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Nondimensional quantities

  1. Jan 20, 2012 #1
    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. jcsd
  3. Jan 20, 2012 #2

    HallsofIvy

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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.
     
  4. Jan 20, 2012 #3
    That is probably right. I was just listing it how the book wrote it.

    How were this substitutions figured out though?
     
  5. Jan 20, 2012 #4
    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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