Coexistence of competing species with flocculation in an unstirred chemostat. (English) Zbl 1505.92169
Summary: In this paper, an analysis is given of a mathematical model of two competing species feeding on one single resource growing in the chemostat, and the more superior species can flocculate. We firstly establish the existence and uniqueness of the positive solution to the single-species model with flocculation. In the case that the superior species flocculates, by degree theory the existence of positive steady-state solution is established. It turns out that for small attachment, all species may coexist.
References:
| [1] | Hsu, S. B., Limiting behavior for competing species, SIAM J. Appl. Math., 34, 760-763 (1978) · Zbl 0381.92014 |
| [2] | H.L. Smith, P. Waltman, The Theory of the Chemostat, Dynamics of Microbial Competition, in: Cambridge Studies in Mathematical Biology, vol. 13, Cambridge, 1995. · Zbl 0860.92031 |
| [3] | Hsu, S. B.; Waltman, P., On a system of reaction-diffusion equations arising from competition in an unstirred chemostat, SIAM J. Appl. Math., 53, 1026-1044 (1993) · Zbl 0784.92024 |
| [4] | Wu, J. H., Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Anal., 39, 817-835 (2000) · Zbl 0940.35114 |
| [5] | Shi, J. P.; Wu, Y. X.; Zou, X. F., Coexistence of competing species for intermediate dispersal rates in a reaction-diffusion chemostat model, J. Dynam. Differential Equations, 32, 1085-1112 (2020) · Zbl 1446.35055 |
| [6] | Fekih-Salem, R., Extensions of the chemostat model with flocculation, J. Math. Anal. Appl., 397, 292-306 (2013) · Zbl 1275.92014 |
| [7] | Haegeman, B.; Lobry, C.; Harmand, J., Modeling bacteria flocculation as density-dependent growth, AIChE J., 53, 535-539 (2007) |
| [8] | Fekih-Salem, R.; Sari, T.; Rapaport, A., La flocculation et la coexistence dans le chemostat, (Hassine, M.; Moakher, M., Proceedings of the 5th Conference on Trends in Applied Mathematics. Proceedings of the 5th Conference on Trends in Applied Mathematics, Tunisia, Algeria, Morocco, Sousse, 23-26 April 2011 (2011), Centre de Publication Universitaire: Centre de Publication Universitaire Tunisia), 477-483 |
| [9] | Haegeman, B.; Rapaport, A., How flocculation can explain coexistence in the chemostat, J. Biol. Dyn., 2, 1-13 (2008) · Zbl 1140.92320 |
| [10] | Fekih-Salem, R.; Rapaport, A.; Sari, T., Emergence of coexistence and limit cycles in the chemostat model with flocculation for a general class of functional responses, Appl. Math. Model., 40, 7656-7677 (2016) · Zbl 1471.92223 |
| [11] | Guo, S. B.; Ma, W. B.; Zhao, X. Q., Global dynamics of a time-delayed microorganism flocculation model with saturated functional responses, J. Dynam. Differential Equations, 30, 1247-1271 (2018) · Zbl 1403.37094 |
| [12] | Nie, H.; Hsu, S. B.; Wu, J. H., A competition model with dynamically allocated toxin production in the unstirred chemostat, Commun. Pure Appl. Anal., 16, 1373-1404 (2017) · Zbl 1360.35302 |
| [13] | Jiang, D. H.; Nie, H.; Wu, J. H., Crowding effects on coexistence solutions in the unstirred chemostat, Appl. Anal., 96, 1016-1046 (2017) · Zbl 1365.35071 |
| [14] | Smoller, J., Shock Waves and Reaction-Diffusion Equations (1994), Springer-Verlag: Springer-Verlag New York · Zbl 0807.35002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
