Journal of Applied Mathematics and Computing

, Volume 43, Issue 1–2, pp 387–407 | Cite as

Asymptotic properties and simulations of a stochastic single-species dispersal model under regime switching

Original Research


Taking both white noise and colored environmental noise into account, a single-species logistic model with population’s nonlinear diffusion among two patches is proposed and investigated. The sufficient conditions of the existence of positive solutions, stochastic permanence, persistence in mean and extinction are established. Moreover, we use an example and simulation figures to illustrate our main results.


Stochastic permanence Persistent in mean Extinction Environment noise 

Mathematics Subject Classification

34F05 34E10 60H10 60H20 


  1. 1.
    Takeuchi, Y.: Cooperative system theory and global stability of diffusion models. Acta Appl. Math. 14, 49–57 (1989) MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Wang, W., Chen, L.: Global stability of a population dispersal in a two-patch environment. Dyn. Syst. Appl. 6, 207–216 (1997) MATHGoogle Scholar
  3. 3.
    Allen, L.: Persistence and extinction in single-species reaction-diffusion models. Bull. Math. Biol. 45, 209–227 (1983) MathSciNetMATHGoogle Scholar
  4. 4.
    Lu, Z., Takeuchi, Y.: Global asymptotic behavior in single-species discrete diffusion systems. J. Math. Biol. 32, 67–77 (1993) MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Allen, L.: Persistence, extinction, and critical patch number for island populations. Bull. Math. Biol. 65, 1–12 (1987) Google Scholar
  6. 6.
    Gard, T.: Persistence in stochastic food web models. Bull. Math. Biol. 46, 357–370 (1984) MathSciNetMATHGoogle Scholar
  7. 7.
    Liu, M., Wang, K.: Persistence and extinction in stochastic non-autonomous logistic systems. J. Math. Anal. Appl. 375, 443–457 (2011) MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Mao, X., Yuan, C., Zou, J.: Stochastic differential delay equations of population dynamics. J. Math. Anal. Appl. 304, 296–320 (2005) MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Jiang, D., Shi, N.: A note on non-autonomous logistic equation with random perturbation. J. Math. Anal. Appl. 303, 164–172 (2005) MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ji, C., Jiang, D., Liu, H., Yang, Q.: Existence, uniqueness and ergodicity of positive solution of mutualism system with stochastic perturbation. Math. Probl. Eng. 2010, 684926 (2010) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Luo, Q., Mao, X.: Stochastic population dynamics under regime switching. J. Math. Anal. Appl. 334, 69–84 (2007) MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Li, X., Gray, A., Jiang, D., Mao, X.: Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching. J. Math. Anal. Appl. 376, 11–28 (2011) MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Du, N., Kon, R., Sato, K., Takeuchi, Y.: Dynamical behavior of Lotka-Volterra competition systems: non-autonomous bistable case and the effect of telegraph noise. J. Comput. Appl. Math. 170, 399–422 (2004) MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Stakin, M.: The dynamics of a population in a Markovian environment. Ecology 59, 249–256 (1987) Google Scholar
  15. 15.
    Mao, X., Marion, G., Renshaw, E.: Environmental Brownian noise suppresses explosions in populations dynamics. Stoch. Process. Appl. 97, 95–110 (2002) MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Li, X., Jiang, D., Mao, X.: Population dynamical behavior of Lotka-Volterra system under regime switching. J. Comput. Appl. Math. 232, 427–448 (2009) MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Mao, X.: Differential Equations and Applications. Horwood, Chichester (1997) MATHGoogle Scholar
  18. 18.
    Mao, X., Yuan, C.: Stochastic Differential Equations with Markovian Switching. Imperial College Press, London (2006) CrossRefMATHGoogle Scholar
  19. 19.
    Highm, D.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43, 525–546 (2001) MathSciNetCrossRefGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2013

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsNortheast Normal UniversityChangchunP.R. China
  2. 2.School of ScienceChangchun UniversityChangchunP.R. China
  3. 3.School of Mathematics, Statistics and Applied MathematicsNational University of IrelandGalwayIreland

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