Applied Mathematics & Information Sciences

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This study proposes optimal control problems with two different biological dynamics: a compensation model and a critical depensation model. The static equilibrium reference points of the models are defined and discussed. Also, bifurcation analyses on the models show the existence of transcritical and saddle-node bifurcations for the compensation and critical depensation models respectively. Pontyagin’s maximum principle is employed to determine the necessary conditions of the model. In addition, sufficiency conditions that guarantee the existence and uniqueness of the optimality system are defined. The characterization of the optimal control gives rise to both the boundary and interior solutions, with the former indicating that the resource should be harvested if and only if the value of the net revenue per unit harvest (due to the application of up to the maximum fishing effort) is at least the value of the shadow price of fish stock. Numerical simulations with empirical data on the sardinella are carried out to validate the theoretical results.

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