Alzahrani Ebraheem O, Davidson Fordyce A, Dodds Niall
Division of Mathematics, University of Dundee, Dundee, DD1 4HN, Scotland, UK.
J Math Biol. 2012 Dec;65(6-7):1101-24. doi: 10.1007/s00285-011-0490-9. Epub 2011 Nov 16.
In this paper, we discuss a class of bistable reaction-diffusion systems used to model the competitive interaction of two species. The interactions are assumed to be of classic "Lotka-Volterra" type and we will consider a particular problem with relevance to applications in population dynamics: essentially, we study under what conditions the interplay of relative motility (diffusion) and competitive strength can cause waves of invasion to be halted and reversed. By establishing rigorous results concerning related degenerate and near-degenerate systems, we build a picture of the dependence of the wave speed on system parameters. Our results lead us to conjecture that this class of competition model has three "zones of response". In the central zone, varying the motility can slow, halt and reverse invasion. However, in the two outer zones, the direction of invasion is independent of the relative motility and is entirely determined by the relative competitive strengths. Furthermore, we conjecture that for a large class of competition models of the type studied here, the wave speed is an increasing function of the relative motility.
在本文中，我们讨论一类用于模拟两个物种竞争相互作用的双稳反应扩散系统。假设相互作用为经典的“Lotka-Volterra”类型，并且我们将考虑一个与种群动态应用相关的特定问题：本质上，我们研究在何种条件下相对迁移率（扩散）和竞争强度的相互作用会导致入侵波停止并逆转。通过建立关于相关退化和近退化系统的严格结果，我们构建了波速对系统参数依赖性的图景。我们的结果使我们推测这类竞争模型有三个“响应区域”。在中心区域，改变迁移率可以减缓、停止并逆转入侵。然而，在两个外部区域，入侵方向与相对迁移率无关，完全由相对竞争强度决定。此外，我们推测对于这里研究的这类竞争模型中的一大类，波速是相对迁移率的增函数。
