Global Dynamics, Bifurcation Analysis, and Chaos in a Discrete Kolmogorov Model with Piecewise-Constant Argument

The local behavior with topological classifications, bifurcation analysis, chaos control, boundedness, and global attractivity of the discrete-time Kolmogorov model with piecewise-constant argument are investigated. It is explored that Kolmogorov model has trivial and two semitrival fixed points for all involved parameters, but it has an interior fixed point under definite parametric condition. Then, by linear stability theory, local dynamics with different topological classifications are investigated around trivial, semitrival, and interior fixed points. Further for the discrete Kolmogorov model, existence of periodic points is also investigated. It is also investigated the occurrence of bifurcations at interior fixed point and proved that at interior fixed point, there exists no bifurcation, except flip bifurcation by bifurcation theory. Next, feedback control method is utilized to stabilize chaos existing in discrete Kolmogorov model. Boundedness and global attractivity of the discrete Kolmogorov model are also investigated. Finally, obtained results are numerically verified.

Numerical Simulation of a Class of Three-Dimensional Kolmogorov Model with Chaotic Dynamic Behavior by Using Barycentric Interpolation Collocation Method

This paper numerically simulates three-dimensional Kolmogorov model with chaotic dynamic behavior by barycentric Lagrange interpolation collocation method. Some numerical examples are studied for finding some new chaotic behaviors and demonstrating some existing chaotic dynamic behaviors of the Kolmogorov model. Results obtained by the present method indicate that the method has merits of small operations and good numerical stability.