Question 3.9.8: Finding the Maximum Rate of Population Growth Suppose that a...
Finding the Maximum Rate of Population Growth
Suppose that a population grows according to the equation p^{\prime}(t)=2 p(t)[1-p(t)] (the logistic equation with r =2). Find the population for which the growth rate is a maximum. Interpret this point graphically.
To clarify the problem, we write the population growth rate as
f(p)=2 p(1-p).
Our aim is then to find the population p ≥ 0 that maximizes f (p). We have
\begin{aligned}f^{\prime}(p) & =2(1)(1-p)+2 p(-1) \\& =2(1-2 p)\end{aligned}and so, the only critical number is p=\frac{1}{2} \text {. Notice that the graph of } y=f(p) is a parabola opening downward and hence, the critical number must correspond to the absolute maximum. In Figure 3.104, observe that the height p=\frac{1}{2} corresponds to the portion of the graph with maximum slope. Also, notice that this point is an inflection point on the graph. We can verify this by noting that we solved the equation f^{\prime}(p)=0, where f(p) \text { equals } p^{\prime}(t) \text {. Therefore, } p=\frac{1}{2} is the p-value corresponding to the solution of p^{\prime \prime}(t)=0. This fact can be of value to population biologists. If they are tracking a population that reaches an inflection point, then (assuming that the logistic equation gives an accurate model) the population will eventually double in size.
