Abstract
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.
Similar content being viewed by others
References
Chen, L.: Mathematical ecology modeling and research methods. Science Press, Beijing (1988)
Murray, J.: Mathematical Biology. Springer, Berlin (1989)
Takeuchi, Y.: Global dynamical properties of Lotka–Volterra systems. World Scientific, Singapore (1996)
Aiello, W., Freedman, H.: A time delay model of single species growth with stage structure. Math. Biosci. 101, 139–156 (1990)
Song, X., Chen, L.: Optimal harvesting and stability for a two-species competitive system with stage structure. Math. Biosci. 170, 173–186 (2001)
Wang, W., Chen, L.: A predator–prey system with stage structure for predator. Comput. Math. Appl. 33, 83–91 (1997)
Georgescu, P., Hsieh, Y.: Global dynamics of a predator–prey model with stage structure for predator. SIAM J. Appl. Math. 67, 1379–1395 (2006)
Cui, J., Chen, L., Wang, W.: The effect of dispersal on population growth with stage-structure. Comput. Math. Appl. 39, 91–102 (2000)
Xu, R., Ma, Z.: The effect of stage-structure on the permanence of a predator–prey system with time delay. Appl. Math. Comput. 189, 1164–1177 (2007)
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)
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)
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)
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)
Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1976)
Hale, J., Waltman, P.: Persistence in infinite-dimensional systems. SIAM J. Math. Anal. 20, 388–395 (1989)
Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, New York (1993)
Beretta, E., Kuang, Y.: Geometric stability switch criteria in delay differential systems with delay dependent parameters. SIAM J. Math. Anal. 33, 1144–1165 (2002)
Acknowledgments
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).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Song, Y., Xiao, W. & Qi, X. Stability and Hopf bifurcation of a predator–prey model with stage structure and time delay for the prey. Nonlinear Dyn 83, 1409–1418 (2016). https://doi.org/10.1007/s11071-015-2413-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-015-2413-6