Bifurcation analysis and sliding mode control of a singular piecewise-smooth prey–predator model with distributed delay

Abstract

In this paper, the piecewise-smooth functional response function and distributed delay are used to describe the memory effect of predators and capture law when the abundance of prey changes greatly in ecosystems more realistically. A singular piecewise-smooth prey–predator model with distributed delay is studied. Considering the growth and loss rate of the predator much smaller than that of the prey, the model is described by a fast–slow system that mathematically leads to a singular perturbation problem. The dynamic behavior of the fast–slow system with distributed delay, piecewise smooth is novel and interesting. The system undergoes a Hopf bifurcation where the interior equilibrium becomes unstable leading to a stable limit cycle. As the perturbation parameter decreases, the co-existence equilibrium has a transition from the unstable node to the stable node which leads multiple relaxation oscillations occurring. This study reveals the occurrence of boundary equilibrium bifurcations, enriching the understanding of predator–prey dynamics. In addition, a sliding mode controller is designed in the fast–slow predator–prey system to make the periodic trajectory tend to the internal equilibrium point. Taking the predator–prey relationship between insect and bird as an example, numerical simulations are provided to verify the theoretical results.