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Adaptation in a heterogeneous environment I: persistence versus extinction


Understanding how a diversity of plants in agroecosystems affects the adaptation of pathogens is 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-patch 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.

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Fig. 1

Code availability

The solution of the PDE was computed using the method of lines coupled with the Runge-Kutta ODE solver \(\text {Matlab}^\circledR \) ode45.


  1. The breeding value for a phenotypic trait is usually defined as the total additive effect of its genes on that trait (Falconer and Mackay 1996) and is independent of the environmental conditions, given the genotype. For simplicity and consistency with other modeling studies, we will call \(\mathbf {x}\) the ‘phenotype’ in the following, although it represents breeding values.

  2. We use in the last term the fact that the proportionality factor between \(\widetilde{u}_i\) and \(u_i\) is the same for \(i\in \{1,2\}\).

  3. Notice that in formula (36) of the proof of Theorem 2, N(t) was the population size of the solution for growth functions of the first type (5), whereas here this population size is called \(\widetilde{N}(t)\).


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The authors are grateful to the anonymous referees for their valuable comments and suggestions, which led to significant improvements of the manuscript.


The project leading to this publication has received funding from Excellence Initiative of Aix-Marseille University - A*MIDEX, a French “Investissements d’Avenir” programme, and from the ANR project RESISTE (ANR-18-CE45-0019).

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Correspondence to François Hamel.

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Hamel, F., Lavigne, F. & Roques, L. Adaptation in a heterogeneous environment I: persistence versus extinction. J. Math. Biol. 83, 14 (2021).

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  • Mutation
  • Selection
  • Migration
  • Heterogeneous environment
  • Persistence
  • Extinction

Mathematics Subject Classification

  • 35B30
  • 35B40
  • 35K40
  • 35Q92
  • 92D25