Journal of applied mathematics & informatics

한국전산응용수학회 (The Korean Society for Computational and Applied Mathematics)
권호 표지
DOI QR코드

DOI QR Code

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)
  • 투고 : 2012.05.12
  • 심사 : 2012.10.29
  • 발행 : 2013.05.30
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

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.

키워드

참고문헌

  1. R. Arditi, L. Ginzburg, Coupling in predator-prey dynamics: ratio-dependence, J. Theor. Biol. 139 (1989), 311-326. https://doi.org/10.1016/S0022-5193(89)80211-5
  2. R. Armstrong, R. McGehee, Competitive exclusion, Amer. Natur. 115 (1980), 151-170. https://doi.org/10.1086/283553
  3. 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.
  4. P. De Leenheer, H. Smith, Feedback control for the chemostat, J. Math. Biol. 46 (2003), 48-70. https://doi.org/10.1007/s00285-002-0170-x
  5. J. Flegr, Two distinct types of natural selection in turbidostat-like and chemostat-like ecosystems, J. Theor. Biol. 188 (1997), 121-126. https://doi.org/10.1006/jtbi.1997.0458
  6. J. Grover, Resource Competition, Chapman and Hall, 1997.
  7. S. Hansen, S. Hubbell, Single-nutrient microbial competition: Agreement between experimental and theoretical forecast outcomes, Science 207 (1980), 1491-1493. https://doi.org/10.1126/science.6767274
  8. S. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math. 34 (1978), 760-763. https://doi.org/10.1137/0134064
  9. 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. https://doi.org/10.1137/0132030
  10. 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. https://doi.org/10.1016/j.chaos.2005.09.077
  11. 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. https://doi.org/10.1142/S1793524508000163
  12. R. Levins, Coexistence in a variable environment, Amer. Natur. 114 (1979), 765-783. https://doi.org/10.1086/283527
  13. B. Li, Competition in a turbidostat for an inhibitory nutrient, J. Biol. Dynam. 2 (2008), 208-220. https://doi.org/10.1080/17513750802018345
  14. 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. https://doi.org/10.1016/j.jtbi.2009.07.016
  15. 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. https://doi.org/10.1142/S1793524508000151
  16. 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. https://doi.org/10.1016/j.apm.2008.06.001
  17. P. Simeonov, D. Bainov, Orbital stability of periodic solutions autonomous systems with impulse effect, Int. J. Syst. Sci. 19 (1988),2562-85.
  18. S. Tang, L. Chen, Modelling and analysis of integrated management strategy, Discrete contin. Dyn. Syst. (Series B) 4 (2004), 759-768.
  19. D. Tilman, Resource competition and Community Structure, Princeton U. P., Princeton, N.J., 1982.
  20. 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. https://doi.org/10.1137/0152012
  21. 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.

피인용 문헌

  1. Dynamics of microorganism cultivation with delay and stochastic perturbation vol.101, pp.1, 2013, https://doi.org/10.1007/s11071-020-05718-z