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Global Dynamics of a Tri-trophic Food Chain Model

U. K. Pahari, Chaity Ganguli, T.K. Kar, Uttam Das


In this paper we study the dynamical behavior of a tritrophic food chain model in presence of harvesting. In this model there is one predator feeding on the prey and a second predator called super predator feeding on the first predator. This means that the first predator plays a hybrid role: it acts as both predator and prey. A time delay is introduced to the functional response term involved with the growth equation of the first predator. The effect of
super predator and delay on the stability of the system is investigated and the time delay is regarded as bifurcation parameter. Local stability of the equilibrium points are investigated.
Global stability and bifurcation analysis under different conditions are also investigated. Moreover, we show that Hopf bifurcation occurs when time delay crosses some critical values. By using the normal form method and center manifold theorem, we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Numerical simulations are carried out to support the analytical findings.


Time delay, harvesting, stability, Hopf bifurcation, periodic solution.

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