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Competitive Exclusion in a General Multi-species Chemostat Model with Stochastic Perturbations

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Abstract

Based on the fact that the continuous culture of microorganisms in a chemostat is subject to environmental noises, we present and analyze a stochastic competition chemostat model with general monotonic response functions and differential removal rates. The existence and boundedness of the unique positive solution are first obtained. By defining a stochastic break-even concentration for every species, we prove that at most one competitor survives in the chemostat and the winner has the smallest stochastic break-even concentration, provided its response function satisfies a technical assumption. That is to say, the competitive exclusion principle holds for the stochastic competition chemostat model. Furthermore, we find that the noise experienced by one species is adverse to its growth while may be favorable for the growth of other one species. Namely, the destinies can be exchanged between two microorganism species in the chemostat due to the environmental noise.

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References

  • Beddington JR, May RM (1977) Harvesting natural populations in a randomly fluctuating environment. Science 197:463–465

    Article  Google Scholar 

  • Butler GJ, Wolkowicz GSK (1985) A mathematical model of the chemostat with a general class of functions describing nutrient uptake. SIAM J Appl Math 45:138–151

    Article  MathSciNet  Google Scholar 

  • Campillo F, Joannides M, Larramendy-Valverde I (2011) Stochastic modeling of the chemostat. Ecol Model 222:2676–2689

    Article  Google Scholar 

  • De Leenheer P, Li B, Smith H (2003) Competition in the chemostat: some remarks. Can Appl Math Q 11:229–248

    MathSciNet  MATH  Google Scholar 

  • Ellermeyer SF (1994) Competition in the chemostat: global asymptotic behavior of a model with delayed response in growth. SIAM J Appl Math 54:456–465

    Article  MathSciNet  Google Scholar 

  • Grasman J, Gee MD, Herwaarden OAV (2005) Breakdown of a chemostat exposed to stochastic noise. J Eng Math 53:291–300

    Article  MathSciNet  Google Scholar 

  • Hardin G et al (1960) The competitive exclusion principle. Science 131:1292–1297

    Article  Google Scholar 

  • Higham DJ (2001) An algorithmic introduction to numerical simulation of stochastic diferential equations. SIAM Rev 43:525–546

    Article  MathSciNet  Google Scholar 

  • Hsu SB, Hubbell S, Waltman P (1977) A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms. SIAM J Appl Math 32:366–383

    Article  MathSciNet  Google Scholar 

  • Imhof L, Walcher S (2005) Exclusion and persistence in deterministic and stochastic chemostat models. J Differ Equ 217:26–53

    Article  MathSciNet  Google Scholar 

  • Ji C, Jiang D, Shi N (2011) A note on a predator-prey model with modified Leslie–Gower and Holling-type II schemes with stochastic perturbation. J Math Anal Appl 377:435–440

    Article  MathSciNet  Google Scholar 

  • Kutoyants DYA (2003) Statistical inference for ergodic diffusion processes. Springer, London

    MATH  Google Scholar 

  • Li B (1998) Global asymptotic behaviour of the chemostat: general response functions and different removal rates. SIAM J Appl Math 59:411–422

    Article  Google Scholar 

  • Liu S, Wang X, Wang L, Song H (2014) Competitive exclusion in delayed chemostat models with differential removal rates. SIAM J Appl Math 74:634–648

    Article  MathSciNet  Google Scholar 

  • Mao X (2007) Stochastic differential equations and applications, 2nd edn. Horwood Publishing, Chichester

    MATH  Google Scholar 

  • Smith HL, Waltman P (1995) The theory of the chemostat: dynamics of microbial competition. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Wang L, Wolkowicz GSK (2006) A delayed chemostat model with general nonmonotone response functions and differential removal rates. J Math Anal Appl 321:452–468

    Article  MathSciNet  Google Scholar 

  • Wang L, Jiang D, O’Regan D (2016) The periodic solutions of a stochastic chemostat model with periodic washout rate. Commun Nonlinear Sci Numer Simul 37:1–13

    Article  MathSciNet  Google Scholar 

  • Wang L, Jiang D, Wolkowicz GSK (2020) Global asymptotic behavior of a multi-species stochastic chemostat model with discrete delays. J Dyn Differ Equ 32:849–872

    Article  MathSciNet  Google Scholar 

  • Wang W, Ma W, Feng Z (2020) Global dynamics and travelling waves for a periodic and diffusive chemostat model with two nutrients and one microorganism. Nonlinearity 33:4338–4380

    Article  MathSciNet  Google Scholar 

  • Wolkowicz GSK, Lu Z (1992) Global dynamics of a mathematical model of competition in the chemostat: general response functions and differential death rates. SIAM J Appl Math 52:222–233

    Article  MathSciNet  Google Scholar 

  • Wolkowicz GSK, Xia H (1997) Global asymptotic behavior of a chemostat model with discrete delays. SIAM J Appl Math 57:1019–1043

    Article  MathSciNet  Google Scholar 

  • Wolkowicz GSK, Xia H, Ruan S (1997) Competition in the chemostat: a distributed delay model and its global asymptotic behavior. SIAM J Appl Math 57:1281–1310

    Article  MathSciNet  Google Scholar 

  • Wolkowicz GSK, Xia H, Wu J (1999) Global dynamics of a chemostat competition model with distributed delay. J Math Biol 38:285–316

    Article  MathSciNet  Google Scholar 

  • Xu C, Yuan S (2015) An analogue of break-even concentration in an simple stochastic chemostat model. Appl Math Lett 48:62–68

    Article  MathSciNet  Google Scholar 

  • Xu C, Yuan S (2016) Competition in the chemostat: a stochastic multi-species model and its asymptotic behavior. Math Biosci 280:1–9

    Article  MathSciNet  Google Scholar 

  • Xu C, Yuan S, Zhang T (2018) Average break-even concentration in a simple chemostat model with telegraph noise. Nonlinear Anal-Hybrid 29:373–382

    Article  MathSciNet  Google Scholar 

  • Yu X, Yuan S (2020) Asymptotic properties of a stochastic chemostat model with two distributed delays and nonlinear perturbations. Discrete Contin Dyn Syst B 25:2373–2390

    Article  MathSciNet  Google Scholar 

  • Yuan S, Wu D, Lan G, Wang H (2020) Noise-induced transitions in a nonsmooth producerCgrazer model with stoichiometric constraints. Bull Math Biol 82:55

    Article  Google Scholar 

  • Zhang Q, Jiang D (2016) Competitive exclusion in a stochastic chemostat model with Holling type II functional response. J Math Chem 54:777–791

    Article  MathSciNet  Google Scholar 

  • Zhang T, Chen Z, Han M (2014) Dynamical analysis of a stochastic model for cascaded continuous flow bioreactors. J Math Chem 52:1441–1459

    Article  MathSciNet  Google Scholar 

  • Zhao D, Yuan S (2016) Critical result on the break-even concentration in a single-species stochastic chemostat model. J Math Anal Appl 434:1336–1345

    Article  MathSciNet  Google Scholar 

  • Zhao Y, Yuan S, Ma J (2015) Survival and stationary distribution analysis of a stochastic competitive model of three species in a polluted environment. Bull Math Biol 77:1285–1326

    Article  MathSciNet  Google Scholar 

  • Zhao S, Yuan S, Wang H (2020) Threshold behavior in a stochastic algal growth model with stoichiometric constraints and seasonal variation. J Differ Equ 268:5113–5139

    Article  MathSciNet  Google Scholar 

  • Zhu C, Yin G (2007) Asymptotic properties of hybrid diffusion systems. SIAM J Control Optim 46:1155–1179

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China

    Chaoqun Xu & Sanling Yuan

  2. School of Mathematical Science, Jiangsu University, Zhenjiang, 212013, Jiangsu, China

    Chaoqun Xu

  3. Department of Mathematics, Swinburne University of Technology, Hawthorn, VIC, 3122, Australia

    Tonghua Zhang

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  1. Chaoqun Xu
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  2. Sanling Yuan
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  3. Tonghua Zhang
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Correspondence to Sanling Yuan.

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Research is supported by the National Natural Science Foundation of China (11801224, 11671260, 12071293), and Natural Science Foundation of Jiangsu Province (BK20180856).

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Xu, C., Yuan, S. & Zhang, T. Competitive Exclusion in a General Multi-species Chemostat Model with Stochastic Perturbations. Bull Math Biol 83, 4 (2021). https://doi.org/10.1007/s11538-020-00843-7

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