The improved results on the stochastic Kolmogorov system with time-varying delay

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July 2015, 20(5): 1481-1497. doi: 10.3934/dcdsb.2015.20.1481

The improved results on the stochastic Kolmogorov system with time-varying delay

1.	School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
2.	School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074

Received November 2013 Revised December 2014 Published May 2015

Abstract
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This paper discusses the stochastic Kolmogorov system with time-varying delay. Under two classes of sufficient conditions, this paper solves the non-explosion, the moment boundedness and the polynomial pathwise growth simultaneously. This is an important improvement for the existing results, since the moment boundedness and the polynomial pathwise growth do not imply each in general. Moreover, these two classes of conditions only depends on the parameters of the system and are easier to be used. Finally, a two-dimensional Komogorov model is examined to illustrate the efficiency of our result.

Keywords: non-explosion, time-varying delay, polynomial pathwise growth., moment boundedness, Stochastic Kolmogorov system.

Mathematics Subject Classification: Primary: 60H10, 34A34, 34D40, 34K5.

Citation: Yangzi Hu, Fuke Wu. The improved results on the stochastic Kolmogorov system with time-varying delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1481-1497. doi: 10.3934/dcdsb.2015.20.1481

References:

[1]	A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model,, Journal of Mathematical Analysis and Applications, 292 (2004), 364. doi: 10.1016/j.jmaa.2003.12.004. Google Scholar
[2]	A. Bahar and X. Mao, Stochastic delay population dynamics,, International Journal of Pure and Applied Mathematics, 11 (2004), 377. Google Scholar
[3]	A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, SIAM, (1994). doi: 10.1137/1.9781611971262. Google Scholar
[4]	Y. Hu and F. Wu, Stochastic Kolmogorov-type population dynamics with infinite distributed delays,, Acta Applicandae Mathematicae, 110 (2010), 1407. doi: 10.1007/s10440-009-9517-2. Google Scholar
[5]	Y. Hu and C. Huang, Lasalle method and general decay stability of stochastic neural networks with mixed delays,, Journal of Applied Mathematics and Computing, 38 (2012), 257. doi: 10.1007/s12190-011-0477-0. Google Scholar
[6]	Y. Hu and C. Huang, Existence results and the momentestimate for nonlocal stochastic differential equations with time-varying delay,, Nonlinear Analysis, 75 (2012), 405. doi: 10.1016/j.na.2011.08.042. Google Scholar
[7]	Y. Hu, F. Wu and C. Huang, Stochastic Lotka-Volterra models with multiple delays,, Journal of Mathematical Analysis and Applications, 375 (2011), 42. doi: 10.1016/j.jmaa.2010.08.017. Google Scholar
[8]	M. Jovanović and M. Vasilova, Dynamics of non-autonomous stochastic Gilpin-Ayala competition model with time-varying delays,, Applied Mathematics and Computation, 219 (2013), 6946. doi: 10.1016/j.amc.2012.12.073. Google Scholar
[9]	X. Mao, G. Marion and E. Renshaw, Environmental noise supresses explosion in population dynamics,, Stochastic Process and their Applications, 97 (2002), 95. doi: 10.1016/S0304-4149(01)00126-0. Google Scholar
[10]	X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Swithching,, Imperial Collage Press, (2006). doi: 10.1142/p473. Google Scholar
[11]	X. Mao, C. Yuan and J. Zhou, Stochatic differential delay equations of population dynamics,, Journal of Mathematical Analysis and Applications, 304 (2005), 296. doi: 10.1016/j.jmaa.2004.09.027. Google Scholar
[12]	S. Pang, F. Deng and X. Mao, Asymptotic properties of stochastic population dynamics,, Dynamics of Continuous Discrete and Impulsive Systems Series A, 15 (2008), 603. Google Scholar
[13]	F. Wu and S. Hu, Stochastic Kolmogorov-type population dynamics with variable delay,, Stochastic Models, 25 (2009), 129. doi: 10.1080/15326340802646286. Google Scholar
[14]	F. Wu and Y. Xu, Stochastic Lotka-Volterra population dynamics with infinite delay,, SIAM Journal on Applied Mathematics, 70 (2009), 641. doi: 10.1137/080719194. Google Scholar
[15]	F. Wu, Unbounded delay stochastic functional Kolmogorov-type system,, Proceedings of the Royal Society of Edinburgh: Section A, 140 (2010), 1309. doi: 10.1017/S0308210509000237. Google Scholar
[16]	F. Wu and Y. Hu, Existence and uniqueness of global positive solutions to the stochastic functional Kolmogorov-type system,, IMA Journal of Applied Mathematics, 75 (2010), 317. doi: 10.1093/imamat/hxq025. Google Scholar
[17]	F. Wu, S. Hu and Y. Liu, Positive solution and its asymptotic behaviour of stochastic functional Kolmogorov-type system,, Journal of Mathematical Analysis and Applications, 364 (2010), 104. doi: 10.1016/j.jmaa.2009.10.072. Google Scholar

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References:

[1]	A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model,, Journal of Mathematical Analysis and Applications, 292 (2004), 364. doi: 10.1016/j.jmaa.2003.12.004. Google Scholar
[2]	A. Bahar and X. Mao, Stochastic delay population dynamics,, International Journal of Pure and Applied Mathematics, 11 (2004), 377. Google Scholar
[3]	A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, SIAM, (1994). doi: 10.1137/1.9781611971262. Google Scholar
[4]	Y. Hu and F. Wu, Stochastic Kolmogorov-type population dynamics with infinite distributed delays,, Acta Applicandae Mathematicae, 110 (2010), 1407. doi: 10.1007/s10440-009-9517-2. Google Scholar
[5]	Y. Hu and C. Huang, Lasalle method and general decay stability of stochastic neural networks with mixed delays,, Journal of Applied Mathematics and Computing, 38 (2012), 257. doi: 10.1007/s12190-011-0477-0. Google Scholar
[6]	Y. Hu and C. Huang, Existence results and the momentestimate for nonlocal stochastic differential equations with time-varying delay,, Nonlinear Analysis, 75 (2012), 405. doi: 10.1016/j.na.2011.08.042. Google Scholar
[7]	Y. Hu, F. Wu and C. Huang, Stochastic Lotka-Volterra models with multiple delays,, Journal of Mathematical Analysis and Applications, 375 (2011), 42. doi: 10.1016/j.jmaa.2010.08.017. Google Scholar
[8]	M. Jovanović and M. Vasilova, Dynamics of non-autonomous stochastic Gilpin-Ayala competition model with time-varying delays,, Applied Mathematics and Computation, 219 (2013), 6946. doi: 10.1016/j.amc.2012.12.073. Google Scholar
[9]	X. Mao, G. Marion and E. Renshaw, Environmental noise supresses explosion in population dynamics,, Stochastic Process and their Applications, 97 (2002), 95. doi: 10.1016/S0304-4149(01)00126-0. Google Scholar
[10]	X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Swithching,, Imperial Collage Press, (2006). doi: 10.1142/p473. Google Scholar
[11]	X. Mao, C. Yuan and J. Zhou, Stochatic differential delay equations of population dynamics,, Journal of Mathematical Analysis and Applications, 304 (2005), 296. doi: 10.1016/j.jmaa.2004.09.027. Google Scholar
[12]	S. Pang, F. Deng and X. Mao, Asymptotic properties of stochastic population dynamics,, Dynamics of Continuous Discrete and Impulsive Systems Series A, 15 (2008), 603. Google Scholar
[13]	F. Wu and S. Hu, Stochastic Kolmogorov-type population dynamics with variable delay,, Stochastic Models, 25 (2009), 129. doi: 10.1080/15326340802646286. Google Scholar
[14]	F. Wu and Y. Xu, Stochastic Lotka-Volterra population dynamics with infinite delay,, SIAM Journal on Applied Mathematics, 70 (2009), 641. doi: 10.1137/080719194. Google Scholar
[15]	F. Wu, Unbounded delay stochastic functional Kolmogorov-type system,, Proceedings of the Royal Society of Edinburgh: Section A, 140 (2010), 1309. doi: 10.1017/S0308210509000237. Google Scholar
[16]	F. Wu and Y. Hu, Existence and uniqueness of global positive solutions to the stochastic functional Kolmogorov-type system,, IMA Journal of Applied Mathematics, 75 (2010), 317. doi: 10.1093/imamat/hxq025. Google Scholar
[17]	F. Wu, S. Hu and Y. Liu, Positive solution and its asymptotic behaviour of stochastic functional Kolmogorov-type system,, Journal of Mathematical Analysis and Applications, 364 (2010), 104. doi: 10.1016/j.jmaa.2009.10.072. Google Scholar

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