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.

了解农业生态系统中植物的多样性如何影响病原体的适应是农业生态学的一个关键问题。我们分析了 PDE 系统，该系统描述了两个表型结构种群在突变、选择和迁移的影响下，在双补丁环境中的适应动态，每个补丁都与不同的表型优化相关。我们考虑两种依赖于n的增长函数-维表型特征：局部和线性或非局部非线性。在这两种情况下，我们都获得了存在性和唯一性结果以及基于主特征值符号的解的长时间行为（持久性或消光性）的表征。我们表明，两种环境之间的迁移会降低持久性的机会，在某些情况下存在“致命迁移阈值”，高于该阈值的持久性是不可能的。与基于个体的随机模拟的比较表明 PDE 方法准确地捕获了该阈值。我们的结果说明了品种混合物对疾病预防和控制的重要性。