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.

We investigate global dynamics of the following systems of difference equations , , , where the parameters , , , , , and are positive numbers and the initial conditions and 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.

The proven safe operation and the high availability of aerial ropeways is mainly thanks to their design. This is based on the manufacturer’s extensive experience as well as the strict application of the relevant rules and norms. In that regard, this paper describes an analysis of the haulage ropes on ropeways in case of accidental loads. In solving this problem, analyzed ropeway system with sufficient accuracy was modeled as a system with three degrees of freedom. For solving differential equations we have used the software “Wolfram Mathematica”. At the end of the paper, we discuss the results of the dynamic forces in the haulage ropes in a certain time interval, and the results of the safety factors which in this case are sufficient to ensure reliable operation of the system.

We investigate global dynamics of the following systems of difference equations: {xn+1=b1xn2A1+yn2,yn+1=a2+c2yn2xn2,n=0,1,2,…, where the parameters b1, a2, A1, c2 are positive numbers and the initial condition y0 is an arbitrary nonnegative number and x0 is a positive number. We show that this system has rich dynamics which depends on the part of a parametric space. We find precisely the basins of attraction of all attractors including the points at ∞. MSC:39A10, 39A30, 37E99, 37D10.

This paper presents sufficient conditions for the existence of solutions for certain classes of Cauchy’s solutions of the Lagerstrom equation as well as their behavior. Behavior of integral curves in the neighborhoods of an arbitrary or integral curve are considered. The obtained results contain the answer to the question on approximation of solutions whose existence is established. The errors of the approximation are defined by functions that can be sufficiently small. The theory of qualitative analysis of differential equations and topological retraction method are used.