发表机构
School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究非局部种群方程中空间异质性对唯一正稳态线性稳定性的影响，建立了大扩散和小扩散条件下的稳定性，并构造不稳定稳态及Hopf分支，表明唯一性不蕴含稳定性。
AI 中文摘要
本文研究空间异质性对非局部种群方程唯一正稳态线性稳定性的影响。主要困难在于，由于非局部项的存在，线性化算子不必是自伴的，也不必满足最大值原理，因此基于主特征值的通常稳定性论证不能直接应用。我们建立了充分大扩散情形下的线性稳定性，并在局部增长率最大值集合的适当条件下，建立了充分小扩散情形下的线性稳定性。在一维空间中，我们还构造了不稳定的正稳态，并在充分小扩散下获得Hopf分支。保持局部系数不变，通过改变非局部积分中的权重，我们得到超临界或亚临界的Hopf分支。我们的结果表明，正稳态的唯一性并不意味着其线性稳定性。
英文摘要
This paper is concerned with the effect of spatial heterogeneity on the linear stability of unique positive steady states for a nonlocal population equation. The main difficulty is that, because of the nonlocal term, the linearized operator need not be self-adjoint or satisfy the maximum principle, so the usual stability arguments based on the principal eigenvalue do not apply directly. We establish linear stability for sufficiently large diffusion and, under suitable conditions on the maximum set of the local growth ratio, for sufficiently small diffusion. In one space dimension, we also construct unstable positive steady states and obtain Hopf bifurcations for sufficiently small diffusion. Keeping the local coefficients fixed, we obtain either a supercritical or a subcritical Hopf bifurcation by changing the weight in the nonlocal integral. Our results show that uniqueness of a positive steady state does not imply its linear stability.
