Bifurcations and stability of a discrete singular bioeconomic system

The paper analyzes the stability and bifurcations of a discrete singular bioeconomic system in the closed first quadrant $R_{+}^{3}$ . First, applying the Poincaré scheme to a differential-algebraic predator–prey system where the economic interest of harvesting is taken into account, a discrete singular bioeconomic system is proposed. Then, local stability and the existing conditions of the flip bifurcation and Neimark–Sacker bifurcation around the interior equilibria of the proposed model are discussed by using the normal form of the discrete singular bioeconomic system, the center manifold theorem and the bifurcation theory, when choosing the step size δ as the parameter of the bifurcation. Finally, the results are illustrated and the complex dynamical behaviors are exhibited by computer numerical simulations.