Taiwanese Journal of Mathematics

Bifurcation and Stability for the Unstirred Chemostat Model with Beddington-DeAngelis Functional Response

Shanbing Li, Jianhua Wu, and Yaying Dong

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In this paper, we consider a basic $N$-dimensional competition model in the unstirred chemostat with Beddington-DeAngelis functional response. The bifurcation solutions from a simple eigenvalue and a double eigenvalue are obtained respectively. In particular, for the double eigenvalue, we establish the existence and stability of coexistence solutions by the techniques of space decomposition and Lyapunov-Schmidt procedure. Moreover, we describe the global structure of these bifurcation solutions.

Article information

Taiwanese J. Math., Volume 20, Number 4 (2016), 849-870.

First available in Project Euclid: 1 July 2017

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Zentralblatt MATH identifier

Primary: 35K55: Nonlinear parabolic equations 35K57: Reaction-diffusion equations 35B40: Asymptotic behavior of solutions

chemostat double eigenvalue bifurcation stability


Li, Shanbing; Wu, Jianhua; Dong, Yaying. Bifurcation and Stability for the Unstirred Chemostat Model with Beddington-DeAngelis Functional Response. Taiwanese J. Math. 20 (2016), no. 4, 849--870. doi:10.11650/tjm.20.2016.5482. https://projecteuclid.org/euclid.twjm/1498874494

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  • J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol. 44 (1975), no. 1, 331–340.
  • G. J. Butler and G. S. K. Wolkowicz, A mathematical model of the chemostat with a general class of functions describing nutrient uptake, SIAM J. Appl. Math. 45 (1985), no. 1, 138–151.
  • M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis 8 (1971), no. 2, 321–340.
  • E. N. Dancer, On the structure of solutions of non-linear eigenvalue problems, Indiana Univ. Math. J. 23 (1974), 1069–1076.
  • D. L. DeAngelis, R. A. Goldstein and R. V. ONeill, A model for trophic interaction, Ecology 56 (1975), 881–892.
  • L. Dung and H. L. Smith, A parabolic system modeling microbial competition in an unmixed bio-reactor, J. Differential Equations 130 (1996), no. 1, 59–91.
  • G. Guo, J. Wu and Y. Wang, Bifurcation from a double eigenvalue in the unstirred chemostat, Appl. Anal. 92 (2013), no. 7, 1449–1461.
  • D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics 840, Springer-Verlag, Berlin, New York, 1981.
  • S.-B. Hsu and P. Waltman, On a system of reaction-diffusion equations arising from competition in an unstirred Chemostat, SIAM J. Appl. Math. 53 (1993), no. 4, 1026–1044.
  • ––––, A survey of mathematical models of competition with an inhibitor, Math. Biosci. 187 (2004), no. 1, 53–91.
  • M. Ito, Global aspect of steady-states for competitive-diffusive systems with homogeneous Dirichlet conditions, Phys. D 14 (1984), no. 1, 1–28.
  • T. Kato, Perturbation Theory for Linear Operators, Second edition, Springer-Verlag, Berlin, New York, 1976.
  • J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Chapman & Hall/CRC Research Notes in Mathematics 426, Chapman & Hall/CRC, Boca Raton, FL, 2001.
  • J. Monod, Recherces sur la Croissance des Cultures Bacteriennes, Paris: Hermann et Cie, 1942.
  • H. Nie and J. Wu, A system of reaction-diffusion equations in the unstirred chemostat with an inhibitor, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 16 (2006), no. 4, 989–1009.
  • ––––, Uniqueness and stability for coexistence solutions of the unstirred chemostat model, Appl. Anal. 89 (2010), no. 7, 1141–1159.
  • ––––, Coexistence of an unstirred chemostat model with Beddington-DeAngelis functional response and inhibitor, Nonlinear Anal. Real World Appl. 11 (2010), no.`5, 3639–3652.
  • ––––, Multiplicity results for the unstirred chemostat model with general response functions, Sci. China Math. 56 (2013), no. 10, 2035–2050.
  • H. Nie, J. H. Wu, Y. E. Wang and Z. G. Wang, Dynamics of Reaction-Diffusion Models (in Chinese), Beijing: Science Press, 2013.
  • Z. Qui, J. Yu and Y. Zou, The asymptotic behavior of a chemostat model with the Beddington-DeAngelis functional response, Math. Biosci. 187 (2004), no. 2, 175–187.
  • P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), no. 3, 487–513.
  • H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University, Cambridge, 1995.
  • J. W.-H. So and P. Waltman, A nonlinear boundary value problem arising from competition in the chemostat, Appl. Math. Comput. 32 (1989), no. 2-3, 169–183.
  • O. Tagashira, Permanent coexistence in chemostat models with delayed feedback control, Nonlinear Anal. Real World Appl. 10 (2009), no. 3, 1443–1452.
  • Y. E. Wang and J. H. Wu, Existence and stability of positive steady-state solutions for a chemostat model, J. Shaanxi Normal Univ. Nat. Sci. Ed. 34 (2006), no. 2, 9–12.
  • Y. Wang, J. Wu and G. Guo, Coexistence and stability of an unstirred chemostat model with Beddington-DeAngelis function, Comput. Math. Appl. 60 (2010), no. 8, 2497–2507.
  • Y. Wang, J. Wu and H. Nie, The global bifurcation and asymptotic behavior of a competition model in the unstirred chemostat, Acta Math. Sinica (Chin. Ser.) 54 (2011), 397–408.
  • G. S. K. Wolkowicz and Z. Q. Lu, Global dynamics of a mathematical model of competition in the chemostat: general response functions and differential death rates, SIAM J. Appl. Math. 52 (1992), no. 1, 222–233.
  • J. H. Wu, Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Anal. 39 (2000), no. 7, Ser. A: Theory Methods, 817–835.
  • J. Wu, H. Nie and G. S. K. Wolkowicz, The effect of inhibitor on the plasmid-bearing and plasmid-free model in the unstirred chemostat, SIAM J. Math. Anal. 38 (2007), no. 6, 1860–1885.
  • Y. Yamada, Stability of steady states for prey-predator diffusion equations with homogeneous Dirichlet conditions, SIAM J. Math. Anal. 21 (1990), no. 2, 327–345.
  • Q. X. Ye and Z. Y. Li, Introduction to Reaction-Diffusion Equations (in Chinese), Beijing, Science Press, 1990.
  • S. Zhang and D. Tan, Study of a chemostat model with Beddington-DeAngelis functional response and pulsed input and washout at different times, J. Math. Chem. 44 (2008), no. 1, 217–227.
  • S. Zhang, D. Tan and L. Chen, Chaotic behavior of a chemostat model with Beddington-DeAngelis functional response and periodically impulsive invasion, Chaos Solitons Fractals 29 (2006), no. 2, 474–482.
  • H. Zhang, P. Georgescu, J. J. Nieto and L.-S. Chen, Impulsive perturbation and bifurcation of solutions for a model of chemostat with variable yield, Appl. Math. Mech. (English Ed.) 30 (2009), no. 7, 933–944.