Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada.
School of Mathematics and Statistics, University of Glasgow, Glasgow, UK.
Math Biosci Eng. 2022 Jan 7;19(3):2506-2537. doi: 10.3934/mbe.2022116.
Diffusion-driven instability and Turing pattern formation are a well-known mechanism by which the local interaction of species, combined with random spatial movement, can generate stable patterns of population densities in the absence of spatial heterogeneity of the underlying medium. Some examples of such patterns exist in ecological interactions between predator and prey, but the conditions required for these patterns are not easily satisfied in ecological systems. At the same time, most ecological systems exist in heterogeneous landscapes, and landscape heterogeneity can affect species interactions and individual movement behavior. In this work, we explore whether and how landscape heterogeneity might facilitate Turing pattern formation in predator-prey interactions. We formulate reaction-diffusion equations for two interacting species on an infinite patchy landscape, consisting of two types of periodically alternating patches. Population dynamics and movement behavior differ between patch types, and individuals may have a preference for one of the two habitat types. We apply homogenization theory to derive an appropriately averaged model, to which we apply stability analysis for Turing patterns. We then study three scenarios in detail and find mechanisms by which diffusion-driven instabilities may arise even if the local interaction and movement rates do not indicate it.
扩散驱动不稳定性和图灵模式形成是一种众所周知的机制，通过这种机制，物种的局部相互作用，加上随机的空间运动，可以在没有底层介质空间异质性的情况下产生稳定的种群密度模式。在捕食者和猎物之间的一些生态相互作用中存在着这样的模式，但这些模式所需的条件在生态系统中不容易满足。同时，大多数生态系统存在于异质景观中，景观异质性会影响物种相互作用和个体运动行为。在这项工作中，我们探讨了景观异质性是否以及如何促进捕食者-猎物相互作用中的图灵模式形成。我们在一个无限的斑块状景观上为两种相互作用的物种建立了反应扩散方程，该景观由两种周期性交替的斑块类型组成。斑块类型之间的种群动态和运动行为不同，个体可能对两种生境类型中的一种有偏好。我们应用均匀化理论推导出一个适当的平均模型，并对其进行图灵模式稳定性分析。然后，我们详细研究了三个场景，并找到了即使局部相互作用和运动速度没有指示，扩散驱动不稳定性也可能出现的机制。
