Stability and Bifurcation in a Food Chain Model with the Allee Effect

This chapter describes the effects of specific intra-predator competition in a three-nutrient food chain model of a prey-dependent interacting population under Allee effect. We present a detailed mathematical analysis of the proposed food chain model to illustrate some ecological phenomenon and process under realistic environmental impact. In the context of your food chain model, the Allee effect may influence the behavior of prey, predator and super-predator populations, as their interactions are likely to be affected by changes in population density. We observe the biologically feasible equilibrium points of system and describe the behavior of system around each equilibrium point. Stability (local and global) and bifurcation (transcritical, saddle-node and Hopf-Andronov) analyses of the model are obtained. Global stability analysis is obtained through constructing particular Lyapunov functions. Numerical simulations show the relationship between prey, predator and super predator at different equilibrium points. In addition, numerical simulations with experimental parameter values confirm the analytical results and show the existence of bifurcation. Finally, a discussion of the analytical and numerical results is presented as the conclusion of this chapter.