Survival Threshold And Bifurcation Analysis Of Two Types Of Stochastic Predator-prey Models

This paper focuses on the persistence,extinction and dynamic bifurcations of stochastic predator-prey models.The survival threshold and dynamic bifurcations of models are studied by discussing the stability of the invariant probability measures on the invariant sets.In the second chapter,we consider a two-dimensional stochastic rate-dependent predator-prey model with intra-species competition.Firstly,the chapter prove that a positive invariant set is composed of four invariant subsets.By analyzing the stability of the ergodic invariant probability measures on each invariant subset,we further threshold the stochastic persistence and extinction of the two populations.Secondly,the dynamic bifurcation phenomena of the populations are analyzed by using the dynamic bifurcation theory in stochastic bifurcation.Finally,we illustrate the survivability and stochastic bifurcations of the two populations through examples and numerical simulations.In the third chapter,we study the persistence,extinction and dynamic bifurcations of the stochastic predator-prey model of two species competing for one prey on the basis of the two-dimensional model.The two-dimensional model studied previously exists as part of the system boundaries.Due to the increasing boundary conditions of three-dimensional model,there are not only one-dimensional boundaries but also two-dimensional boundaries,so the situation becomes more complex.Firstly,we prove that a positive invariant set is composed of eight invariant subsets.By analyzing the stability of the ergodic invariant probability measures on each invariant subset,the threshold of the stochastic persistence and extinction of the three populations are carried out.Secondly,the dynamic bifurcation phenomena of populations is analyzed by means of dynamic bifurcation theory in stochastic bifurcation.Finally,we illustrate the survivability and stochastic bifurcations of the three populations through examples and numerical simulations.In this paper,the persistence,extinction and dynamic bifurcations of two stochastic predator-prey models are studied to provide a theoretical scheme for the persistence and extinction of biological populations in ecosystems.