Department of Applied Mathematics, National Pingtung University, Pingtung, Taiwan 900.
Math Biosci Eng. 2019 Jan 30;16(2):909-946. doi: 10.3934/mbe.2019043.
A system of two competing species μ and ν that diffuse over a two-patch environment is investigated. When u-species has smaller birth rate in the first patch and larger birth rate in the second patch than v-species, and the average birth rate for u-species is larger than or equal to v-species, it was shown in a previous publication that two species coexist in a slow diffusion environment, whereas u-species drives v-species into extinction in a fast diffusion environment. In this paper, we analyze global dynamics and bifurcations for the same model with identical order of birth rates, but with opposite order of average birth rates, i.e., the average birth rate of u-species is less than that of v-species. We observe richer dynamics with two scenarios, depending on the relative difference between the variation in the birth rates of v-species on two patches and the variation in the average birth rates of two species. When the variation in average birth rates is relatively large, there is no stability switch for the semitrivial equilibria. On the other hand, such a stability switch takes place when the variation in average birth rates is relatively mild. In both cases, v-species, with larger average birth rate, prevails in a fast diffusion environment, whereas in a slow diffusion environment, the two species can coexist or u-species that has the greatest birth rate among both species and patches will persist and drive v-species to extinction.
研究了两种竞争物种μ和ν在双斑块环境中扩散的系统。当 u 物种在第一个斑块中的出生率较小，在第二个斑块中的出生率较大，而 u 物种的平均出生率大于或等于 v 物种时，先前的研究表明，在缓慢扩散环境中两种物种共存，而在快速扩散环境中 u 物种会使 v 物种灭绝。在本文中，我们分析了相同模型的全局动力学和分支，这些模型具有相同的出生率阶数，但平均出生率的顺序相反，即 u 物种的平均出生率小于 v 物种的平均出生率。我们观察到了两种情况下更丰富的动力学，这取决于 v 物种在两个斑块上的出生率变化和两种物种的平均出生率变化之间的相对差异。当平均出生率的变化相对较大时，半平凡平衡点没有稳定性切换。另一方面，当平均出生率的变化相对较小时，这种稳定性切换就会发生。在这两种情况下，具有较大平均出生率的 v 物种在快速扩散环境中占优势，而在缓慢扩散环境中，两种物种可以共存，或者两种物种和斑块中出生率最大的 u 物种将继续存在并使 v 物种灭绝。
