Abstract
In this paper, a single reaction-diffusion population model with memory effect and the heterogeneity of the environment, equipped with the Neumann boundary, is considered. The global existence of a spatial nonhomogeneous steady state is proved by the method of super and subsolutions, which is linearly stable for relatively small memory-induced diffusion. However, after the memory-induced diffusion rate exceeding a critical value, spatial inhomogeneous periodic solution can be generated through Hopf bifurcation, if the integral of intrinsic growth rate over the domain is negative. Such phenomenon will never happen, if only memory-induced diffusion or spatially heterogeneity is presented, and therefore must be induced by their joint effects. This indicates that the memory-induced diffusion will bring about spatial-temporal patterns in the overall hostile environment. When the integral of intrinsic growth rate over the domain is positive, it turns out that the steady state is still linearly stable. Finally, the possible dynamics of the model is also discussed, if the boundary condition is replaced by Dirichlet condition.
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References
Adams, R., Fournier, J.: Sobolev Spaces. Elsevier, London (2003)
An, Q., Wang, C., Wang, H.: Analysis of a spatial memory model with nonlocal maturation delay and hostile boundary condition. Discrete Contin. Dyn. Syst. 40(10), 5845–5868 (2020)
Busenberg, S., Huang, W.: Stability and Hopf bifurcation for a population delay model with diffusion effects. J. Differ. Equ. 124(1), 80–107 (1996)
Cantrell, R., Cosner, C.: The effects of spatial heterogeneity in population dynamics. J. Math. Biol. 29(4), 315–338 (1991)
Cantrell, R., Cosner, C.: On the effects of spatial heterogeneity on the persistence of interacting species. J. Math. Biol. 37(2), 103–145 (1998)
Cantrell, R., Cosner, C.: Spatial Ecology Via Reaction-Diffusion Equations. Wiley, London (2003)
Chen, S., Lou, Y., Wei, J.: Hopf bifurcation in a delayed reaction-diffusion-advection population model. J. Differ. Equ. 264(8), 5333–5359 (2018)
Chen, S., Shi, J.: Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect. J. Differ. Equ. 253(12), 3440–3470 (2012)
Chen, S., Wei, J., Zhang, X.: Bifurcation analysis for a delayed diffusive logistic population model in the advective heterogeneous environment. J. Dynam. Differ. Equ. 32(2), 823–847 (2020)
Chen, S., Yu, J.: Stability analysis of a reaction-diffusion equation with spatiotemporal delay and Dirichlet boundary condition. J. Dynam. Differ. Equ. 28(3–4), 857–866 (2016)
Chen, S., Yu, J.: Stability and bifurcations in a nonlocal delayed reaction-diffusion population model. J. Differ. Equ. 260(1), 218–240 (2016)
Crandall, P., Rabinowitz, M.: Bifurcation, perturbation of simple eigenvalues, and linearized stability. Arch. Ration. Mech. Anal. 52(2), 161–180 (1973)
DeAngelis, D., Ni, W., Zhang, B.: Dispersal and spatial heterogeneity: single species. J. Math. Biol. 72(1–2), 239–254 (2016)
Ding, W., Finott, H., Lenhart, S., Lou, Y., Ye, Q.: Optimal control of growth coefficient on a steady-state population model. Nonlinear Anal. Real World Appl. 11(2), 688–704 (2010)
Du, Y., Shi, J.: Some recent results on diffusive predator-prey models in spatially heterogeneous environment. In: Nonlinear Dynamics and Evolution Equations, AMS Fields Institute Communications, American Mathematical Society, vol. 48, pp. 95–135. Providence, RI (2006)
Fagan, W., Lewis, M., Auger-Méthé, M., Avgar, T., Benhamou, S., Breed, G., LaDage, L., Schlägel, D., Tang, W., Papastamatiou, Y., Forester, J., Mueller, T.: Spatial memory and animal movement. Ecol. Lett. 16(10), 1316–1329 (2014)
Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001)
Guo, S.: Stability and bifurcation in a reaction-diffusion model with nonlocal delay effect. J. Differential Equations 259(4), 1409–1448 (2015)
Guo, S., Ma, L.: Stability and bifurcation in a delayed reaction-diffusion equation with Dirichlet boundary condition. J. Nonlinear Sci. 26(2), 545–580 (2016)
He, X., Lam, K., Lou, Y., Ni, W.: Dynamics of a consumer-resource reaction-diffusion model: homogeneous versus heterogeneous environments. J. Math. Biol. 78(6), 1605–1636 (2019)
He, X., Ni, W.: The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system I: Heterogeneity vs. homogeneity. J. Differential Equations 254(2), 528–546 (2013)
He, X., Ni, W.: Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources II. Calc. Var. Partial Differ. Equ. 55(2), 25 (2016)
Lam, K., Ni, W.: Uniqueness and complete dynamics in the heterogeneous competition-diffusion systems. SIAM J. Appl. Math. 72(6), 1695–1712 (2012)
Lam, K., Ni, W.: Advection-mediated competition in general environments. J. Differ. Equ. 257(9), 3466–3500 (2014)
Lou, Y.: On the effects of migration and spatial heterogeneity on single and multiple species. J. Differential Equations 223(2), 400–426 (2006)
Lou, Y.: Some challenging mathematical problems in evolution of dispersal and population dynamics. In: Tutorials in Mathematical Biosciences. IV, Mathematical Biosciences Subseries, vol. 1922, pp. 171–205. Springer, Berlin (2008)
Lou, Y., Martinez, S., Poláčik, P.: Loops and branches of coexistence states in a Lotka-Volterra competition model. J. Differ. Equ. 230(2), 720–742 (2006)
Lou, Y., Wang, B.: Local dynamics of a diffusive predator-prey model in spatially heterogeneous environment. J. Fixed Point Theory Appl. 19(1), 755–772 (2017)
Lou, Y., Yanagida, E.: Minimization of the principal eigenvalue for an elliptic boundary value problem with indefinite weight and applications topopulation dynamics. Japan J. Indust. Appl. Math. 23(3), 275–292 (2006)
Murray, J.: Mathematical Biology II. Springer, Berlin (2003)
Ni, W.: The mathematics of diffusion. SIAM CBMS Reg. Conf. Ser. Appl. Math. 6, 10 (2011)
Pao, C.: Quasilinear parabolic and elliptic equations with nonlinear boundary conditions. Nonlinear Anal. 66(3), 639–662 (2007)
Shi, J., Wang, C., Wang, H.: Diffusive spatial movement with memory and maturation delays. Nonlinearity 32(9), 3188–3208 (2019)
Shi, J., Wang, C., Wang, H., Yan, X.: Diffusive spatial movement with memory. J. Dynam. Differ. Equ. 32(2), 979–1002 (2020)
Shi, Q., Shi, J., Song, Y.: Hopf bifurcation in a reaction-diffusion equation with distributed delay and Dirichlet boundary condition. J. Differ. Equ. 263(10), 6537–6575 (2017)
Shi, Q., Shi, J., Song, Y.: Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity. Discrete Contin. Dyn. Syst. Ser. B 24(2), 467–486 (2019)
Song, Y., Wu, S., Wang, H.: Spatiotemporal dynamics in the single population model with memory-based diffusion and nonlocal effect. J. Differ. Equ. 267(11), 6316–6351 (2019)
Su, Y., Wei, J., Shi, J.: Hopf bifurcations in a reaction-diffusion population model with delay effect. J. Differential Equations 247(4), 1156–1184 (2009)
Su, Y., Wei, J., Shi, J.: Hopf bifurcation in a diffusive logistic equation with mixed delayed and instantaneous density dependence. J. Dynam. Differ. Equ. 24(4), 897–925 (2012)
Yan, X., Li, W.: Stability of bifurcating periodic solutions in a delayed reaction-diffusion population model. Nonlinearity 23(6), 1413–1431 (2010)
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Wang, Y., Fan, D. & Wang, C. Dynamics of A Single Population Model with Memory Effect and Spatial Heterogeneity. J Dyn Diff Equat 34, 1433–1452 (2022). https://doi.org/10.1007/s10884-021-10010-8
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DOI: https://doi.org/10.1007/s10884-021-10010-8