Can somebody explain to me this integrodifferential equation?

end3r7

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. =)