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Math Help - Prey/Predator Model (need help asap please)

  1. #1
    Junior Member
    Joined
    Mar 2007
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    Prey/Predator Model (need help asap please)

    Hey I need help with a question regarding a prey/predator model. The question is in 2 parts (paraphrasing them). (i) if there are no predators, what is the solution (e.g will the population exponentially grow/decline). (ii) if there are no prey, what is the solution (e.g will the population exponentially grow/decline).

    dx/dt = 0.3x - xy/100 <--- prey

    dy/dt = 15y(1-y/15) +25xy <--- predator

    My solutions.

    (i) dx/dt = 0.3x - xy/100
    dx/dt = 0.3x (y=0)

    dx/x = 0.3 dt

    integrating

    ln(x) = 0.3t + c
    x= ke^0.3t (k = e^c)

    therefore the population is increasing exponentially.

    (ii) dy/dt = 15y(1-y/15) +25xy
    dy/dt = 15y(1-y/15) (x=0)
    dy/dt = 15y - y^2
    dy/dt = -y(-15 + y)
    dy/y(-15+y) = -1 dt

    integrating from this point is where I'm finding difficulty. I've attempted it, but I'm pretty sure I'm wrong.

    (1/y - 1/(-15+y))dy = -1 dt
    ln(y) - ln(-15+y) = -t + c
    ln(y/(-15+y)) = -t + c
    y/(-15+y) = ke^-t (k = e^c)
    y = (ke^-t)(-15 + y)

    the last line doesn't make sense to me, because I was trying to get y = something that doesn't include a 'y' in it... can someone please help me?
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  2. #2
    Junior Member
    Joined
    Mar 2007
    Posts
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    Also I'd like to ask another question related to this. If you've got this equation:

    dR/dt = 0.4R

    and you integrate it.

    dR/R = 0.4 dt

    ln(R) = 0.4t + c

    R(t) = ke^0.4t (k = e^c)

    Now lets just say I start with a 8 R's:

    So dR/dt = 0.4(8)
    = 3.2

    (1) R = 8 + 3.2 = 11.2

    Now if I plug the same values into R(t):

    k = 8
    t = 0

    R(t) = 8e^0.4(0) = 8

    I get 8 which seems correct.

    now If I put t = 1, I get:

    R(t) = 8e^0.4(1) = 11.9346 (4 d.p)

    Now according to my (1) I should get 11.2 not 11.9346. Why is this?

    Also I'd like to know why R(t) = k(1.4)^t works?

    From what I understand, R(t) = ke^0.4t is the solution, not R(t) = k(1.4)^t.

    Can someone please help me with this. I'm very lost
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