2.5.1 A Difference Equation and a Differential Equation

We return to our examination of models of population growth. For our first model we make the assumption that populations tend to grow at a rate proportional to their numbers. In particular, this says that the more individuals there are in a population, the faster the population grows. However, “proportional” says something more specific about the rate of growth: For a given time step $\Delta t$, the mathematical interpretation of this biological principle is

where \(k\) is a constant and \(P\) is the population at the start of a time interval of length $\Delta t$. Because of the differences, $\Delta P$ and $\Delta t$, and the assertion of equality appearing in this formula, it is called a difference equation.

Note 1 – A prior example of a difference equation

For some populations — bears, for example — $\Delta t$ is essentially constant. That is, there are fixed times at which reproduction is possible. If we assume that conditions are so favorable for our population that few or no deaths occur in the time frame of interest, and that there are no migrations in or out of the population, then reproduction accounts for all the change in the population, and we may reasonably assume that $\Delta t$ is constant, that it measures the smallest interval between reproduction times. On the other hand, there are biological populations — bacteria and humans, for example — in which reproduction can and does occur at any time. In these cases there is no smallest value $\Delta t$ can assume.

Now let's examine this growth equation,

under conditions of continuous change. The equation says that, no matter how small $\Delta t$ is, the average rate of change of \(P\) is proportional to \(P\). We need to state that a little more carefully: The proportionality constant \(k\) actually depends on the time step (but not on the population). For each fixed $\Delta t$, no matter how small $\Delta t$ is, the average rate of change of \(P\) is proportional to \(P\). If we let $\Delta t\to 0$, we have already seen that \((\Delta P) / (\Delta t)\) approaches the instantaneous rate of change of \(P\) .

Of the two factors on the right, only the \(k\) can change as $\Delta t$ changes; \(P\) stands for the population at the start of a time interval, and that is independent of the length of the interval. Thus, the instantaneous rate of change of \(P\) must also be proportional to \(P\), i.e.,

where \(K\) is whatever \(k\) approaches as $\Delta t$ approaches zero.

Note 2 – Initial growth rate

Since the differences in

have become differentials in

we call the latter a differential equation.

Note 3 – Terminology

To summarize, our model of natural biological growth is the difference equation if the reproductive times are discrete, the differential equation if reproduction is occurring continuously. As we consider influences that hinder or enhance natural growth rates, we will modify the models accordingly. Our immediate goal is to determine the implications of the differential equation, i.e., to solve the equation in order to find functions that can represent natural populations.