Exponential growth
Exponential growth is the simplest continuous model of population change. It assumes that each individual contributes to population growth at the same constant per-capita rate.
The model
\[\frac{dN}{dt}=rN,\qquad N(0)=N_0.\]Here \(N(t)\) is population size, \(r\) is the constant per-capita growth rate, and \(N_0\) is the initial population.
Solution
\[N(t)=N_0e^{rt}.\]If \(r>0\), the population grows exponentially. If \(r<0\), it declines exponentially. If \(r=0\), the population remains constant.
Doubling time. For \(r>0\), the time required for the population to double is \[T_d=\frac{\ln 2}{r}.\]
Biological interpretation
The model assumes unlimited resources and no density effects. It is therefore most appropriate for short periods or low-density populations where competition is weak.
Key idea. Exponential growth gives a baseline model in which the per-capita growth rate is constant and total growth is proportional to population size.