Stability and Hopf bifurcation of a delayed ratio-dependent predator-prey system

• Published in: Acta mechanica Sinica Vol. 27; no. 2; pp. 285 - 296
• Main Authors: Wang, Wan-Yong, Pei, Li-Jun
• Format: Journal Article
• Language: English
• Published: Heidelberg The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences 01.04.2011
• Subjects: Classical and Continuum Physics; Computational Intelligence; Criteria; Delay; Engineering; Engineering Fluid Dynamics; Gestation; Hopf bifurcation; Hopf分岔; Mathematical models; Predators; Research Paper; Stability; Theoretical and Applied Mechanics; Time delay; 动力学关系; 局部稳定性; 捕食模型; 捕食系统; 时间延迟; 正平衡点; 比率依赖; Time delays; Stability; Center manifold; Hopf bifurcation; Normal form
• ISSN: 0567-7718; 1614-3116
• DOI: 10.1007/s10409-011-0424-1

Since the ratio-dependent theory reflects the fact that predators must share and compete for food, it is suitable for describing the relationship between predators and their preys and has recently become a very important theory put forward by biologists. In order to investigate the dynamical relationship between predators and their preys, a so-called Michaelis-Menten ratio-dependent predator-prey model is studied in this paper with gestation time delays of predators and preys taken into consideration. The stability of the positive equilibrium is investigated by the Nyquist criteria, and the existence of the local Hopf bifurcation is analyzed by employing the theory of Hopf bifurcation. By means of the center manifold and the normal form theories, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. The above theoretical results are validated by numerical simulations with the help of dynamical software WinPP. The results show that if both the gestation delays are small enough, their sizes will keep stable in the long run, but if the gestation delays of predators are big enough, their sizes will periodically fluc-tuate in the long term. In order to reveal the effects of time delays on the ratio-dependent predator-prey model, a ratiodependent predator-prey model without time delays is considered. By Hurwitz criteria, the local stability of positive equilibrium of this model is investigated. The conditions under which the positive equilibrium is locally asymptotically stable are obtained. By comparing the results with those of the model with time delays, it shows that the dynamical behaviors of ratio-dependent predator-prey model with time delays are more complicated. Under the same conditions, namely, with the same parameters, the stability of positive equilibrium of ratio-dependent predator-prey model would change due to the introduction of gestation time delays for predators and preys. Moreover, with the variation of time delays, the positive equilibrium of the ratio-dependent predator-prey model subjects to Hopf bifurcation.