Qualitative dynamics and optimal control of a delayed algae–fish bioeconomic system with nitrate recycling

Abstract

A bioeconomic delay model is developed for an algae–fish ecosystem in which the time delay τ represents the lag required for nutrients released by fish decomposition to become bioavailable. The model incorporates logistic growth for both populations together with a Holling type-II functional response to describe trophic interactions. The well-posedness of the system is established by proving the existence, uniqueness, positivity, and boundedness of solutions. The local stability of biologically relevant equilibria is then analyzed, including the boundary equilibrium associated with algal extinction and the interior equilibrium corresponding to species coexistence. The influence of the delay parameter on system stability is investigated, and explicit conditions for the occurrence of Hopf bifurcations are derived. Numerical simulations illustrate the destabilizing impact of increasing delay and support the analytical results. Finally, an optimal control framework is introduced to address long-term resource management, and Pontryagin's maximum principle is applied to characterize harvesting strategies that balance economic performance with ecological sustainability.