The Dynamics Of Stochastic Species Models

Population ecology studied the development law of the species,and its research methods is using mathematical models to predict the change of each species.In this paper,we consider the dynamical behavior of the predator-prey systems with functional responses and Gilpin-Ayala competition system under environmental noise.The regime switching which is a finite-state and continuous-time Markov chain has been introduced to population model.The telegraph noise can be illustrated as a switching between two or more regimes of environment,which differ by factors such as dry or rainy season.There exists white noise in the environment,which will lead to various species in the ecosystem are subject to various forms of random interference,such as earthquake,tsunami,temperature and humidity.Under the effect of environmental white noise and color noise,we study the dynamical behaviors of the systems from three aspects as follows:1.This section addresses the dynamical behaviors of the predator-prey systems with functional response by white noise.We prove that existence of the positive solution to the systems by the comparison theorem of stochastic equation and derive the conditions for the species to be extinction or permanence.We discuss the system with the linear and nonlinear perturbation.We study the optimal harvesting strategy of the stochastic modified Leslie-Gower and Holling-type II model.Then the sufficient and necessary criteria for the existence of optimal harvesting policy are shown employing the property of stationary distribution.The optimal harvesting effort and the maximum of expectation of sustainable yield are obtained as well.2.Dynamics of ratio-dependent predator-prey systems with functional responses in a random environment are studied.Two models in this section are considered: the stochastic Holling-type III ratio-dependent predator-prey system and the stochastic Michaelis-Menten-type ratio-dependent predator-prey system.Through construction of Lyapunov function it is proved that the system admits a stationary distribution under the some parametric conditions3.On stochastic Gilpin-Ayala competition system with periodic parameter functions and Markov switching.Utilizing Has' minskii's theory on ergodicity and Markov regime switching,we derive that there exists a unique stationary distribution which is ergodic in the case of persistence under the color and white noise.Based on Has' minskii's theory on periodic Markov processes,we obtain the existence for the non-trivial periodic solution of the system.