Bifurcation Analysis and Chaos of a Discrete-Time Kolmogorov Model

Abstract

In this paper, we explore local dynamical characteristics with different topological classifications at fixed points, bifurcations and chaos in the discrete Kolmogorov model. More precisely, we investigate the existence of trivial, boundary and interior fixed points of the discrete Kolmogorov model by algebraic techniques. We prove that for all involved parameters, the discrete Kolmogorov model has trivial and two boundary fixed points, and the interior fixed point under specific parametric condition. Further we explore the local dynamics with topological classifications at fixed points and existence of periodic points of the discrete Kolmogorov model simultaneously. We also explore the occurrence of bifurcation at fixed points and prove that at boundary points there exists no flip bifurcation but it occurs at the interior fixed point. Moreover, we utilize feedback control method to stabilize chaos appears in the Kolmogorov model. Finally, we present numerical simulations to verify corresponding theoretical results and also reveal some new dynamics.