Article Contents
Article Contents

# Analysis of a spatial memory model with nonlocal maturation delay and hostile boundary condition

The first author's research is supported by Startup Foundation for Introducing Talent of NUIST 1411111901023 and Natural Science Foundation of Jiangsu Province of China. The second author's research is supported by Chinese NSF grants 11671110 and Heilongjiang NSF LH2019A010. The third author's research is partially supported by an NSERC grant

• 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.

Mathematics Subject Classification: Primary: 35B32, 35B35; Secondary: 92B05.

 Citation:

• Figure 1.  Triangle formed by $P_0(h)$, $P_1(h) e^{-\mathrm{i}\theta_1}$ and $P_2(h)e^{-\mathrm{i}\theta_2}$

Figure 2.  Regions in the $(\rho_1, \rho_2)$ plane that satisfy both (A1) and (A3): (a) $d>0$ and (b) $d<0$

Figure 3.  The area painted green is the connected region I of $(\lambda, h)$

Figure 4.  Approximation of the crossing curve $\mathcal{T}_\lambda$. Here $\lambda = 1.1$, $a = 0.3$, $b = 0.5$, $d = 2$. Two crossing curves of (43) are plotted in the top right corner, and one of it (in the red box) is enlarged in the figure

Figure 5.  Let $\lambda = 1.1$, $a = 0.3$, $b = 0.5$, $d = 2$. (a, c) When $(\tau, \sigma)$ are located at the left side of crossing curve, the positive steady state is stable. (b) A stable spatially inhomogeneous periodic solution is generated, when $(\tau, \sigma)$ passes through the crossing curve

Figure 6.  The crossing curves of (43) for other choices of parameters. Here, $\lambda = 1.1$

Figure 7.  Let $\lambda = 1.1$, $a = -0.3$, $b = 0.5$, $d = 0.2$, the positive steady state $u_\lambda$ is still stable for sufficient large $\tau = \sigma = 100$

•  [1] R. A. Adams and  J. J. F. Fournier,  Sobolev Spaces, 2nd edition, Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003. [2] N. F. Britton, Spatial structures and periodic traveling waves in an integro-differential reaction diffusion population model, SIAM J. Anal. Math., 50 (1990), 1663-1688.  doi: 10.1137/0150099. [3] S. Busenberg and W. Huang, Stability and Hopf bifurcation for a population delay model with diffusion effects, J. Differential Equations, 124 (1996), 80-107.  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. doi: 10.1002/0470871296. [5] S. Chen and J. Shi, Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect, J. Differential Equations, 253 (2012), 3440-3470.  doi: 10.1016/j.jde.2012.08.031. [6] S. Chen, J. Wei and X. Zhang, Bifurcation analysis for a delayed diffusive logistic population model in the advective heterogeneous environment, J. Dyn. Differ. Equ., 32 (2020), 823-847.  doi: 10.1007/s10884-019-09739-0. [7] S. Chen and J. Yu, Stability and bifurcations in a nonlocal delayed reaction-diffusion population model, J. Differential Equations, 260 (2016), 218-240.  doi: 10.1016/j.jde.2015.08.038. [8] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2. [9] J. Crank,  The Mathematics of Diffusion, Second edition. Clarendon Press, Oxford, 1975. [10] J. Furter and M. Grinfeld, Local vs. non-local interactions in population dynamics, J. Math. Biol., 27 (1989), 65-80.  doi: 10.1007/BF00276081. [11] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. [12] K. Gu, S. Niculescu and J. Chen, On stability crossing curves for general systems with two delays, J. Math. Anal. Appl., 311 (2005), 231-253.  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 (2015), 1409-1448.  doi: 10.1016/j.jde.2015.03.006. [14] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3. [15] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5. [16] Y. Lou and F. Lutscher, Evolution of dispersal in open advective environments, J. Math. Biol., 69 (2014), 1319-1342.  doi: 10.1007/s00285-013-0730-2. [17] J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer-Verlag, New York, 2003. doi: 10.1007/b98869. [18] A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspective, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4978-6. [19] J. Shi, C. Wang and H. Wang, Diffusive spatial movement with memory and maturation delays, Nonlinearity, 32 (2019), 3188-3208.  doi: 10.1088/1361-6544/ab1f2f. [20] J. Shi, C. Wang, H. Wang and X. Yan, Diffusive spatial movement with memory, J. Dynam. Differential Equations, 32 (2020), 979-1002.  doi: 10.1007/s10884-019-09757-y. [21] Y. Song, S. Wu and H. Wang, Spatiotemporal dynamics in the single population model with memory-based diffusion and nonlocal effect, J. Differential Equations, 267 (2019), 6316-6351.  doi: 10.1016/j.jde.2019.06.025. [22] Y. Su, J. Wei and J. Shi, Hopf bifurcation in a reaction-diffusion population model with delay effect, J. Differential Equations, 247 (2009), 1156-1184.  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 (2010), 1516-1537.  doi: 10.1080/03605300903473426. [24] J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Sciences, 119. Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. [25] X. P. Yan and W. T. Li, Stability of bifurcating periodic solutions in a delayed reaction diffusion population model, Nonlinearity, 23 (2010), 1413-1431.  doi: 10.1088/0951-7715/23/6/008.

Figures(7)