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Competition in the gradostat

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References

  1. Butler, G., Waltman, P.: Persistence in dynamical systems. J. Diff. Eq. 63, 255–263 (1986)

    Google Scholar 

  2. Butler, G., Freedman, H. I., Waltman, P.: Uniformly persistent systems. Proc. A.M.S. 96, 425–430 (1986)

    Google Scholar 

  3. Coppel, W. A.: Stability and asymptotic behavior of differential equations. Boston: Heath 1965

    Google Scholar 

  4. Frederickson, A. G., Stephanopoulos, G.: Microbial competetion. Science 213, 972–979 (1981)

    Google Scholar 

  5. Hansen, S. R., Hubbell, S. P.: Single nutrient microbial competition; agreement between experimental and theoretical forecast outcomes. Science 20, 1491–1493 (1980)

    Google Scholar 

  6. Herbert, D., Elsworth, R., Telling, R. C.: The continuous culture of bacteria: a theoretical and experimental study. J. Gen. Microbiol. 14, 601–622 (1956)

    Google Scholar 

  7. Hirsch, M.: Systems of differential equations which are competitive or cooperative I: limit sets. SIAM J. Appl. Math. 13, 167–179 (1982)

    Google Scholar 

  8. Hirsch, M.: Systems of differential equations which are competitive or cooperative II: convergence almost everywhere. SIAM J. Math. Anal. 16, 423–439 (1985)

    Google Scholar 

  9. Hirsch, M.: The dynamical systems approach to differential equations. Bull. A.M.S. 11, 1–64 (1984)

    Google Scholar 

  10. Hsu, S. B.: Limiting behavior for competing species. SIAM J. Appl. Math. 34, 760–763 (1978)

    Google Scholar 

  11. Hsu, S. B., Hubbell, S. P., Waltman, P.: A mathematical theory for single nutrient competition in continuous cultures of microorganisms. SIAM J. Appl. Math. 32, 366–383 (1977)

    Google Scholar 

  12. Jannash, H. W., Mateles, R. T.: Experimental bacterial ecology studied in continuous culture. Adv. Microbial Physiol. 11, 165–212 (1974)

    Google Scholar 

  13. Lovitt, R. W., Wimpenny, J. W. T.: The gradostat: a bidirectional compound chemostat and its applications in microbiological research. Gen. Microbiol. 127, 261–268 (1981)

    Google Scholar 

  14. Matano, H.: Existence of nontrivial unstable sets for equilibrium of strongly order preserving systems. J. Fac. Sci. Univ. Tokyo. 30, 645–673 (1984)

    Google Scholar 

  15. Smith, H.: Systems of ordinary differential equations which generate an order preserving flow: a survey of results. SIAM Rev., to appear

  16. Tang, B.: Mathematical investigations of growth of microorganisms in the gradostat. J. Math. Biol. 23, 319–339 (1986).

    Google Scholar 

  17. Taylor, P. A., Williams, J. L.: Theoretical studies on the coexistence of competing species under continuous-flow conditions. Can. J. Microbiol. 21, 90–98 (1975)

    Google Scholar 

  18. Veldcamp, H.: Ecological studies with the chemostat. Adv. Microbial Ecol. 1, 59–95 (1977)

    Google Scholar 

  19. Waltman, P., Hubbell, S. P., Hsu, S. P.: Theoretical and experimental investigations of microbial competition in continous culture. In: Burton, T. (ed.) Modeling and differential equations in biology. New York: Marcel Dekker 1980

    Google Scholar 

  20. Waltman, P: Competition models in population biology. Philadelphia: SIAM 1983

    Google Scholar 

  21. Wimpenny, J. W. T., Lovitt, R. W.: The investigation and analysis of heterogeneous environments using the gradostat. In: Grainger, J. M., Lynch, J. M. (ed.) Microbiological methods for environmental biotechnology. Orlando: Academy Press 1984

    Google Scholar 

  22. Yorke, J. A.: Periods of periodic solutions and the Lipschitz constant. Proc. Am. Math. Soc. 22, 509–512 (1969)

    Google Scholar 

  23. Young, D. M.: Iterative solution of large linear systems. New York: Academic Press 1971

    Google Scholar 

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Research partially supported by the Emory University Research Fund

Research partially supported by the Deutsche Forschungsgemeinschaft

Research supported by NSF grants MCS-8401746 and DMS-8601398

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Jäger, W., So, J.W.H., Tang, B. et al. Competition in the gradostat. J. Math. Biology 25, 23–42 (1987). https://doi.org/10.1007/BF00275886

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