2018/08/23 by Michel Benaı̈m, Sebastian J. Schreiber, Benaïm, Michel +1 · 1 citation
Medicine · Biochemistry, Genetics and Molecular Biology · Social Sciences · #Mathematical and Theoretical Epidemiology and Ecology Models #Evolution and Genetic Dynamics #Evolutionary Game Theory and Cooperation
paper · pdf · doi:10.48550/arxiv.1808.07888
The dynamics of species' densities depend both on internal and external\nvariables. Internal variables include frequencies of individuals exhibiting\ndifferent phenotypes or living in different spatial locations. External\nvariables include abiotic factors or non-focal species. These internal or\nexternal variables may fluctuate due to stochastic fluctuations in\nenvironmental conditions. We prove theorems for stochastic persistence and\nexclusion for stochastic ecological difference equations accounting for\ninternal and external variables. Specifically, we use a stochastic analog of\naverage Lyapunov functions to develop sufficient and necessary conditions for\n(i) all population densities spending little time at low densities, and (ii)\npopulation trajectories asymptotically approaching the extinction set with\npositive probability. For (i) and (ii), respectively, we provide quantitative\nestimates on the fraction of time that the system is near the extinction set,\nand the probability of asymptotic extinction as a function of the initial state\nof the system. Furthermore, we provide lower bounds for the expected time to\nescape neighborhoods of the extinction set. To illustrate the applicability of\nour results, we analyze stochastic models of evolutionary games, Lotka-Volterra\ndynamics, trait evolution, and spatially structured disease dynamics. Our\nanalysis of these models demonstrates environmental stochasticity facilitates\ncoexistence of strategies in the hawk-dove game, but inhibits coexistence in\nthe rock-paper-scissors game and a Lotka-Volterra predator-prey model.\nFurthermore, environmental fluctuations with positive auto-correlations can\npromote persistence of evolving populations and persistence of diseases in\npatchy landscapes. While our results help close the gap between the persistence\ntheories for deterministic and stochastic systems, we highlight challenges for\nfuture research.\n