A prey-predator model with harvesting for fishery resource with reserve area

Abstract

Considering that over exploitation would result in the extinction of the population, we propose and investigate a Holling II functional response prey-predator model with harvesting for fishery resource in a two-patch environment: a free fishing zone (patch 1) and a reserve zone (patch 2) where fishing is strictly prohibited. First, the presence of harvesting can impact the existence of equilibria. Further, stability criteria of the model is analyzed both from local and global point of view. Our results indicate that so long as the prey population in the reserved zone does not extinct, the both prey always exist, that is marine reserves should ensure the sustainability of system. Thus, marine reserves not only protect species inside the reserve area but they can also increase fish abundance in adjacent areas. Next, the existence of bionomic equilibrium and the optimal harvesting policy are discussed. The present value of revenues is maximized by using Pontryagin’s maximum principle. It is established that an infinite discount rate leads to complete dissipation of economic rent. Finally, some numerical simulations are given to illustrate our results.