Computer Algebra Recipes

156 CHAPTER 4. NONLINEAR ODE MODELS
second. With the initial population density N0 =0:5329 known, the data from
the ¯rst experiment were used to ¯nd the parameter values b and r to use in
the logistic-model solution for each yeast variety. Using a least-squares ¯tting
approach, Gause found that for the kephir, r =0:0607 and b =5:80, while for
the cerevisiae r =0:2183 and b =13:0. The ¯rst experiment was carried out
over a time period of 160 hours. Heather decides to plot the logistic curve for
the kephir, so enters the parameter values,
> b:=5.80: No:=0.5329: r:=0.0607: yeastdensity;
N (t) =
3:090820
5:2671 e (¡0:0607 t) +0:5329
and applies the plot command, adding a title to the picture.
> plot(rhs(yeastdensity),t=0..160,labels=["t","A"],
color=blue,thickness=2,tickmarks=[2,4],
title="Amount of yeast (A) vs. hours (t)");
Amount of yeast (A) vs. hours (t)
5
4
A
3
2
1
0 100
t
Figure 4.2: Growth curve for the yeast kephir.
Thelogisticgrowthcurveofthekephir population is shown in Figure 4.2. The
plot for the cerevisiae would be similar in appearance.
Gause's second experiment studied the competition between the two yeasts
for the available nutrient when they were grown together in the same test tube.
To explore this case Heather labels the kephir population density and parameters
with the subscript k, while the subscript c is used for the cerevisiae. To
account for the interaction between the two species of yeast, Gause assumed
that the probability of an interaction is proportional to the product of the
population densities, i.e., to NkNc, and that the competition for nutrient was
detrimental to both populations. Accordingly, Gause described the growth of