1.1: The Malthusian Growth Model
Let \(N(t)\) be the number of individuals in a population at time \(t\), and let \(b\) and \(d\) be the average per capita birth rate and death rate, respectively. In a short time \(\Delta t\), the number of births in the population is \(b \Delta t N\), and the number of deaths is \(d \Delta t N\). An equation for \(N\) at time \(t+\Delta t\) is then determined to be
\[N(t+\Delta t)=N(t)+b \Delta t N(t)-d \Delta t N(t) \nonumber \]
which can be rearranged to
\[\dfrac{N(t+\Delta t)-N(t)}{\Delta t}=(b-d) N(t) \nonumber \]
and as \(\Delta t \rightarrow 0\)
\[\dfrac{d N}{d t}=(b-d) N \text {. } \nonumber \]
With an initial population size of \(N_{0}\), and with \(r=b-d\) positive, the solution for \(N=N(t)\) grows exponentially:
\[N(t)=N_{0} e^{r t} \nonumber \]
With population size replaced by the amount of money in a bank, the exponential growth law also describes the growth of an account under continuous compounding With interest rate \(r\).