AI 中文总结
本文研究空间异质性对一维捕食者-被捕食者反应-扩散系统正平衡态稳定性的影响，通过构造光滑系数族展示空间异质性可导致唯一正平衡态失稳，并产生Hopf分支的时间周期解。
AI 中文摘要
本文研究空间异质性如何影响具有Neumann边界条件的一维捕食者-被捕食者反应-扩散系统中正平衡态的稳定性。我们的例子表明，即使正平衡态是唯一的，空间异质性也可能导致不稳定。对于这里构造的光滑、空间异质性系数族，当反应系数中的一个参数变化时，该平衡态失去线性稳定性。在临界参数值处，一对简单的共轭特征值横截穿过虚轴，Hopf分支产生一个严格正、非平稳的经典时间周期解的局部分支。这些例子包括两个具有空间集中被捕食者的族，其中一个捕食者内禀增长率处处为正，另一个处处为负。第三个族具有空间集中的捕食者种群和处处为负的捕食者内禀增长率。
英文摘要
In this paper, we investigate how spatial heterogeneity affects the stability of positive equilibria in one-dimensional predator--prey reaction--diffusion systems with Neumann boundary conditions. Our examples show that spatial heterogeneity can lead to instability even when the positive equilibrium is unique. For the smooth, spatially heterogeneous coefficient families constructed here, this equilibrium loses linear stability as a parameter in the reaction coefficients varies. At the critical parameter value, a simple conjugate pair of eigenvalues crosses the imaginary axis transversely, and a Hopf bifurcation produces a local branch of strictly positive, nonstationary classical time-periodic solutions. The examples include two families with spatially concentrated prey, one with an everywhere positive and the other with an everywhere negative predator intrinsic growth rate. A third family has a spatially concentrated predator population and an everywhere negative predator intrinsic growth rate.
