I mixed populations of two species of yeast j exp

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)

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