On a cannibalistic predator–prey model with prey defense and diffusion

Abstract

Protective instincts of prey can strongly influence the development of species and also reduces the risk of predation. Cannibalism is frequent in predators and it has strong influence in the evolution of predator species. The consequences of cannibalism is largely neglected when studying the evolution of phenotype prey defenses in predator–prey systems. Motivated by this biological fact, we propose a predator–prey model that involves (a) prey defense; (b) predator cannibalism; and (c) diffusion. We discuss conditions for existence of all biological feasible equilibrium points for non-spatial system. Kolmogorov analysis is performed to examine appearance of limit cycle and chaos in non-spatial system. We obtain local and global stability conditions around all biological feasible equilibrium points. The direction and stability of the Hopf-bifurcation is determined at the vicinity of the concomitant population state of the non-spatial system. In order to study spatio-temporal dynamics, we derive local stability conditions of diffusive system and obtain conditions for the occurrence of Turing instability. Numerical simulations are performed to explore the cases which are beyond the range of analytical methods. Bifurcation diagrams show that non-spatial system has rich dynamics. Various spatial structures such as stripe, spot, rhombus, irregular and target wave patterns of interacting species through Turing and Hopf-Turing instability in two dimensional spatial domain are portrayed and analysed at finite length in order to substantiate the applicability of the present model. Numerical results reveal the impact of drivers of pattern formation in predator–prey systems which provides a better understanding of complex predator–prey interaction.