DYNAMICS OF A FISHERY MODEL WITH CONTINUOUS THRESHOLD HARVESTING POLICY AND ITS LEVERAGE FOR CONSERVATION AND MANAGEMENT

Abstract

There is a global decline in marine fish abundance due to unsustainable harvesting. An effective harvesting policy can protect the overfished population from possible extinction. In this study, we used a mathematical model characterized by density-dependent refuge protection for herbivorous fish, exhibiting an anti-predator response in presence of a generalist invasive fish. The anti-predator behavior entails predator density-dependent reduced fecundity of the herbivorous fish. The model assumes a continuous threshold harvesting policy (CTHP) for the herbivorous fish and uses the catch-per-unit-effort (CPUE) hypothesis for harvesting the invasive fish. The CTHP allows harvesting of the herbivorous fish only when the density of the herbivorous fish exceeds a specified threshold value, thus ensuring the long-term sustainability of the herbivorous fish stock. The existence and stability of steady-state solutions and the bifurcations of the model are investigated. Our study reveals that the level of apprehension of the herbivorous fish and fishing efforts will play a significant role in the stability of the system. We examine the existence of the bionomic equilibrium and then study the dynamic optimization of the harvesting policy by employing Pontryagin’s maximum principle. We discuss different subsidies and tax policies for the effective management of a sustainable fishery. We use numerical simulations to compare the revenues corresponding to the harvest policies based on maximum sustainable yield (MSY), maximum economic yield (MEY), and optimal sustainable yield (OSY) for inferring an ecologically sustainable and economically viable harvesting policy.