Setting up a differential equation where population growth rate $\propto$ population size.

Question

I'm having a hard time with this one problem:

A populations growth rate is proportional to the population size, when the population is below a certain threshold. The proportionality factor is $0.043$ in this area (that is, when it is below said "limit"). Set up a differential equation that (under the said limit) can be used to determine the expression of the population as a function of time.

How do you make a differential equation when it has to be under a threshhold?

Answer 1

As @lulu mentioned, you are only asked to find the differential equation below the threshold. Therefore, you don't need to consider what happens above the threshold and you can simply ignore this fact.

---

Let us analyse the important parts of the problem:

A populations growth rate is proportional to the population size.

Let $P$ be the population. Now, we can represent population growth rate in mathematical notation as $\frac{dP}{dt}$. Do you understand why? Therefore, in mathematical notation, we have: $$\frac{dP}{dt} \propto P$$

The proportionality factor is $0.043$ in this area.

Recall the definition of the proportionality constant, then you should have it.