Mathematical Studies of Population Models in Stochastic Environments

Abstract

This dissertation is devoted to the study of population models in stochastic environments. We will investigate a two-species lottery model in non-stationary stochastic environment, an $N-$species lottery model in stationary stochastic environment and an age-structured model in random environment.

First, a two-species lottery competition model with non-stationary environmental parameters is studied. We start with viewing the classical discrete lottery model as a Markov process. Then a diffusion process that represents the fraction of sites occupied by adults of the species, as the limit of the Markov process, is derived. A non-autonomous stochastic differential equation that describes the diffusion process, as well as a Fokker-Planck equation on its transitional probability are developed. Existence, uniqueness, and dynamics of solutions for the resulting stochastic differential equation and Fokker-Planck equation are investigated, from which sufficient conditions for coexistence are established. Numerical simulations are presented to illustrate the theoretical results. Furthermore, a lottery competition model with $N \geq 2$ species in stationary stochastic environments is studied under the assumption that the environmental parameters are i.i.d.. We establish a system of nonlinear SDEs as the diffusion approximation for the discrete lottery model. Then the existence and uniqueness of the well-posed global solutions, along with asymptotic behaviors for the SDE system are investigated. Especially, sufficient conditions under which extinction and persistence occur are constructed, respectively.

Finally, a random age-structured model with random nonlinear birth rate is formulated. Its mathematical theories including well-posedness, co-cycle property and long time dynamics of the solution are developed. The emphasis is given to the asymptotic smoothness and the bounded dissipativeness of the solution to the model, which implies the existence of the random pullback attractor.