Permanence and global attractivity for competitive Lotka-Volterra systems with delay.

*(English)*Zbl 0809.92025Stability of the Lotka-Volterra delay system has been studied by a lot of authors. Most of the papers consider the situation at which undelayed intraspecific competition is present. In these cases, either a Lyapunov- Razumikhin functional is used or comparison theorems can be applied to obtain global attractivity of a positive equilibrium point. Essentially, the point is a global attractor if the undelayed intraspecific competition dominates over the delayed intra- (and inter-) specific competition.

In this paper, by considering a similar continuous functional to the one used by W. Wang and Z. Ma [J. Math. Anal. Appl. 158, No. 1, 256-268 (1991; Zbl 0731.34085)] and by adopting a modified approach of E. M. Wright [J. Reine Angew. Math. 194, 66-87 (1955; Zbl 0064.342)], we extend J. K. Hale and P. Waltman’s results [SIAM J. Math. Anal. 20, No. 2, 388-395 (1989; Zbl 0692.34053); Lect. Notes Math. 1475, 31-40 (1991; Zbl 0756.34054)] in the following sense: their conditions for permanence of a two-species competition delayed system are actually sufficient for its positive equilibrium point to be a global attractor. We also give a sufficient condition for permanence of the system which is weaker than the one obtained by Hale and Waltman. Our global attractivity condition (which is identical to Hale and Waltman’s permanence condition) is also much weaker than K. Gopalsamy’s [Int. J. Syst. Sci. 21, No. 9, 1841-1853 (1990; Zbl 0708.93071)] and is optimal in the sense that if there is no delay in the system, our condition is sufficient and necessary for the system to have a globally stable equilibrium point.

In this paper, by considering a similar continuous functional to the one used by W. Wang and Z. Ma [J. Math. Anal. Appl. 158, No. 1, 256-268 (1991; Zbl 0731.34085)] and by adopting a modified approach of E. M. Wright [J. Reine Angew. Math. 194, 66-87 (1955; Zbl 0064.342)], we extend J. K. Hale and P. Waltman’s results [SIAM J. Math. Anal. 20, No. 2, 388-395 (1989; Zbl 0692.34053); Lect. Notes Math. 1475, 31-40 (1991; Zbl 0756.34054)] in the following sense: their conditions for permanence of a two-species competition delayed system are actually sufficient for its positive equilibrium point to be a global attractor. We also give a sufficient condition for permanence of the system which is weaker than the one obtained by Hale and Waltman. Our global attractivity condition (which is identical to Hale and Waltman’s permanence condition) is also much weaker than K. Gopalsamy’s [Int. J. Syst. Sci. 21, No. 9, 1841-1853 (1990; Zbl 0708.93071)] and is optimal in the sense that if there is no delay in the system, our condition is sufficient and necessary for the system to have a globally stable equilibrium point.

##### MSC:

92D40 | Ecology |

34D45 | Attractors of solutions to ordinary differential equations |

92D25 | Population dynamics (general) |

##### Keywords:

global stability; Lotka-Volterra delay system; positive equilibrium point; global attractor
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\textit{Z. Lu} and \textit{Y. Takeuchi}, Nonlinear Anal., Theory Methods Appl. 22, No. 7, 847--856 (1994; Zbl 0809.92025)

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##### References:

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