Chaotic response of a Monod-Haldane food-web model system

The present theoretical study is concerned with the dynamics of a tri-trophic predator-prey interactions with special inclusion of refuge and intra-specific competition into the system. In view of the potentiality of group defense for both the prey and the predator species, the relevant Monod-Haldane functional response is introduced here. As a preliminary requirement, the present system is checked for its uniform boundedness, dissipativeness and permanence properties as well. Diverse criteria are generated for the existence of feasible equilibria of the system and accordingly parametric constraints for stability and instability of the system around the positive equilibrium are realized. The analysis of Kolmogorov and Hopf-bifurcation for the proposed model system is not ruled out however from the present concern. The numerical simulations exhibit the chaotic and periodic oscillatory behaviour of the system under certain range of parameter values. One may note how the bifurcation scenario about the coexistence equilibrium point recognizes the complexity of the dynamical response of the three-species food web model. Special emphasis is paid to the chaotic dynamics produced by the half-saturation constant. The present numerical simulations reveal that the three-species tri-trophic food chain model experiences chaotic response from a stable focus in the event of diminishing value of the half-saturation constant. The exploration of various categories of bifurcations corresponding to both the codimension 1 and codimension 2 at this juncture is however, not ruled out from the present study. The individual influence of refuge and the intra-specific competition on the proposed dynamics of the system is also duly recorded.