Relaxation Oscillations and Dynamical Properties in a Time Delay Slow–Fast Predator–Prey Model with a Piecewise Smooth Functional Response

• Published in: Mathematics (Basel) Vol. 10; no. 9; p. 1498
• Main Authors: Qian, Youhua, Peng, Yuhui, Wang, Yufeng, Lin, Bingwen
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.05.2022
• Subjects: Behavior; Dynamic stability; entry–exit function; Equilibrium; Food science; geometric singular perturbation theory; Mathematical models; Perturbation theory; Predation; Predator-prey simulation; Predators; relaxation oscillation cycle; Relaxation oscillations; Singular perturbation; slow–fast predator–prey model; System effectiveness; time delay; Time delay systems
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math10091498

In the past few decades, the predator–prey model has played an important role in the dynamic behavior of populations. Many scholars have studied the stability of the predator–prey system. Due to the complex influence of time delay on the dynamic behavior of systems, time-delay systems have garnered wide interest. In this paper, a classical piecewise smooth slow–fast predator–prey model is considered. The dynamic properties of the system are analyzed by linearization. The existence and uniqueness of the relaxation oscillation are then proven through the geometric singular perturbation theory and entry–exit function. Finally, a stable limit cycle is obtained. A numerical simulation verifies our results for the systems and shows the effectiveness of the method in dealing with time delays.