×

Stochastic Gilpin-Ayala competition model with infinite delay. (English) Zbl 1206.92070

Summary: We study the stochastic M. E. Gilpin and F. J. Ayala [Proc. Natl. Acad. Sci. USA 70, 3590–3593 (1973; Zbl 0272.92016)] competition model with an infinite delay. We verify that the environmental noise included in the model does not only provide a positive global solution (there is no explosion in a finite time), but this solution is also stochastically ultimately bounded. We obtain certain asymptotic results regarding a large time behavior.

MSC:

92D40 Ecology
34F05 Ordinary differential equations and systems with randomness
34K60 Qualitative investigation and simulation of models involving functional-differential equations

Citations:

Zbl 0272.92016
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Lotka, A., Elements of Physical Biology (1924), Williams and Wilkins: Williams and Wilkins Baltimore, Md · JFM 51.0416.06
[2] Volterra, V., Lecons Sur la Theorie Mathematique de la Lutte pour la Vie (1931), Gauthier-Villars: Gauthier-Villars Paris · JFM 57.0466.02
[3] He, X.; Gopalsamy, K., Persistence, attractivity, and delay in facultative mutualism, J. Math. Anal. Appl., 215, 154-173 (1997) · Zbl 0893.34036
[4] 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
[5] Gilpin, M. E.; Ayala, F. J., Global models of growth and competition, Proc. Natl. Acad. Sci. USA, 70, 3590-3593 (1973) · Zbl 0272.92016
[6] Fan, M.; Wang, K., Global periodic solutions of a generalized \(n\)-species Gilpin-Ayala competition model, Comput. Math. Appl., 40, 1141-1151 (2000) · Zbl 0954.92027
[7] Mao, X.; Marion, G.; Renshaw, E., Environmental Brownian noise suppresses explosions in population dynamics, Stoch. Process. Appl., 97, 95-110 (2002) · Zbl 1058.60046
[8] Mao, X.; Sabanis, S.; Renshaw, E., Asymptotic behaviour of the stochastic Lotka-Volterra model, J. Math. Anal. Appl., 287, 141-156 (2003) · Zbl 1048.92027
[9] Du, N. H.; Sam, V. H., Dynamics of a stochastic Lotka-Volterra model perturbed by white noise, J. Math. Anal. Appl., 324, 82-97 (2006) · Zbl 1107.92038
[10] Lian, B.; Hu, S., Asymptotic behaviour of the stochastic Gilpin-Ayala competition models, J. Math. Anal. Appl., 339, 419-428 (2008) · Zbl 1195.34083
[11] Vasilova, M.; Jovanović, M., Dynamics of Gilpin-Ayala competition model with random perturbation, Filomat, 24, 1, 101-113 (2010) · Zbl 1299.60075
[12] Bahar, A.; Mao, X., Stochastic delay Lotka-Volterra model, J. Math. Anal. Appl., 292, 364-380 (2004) · Zbl 1043.92034
[13] Wan, L.; Zhou, Q., Stochastic Lotka-Volterra model with infinite delay, Statist. Probab. Lett., 79, 698-706 (2009) · Zbl 1159.92321
[14] Mao, X.; Yuan, C.; Zou, J., Stochastic differential delay equations of population dynamics, J. Math. Anal. Appl., 304, 296-320 (2005) · Zbl 1062.92055
[15] Lian, B.; Hu, S., Stochastic delay Gilpin-Ayala competition models, Stoch. Dyn., 6, 4, 561-576 (2006) · Zbl 1117.34079
[16] Yan, J., Global positive periodic solutions of periodic \(n\)-species competition systems, J. Math. Anal. Appl., 356, 288-294 (2009) · Zbl 1177.34056
[17] Mao, X., Stochastic Differential Equations and Applications (2008), Horwood: Horwood Chichester
[18] Mao, X., Exponential Stability of Stochastic Differential Equations (1994), Marcel Dekker: Marcel Dekker New York · Zbl 0851.93074
[19] Sinclair, A. R.E., Mammal population regulation, keystone processes and ecosystem dynamics, Phil. Trans. R. Soc. Lond. B, 358, 1729-1740 (2003)
[20] Kloeden, P. E.; Platen, E., Numerical Solution of Stochastic Differential Equations (1995), Springer: Springer Berlin · Zbl 0858.65148
[21] Song, Y.; Baker, C. T.H., Qualitative behaviour of numerical approximations to Volterra integro-differential equations, J. Comput. Appl. Math., 172, 101-115 (2004) · Zbl 1059.65129
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.