Multi-part Gompertz growth problem

**drew.walker** · September 13th 2009, 09:38 PM

Growth of an individual chicken is modelled by the Gompertz differential equation $W'(t) = \mu_0 e^{-Dt}W$ . This gives the chicken's weight $W$ (in kilograms) in terms of constants $\mu_0$ and $D$ . Time $t \ge 0$ is days after reaching a weight of 0.25 kilograms.

i. What aspect of the chicken growth does the term $e^{-Dt}$ describe?

I have no idea what aspect this describes. Can anyone point me in the right direction about how to think about this sort of problem?

ii. Use separation of variables to find the general solution to this differential equation.

I think I can attempt this one. I'll give it a go and post my working. Essentially I'm finding an equation for $W(t)$ though right?

iii. In the case that $\mu = 0.068$ and $D = 0.022$ , find the time at which the chicken weighs 4 kilograms. What meanings and units do these constants have?

Is this simply a matter of calculating t using the equation from ii and substituting in the specified variables?

iv. Does the chicken have a maximum (or asymptotic) weight, and what is it?

Think I can work this one out once I have ii and iii.

v. At what time does the value of $W'(t)$ have a maximum? This gives the maximum growth rate of the chicken.

Think I can work this one out too. Will post working.

**shawsend** · September 15th 2009, 11:17 PM

Apparently you're having some problems with this.

(i) If you write it as:

$\frac{dW}{dt}=\mu_0 e^{-Dt}W(t)$

Then the rate at which the weight is changing is dependent on an exponentially decaying factor of the current weight right? As time proceeds, the rate at which the weight is changing must continually decrease because of this exponential factor. But that factor never goes to zero so I'd expect the weight to asymptotically approach some limiting value.

(ii) When you separate variables and integrate you'll get:

$\int_{W_0}^{W} \frac{dW}{W}=\int_{0}^{t} \mu_0 e^{-Dt}dt$

You can do that right? Integrate it and obtain the function $W(t)$ then try and do the rest.

**drew.walker** · September 15th 2009, 11:25 PM

Yeah, so if I integrate that I end up with:

$<br /> lnW = \frac{-\mu_0e^{-Dt}}{D}<br />$

Which I think is correct, but how can I make W the subject of this equation? I know I should know this stuff, but I'm very rusty.

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