zbMATH — the first resource for mathematics

Asymptotic stability of competitive systems with delays and impulsive perturbations. (English) Zbl 1153.34044
The authors consider a general impulsive nonautonomous Lotka-Volterra system of integro-differential equations with infinite delay. The impulses are realized at fixed moments of time. By using the comparison principle and the Lyapunov method, they obtain sufficient conditions for uniform stability and asymptotic stability of solutions.

34K20 Stability theory of functional-differential equations
34K45 Functional-differential equations with impulses
Full Text: DOI
[1] S. Ahmad, A.C. Lazer, On persistence and extinction of species, Centro International de Matematica 20, Lisbon, Portugal, 2002, pp. 1-12
[2] Ahmad, S.; Lazer, A.C., Average conditions for global asymptotic stability in a nonautonomous lotka – volterra system, Nonlinear anal., 40, 37-49, (2000) · Zbl 0955.34041
[3] Ahmad, S.; Rao, M. Rama M., Asymptotic periodic solutions of N-competing species problem with time delay, J. math. anal. appl., 186, 559-571, (1994) · Zbl 0818.45004
[4] Ahmad, S.; Stamova, I.M., Partial persistence and extinction in N-dimensional competitive systems, Nonlinear anal., 60, 821-836, (2005) · Zbl 1071.34046
[5] Fu, X.; Zhang, L., On boundedness of the solutions of impulsive integro-differential systems with fixed moments of impulse effects, Acta math. sci., 17, 219-229, (1997) · Zbl 0881.45005
[6] Gopalsamy, K., Global asymptotic stability in periodic integro-differential systems, Tohoku math. J., 37, 323-332, (1985) · Zbl 0587.45013
[7] Gopalsamy, K.; Weng, P., Global attractivity and level crossing in model of hematopoiesis, Bull. inst. math. acad. sin., 22, 4, 341-360, (1994) · Zbl 0829.34067
[8] Gopalsamy, K.; Zhang, B.G., On delay differential equations with impulses, J. math. anal. appl., 139, 110-122, (1989) · Zbl 0687.34065
[9] Lisena, B., Global attractive periodic models of predator – prey type, Nonlinear anal., 6, 133-144, (2005) · Zbl 1097.34029
[10] Samoilenko, A.M.; Perestyuk, N.A., Differential equations with impulse effect, (1987), Visca Skola Kiev, (in Russian) · Zbl 0837.34003
[11] Stamov, G.T.; Stamova, I.M., Second method of Lyapunov and existence of integral manifolds for impulsive differential-difference equations, J. math. anal. appl., 258, 371-379, (2001) · Zbl 0982.34068
[12] Stamova, I.M., Lyapunov method for boundedness of solutions of nonlinear impulsive functional differential equations, Dynam. systems appl., 14, 561-568, (2005) · Zbl 1096.34059
[13] Tineo, A., Necessary and sufficient conditions for extinction of one species, Adv. nonlinear stud., 5, 1, 57-71, (2005) · Zbl 1091.34028
[14] Zanolin, F., Permanence and positive periodic solutions for Kolmogorov competing species systems, Results math., 21, 1-2, 224-250, (1992) · Zbl 0765.92022
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.