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Can somebody explain to me this integrodifferential equation?

  1. Jul 30, 2007 #1
    It's a volterra equation with a continous delay

    *I is negative infinity, couldn't figure out how to write it

    [tex]
    \dot{x} = rx(t)[1 - K^-1\int_{I}^{t} k(t-s)x(s)\,ds]
    [/tex]

    The part in parenthesis is the density dependent factor, but I don't understand how the integral works exactly. I know the function k(t) is a weightfactor which says how much weight should be given to past populations.

    So let's see if I get it, feel free to yell at me (AKA reply in CAPS) if I am wrong.

    k(t-s)x(s) is same as the function k(t) shifted to the right by s multiplied by a scalar (here x(s) represents a population at time t=s).
    Thus the integral is just gonna be the summation of all these functions from initial up to current time. This is gonna be a function in 't'.

    If the max of the kernel occurs at zero then there is almost no delay, right?
    Cuz it follows that, say, k(t-s)x(s) will contribute the most to the resulting integral when s=t=now.

    On the other hand, if max is at t=T then the major contributor will be when t-s = T, or s = t-T, that is, T generations ago.

    If anybody wants to add/correct anything, feel more than free. =)
     
  2. jcsd
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