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Nondimensional quantities

by Dustinsfl
Tags: nondimensional, quantities
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Dustinsfl
#1
Jan20-12, 11:48 AM
P: 629
The DE is Insect Outbreak Model: Spruce Budworn with Ludwig's predation model

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

[itex]r_B[/itex] is the linear birth rate

[itex]K_B[/itex] is the carrying capacity

The last term is predation

[itex]A[/itex] is the threshold where predation is switched on

[itex]A,K_B,N,r_B[/itex] has the dimension [itex](\text{time})^{-1}[/itex]

[itex]B[/itex] has the dimension [itex]N(\text{time})^{-1}[/itex]

Nondimensional quantities

[tex]u=\frac{N}{A}, \ r=\frac{Ar_B}{B}, \ q=\frac{K_B}{A}, \ \tau=\frac{Bt}{A}[/tex]

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.
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HallsofIvy
#2
Jan20-12, 11:57 AM
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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 rB have dimensions of (time)-1 so their product has dimension (time)-2, canceling the dimensions of B.
Dustinsfl
#3
Jan20-12, 12:02 PM
P: 629
Quote Quote by HallsofIvy View Post
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 rB 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
#4
Jan20-12, 01:56 PM
P: 629
Nondimensional quantities

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

[tex]\frac{du}{dt}=ru\left(1-\frac{u}{q}\right)-\frac{u^2}{1+u^2}[/tex]

From the substitution, I actually obtain:

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

How can I manipulate that into the correct answer?

Or is there a mistake somewhere?


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