Analysis Of Three Stochastic Predator-prey Systems

In biomathematics, deterministic models are often used to solve those questions appearing in biology and ecology, on the main basis of assuming the individual quan-tity is big enough, by virtue of large number law, the system will show stationary statistical rules. However, in the real natural world, randomness and contingency is one of the necessary characteristics of ecological system, especially when the sample space is a bit small, for example, researching the engendered species or the extinction of population, the influence caused by random changes of environment to ecological system is very important, we can’t consider it as a deterministic system, ignore the stochastic perturbation of environment. In this paper, we mainly discuss three eco-logical systems, study the deterministic systems and their corresponding stochastic systems, respectively. In view of comparison and analysis about the results, we find it is very necessary to consider environmental random perturbations in ecological systems.This paper is divided into five chapters, the first chapter introduces the back-ground, basic concepts and main theorems; In chapter two, a deterministic prey-predator system with disease in the predator and its corresponding stochastic sys-tem are proposed and studied, the stability of equilibrium for deterministic system is discussed, by discussing whether the characteristic roots of Jacobi matrix corre-sponding to equilibrium have negative real parts; By virtue of analysis of Lyapunov function, we discuss the existence of positive solution for stochastic system; Although stochastic system do not have equilibrium, we can not calculate its global stability, stationary distribution of solution can be derived, which can be seen as stable in stochastic sense; Without considering random perturbation of environment, from the above conclusions, the global asymptotic stability of positive equilibrium for deterministic system is achieved; By constructing Lyapunov function, the asymp-totic behavior of solution for stochastic system fluctuating around the disease-free equilibrium and boundary equilibrium of deterministic system is discussed, thus, the global asymptotic stability of boundary equilibrium for deterministic system is obtained; The results show that, for the deterministic system, as long as the positive equilibrium exists, it is global asymptotic stable, that is, the disease has always been popular; When the intensity of environmental white noise is small, the property of solution for stochastic system is similar to the solution of deterministic system, there is a stationary distribution around the positive equilibrium; However, suffering big white noise, the solution of stochastic system will have a big change, oscillating around disease-free equilibrium of deterministic system, the change for infectious disease model, has the vital significance, thus considering random effects of the environment is very necessary.In chapter three, a prey-predator model with disease in predator and Beddington-DeAngelis functional response is raised. By analyzing the corresponding characteris-tic roots, the stability of four non-negative equilibria is derived; Without considering the random factors of environment, based on the conclusions of the fourth chapter, the global asymptotic stable result about boundary, disease-free and positive equi-libria of deterministic system can be obtained; Applying comparison theorem, we get conditions of permanence for deterministic system; The fourth chapter studies two stochastic systems corresponding to the deterministic system in chapter three, for the first stochastic system, we discuss the global solution by constructing Lyapunov function; Using comparison theorem of stochastic differential equation, an estimate of the solution for stochastic system can be formed; By virtue of some skills such as enlarging, reducing, and so on, the persistence in the time mean and extinction for prey and predator populations is discussed, here the threshold value of persistence in the mean and extinction is considered, the results compare with permanence in chapter three, we can find that the environment random disturbance, to some exten-t, reduces the size of population; Constructing Lyapunov function, the oscillation of solution around boundary equilibrium and disease-free equilibrium is discussed, and the amplitude of oscillation is closely related to the intensity of environmental white noise; The stationary distribution of solution is also studied, the solution of stochas-tic system fluctuates around the positive equilibrium of deterministic system; For the second stochastic system, the stochastic global asymptotic stability is derived; Comparing the conclusions in the third chapter with that in the fourth chapter, we find that the solution of corresponding stochastic system fluctuates around the equi-librium, when one equilibrium of deterministic system is global asymptotic stable, and the amplitude of fluctuation is related to the intensity of environmental white noise. In the fifth chapter, a non-autonomous stochastic system is proposed, with Beddington-DeAngelis functional response, we consider the existence of global solu-tion, using a similar method; By showing moment boundedness of solution, stochas-tic ultimate boundedness of solution can be derived; Adopting the comparative method, the persistence in the mean and extinction for prey and predator popula-tions is discussed, respectively, these results can be used to estimate the extinction risk of population; In the final part of every chapter, numerical simulations are given to verify our conclusions.