A model of a fishery with fish storage and variable price involving delay equations

Abstract

We propose a bio-economic model of a fishery describing the variations of the fish stock, the fishing effort and the price of the resource on the market supposed to depend on supply and demand. The originality of this model comes from taking into account the storage of part of the resource for a certain time before being put up for sale on the market. Taking into account the supposedly fast price dynamics compared to the other mechanisms involved and after integration of the stock equation, the system is reduced to a system of two delayed differential equations. The qualitative analysis of the model is carried out with the search for equilibrium points and the study of their stability. The study shows the existence of a catastrophic equilibrium corresponding to the extinction of the resource and one or two sustainable fishery equilibrium points that can coexist under certain conditions. The model shows that storing part of the resource makes it possible to avoid a catastrophic situation with the extinction of the fish stock and to stabilize the fishery in the long term. The study also shows that the price variation of the resource has a stabilizing effect by avoiding the appearance of periodic solutions associated with a stable limit cycle surrounding a sustainable fishery equilibrium point resulting from a Hopf bifurcation, which is contrary to the case without price variation where this is possible.

Keywords: Aggregation of variables; Fishery management; Nonlinear demand; Stock-effort model; Varying price.