The threshold of stochastic chemostat model with Monod-Haldane response function

Volume 10, Issue 8, pp 4364--4371 http://dx.doi.org/10.22436/jnsa.010.08.29

Publication Date: August 24, 2017 Submission Date: February 21, 2017

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Authors

Zhongwei Cao - Department of Applied Mathematics, Jilin University of Finance and Economics, Changchun 130117, P. R. China. Liya Liu - College of Science, China University of Petroleum, Qingdao 266580, Shandong Province, P. R. China.

Abstract

This paper deals with problem of a stochastic chemostat model with Monod-Haldane response function. Firstly, we confirm the truth of the existence and uniqueness of the positive solution to the system. Then, we show the condition for the microorganism to be extinct. Moreover, we investigate there is a stationary distribution of this stochastic system and finally, we derive the expression for its invariant density.

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Keywords

• Stochastic chemostat model
• threshold
• extinction
• persistence
• stationary distribution.

MSC

•  60H10
•  93E03
•  93E15

References

• [1] L. Arnold, Stochastic differential equations: theory and applications, Translated from the German, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney (1972)


• [2] L. S. Chen, J. Chen, Nonlinear biodynamical system, Science Press, Beijing (1993)
  • Google Scholar


• [3] D. Herbert, R. Elsworth, R. C. Telling, The continuous culture of bacteria; a theoretical and experimental study, Microbiology, 14 (1956), 601–622.
  • View Article
  • Google Scholar


• [4] T. W. James, Continuous culture of microorganisms, Ann. Rev. Microbiol., 15 (1961), 27–46.
  • View Article
  • Google Scholar


• [5] I. Karatzas, S. E. Shreve, Brownian motion and stochastic calculus, Graduate Texts in Mathematics, Springer-Verlag, New York (1988)
  • Google Scholar


• [6] A. Y. Kutoyants, Statistical inference for ergodic diffusion processes, Springer Series in Statistics, Springer-Verlag London, Ltd., London (2004)
  • View Article
  • MathSciNet
  • Google Scholar


• [7] Y.-G. Lin, D.-Q. Jiang, P.-Y. Xia, Long-time behavior of a stochastic SIR model, Appl. Math. Comput., 236, 1–9. (2014)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] 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


• [9] M. Liu, C.-Z. Bai, Optimal harvesting of a stochastic mutualism model with Lévy jumps, Appl. Math. Comput., 276 (2016), 301–309.
  • View Article
  • MathSciNet
  • Google Scholar


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


• [11] 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


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


• [13] 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


• [14] J.-L. Lv, K. Wang, Almost sure permanence of stochastic single species models, J. Math. Anal. Appl., 422 (2015), 675–683.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


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


• [16] J. Monod, La technique de culture continue: theorie et applications, Annales de I Institut Pasteur, 79 (1950), 390–410.
  • Google Scholar


• [17] A. Novick, L. Szilard, Description of the chemostat, Science, 112 (1950), 715–716.
  • View Article
  • Google Scholar


• [18] B. Øksendal, Stochastic differential equations, An introduction with applications, Sixth edition, Universitext, Springer-Verlag, Berlin (2003)
  • View Article
  • Google Scholar


• [19] C.-Q. Xu, S.-L. Yuan, An analogue of break-even concentration in a simple stochastic chemostat model, Appl. Math. Lett., 48 (2015), 62–68.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] Q.-S. Yang, D.-Q. Jiang, A note on asymptotic behaviors of stochastic population model with Allee effect, Appl. Math. Model., 35 (2011), 4611–4619.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH