[From Bill Powers (950104.1635 MST)]

Bruce Abbott (960104.1430) --

A very interesting model you have come up with. I admire your

mathematical bravery! But jeez, Bruce, you are going to give me a brain

hernia if you keep making me try to remember how to solve differential

equations.

(1) R = B/n environment function (ratio schedule)

(2) F = F0 + (m*R - M)*t food amount at time t

(3) p = F perceptual signal

(4) e = Fr - F error signal

(5) B = g * e output function

For a simulation solution, the second line should be cast as an

integral:

F := F + (m*R - M) * dt,

with F initialized to F0. This says that in one iteration, the amount of

F increases by the amount m*R, and decreases by the amount M, times the

time per iteration, dt.

Put in terms of the calculus, the second line would be written

dF/dt = m*R - M.

The constant of integration, F0, will be introduced when you integrate

the combined equations to solve them.

Solving for dF/dt, we have

dF/dt = m*g*(Fr - F)/n - M.

If we set k = -m*g/n and

a = (m*g/n)*Fr - M,

this equation becomes a recognizable form,

dF/dt = kF + a.

The general solution with a parameter A to be determined is

(6) F = A*exp(k*t) - a/k

Putting the equivalents of k and a back into equation (6) we get

F = A*exp(-m*g*t/n) + Fr - M*n/(m*g)

At t = 0 (where F = F0) we have

F0 = A + Fr - M*n/(m*g), or

A = F0 - Fr + M*n/(m*g)

This leads to the final solution for F:

F = F0*exp(-m*g*t/n) + (Fr - M*n/(m*g)]*[(1 - exp(-m*g*t/n)]

That's the best I can do. I hope I did this right. The only way to check

the answer is to run a simulation and see if the analytical solution

matches the simulation output. I'll try to do that. If anybody else on

the net cares to check my solution I would be grateful. I've spent too

many years avoiding having to do this kind of math.

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I don't know if I can express intelligibly what your mistake was in step

2. The basic problem is that you presented an equation for F as if the

effect of R simply increases linearly with time. But R depends on F, so

the real solution will not be a linear function of time. You have to

approach the solution in increments of dt, and as you can see the result

is a different form of the equation for F. Some of your numerical

results may be right, since you used the balance between mR and m to

find them. But some others may be wrong -- we'll see.

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Best,

Bill P.