Chaos, Bifurcation and Stability analysis of Trophic Level Prey Predator System

Abstract

Mathematical models constructed with difference equations are ideal and justified even when the size of the populations is small or when the constructed model considers non-overlapping populations. In addition, difference equation systems exhibit the complex behavior and intriguing dynamics. A three species competition model in discrete time space is considered for study in this article. Existence of a fixed point and its uniqueness is established. Local stability conditions at coexisting equilibrium state of the system are derived with respect to parameter. Analytical conditions for the existence of Neimark-Sacker Bifurcation (NSB) and Period-Doubling Bi-furcation (PDB) are obtained and numerically verified with simulations. The article also implements the hybrid method to control the occurrence of chaos in the system. Validity of the theoretical results are demonstrated numerically by phase plane analysis and bifurcation diagrams. Complex behavior exhibited by the system are illustrated.