Stability and bifurcation analysis on a three-species food chain system with two delays

The present paper deals with a three-species Lotka–Volterra food chain system with two discrete delays. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the positive equilibrium and existence of local Hopf bifurcations are investigated. Furthermore, by using the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction of bifurcations and the stability of bifurcating periodic solutions. Finally, to verify our theoretical predictions, some numerical simulations are also included at the end of this paper.

Research highlights
► Three-species Lotka-Volterra food chain model with two discrete delays is considered.
► Stability and Hopf bifurcation are investigated by applying the linearization method, normal form theory and center manifold reduction for retarded functional differential equations.
► Simulation figures support the analytical findings.