Persistence and stability of a stage-structured predator-prey model with time delays.

*(English)*Zbl 1064.92049From the introduction: An important problem in predator-prey theory and related topics in mathematical ecology concerns the global stability of ecological systems with time delays. However, most of the global stability or convergence results appearing so far for delayed ecological systems require that the instantaneous negative feedbacks dominate both delayed feedbacks and interspecific interactions. Such a requirement is rarely met in real systems since feedbacks are generally delayed. This leads to the standing question: Under what conditions will the global stability of a nonnegative steady state of a delay differential system persist when time delays involved in some part of the negative feedbacks are small enough? Y. Kuang [J. Differ. Equations 119, No. 2, 503–532 (1995; Zbl 0828.34066)] presented a partial answer to this open question for Lotka-Voterra-type systems.

Motivated by works of W. G. Aiello and H. I. Freedman [Math. Biosci. 101, 139–153 (1990; Zbl 0719.92017)], W. Wang et al. [J. Math. Anal. Appl. 262, No. 2, 499–528 (2001; Zbl 0997.34069)], and Y. Kuang [op. cit.], we study the combined effects of stage structure for predators and time delays due to negative feedback of the prey on the global dynamics of predator-prey models.

Motivated by works of W. G. Aiello and H. I. Freedman [Math. Biosci. 101, 139–153 (1990; Zbl 0719.92017)], W. Wang et al. [J. Math. Anal. Appl. 262, No. 2, 499–528 (2001; Zbl 0997.34069)], and Y. Kuang [op. cit.], we study the combined effects of stage structure for predators and time delays due to negative feedback of the prey on the global dynamics of predator-prey models.

##### MSC:

92D40 | Ecology |

34K20 | Stability theory of functional-differential equations |

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

92D25 | Population dynamics (general) |

##### Software:

dde23
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\textit{R. Xu} et al., Appl. Math. Comput. 150, No. 1, 259--277 (2004; Zbl 1064.92049)

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##### References:

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