Stability Analysis of a Fishery Model with Nonlinear Variation in Market price
A fishery model with price dependent harvesting is developed, by formulating a system of three differential equations describing the dynamics of the model. In order to reduce the dimensions of the system, it is aggregated to a system of two equations to ease analysis while preserving the necessary dynamics. Aggregation is attained by approximate aggregation where the system of three equations is replaced by a system of two equations through justified simplification in which evolution of the price is taken to evolve relatively faster than the evolution of the fish stock and the fishing effort. Local stability analysis is carried out and from the analysis, three cases of equilibrium solutions are obtained: unstable equilibrium, stable equilibrium and the co-existence of three strictly positive equilibria where two are stable, separated by a saddle. In the case of the co-existence of two stable equilibrium points, market prices are compared and a prediction of two kinds of fishery: an over-exploited fishery where the fishery supports a large economic activity but risks extinction and an under-exploited fishery where the stock is maintained at a large level far from extinction but the fishery only supports a small economic activity.
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