Abstract
We investigate global dynamics of the following systems of difference equations xn+1=xn/A1+B1xn+C1yn, yn+1=yn2/A2+B2xn+C2yn2, n=0,1,…, where the parameters A1, A2, B1, B2, C1, and C2 are positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers. This system is a version of the Leslie-Gower competition model for two species. We show that this system has rich dynamics which depends on the part of parametric space.
Highlights
In this paper we study the global dynamics of the following rational system of difference equations: xn+1 = A1 xn + B1xn + C1yn, yn+1 A2 yn2 + B2xn + C2yn2, (1)n = 0, 1, . . . , where the parameters A1, A2, B1, B2, C1, and C2 are positive numbers and initial conditions x0 and y0 are arbitrary nonnegative numbers.System (1) is a competitive system, and our results are based on recent results about competitive systems in the plane; see [1]
We investigate global dynamics of the following B2xn + C2yn2), n = 0, 1, . . ., where the parameters systems A1, A2, of difference equations xn+1 = B1, B2, C1, and C2 are positive xn/(A1 + B1xn + C1yn), yn+1 = yn2/(A2 + numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers
This system is a version of the Leslie-Gower competition model for two species. We show that this system has rich dynamics which depends on the part of parametric space
Summary
In this paper we study the global dynamics of the following rational system of difference equations: xn+1. Abstract and Applied Analysis scenario with nine equilibrium points system (3) exhibits both competitive exclusion and competitive coexistence as well as the Allee effect Another system with quadratic terms is xn+1. Which exhibits nine dynamic scenarios and whose dynamics is very similar to the corresponding system without quadratic terms considered in [7] It seems that an introduction of quadratic terms in equations of the Leslie-Gower model (2) generates the Allee effect. When A1 < 1, there exist 11 regions of parameters with different global dynamics In nine of these regions the global dynamics is in competitive exclusion case, which means that all solutions converge to one of the equilibrium points on the axes and in only two situations we have competitive coexistence case, which means that the interior equilibrium points have substantial basin of attraction.
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