474 MHR • Unit 5 Population Dynamics
offspring would be born. Thus B=bNand by the
same reasoning D=dN. Using these concepts, a
more generalized version of the population growth
equation becomes:
∆N
∆t
=bN−dN=(b−d)NThis equation can be simplified even further.
What ecologists are really interested in is the
overall change in population size — the balance of
births and deaths. They therefore calculate r, the
average per capita growth rateof the population
as r=b−d. The value of rindicates whether a
population is growing (when r> 0 ) or declining
(when r< 0 ). The size of a population is stable
when r= 0. When this happens, births and deaths
are still occurring in a population, but the per capita
birth rate exactly balances the per capita death rate.
Using r(the population growth rate), the equation
describing change in population size can be
rewritten as:
∆N
∆t
=rNIn words, this means that the change in the size
of a population that occurs over some period of
time (∆N/∆t) depends on the growth rate of the
population (r) and on the size of the population (N).
This makes sense. Populations with higher growth
rates will grow faster, as will larger populations
(in which there are more individuals available to
produce offspring). Most ecologists actually use a
slightly different form of this equation, which uses
the symbols of calculus: dN/dt=rN. This means
essentially the same thing as the equation above.
The only difference is that here drepresents a very
small change (a very small ∆) and dN/dtis thus
the very small change in population size that occurs
over a very short time interval — an “instant.” In
this case, ris referred to as the instantaneous rate
of growth.
The formula dN/dt=rNdescribes the growth of
populations in ideal conditions, where supplies of
all necessary resources (for example, food, water,
space, shelter) are abundant and there are no limits
to population growth. Under these conditions, a
population can reach its biotic potential— the
highest rvalue (the highest possible per capita
growth rate) possible for that population. Whether
the biotic potential of a population is high or low
depends on many factors. These factors include the
number of offspring usually produced at each
reproductive event, how often each individual
reproduces, and at what age individuals start toreproduce (also referred to as the generation time
of the species).
The type of population growth that occurs in an
unlimited environment is termed exponential
growth. A graph of population size versus time for
a population showing this type of unlimited
growth would be an exponential curve — a curve
that rises slowly at first, then shoots up rapidly and
keeps on going, apparently forever (as shown in
Figure 14.10). The shape of this curve reflects what
you already know about changes in the size of a
population over time. Such changes depend on
both the growth rate (r) and the size (N) of the
population. When a population is just starting
out and Nis small, the population grows slowly
because there are relatively few individuals
reproducing. However, once the population reaches
a certain size, it starts to grow faster and faster as
ever-increasing numbers of individuals reproduce.
The result is often referred to as a J-shaped curve.Figure 14.10This figure shows an exponential or J-shaped
growth curve.Predicting Future Population Size
If both the current size (N 0 for “Nat time zero,”
which refers to the time at which an investigation
began) and the growth rate (r) are known, it is
possible not only to calculate how fast the
population is growing (or declining), but also to
predict what size it will be at some time in the
future. Doing this, however, requires looking more
closely at how and when new individuals are
added to an exponentially growing population.
Imagine a species of bacteria that divides to
produce two new cells every 30 min. If thePopulation size (N
)
Time