Biology Reference
In-Depth Information
700
600
500
400
300
200
100
0
0
2
4
6
8
10
12
14
16
18
Time
FIGURE 1-6.
A logistic curve. The solution of the logistic equation (1-12) for P(0)¼ 5, K ¼ 660, and a ¼ 0.7 (solid
line). The dashed line corresponds to the carrying capacity K.
trajectory describes the evolution of the function quantity in time. The
graph of the solution given by Eq. (1-12), for the special case of P(0)
¼
5,
K
¼
660, and a
¼
0.7, is shown in Figure 1-6.
E
XERCISE
1-6
Consider the solution of the logistic model (1-13). What happens to
P(t) as time gets very large (t
!1
)? Consider the following cases
separately:
(a) P(0)
¼
0,
(b) 0 <P(0) <K,
(c) P(0)
¼
K, and
(d) P(0)
>
K.
It is gratifying that the solution (1-13) of our modified model produces the
distinctive sigmoidal (S-shaped) curve exhibited by the yeast growth data
in Figure 1-4. Comparing the model predictions with the actual data is also
encouraging. Using the value of a
0.543 calculated in Exercise 1-3
as the per capita growth rate during the initial growth phase
(which is the inherent per capita growth rate for the logistic model) and a
value for the carrying capacity K
¼
¼
660, estimated from the data,
we obtain the graph in Figure 1-7.
We note that function (1-13) is just one of many different functions
exhibiting S-shaped trajectories like the one in Figure 1-7. Such
trajectories are often given the generic name ''logistic curves,'' a term