A predator–prey system with stage structure and time delay for the prey is investigated. By analyzing the corresponding characteristic equations, the local stability of a positive equilibrium and two boundary equilibria of the system is discussed, respectively. By using persistence theory on infinite dimensional systems and comparison argument, respectively, sufficient conditions are obtained for the global stability of the positive equilibrium and one of the boundary equilibria of the proposed system. Further, the existence of a Hopf bifurcation at the positive equilibrium is studied. Numerical simulations are carried out to illustrate the main results.
Predator–prey model Stage structure Time delay Local and global stability Hopf bifurcation
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This work was supported by the Natural Science Foundation of Liaoning Province of China (LN2014160) and the Ph.D. Startup Funds of Liaoning Province of China (20141137).
Chen, L.: Mathematical ecology modeling and research methods. Science Press, Beijing (1988)Google Scholar
Xu, R.: Global stability and Hopf bifurcation of a predator–prey model with stage structure and delayed predator response. Nonlinear Dyn. 67, 1683–1693 (2012)zbMATHCrossRefGoogle Scholar
Xu, R., Ma, Z.: Stability and Hopf bifurcation in a predator–prey model with stage structure for the predator. Nonlinear Anal. Real World Appl. 9, 1444–1460 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
Sun, X., Huo, H., Xiang, H.: Bifurcation and stability analysis in predator–prey model with a stage structure for predator. Nonlinear Dyn. 58, 497–513 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
Wang, F., Kuang, Y., Ding, C., et al.: Stability and bifurcation of a stage-structured predator–prey model with both discrete and distributed delays. Chaos Solitons Fract. 46, 19–27 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1976)Google Scholar
Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, New York (1993)zbMATHGoogle Scholar
Beretta, E., Kuang, Y.: Geometric stability switch criteria in delay differential systems with delay dependent parameters. SIAM J. Math. Anal. 33, 1144–1165 (2002)zbMATHMathSciNetCrossRefGoogle Scholar