The simplest population growth law is given by
where is the population and is a constant dependent on the type of population in question. A more involved growth law in which the population is subject to a maximum constraint , gives an equation for population growth of
where is a constant, or possibly a function of time.
Express the population as a function of in each of these growth laws (with and both constant). Prove that the result in the second growth law can be expressed as:
where is a constant and is the time at which .
Proof. For the first growth law,
This has solutions of the form
where at time .
For the second, more complicated, growth law we have
This is a Bernoulli equation (as seen in this exercise, Section 8.5 Exercise #13) so we know where is the unique solution to
Thus, we are looking for the unique solution of
Using Theorem 8.3 (page 310 of Apostol) for the solutions of first-order linear differential equations, we have
So, if we have , then and , so