June  2013, 33(6): 2495-2522. doi: 10.3934/dcds.2013.33.2495

Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations

1. 

School of Mathematical Science, Huaiyin Normal University, Huaian 223300, China

2. 

Department of Mathematics, Harbin Institute of Technology, Weihai 264209, China

Received  March 2011 Revised  October 2012 Published  December 2012

This paper is concerned with two $n$-species stochastic cooperative systems. One is autonomous, the other is non-autonomous. For the first system, we prove that for each species, there is a constant which can be represented by the coefficients of the system. If the constant is negative, then the corresponding species will go to extinction with probability 1; If the constant is positive, then the corresponding species will be persistent with probability 1. For the second system, sufficient conditions for stochastic permanence and global attractivity are established. In addition, the upper- and lower-growth rates of the positive solution are investigated. Our results reveal that, firstly, the stochastic noise of one population is unfavorable for the persistence of all species; secondly, a population could be persistent even the growth rate of this population is less than the half of the intensity of the white noise.
Citation: Meng Liu, Ke Wang. Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2495-2522. doi: 10.3934/dcds.2013.33.2495
References:
[1]

X. Abdurahman and Z.Teng, Persistence and extinction for general nonautonomous n-species Lotka-Volterra cooperative systems with delays,, Stud. Appl. Math., 118 (2007), 17.  doi: 10.1111/j.1467-9590.2007.00362.x.  Google Scholar

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S. Ahmad and A. C. Lazer, Average conditions for global asymptotic stability in a nonautonomous Lotka-Volterra system,, Nonlinear Anal., 40 (2000), 37.  doi: 10.1016/S0362-546X(00)85003-8.  Google Scholar

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A. Bahar and X. Mao, Stochastic delay population dynamics,, International J. Pure and Applied in Math., 11 (2004), 377.   Google Scholar

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A. Berman and R. J. Plemmons, "Nonnegative Matrices in the Mathematical Science,", Academic Press, (1979).   Google Scholar

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S. Cheng, Stochastic population systems,, Stoch. Anal. Appl., 27 (2009), 854.  doi: 10.1080/07362990902844348.  Google Scholar

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D. J. Higham, An algorithmic introduction to numerical simulation of stochastic diffrential equations,, SIAM Rev., 43 (2001), 525.  doi: 10.1137/S0036144500378302.  Google Scholar

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Y. Hu, F. Wu and C. Huang, Stochastic Lotka-Volterra models with multiple delays,, J. Math. Anal. Appl., 375 (2011), 42.  doi: 10.1016/j.jmaa.2010.08.017.  Google Scholar

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D. Jiang, N. Shi and X. Li, Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation,, J. Math. Anal. Appl., 340 (2006), 588.  doi: 10.1016/j.jmaa.2007.08.014.  Google Scholar

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[21]

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X. Li, D. Jiang and X. Mao, Population dynamical behavior of Lotka-Volterra system under regime switching,, J. Comput. Appl. Math., 232 (2009), 427.  doi: 10.1016/j.cam.2009.06.021.  Google Scholar

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M. Liu and K. Wang, Survival analysis of a stochastic cooperation system in a polluted environment,, J. Biol. Syst., 19 (2011), 183.  doi: 10.1142/S0218339011003877.  Google Scholar

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M. Liu and K. Wang, Persistence and extinction in stochastic non-autonomous logistic systems,, J. Math. Anal. Appl., 375 (2011), 443.  doi: 10.1016/j.jmaa.2010.09.058.  Google Scholar

[25]

M. Liu, K. Wang and Q. Wu, Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle,, Bull. Math. Biol., 73 (2011), 1969.  doi: 10.1007/s11538-010-9569-5.  Google Scholar

[26]

G. Lu, Z. Lu and X. Lian, Delay effect on the permanence for Lotka-Volterra cooperative systems,, Nonlinear Anal. Real World Appl., 11 (2010), 2810.  doi: 10.1016/j.nonrwa.2009.10.005.  Google Scholar

[27]

Z. Lu and Y. Takeuchi, Permanence and global stability for cooperative Lotka-Volterra diffusion systems,, Nonlinear Anal., 19 (1992), 963.  doi: 10.1016/0362-546X(92)90107-P.  Google Scholar

[28]

Q. Luo and X. Mao, Stochastic population dynamics under regime switching II,, J. Math. Anal. Appl., 355 (2009), 577.   Google Scholar

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

[30]

X. Mao and C. Yuan, "Stochastic Differential Equations with Markovian Switching,", Imperial College Press, (2006).   Google Scholar

[31]

J. Pan, Z. Jin and Z. Ma, Thresholds of survival for an n-dimensional Volterra mutualistic system in a polluted environment,, J. Math. Anal. Appl., 252 (2000), 519.  doi: 10.1006/jmaa.2000.6853.  Google Scholar

[32]

S. Pang, F. Deng and X. Mao, Asymptotic properties of stochastic population dynamics,, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 15 (2008), 603.   Google Scholar

[33]

H. L. Smith, On the asymptotic behavior of a class of deterministic models of cooperating species,, SIAM J. Math. Anal., 46 (1986), 368.  doi: 10.1137/0146025.  Google Scholar

[34]

F. Wu and Y. Hu, Stochastic Lotka-Volterra system with unbounded distributed delay,, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 275.  doi: 10.3934/dcdsb.2010.14.275.  Google Scholar

[35]

J. Zhao and J. Jiang, Average conditions for permanence and extinction in nonautonomous Lotka-Volterra system,, J. Math. Anal. Appl., 229 (2004), 663.  doi: 10.1016/j.jmaa.2004.06.019.  Google Scholar

[36]

J. Zhao, J. Jiang and A. Lazer, The permanence and global attractivity in a nonautonomous Lotka-Volterra system,, Nonlinear Anal. Real World Appl., 5 (2004), 265.  doi: 10.1016/S1468-1218(03)00038-5.  Google Scholar

show all references

References:
[1]

X. Abdurahman and Z.Teng, Persistence and extinction for general nonautonomous n-species Lotka-Volterra cooperative systems with delays,, Stud. Appl. Math., 118 (2007), 17.  doi: 10.1111/j.1467-9590.2007.00362.x.  Google Scholar

[2]

S. Ahmad, Extinction of species in nonautonomous Lotka-Volterra system,, Proc. Am. Math. Soc., 127 (1999), 2905.  doi: 10.1090/S0002-9939-99-05083-2.  Google Scholar

[3]

S. Ahmad and A. C. Lazer, Average conditions for global asymptotic stability in a nonautonomous Lotka-Volterra system,, Nonlinear Anal., 40 (2000), 37.  doi: 10.1016/S0362-546X(00)85003-8.  Google Scholar

[4]

E. S. Allman and J. A. Rhodes, "Mathematical Models in Biology: An Introduction,", Cambridge University Press, (2004).   Google Scholar

[5]

A. Bahar and X. Mao, Stochastic delay population dynamics,, International J. Pure and Applied in Math., 11 (2004), 377.   Google Scholar

[6]

I. Barbalat, Systems dequations differentielles d'osci d'oscillations nonlineaires,, Revue Roumaine de Mathematiques Pures et Appliquees, 4 (1959), 267.   Google Scholar

[7]

A. Berman and R. J. Plemmons, "Nonnegative Matrices in the Mathematical Science,", Academic Press, (1979).   Google Scholar

[8]

S. Cheng, Stochastic population systems,, Stoch. Anal. Appl., 27 (2009), 854.  doi: 10.1080/07362990902844348.  Google Scholar

[9]

B. S. Goh, Stability in models of mutualism,, Amer. Natural, 113 (1979), 261.  doi: 10.1086/283384.  Google Scholar

[10]

K. Golpalsamy, "Stability and Oscillations in Delay Differential Equations of Population Dynamics,", Kluwer Academic, (1992).   Google Scholar

[11]

T. G. Hallam and Z. Ma, Persistence in population models with demographic fluctuations,, J. Math. Biol., 24 (1986), 327.  doi: 10.1007/BF00275641.  Google Scholar

[12]

X. He and K. Gopalsamy, Persistence, attractivity, and delay in facultative mutualism,, J. Math. Anal. Appl., 215 (1997), 154.  doi: 10.1006/jmaa.1997.5632.  Google Scholar

[13]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic diffrential equations,, SIAM Rev., 43 (2001), 525.  doi: 10.1137/S0036144500378302.  Google Scholar

[14]

Y. Hu, F. Wu and C. Huang, Stochastic Lotka-Volterra models with multiple delays,, J. Math. Anal. Appl., 375 (2011), 42.  doi: 10.1016/j.jmaa.2010.08.017.  Google Scholar

[15]

V. Hutson and K. Schmitt, Permanence and the dynamics of biological systems,, Math. Biosci., 111 (1992), 1.  doi: 10.1016/0025-5564(92)90078-B.  Google Scholar

[16]

D. Jiang, N. Shi and X. Li, Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation,, J. Math. Anal. Appl., 340 (2006), 588.  doi: 10.1016/j.jmaa.2007.08.014.  Google Scholar

[17]

J. Jiang, On the global stability of cooperative systems,, Bull. Lond. Math. Soc., 26 (1994), 455.  doi: 10.1112/blms/26.5.455.  Google Scholar

[18]

I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus,", Springer-Verlag, (1991).  doi: 10.1007/978-1-4612-0949-2.  Google Scholar

[19]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Academic Press, (1993).   Google Scholar

[20]

X. Li, A. Gray, D. Jiang and X. Mao, Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching,, J. Math. Anal. Appl., 376 (2011), 11.  doi: 10.1016/j.jmaa.2010.10.053.  Google Scholar

[21]

X. Li and X. Mao, Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation,, Discrete Contin. Dyn. Syst., 24 (2009), 523.  doi: 10.3934/dcds.2009.24.523.  Google Scholar

[22]

X. Li, D. Jiang and X. Mao, Population dynamical behavior of Lotka-Volterra system under regime switching,, J. Comput. Appl. Math., 232 (2009), 427.  doi: 10.1016/j.cam.2009.06.021.  Google Scholar

[23]

M. Liu and K. Wang, Survival analysis of a stochastic cooperation system in a polluted environment,, J. Biol. Syst., 19 (2011), 183.  doi: 10.1142/S0218339011003877.  Google Scholar

[24]

M. Liu and K. Wang, Persistence and extinction in stochastic non-autonomous logistic systems,, J. Math. Anal. Appl., 375 (2011), 443.  doi: 10.1016/j.jmaa.2010.09.058.  Google Scholar

[25]

M. Liu, K. Wang and Q. Wu, Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle,, Bull. Math. Biol., 73 (2011), 1969.  doi: 10.1007/s11538-010-9569-5.  Google Scholar

[26]

G. Lu, Z. Lu and X. Lian, Delay effect on the permanence for Lotka-Volterra cooperative systems,, Nonlinear Anal. Real World Appl., 11 (2010), 2810.  doi: 10.1016/j.nonrwa.2009.10.005.  Google Scholar

[27]

Z. Lu and Y. Takeuchi, Permanence and global stability for cooperative Lotka-Volterra diffusion systems,, Nonlinear Anal., 19 (1992), 963.  doi: 10.1016/0362-546X(92)90107-P.  Google Scholar

[28]

Q. Luo and X. Mao, Stochastic population dynamics under regime switching II,, J. Math. Anal. Appl., 355 (2009), 577.   Google Scholar

[29]

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

[30]

X. Mao and C. Yuan, "Stochastic Differential Equations with Markovian Switching,", Imperial College Press, (2006).   Google Scholar

[31]

J. Pan, Z. Jin and Z. Ma, Thresholds of survival for an n-dimensional Volterra mutualistic system in a polluted environment,, J. Math. Anal. Appl., 252 (2000), 519.  doi: 10.1006/jmaa.2000.6853.  Google Scholar

[32]

S. Pang, F. Deng and X. Mao, Asymptotic properties of stochastic population dynamics,, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 15 (2008), 603.   Google Scholar

[33]

H. L. Smith, On the asymptotic behavior of a class of deterministic models of cooperating species,, SIAM J. Math. Anal., 46 (1986), 368.  doi: 10.1137/0146025.  Google Scholar

[34]

F. Wu and Y. Hu, Stochastic Lotka-Volterra system with unbounded distributed delay,, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 275.  doi: 10.3934/dcdsb.2010.14.275.  Google Scholar

[35]

J. Zhao and J. Jiang, Average conditions for permanence and extinction in nonautonomous Lotka-Volterra system,, J. Math. Anal. Appl., 229 (2004), 663.  doi: 10.1016/j.jmaa.2004.06.019.  Google Scholar

[36]

J. Zhao, J. Jiang and A. Lazer, The permanence and global attractivity in a nonautonomous Lotka-Volterra system,, Nonlinear Anal. Real World Appl., 5 (2004), 265.  doi: 10.1016/S1468-1218(03)00038-5.  Google Scholar

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