Adaptation in a heterogeneous environment. I: Persistence versus extinction

TL;DR

This paper models pathogen adaptation in heterogeneous environments using PDE systems, revealing how migration influences persistence and extinction, with implications for disease management in agroecosystems.

Contribution

It introduces a PDE framework for analyzing pathogen adaptation across two patches, highlighting the impact of migration and phenotypic dynamics on persistence.

Findings

Migration reduces pathogen persistence chances.

A lethal migration threshold exists beyond which extinction occurs.

PDE results align with stochastic simulations.

Abstract

Understanding how a diversity of plants in agroecosystems affects the adaptation of pathogens in a key issue in agroecology. We analyze PDE systems describing the dynamics of adaptation of two phenotypically structured populations, under the effects of mutation, selection and migration in a two-patches environment, each patch being associated with a different phenotypic optimum. We consider two types of growth functions that depend on the n--dimensional phenotypic trait: either local and linear or nonlocal nonlinear. In both cases, we obtain existence and uniqueness results as well as a characterization of the large-time behaviour of the solution (persistence or extinction) based on the sign of a principal eigenvalue. We show that migration between the two environments decreases the chances of persistence, with in some cases a 'lethal migration threshold' above which persistence is not possible. Comparison with stochastic individual-based simulations shows that the PDE approach accurately captures this threshold. Our results illustrate the importance of cultivar mixtures for disease prevention and control.