\[ \newcommand{\euler}{\mathrm{e}} \newcommand{\d}{\text d} \newcommand{\dt}{\d t} \newcommand{\dN}{\d N} \newcommand{\dP}{\d P} \]
There are difference equation form and differential equation form. The difference form is \[ N_{t+T} = \lambda N_t ~~\Rightarrow~~ N_t = N_0\lambda^{t/T} ~~\Rightarrow~~ \lg N_t = \lg N_0 + \frac{t}{T}\lg\lambda \]
where \(\lambda\), also denoted as \(R_0\) is the finite rate of increase of population, \(T\) is the generation time. if we denote \(n:=t/T\) where \(n\) is the number of generation, the formula becomes \[ N_t = N_0\lambda^n \Rightarrow \lg N_t = \lg N_0 + n\lg \lambda \] The differential form is \[ \frac{\dN}{\dt} = rN ~~\Rightarrow~~ N(t) = N_0\euler^{rt} \] where \(r\) is the instant growth rate.
Comparing the two form we have \(\lambda = \euler^{rT}\), or \(r = \dfrac{\ln \lambda}{T}\).
Physical interpretation of parameters \(\lambda ,r\):
| \(\lambda\) | \(r\) | |
|---|---|---|
| Population increase | \(>1\) | \(>0\) |
| Population stable | \(1\) | \(0\) |
| Population decrease | \((0,1)\) | \(<0\) |
| Population extinct | \(0\) | \(\rightarrow -\infty\) |
\[ \frac{\dN}{\dt} = rN\left( 1 - \frac{N}{K} \right) \]
The physical interpretation of \(\left(1-\dfrac{N}{K}\right)\) is the left living space of population.