Competition in the Chemostat: A Distributed Delay Model and Its Global Asymptotic Behavior

In this paper, we propose a two species competition model in a chemostat that uses a distributed delay to model the lag in the process of nutrient conversion and study the global asymptotic behavior of the model. The model includes a washout factor over the time delay involved in the nutrient conversion, and hence the delay is distributed over the species concentrations as well as over the nutrient concentration (using the gamma distribution). The results are valid for a very general class of monotone growth response functions.

By using the linear chain trick technique and the fluctuation lemma, we completely determine the global limiting behavior of the model, prove that there is always at most one survivor, and give a criterion to predict the outcome that is dependent upon the parameters in the delay kernel. We compare these predictions on the qualitative outcome of competition introduced by including distributed delay in the model with the predictions made by the corresponding discrete delay model, as well as with the corresponding no delay ODEs model. We show that the discrete delay model and the corresponding ODEs model can be obtained as limiting cases of the distributed delay models. Also, provided that the mean delays are small, the predictions of the delay models are almost identical with the predictions given by the ODEs model. However, when the mean delays are significant, the predictions given by the delay models concerning which species wins the competition and avoids extinction can be different from each other or from the predictions of the corresponding ODEs model. By varying the parameters in the delay kernels, we find that the model seems to have more potential to mimic reality. For example, computer simulations indicate that the larger the mean delay of the losing species, the more quickly that species proceeds toward extinction.

  • [1]  Google Scholar

  • [2]  Google Scholar

  • [3]  E. Beretta and  and Y. Takeuchi, Qualitative properties of chemostat equations with time delay: Boundedness, local and global asymptotic stability, Diff. Eqns. and Dynamical Systems, 2 (1994), pp. 19–40. Google Scholar

  • [4]  Google Scholar

  • [5]  S. N. Busenberg and  and C. C. Travis, On the use of reducible‐functional differential equations in biological models, J. Math. Anal. Appl., 89 (1982), pp. 46–66. jma JMANAK 0022-247X J. Math. Anal. Appl. CrossrefISIGoogle Scholar

  • [6]  A. W. Bush and  and A. E. Cook, The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater, J. Theoret. Biol., 63 (1975), pp. 385–396. jtb JTBIAP 0022-5193 J. Theor. Biol. CrossrefISIGoogle Scholar

  • [7]  G. J. Butler and  and G. S. K. Wolkowicz, A mathematical model of the chemostat with a general class of functions describing nutrient uptake, SIAM J. Appl. Math., 45 (1985), pp. 138–151. smj SMJMAP 0036-1399 SIAM J. Appl. Math. LinkISIGoogle Scholar

  • [8]  J. Caperon, Time lag in population growth response of Isochrysis galbana to a variable nitrate environment, Ecology, 50 (1969), pp. 188–192. 8yr ECOLAR 0012-9658 Ecology CrossrefISIGoogle Scholar

  • [9]  H. A. Caswell, A simulation study of a time lag population model, J. Theoret. Biol., 34 (1972), pp. 419–439. jtb JTBIAP 0022-5193 J. Theor. Biol. CrossrefISIGoogle Scholar

  • [10]  K. L. Cooke and  and Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl., 86 (1982), pp. 592–627. jma JMANAK 0022-247X J. Math. Anal. Appl. CrossrefISIGoogle Scholar

  • [11]  A. Cunningham and  and R. M. Nisbet, Time lag and co‐operativity in the transient growth dynamics of microalgae, J. Theoret. Biol., 84 (1980), pp. 189–203. jtb JTBIAP 0022-5193 J. Theor. Biol. CrossrefISIGoogle Scholar

  • [12]  Google Scholar

  • [13]  K. Dietz, L. Molineaux and , and A. Thomas, A malaria model tested in the African savannah, Bull. Wld. Hlth. Org., 50 (1974), pp. 347–357. 3vy ZZZZZZ 0042-9686 Bull. World Health Organ ISIGoogle Scholar

  • [14]  M. R. Droop, Vitamin B12 and marine ecology. IV. The kinetics of uptake, growth, and inhibition in Monochrysis lutheri, J. Marine Biol. Assoc. UK, 48 (1968), pp. 689–733. jnh JMBAAK 0025-3154 J. Mar. Biol. Assoc. U.K. CrossrefISIGoogle Scholar

  • [15]  S. F. Ellermeyer, Competition in the chemostat: Global asymptotic behavior of a model with delayed response in growth, SIAM J. Appl. Math., 54 (1994), pp. 456–465. smj SMJMAP 0036-1399 SIAM J. Appl. Math. LinkISIGoogle Scholar

  • [16]  H. I. Freedman, J. So and , and P. Waltman, Coexistence in a model of competition in the chemostat incorporating discrete delay, SIAM J. Appl. Math., 49 (1989), pp. 859–870. smj SMJMAP 0036-1399 SIAM J. Appl. Math. LinkISIGoogle Scholar

  • [17]  Google Scholar

  • [18]  Google Scholar

  • [19]  Google Scholar

  • [20]  S. R. Hansen and  and S. P. Hubbell, Single‐nutrient microbial competition: Qualitative agreement between experimental and theoretically forecast outcomes, Science, 207 (1980), pp. 1491–1493. c9t ZZZZZZ 1095-9203 Science CrossrefISIGoogle Scholar

  • [21]  D. Herbert, R. Elsworth and , and R. C. Telling, The continuous culture of bacteria; a theoretical and experimental study, J. Gen. Microbiol., 14 (1956), pp. 601–622. jgn JGMIAN 0022-1287 J. Gen. Microbiol. CrossrefGoogle Scholar

  • [22]  Google Scholar

  • [23]  W. M. Hirsch, H. Hanisch and , and J. P. Gabriel, Differential equation models of some parasitic infections: methods for the study of asymptotic behavior, Comm. Pure Appl. Math., 38 (1985), pp. 733–753. cpa CPMAMV 0010-3640 Commun. Pure Appl. Math. CrossrefISIGoogle Scholar

  • [24]  S.‐B. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), pp. 760–763. smj SMJMAP 0036-1399 SIAM J. Appl. Math. LinkISIGoogle Scholar

  • [25]  S.‐B. Hsu, S. P. Hubbell and , and P. Waltman, A mathematical theory for single‐nutrient competition in continuous culture of micro‐organisms, SIAM J. Appl. Math., 32 (1977), pp. 366–383. smj SMJMAP 0036-1399 SIAM J. Appl. Math. LinkISIGoogle Scholar

  • [26]  S.‐B. Hsu, P. Waltman and , and S. F. Ellermeyer, A remark on the global asymptotic stability of a dynamical system modeling two species competition, Hiroshima Math. J., 24 (1994), pp. 435–445. a8v HMTJAD 0018-2079 Hiroshima Math. J. CrossrefGoogle Scholar

  • [27]  Google Scholar

  • [28]  Google Scholar

  • [29]  Google Scholar

  • [30]  B. R. Levin, F. M. Stewart and , and L. Chao, Resource‐limited growth, competition, and predation: A model and experimental studies with bacteria and bacteriophages, Amer. Natur., 111 (1977), pp. 3–24. anl AMNTA4 0003-0147 Am. Nat. CrossrefISIGoogle Scholar

  • [31]  N. MacDonald, Time lag in simple chemostat models, Biotechnol. Bioengrg., 18 (1976), pp. 805–812. btb BIBIAU 0006-3592 Biotechnol. Bioeng. CrossrefISIGoogle Scholar

  • [32]  Google Scholar

  • [33]  Google Scholar

  • [34]  Google Scholar

  • [35]  Google Scholar

  • [36]  Google Scholar

  • [37]  J. Monod, La technique de la culture continue: Théorie et applications, Ann. Inst. Pasteur, 79 (1950), pp. 390–401. 2h5 ZZZZZZ 0020-2444 Ann. Inst. Pasteur (Paris) Google Scholar

  • [38]  S. Ruan and  and G. Wolkowicz, Bifurcation analysis of a chemostat model with a distributed delay, J. Math. Anal. Appl., 204 (1996), pp. 786–812. jma JMANAK 0022-247X J. Math. Anal. Appl. CrossrefISIGoogle Scholar

  • [39]  Google Scholar

  • [40]  Google Scholar

  • [41]  M. L. Stephens and  and G. Lyberatos, Effect of cycling on final mixed culture fate, Biotechnol. Bioengrg., 29 (1987), pp. 672–678. btb BIBIAU 0006-3592 Biotechnol. Bioeng. CrossrefISIGoogle Scholar

  • [42]  C. Strobeck, N species competition, Ecology, 54 (1973), pp. 650–654. 8yr ECOLAR 0012-9658 Ecology CrossrefISIGoogle Scholar

  • [43]  H. R. Thieme, Convergence results and a Poincaré‐Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), pp. 755–763. jmd JMBLAJ 0303-6812 J. Math. Biol. CrossrefISIGoogle Scholar

  • [44]  T. F. Thingstad and  and T. I. Langeland, Dynamics of chemostat culture: The effect of a delay in cell response, J. Theoret. Biol., 48 (1974), pp. 149–159. jtb JTBIAP 0022-5193 J. Theor. Biol. CrossrefISIGoogle Scholar

  • [45]  Google Scholar

  • [46]  Google Scholar

  • [47]  G. S. K. Wolkowicz and  and Z. Lu, Global dynamics of a mathematical model of competition in the chemostat: general response functions and differential death rates, SIAM J. Appl. Math., 52 (1992), pp. 222–233. smj SMJMAP 0036-1399 SIAM J. Appl. Math. LinkISIGoogle Scholar

  • [48]  G. Wolkowicz and  and H. Xia, Global asymptotic behavior of a chemostat model with discrete delays, SIAM J. Appl. Math., 57 (1997), pp. 1019–1043. smj SMJMAP 0036-1399 SIAM J. Appl. Math. LinkISIGoogle Scholar