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

Therefore,

So, if we have , then and , so

where

Rori- wondering if you could explain how you got T0=T1 at the end. Is this just the obvious choice to remove the exponential? Thanks.

Okay think I got it now- T1 was just used for the initial condition ,a, and M/2 ,a constant value, is set as x(a)=b.