Extinction in Two-Species Nonlinear Discrete Competitive System

Abstract

We propose a nonlinear discrete system of two species with the effect of toxic substances. By constructing a suitable Lyapunov-type function, we obtain the sufficient conditions which guarantee that one of the components will be driven to extinction while the other will be globally attractive with any positive solution of a discrete equation. Two examples together with their numerical simulations illustrate the feasibility of our main results. The results not only improve but also complement some known results.

Extinction of a two species non-autonomous competitive system with Beddington-DeAngelis functional response and the effect of toxic substances

“…For more work on competitive system with toxic substance, one could refer to [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][35][36][37][38] and the references cited therein. On the other hand, based on the traditional Lotka-Volterra competition model, some scholars argued that the more appropriate competition model should with nonlinear inter-inhibition terms.…”

Permanence of a nonlinear competition model with delay

“…On the other hand, during the past decades, many scholars argued that nonlinear population model is more appropriate then the Logistic type model, and they investigated the extinction, persistent, and stability property of the nonlinear population models ( [22]- [30]). For example, Chen and Shi [27] investigated the following nonlinear model: 6) where i = 1, 2, ..., n, j = 1, 2, ..., m, x i (t) denotes the density of prey species X i at time t, y j (t)…”

“…The success of [22]- [30] motivated us to proposed the system (1.1). The aim of this paper is, by further developing the analysis technique of [2,25], to obtain a set of sufficient conditions to ensure the permanence of the system (1.1).…”

Permanence of a nonlinear mutualism model with time varying delay

“…After that, by constructing some suitable Lyapunov functional, they also obtained a set of sufficient conditions which ensure the global attractivity of the positive periodic solution. After the works of [1,13,14,21,24,27,28], many scholars ( [4,7,8,12,16,23,31,32,[34][35][36]) done works on this direction. For example, Chen [12] further incorporated the feedback control variables to the system (1.3) and investigated the persistent property of the system.…”