# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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), 17-43. doi: 10.1111/j.1467-9590.2007.00362.x. [2] S. Ahmad, Extinction of species in nonautonomous Lotka-Volterra system, Proc. Am. Math. Soc., 127 (1999), 2905-2910. doi: 10.1090/S0002-9939-99-05083-2. [3] S. Ahmad and A. C. Lazer, Average conditions for global asymptotic stability in a nonautonomous Lotka-Volterra system, Nonlinear Anal., 40 (2000), 37-49. doi: 10.1016/S0362-546X(00)85003-8. [4] E. S. Allman and J. A. Rhodes, "Mathematical Models in Biology: An Introduction," Cambridge University Press, 2004. [5] A. Bahar and X. Mao, Stochastic delay population dynamics, International J. Pure and Applied in Math., 11 (2004), 377-400. [6] I. Barbalat, Systems dequations differentielles d'osci d'oscillations nonlineaires, Revue Roumaine de Mathematiques Pures et Appliquees, 4 (1959), 267-270. [7] A. Berman and R. J. Plemmons, "Nonnegative Matrices in the Mathematical Science," Academic Press, New York, 1979. [8] S. Cheng, Stochastic population systems, Stoch. Anal. Appl., 27 (2009), 854-874. doi: 10.1080/07362990902844348. [9] B. S. Goh, Stability in models of mutualism, Amer. Natural, 113 (1979), 261-275. doi: 10.1086/283384. [10] K. Golpalsamy, "Stability and Oscillations in Delay Differential Equations of Population Dynamics," Kluwer Academic, Dordrecht, 1992. [11] T. G. Hallam and Z. Ma, Persistence in population models with demographic fluctuations, J. Math. Biol., 24 (1986), 327-339. doi: 10.1007/BF00275641. [12] X. He and K. Gopalsamy, Persistence, attractivity, and delay in facultative mutualism, J. Math. Anal. Appl., 215 (1997), 154-173. doi: 10.1006/jmaa.1997.5632. [13] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic diffrential equations, SIAM Rev., 43 (2001), 525-546. doi: 10.1137/S0036144500378302. [14] Y. Hu, F. Wu and C. Huang, Stochastic Lotka-Volterra models with multiple delays, J. Math. Anal. Appl., 375 (2011), 42-57. doi: 10.1016/j.jmaa.2010.08.017. [15] V. Hutson and K. Schmitt, Permanence and the dynamics of biological systems, Math. Biosci., 111 (1992), 1-71. doi: 10.1016/0025-5564(92)90078-B. [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), 588-597. doi: 10.1016/j.jmaa.2007.08.014. [17] J. Jiang, On the global stability of cooperative systems, Bull. Lond. Math. Soc., 26 (1994), 455-458. doi: 10.1112/blms/26.5.455. [18] I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus," Springer-Verlag, Berlin, 1991. doi: 10.1007/978-1-4612-0949-2. [19] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Academic Press, Boston, 1993. [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), 11-28. doi: 10.1016/j.jmaa.2010.10.053. [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), 523-545. doi: 10.3934/dcds.2009.24.523. [22] X. Li, D. Jiang and X. Mao, Population dynamical behavior of Lotka-Volterra system under regime switching, J. Comput. Appl. Math., 232 (2009), 427-448. doi: 10.1016/j.cam.2009.06.021. [23] M. Liu and K. Wang, Survival analysis of a stochastic cooperation system in a polluted environment, J. Biol. Syst., 19 (2011), 183-204. doi: 10.1142/S0218339011003877. [24] M. Liu and K. Wang, Persistence and extinction in stochastic non-autonomous logistic systems, J. Math. Anal. Appl., 375 (2011), 443-457. doi: 10.1016/j.jmaa.2010.09.058. [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), 1969-2012. doi: 10.1007/s11538-010-9569-5. [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), 2810-2816. doi: 10.1016/j.nonrwa.2009.10.005. [27] Z. Lu and Y. Takeuchi, Permanence and global stability for cooperative Lotka-Volterra diffusion systems, Nonlinear Anal., 19 (1992), 963-975. doi: 10.1016/0362-546X(92)90107-P. [28] Q. Luo and X. Mao, Stochastic population dynamics under regime switching II, J. Math. Anal. Appl., 355 (2009), 577-593. [29] X. Mao, S. Sabais and E. Renshaw, Asymptotic behavior of stochastic Lotka-Volterra model, J. Math. Anal. Appl., 287 (2003), 141-156. doi: 10.1016/S0022-247X(03)00539-0. [30] X. Mao and C. Yuan, "Stochastic Differential Equations with Markovian Switching," Imperial College Press, 2006. [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), 519-531. doi: 10.1006/jmaa.2000.6853. [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), 603-620. [33] H. L. Smith, On the asymptotic behavior of a class of deterministic models of cooperating species, SIAM J. Math. Anal., 46 (1986), 368-375. doi: 10.1137/0146025. [34] F. Wu and Y. Hu, Stochastic Lotka-Volterra system with unbounded distributed delay, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 275-288. doi: 10.3934/dcdsb.2010.14.275. [35] J. Zhao and J. Jiang, Average conditions for permanence and extinction in nonautonomous Lotka-Volterra system, J. Math. Anal. Appl., 229 (2004), 663-675. doi: 10.1016/j.jmaa.2004.06.019. [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), 265-276. doi: 10.1016/S1468-1218(03)00038-5.

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), 17-43. doi: 10.1111/j.1467-9590.2007.00362.x. [2] S. Ahmad, Extinction of species in nonautonomous Lotka-Volterra system, Proc. Am. Math. Soc., 127 (1999), 2905-2910. doi: 10.1090/S0002-9939-99-05083-2. [3] S. Ahmad and A. C. Lazer, Average conditions for global asymptotic stability in a nonautonomous Lotka-Volterra system, Nonlinear Anal., 40 (2000), 37-49. doi: 10.1016/S0362-546X(00)85003-8. [4] E. S. Allman and J. A. Rhodes, "Mathematical Models in Biology: An Introduction," Cambridge University Press, 2004. [5] A. Bahar and X. Mao, Stochastic delay population dynamics, International J. Pure and Applied in Math., 11 (2004), 377-400. [6] I. Barbalat, Systems dequations differentielles d'osci d'oscillations nonlineaires, Revue Roumaine de Mathematiques Pures et Appliquees, 4 (1959), 267-270. [7] A. Berman and R. J. Plemmons, "Nonnegative Matrices in the Mathematical Science," Academic Press, New York, 1979. [8] S. Cheng, Stochastic population systems, Stoch. Anal. Appl., 27 (2009), 854-874. doi: 10.1080/07362990902844348. [9] B. S. Goh, Stability in models of mutualism, Amer. Natural, 113 (1979), 261-275. doi: 10.1086/283384. [10] K. Golpalsamy, "Stability and Oscillations in Delay Differential Equations of Population Dynamics," Kluwer Academic, Dordrecht, 1992. [11] T. G. Hallam and Z. Ma, Persistence in population models with demographic fluctuations, J. Math. Biol., 24 (1986), 327-339. doi: 10.1007/BF00275641. [12] X. He and K. Gopalsamy, Persistence, attractivity, and delay in facultative mutualism, J. Math. Anal. Appl., 215 (1997), 154-173. doi: 10.1006/jmaa.1997.5632. [13] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic diffrential equations, SIAM Rev., 43 (2001), 525-546. doi: 10.1137/S0036144500378302. [14] Y. Hu, F. Wu and C. Huang, Stochastic Lotka-Volterra models with multiple delays, J. Math. Anal. Appl., 375 (2011), 42-57. doi: 10.1016/j.jmaa.2010.08.017. [15] V. Hutson and K. Schmitt, Permanence and the dynamics of biological systems, Math. Biosci., 111 (1992), 1-71. doi: 10.1016/0025-5564(92)90078-B. [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), 588-597. doi: 10.1016/j.jmaa.2007.08.014. [17] J. Jiang, On the global stability of cooperative systems, Bull. Lond. Math. Soc., 26 (1994), 455-458. doi: 10.1112/blms/26.5.455. [18] I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus," Springer-Verlag, Berlin, 1991. doi: 10.1007/978-1-4612-0949-2. [19] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Academic Press, Boston, 1993. [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), 11-28. doi: 10.1016/j.jmaa.2010.10.053. [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), 523-545. doi: 10.3934/dcds.2009.24.523. [22] X. Li, D. Jiang and X. Mao, Population dynamical behavior of Lotka-Volterra system under regime switching, J. Comput. Appl. Math., 232 (2009), 427-448. doi: 10.1016/j.cam.2009.06.021. [23] M. Liu and K. Wang, Survival analysis of a stochastic cooperation system in a polluted environment, J. Biol. Syst., 19 (2011), 183-204. doi: 10.1142/S0218339011003877. [24] M. Liu and K. Wang, Persistence and extinction in stochastic non-autonomous logistic systems, J. Math. Anal. Appl., 375 (2011), 443-457. doi: 10.1016/j.jmaa.2010.09.058. [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), 1969-2012. doi: 10.1007/s11538-010-9569-5. [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), 2810-2816. doi: 10.1016/j.nonrwa.2009.10.005. [27] Z. Lu and Y. Takeuchi, Permanence and global stability for cooperative Lotka-Volterra diffusion systems, Nonlinear Anal., 19 (1992), 963-975. doi: 10.1016/0362-546X(92)90107-P. [28] Q. Luo and X. Mao, Stochastic population dynamics under regime switching II, J. Math. Anal. Appl., 355 (2009), 577-593. [29] X. Mao, S. Sabais and E. Renshaw, Asymptotic behavior of stochastic Lotka-Volterra model, J. Math. Anal. Appl., 287 (2003), 141-156. doi: 10.1016/S0022-247X(03)00539-0. [30] X. Mao and C. Yuan, "Stochastic Differential Equations with Markovian Switching," Imperial College Press, 2006. [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), 519-531. doi: 10.1006/jmaa.2000.6853. [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), 603-620. [33] H. L. Smith, On the asymptotic behavior of a class of deterministic models of cooperating species, SIAM J. Math. Anal., 46 (1986), 368-375. doi: 10.1137/0146025. [34] F. Wu and Y. Hu, Stochastic Lotka-Volterra system with unbounded distributed delay, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 275-288. doi: 10.3934/dcdsb.2010.14.275. [35] J. Zhao and J. Jiang, Average conditions for permanence and extinction in nonautonomous Lotka-Volterra system, J. Math. Anal. Appl., 229 (2004), 663-675. doi: 10.1016/j.jmaa.2004.06.019. [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), 265-276. doi: 10.1016/S1468-1218(03)00038-5.
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