Biological control in a simple ecological model via subcritical Hopf and Bogdanov-Takens bifurcations

Parasitoids are ubiquitous in nature and they play a fundamental role in the equilibrium of ecosystems. Their interactions with other species are particularly crucial for agricultural activities, since they act as a biological control of invasive hosts. In this paper we study Hopf, generalized Hopf and nondegenerate Bogdanov-Takens bifurcations in a simple model which describes the biological control of invasive species through predation by generalist parasitoids. We assume logistic growth for the generalist parasitoid in the absence of host species, and their interaction is described by a Holling type II functional response. We determine a geometric condition which characterizes local bifurcations associated with the appearance of periodic orbits. Further, we find conditions on the parameters which yield to the extinction of the host species, independently of the initial conditions and the charge capacity of the generalist parasitoid. Our results are supported by numerical examples. (C) 2022 Elsevier Ltd. All rights reserved.

Automated summary

Parasitoids play a fundamental role in the equilibrium of ecosystems and are crucial for agricultural activities as they serve as a biological control of invasive hosts. This paper studies Hopf, generalized Hopf, and nondegenerate Bogdanov-Takens bifurcations in a simple model of biological control through predation by generalist parasitoids. The authors determine geometric and parameter conditions associated with the appearance of periodic orbits and the extinction of host species. Numerical examples support the results.