Dynamics of a fishery system in a patchy environment with nonlinear harvesting

Abstract

In this paper, a stock‐effort dynamical model with two fishing zones is discussed. The nonlinear harvesting function is assumed depending upon stock size as well as fishing effort. The migration of fish is considered between two zones. The harvesting vessels also move between zones to increase their revenue. The movements of fish and fishing vessels between zones are assumed to take place at a faster time scale as compared with processes involving growth and harvesting occurring at a slow time scale. The aggregated model is obtained for total fish stock and fishing effort. This aggregated (reduced) model is analyzed analytically as well as numerically. Biological and bionomic equilibria of the system are obtained, and criteria for local stability or instability of the system are derived. The impact of levels of taxation T on the fish population and on the revenue earned by the fishery is investigated. An optimal harvesting policy is also discussed using the Pontryagin's maximum principle. The aggregated model also exhibits Hopf and transcritical bifurcation with respect to the bifurcation parameter tax T. Numerical simulations are presented to illustrate the results.