Numerical simulation and optimal fishing effort for a fishery with total allowable catch (TAC)

Fish stocks in the developing world are often depleted as a result of the application of excessive fishing effort on the part of the fishermen. A bioeconomic model with logistic growth, proportional harvesting and quadratic costs is proposed to study the effect of overcapacity on a marine fishery. Also incorporated into the model is an isoperimetric constraint to account for the annual total allowable catch (TAC). Pontryagin s maximum principle is employed to determine the necessary conditions for optimality of the model. Additionally, the sufficiency conditions that guarantee the existence and uniqueness of the optimality system are discussed. Furthermore, the relationship between the shadow price of fish stock, the shadow price of the total allowable catch and the marginal net revenue as it relates to the optimal fishing effort is explored. Numerical simulation with empirical data on the Ghana sardinella fishery is performed to validate the theoretical results. The findings of the study indicate that for a TAC equal to the maximum sustainable yield (MSY), the average fishing effort should not exceed 95\% of the MSY effort, provided that the initial stock size is exactly 55% of the carrying capacity.