대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제54권2호
  • /
  • Pages.383-389
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  • 2017
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  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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GLOBAL STABILITY OF THE POSITIVE EQUILIBRIUM OF A MATHEMATICAL MODEL FOR UNSTIRRED MEMBRANE REACTORS

  • Song, Yongli (Department of Mathematics Hangzhou Normal University) ;
  • Zhang, Tonghua (Department of Mathematics Swinburne University of Technology)
  • 투고 : 2015.07.14
  • 발행 : 2017.03.31
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초록

This paper devotes to the study of a diffusive model for unstirred membrane reactors with maintenance energy subject to a homogeneous Neumann boundary condition. It shows that the unique constant steady state is globally asymptotically stable when it exists. This result further implies the non-existence of the non-uniform steady state solution.

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참고문헌

  1. K. Hattaf and N. Yousfi, Global stability for reaction-diffusion equations in biology, Comput. Math. Appl. 66 (2013), no. 8, 1488-1497. https://doi.org/10.1016/j.camwa.2013.08.023
  2. D. Henry, Geometric theory of semilinear parabolic equations, in: Lecture Notes in Mathematics, 3 ed., vol. 840, Springer-Verlag, Berlin, New York, 1993.
  3. S.-B. Hsu, A survey of constructing lyapunov functions for mathematical models in population biology, Taiwanese J. Math. 9 (2005), no. 2, 151-173. https://doi.org/10.11650/twjm/1500407791
  4. S.-B. Hsu, J. Jiang, and F.-B. Wang, On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat, J. Differential Equations 248 (2010), no. 10, 2470-2496. https://doi.org/10.1016/j.jde.2009.12.014
  5. J. H. Merkin, V. Petrov, S. K. Scott, and K. Showalter, Wave-induced chaos in a continuously fed unstirred reactor, J. Chemical Soc. Faraday Transactions 92 (1996), no. 16, 2911-2918. https://doi.org/10.1039/ft9969202911
  6. M. I. Nelson, T. B. Kerr, and X. Chen, A fundamental analysis of continuous flow bioreactor and membrane reactor models with death and maintenance included, Asia-Pacific J. of Chemical Engineering 3 (2008), 70-80. https://doi.org/10.1002/apj.106
  7. R. Peng and M. Wang, Global stability of the equilibrium of a diffusive holling-tanner prey-predator model, Appl. Math. Lett. 20 (2007), no. 6, 664-670. https://doi.org/10.1016/j.aml.2006.08.020
  8. H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 2008.
  9. M. Wang and P. Y. H. Pang, Global asymptotical stability of positive steady states of a diffusive ratio-dependent prey-predator model, Appl. Math. Lett. 21 (2008), 1215-1220. https://doi.org/10.1016/j.aml.2007.10.026
  10. T. Zhang, Global analysis of continuous flow bioreactor and membrane reactor models with death and maintenance, J. Math. Chemistry 50 (2012), 2239-2247. https://doi.org/10.1007/s10910-012-0027-5
  11. T. Zhang and H. Zang, Delay-induced turing instability in reaction-diffusion equations, Phys. Rev. E 90 (2014), 052908. https://doi.org/10.1103/PhysRevE.90.052908