Persistence Of Stochastic Mutualism System And MLE Of Its Parameters

The study about mathematical biology become more and more important in the 21thcentury. The intersectant domain will be researched mainly between mathematical biologyand other subjects. Compared with the determinate mathematical biology model, speciesecology systems are often subject to environmental noise, it is important to discoverwhether the presence of a such noise affects these results that we have obtained. Moreoverthe estimating of the parameters by applying statistical methods will become an importantdisscussion with applying stochastic differential equation extensively in the mathematicalbiology.This paper discusses a randomized 2-species Lotka-Volterra mutualism systemwhere r_i, a_ijσ_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 equa-tion does not explode to infinity in a finite time. In addition, the existence, uniqueness,persistence of the positive solutions of the system are studied.In general, the growth rate, death rate and the strength of white noise in the systemare unknown. Making use of statistics method to estimate the parameters in Biologyicalmodels has become a new research topic, however the problem estimating parameters inBiologyical model is totally different from the problem in finance . In the last section wegive the MLE of the parameters in system.This paper is composed of two parts. In the first chapter, we introduce the historicalbackground 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 solutionsto equation (1.2) are studied which is fundamental for the subsequent developments. Insection 2.2, we study the persistence of the solution by using inequality of Cheb andthe boundedness of p-th moments of x_i and 1/x_i. At last, since the parameters areusually unknown, so we give the MLEs of parameters. Simulation results show that theperformance of MLEs is fit well.