Computer Algebra Recipes

4.1. FIRST-ORDER MODELS 153
tion density at time t>0andk>0 the rate of reproduction. If k is taken to
be a constant, the following linear yeast equation, YE, results:
> YE:=diff(N(t),t)=k*N(t);
YE := d
N (t) =k N (t)
dt
This growth equation is historically known as Malthus's law, named in honor
of Thomas Malthus (1766{1834), who published a pamphlet (entitled Essay on
Population) on population growth in 1798. Although Heather can solve this
simple linear ODE in her head, she lets Maple generate the solution, subject to
the initial condition N (0) = No.
> ic:=N(0)=No;
ic := N (0) = No
> sol:=dsolve(fYE,icg,N(t));
sol := N (t) =No e (kt)
Malthus's law leads to exponential growth of the yeast population density.
Having read The Andromeda Strain, Heather is curious as to how many
Escherichia coli there would be at the end of 24 hours if the exponential solution
prevailed. According to the Crichton quotation, the doubling time is
20 minutes, or 1=3 of an hour. In the following command line, Heather takes
N(t =1=3)=No = 2 in the exponential solution,
> eq:=2=exp(k/3); #time in hours
k
(
eq := 2 = e 3 )
and numerically solves for the reproductive rate constant, labeled k1 .
> k1:=fsolve(eq,k);
k1 := 2:079441542
Using the Malthus solution, at the end of 24 hours the number of E. coli would
have grown from a single bacterium (No =1),
> Number:=1*exp(k1*24); #in 24 hours
Number := 0:4722366529 1022 to about 1022 bacteria. Talk about explosive growth! Heather wonders how
accurate the Malthus solution is. After all, she need not have formulated the E.
coli growth as a continuous-time ODE, but instead could calculate the growth
directly. In a 24-hour period there would be 24 £ 3 = 72 doublings. At the end
of 24 hours, the number of E. coli is given by the output of the following line:
> Number2:=2^(24.0*3);
Number2 := 0:4722366483 1022 The two numbers di®er only in the eighth decimal place.
Although 1022 E. coli is a large number, Heather notes that since an E. coli
bacterium weighs about 10 ¡12 gm, their total weight is only 10 10 gm, much less
than the 6 £ 1027 gm weight of the earth. Crichton's conclusion seems wrong.