具有空间异质性和非线性边界条件的基于记忆的扩散模型的动力学
Dynamics of a memory-based diffusion model with spatial heterogeneity and nonlinear boundary condition
AI总结：
本文研究具有记忆效应、空间异质性和非线性边界条件的单种群扩散模型，利用隐函数定理和分支分析发现空间非常正稳态解和 Hopf 分支，并证明记忆延迟会导致模型发生单一稳定性切换。
摘要：
本文研究具有记忆效应和非线性边界条件的空间异质单种群模型的动力学。借助隐函数定理和 Lyapunov-Schmidt reduction，分别从两个平凡解中产生出空间非常正稳态解。通过分支分析，发现与某一空间非常正稳态相关联的 Hopf 分支会发生。这些结果补充了现有研究。具体而言，发现在空间异质性与非线性边界条件的相互作用下，当记忆项强于内部反应项与边界项的相互作用时，随着延迟记忆值的增大，基于记忆的扩散模型会发生从稳定到不稳定的单一稳定性切换。因此，记忆延迟会导致此类基于记忆的扩散模型发生单一稳定性切换，进而引发模型中的 Hopf 分支。
英文摘要：
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.
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