I Mixed populations of two species of yeast J Exp Biol 9 p 389 Joseph M Mahaffy

I mixed populations of two species of yeast j exp

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I. Mixed populations of two species of yeast, J. Exp. Biol. 9 , p. 389. Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Equ — (56/68)
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Introduction Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of Systems of DEs Model of Glucose and Insulin Control Glucose Tolerance Test Competition Model Monoculture Models 1 Monoculture Model: Previous slide gave data for monocultures, which should satisfy logistic growth model dY dt = rY 1 - Y M , Y (0) = Y 0 , which has the solution Y ( t ) = MY 0 Y 0 + ( M - Y 0 ) e - rt Use MatLab to fit parameters to the data, and the results for Saccharomyces cerevisiae are r = 0 . 25864 M = 12 . 742 Y 0 = 1 . 2343 The results for Schizosaccharomyces kephir are r = 0 . 057443 M = 5 . 8802 Y 0 = 0 . 67805 These models show that S. cerevisiae grows much faster than S. kephir Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Equations: — (57/68) Introduction Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of Systems of DEs Model of Glucose and Insulin Control Glucose Tolerance Test Competition Model Monoculture Models 2 Monoculture Models and Data: Y c ( t ) = 12 . 742 1 + 9 . 3230 e - 0 . 25864 t and Y k ( t ) = 5 . 8802 1 + 7 . 6723 e - 0 . 057443 t Graphs show the best fitting logistic models for the two species with the Gause experiment data 0 5 10 15 20 25 30 35 40 45 50 0 2 4 6 8 10 12 14 t (hr) Volume (yeast) Saccharomyces cerevisiae 0 20 40 60 80 100 120 140 0 1 2 3 4 5 6 7 t (hr) Volume (yeast) Schizosaccharomyces kephir Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Equ — (58/68) Introduction Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of Systems of DEs Model of Glucose and Insulin Control Glucose Tolerance Test Competition Model Competition Experiment Competition Experiment: G. F. Gause ran experiments (same nutrient conditions) mixing the cultures of S. cerevisiae and S. kephir Table combining two experimental studies of the mixed culture t (hr) 0 1.5 9 10 18 18 23 Y c 0.375 0.92 3.08 3.99 4.69 5.78 6.15 Y k 0.29 0.37 0.63 0.98 1.47 1.22 1.46 t (hr) 25.5 27 38 42 45.5 47 Y c 9.91 9.47 10.57 7.27 9.88 8.3 Y k 1.11 1.225 1.1 1.71 0.96 1.84 The data show the populations are increasing, but the S. cerevisiae population is significantly below the carrying capacity If two species compete for a single resource, then 1. Competitive Exclusion - one species out competes the other and becomes the only survivor 2. Coexistence - both species coexist around a stable equilibrium Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Equations: — (59/68) Introduction Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of Systems of DEs Model of Glucose and Insulin Control Glucose Tolerance Test Competition Model Competition Model Competition Model: Assume a competition model of the form dY c dt = a 1 Y c - a 2 Y 2 c - a 3 Y c Y k = f 1 ( Y c , Y k ) dY k dt = b 1 Y k - b 2 Y 2 k - b 3 Y k Y c = f 2 ( Y c , Y k ) First terms with a 1 and b 1 represent the exponential or Malthusian growth at low densities The terms a 2 and b 2 represent intraspecies competition from crowding by the same species The terms a 3 and b 3 represent
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