Catch-to-stock dependence: the case of small pelagic fish with
bounded fishing effort
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Small pelagic fish (such as herring, anchovies, capelin, smelts, sardines or pilchards) is characterized by high reproduction rate and rather short life-cycle. Additionally, pelagic fish stock have strong recurrent cycles of fish abundance and scarcity and may provide high catch yields per unit of fishing effort even within the scarcity periods. The latter may provoke a collapse of fish stock since our abilities to predict their periods of abundance and/or scarceness are very limited. Empirical evidence and biological characteristics of pelagic fish suggest that, in contradiction with traditional fishery models, marginal catch of pelagic species does not react in linear way to changes in stock level. In this presentation, we allow non-linearity in catch-to-stock parameter and propose another variant of single-stock harvesting economic model focusing on the dependence of stationary solutions upon such non-linear parameter. Our principal interest consists in finding an optimal fishing effort leading to stationary solutions that prevent fishing collapse and help to avoid the species extinction. To do so, we first formulate a social planner’s problem in terms of optimal control for infinite horizon, then analyze its formal solution by applying the Pontryagin’s maximum principle and finally revise a possibly of a singular arc appearance. In conclusion, we also examine some core properties of station- ary equilibrium reachable by means of a singular optimal control and prove the existence and uniqueness of steady states under some additional assumptions. |
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