Dynamics Of Some Higher Order Rational Difference Equations

Difference equations (as well as systems of difference equations) are mathematical tools to model the discrete time evolution of natural and social systems. In the past few decades, numerous difference equations and systems of difference equations have been proposed to characterize phenomena occurring in diverse areas such as physics, chemistry, biology, medical sciences, economics, ecology, engineering, military sciences, social sciences, etc..As a large class of simple-looking nonlinear difference equations, rational difference equations not only have been successfully applied to a wide range of areas, but are theoretically important. Indeed, previous researches showed that rational difference equations can exhibit complex and fascinating dynamic properties such as equilibria, periodic solutions, and chaos.This thesis addresses the dynamics of higher-order nonlinear rational difference equations, with emphasis on their asymptotic stability, global asymptotic stability, persistence, and boundedness. The materials are organized in the following way.In Chapter 1, several typical applications of difference equations are given. Then the state-of-the-art of some kinds of rational difference equations are reviewed. On this basis, the main contributions of this thesis are briefly mentioned.Chapter 2 gives basic notions and notations that will be used in our work.Chapter 3 is devoted to the study of the difference equation with positive parameters and positive initial conditions. It is concluded that(1) if p > q or p≤q < 1 + 4p, the positive equilibrium of this equation is global attractor.(2) if q≥1 + 4p, there exists N≥0 such that 1≥xn≥p /q for all n≥N.Chapter 4 focuses on the difference equation with positive parameters and positive initial conditions. The main results are presented as follows:(1) Every solution of this equation satisfies min{1, p / q} < xn < max{1, p / q}, n≥1. (2) The positive equilibrium of the equation is a global attractor if one of the following two conditions is satisfied: (a) Either p > q≥1 or 1≥p > q or (1 + 3 q ) /(1 ? q )≥p > 1> q. (b) Either q > p≥1 or 1≥q > p or (1 + 3 p ) /(1 ? p )≥q > 1> p.Chapter 5 deals with the difference equation with positive parameters and positive initial conditions. The main results are as follows: (1) Every solution of this equation satisfies M≥x n≥p /(1 + (1 + r ) M) for all n≥0. (2) The positive equilibrium of the equation is a global attractor if 0 < p≤q and one of the following three conditions is satisfied, (a) q≤1; (b) 0 < r≤1; (c) r > 1 and ( q ? 1) 2( r ? 1)≤4p. (3) The positive equilibrium is a global attractor if q < p < q + q 2r and the following conditions are satisfied: (a) There exists N≥0 such that ( p ? q ) / qr < x n< q for all n≥N (b) One of the following three conditions holds: (b1) q≤1; ( b2 ) r≤1; ( b3 ) r > 1 and ( q ? 1) 2( r ? 1)≤4p. (4) The positive equilibrium of the equation is a global attractor if p≥q + q 2r.Chapter 6 is focused on the difference equation with positive parameters and positive initial conditions. The main results are as follows: (1) If p < q + 1, the positive equilibrium of the equation is a global attractor. (2) If p = q + 1 and s is odd, the positive equilibrium of the equation is a global attractor. (3) If p > q + 1 and s, t satisfy v = 3v/s for each v = 1, 2,…, every positive solutions of the equation is bounded.In Chapter 7, we study the following system with positive parameters and positive initial conditions. The following results are established: (1) Every positive solutions of the equation is bounded if A > 1. (2) The positive equilibrium of the equation is asymptotically stable if A > 2/ 3. (3) Every positive solutions of the equation converges to the positive equilibrium if A > 2.In Chapter 8, we make a summary of our works as well as the outloolk for the next works.