Chemostats and epidemics: Competition for nutrients/hosts

  • Received: 01 February 2013 Accepted: 29 June 2018 Published: 01 August 2013
  • MSC : Primary: 92D15, 92D25, 92D30; Secondary: 34D23, 37B25, 93D30.

  • In a chemostat, several species compete for the same nutrient, whilein an epidemic, several strains of the same pathogen may competefor the same susceptible hosts. As winner, chemostat models predict the specieswith the lowest break-even concentration, while epidemicmodels predict the strain with the largest basic reproduction number.We show that these predictions amount to the same if the per capitafunctional responses of consumer species to the nutrient concentration or ofinfective individuals to the density of susceptibles are proportional to eachother but that they are different if the functional responses are nonproportional.In the second case, the correct prediction is given by the break-even concentrations.In the case of nonproportional functional responses, we add a class for which the prediction does not only rely on local stability and instability of one-species (strain) equilibriabut on the global outcome of the competition. We also review some results fornonautonomous models.

    Citation: Hal L. Smith, Horst R. Thieme. Chemostats and epidemics: Competition for nutrients/hosts[J]. Mathematical Biosciences and Engineering, 2013, 10(5&6): 1635-1650. doi: 10.3934/mbe.2013.10.1635

    Related Papers:

  • In a chemostat, several species compete for the same nutrient, whilein an epidemic, several strains of the same pathogen may competefor the same susceptible hosts. As winner, chemostat models predict the specieswith the lowest break-even concentration, while epidemicmodels predict the strain with the largest basic reproduction number.We show that these predictions amount to the same if the per capitafunctional responses of consumer species to the nutrient concentration or ofinfective individuals to the density of susceptibles are proportional to eachother but that they are different if the functional responses are nonproportional.In the second case, the correct prediction is given by the break-even concentrations.In the case of nonproportional functional responses, we add a class for which the prediction does not only rely on local stability and instability of one-species (strain) equilibriabut on the global outcome of the competition. We also review some results fornonautonomous models.


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    [1] J. Math. Biol., 47 (2003), 153-168.
    [2] Discrete and Continuous Dynamical Systems Series B, 5 (2005), 175-188.
    [3] Disc. Cont. Dyn. Syst. Ser. B, 8 (2007), 1-17.
    [4] Parasitology, 85 (1982), 411-426.
    [5] Can. Appl. Math. Q., 11 (2003), 107-142.
    [6] Amer. Natur., 115 (1980), 151-170.
    [7] in "Microbial Population Dynamics" (ed. M. J. Bazin), CRC Series in Mathematical Models in Microbiology, CRC Press, Boca Raton, FL, (1982), 1-32.
    [8] J. Math. Biol., 51 (2005), 458-490.
    [9] Math. Biosci., 118 (1993), 127-180.
    [10] Nonlinear Analysis, 47 (2001), 4107-4115.
    [11] Ann. Bot. London, 19 (1905), 281-295.
    [12] Amer. Nat., 145 (1995), 855-887.
    [13] J. Math. Biol., 27 (1989), 179-190.
    [14] SIAM J. Appl. Math., 45 (1985), 138-151.
    [15] Lecture Notes in Biomathematics, 97, Springer-Verlag, Berlin, 1993.
    [16] Math. Biosci., 42 (1978), 43-61.
    [17] in "Mathematical Population Dynamics. Analysis of Heterogeneity. Vol. One. Theory of Epidemics" (eds. O. Arino, D. Axelrod, M. Kimmel and M. Langlais), Wuerz, Winnipeg, (1995), 33-50.
    [18] J. Math. Biol., 10 (1980), 385-400.
    [19] J. Math. Biol., 11 (1981), 319-335.
    [20] J. Biol. Systems, 5 (1997), 325-339.
    [21] in "Mathematical Modelling of Population Dynamics," Banach Center Publications, 63, Polish Acad. Sci., (2004), 47-86.
    [22] Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2013.
    [23] in "Adaptive Dynamics of Infectious Diseases: In Pursuit of Virulence Management" (eds. U. Dieckmann, J. A. J. Metz, M. W. Sabelis and K. Sigmund), International Institute for Applied Systems Analysis, Cambridge University Press, Cambridge, (2002), 10-25.
    [24] Math. Model. Nat. Phenom., 2 (2007), 55-73.
    [25] J. Math. Biol., 31 (1993), 513-527.
    [26] SIAM J. Appl. Math., 67 (2006/07), 337-353.
    [27] Amer. Nat., 111 (1977), 135-142.
    [28] Math. Biosci. Engin., 3 (2006), 513-525.
    [29] Proc. Amer. Math. Soc., 136 (2008), 2793-2802.
    [30] in "Modeling and Dynamics of Infectious Diseases" (eds. Z. Ma, Y. Zhou and J. Wu), Ser. Contemp. Appl. Math. CAM, 11, Higher Ed. Press, Beijing, (2009), 268-288.
    [31] Comm. Pure Appl. Math., 38 (1985), 733-753.
    [32] SIAM J. Appl. Math., 34 (1978), 760-763.
    [33] SIAM J. App. Math., 32 (1977), 366-383.
    [34] J. Math. Biol., 9 (1980), 115-132.
    [35] Taiwanese J. Math., 9 (2005), 151-173.
    [36] SIAM J. Appl. Math., 67 (2006), 260-278.
    [37] Math. Biosci., 209 (2007), 51-75.
    [38] Yale University Press, New Haven, 1955.
    [39] Math. Med. Biol., 21 (2004), 75-83.
    [40] Bull. Math. Biol., 68 (2006), 615-626.
    [41] Bull. Math. Biol., 69 (2007), 1871-1886.
    [42] Math. Med. Biol., 26 (2009), 225-239.
    [43] MMB IMA, 22 (2005), 113-128.
    [44] Appl. Math. Letters, 15 (2002), 955-960.
    [45] SIAM J. Appl. Math., 59 (1999), 411-422.
    [46] SIAM J. Appl. Math., 70 (2010), 2434-2448.
    [47] J. Differential Eqns., 248 (2010), 1-20.
    [48] J. Math. Anal. Appl., 361 (2010), 38-47.
    [49] J. Austral. Math. Soc. Ser. B, 34 (1993), 282-295.
    [50] Math. Biosci. Eng., 7 (2010), 675-685.
    [51] Applicable Analysis, 89 (2010), 1109-1140.
    [52] J. Biol. Dyn., 3 (2009), 235-251.
    [53] Math. Biosci., 207 (2007), 58-77.
    [54] Math. Biosci. Eng., 3 (2006), 603-614.
    [55] J. Math. Anal. Appl., 338 (2008), 518-535.
    [56] Math. Biosci. Engin., 6 (2009), 603-610.
    [57] Nonlinear Anal. RWA, 11 (2010), 55-59.
    [58] Nonlinear Anal. RWA, 11 (2010), 3106-3109.
    [59] Math. Biosci. Engin., 7 (2010), 837-850.
    [60] IMA J. Math. Appl. Med. Biol., 16 (1999), 307-317.
    [61] Evolutionary Ecology Research, 10 (2008), 629-654.
    [62] Discr. Contin. Dyn. Syst. B, 6 (2006), 225-235.
    [63] Macmillan, New York, 1979.
    [64] SIAM J. Appl. Math., 58 (1998), 170-192.
    [65] Math. Biosci. Eng., 8 (2011), 827-840.
    [66] SIAM J. Appl. Math., 40 (1981), 498-522.
    [67] Cambridge Studies in Mathematical Biology, 13, Cambridge University Press, Cambridge, 1995.
    [68] in "Mathematical Studies on Human Disease Dynamics" (eds. Abba B. Gumel, Carlos Castillo-Chavez, Ronald E. Mickens and Dominic P. Clemence), Contemporary Mathematics, 410, Amer. Math. Soc., Providence, RI, (2006), 367-389.
    [69] Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2003.
    [70] J. Differential Eqns., 250 (2011), 3772-3801.
    [71] in "Mathematics for Life Sciences and Medicine" (eds. Y. Takeuchi, Y. Iwasa and K. Sato), Biol. Med. Phys. Biomed. Eng., Springer, Berlin, (2007), 123-153.
    [72] Gauthier-Villars, Paris, 1931.
    [73] Proc. Nat. Acad. Sci., 31 (1945), 24-34; Part II, Proc. Nat. Acad. Sci., 31 (1945), 109-116.
    [74] Math. Biosci., 93 (1989), 249-268.
    [75] Rocky Mountain Journal of Mathematics, 25 (1995), 515-543.
    [76] SIAM J. Appl. Math., 52 (1992), 222-233.
    [77] SIAM J. Appl. Math., 57 (1997), 1019-1043.
    [78] Differential Integral Equations, 11 (1998), 465-491.
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