# Autonomous ODE equilibrium solutions - are they always asymptotic?

I'm a beginner to ODEs (and to math in general, pretty much. I'm mostly equipped with high-school level math currently). I'm studying the Logistic Growth differential equations:

$$y' = ay(1 - y)$$

Where $$y$$ is a value between $$0$$ and $$1$$ - the proportion of some quantity (e.g. an animal population) in relation to its maximum capacity. $$a$$ is some (positive?) constant.

I understand that this equation has two equilibrium solutions: $$y = 0$$ and $$y = 1$$.

When $$y = 1$$, it makes sense that $$y'$$ will be $$0$$, because that means we have reached the maximal capacity of the environment in question, so no more growth can happen.

However, I'm trying to understand whether equilibrium points (generally in autonomous ODEs) are necessarily asymptotic. Assuming the initial condition is not $$y = 1$$, is it possible for $$y$$ to ever actually reach $$y = 1$$? Or can we only ever get asymptotically close?

• Check the various uniqueness statements, in nice systems solutions never cross or branch in or out. There are "not-so-nice" equations where this happens, an easy study are the Clairaut DE. Jun 30, 2023 at 9:11

One of these theorems says that if the state function $$f$$ and the derivative with respect to the unknown (in your case $$y$$) are continuous in a certain open set, then the solution exists and is locally unique around an initial value in that set.
In the logistic equation, these properties are satisfied and since $$y=1$$ is a solution, any other solution cannot cross (not even touch) the line $$y=1$$, else it would violate the uniqueness property. So, you'll get asymptotically close to that solution.