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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?

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    $\begingroup$ 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. $\endgroup$ Jun 30, 2023 at 9:11

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To answer your question one needs to understand the space of solutions of an ODE. As hinted below in your question, many mathematical results prove the existence (e.g Peano) and uniqueness (e.g. Picard) of a solution under certain mild conditions.

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.

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  • $\begingroup$ Peano theorem and Picard theorem are two different theorems, with very important differences in their proofs. $\endgroup$
    – Artem
    Jul 2, 2023 at 18:29
  • $\begingroup$ I expressed myself in the wrong way, my bad. I'll modify the answer to avoid misunderstandings, thanks for pointing it out. Still, since the "Picard-Lindelof-Cauchy-Lipschitz" theorem has the existence part in it, at times people may also add the name "Peano" in front. A rare convention that is curious to know :) $\endgroup$
    – Sandro
    Jul 2, 2023 at 20:00
  • $\begingroup$ Not really. These people, like yourself, are simply wrong. $\endgroup$
    – Artem
    Jul 2, 2023 at 20:01
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    $\begingroup$ I honestly didn't know that it/I was wrong since I was introduced to those theorems for the first time this way. Now, I know that and I will not make this mistake again. Thanks. $\endgroup$
    – Sandro
    Jul 2, 2023 at 20:08

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