Controlling chaos and pattern formation study in a tritrophic food chain model with cannibalistic intermediate predator

The present study constitutes the first systematic investigation on the effect of cannibalism on a mid-level predator in a tritrophic food chain model. Moreover, it discusses the role of cannibalism in chaos control and pattern formation. The proposed food chain consists prey, cannibalistic mid predator and generalist top predator, where the mid-level predator consumes its own species to survive. This act of consuming the conspecifics is called cannibalism, which is an important regulating factor that maintains population equilibrium. In this work, we have done a detailed dynamical study to understand the effect of cannibalism. In the case of temporal model, local stability and Hopf bifurcation analysis are carried out. The model system undergoes a Hopf bifurcation with respect to the cannibalism parameter. Further, we have analyzed the spatially extended version of the temporal model. Turing instability and Hopf bifurcation are proved to ensure the existence of Turing patterns. We have presented the role of cannibalism via several numerical experiments. Chaotic dynamics is observed in the temporal model by increasing the growth rate of the prey. Diffusion-induced chaos is also observed in the spatially extended model system. This study infers that chaotic dynamics are successfully controlled through cannibalism and competition parameters in temporal and extended model systems. It is noticed that the increased cost of cannibalism promotes the occurrence of Turing patterns. The diffusivity of the top predator being a sensitive parameter induces various interesting Turing patterns such as mixed spot-stripe, labyrinth, stripes and mesh, as seen from this work. Overall, cannibalism can be seen as the most sensitive factor for change in the dynamics of the food chain model for a balanced ecosystem.