The Predator-prey System With Random Perturbation And Maximum Likelihood Estimations Of Its Parameters

The research of mathematical biology model has got the extensive application through the development of a century. The study about mathematical biology become more and more important in the 21th century. The intersectant domain will be researched mainly between mathematical biology and other subjects. Compared with the determinate mathematical biology model, species ecology systems are often subject to environmental noise. It is important to discover whether the presence of a such noise affects these results that we have obtained by applying the stochastic differential equation. Moreover, the estimating of the parameters by applying statistical methods will become an important disscussion with applying stochastic differential equation extensively in the mathematical biology .This paper discusses a randomized 2-species Lotka-Volterra mutualism systemwhere T_i,, a_ij, σ_i > 0 (i, j = 1, 2), and B_i(t) are independent standard Brownian motions, i = 1, 2. We show that the positive solution of the associated stochastic differential equation does not explode to infinity in a finite time. In addition, the existence, uniqueness, persistence and global asymptotic stability in mean of the positive solutions of the system are studied.In general, the growth rate, death rate and the strength of white noise in the system are unknown. Making use of statistics method to estimate the parameters in Biologyical models has become a new research topic. In the last section we. give the MLE of the parameters in system.This paper is composed of two parts. In the first chapter, we introduce the historical background of the problems which will be investigated and the main results of this paper. In the second chapter, in section 2.1, the existence and the uniqueness of positive solutions to equation (1.1-2) are studied which is fundamental for the subsequent developments. In section 2.2, we show that the solutions of Eq (1-1.2) will remain in the positive cone R₊². This nice positive property provides us with a great opportunity to discuss how the solutions vary in R₊² in more detail. In this section we will show that the solution of Eq(1.1.2) is between the positive solutions of two different stochastic differential equations. In section 2.3, we study the persistence of the solution, and we give its boundedness. In section 2.4, we give global asymptotic stability in mean of the positive solutions. At last,since the parameters are usually unknown, so we give the MLEs of parameters. Simulation results show that the performance of MLEs is fit well.