##
**Permanence and global attractivity for competitive Lotka-Volterra systems with delay.**
*(English)*
Zbl 0809.92025

Stability 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
PDF
BibTeX
XML
Cite

\textit{Z. Lu} and \textit{Y. Takeuchi}, Nonlinear Anal., Theory Methods Appl. 22, No. 7, 847--856 (1994; Zbl 0809.92025)

Full Text:
DOI

### References:

[1] | Freedman, H.I.; Rao, V.S.H., Stability criteria for a system involving two time delays, SIAM J. appl. math., 46, 552-560, (1986) · Zbl 0624.34066 |

[2] | Gopalsamy, K., Time lags and global stability in two-species competition, Bull. math. biol., 42, 729-737, (1980) · Zbl 0453.92014 |

[3] | Gopalsamy, K., Harmless delays in model systems, Bull. math. biol., 45, 295-309, (1983) · Zbl 0514.34060 |

[4] | Kuang, Y., Global stability for a class of nonlinear nonautonomous delay equations, Nonlinear analysis, 17, 627-634, (1991) · Zbl 0766.34053 |

[5] | Leung, A., Conditions for global stability concerning a prey-predator model with delay effects, SIAM J. appl. math., 36, 281-286, (1979) · Zbl 0418.92015 |

[6] | Martin, R.H.; Smith, H.L., Reaction-diffusion systems with time delays: monotonicity, invariance, comparison and convergence, J. reine angew. math., 413, 1-35, (1991) · Zbl 0709.35059 |

[7] | Serifert, G., On a delay-differential equation for single species populations, Nonlinear analysis, 11, 1051-1059, (1987) |

[8] | Shukla, V.P., Conditions for global stability of two-species population models with discrete time delay, Bull. math. biol., 45, 793-805, (1983) · Zbl 0524.92017 |

[9] | Kuang, Y.; Smith, H.L.; Martin, R.H., Global stability for infinite-delay, dispersive lotka – volterra system: weakly interacting populations in nearly identical patches, J. dynam. diff. eqns, 3, 339-360, (1991) · Zbl 0731.92029 |

[10] | Wright, E.M., A non-linear difference-differential equation, J. reine angew. math., 194, 66-87, (1955) · Zbl 0064.34203 |

[11] | Gopalsamy, K., Stability criteria for the linear system X(t) + A(t)X(t − τ) = 0 and an application to a non-linear system, Int. J. systems sci., 21, 1841-1853, (1990) · Zbl 0708.93071 |

[12] | Hale, J.K.; Waltman, P., Persistence in infinite-dimensional systems, SIAM J. math. analysis, 20, 388-395, (1989) · Zbl 0692.34053 |

[13] | Waltman, P., A brief survey of persistence in dynamical systems, (), 31-40 · Zbl 0756.34054 |

[14] | Wang, W.D.; Ma, Z.E., Harmless delays for uniform persistence, J. math. analysis applic., 158, 256-268, (1991) · Zbl 0731.34085 |

[15] | Shibata, A.; Saito, N., Time delays and chaos in two competing species, Math. biosci., 51, 199-211, (1980) · Zbl 0455.92011 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.