JAMP  Vol.5 No.1 , January 2017
On a Hibernation Plankton-Nutrient Chemostat Model with Delayed Response in Growth
Abstract: In this paper, a hibernation plankton-nutrient chemostat model with delayed response in growth is considered. By using the stroboscopic map and the theorem of impulsive delay differential equation, a plankton-extinction boundary periodic solution is obtained. The sufficient conditions on the permanence and globally attractive of the chemostat system are also obtained. Our main results reveal that the delayed response in growth plays an important role on the dynamical behaviors of system.
Cite this paper: Ma, J. and Rehim, M. (2017) On a Hibernation Plankton-Nutrient Chemostat Model with Delayed Response in Growth. Journal of Applied Mathematics and Physics, 5, 45-58. doi: 10.4236/jamp.2017.51007.

[1]   Bush, A.W. and Somolinas, A.E. (1975) The Effect of Time Delay and Growth Rate Inhibition in the Bacterial Treatment of Wastewater. Journal of Theoretical Biology, 63, 385-395.

[2]   Smith, H. and Waltman, P. (1999) Perturbation of a Globally Stable Ateady State. Proceedings of the AMS, 127, 447-453.

[3]   Hsu, S.B., Waltman, P. and Ellermeyer, S.F. (1994) A Remark on the Global Asymptotic Stability of a Dynamical System Modeling Two Species Competition. Hiroshima Mathematical Journal, 24, 435-445.

[4]   Ellermeyer, S., Hendrix, J. and Ghoochan, N. (2003) A Theoretical and Empirical Investigation of Delayed Growth Response in the Continuous Culture of Bacteria. Journal of Theoretical Biology, 222, 485-494.

[5]   Bulert, G.L., Hsu, S.B. and Waltman, P. (1985) A Mathematical Model of the Chemostat with Periiodic Washout Rate. SIAM Journal on Applied Mathematics, 45, 435-449.

[6]   Bainov, D. and Simeonov, P. (1993) Impulsive Differential Equations: Periodic Solutions and Applications. Longman, Harlow.

[7]   Jiao, J. and Chen, L. (2008) Global Attractivity of a Stage-Structure Variable Coefficients Predator-Prey System with Time Delay and Impulsive Perturbations on Predators. International Journal of Biomathematics, I, 197-208.

[8]   Smith, R.J. and Wolkowicz, G.S.K. (2004) Analysis of a Model of the Nutrient Driven Self-Cycling Fermentation Process. Dynamics of Continuous, Discrete and Impulsive Systems Series B, 11, 239-265.

[9]   Wang, L.M., Chen, L.S., et al. (2009) Impulsive Diffusion in Single Species Model. Chaos, Solitons & Fractals, 33, 1213-1219.

[10]   Wang, L.C.H. and Lee, T.F. (1996) Torpor and Hibernation in Manmals: Metabolic, Physiological and Chemical Adaptations. In: Fregley, M.J., Blatteis, C.M., Eds., Handbook of Physiology: Environmental Physiology, Oxford University Press, New York, 507-532.

[11]   Staples, J.F. and Brown, J.C.L. (2008) Mitochondrial Metabolism in Hibernation and daily Torpor: A Review. Journal of Comparative Physiology B, 178, 811-827.

[12]   Levin, S.A. and Segel, L.A. (1976) An Hypothesis for the Origin of Planktonic Patchiness. Nature, 259, 659.

[13]   Okubo, A. (1980) Diffusion and Ecological Problems: Mathematical Models. Springer-Verlag, Berlin.

[14]   Ruan, S. (1998) Turing Instability and Traveling Waves in Diffusive Plankton Models with Delayed Nutrient Recyling. IMA Journal of Applied Mathematics, 61, 15-32.

[15]   Jiao, J., Chen, L. and Cai, S. (2016) Dynamics of a Plankton-Nutrient Chemostat Model with Hibernation and It Described by Impulsive Switched Systems. Journal of Applied Mathematics and Computing, 35, 211-227.

[16]   Zhang, T., Ma, W. and Meng, X. (2015) Dynamical Analysis of a Continuous-Culture and Harvest Chemostat Model with Impulsive Effect. Journal of Biological Systems, 23, Article ID: 1550028, 21 p.

[17]   Jiao, J. and Chen, L. (2009) Dynamical Analysis of a Chemostat Model with Delayed Response in Growth and Pulse Input in Polluted Environment. Journal of Mathematical Chemistry, 46, 502-513.