Biology Reference
In-Depth Information
U.S. population in the year 1875. We can calculate values for the
U.S. population in 1870 or in 1880, but not for the intermediate years
(although such values could be interpolated). More importantly, our
model has the added limitation that it does not capture change as it
occurs over time and instead assumes that the changes are compounded
at the end of each unit of time. This certainly is not how the size of the
U.S. population changes. New births, as well as deaths, occur in the
United States practically every minute (actually, on average, every
8 seconds, according to current U.S. Census Bureau data), so the
population changes almost continuously. A useful model should be
capable of capturing the instantaneous dynamics of the population and
should assume that every time instant is equally likely to be a time of
change in the population size.
When studying populations of some other living organisms, however,
using discrete models may be more realistic if the organisms reproduce in
a synchronized manner. For example, annual flowers die in the fall and
their offspring appear in the spring, bears have their cubs in midwinter,
and deer have their fawns in the spring. In the laboratory, cell biologists
have learned much about the control of the cell cycle through the artificial
synchronization of cell division. When modeling these kinds of
phenomena, it is more appropriate to consider discrete models.
III. A CONTINUOUS POPULATION GROWTH MODEL
What modifications would be necessary to build a continuous
population growth model? Continuous mathematics has calculus as one
of its essential components, and measuring rates of instantaneous
change is one of the fundamental uses of calculus. Mathematically, an
instantaneous rate of change is represented by the derivative of the
function that describes how a given quantity changes with time. Thus, if
P(t) denotes the U.S. population at time t, then the instantaneous rate of
change of the population can be expressed by the derivative dP(t)/dt or
P
0
(t).
We are now ready to express our major hypothesis that there is a linear
dependence between the rate of change in population size and the population size
itself. In the language of calculus:
dP
ð
t
Þ
¼
rP
ð
t
Þ:
(1-2)
dt
The left-hand side of this equation gives the (instantaneous) rate of
change for P(t) at time t. The right-hand side expresses this rate as a
fraction (r) of the current population size P(t). Notice that this model
represents exactly the same hypothesis as before. The only reason
Eq. (1-1) looks different from Eq. (1-2) is that they state our hypothesis in
two different languages—Eq. (1-1) uses the language of discrete
