2019/10/21 by Jean‐Baptiste Burie, Burie, Jean-Baptiste, Arnaud Ducrot +5
Medicine · Biochemistry, Genetics and Molecular Biology · Mathematics · #Mathematical and Theoretical Epidemiology and Ecology Models #Evolution and Genetic Dynamics #Mathematical Biology Tumor Growth
paper · pdf · doi:10.48550/arxiv.1910.09385
In this work we consider an epidemic system modelling the evolution of a\nspore-producing pathogen within a multi-host population of plants. Here we\nfocus our analysis on the study of the stationary states. We first discuss the\nexistence of such nontrivial states by using the theory of global attractors.\nThen we introduce a small parameter epsilon that characterises the width of the\nmutation kernel, and we describe the asymptotic shape of steady states with\nrespect to epsilon. In particular, we show that the distribution of spores\nconverges to the singular measure concentrated on the maxima of fitness of the\npathogen in each plant population. This asymptotic description allows us to\nshow the local stability of each of the positive steady states in the regime of\nnarrow mutations, from which we deduce a uniqueness result for the nontrivial\nstationary states by means of a topological degree argument. These analyses\nrely on a careful investigation of the spectral properties of some non-local\noperators.\n