Hopf bifurcation in a grazing system with two delays

Life in the Sahelian zone is very difficult due to lack of resources on a permanent basis. This scarcity of resources leads the pastor, with their flock to organize themselves to better resist the difficult conditions. Pastoralism is of paramount importance for this region. There is an urgent need to understand this way of life in order to help pastors to better organize themselves. In this paper, we propose mathematical models that consist of three food trophic chains. We consider and formalize, in a given location, interactions between browsers, grazers and forage resources. We first analyze the resulting system without delays and in a second time take into account two positive and discrete time-delays. In our model, delays denote an average time required in order that food consumed by browsers and grazers are beneficial for them. Qualitative analysis of the delays-free model shows several equilibria as well as various situations of multistability. Considering delay as parameter, we investigate the effect of delay on the stability of equilibria. It is found that the delays can lead the system dynamic behavior to exhibit stability switches, and Hopf bifurcation occurs when the delay crosses some critical value. By applying the normal form theory and the center manifold theorem, the explicit formula which determines the stability and direction of the bifurcating periodic solutions is determined. Finally based on a nonstandard numerical scheme, we provide some numerical simulations in order to illustrate our qualitative results and support discussions.