Adaptation of an asexual population with environmental changes

Because of mutations and selection, pathogens can manage to resist to drugs. However, the evolution of an asexual population (e.g., viruses, bacteria and cancer cells) depends on some external factors (e.g., antibiotic concentrations), and so understanding  the impact of the environmental changes is an important issue. This paper is devoted to model this problem with a nonlocal diffusion PDE, describing the dynamics of such a phenotypically structured population, in a changing environment. The large-time behaviour of this model, with particular forms of environmental changes (linear or periodically fluctuations), has been previously developed. A new mathematical approach (limited to isotropic mutations) has been developed recently for this problem, considering a very general form of environmental variations, and giving an analytic description of the full trajectories of adaptation. However, recent studies have shown that an anisotropic mutation kernel can change the evolutionary dynamics of the population: some evolutive plateaus can appear. Thus the aim of this paper is to mix the two previous studies, with an anisotropic mutation kernel, and a changing environment. The main idea is to study a multivariate distribution of (2n) "fitness components". Its generating function solves a transport equation, and describes the distribution of fitness at any time.