Capacity Constraints and Semisingular Optimal Controls in a Predator-Prey Fishery

We study the optimal management of a two-species predator-prey fishery under
explicit capacity constraints on harvesting effort. Unlike classical models that assume
unconstrained controls, we treat prey and predator capacity limits as bifurcation parameters and show how they reshape both steady states and transition dynamics. Our analysis provides three main contributions. First, we deliver a complete classification of steady state regimes – singular, semisingular, and bang-bang – together with local stability via phase-plane methods. Second, we derive a methodological result: segments of the singular loci, where costates equal marginal revenues, constitute optimal approach paths, and we obtain closed-form feedback expressions for the associated semisingular controls. Third, we uncover sharp policy shifts and path dependence induced by capacity constraints: when prey capacity binds, predator control becomes a compensating strategy; when predator capacity binds, prey harvesting emerges as leverage, and prey closures can arise endogenously. These findings provide valuable extensions to the bioeconomic Gordon–Schaefer model with Lotka–Volterra dynamics.