Stability Analysis of Equilibrium Point of Population Competitive Model

In this paper, the stability of equilibrium points of two species competitive model are studied. On the base of logistic model, the competitive model of two species living in the same natural environment is considered. Firstly, the four equilibrium points of the two species competition model are given, and then stability of each equilibrium point is judged by ordinary differential equation stability theory. Secondly, according to the characteristics of the model, the phase plane is divided into several regions. Finally, matlab programs are wrote, and the numerical solutions of two species competition model are obtained for several different initial conditions, and the graphics of corresponding phase trajectory are drew, and the stability of equilibrium points are analyzed. Introduction Combination of mathematics and ecology provides an effective tool to understand, explain and predict the ecological phenomenon. In the specific ecological environment, generally there are multiple species populations, and the increase or decrease of the quantity of each population is not only affected by their self limiting law, inevitably will be restricted by other populations. In order to compete for the same limited sources of food and living space, there are the mutual competition, and the interdependence or the law of the jungle between two or more than two populations live in the same environment. The result of competition is that the two populations coexist or the weak competitive populations become extinct, while the competitive population can survive and reach the maximum amount of environmental capacity (see [1-10]). In this paper, we first consider the logistic model of single species, and then consider the model of two competition species living the same natural environment. According to the method for solving equilibrium points of the differential equations, we obtain four equilibrium points of two species competitive model. Then by means of stability theory on the ordinary differential equation, we judge the stability of every equilibrium point. According to the characteristics of the model, we divide the phase plane into several regions. In order to obtain the numerical solution of the two species competitive model under different initial conditions, we write corresponding matlab program, and draw the corresponding to the phase trajectory diagram and analysis of equilibrium stability. Competitive Model of Two Populations When population A and population B live alone in a natural environment, the change rate of a population obeys the law of Logistic. Let ) ( ), ( 2 1 t y t y denote the number of population A and population B, 2 1,m m their intrinsic growth rate, and 2 1, N N their maximum capacity. Specifically, the number of populations of A satisfies the following differential equation (see [1]) ) 1 ( ) ( 1 1 1 1 ' 1 N y y m t y   (1) International Conference on Education, Management and Computer Science (ICEMC 2016) © 2016. The authors Published by Atlantis Press 540 where ) 1 ( 1 1 N y  reflect the blocking effect of A on the growth of its own due to the consumption of limited resources, 1 1 N y can be explained to amount of food which unit number A (be relative to 1 N ) consume. When two populations live in the same natural environment, the number of populations of A and B satisfy the following differential equation (see [1])