Period-doubling and Neimark–Sacker bifurcations of plant–herbivore models

Abstract

The interaction between plants and herbivores plays a vital role for understanding community dynamics and ecosystem function given that they are the critical link between primary production and food webs. This paper deals with the qualitative nature of two discrete-time plant–herbivore models. In both discrete-time models, function for plant-limitation is of Ricker type, whereas the effect of herbivore on plant population and herbivore population growth rate are proportional to functional responses of type-II and type-III. Furthermore, we discuss the existence of equilibria and parametric conditions of topological classification for these equilibria. Our analysis shows that positive steady states of both discrete-time plant–herbivore models undergo flip and Hopf bifurcations. Moreover, we implement a hybrid control strategy, based on parameter perturbation and state feedback control, for controlling chaos and bifurcations. Finally, we provide some numerical simulations to illustrate theoretical discussion.