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

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 and spatial heterogeneous environment. 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 as well as the change of population spatial diffusion and gestation delay. Local stability analysis of the proposed model without gestation delay and diffusion 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. It is also shown that population spatial diffusion and harvesting have a stabilizing effect on population dynamics. Finally, numerical simulations are carried out to show consistency with theoretical analysis obtained in this paper.