2015/09/07 by Hélène Leman, Leman, Helene · 1 citation
Biochemistry, Genetics and Molecular Biology · Medicine · Social Sciences · #Analysis of PDEs (math.AP) #Evolution and Genetic Dynamics #Evolutionary Game Theory and Cooperation #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.1509.02022
openalex publication_date 2015/09/07 · openalex created_date 2022/11/29 · openalex updated_date 2026/07/28
We consider an individual-based spatially structured population for Darwinian\nevolution in an asexual population. The individuals move randomly on a bounded\ncontinuous space according to a reflected brownian motion. The dynamics\ninvolves also a birth rate, a density-dependent logistic death rate and a\nprobability of mutation at each birth event. We study the convergence of the\nmicroscopic process when the population size grows to +\∞ and the\nmutation probability decreases to 0. We prove a convergence towards a jump\nprocess that jumps in the infinite dimensional space of the stable spatial\ndistributions. The proof requires specific studies of the microscopic model.\nFirst, we examine the large deviation principle around the deterministic large\npopulation limit of the microscopic process. Then, we find a lower bound on the\nexit time of a neighborhood of a stationary spatial distribution. Finally, we\nstudy the extinction time of the branching diffusion processes that approximate\nsmall size populations.\n