Biology Reference
In-Depth Information
2. Describe the conjecture, as well as possible, in quantitative and analytical
terms. This phase may involve statistics, mathematical formulations,
and follow-up analyses. Statisticians and mathematicians
will usually carry out this phase in close collaboration with
biologists. This process often leads to clarifying and refining the
hypothesis.
3. Test the refined hypothesis on several data sets. Consider the limitations
of previous experiments, and design your own new data collection
in order to address them. Formulate your refined conjectures.
Each of these steps can sometimes be carried out and thoroughly
explored within hours or days. In other cases, it may take much longer.
Charles Darwin, for example, took several decades to systematically
collect data for his famous On the Origin of Species by Means of Natural
Selection.
When applying the steps outlined above to the growth of populations,
our hypothesis passed the ''expert opinion test,'' but only conditionally.
We learned that the rate of growth of populations might, indeed, be
proportional to the size of the population, but only during the initial phases
of their growth. This phase could be characterized as a period during
which an abundance of resources allows for unfettered growth. During
later phases, the growth of the population might be impeded by
competition or a shortage of resources. So our hypothesis had potential
and, in fact, it seemed reasonable that the period from 1800 to 1860 was an
''initial phase of growth'' for the U.S. population. However, the model
developed on our general hypothesis had its limitations—not a big
surprise, given that it was our first model. We also began to understand
some of the rationale for these limitations. We decided, nevertheless, to
move on to describing our hypothesis quantitatively and analytically.
Denoting the U.S. population at the end of the n-th decade by P
n
(where n
can take the integer values 0, 1, 2, 3,
). We can express the change in
population size from the beginning of one decade to the next by P
n
...
P
n
1
,
for n =1,2,3,
. The conjectured linear relationship between the rate of
change of population and the population size itself then means that the two
quantities are proportional. Thus, there is a constant k such that the
relationship
...
P
n
P
n
1
¼
kP
n
1
(1-1)
is satisfied for any value n
¼
1, 2, 3,
. In particular, for n
¼
1, we have
...
P
1
kP
1
; etc. Notice that the constant of
proportionality k can be interpreted as the net per capita rate of change
(also referred to as the net per capita growth rate) for the population. The
left-hand side of our model represents the change per decade, and the
right-hand side expresses this change as a multiple (k) of the population
size in the beginning of the decade.
P
0
¼
kP
0
; for n
¼
2, P
2
P
1
¼
