Stability and Bifurcation Analysis of a Three-Species Food Chain Model with Delay

In the present paper, we study the effect of gestation delay on a tri-trophic food chain model with Holling type-II functional response. The essential mathematical features of the proposed model are analyzed with the help of equilibrium analysis, stability analysis, and bifurcation theory. Considering time-delay as the bifurcation parameter, the Hopf-bifurcation analysis is carried out around the coexisting equilibrium. The direction of Hopf-bifurcation and the stability of the bifurcating periodic solutions are determined by applying the normal form theory and center manifold theorem. We observe that if the magnitude of the delay is increased, the system loses stability and shows limit cycle oscillations through Hopf-bifurcation. The system also shows the chaotic dynamics via period-doubling bifurcation for further enhancement of time-delay. Our analytical findings are illustrated through numerical simulations.