A delayed predator–prey model with strong Allee effect in prey population growth

• Published in: Nonlinear dynamics Vol. 68; no. 1-2; pp. 23 - 42
• Main Authors: Pal, Pallav Jyoti, Saha, Tapan, Sen, Moitri, Banerjee, Malay
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.04.2012; Springer Nature B.V
• Subjects: Automotive Engineering; Canonical forms; Centre manifold theory; Classical Mechanics; Computer simulation; Control; Delay; Differential equations; Dynamical Systems; Engineering; Hopf bifurcation; Mathematical analysis; Mathematical models; Mechanical Engineering; Nonlinear dynamics; Original Paper; Population growth; Predator-prey simulation; Stability; Vibration; Local Hopf bifurcation; Time delay; Predator–prey model; Allee effect; Global continuation
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-011-0201-5

In this paper, we consider a delayed predator-prey system with intraspecific competition among predator and a strong Allee effect in prey population growth. Using the delay as bifurcation parameter, we investigate the stability of coexisting equilibrium point and show that Hopf-bifurcation can occur when the discrete delay crosses some critical magnitude. The direction of the Hopf-bifurcating periodic solution and its stability are determined by applying the normal form method and the centre manifold theory. In addition, special attention is paid to the global continuation of local Hopf bifurcations. Using the global Hopf-bifurcation result of Wu ({Trans. Am. Math. Soc.} 350:4799–4838, 1998 ) for functional differential equations, we establish the global existence of periodic solutions. Numerical simulations are carried out to validate the analytical findings.