Abstract
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, \(\lambda _H\), and we study the dependency of \(\lambda _H\) 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 \(H\ge 3\), compared to \(H=1\) or \(H=2\), 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.
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All Matlab source code used to generate the analyses performed here is available at https://doi.org/10.17605/OSF.IO/QAV2M.
Notes
In the sequel, we say that \(\varPsi \in L^2(\mathbb {R}^n)^H\) is normalised whenever \(\int _{\mathbb {R}^{n}}\Vert \varPsi (\textbf{x})\Vert ^2\,\textrm{d}\textbf{x}=1.\)
Notice that the inequalities \(0\le \lambda _H'(0)<1\) are coherent with Corollary 1.
The authors are grateful to L. Dupaigne, F. Murat and an anonymous referee for helpful bibliographical comments on this point.
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Funding
This work has received funding from the French ANR RESISTE (ANR-18-CE45-0019) and DEEV (ANR-20-CE40-0011-01) projects, from Excellence Initiative of Aix-Marseille Université - A*MIDEX, a French “Investissements d’Avenir” program, and from the région Normandie BIOMA-NORMAN (21E04343) project. The authors acknowledge support of the Institut Henri Poincaré (UAR 839 CNRS-Sorbonne Université) and LabEx CARMIN (ANR-10-LABX-59-01).
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Alfaro, M., Hamel, F., Patout, F. et al. Adaptation in a heterogeneous environment II: to be three or not to be. J. Math. Biol. 87, 68 (2023). https://doi.org/10.1007/s00285-023-01996-4
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DOI: https://doi.org/10.1007/s00285-023-01996-4