Modeling and analysis in a prey-predator system with commercial harvesting and double time delays

A differential-algebraic prey-predator system with commercial harvesting on predator is proposed, where maturation delay for prey and gestation delay for predator are considered. Since commercial harvesting is dynamically influenced by variation of economic interest, we will investigate combined dynamic effects of double time delays and economic interest on population dynamics. Positivity of solutions and uniform persistence of system are studied. In the absence of time delay, by taking economic interest as bifurcation parameter, existence of singularity induced bifurcation is investigated based on differential-algebraic system theory. State feedback controllers are designed to eliminate singularity induced bifurcation and stabilize the proposed system around corresponding interior equilibrium. In the presence of double time delays, by analyzing associated characteristic transcendental equation, it is found that interior equilibrium loses local stability when double time delays cross corresponding critical values. According to Hopf bifurcation theorem for functional differential equation, existence of Hopf bifurcation is investigated as local stability switches. Based on normal form theory and center manifold theorem, directions of Hopf bifurcation and stability of the bifurcating periodic solutions are studied. Numerical simulations are carried out to show consistency with theoretical analysis.