×

Stochastic Lotka-Volterra model with infinite delay. (English) Zbl 1159.92321

Summary: A stochastic Lotka-Volterra system with infinite delay is studied. We show that the solution of such a system is a positive solution without explosion and give conditions to guarantee stochastic ultimate boundedness of the solutions.

MSC:

92D40 Ecology
34K50 Stochastic functional-differential equations
34K60 Qualitative investigation and simulation of models involving functional-differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ahmad, A.; Rao, M. R.M., Asymptotically periodic solutions of \(n\)-competing species problem with time delay, J. Math. Anal. Appl., 186, 557-571 (1994) · Zbl 0818.45004
[2] Bereketoglu, H.; Gyori, I., Global asymptotic stability in a nonautonomous Lotka-Volterra type system with infinite delay, J. Math. Anal. Appl., 210, 279-291 (1997) · Zbl 0880.34072
[3] Bahar, A.; Mao, X., Stochastic delay Lotka-Volterra tmodel, J. Math. Anal. Appl., 292, 364-380 (2004) · Zbl 1043.92034
[4] Freedman, H. I.; Ruan, S., Uniform persistence in functional differential equations, J. Differential Equations, 115, 173-192 (1995) · Zbl 0814.34064
[5] Gopalsamy, K., Global asymptotic stability in Volterra’s population systems, J. Math. Biol., 19, 157-168 (1984) · Zbl 0535.92020
[6] He, X.; Gopalsamy, K., Persistence, attractivity, and delay in facultative mutualism, J. Math. Anal. Appl., 215, 154-173 (1997) · Zbl 0893.34036
[7] Kolmanovskii, V.; Myshkis, A., Applied Theory of Functional Differential Equations (1992), Kluwer Academic · Zbl 0917.34001
[8] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press: Academic Press Boston · Zbl 0777.34002
[9] Kuang, Y.; Smith, H. L., Global stability for infinite delay Lotka-Volterra type systems, J. Differential Equations, 103, 221-246 (1993) · Zbl 0786.34077
[10] Mao, X., (Stability of Stochastic Differential Equation with respect to Semimartingales. Stability of Stochastic Differential Equation with respect to Semimartingales, Pitman Research Notes in Mathematics Series, vol. 251 (1991), Longman Scientific and Technical) · Zbl 0724.60059
[11] Mao, X., Exponential Stability of Stochastic Differential Equations (1994), Marcel Dekker: Marcel Dekker New York · Zbl 0851.93074
[12] Mao, X., Stochastic Differential Equations and Applications (1997), Horwood Publishing · Zbl 0874.60050
[13] Mao, X.; Yuan, C.; Zou, J., Stochastic differential delay equations of population dynamics, J. Math. Anal. Appl., 304, 296-320 (2005) · Zbl 1062.92055
[14] Teng, Z.; Yu, Y., Some new results of nonautonomous Lotka-Volterra competitive systems with delays, J. Math. Anal. Appl., 241, 254-275 (2000) · Zbl 0947.34066
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.