Better population models than the logistic equation
It is not complicated to make your own model of population growth

You can make a model with any kind of function, it is not super hard. It just happen that logistic growth offer quite a good match to observations. You can have exponential growth, linear growth, quadratic growth, ...

However, there are more possible behaviours behind those models than what you think!

The behaviour that you describe is totally possible with a discrete time logistic model. In short, consider the recursion equation $n_{t+1} = f(n_t)$. Consider also $\hat n$ as being an equilibrium of your system (typically n=0 or n=k, where k is the carrying capacity). In other words, $\hat n$ satisfies the equation $\hat n = f(\hat n)$. And finally consider the statistics $\lambda$ which is equal to the derivative of $f(n)$ over $n$ evaluated at the equilibrium $\hat n$. If $\lambda &lt; 0$, then you have oscillatory behaviour (and therefore overshoot) at the vicinity of the equilibrium $\hat n$. Using the correct parameters, the logistics equation gives you a lambda smaller than 0 (typically between $-1 &gt; \lambda &gt; 0$), and therefore you can have the oscillatory behaviour you describe.

If you have any trouble to make these basic analyses of those models, I strongly recommend that book.