Nondimensionalization of the logistic equation.

Question

In the process of studying nonlinear dynamics by Strogatz, I saw how he did simplify the model for an insect outbreak with the use of nondimensionalization. So as an exercise I picked an equation and tried to nondimensionalize it. That equation happened to be the logistic equation for population growth.

Here is logistic equation for population gowth with two parameters $R$ and $K$:

$\frac{dN}{dt} = RN(1-\frac{N}{K})$, and

Here is the process by which I nondimensionalized it:

I defined $\tilde N = \frac{N}{N_s}$ and $\tilde t = \frac{t}{t_s}$. Therefore $N = N_s\tilde N$ and $t = t_s\tilde t$.

Now substituting the new variables for $N$ and $t$ we get:

$$\frac{N_s}{t_s}\frac{d\tilde N}{d\tilde t} = RN_s\tilde N(1-\frac{N_s\tilde N}{K}) = RN_s\tilde N - \frac{R{N_s}^2\tilde N^2}{K} \Rightarrow$$

$$\frac{d\tilde N}{d\tilde t} = Rt_s\tilde N - \frac{Rt_sN_s\tilde N^2}{K}.$$

Now we let $Rt_s = 1$ and therefore $t_s=\frac{1}{R}$. We also let $\frac{Rt_sN_s}{K}=1$ and therefore $N_s = K$. Substituting the new values for $t_s$ and $N_s$ we get:

$$\frac{d\tilde N}{d\tilde t} = \tilde N - \tilde N^2.$$

In the resulting equation from nondimensionaliztion there are no parameters. So here are my questions:

1. Did I do it correctly?
2. If I did, since there are no parameters, how am I supposed to analyze the resulting equation which is simply a $2D$ curve?

Thank you in advance.

Answer 1

You've imposed $t_s=R^{-1},\,N_s=K$; that's fine. What you need to do next is work out how $\tilde{N}$ varies with $t$. You should find a constant $\tilde{t}_0$ exists for which $\tilde{t}-\tilde{t}_0=\frac{1}{1+\exp -\tilde{N}}$.