# Nondimensional quantities

by Dustinsfl
Tags: nondimensional, quantities
 P: 629 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.
 P: 629 Nondimensional quantities 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?