Bifurcations of a Class of Discrete-Time Singular Biological Economic Models

Introduction

It is well known that the Lotka-Volterra prey-predatormodel is one of the fundamental population models. A predator-prey interaction was first described by two pioneers, Lotka [8] and Volterra [12], in two independent works. The research dealing with interspecific interactions hasmainly focused on continuous prey-predatormodels of two variables. However, certain species, including many species of insects, have no overlap between successive generations and so their population evolves in discrete time-steps. Such populations can be modeled by difference equations, otherwise known as discrete dynamical systems or (iterative)maps. There is now a considerable literature on discrete ecosystem models, e.g., see [6, 9, 10] and references therein. Some works by Danca et al. [2], Jing and Yang [5], Liu and Xiao [7], and Agiza et al. [1] showed that, for the discrete-time prey-predator models, the dynamics can produce a much richer set of patterns than those observed in continuous-time models. Also, Summers et al. have examined four typical discrete-time ecosystem models under the effects of periodic forcing [11]. They found that a system that has simplistic behavior in its unforced state can assume chaotic behaviorwhen subjected to periodic forcing, dependent on the values chosen for the controlling parameters.