Mechanisms of stable species coexistence in food chain systems: Strength of odor disturbance and group defense

Odor disturbance and group defense generally play an important role in ecosystem stability. By incorporating odor disturbances and group defenses into the traditional three-species food chain system, where the preys are disturbed by the odor of predators, and group together to defend. We proved the boundedness of solutions of the system, and discussed the existence of equilibrium points. By Lyapunov’s first method and Routh-Hurwitz criterion, the sufficient conditions for stability of equilibrium points are obtained. It is found that, with the increasing of group defense, the density of the preys increases, while the odor disturbance of predators decreases the density of preys. Interestingly, the group defense induces a transition from a boundary equilibrium point to a positive equilibrium one, or from no Hopf-bifurcation to a Hopf-bifurcation. Furthermore, we demonstrated that the system undergoes a saddle-node bifurcation at $m_c=0.3362$, below which the system has a stable node and a saddle point, suggesting that the less odor disturbance avail system stability; while above which the system has no equilibrium point, indicating that the greater odor disturbance causes the collapse of the system. To reveal the underlying mechanisms of the interplay of odor disturbance and group defense, the bifurcation diagrams of the preys with two factors are given respectively.