Harmless delays for uniform persistence.

*(English)*Zbl 0731.34085The authors consider a predator-prey system of Lotka-Volterra type with a finite number of discrete delays. They give sufficient conditions that the system either be uniformly persistent or not persistent. These conditions are the same as the corresponding conditions for the system with all delays set equal to zero and are independent of the magnitude of the delays. Hence uniform persistence of a system is not affected by the time delays. The results are obtained by construction of suitable Lyapunov functionals.

If the condition for persistence is satisfied, the system possesses an equilibrium. In this case, the average of the solution approaches the equilibrium.

Finally, the authors give an example of a system in which the stability of the equilibrium changes as the delay varies, but the persistence of the system is unaffected.

If the condition for persistence is satisfied, the system possesses an equilibrium. In this case, the average of the solution approaches the equilibrium.

Finally, the authors give an example of a system in which the stability of the equilibrium changes as the delay varies, but the persistence of the system is unaffected.

Reviewer: A.Hausrath (Boise)

##### MSC:

34K20 | Stability theory of functional-differential equations |

92D25 | Population dynamics (general) |

##### Keywords:

harmless delays; predator-prey system; Lotka-Volterra type; uniform persistence; Lyapunov functionals
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\textit{W. Wendi} and \textit{M. Zhien}, J. Math. Anal. Appl. 158, No. 1, 256--268 (1991; Zbl 0731.34085)

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