Dynamics and bifurcations of a Lotka-Volterra population model

Abstract

This paper investigates the dynamics and stability properties of a discrete-time Lotka-Volterra type system. We first analyze stability of the fixed points and the existence of local bifurcations. Our analysis shows the presence of rich variety of local bifurcations, namely, stable fixed points; in which population numbers remain constant, periodic cycles; in which population numbers oscillate among a finite number of values; quasi-periodic cycles; which are constraint to stable attractor called invariant closed curve, and chaos, where population numbers change erratically. Our study is based on the numerical continuation method under variation of one and two parameters and computing different bifurcation curves of the system and its iterations. For the all codimension 1 and codimension 2 bifurcation points, we compute the corresponding normal form coefficients to reveal criticality of the corresponding bifurcations as well as to identify different bifurcation curves which emerge around the corresponding bifurcation point. In particular we compute a dense array of resonance Arnol'd tongue corresponding to quasi-periodic invariant circles rooted in weakly resonant Neimark-Sacker associated to multiplier with frequency . We further perform numerical simulations to characterize qualitatively different dynamical behaviors within each regime of parameter space.