A model for population growth is given by:
$$\frac{dN}{dt} = f(N) = r N \left( 1-\frac{N}{K} \right) \left( \frac{N}{U}-1 \right) $$
where $r,\ U,$ and $K$ are positive parameters and $U < K$. (a) Find all equilibrium solutions of this equation and classify each as stable, semistable, or unstable using calculus.
Please see the picture for the question at hand. Currently working my way through some questions and I've got a bit stuck here. I seem to have a logistic model multiplied by the explosion extinction model. I can easily identify what the equilibrium solutions are but how would I go about grading the stability via calculus and not a directionfield chart?
It's a lot simpler than it looks. Since you've already known that $0, K, U$ are equilibria, where $0<K<U$. Also, since $N$ represents a population, we'll only consider $N \ge 0$.
For $0 < N < K$, $f(N) < 0$ meaning $N$ will decrease to $0$.
For $K < N < U$, $f(N) > 0$ meaning $N$ will increase.
For $N > U$, $f(N) < 0$ meaning $N$ will decrease to $U$.
This establishes stability for $0, K$ and $N$. This method actually reveals the meaning behind the values of $K$ and $N$.
Alternatively, you can take $f'(N)$ like one of the comments suggested. But this is such a nice expression, why make it ugly to study just the mathematics part?