Complex Dynamics and Periodic Feedback Control in a Discrete-Time Predator-Prey System

The dynamics of a discrete-time predator-prey system is investigated in the closed first quadrant R
₊

²
. First, the existence and local stability of the fixed points in the closed first quadrant R
₊

²
are discussed in detail. Then, by using center manifold theorem and bifurcation theory, it is proved rigorously that when the bifurcation parameters vary in the small neighborhood of the corresponding bifurcation set, flip bifurcation and Hopf bifurcation can emerge near the unique positive fixed point of the system. To suppress the undesirable chaos induced by the period-doubling bifurcation and stabilize the unstable period-1 point embedded in the chaotic attractor, periodic feedback control scheme is proposed. For this system, it is easy to verify that 1 is not the eigenvalue of the Jacobian matrix near the period-1 point, hence, the gain matrix can be obtained analytically. Numerical simulations are presented to illustrate our results with the theoretical analysis and show the effect of the control method.