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**Global stability and Hopf bifurcation of a predator-prey model with stage structure and delayed predator response.**
*(English)*
Zbl 1242.92063

Summary: A Holling type predator-prey model with stage structure for the predator and a time delay due to the gestation of the mature predator is investigated. By analyzing the characteristic equations, the local stability of a predator-extinction equilibrium and a coexistence equilibrium of the model is addressed and the existence of Hopf bifurcations at the coexistence equilibrium is established. By means of the persistence theory on infinite dimensional systems, it is proven that the system is permanent if the coexistence equilibrium is feasible. By using Lyapunov functionals and the LaSalle invariance principle, it is shown that the predator-extinction equilibrium is globally asymptotically stable when the coexistence equilibrium is not feasible, and sufficient conditions are derived for the global stability of the coexistence equilibrium. Numerical simulations are carried out to illustrate the main theoretical results.

### MSC:

92D40 | Ecology |

34K60 | Qualitative investigation and simulation of models involving functional-differential equations |

34K18 | Bifurcation theory of functional-differential equations |

34K20 | Stability theory of functional-differential equations |

65C20 | Probabilistic models, generic numerical methods in probability and statistics |

37N25 | Dynamical systems in biology |

34C23 | Bifurcation theory for ordinary differential equations |

Full Text:
DOI

### References:

[1] | Aiello, W.G., Freedman, H.I.: A time delay model of single species growth with stage structure. Math. Biosci. 101, 139–156 (1990) · Zbl 0719.92017 |

[2] | Bartlett, M.S.: On theoretical models for competitive and predatory biological systems. Biometrika 44, 27–42 (1957) · Zbl 0080.36301 |

[3] | Beretta, E., Kuang, Y.: Global analyses in some delayed ratio-dependent predator-prey systems. Nonlinear Anal. TMA 32, 381–408 (1998) · Zbl 0946.34061 |

[4] | Chen, L., Jing, Z.: The existence and uniqueness of limit cycles in differential equations modelling the predator-prey interaction. Chin. Sci. Bull. 9, 521–523 (1984) |

[5] | Cushing, J.M.: Integro-differential Equations and Delay Models in Population Dynamics. Springer, Heidelberg (1977) · Zbl 0363.92014 |

[6] | Georgescu, P., Hsieh, Y.: Global dynamics of a predator-prey model with stage structure for predator. SIAM J. Appl. Math. 67, 1379–1395 (2006) · Zbl 1120.92045 |

[7] | Goh, B.S.: Global stability in two species interactions. J. Math. Biol. 3, 313–318 (1976) · Zbl 0362.92013 |

[8] | Gopalsamy, K.: Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer Academic, Dordrecht/Norwell (1992) · Zbl 0752.34039 |

[9] | Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1976) · Zbl 0375.34047 |

[10] | Hale, J., Waltman, P.: Persistence in infinite-dimensional systems. SIAM J. Math. Anal. 20, 388–395 (1989) · Zbl 0692.34053 |

[11] | Hastings, A.: Global stability in two species systems. J. Math. Biol. 5, 399–403 (1978) · Zbl 0382.92008 |

[12] | Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, New York (1993) · Zbl 0777.34002 |

[13] | MacDonald, N.: Time Lags in Biological Models. Springer, Heidelberg (1978) · Zbl 0403.92020 |

[14] | Song, X., Chen, L.: Optimal harvesting and stability for a two-species competitive system with stage structure. Math. Biosci. 170, 173–186 (2001) · Zbl 1028.34049 |

[15] | Wang, W., Chen, L.: A predator-prey system with stage structure for predator. Comput. Math. Appl. 33, 83–91 (1997) |

[16] | Wang, W.: Global dynamics of a population model with stage structure for predator. In: Chen, L., et al. (eds.) Advanced topics in Biomathematics, Proceeding of the International Conference on Mathematical Biology, pp. 253–257. World Scientific, Singapore (1997) · Zbl 0986.92026 |

[17] | Wangersky, P.J., Cunningham, W.J.: Time lag in prey-predator population models. Ecology 38, 136–139 (1957) |

[18] | Xiao, Y., Chen, L.: Global stability of a predator-prey system with stage structure for the predator. Acta Math. Sin. Engl. Ser. 20, 63–70 (2004) · Zbl 1062.34056 |

[19] | Zhang, X., Chen, L., Neumann, Avidan U.: The stage-structured predator-prey model and optimal havesting policy. Math. Biosci. 168, 201–210 (2000) · Zbl 0961.92037 |

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