Study Of Two Stochastic Population Systems

Ecological systems in the real world are inevitably affected by environmental noises. So it is important to investigate the stochastic mathematical models for ecology to reveal the effect of the environmental noises on ecological systems. In this paper, we devote to investigating the dynamical properties of two stochastic population systems by using the theoretical knowledge of stochastic differential equations. Moreover, we present several specific examples and their corresponding numerical simulations to verify our theoretical results.The whole paper has been divided into four chapters:The first chapter introduces the research backgrounds, significances and status as well as the main works that we have done in this thesis.The second chapter briefly introduces some preliminaries, such as probability theory, stochastic processes, stochastic calculus and stochastic differential equations etc.In the third chapter, a non-autonomous single-species model subject to impulsive and stochastic perturbations is investigated. By using the relevant preliminaries and Lyapunov function method, we obtain the existence and uniqueness of the globally positive solution. Based on the above property, we further discuss some important dynamical properties, including the stochastic permanence, extinction, asymptotic pathwise estimation and global attractivity etc. Moreover, we present several specific examples and their corresponding numerical simulations.In the fourth chapter, an autonomous stochastic competitive system is investigated. By applying the comparison theorem of stochastic differential equation and the ?Ito formula, the moment inequality for stochastic integrals, the Chebyshev inequality etc. and constructing a Lyapunov function, we discuss the existence and uniqueness of the globally positive solution, stochastically ultimate boundedness and the global attractivity. Moreover, we present the specific example and its corresponding numerical simulation.