Browse > Article
http://dx.doi.org/10.14317/jami.2013.385

FEEDBACK CONTROL FOR A TURBIDOSTAT MODEL WITH RATIO-DEPENDENT GROWTH RATE  

Hu, Xiaoyu (Department of Mathematics, Hubei University for Nationalities)
Li, Zuxiong (Department of Mathematics, Hubei University for Nationalities)
Xiang, Xingguo (Department of Mathematics, Hubei University for Nationalities)
Publication Information
Journal of applied mathematics & informatics / v.31, no.3_4, 2013 , pp. 385-398 More about this Journal
Abstract
In this paper, a turbidostat model with ratio-dependent growth rate and impulsive state feedback control is considered. We obtain sufficient conditions of the globally asymptotically stable of the system without impulsive state feedback control. We also obtain that the system with impulsive state feedback control has periodic solution of order one. Sufficient conditions for existence and stability of periodic solution of order one are given. In some cases, it is possible that the system exists periodic solution of order two. Our results show that the control measure is effective and reliable.
Keywords
turbidostat; impulsive effect; state feedback control; globally asymptotical stability; periodic solution;
Citations & Related Records
연도 인용수 순위
  • Reference
1 X. Meng, L. Chen, Permanence and global stability in an impulsive Lotka-Volterra N-Species competitive system with both discrete delays and continuous delays, Int. J. Biomath. 1 (2008), 179-196.   DOI
2 R. Shi, L. Chen, A predator-prey model with disease in the prey and two impulses for integrated pest management, Appl. Math. Modelling 33 (2009), 2248-2256.   DOI   ScienceOn
3 P. Simeonov, D. Bainov, Orbital stability of periodic solutions autonomous systems with impulse effect, Int. J. Syst. Sci. 19 (1988),2562-85.
4 S. Tang, L. Chen, Modelling and analysis of integrated management strategy, Discrete contin. Dyn. Syst. (Series B) 4 (2004), 759-768.
5 D. Tilman, Resource competition and Community Structure, Princeton U. P., Princeton, N.J., 1982.
6 G. Wolkowicz, Z. Lu, Global dynamics of a mathematical model of competition in the chemostat: general response function and differential death rates, SIAM J. Appl. Math. 52 (1992), 222-233.   DOI   ScienceOn
7 G. Zeng, Existence of periodic solution of order one of state-depended impulsive differnetial equations and its apllication in pest control, J. Biomath. (in China) 22 (2007), 652-660.
8 G. Butler, G. Wolkowicz, A mathematical model of the chemostat with a general class of functions describing nutrient uptake, SIAM J. Appl. Math. 45 (1995), 138-151.
9 P. De Leenheer, H. Smith, Feedback control for the chemostat, J. Math. Biol. 46 (2003), 48-70.   DOI
10 J. Flegr, Two distinct types of natural selection in turbidostat-like and chemostat-like ecosystems, J. Theor. Biol. 188 (1997), 121-126.   DOI   ScienceOn
11 J. Grover, Resource Competition, Chapman and Hall, 1997.
12 S. Hansen, S. Hubbell, Single-nutrient microbial competition: Agreement between experimental and theoretical forecast outcomes, Science 207 (1980), 1491-1493.   DOI
13 S. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math. 34 (1978), 760-763.   DOI   ScienceOn
14 S. Hsu, S. Hubbell, P. Waltman, A mathematical theory of single-nutrient competition in continuous cultures of micro-organisms, SIAM J. Appl. Math. 32 (1977), 366-383.   DOI   ScienceOn
15 G. Jiang, Q. Lu, L. Qian, Complex dynamics of a Holling II prey-predator system with state feedback control, Chaos, Soliton and Fractal 31 (2007), 448-461.   DOI   ScienceOn
16 J. Jiao, L. Chen, Global attractivity of a stage-structure variable coefficients predator-prey system with time delay and impulsive perturbations on predators, Int. J. Biomath. 1 (2008), 197-208.   DOI
17 R. Arditi, L. Ginzburg, Coupling in predator-prey dynamics: ratio-dependence, J. Theor. Biol. 139 (1989), 311-326.   DOI
18 R. Levins, Coexistence in a variable environment, Amer. Natur. 114 (1979), 765-783.   DOI   ScienceOn
19 B. Li, Competition in a turbidostat for an inhibitory nutrient, J. Biol. Dynam. 2 (2008), 208-220.   DOI   ScienceOn
20 Z. Li, T.Wang, L. Chen, Periodic solution of a chemostat model with Beddington-DeAnglis uptake function and impulsive state feedback control, J. Theoret. Biol. 261 (2009), 23-32.   DOI   ScienceOn
21 R. Armstrong, R. McGehee, Competitive exclusion, Amer. Natur. 115 (1980), 151-170.   DOI   ScienceOn