On a non-autonomous stochastic Lotka-Volterra competitive system

Volume 10, Issue 6, pp 3099--3108 http://dx.doi.org/10.22436/jnsa.010.06.24

Publication Date: June 25, 2017 Submission Date: March 06, 2017

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Authors

Meiling Deng - School of Mathematical Science, Huaiyin Normal University, Huaian 223300, P. R. China.

Abstract

In this paper, we consider a general non-autonomous Lotka-Volterra competitive model with random perturbations. Sufficient conditions for stochastic permanence and extinction are established. Particularly, when these conditions are applied to a stochastic logistic equation, these conditions are sufficient and necessary. Some figures are also worked out to illustrate the main results. Some recent results are extended. Moreover, our results reveal that different types of stochastic noises have different effects on the permanence and extinction of the population.

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Keywords

• Competitive system
• random perturbations
• permanence
• extinction.

MSC

•  92D25
•  60H10
•  60H30

References

• [1] S. Ahmad, On the nonautonomous Volterra-Lotka competition equations, Proc. Amer. Math. Soc., 117 (1993), 199–204.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] S. Ahmad, Extinction of species in nonautonomous Lotka-Volterra systems, Proc. Amer. Math. Soc., 127 (1999), 2905– 2910.
  • View Article
  • MathSciNet
  • Google Scholar


• [3] S. Ahmad, I. M. Stamova, Asymptotic stability of an N-dimensional impulsive competitive system, Nonlinear Anal. Real World Appl., 8 (2007), 654–663.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] A. Bahar, X.-R. Mao, Stochastic delay Lotka-Volterra model, J. Math. Anal. Appl., 292 (2004), 364–380.
  • View Article
  • Google Scholar


• [5] C.-Z. Bai, Multiplicity of solutions for a class of non-local elliptic operators systems, Bull. Korean Math. Soc., 54 (2017), 715–729.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] M. Bandyopadhyay, J. Chattopadhyay, Ratio-dependent predator-prey model: effect of environmental fluctuation and stability, Nonlinearity, 18 (2005), 913–936.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] J.-H. Bao, X.-R. Mao, G. Yin, C.-G. Yuan, Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal., 74 (2011), 6601–6616.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] Y.-L. Cai, Y. Kang, M. Banerjee, W.-M. Wang, A stochastic SIRS epidemic model with infectious force under intervention strategies, J. Differential Equations, 259 (2015), 7463–7502.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] Y.-L. Cai, Y. Kang, M. Banerjee, W.-M. Wang, A stochastic epidemic model incorporating media coverage, Commun. Math. Sci., 14 (2016), 893–910.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] N. H. Du, V. H. Sam, Dynamics of a stochastic Lotka-Volterra model perturbed by white noise, J. Math. Anal. Appl., 324 (2006), 82–97.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] T. C. Gard, Persistence in stochastic food web models, Bull. Math. Biol., 46 (1984), 357–370.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] T. C. Gard, Stability for multispecies population models in random environments, Nonlinear Anal., 10 (1986), 1411–1419.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] K. Gopalsamy, Stability and oscillations in delay differential equations of population dynamics, Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht (1992)
  • View Article
  • MathSciNet


• [14] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525–546.
  • View Article
  • MathSciNet
  • Google Scholar


• [15] J. Hou, Z.-D. Teng, S.-J. Gao, Permanence and global stability for nonautonomous N-species Lotka-Valterra competitive system with impulses, Nonlinear Anal. Real World Appl., 11 (2010), 1882–1896.
  • View Article
  • MathSciNet
  • Google Scholar


• [16] D.-Q. Jiang, C.-Y. Ji, X.-Y. Li, D. O’Regan, Analysis of autonomous Lotka-Volterra competition systems with random perturbation, J. Math. Anal. Appl., 390 (2012), 582–595.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] Y. Kuang, Delay differential equations with applications in population dynamics, Mathematics in Science and Engineering, Academic Press, Inc., Boston, MA (1993)
  • MathSciNet
  • Google Scholar


• [18] X.-Y. Li, X.-R. Mao, Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation, Discrete Contin. Dyn. Syst., 24 (2009), 523–545.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [19] M. Liu, C.-Z. Bai, Optimal harvesting of a stochastic logistic model with time delay, J. Nonlinear Sci., 25 (2015), 277–289.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] M. Liu, C.-Z. Bai, Analysis of a stochastic tri-trophic food-chain model with harvesting, J. Math. Biol., 73 (2016), 597–625.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] M. Liu, C.-Z. Bai, Dynamics of a stochastic one-prey two-predator model with Lévy jumps, Appl. Math. Comput., 284 (2016), 308–321.
  • View Article
  • MathSciNet
  • Google Scholar


• [22] M. Liu, C.-Z. Bai, Optimal harvesting of a stochastic delay competitive model, Discrete Contin. Dyn. Syst., 22 (2017), 1493–1508.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [23] M. Liu, C.-Z. Bai, Y. Jin, Population dynamical behavior of a two-predator one-prey stochastic model with time delay, Discrete Contin. Dyn. Syst., 37 (2017), 2513–2538.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [24] S.-Q. Liu, L.-S. Chen, Z.-J. Liu, Extinction and permanence in nonautonomous competitive system with stage structure, J. Math. Anal. Appl., 274 (2002), 667–684.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] M. Liu, M. Fan, Permanence of stochastic Lotka-Volterra systems, J. Nonlinear Sci., 27 (2017), 425–452.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] M. Liu, M. Fan, Stability in distribution of a three-species stochastic cascade predator-prey system with time delays, IMA J. Appl. Math., 82 (2017), 396–423.
  • View Article
  • MathSciNet
  • Google Scholar


• [27] M. Liu, K. Wang, Persistence and extinction in stochastic non-autonomous logistic systems, J. Math. Anal. Appl., 375 (2011), 443-457.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [28] M. Liu, K. Wang, Stochastic Lotka-Volterra systems with Lévy noise, J. Math. Anal. Appl., 410 (2014), 750–763.
  • View Article
  • Google Scholar


• [29] M. Liu, K. Wang, Q. Wu, Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle, Bull. Math. Biol., 73 (2011), 1969–2012.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [30] Q. Luo, X.-R. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl., 334 (2007), 69–84.
  • View Article
  • Google Scholar


• [31] X.-R. Mao, Stochastic differential equations and their applications, Horwood Publishing Series in Mathematics & Applications, Horwood Publishing Limited, Chichester (1997)
  • Google Scholar


• [32] X.-R. Mao, G. Marion, E. Renshaw, Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Process. Appl., 97 (2002), 95–110.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [33] X.-R. Mao, S. Sabanis, E. Renshaw, Asymptotic behaviour of the stochastic Lotka-Volterra model, J. Math. Anal. Appl., 287 (2003), 141–156.
  • View Article
  • MathSciNet
  • Google Scholar


• [34] R. M. May, Stability and complexity in model ecosystems, Princeton Univ. Press, Princeton (2001)
  • Google Scholar


• [35] F. Montesf de Oca, M. L. Zeeman, Extinction in nonautonomous competitive Lotka-Volterra systems, Proc. Amer. Math. Soc., 124 (1996), 3677–3687.
  • View Article
  • MathSciNet
  • Google Scholar


• [36] S.-L. Pang, F.-Q. Deng, X.-R. Mao, Asymptotic properties of stochastic population dynamics, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 15 (2008), 603–620.
  • MathSciNet
  • Google Scholar


• [37] Z.-D. Teng, On the non-autonomous Lotka-Volterra N-species competing systems, Appl. Math. Comput., 114 (2000), 175–185.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [38] J.-D. Zhao, J.-F. Jiang, A. C. Lazer, The permanence and global attractivity in a nonautonomous Lotka-Volterra system, Nonlinear Anal. Real World Appl., 5 (2004), 265–276.
  • View Article
  • MathSciNet
  • Google Scholar


• [39] Y. Zhu, M. Liu, Permanence and extinction in a stochastic service-resource mutualism model, Appl. Math. Lett., 69 (2017), 1–7.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [40] C. Zhu, G. Yin, On competitive Lotka-Volterra model in random environments, J. Math. Anal. Appl., 357 (2009), 154–170.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH