6.15 Behaviour of solutions

The logistic equation is often used to model a biological population, where $N$ indicates the number of individuals at time $t$. Let us examine the behaviour of the various solutions to the equation. system, it is sensible to examine the behaviour of the solutions obtained, to see if it matches the expected behaviour of the physical system.

For any value of $C^{\prime}$ the denominator $1+C^{\prime}e^{-rt}$ tends to ${1}$ as $t$ tends to infinity, so that $N(t)$ tends towards $K$ in the long-term. This suggests that $K$ is a ‘theoretical maximum’ (sustainable) population, sometimes called the carrying capacity. If $C^{\prime}$ is positive then $N(t)$ starts off smaller than $K$ and increases towards the limit; while if $C^{\prime}$ is negative then the population is initially larger than the carrying capacity.

Although it can be tempting to think that the method we applied will produce all possible solutions of the equation, note that there are two solutions which we missed. One of these is given by allowing $C^{\prime}$ to be zero, so that $N(t)$ is constant and equal to $K$.

Question: what is the other missing solution? Can we have $C^{\prime}\leq-1$?