Role of cannibalism in controlling chaos in Lesile-Gower-type tritrophic food chain model

Abstract

In this article, we have proposed and analyzed a cannibalistic Lesile-Gower-type tritrophic food chain model. We assume cannibalism in the intermediate predator population. Local stability of the system around the biologically feasible equilibria are studied. Results of the global boundedness of solutions to the model system under certain parametric restrictions are derived. We have proved that solutions can blow-up in finite time, for certain initial conditions, even under the parametric restrictions. We have shown that blow-up in the top predator population can be avoided under certain conditions on parameters and initial conditions. The existence of Hopf bifurcation is investigated by choosing the appropriate bifurcation parameter. Furthermore, applying the normal form theory and central manifold theorem, the direction of Hopf bifurcation and the stability conditions are derived. An extensive numerical experiments are performed to observe the global dynamics of the system. From our numerical results it is observed that the chaotic dynamics appears through different non-linear dynamics such as limit cycles and period-doubling for variation of $a_0$. It is noticed that both cannibalism and density dependent mortality play important role in controlling chaotic dynamics. Thus cannibalism and density-dependent mortality parameter may be considered as bio-controlling parameters for controlling chaotic dynamics in food chain model.