Document Type: Regular Paper
Department of Applied Mathematics, Shahrekord University, P. O. Box 115, Shahrekord, Iran
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.