Dynamics And Optimal Harvesting Strategy Of Two Stochastic Population Models With Time Delays

Population ecology is a significant method to investigate the quantitative relation between populations and habitat environment, and dynamic models are important means to describe the variation of species,which can be used to explain and predict asymptotic properties of population dynamics. The introduction of Lotka-Volterra model starts a new chapter for the study of biomathematics. After that, many experts and scholars have been devoted to population ecology, so that it has developed rapidly. In the early years, researchers usually employ deterministic model to simulate population growth rules. However, there are some uncertainties in the environment, so it is closer to the reality of the change of species by using stochastic differential equations. In addition, the study of optimal control strategy provides strong rationale for the exploitation and protection of the environment.This paper investigates the dynamics and optimal harvesting strategy of two stochastic population models with time delays, and it consists of four chapters, as follows:In chapter one, we briefly introduce the significance of mathematical biology, the development of population ecology and some definitions and theorems related to ordinary differential equation,stochastic differential equation and retarded functional differential equation.In chapter two, taking the stochastic effects on growth rate and harvesting effort into account, we propose a stochastic delay model of species in two habitats. Firstly, by using the stochastic analysis theory and differential inequality technology, we obtain sufficient conditions for persistence in the mean and extinction of the species. Afterwards, we prove that the system is asymptotically stable in distribution by using ergodic method. Furthermore, the optimal harvesting effort and the maximum of expectation of sustainable yield (ESY) are derived from Hessian matrix method and optimal harvesting theory of differential equations. Finally, some numerical simulations are provided to illustrate the theoretical results.Chapter three builds a stochastic competition system in a polluted environment driven by Brownian movement and Levy jumps. We investigate how the noises affect extinction and persistence in the mean of the population and obtain the optimal harvesting effort and dynamics of the stochastic delay competition model. The positive equilibrium point of deterministic competition system exists under some conditions.However, population quantities of stochastic system are away from the positive equilibrium point caused by random disturbance. If the intensity of noise is sufficiently large, the species may suffer extinction, whereas the prosperity of permanence can be preserved under noises with small intensity. That is to say, environment disturbance tends to have negative effects on the persistence of population.In chapter four, we summarize the main work of this paper and discuss the content and results. In addition, some future research directions are prospected.