Abstract
A system of reaction-diffusion equations arising from the unstirred chemostat model with ratio-dependent function is considered. The asymptotic behavior of solutions is given and all positive steady-state solutions to this model lie on a single smooth solution curve. It turns out that the ratio-dependence effect will not affect the dynamics, compared with (Hsu and Waltman in SIAM J. Appl. Math. 53(4):1026–1044, 1993) and (Nie and Wu in Sci. China Math. 56(10):2035–2050, 2013).
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References
Smith, H.L., Waltman, P.: The Theory of the Chemostat: Dynamics of Microbial Competition, vol. 13. Cambridge University Press, Cambridge (1995)
Waltman, P.: Competition Models in Population Biology. SIAM, Philadelphia (1983)
Hsu, S.B., Waltman, P.: On a system of reaction-diffusion equations arising from competition in an unstirred chemostat. SIAM J. Appl. Math. 53(4), 1026–1044 (1993)
Monod, J.: Recherches sur la croissance des cultures bactériennes. Hermann, Paris (1942)
Andrews, J.F.: A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates. Biotechnol. Bioeng. 10(6), 707–723 (1968)
Sokol, W., Howell, J.A.: Kinetics of phenol oxidation by washed cells. Biotechnol. Bioeng. 23(9), 2039–2049 (1981)
Wang, Y., Wu, J., Guo, G.: Coexistence and stability of an unstirred chemostat model with Beddington-Deangelis function. Comput. Math. Appl. 60(8), 2497–2507 (2010)
Nie, H., Wu, J.: Coexistence of an unstirred chemostat model with Beddington-Deangelis functional response and inhibitor. Nonlinear Anal., Real World Appl. 11(5), 3639–3652 (2010)
Yang, W., Li, Y., Wu, J., Li, H.: Dynamics of a food chain model with ratio-dependent and modified Leslie-Gower functional responses. Discrete Contin. Dyn. Syst., Ser. B 20(7), 2269–2290 (2015)
De Figueiredo, D.G., Gossez, J.P.: Strict monotonicity of eigenvalues and unique continuation. Commun. Partial Differ. Equ. 17(1–2), 339–346 (1992)
Wu, J.: Global bifurcation of coexistence state for the competition model in the chemostat. Nonlinear Anal., Real World Appl. 39(7), 817–835 (2000)
Baxley, J.V., Robinson, S.B.: Coexistence in the unstirred chemostat. Appl. Math. Comput. 89(1–3), 41–65 (1998)
Smoller, J.: Shock Waves and Reaction-Diffusion Equations, vol. 258. Springer, Berlin (1994)
Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Springer, Berlin (1984)
Nie, H., Wu, J.: Multiplicity results for the unstirred chemostat model with general response functions. Sci. China Math. 56(10), 2035–2050 (2013)
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The authors gratefully acknowledge the referee for his/her comments and suggestions, which have significantly improved not only the presentation but the overall quality of the paper.
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The work has been partially supported by the Natural Science Foundation of China (Grant Nos.: 11771262, 61872227), and the Natural Science Basic Research Plan in Shaanxi Province of China (No. 2018JQ1021).
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Yang, W. Existence and Asymptotic Behavior of Solutions for the Unstirred Chemostat Model with Ratio-Dependent Function. Acta Appl Math 166, 223–232 (2020). https://doi.org/10.1007/s10440-019-00264-2
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DOI: https://doi.org/10.1007/s10440-019-00264-2