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

Qi An; Chuncheng Wang; Hao Wang

doi:10.3934/dcds.2020249

Article Contents

2020, Volume 40, Issue 10: 5845-5868. Doi: 10.3934/dcds.2020249

This issue Previous Article Existence of positive solutions of Schrödinger equations with vanishing potentials Next Article Extended symmetry groups of multidimensional subshifts with hierarchical structure

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

Received: September 30, 2019

Revised: February 29, 2020

Published: June 2020

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

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Abstract

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.

Keywords:

Memory-based reaction-diffusion equation,
Dirichlet boundary condition,
two delays,
inhomogeneous steady state,
inhomogeneous periodic solution,
Hopf bifurcation.

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

Citation:

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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} $

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

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

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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

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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

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

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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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