October  2020, 40(10): 5845-5868. doi: 10.3934/dcds.2020249

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

1. 

School of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China

2. 

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, Jiangsu 210044, China

3. 

Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, T6G 2G1, Canada

* Corresponding author: wangchuncheng@hit.edu.cn (Chuncheng Wang)

Received  October 2019 Revised  March 2020 Published  June 2020

Fund Project: 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.

Citation: Qi An, Chuncheng Wang, Hao Wang. Analysis of a spatial memory model with nonlocal maturation delay and hostile boundary condition. Discrete & Continuous Dynamical Systems - A, 2020, 40 (10) : 5845-5868. doi: 10.3934/dcds.2020249
References:
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show all references

References:
[1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

S. ChenJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9] J. Crank, The Mathematics of Diffusion, Second edition. Clarendon Press, Oxford, 1975.   Google Scholar
[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.  Google Scholar

[11]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. Google Scholar

[12]

K. GuS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer-Verlag, New York, 2003. doi: 10.1007/b98869.  Google Scholar

[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.  Google Scholar

[19]

J. ShiC. Wang and H. Wang, Diffusive spatial movement with memory and maturation delays, Nonlinearity, 32 (2019), 3188-3208.  doi: 10.1088/1361-6544/ab1f2f.  Google Scholar

[20]

J. ShiC. WangH. Wang and X. Yan, Diffusive spatial movement with memory, J. Dynam. Differential Equations, 32 (2020), 979-1002.  doi: 10.1007/s10884-019-09757-y.  Google Scholar

[21]

Y. SongS. 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.  Google Scholar

[22]

Y. SuJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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 $
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