Nondimensional substitutions for Insect Outbreak Model

[Dustinsfl]

The DE is Insect Outbreak Model: Spruce Budworn with Ludwig's predation model

$$\frac{dN}{dt}=r_BN\left(1-\frac{N}{K_B}\right)-\frac{BN^2}{A^2+N^2}$$

$r_B$ is the linear birth rate

$K_B$ is the carrying capacity

The last term is predation

$A$ is the threshold where predation is switched on

$A,K_B,N,r_B$ has the dimension $(\text{time})^{-1}$

$B$ has the dimension $N(\text{time})^{-1}$

Nondimensional quantities

$$u=\frac{N}{A}, \ r=\frac{Ar_B}{B}, \ q=\frac{K_B}{A}, \ \tau=\frac{Bt}{A}$$

How were this substitutions decided on?

I see that u,q is nondimensional since they cancel, but r and tau I don't get it.

 

[HallsofIvy]

I'm not sure I understand what you mean when you say that "B has dimension N(time)^-1". Since you have said that N has dimensions of (time)^-1 itself, do you mean that B has dimensions of (time)^-2? If so then Bt has dimensions of (time)^-1, the same as A and so Bt/A is dimensionless. Also, both A and r_B have dimensions of (time)^-1 so their product has dimension (time)^-2, canceling the dimensions of B.

 

[Dustinsfl]

I'm not sure I understand what you mean when you say that "B has dimension N(time)^-1". Since you have said that N has dimensions of (time)^-1 itself, do you mean that B has dimensions of (time)^-2? If so then Bt has dimensions of (time)^-1, the same as A and so Bt/A is dimensionless. Also, both A and r_B have dimensions of (time)^-1 so their product has dimension (time)^-2, canceling the dimensions of B.

That is probably right. I was just listing it how the book wrote it.

How were this substitutions figured out though?

 

[Dustinsfl]

Additionally, when I make the substitution, I should obtain:

$$\frac{du}{dt}=ru\left(1-\frac{u}{q}\right)-\frac{u^2}{1+u^2}$$

From the substitution, I actually obtain:

$$uBr\left(1-\frac{u}{q}\right)-\frac{A^3\tau N^2}{t(u+A^2N^2}$$

How can I manipulate that into the correct answer?

Or is there a mistake somewhere?