This is really a small cluster of questions on the same thing.
I was working with this problem to do with the temperature inside a cooler. The idea was that the rate of change of temperature inside the cooler was proportional to the difference in the inside and outside temperatures. So
$x'(t) = k(z(t))$
where z(t) is the difference in temps at time t.
But then I learned that this was only good for approximations around t=0, since it's a linearisation of
$x'(t) = f(z(t))$
where $f$ is some function on the temperature difference. If I wanted something more than an approximation and didn't want to linearise, what's a good choice of function for $f$?
Also, I could have built a function around the ratio of the inside to outside temps. What's the criteria for choosing between the two choices: the ratio and the difference?
Also, if I didn't know about logistic functions, and I'm building the DE from scratch, where do I begin so that a logistic function 'emerges'; what's the difference in the assumptions I make between the sum model and the ratio model that leads to a logistic equation?
If this question is too messy, I'll happily delete it.