Stability Bifurcation Analysis of a Fishery Model with Nonlinear Variation in Market Price

Abstract

We develop a fishery model with price dependent harvesting by formulating a system of three differential equations describing its dynamics. Assuming that the price of the harvested fish on the market evolves relatively faster than the evolution of the fish stock and the fishing effort, we apply approximate aggregation to reduce the system of equations from three to two. From the stability analysis of the aggregated model, we show the co-existence of three strictly positive equilibria where two are stable and are separated by a saddle. The two stable equilibtia represent two kinds of fishery namey; an over-exploited fishery where the fishery supports a large economic activity but risks extinction and an 338 Makwata Harun Omukuba, Adu A.M. Wasike and Lucy Chikamai under-exploited fishery where the stock is maintained at a large level far from extinction but the fishery only supports a small economic activity. Mathematics Subject Classification: 93A30, 92B05, 34C23 Keywords: Aggregation, Equilibrium points, Local stability