Chapter 6
Antidifferentiation
6.3 The Logistic Growth Differential Equation
6.3.4 Exponential Approach to Stability
We pause to reflect on a theme that has already occurred in several apparently different contexts.
In radioactive decay problems, we found that a function representing the quantity of a substance present at time `t` had the form `Ae^(-kt)`, an exponential with a negative exponent, and its values therefore decreased to zero as `t` became large.
With Newton's Law of Cooling, the solution function representing temperature had two terms: a constant, which was the ambient temperature, and an exponential that resembled radioactive decay. Thus, as time became large, the exponential term decreased to zero, and the temperature decreased to the ambient temperature.
For falling bodies with air resistance proportional to velocity, the computation was similar to Newton's Law of Cooling, but the factor multiplying the exponential term was negative. Thus the solution for velocity as a function of time,
had a constant term representing terminal velocity and a decreasing exponential term representing how much faster the object had to go to reach terminal velocity.
With the logistic model, we see a similar phenomenon, an approach as time goes on toward a stable population level, namely, the maximum supportable population `M`. We also see exponential functions in the solution
but the approach toward `M` is not immediately evident in this algebraic form of the solution — in fact, the coefficient of each exponential is `Mk`, a quantity that is clearly positive. These exponentials don't decrease with time, they grow — and at a very rapid rate (exponentially!). However, since there is a rapidly growing term in both numerator and denominator, the eventual behavior as `t` becomes large is indeterminate — another phenomenon we have encountered frequently.
On several occasions we have stressed the importance of changing the form of algebraic expressions as a problem-solving step or a way to gain a new insight. We use this approach here to mold our logistic solution formula into a form in which we can see that the population really approaches the maximum supportable population.
The exponentials grow when we think they should decrease? Let's change them into exponentials that decrease! How? Multiply and divide by the appropriate decreasing exponential. Here's the calculation:
Thus exactly the same function also can be described in a more revealing form:
The only exponential in this equation has a negative exponent, so it decreases to zero as time becomes large. And, as that term in the denominator approaches zero, `P` approaches `(P_0M)/P_0=M`. Hence we see in this algebraic representation of the solution the approaching behavior we saw previously in the numerical and graphical representations.
