\section{The Logistic Equation}
The so called logistic equation, that we discuss in this section, arises in the analysis of the growth of a population. We first analyze a sitauation where there are no predators and there is an unlimited supply of food. We consider a population of rabbits. Assume that on the average each rabbit gives birth to $\sigma$ new rabbits in one unit of time, and that no rabbit dies. Let $N(k)$ be the number of rabbits at time $t=k$, then the change in population over a time period, between $k$ and $k+1$ is
\[
N(k+1)-N(k)=\sigma N(k).
\]
Thus,
\[
N(k+1)=(1+\sigma)N(k).
\]
Given the number $N(0)$ of rabbits at time $k=0$, the number of rabbits at time $k$ is
\[
N(k)=(1+\sigma)^k N(0).
\]

We now try to make the above model more realistic. We assume, as before, that the number of births is proportional to the size of the population. Let $b$ be the birth rate, then the number of births at time $k$ is $b N(k)$. We also assume that the number of deaths is proportional to the popualtion size. Let $d$ be the death rate. Then, the number of deaths at time $k$ is $d N(k)$. The change in population in a time period between times $k$ and $k+1$ is
\[
N(k+1)-N(k)=bN(k)-dN(k).
\]
We rewrite the above equation as
\[
N(k+1)=(1+b-d)N(k).
\]
Let $\sigma=b-d$. then the above equation has the same form as the previou equation. In both cases, the popualation grows exponentially. We refer to the constant $\sigma$ as the {\em unrestricted growth rate\/}. Over short periods of time, the above equations may be sufficient for our purposes. Over long periods of time, the growth rates may change as the size of population changes.

We now alter the above exponential models by introducing the concept of satuaration. We first assume that the environemnt of the population can only support a certain number of species. We denote this number by $L$, and call it the {\em carrying capacity\/}. Thus, if $N(k)>L$, then there will not be enough food or space available, and more species will die than are born. In such a situation the growth rate should be negative. If the population at time $k$ is $N(k) < L$, then the the growth rate should be positive. Finally, if $N(k)$ is much smaller than the carrying capacity, then the growth rate should be closed to the unrestricted growth rate $\sigma$. A possible function that satisfies the above requirements has the form
\[
\sigma \left( 1-\frac{N(k)}{L}\right).
\]
The change in the size of population then can be described as
\[
N(k+1)-N(k)=\sigma \left( 1-\frac{N(k)}{L}\right),
\]
or equivalently
\[
N(k+1)=(1+\sigma)N(k)-\frac{\sigma}{L}N^2(k).
\]
The above equation is called the {\em logistic equation\/}. The term $\frac{\sigma}{L}N^2(k)$ is referred to as the {\em damping term\/}, because it dampens the growth of the population.