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Analysis of a spatial memory model with nonlocal maturation delay and hostile boundary condition. (English) Zbl 1447.35036

Summary: In this paper, we propose and investigate a memory-based reaction-diffusion equation with nonlocal maturation delay and homogeneous Dirichlet boundary condition. We first study the existence of the spatially inhomogeneous steady state. By analyzing the associated characteristic equation, we obtain sufficient conditions for local stability and Hopf bifurcation of this inhomogeneous steady state, respectively. For the Hopf bifurcation analysis, a geometric method and prior estimation techniques are combined to find all bifurcation values because the characteristic equation includes a non-self-adjoint operator and two time delays. In addition, we provide an explicit formula to determine the crossing direction of the purely imaginary eigenvalues. The bifurcation analysis reveals that the diffusion with memory effect could induce spatiotemporal patterns which were never possessed by an equation without memory-based diffusion. Furthermore, these patterns are different from the ones of a spatial memory equation with Neumann boundary condition.

MSC:

35B32 Bifurcations in context of PDEs
35B36 Pattern formations in context of PDEs
35K58 Semilinear parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35R09 Integro-partial differential equations
92B05 General biology and biomathematics
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[1] N. F. Britton, Spatial structures and periodic traveling waves in an integro-differential reaction diffusion population model, SIAM J. Anal. Math., 50, 1663-1688 (1990) · Zbl 0723.92019 · doi:10.1137/0150099
[2] N. F. Britton, Spatial structures and periodic traveling waves in an integro-differential reaction diffusion population model, SIAM J. Anal. Math., 50, 1663-1688 (1990) · Zbl 0854.35120 · doi:10.1137/0150099
[3] S. Busenberg; W. Huang, Stability and Hopf bifurcation for a population delay model with diffusion effects, J. Differential Equations, 124, 80-107 (1996) · Zbl 1059.92051 · doi:10.1006/jdeq.1996.0003
[4] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2003. · Zbl 1256.35177 · doi:10.1016/j.jde.2012.08.031
[5] S. Chen; J. Shi, Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect, J. Differential Equations, 253, 3440-3470 (2012) · Zbl 1439.35040 · doi:10.1016/j.jde.2012.08.031
[6] S. Chen; J. Wei; X. Zhang, Bifurcation analysis for a delayed diffusive logistic population model in the advective heterogeneous environment, J. Dyn. Differ. Equ., 32, 823-847 (2020) · Zbl 1325.35003 · doi:10.1007/s10884-019-09739-0
[7] S. Chen; J. Yu, Stability and bifurcations in a nonlocal delayed reaction-diffusion population model, J. Differential Equations, 260, 218-240 (2016) · Zbl 0219.46015 · doi:10.1016/j.jde.2015.08.038
[8] M. G. Crandall; P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8, 321-340 (1971) · Zbl 0714.92012 · doi:10.1016/0022-1236(71)90015-2
[9] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. · Zbl 1042.35002
[10] J. Furter; M. Grinfeld, Local vs. non-local interactions in population dynamics, J. Math. Biol., 27, 65-80 (1989) · Zbl 1087.34052 · doi:10.1007/BF00276081
[11] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. · Zbl 1323.35082 · doi:10.1016/j.jde.2015.03.006
[12] K. Gu; S. Niculescu; J. Chen, On stability crossing curves for general systems with two delays, J. Math. Anal. Appl., 311, 231-253 (2005) · Zbl 1161.92003 · doi:10.1016/j.jmaa.2005.02.034
[13] S. Guo, Stability and bifurcation in a reaction-diffusion model with nonlocal delay effect, J. Differential Equations, 259, 1409-1448 (2015) · Zbl 1170.92306 · doi:10.1016/j.jde.2015.03.006
[14] T. Hillen; K. J. Painter, A user’s guide to PDE models for chemotaxis, J. Math. Biol., 58, 183-217 (2009) · Zbl 1307.35144 · doi:10.1007/s00285-008-0201-3
[15] E. F. Keller; L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26, 399-415 (1970) · Zbl 1006.92002 · doi:10.1016/0022-5193(70)90092-5
[16] Y. Lou; F. Lutscher, Evolution of dispersal in open advective environments, J. Math. Biol., 69, 1319-1342 (2014) · Zbl 1027.92022 · doi:10.1007/s00285-013-0730-2
[17] J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer-Verlag, New York, 2003. · Zbl 1419.35114 · doi:10.1088/1361-6544/ab1f2f
[18] A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspective, Springer-Verlag, New York, 2001. · Zbl 1439.92215 · doi:10.1007/s10884-019-09757-y
[19] J. Shi; C. Wang; H. Wang, Diffusive spatial movement with memory and maturation delays, Nonlinearity, 32, 3188-3208 (2019) · Zbl 1423.35027 · doi:10.1088/1361-6544/ab1f2f
[20] J. Shi; C. Wang; H. Wang; X. Yan, Diffusive spatial movement with memory, J. Dynam. Differential Equations, 32, 979-1002 (2020) · Zbl 1203.35029 · doi:10.1007/s10884-019-09757-y
[21] Y. Song; S. Wu; H. Wang, Spatiotemporal dynamics in the single population model with memory-based diffusion and nonlocal effect, J. Differential Equations, 267, 6316-6351 (2019) · Zbl 1290.35139 · doi:10.1016/j.jde.2019.06.025
[22] Y. Su; J. Wei; J. Shi, Hopf bifurcation in a reaction-diffusion population model with delay effect, J. Differential Equations, 247, 1156-1184 (2009) · Zbl 1203.35029 · doi:10.1016/j.jde.2009.04.017
[23] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35, 1516-1537 (2010) · Zbl 1198.37080 · doi:10.1080/03605300903473426
[24] J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Sciences, 119. Springer-Verlag, New York, 1996. · Zbl 0870.35116
[25] X. P. Yan; W. T. Li, Stability of bifurcating periodic solutions in a delayed reaction diffusion population model, Nonlinearity, 23, 1413-1431 (2010) · Zbl 1198.37080 · doi:10.1088/0951-7715/23/6/008
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