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November  2014, 13(6): 2693-2712. doi: 10.3934/cpaa.2014.13.2693

Asymptotic behavior of Kolmogorov systems with predator-prey type in random environment

Nguyen Huu Du 1, and  Nguyen Hai Dang 2,

1. 

Department of Mathematics, Mechanics and Informatics, Hanoi National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam
2. 

Department of Mathematics, Wayne State University, Detroit, MI 48202, United States

Received  October 2013 Revised  March 2014 Published  July 2014

The paper is concerned with the asymptotic behavior of two species population whose densities are described by Kolmogorov systems of predator-prey type in random environment. We study the omega-limit set and find conditions ensuring the existence and attractivity of a stationary density. Some applications to the predator-prey model with Beddington-DeAngelis functional response are considered to illustrate our results.
Keywords: telegraph noise, stationary distribution, Kolmogorov systems of predator-prey type, piecewise deterministic Markov process..
Mathematics Subject Classification: Primary: 34C12, 60H10; Secondary: 92D2.
Citation: Nguyen Huu Du, Nguyen Hai Dang. Asymptotic behavior of Kolmogorov systems with predator-prey type in random environment. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2693-2712. doi: 10.3934/cpaa.2014.13.2693
References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, Heidelberg, New York, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

P. Auger, N. H. Du and N. T. Hieu, Evolution of Lotka-Volterra predator-prey systems under telegraph noise, Math. Biosci. Eng., 6 (2009), 683-700. doi: 10.3934/mbe.2009.6.683.

[3]

A. D. Bazykin, Nonlinear Dynamics of Interacting Populations, World Scientific, Singapore, 1998. doi: 10.1142/9789812798725.

[4]

A. Bobrowski, T. Lipniacki, K. Pichor and R. Rudnicki, Asymptotic behavior of distributions of mRNA and protein levels in a model of stochastic gene expression, J. Math. Anal. Appl., 333 (2007), 753-769. doi: 10.1016/j.jmaa.2006.11.043.

[5]

Z. Brzeniak, M. Capifiski and F. Flandoli, Pathwise global attractors for stationary random dynamical systems, Probab. Theory Relat. Fields, 95 (1993), 87-102 doi: 10.1007/BF01197339.

[6]

N. H. Dang, N. H. Du and T. V. Ton, Asymptotic behavior of predator-prey systems perturbed by white noise, Acta Appl. Math., 115 (2011), 351-370. doi: 10.1007/s10440-011-9628-4.

[7]

N. H. Du and N. H. Dang, Dynamics of Kolmogorov systems of competitive type under the telegraph noise, J. Differential Equations, 250 (2011), 386-409. doi: 10.1016/j.jde.2010.08.023.

[8]

N. H. Du, R. Kon, K. Sato and Y. Takeuchi, Dynamical behavior of Lotka - Volterra competition systems: Non autonomous bistable case and the effect of telegraph noise, J. Comput. Appl. Math, 170 (2004), 399-422. doi: 10.1016/j.cam.2004.02.001.

[9]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Relat. Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.

[10]

I. I Gihman and A. V. Skorohod, The Theory of Stochastic Processes, Springer-Verlag, Berlin, Heidelberg, New York, 1979.

[11]

T. W. Hwang, Global analysis of the predator-prey system with Beddington DeAngelis functional response, J. Math. Anal. Appl., 281 (2003), 395-401.

[12]

T. W. Hwang, Uniqueness of limit cycles of the predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 290 (2004), 113-122. doi: 10.1016/j.jmaa.2003.09.073.

[13]

C. Ji and D. Jiang, Dynamics of a stochastic density dependent predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 381 (2011), 441-453. doi: 10.1016/j.jmaa.2011.02.037.

[14]

C. Ji, D. Jiang and N. Shi, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation, J. Math. Anal. Appl., 359 (2009), 482-498. doi: 10.1016/j.jmaa.2009.05.039.

[15]

Q. Luo and X. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl., 334 (2007), 69-84 . doi: 10.1016/j.jmaa.2006.12.032.

[16]

Q. Luo and X. Mao, Stochastic population dynamics under regime switching. II, J. Math. Anal. Appl., 355 (2009), 577-593. doi: 10.1016/j.jmaa.2009.02.010.

[17]

X. Mao, Attraction, stability and boundedness for stochastic differential delay equations, Nonlinear Analysis: Theory, Methods and Applications, 47 (2001), 4795-4806. doi: 10.1016/S0362-546X(01)00591-0.

[18]

X. Mao, S. Sabanis and E. Renshaw, Asymptotic behaviour of the stochastic Lotka-Volterra model, J. Math. Anal. Appl., 287 (2003), 141-156. doi: 10.1016/S0022-247X(03)00539-0.

[19]

L. Michael, Conservative Markov processes on a topological space, Israel J. Math., 8 (1970), 165-186.

[20]

J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, Heidelberg, 2002. doi: 10.1007/b98869.

[21]

K. Pichór and R. Rudnicki, Continuous Markov semigroups and stability of transport equations, J. Math. Anal. Appl., 249 (2000), 668-685. doi: 10.1006/jmaa.2000.6968.

[22]

R. Rudnicki, K. Pichór and M. Tyran-Kaminska, Markov semigroups and their applications, in Dynamics of Dissipation (P. Garbaczewski and R. Olkiewicz Eds), Lecture Note in Physics, Vol 587, Springer, Berlin, 2002, 215-238.

[23]

Y. Takeuchi, N. H. Du, N. T. Hieu and K. Sato, Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment, J. Math. Anal. Appl., 323 (2006), 938-957. doi: 10.1016/j.jmaa.2005.11.009.

[24]

C. Yuan and X. Mao, Attraction and stochastic asymptotic stability and boundedness of stochastic functional differential equations with respect to semimartingales, Stochastic Analysis and Applications, 24 (2006), 1169-1184. doi: 10.1080/07362990600958937.

[25]

C. Zhu and G. Yin, On competitive Lotka-Volterra model in random environments, Journal of Mathematical Analysis and Applications, 357 (2009), 154-170. doi: 10.1016/j.jmaa.2009.03.066.

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, Heidelberg, New York, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

P. Auger, N. H. Du and N. T. Hieu, Evolution of Lotka-Volterra predator-prey systems under telegraph noise, Math. Biosci. Eng., 6 (2009), 683-700. doi: 10.3934/mbe.2009.6.683.

[3]

A. D. Bazykin, Nonlinear Dynamics of Interacting Populations, World Scientific, Singapore, 1998. doi: 10.1142/9789812798725.

[4]

A. Bobrowski, T. Lipniacki, K. Pichor and R. Rudnicki, Asymptotic behavior of distributions of mRNA and protein levels in a model of stochastic gene expression, J. Math. Anal. Appl., 333 (2007), 753-769. doi: 10.1016/j.jmaa.2006.11.043.

[5]

Z. Brzeniak, M. Capifiski and F. Flandoli, Pathwise global attractors for stationary random dynamical systems, Probab. Theory Relat. Fields, 95 (1993), 87-102 doi: 10.1007/BF01197339.

[6]

N. H. Dang, N. H. Du and T. V. Ton, Asymptotic behavior of predator-prey systems perturbed by white noise, Acta Appl. Math., 115 (2011), 351-370. doi: 10.1007/s10440-011-9628-4.

[7]

N. H. Du and N. H. Dang, Dynamics of Kolmogorov systems of competitive type under the telegraph noise, J. Differential Equations, 250 (2011), 386-409. doi: 10.1016/j.jde.2010.08.023.

[8]

N. H. Du, R. Kon, K. Sato and Y. Takeuchi, Dynamical behavior of Lotka - Volterra competition systems: Non autonomous bistable case and the effect of telegraph noise, J. Comput. Appl. Math, 170 (2004), 399-422. doi: 10.1016/j.cam.2004.02.001.

[9]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Relat. Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.

[10]

I. I Gihman and A. V. Skorohod, The Theory of Stochastic Processes, Springer-Verlag, Berlin, Heidelberg, New York, 1979.

[11]

T. W. Hwang, Global analysis of the predator-prey system with Beddington DeAngelis functional response, J. Math. Anal. Appl., 281 (2003), 395-401.

[12]

T. W. Hwang, Uniqueness of limit cycles of the predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 290 (2004), 113-122. doi: 10.1016/j.jmaa.2003.09.073.

[13]

C. Ji and D. Jiang, Dynamics of a stochastic density dependent predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 381 (2011), 441-453. doi: 10.1016/j.jmaa.2011.02.037.

[14]

C. Ji, D. Jiang and N. Shi, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation, J. Math. Anal. Appl., 359 (2009), 482-498. doi: 10.1016/j.jmaa.2009.05.039.

[15]

Q. Luo and X. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl., 334 (2007), 69-84 . doi: 10.1016/j.jmaa.2006.12.032.

[16]

Q. Luo and X. Mao, Stochastic population dynamics under regime switching. II, J. Math. Anal. Appl., 355 (2009), 577-593. doi: 10.1016/j.jmaa.2009.02.010.

[17]

X. Mao, Attraction, stability and boundedness for stochastic differential delay equations, Nonlinear Analysis: Theory, Methods and Applications, 47 (2001), 4795-4806. doi: 10.1016/S0362-546X(01)00591-0.

[18]

X. Mao, S. Sabanis and E. Renshaw, Asymptotic behaviour of the stochastic Lotka-Volterra model, J. Math. Anal. Appl., 287 (2003), 141-156. doi: 10.1016/S0022-247X(03)00539-0.

[19]

L. Michael, Conservative Markov processes on a topological space, Israel J. Math., 8 (1970), 165-186.

[20]

J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, Heidelberg, 2002. doi: 10.1007/b98869.

[21]

K. Pichór and R. Rudnicki, Continuous Markov semigroups and stability of transport equations, J. Math. Anal. Appl., 249 (2000), 668-685. doi: 10.1006/jmaa.2000.6968.

[22]

R. Rudnicki, K. Pichór and M. Tyran-Kaminska, Markov semigroups and their applications, in Dynamics of Dissipation (P. Garbaczewski and R. Olkiewicz Eds), Lecture Note in Physics, Vol 587, Springer, Berlin, 2002, 215-238.

[23]

Y. Takeuchi, N. H. Du, N. T. Hieu and K. Sato, Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment, J. Math. Anal. Appl., 323 (2006), 938-957. doi: 10.1016/j.jmaa.2005.11.009.

[24]

C. Yuan and X. Mao, Attraction and stochastic asymptotic stability and boundedness of stochastic functional differential equations with respect to semimartingales, Stochastic Analysis and Applications, 24 (2006), 1169-1184. doi: 10.1080/07362990600958937.

[25]

C. Zhu and G. Yin, On competitive Lotka-Volterra model in random environments, Journal of Mathematical Analysis and Applications, 357 (2009), 154-170. doi: 10.1016/j.jmaa.2009.03.066.

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