Biology Reference
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a
x 100 , x 101 , x 102 , ...
1.50
1, 1, 1, 1,
...
2.10
0.823735, 1.12864, 0.823735, 1.12864, 0.823735, 1.12864, 0.823735, 1.12864,
...
2.50
1.225000, 0.535948, 1.157720, 0.701238, 1.225000, 0.535948, 1.157720, 0.701238,
...
TABLE 1-5.
Long-term behavior depends upon the value of parameter a.
E XERCISE 1-12
Consider the following population model:
2
P n þ 1
¼
P n
:
(1-27)
P K
1
þ
where K is the carrying capacity.
¼
¼
(a) Show that P
0 and P
K are the equilibrium states.
(b) Show that if 0 <P(0) <K, the population will be increasing.
(c) Assuming that lim P n exists as n
!1
, show that, under the
conditions of part (b), lim P n ¼
K.
IX. A POPULATION GROWTH MODELWITH DELAY
Despite substantial improvement over the ''unlimited'' population
growth model dP
the logistic growth model (1-12) has one major
drawback—replacing r with the factor r
dt ¼
rP
ð
t
Þ;
only
provides a mechanism for the net per capita growth rate to adjust itself
based on current population size. The logistic model (1-24) is entirely
based on the present and disregards, to a large extent, the past. In reality,
certain delay effects are essential, although this logistic model does not
account for them.
ð
P
ð
t
ÞÞ ¼
a
ð
1
P
ð
t
Þ=
K
Þ
Just as we need to appreciate the logistic model's merits, we also need to
understand its limitations. In Section IV, we found that the logistic
model described yeast and bacterial growth with great accuracy. It may
be less successful, however, in describing the growth of populations of
more complex organisms. For example, populations of the water flea
Daphnia have been observed to oscillate when cultures are maintained at
25 C (Pratt [1943]), as shown in Figure 1-21.
To recognize the effect caused by delay, notice, for example, that because
the gestational age of newborn human babies is about 9 months, the
number of babies born on January 1, 2005 was generally determined
9 months earlier, on April 1, 2004. In order to refine our model, we need
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