In general, polynomial difference equations and polynomial maps in the plane have been studied in both the real and complex domains (see [8, 9]). First results on quadratic polynomial difference equation have been obtained in [1, 2] but these results gave us only a part of the basins of attraction of equilibrium points and period-two solutions. In [4], the general second order difference equation is completely investigated and described the regions of initial conditions in the first quadrant for which all solutions tend to equilibrium points, period-two solutions, or the point at infinity, except for the case of infinitely many period-two solutions. In [3], case of infinitely many period-two solutions is completely investigated. Our results are based on the theorems which hold for monotone difference equations. Our principal tool is the theory of monotone maps, and in particular cooperative maps, which guarantee the existence and uniqueness of the stable and unstable invariant manifolds for the fixed points and periodic points (see [5]). Consider the difference equation

In this paper we will observe the model of competitive types and it will be analyzed using the nullcline method. It will be shown that this model has four points of equilibria, which are stable or unstable depending on the parameters a and b. The local stability of these points was investigated and global dynamics was determined using nullcline methods, that is, the bases of attraction of these points were shown.

Here we examine the behavior of a rational Lotka - Volterra model which is a modification of the ordinary polynomial case. We find nonnegative equilibrium points and define conditions in the parametric space for the stable positive equilibrium point. We also prove existence of the stable limit cycle in the case of the unstable positive equilibrium point.

In this paper will be observed the population dynamics of a three-species Lotka-Volterra model: two predators and their prey. This simplified model yields a more complicated dynamical system than classic Lotka-Volterra model. We will give the conditions under which one of the predators becomes extinct and when the coexistence between predators is possible. Given will be sufficient conditions for the existence of solutions for certain classes of Cauchy’s solutions of Lotka-Volterra model. The behavior of integral curves in the neighborhoods of an arbitrary integral curves will be considered.