Dynamics and Bifurcation for One Non-linear System

In this paper, we observed the ordinary differential equation (ODE) system and determined the equilibrium points. To characterize them, we used the existing theory developed to visualize the behavior of the system. We describe the bifurcation that appears, which is characteristic of higher-dimensional systems, that is when a fixed point loses its stability without colliding with other points. Although it is difficult to determine the whole series of bifurcations that lead to chaos, we can say that it is a common opinion that it is precisely the Hopf bifurcation that leads to chaos when it comes to situations that occur in applications. Here, subcritical and supercritical bifurcation occurs, and we can say that subcritical bifurcation represents a much more dramatic situation and is potentially more dangerous than supercritical bifurcation, technically speaking. Namely, bifurcations or trajectories jump to a distant attractor, which can be a fixed point, limit cycle, infinity, or in spaces with three or more dimensions, a foreign attractor

Application of the Nullcline Method to a Certain Model of Competitive Species

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

Lotka-Volterra Model with Two Predators and Their Prey

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

Coexistence between predator and prey in the modified Lotka - Volterra model

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