American Institute of Mathematical Sciences

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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

Yangzi Hu 1, and  Fuke Wu 2,

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

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 and 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), 364-380. doi: 10.1016/j.jmaa.2003.12.004.

[2]

A. Bahar and X. Mao, Stochastic delay population dynamics, International Journal of Pure and Applied Mathematics, 11 (2004), 377-400.

[3]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, PA, 1994. doi: 10.1137/1.9781611971262.

[4]

Y. Hu and F. Wu, Stochastic Kolmogorov-type population dynamics with infinite distributed delays, Acta Applicandae Mathematicae, 110 (2010), 1407-1428. doi: 10.1007/s10440-009-9517-2.

[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), 257-278. doi: 10.1007/s12190-011-0477-0.

[6]

Y. Hu and C. Huang, Existence results and the momentestimate for nonlocal stochastic differential equations with time-varying delay, Nonlinear Analysis, 75 (2012), 405-416. doi: 10.1016/j.na.2011.08.042.

[7]

Y. Hu, F. Wu and C. Huang, Stochastic Lotka-Volterra models with multiple delays, Journal of Mathematical Analysis and Applications, 375 (2011), 42-57. doi: 10.1016/j.jmaa.2010.08.017.

[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), 6946-6964. doi: 10.1016/j.amc.2012.12.073.

[9]

X. Mao, G. Marion and E. Renshaw, Environmental noise supresses explosion in population dynamics, Stochastic Process and their Applications, 97 (2002), 95-110. doi: 10.1016/S0304-4149(01)00126-0.

[10]

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Swithching, Imperial Collage Press, 2006. doi: 10.1142/p473.

[11]

X. Mao, C. Yuan and J. Zhou, Stochatic differential delay equations of population dynamics, Journal of Mathematical Analysis and Applications, 304 (2005), 296-320. doi: 10.1016/j.jmaa.2004.09.027.

[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), 603-620.

[13]

F. Wu and S. Hu, Stochastic Kolmogorov-type population dynamics with variable delay, Stochastic Models, 25 (2009), 129-150. doi: 10.1080/15326340802646286.

[14]

F. Wu and Y. Xu, Stochastic Lotka-Volterra population dynamics with infinite delay, SIAM Journal on Applied Mathematics, 70 (2009), 641-657. doi: 10.1137/080719194.

[15]

F. Wu, Unbounded delay stochastic functional Kolmogorov-type system, Proceedings of the Royal Society of Edinburgh: Section A, 140 (2010), 1309-1334. doi: 10.1017/S0308210509000237.

[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), 317-332. doi: 10.1093/imamat/hxq025.

[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), 104-118. doi: 10.1016/j.jmaa.2009.10.072.

show all references

References:
[1]

A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model, Journal of Mathematical Analysis and Applications, 292 (2004), 364-380. doi: 10.1016/j.jmaa.2003.12.004.

[2]

A. Bahar and X. Mao, Stochastic delay population dynamics, International Journal of Pure and Applied Mathematics, 11 (2004), 377-400.

[3]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, PA, 1994. doi: 10.1137/1.9781611971262.

[4]

Y. Hu and F. Wu, Stochastic Kolmogorov-type population dynamics with infinite distributed delays, Acta Applicandae Mathematicae, 110 (2010), 1407-1428. doi: 10.1007/s10440-009-9517-2.

[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), 257-278. doi: 10.1007/s12190-011-0477-0.

[6]

Y. Hu and C. Huang, Existence results and the momentestimate for nonlocal stochastic differential equations with time-varying delay, Nonlinear Analysis, 75 (2012), 405-416. doi: 10.1016/j.na.2011.08.042.

[7]

Y. Hu, F. Wu and C. Huang, Stochastic Lotka-Volterra models with multiple delays, Journal of Mathematical Analysis and Applications, 375 (2011), 42-57. doi: 10.1016/j.jmaa.2010.08.017.

[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), 6946-6964. doi: 10.1016/j.amc.2012.12.073.

[9]

X. Mao, G. Marion and E. Renshaw, Environmental noise supresses explosion in population dynamics, Stochastic Process and their Applications, 97 (2002), 95-110. doi: 10.1016/S0304-4149(01)00126-0.

[10]

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Swithching, Imperial Collage Press, 2006. doi: 10.1142/p473.

[11]

X. Mao, C. Yuan and J. Zhou, Stochatic differential delay equations of population dynamics, Journal of Mathematical Analysis and Applications, 304 (2005), 296-320. doi: 10.1016/j.jmaa.2004.09.027.

[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), 603-620.

[13]

F. Wu and S. Hu, Stochastic Kolmogorov-type population dynamics with variable delay, Stochastic Models, 25 (2009), 129-150. doi: 10.1080/15326340802646286.

[14]

F. Wu and Y. Xu, Stochastic Lotka-Volterra population dynamics with infinite delay, SIAM Journal on Applied Mathematics, 70 (2009), 641-657. doi: 10.1137/080719194.

[15]

F. Wu, Unbounded delay stochastic functional Kolmogorov-type system, Proceedings of the Royal Society of Edinburgh: Section A, 140 (2010), 1309-1334. doi: 10.1017/S0308210509000237.

[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), 317-332. doi: 10.1093/imamat/hxq025.

[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), 104-118. doi: 10.1016/j.jmaa.2009.10.072.

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