Persistence, attractivity, and delay in facultative mutualism.

*(English)*Zbl 0893.34036The general two-dimensional cooperative Lotka-Volterra system is treated in which the cooperation is “facultative” (i.e. each species can survive on its own following the logistic dynamics) and in which there is a single discrete delay at each term in the per capita growth rates. It is well known that in the absence of delay if the overall intraspecific competition is stronger than the overall mutualism then there is a globally asymptotically stable positive equilibrium. It is proved here that if this condition holds then the system with delay is persistent iff the solutions are ultimately bounded. Also it is proved that under the same conditions if the system is uniformly persistent then for sufficiently small delay the positive equilibrium is a global attractor.

Reviewer: M.Farkas (Budapest)

##### MSC:

37-XX | Dynamical systems and ergodic theory |

92D25 | Population dynamics (general) |

34D45 | Attractors of solutions to ordinary differential equations |

##### Keywords:

Lotka-Volterra system; discrete delay; mutualism; globally asymptotically stable positive equilibrium; solutions; global attractor
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\textit{X.-z. He} and \textit{K. Gopalsamy}, J. Math. Anal. Appl. 215, No. 1, 154--173 (1997; Zbl 0893.34036)

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