Periodic oscillations in a chemostat model with two discrete delays. (English) Zbl 1418.92080
Summary: Periodic oscillations of solutions of a chemostat-type model in which a species feeds on a limiting nutrient are considered. The model incorporates two discrete delays representing the lag in nutrient recycling and nutrient conversion. Through the study of characteristic equation associated with the linearized system, a unique positive equilibrium is found and proved to be locally asymptotically stable under some conditions. Meanwhile, a Hopf bifurcation causing periodic solutions is also obtained. Numerical simulations illustrate the theoretical results.
MSC:
| 92C99 | Physiological, cellular and medical topics |
| 92D40 | Ecology |
| 34K20 | Stability theory of functional-differential equations |
| 34K60 | Qualitative investigation and simulation of models involving functional-differential equations |
References:
| [1] | Egli, T., The ecology and physiological significance of the growth of heterotrophicmicroorganisms with mixtures of substrates, Advances in Microbial Ecology. Advances in Microbial Ecology, Advances in Microbial Ecology, 14, 305-386 (1995), New York, NY, USA: Springer US, New York, NY, USA · doi:10.1007/978-1-4684-7724-5_8 |
| [2] | Smith, H. L.; Waltman, P., The Theory of the Chemostat. The Theory of the Chemostat, Cambridge Studies in Mathematical Biology, 13 (1995), Cambridge, UK: Cambridge University, Cambridge, UK · Zbl 1139.92029 · doi:10.1017/cbo9780511530043 |
| [3] | Hsu, S. B.; Hubbell, S.; Waltman, P., A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, SIAM Journal on Applied Mathematics, 32, 2, 366-383 (1977) · Zbl 0354.92033 · doi:10.1137/0132030 |
| [4] | Waltman, P.; Hubbell, S. P.; Hsu, S. B.; Burton, T., Theoretical and experimental investigations of microbial competition in continuous culture, Modeling and Differential Equations in Biology, 107-152 (1980), New York, NY, USA: Dekker, New York, NY, USA · Zbl 0442.92015 |
| [5] | Yuan, S. L.; Xiao, D. M.; Han, M. A., Competition between plasmid-bearing and plasmid-free organisms in a chemostat with nutrient recycling and an inhibitor, Mathematical Biosciences, 202, 1, 1-28 (2006) · Zbl 1112.34062 · doi:10.1016/j.mbs.2006.04.003 |
| [6] | Nisbet, R. M.; McKinstry, J.; Gurney, W. S. C., A ‘strategic’ model of material cycling in a closed ecosystem, Mathematical Biosciences, 64, 1, 99-113 (1983) · Zbl 0524.92027 · doi:10.1016/0025-5564(83)90030-5 |
| [7] | Ruan, S. G., Persistence and coexistence in zooplankton-phytoplankton-nutrient models with instantaneous nutrient recycling, Journal of Mathematical Biology, 31, 6, 633-654 (1993) · Zbl 0779.92021 · doi:10.1007/BF00161202 |
| [8] | Ruan, S. G., A three-trophic-level model of plankton dynamics with nutrient recycling, The Canadian Applied Mathematics Quarterly, 1, 4, 529-553 (1993) · Zbl 0796.92027 |
| [9] | Whittaker, R. H., Communities and Ecosystems (1975), New York, NY, USA: Macmillan, New York, NY, USA |
| [10] | Bischi, G. I., Effects of time lags on transient characteristics of a nutrient cycling model, Mathematical Biosciences, 109, 2, 151-175 (1992) · Zbl 0748.92014 · doi:10.1016/0025-5564(92)90043-V |
| [11] | He, X.-Z.; Ruan, S., Global stability in chemostat-type plankton models with delayed nutrient recycling, Journal of Mathematical Biology, 37, 3, 253-271 (1998) · Zbl 0935.92034 · doi:10.1007/s002850050128 |
| [12] | Wang, K.; Teng, Z. D.; Zhang, F. Q., Global behaviors of a chemostat model with delayed nutrient recycling and periodically pulsed input, Discrete Dynamics in Nature and Society, 2010 (2010) · Zbl 1208.34131 · doi:10.1155/2010/830951 |
| [13] | Freedman, H. I.; Xu, Y. T., Models of competition in the chemostat with instantaneous and delayed nutrient recycling, Journal of Mathematical Biology, 31, 5, 513-527 (1993) · Zbl 0778.92022 · doi:10.1007/bf00173890 |
| [14] | Beretta, E.; Bischi, G. I.; Solimano, F., Stability in chemostat equations with delayed nutrient recycling, Journal of Mathematical Biology, 28, 1, 99-111 (1990) · Zbl 0665.45006 · doi:10.1007/BF00171521 |
| [15] | Ruan, S. G.; Wolkowicz, G., Persistence in plankton models with delayed nutrient recycling, The Canadian Applied Mathematics Quarterly, 3, 2, 219-235 (1995) · Zbl 0837.92028 |
| [16] | Ruan, S. G., Oscillations in plankton models with nutrient recycling, Journal of Theoretical Biology, 208, 1, 15-26 (2001) · doi:10.1006/jtbi.2000.2196 |
| [17] | de Leenheer, P.; Li, B.; Smith, H. L., Competition in the chemostat: some remarks, Canadian Applied Mathematics Quarterly, 11, 3, 229-248 (2003) · Zbl 1132.92341 |
| [18] | Lu, Z. Q.; Hadeler, K. P., Model of plasmid-bearing, plasmid-free competition in the chemostat with nutrient recycling and an inhibitor, Mathematical Biosciences, 148, 2, 147-159 (1998) · Zbl 0930.92029 · doi:10.1016/s0025-5564(97)10010-4 |
| [19] | Yuan, S.; Zhang, T., Dynamics of a plasmid chemostat model with periodic nutrient input and delayed nutrient recycling, Nonlinear Analysis: Real World Applications, 13, 5, 2104-2119 (2012) · Zbl 1327.92052 · doi:10.1016/j.nonrwa.2012.01.006 |
| [20] | He, X.-Z.; Ruan, S.; Xia, H., Global stability in chemostat-type equations with distributed delays, SIAM Journal on Mathematical Analysis, 29, 3, 681-696 (1998) · Zbl 0916.34064 · doi:10.1137/s0036141096311101 |
| [21] | Caperon, J., Time lag in population growth response of isochrysis galbana to a variable nitrate environment, Ecology, 50, 2, 188-192 (1969) · doi:10.2307/1934845 |
| [22] | Ellermeyer, S. F., Competition in the chemostat: global asymptotic behavior of a model with delayed response in growth, SIAM Journal on Applied Mathematics, 54, 2, 456-465 (1994) · Zbl 0794.92023 · doi:10.1137/s003613999222522x |
| [23] | Xia, H.; Wolkowicz, G. S.; Wang, L., Transient oscillations induced by delayed growth response in the chemostat, Journal of Mathematical Biology, 50, 5, 489-530 (2005) · Zbl 1062.92079 · doi:10.1007/s00285-004-0311-5 |
| [24] | Ruan, S.; Wei, J., On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis, 10, 6, 863-874 (2003) · Zbl 1068.34072 |
| [25] | Wroblewski, J. S.; Richman, J. G., The non-linear response of plankton to wind mixing events—implications for survival of larval northern anchovy, Journal of Plankton Research, 9, 1, 103-123 (1987) · doi:10.1093/plankt/9.1.103 |
| [26] | Hale, J. K.; Verduyn Lunel, S. M., Introduction to Functional-Differential Equations, 99 (1993), New York, NY, USA: Springer, New York, NY, USA · Zbl 0787.34002 · doi:10.1007/978-1-4612-4342-7 |
| [27] | Xia, H.; Wolkowicz, G. S. K.; Wang, L., Transient oscillations induced by delayed growth response in the chemostat, Journal of Mathematical Biology, 50, 5, 489-530 (2005) · Zbl 1062.92079 · doi:10.1007/s00285-004-0311-5 |
| [28] | Meng, X.; Gao, Q.; Li, Z., The effects of delayed growth response on the dynamic behaviors of the Monod type chemostat model with impulsive input nutrient concentration, Nonlinear Analysis: Real World Applications, 11, 5, 4476-4486 (2010) · Zbl 1213.34098 · doi:10.1016/j.nonrwa.2010.05.030 |
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.
