Hopf Bifurcation in a Delayed Herd Harvesting Model and Herbivory Optimization Hypothesis

Pastoral livestock farming has always been a major component of livestock production worldwide, especially in the Sahelian regions of Africa. In the Sahel, pastoral livestock systems are those in which 90% of the dry matter consumed by livestock comes from grazing (Carriere, Impact des systèmes d’élevage pastoraux sur l’environnement en Afrique et en Asie tropicale et sub-tropicale aride et sub-aride. Livestock and the Environment Finding a Balance, Scientific Environmental Monitoring Group, 1996). An important issue in this region of Africa, with very low rainfall, is herd management by pastoralists through the accessible resource. Our goal is to build and analyze a mathematical model that translates the resource–livestock herd interactions in a Sahelian region by taking into account the herd harvesting for various needs (sales, nutrition, etc.). We also take into account a delay reflecting the time required for the transformation of the resource consumed into animal biomass. As a result, we consider in our modeling approach the herbivory optimization hypothesis (Lebon et al., Ecol Model 290, 192–203, 2014; Williamson et al., J Range Manag Archives 42(2), 1989) that herbivores to a certain extent stimulate plant biomass production. We performed a stability analysis of the different equilibria of our model with and without delay. We have found that when we consider delay as a bifurcation parameter, the model undergoes a stability change in the neighborhood of the coexistence equilibrium. As a consequence of this change, a Hopf bifurcation occurs when the delay passes through a critical value reflecting periodic fluctuations between the biomass of the animals and that of the resource. Finally, numerical simulations are presented to illustrate our theoretical results and to support the discussion.