I don't understand the sources I found online, so I'm hoping someone can show me how to do the following: I need to put together a 1st-order DE that models some population and includes birth rate, death rate (which is proportional to the size of the population to account for overcrowding), and harvesting. Say, for instance, I'm modelling a population of fish.
The furthest I can get is this:
$$p'(t) = bp(t)-d(t)p(t)-hp(t)$$
where $p(t)$ is the size of the population at time $t$, $b$ is the constant birth rate, $d(t)$ is the death rate, and $h$ is the constant harvest rate.
Now, I've guessed that the death rate is proportional to the difference between the maximum population and the current population, so:
$$d'(t) = (N-p(t))k$$
Where $N$ is the max. pop., and and $k$ is the constant proportion.
So, do I solve this second equation for $d(t)$ and plug it into my first equation, then solve for $p(t)$? In which case, I'd get
$$p'(t) = bp(t) - \left(kNt-k\int p(t)dt + C_0\right)p(t)-hp(t)$$
And that seems far too ugly to be correct.