The question of determining the behavior of iterates of a function has often come up in natural studies. One area in particular is the study of population dynamics. This is concerned with modelling the ``population'' x(t) or number of individuals in a collection of organisms (that could be anything from single-celled creatures to people) as a function of time t. One approach to this problem is to make assumptions about the changes in x(t) from time to time. We might, for instance, assume that there is a constant ``birth rate'' b meaning that in a population of x individuals we can expect bx new individuals to be born over the course of one unit of time. This would be formulated as a finite-difference equation:
where 
is the population at time n. Iterating this relation
leads to the formula 
and rather gloomy Malthusian
predictions for that population.
Often, the population is thought to be continuously evolving, so that
ideally the measurements should be taken at shorter and
shorter time intervals h. This would lead to a difference
equation
writing x(t) as a function of time t. Rearranging this in the form
with t=nh held fixed and h tending to 0, we see this approximates the differential equation
known as the differential equation of exponential growth. The solution of this equation is a standard feature of every calculus and differential equations course. It is the exponential function
In this case, the discrete and continuous versions of this dynamical system behave in roughly the same way.
What we have not yet accounted for is the ``death rate.'' This might be assumed to be proportional to the population again, which would again lead to a difference equation of exponential growth if the birth rate exceeds the death rate, or of exponential decay if the opposite occurs. However, one might suppose that the death rate is proportional to the number of ``encounters'' between individuals. That is, as more and more individuals come into contact with each other, a greater percentage of the individuals starts to expire. This suggests what's known as the Verhulst model:
or as a differential equation
This equation is known as the Verhulst equation or the logistic equation. Solving this differential equation is a great pleasure of learning separation of variables in a first course in differential equations. Here we will simply state the final answer:
The exponential dies down to 0 as 
. Therefore, this
model predicts a limiting population of b/d no matter what the
initial population is. In this
figure, we show some sample solutions of
this equation with b=0.05 and d=0.001.
Figure 63: Solutions to the logistic equation: the different starting
populations are x(0)=10, 50, and 80.
The solutions to this differential equation all tend to the equilibrium population b/d=50. The very first thing one is shocked to discover about the analogous finite-difference equation is that the solutions can fail to stabilize in very wild ways. We turn to this topic in the next section.