Adaptation in a heterogeneous environment II: to be three or not to be

• Published in: Journal of mathematical biology Vol. 87; no. 5; p. 68
• Main Authors: Alfaro, Matthieu, Hamel, François, Patout, Florian, Roques, Lionel
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2023; Springer Nature B.V; Springer
• Subjects: Adaptation; Agroecology; Analysis of PDEs; Applications of Mathematics; Biodiversity; Biology; Coronaviruses; Eigenvalues; Fixed points (mathematics); Genotype & phenotype; Influenza; Life Sciences; Mathematical analysis; Mathematical and Computational Biology; Mathematical models; Mathematics; Mathematics and Statistics; Migration; Mutation; Optimization; Pathogens; Phenotypes; Population; Populations and Evolution; Species extinction; Zoonoses; 35K40; Spillover; 92D25; Migration; Eigenvalues; 35B30; 35B40; Adaptation; Heterogeneous environment; 35Q92
• ISSN: 0303-6812; 1432-1416
• 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, λ H , and we study the dependency of λ 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 ≥ 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.