Slow–fast analysis of a modified Leslie–Gower model with Holling type I functional response

• Published in: Nonlinear dynamics Vol. 108; no. 4; pp. 4531 - 4555
• Main Authors: Saha, Tapan, Pal, Pallav Jyoti, Banerjee, Malay
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.06.2022; Springer Nature B.V
• Subjects: Automotive Engineering; Bistability; Classical Mechanics; Control; Dynamical Systems; Eigenvalues; Engineering; Equilibrium; Hopf bifurcation; Investigations; Mathematical models; Mathematics; Mechanical Engineering; Orbits; Ordinary differential equations; Original Paper; Perturbation theory; Predator-prey simulation; Predators; Relaxation oscillations; Singular perturbation; Stability analysis; Vibration; Relaxation oscillations; Singular Hopf bifurcation; Non-smooth system; Slow–fast system; Geometric singular perturbation theory; Predator–prey model; Bistability
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-022-07370-1

In this paper, we consider a modified Leslie-type prey–generalist predator system with piecewise–smooth Holling type I functional response. Considering the reproduction rate of the prey population higher than that of the predator, the model becomes a slow–fast system that mathematically leads to a singular perturbation problem. To analyse the stability of the boundary equilibrium on the switching boundary, we use a generalized Jacobian that enables us to investigate how the eigenvalues jump at the boundary point. We investigate the slow–fast system by employing geometric singular perturbation theory and blow-up technique that reveal a wide range of interesting complicated dynamical phenomena. We have studied existence of saddle-node bifurcation, Bogdanov–Takens bifurcation, bistability, singular Hopf bifurcation, canard orbits, multiple relaxation oscillations, saddle-node bifurcation of limit cycle and boundary equilibrium bifurcations. Numerical simulations are carried out to substantiate the analytical results.