Impulsive state feedback control of a predator–prey system with group defense

Abstract

In this paper, the complex dynamics of a predator–prey model with group defense and impulsive state feedback control is studied both theoretically and numerically. We obtain the sufficient conditions for the existence and stability of semi-trivial solution and positive periodic solutions by using the Poincare map and the analogue of the Poincare criterion. Moreover, the theoretical analysis reveals that the positive periodic solution bifurcates from the semi-trivial solution through a fold bifurcation. Numerical simulations are also illustrated, which agree well with our theoretical analysis, but also to exhibit the complex dynamical behaviors, such as the period-2, 3, 4, 6, 8 solutions and chaotic solution, cascade of period-doubling bifurcation and inverse period-doubling bifurcation. Moreover, the superiority of impulsive state feedback control strategy is also exhibited over the impulsive fixed-time control.