Period doubling cascades of prey–predator model with nonlinear harvesting and control of over exploitation through taxation

Abstract

The present study investigates a prey predator type model for conservation of ecological resources through taxation with nonlinear harvesting. The model uses the harvesting function as proposed by Agnew (1979) [1] which accounts for the handling time of the catch and also the competition between standard vessels being utilized for harvesting of resources. In this paper we consider a three dimensional dynamic effort prey–predator model with Holling type-II functional response. The conditions for uniform persistence of the model have been derived. The existence and stability of bifurcating periodic solution through Hopf bifurcation have been examined for a particular set of parameter value. Using numerical examples it is shown that the system admits periodic, quasi-periodic and chaotic solutions. It is observed that the system exhibits periodic doubling route to chaos with respect to tax. Many forms of complexities such as chaotic bands (including periodic windows, period-doubling bifurcations, period-halving bifurcations and attractor crisis) and chaotic attractors have been observed. Sensitivity analysis is carried out and it is observed that the solutions are highly dependent to the initial conditions. Pontryagin’s Maximum Principle has been used to obtain optimal tax policy to maximize the monetary social benefit as well as conservation of the ecosystem.