Qualitative behavior of a fishery system

A global portrait of the phase plane for a fishery model is obtained for any acceptable values of the parameters. Three different structures of the phase plane are recovered. The first predicts an eventual collapse of the fishery. The second predicts an unstable limit cycle and an eventual stability of solutions which start inside the limit cycle. The last structure predicts two possible stable equilibria, one with high catch rate, and the other with no catch. Each structure corresponds to a different domain in the parameter space. The boundaries of these domains are found by solving the relevant differential equation for a saddle-to-saddle separatrix in the phase plane by perturbation methods.