Q2. Consider the logistic growth model described by the differential equation:
$$ \frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right) $$
where:
ā¢ $P(t)$ is the population at time $t$,
ā¢ $r$ is the intrinsic growth rate which is constant (in 1/year),
ā¢ $K$ is the carrying capacity of the environment (the maximum sustainable population).
Nondimensionalize the Logistic Growth Equation by introducing the nondimensional variables:
ā¢ $p^* = \frac{P}{P_0}$ (nondimensional population),
ā¢ $t^* = \frac{t}{T}$ (nondimensional time)
where $T$ is the characteristic time and $P_0$ is the characteristic population. Derive the nondimensional form and identify the characteristic time $T$ and the characteristic population $P_0$ explicitly.
