Question

Solve the initial value Problem
dN/dt = τ (N − N^2/q) N(0) = No, where τ, q are positive. Hence or otherwise, find the solution of the Verhulst model
when t → ∞

Answer 1

Given equation is

$\frac{dN}{dt} = \tau (N - \frac{N^2}{q})$

Integrating the above equation

$\frac{dN}{(\frac{N^2}{q} - N)} = - \tau dt$

Integrating it,

$\int \frac{dN}{(\frac{N^2}{q} - N)} = - \int\tau dt$

Solving it we get,

$ln|1-\frac{q}{N}| = -\tau t + C$

Putting boundary equations, N(0) = N₀

Then we get

$ln|1-\frac{q}{N_0}| = C$

Hence, we can say that

$ln|1-\frac{q}{N}| = -\tau t + ln|1-\frac{q}{N_0}|$

It can be written as

$ln|\frac{N(N_0-q)}{N_0(N-q) }| = \tau t$