Alfaro Matthieu, Hamel François, Patout Florian, Roques Lionel
Univ. Rouen Normandie, LMRS, CNRS, Rouen, France.
INRAE, BioSP, 84914, Avignon, France.
J Math Biol. 2023 Oct 9;87(5):68. doi: 10.1007/s00285-023-01996-4.
We propose a model to describe the adaptation of a phenotypically structured population in a H-patch environment connected by migration, with each patch associated with a different phenotypic optimum, and we perform a rigorous mathematical analysis of this model. We show that the large-time behaviour of the solution (persistence or extinction) depends on the sign of a principal eigenvalue, [Formula: see text], and we study the dependency of [Formula: see text] with respect to H. This analysis sheds new light on the effect of increasing the number of patches on the persistence of a population, which has implications in agroecology and for understanding zoonoses; in such cases we consider a pathogenic population and the patches correspond to different host species. The occurrence of a springboard effect, where the addition of a patch contributes to persistence, or on the contrary the emergence of a detrimental effect by increasing the number of patches on the persistence, depends in a rather complex way on the respective positions in the phenotypic space of the optimal phenotypes associated with each patch. From a mathematical point of view, an important part of the difficulty in dealing with [Formula: see text], compared to [Formula: see text] or [Formula: see text], comes from the lack of symmetry. Our results, which are based on a fixed point theorem, comparison principles, integral estimates, variational arguments, rearrangement techniques, and numerical simulations, provide a better understanding of these dependencies. In particular, we propose a precise characterisation of the situations where the addition of a third patch increases or decreases the chances of persistence, compared to a situation with only two patches.
我们提出了一个模型来描述在通过迁移连接的H斑块环境中表型结构化种群的适应性，每个斑块都与一个不同的表型最优值相关联，并且我们对该模型进行了严格的数学分析。我们表明，解的长期行为（持续存在或灭绝）取决于一个主特征值[公式：见原文]的符号，并且我们研究了[公式：见原文]关于H的依赖性。该分析为增加斑块数量对种群持续存在的影响提供了新的见解，这在农业生态学和理解人畜共患病方面具有重要意义；在这种情况下，我们考虑一个致病种群，斑块对应于不同的宿主物种。跳板效应的出现，即添加一个斑块有助于种群持续存在，或者相反，增加斑块数量对种群持续存在产生有害影响，以一种相当复杂的方式取决于与每个斑块相关联的最优表型在表型空间中的各自位置。从数学角度来看，与[公式：见原文]或[公式：见原文]相比，处理[公式：见原文]时困难的一个重要部分来自于缺乏对称性。我们基于不动点定理、比较原理、积分估计、变分论证、重排技术和数值模拟得到的结果，能更好地理解这些依赖性。特别是，与只有两个斑块的情况相比，我们精确地刻画了添加第三个斑块增加或减少种群持续存在机会的情形。
