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Dynamics of a memory-based diffusion model with spatial heterogeneity and nonlinear boundary condition

Quanli Ji, Ranchao Wu, Tonghua Zhang

Abstract

In this work, we study the dynamics of a spatially heterogeneous single population model with the memory effect and nonlinear boundary condition. By virtue of the implicit function theorem and Lyapunov-Schmidt reduction, spatially nonconstant positive steady state solutions appear from two trivial solutions, respectively. By using bifurcation analysis, the Hopf bifurcation associated with one spatially nonconstant positive steady state is found to occur. The results complement the existing ones. Specifically, it is found that with the interaction of spatial heterogeneity and nonlinear boundary condition, when the memory term is stronger than the interaction of the interior reaction term and the boundary one, the memory-based diffusive model has a single stability switch from stability to instability, with the increase of the delayed memory value. Therefore, the memory delay will lead to a single stability switch of such memory-based diffusive model and consequently the Hopf bifurcation will happen in the model.

Dynamics of a memory-based diffusion model with spatial heterogeneity and nonlinear boundary condition

Abstract

In this work, we study the dynamics of a spatially heterogeneous single population model with the memory effect and nonlinear boundary condition. By virtue of the implicit function theorem and Lyapunov-Schmidt reduction, spatially nonconstant positive steady state solutions appear from two trivial solutions, respectively. By using bifurcation analysis, the Hopf bifurcation associated with one spatially nonconstant positive steady state is found to occur. The results complement the existing ones. Specifically, it is found that with the interaction of spatial heterogeneity and nonlinear boundary condition, when the memory term is stronger than the interaction of the interior reaction term and the boundary one, the memory-based diffusive model has a single stability switch from stability to instability, with the increase of the delayed memory value. Therefore, the memory delay will lead to a single stability switch of such memory-based diffusive model and consequently the Hopf bifurcation will happen in the model.
Paper Structure (7 sections, 20 theorems, 102 equations)

This paper contains 7 sections, 20 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Existence of nontrivial steady states
  3. Local bifurcation from $\Gamma_{0}$
  4. Local bifurcation from $\Gamma_{u_{1}}$
  5. Stability of steady state solutions in Theorem \ref{['th2.1']}
  6. Stability of steady states in Theorem \ref{['th2.2']}
  7. Discussion

Key Result

Lemma 2.1

Suppose that $(A0)$ is satisfied.

Theorems & Definitions (37)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Theorem 2.1
  • Remark 2.1
  • Theorem 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 27 more