A delayed ratio-dependent predator–prey model of interacting populations with Holling type III functional response

• Published in: Nonlinear dynamics Vol. 76; no. 1; pp. 201 - 220
• Main Authors: Pal, Pallav Jyoti, Mandal, Prashanta Kumar, Lahiri, Kaushik Kumar
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.04.2014; Springer Nature B.V
• Subjects: Asymptotic properties; Automotive Engineering; Classical Mechanics; Computer simulation; Control; Dynamical Systems; Engineering; Hopf bifurcation; Logistics; Mathematical analysis; Mathematical models; Mechanical Engineering; Original Paper; Predator-prey simulation; Predators; Qualitative analysis; Stability analysis; Time delay; Time lag; Transformations; Vibration; Ratio-dependence; Time delay; Permanence; Hopf bifurcation; Global stability
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-013-1121-3

In this article, we study a ratio-dependent predator–prey model described by a Holling type III functional response with time delay incorporated into the resource limitation of the prey logistic equation. This investigation includes the influence of intra-species competition among the predator species. All the equilibria are characterized. Qualitative behavior of the complicated singular point (0,0) in the interior of the first quadrant is investigated by means of a blow-up transformation. Uniform persistence, stability, and Hopf bifurcation at the positive equilibrium point of the system are examined. Global asymptotic stability analyses of the positive equilibrium point by the Bendixon–Dulac criterion for non-delayed model and by constructing a suitable Lyapunov functional for the delayed model are carried out separately. We perform a numerical simulation to validate the applicability of the proposed mathematical model and our analytical findings.