Computer Algebra Recipes

4.1. FIRST-ORDER MODELS 157
the two interacting species by the following pair of coupled nonlinear equations:
Nk _ = rk (bk ¡ Nk ¡ ¯kc Nc) Nk=bk; Nc _ = rc (bc ¡ Nc ¡ ¯ck Nk) Nc=bc; (4.3)
with the interaction parameters experimentally determined to be ¯kc =0:439
and ¯ck =3:15. This set of nonlinear ODEs cannot be solved analytically, but
the behavior of the two competing yeast populations can easily be determined
by making a phase-plane portrait of Nk versus Nc.
Heather unassigns k so that it can be used as a subscript to label the kephir
population and sets the infolevel[dsolve] command to zero.
> unassign('k'): infolevel[dsolve]:=0:
The values determined by Gause for the coe±cients in (4.3) are entered,
> r[k]:=0.0607: b[k]:=5.8: beta[kc]:=0.439: r[c]:=0.2183:
b[c]:=13.0: beta[ck]:=3.15:
as well as the system (sys) ofequations.
> sys:=diff(N[k](t),t)=r[k]*(b[k]-N[k]-beta[kc]*N[c])*N[k]/b[k],
diff(N[c](t),t)=r[c]*(b[c]-N[c]-beta[ck]*N[k])*N[c]/b[c];
sys := d
dt Nk(t) =0:01046551724 (5:8 ¡ Nk(t) ¡ 0:439 Nc(t)) Nk(t);
d
dt Nc(t) =0:01679230769 (13:0 ¡ Nc(t) ¡ 3:15 Nk(t)) Nc(t)
Heather takes Nk(0)=Nc(0)=0:5 inthephaseportrait command,
> phaseportrait([sys],[N[k](t),N[c](t)],t=0..250,
[[N[k](0)=0.5,N[c](0)=0.5]],stepsize=0.1,linecolor=blue);
N[c]
8
6
4
2
0
1 2 4 5
N[k]
Figure 4.3: Phase portrait showing competition between kephir and cerevisiae.