2018/09/11 by David Steinsaltz, Steinsaltz, David, Shripad Tuljapurkar +1
Environmental Science · Mathematics · #37H15 #60J05 #92D15 #Ecosystem dynamics and resilience #FOS: Biological sciences #FOS: Mathematics #Point processes and geometric inequalities #Populations and Evolution (q-bio.PE) #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1809.04025
openalex publication_date 2018/09/11 · openalex created_date 2022/08/03 · openalex updated_date 2026/07/28
The growth of a population divided among spatial sites, with migration\nbetween the sites, is sometimes modelled by a product of random matrices, with\neach diagonal elements representing the growth rate in a given time period, and\noff-diagonal elements the migration rate. The randomness of the matrices then\nrepresents stochasticity of environmental conditions. We consider the case\nwhere the off-diagonal elements are small, representing a situation where\nmigration has been introduced into an otherwise sessile meta-population. We\nexamine the asymptotic behaviour of the long-term growth rate. When there is a\nsingle site with the highest growth rate, under the assumption of Gaussian log\ngrowth rates at the individual sites (or having Gaussian-like tails) we show\nthat the behavior near zero is like a power of \ε, and derive upper and\nlower bounds for the power in terms of the difference in the growth rates and\nthe distance between the sites. In particular, when the difference in mean log\ngrowth rate between two sites is sufficiently small, or the variance of the\ndifference between the sites sufficiently large, migration will always be\nfavored by natural selection, in the sense that introducing a small amount of\nmigration will increase the growth rate of the population relative to the\nzero-migration case.\n