| For every choice of positive parameters alpha, beta, gamma, A, B and C, consider the two difference equations xn+1=a+bxn+g xn-1Z+Bxn+Cxn-1 ,n=0,1,2,...,x-1,x 0∈&sqbl0;0,infinity&parr0; E1 and xn+1=a+bxn+g xn-1Bxn+Cxn-1 ,n+0,1,2,...,x-1,x 0∈0,infinity, E2;In this thesis, it is shown that all solutions to Eqns.(E1) and (E2) converge to the positive equilibrium or to a prime period-two solution.;A complete qualitative description of the global behavior of solutions to (El) with nonnegative parameters is also given in this thesis whenever prime period-two solutions exist.;Furthermore, a relation is established between local stability of equilibria and slopes of critical curves of planar maps. Then this result is used to give global behavior for nonnegative solutions of the system of difference equations xn+1=b1xn 1+xn+c1yn +h1yn+1=b2 yn1+yn+c2xn +h2n=0,1,&ldots; ,x0,y0 ∈&sqbl0;0,infinity&parr0;x&sqbl0;0,infinity&parr0; with positive parameters. In particular, it is shown that the system has between one and three equilibria, and that the number of equilibria determines global behavior as follows: if there is only one equilibrium, then it is globally asymptotically stable. If there are two equilibria, then one is a local attractor and the other one is nonhyperbolic. If there are three equilibria, then they are linearly ordered in the south-east ordering of the plane, and consist of a local attractor, a saddle point, and another local attractor. In addition, sufficient conditions are given for the system to have a unique equilibrium. |
