This paper considers predator–prey systems in which the prey can move between source and sink patches. First, we give a complete analysis on global dynamics of the model. Then, we show that when diffusion from the source to sink is not large, the species would coexist at a steady state; when the diffusion is large, the predator goes to extinction, while the prey persists in both patches at a steady state; when the diffusion is extremely large, both species go to extinction. It is derived that diffusion in the system could lead to results reversing those without diffusion. That is, diffusion could change species’ coexistence if non-diffusing, to extinction of the predator, and even to extinction of both species. Furthermore, we show that intermediate diffusion to the sink could make the prey reach total abundance higher than if non-diffusing, larger or smaller diffusion rates are not favorable. The total abundance, as a function of diffusion rates, can be both hump-shaped and bowl-shaped, which extends previous theory. A novel finding of this work is that there exist diffusion scenarios which could drive the predator into extinction and make the prey reach the maximal abundance. Diffusion from the sink to source and asymmetry in diffusion could also lead to results reversing those without diffusion. Meanwhile, diffusion always leads to reduction of the predator’s density. The results are biologically important in protection of endangered species.

本文考虑了捕食者-猎物系统，其中猎物可以在源斑块和汇斑块之间移动。首先，我们对模型的全局动力学进行了完整的分析。然后，我们证明当从源到汇的扩散不大时，物种会以稳定状态共存；当扩散大时，捕食者走向灭绝，而猎物在两个斑块中都以稳定状态存在；当扩散非常大时，两个物种都会灭绝。推导出系统中的扩散可能导致与没有扩散的结果相反的结果。也就是说，扩散可以改变物种的共存，如果不扩散，捕食者的灭绝，甚至两个物种的灭绝。此外，我们表明中间扩散到汇可以使猎物达到比非扩散更高的总丰度，更大或更小的扩散速率都是不利的。作为扩散速率的函数，总丰度可以是驼峰形和碗形，这扩展了先前的理论。这项工作的一个新发现是存在扩散场景，可以使捕食者灭绝并使猎物达到最大数量。从汇到源的扩散和扩散的不对称也可能导致与没有扩散的结果相反的结果。同时，扩散总是导致捕食者密度的降低。研究结果在保护濒危物种方面具有重要的生物学意义。这项工作的一个新发现是存在扩散场景，可以使捕食者灭绝并使猎物达到最大数量。从汇到源的扩散和扩散的不对称也可能导致与没有扩散的结果相反的结果。同时，扩散总是导致捕食者密度的降低。研究结果对保护濒危物种具有重要的生物学意义。这项工作的一个新发现是存在扩散场景，可以使捕食者灭绝并使猎物达到最大数量。从汇到源的扩散和扩散的不对称也可能导致与没有扩散的结果相反的结果。同时，扩散总是导致捕食者密度的降低。研究结果对保护濒危物种具有重要的生物学意义。