Dynamical analysis and control in a differential-algebraic bio-economic model with stage structure

Nowadays, biological resource in prey-predator ecosystem is commercially harvested and sold with aim of achieving economic interest. Furthermore, harvest effort is usually influenced by variation of economic interest of harvesting. In this paper, a delayed differential-algebraic bio-economic model is proposed, which is utilized to investigate interaction and coexistence mechanism of biological population in the harvested ecosystem due to the variation of economic interest of harvesting. Local stability analysis of the proposed model reveals that there is a phenomenon of singularity induced bifurcation due to the variation of economic interest of harvesting, and state feedback controllers are designed to stabilize the proposed model at the interior equilibrium. Furthermore, local stability of the proposed model with gestation delay and diffusion is studied, which reveals that the interior equilibrium loses its stability at some critical values of gestation delay and corresponding cycle occurs. Finally, numerical simulations are carried out to show consistency with theoretical analysis obtained in this paper.