–503.CrossRefGoogle Scholar
Potapov, A. B. & Lewis, M. A. (2004) Climate and competition: the effect of moving range boundaries on habitat invasibility. Bull. Math. Biol. 66, 975–1008.CrossRefGoogle ScholarPubMed
Qiao, S. X., Zhu, J. L. & Wang, J. B. (2021) Asymptotic behaviors of forced waves for the lattice Lotka-Volterra competition system with shifting habitats. Appl. Math. Lett. 118, 107168.CrossRefGoogle Scholar
Roques, L., Roques, A., Berestycki, H. & Kretzschmar, A. (2008) A population facing climate change: joint influences of Allee effects and environmental boundary geometry. Popul. Ecol. 50, 215–225.CrossRefGoogle Scholar
Vo, H. H. (2015) Persistence versus extinction under a climate change in mixed environments. J. Differ. Equations 259, 4947–4988.CrossRefGoogle Scholar
Wang, J. B. & Li, W. T. (2019) Pulsating waves and entire solutions for a spatially periodic nonlocal dispersal system with a quiescent stage. Sci. China Math. 62, 2505–2526.CrossRefGoogle Scholar
Wang, J. B. & Li, W. T. (2020) Wave propagation for a cooperative model with nonlocal dispersal under worsening habitats. Z. Angew. Math. Phys. 71, 147.CrossRefGoogle Scholar
Wang, J. B. & Wu, C. (2021) Forced waves and gap formations for a Lotka-Volterra competition model with nonlocal dispersal and shifting habitats. Nonlinear Anal. Real World Appl. 58, 103208.CrossRefGoogle Scholar
Wang, J. B. & Zhao, X. Q. (2019) Uniqueness and global stability of forced waves in a shifting environment. Proc. Am. Math. Soc. 147, 1467–1481.CrossRefGoogle Scholar
Weinberger, H. F. (1982) Long-time behavior of a class of biological models. SIAM J. Math. Anal. 13, 353–396.CrossRefGoogle Scholar
Weng, P. & Zhao, X. Q. (2006) Spreading speed and traveling waves for a multi-type SIS epidemic model. J. Differ. Equations 229, 270–296.CrossRefGoogle Scholar
Wu, C., Wang, Y. & Zou, X. (2019) Spatial-temporal dynamics of a Lotka-Volterra competition model with nonlocal dispersal under shifting environment. J. Differ. Equations 267, 4890–4921.CrossRefGoogle Scholar
Xu, W. B., Li, W. T. & Lin, G. (2020) Nonlocal dispersal cooperative systems: Acceleration propagation among species. J. Differ. Equations 268, 1081–1105.CrossRefGoogle Scholar
Xu, W. B., Li, W. T. & Ruan, S. (2020) Spatial propagation in an epidemic model with nonlocal diffusion: the influences of initial data and dispersals. Sci. China Math. 63, 2177–2206.CrossRefGoogle Scholar
Xu, W. B., Li, W. T. & Ruan, S. (2021) Spatial propagation in nonlocal dispersal Fisher-KPP equations. J. Funct. Anal. 280, 108957.CrossRefGoogle Scholar
Yagisita, H. (2009) Existence and nonexistence of traveling waves for a nonlocal monostable equation. Publ. Res. Inst. Math. Sci. 45, 925–953.CrossRefGoogle Scholar
Yi, T., Chen, Y. & Wu, J. (2020) Asymptotic propagations of asymptotical monostable type equations with shifting habitats. J. Differ. Equations 269, 5900–5930.CrossRefGoogle Scholar
Yuan, Y., Wang, Y. & Zou, X. (2019) Spatial-temporal dynamics of a Lotka-Volterra competition model with a shifting habitat. Discrete Contin. Dyn. Syst. Ser. B 24, 5633–5671.Google Scholar
Zhang, Z., Wang, W. & Yang, J. (2017) Persistence versus extinction for two competing species under a climate change. Nonlinear Anal. Model. Control 22, 285–302.CrossRefGoogle Scholar
Zhang, G. B. & Zhao, X. Q. (2019) Propagation dynamics of a nonlocal dispersal Fisher-KPP equation in a time-periodic shifting habitat. J. Differ. Equations 268, 2852–2885.CrossRefGoogle Scholar
Zhou, Y. & Kot, M. (2011) Discrete-time growth-dispersal models with shifting species ranges. Theor. Ecol. 4, 13–25.Google Scholar