Mathematical Model Of Complex Behaviour In Three Species Food ?�? Web

Abstract

A biologically feasible continuous time food web model consisting of a bottom prey, a middle predator and top predator which is an extension of Hasting and powell three species food chain model that incorporates the nonlinear functional response was studied. The predation follows modified Holling type II functional responses as the up take for both predators. It is shown that under certain restrictions on the parameter space the model has bounded solutions for all positive initial conditions .Which eventually enters an invariant attracting set as an initial step. Towards the analysis of the model the governing equation have been splined in to kolmogorov type systems. Based on the Jacobean matrix associated to the sub system, the dynamical behavior about the semi-trivial equilibra is analyzed and analytic results show that any one of the boundary prey predator planes has a stable equilibrium point. More over the dynamical of the sub systems in the boundary planes changes as some parameter values vary. Criteria’s for the existence, local stability of the nonnegative equilibra are analyzed algebraically. Based on numerical simulations the system is numerically solved for suitable combinations of parameters in such a manner that Kolomorov conditions are satisfied. It is also observed that the RouthHurwitz criteria the necessary and sufficient conditions for stability are not satisfied. More over the coexistence of the three biological species is observed from the oscillation nature of the three solution curves. Finally it is observed that there is a change on the qualitative behavior of the system when trophic function is of prey dependent than ratio-dependent.