Impact of alarm signals and mutualistic interactions in a food chain model of oxpeckers, zebras, and lions

Abstract

This paper has developed a three-species food chain model among the interactions of oxpeckers, zebras, and lions. This model considers a mutualistic interaction between zebras and oxpeckers. In the absence of lions and other species, it is thought that oxpeckers and zebras grow logistically. It is assumed that the lions eat zebras in a Holling type II functional form. It is also assumed that oxpeckers can alert zebras through some signal if there is a possibility of attack by lions. The equilibrium points of the proposed model are determined and analyzed. The local stability of the model around these points has been investigated. Also, the existence of some local bifurcations, such as saddle–node bifurcation, transcritical bifurcation, and Hopf bifurcation, in the model has been studied. The increase in the alert signal $\left(q\right)$ between oxpeckers and zebras is found to stabilize the model. Also, it is found that the hunting cooperation $\left(p\right)$ among the lions may be beneficial for the stability of the system. It has been observed that the mutualistic relationship between oxpeckers and zebras may increase the stability of the model. Finally, some numerical simulation results have been presented.