Dynamic Analysis Of Two Stochastic Predator-prey Models

Ecological mathematical model is the most important content in the field of biological mathematical research in recent years. We can see, with the natural environment is increasingly explorated, developed, transformed by human, and man-made or naturally occurring environmental disasters have also become more frequently. Therefore, we introduced ecological stochastic perturbation to the study of ecological model, it further improve the accuracy of mathematical ecological system simulate the zoology. In this paper, we consider the dynamical behaviors of a stochastic eco-epidemic model with modified Holling-Tanner type and non-linear incidence and a stochastic ratio-dependent predator-prey system with Lévy jumps.The main content of this paper is organized as follows:Chapter 1 introduces the background development and significance of ecological system and the major work of this paper.Chapter 2 consider the dynamical behaviors of a stochastic eco-epidemic model with modified Holling-Tanner type and non-linear incidence. For the ecological model of infectious diseases, in the base of the predator don’t feed on the prey of infectious diseases, we mainly consider that the function between prey and predator is Holling-Tanner function and the way of the spread of the disease is nonlinear incidence, and introducing stochastic perturbations in the model to study its dynamic behaviors. First, we proved the existence and uniqueness of global positive solutions of the system. Then, studies have yielded the conditions of continued survival and extinction of solutions. Finally, since there is no balance point in stochastic model, we construct a suitable Lyapunov function to studied the asymptotic behavior of the solutions of stochastic models around the deterministic model’s boundary equilibrium and disease-free equilibrium. in addition, using the MATLAB mathematical software corroborated our conclusions.Chapter 3 concerned with a stochastic ratio-dependent predator-prey system with Lévy jumps. Considering the influence of the Lévy process, by using thecomparison theorem of stochastic differential equation and the theory of the generalized It?s formula, we proved that there is a unique positive solution of the stochastic ratio-dependent model. Besides, we conclude the conditions for species to be persistent and extinct. In the end, we introduce some numerical simulations to support the main results of the theorem.Chapter 4 summarizes the work of this paper.