具有记忆效应和空间异质性的单种群模型的动力学
Dynamics of A Single Population Model with Memory Effect and Spatial Heterogeneity
摘要：
本文考虑了具有记忆效应和环境异质性的单种群模型，该模型具有Neumann边界。利用上下解方法证明了空间非齐次平衡态的整体存在性，并且对于较小的记忆扩散，该平衡态是渐近稳定的。当记忆扩散率超过一个临界值后，如果区域上内禀增长率的积分为负，则通过Hopf分支可以产生空间非齐次周期解.这种现象如果只表现为记忆扩散或空间异质性，则是不可能发生的，因此必然是它们共同作用的结果。这表明记忆扩散会在整个敌对环境中产生时空模式。当内禀增长率在区域上的积分为正时，证明了定态仍然是渐近稳定的。最后，讨论了用Dirichlet条件代替边界条件时，模型可能的动力学性质。
英文摘要：
In this paper, a single population model with memory effect and the heterogeneity of the environment, equipped with the Neumann boundary, is considered. The global existence of a spatial nonhomogeneous steady state is proved by the method of upper and lower solutions, which is asymptotically stable for relatively small memorized diffusion. However, after the memorized diffusion rate exceeding a critical value, spatial inhomogeneous periodic solution can be generated through Hopf bifurcation, if the integral of intrinsic growth rate over the domain is negative. Such phenomenon will never happen, if only memorized diffusion or spatially heterogeneity is presented, and therefore must be induced by their joint effects. This indicates that the memorized diffusion will bring about spatial-temporal patterns in the overall hostile environment. When the integral of intrinsic growth rate over the domain is positive, it turns out that the steady state is still asymptotically stable. Finally, the possible dynamics of the model is also discussed, if the boundary condition is replaced by Dirichlet condition.
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