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Nondimensional quantities 
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#1
Jan2012, 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. 


#2
Jan2012, 11:57 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,348

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.



#3
Jan2012, 12:02 PM

P: 629

How were this substitutions figured out though? 


#4
Jan2012, 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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