Assume that $\lim\limits\_{t\to \infty}c(t)=a+b-r$.

Is it true, that the limit of the following differential equation is:

$\lim\limits\_{t\to \infty}\dot p(t)=p(t)(r+\lim\limits\_{t\to \infty}c(t)-a-b)=0$

so that $\lim\limits\_{t\to \infty}p(t)=const.$?

The differential equation is given by:

$\dot p(t)=p(t)(r+c(t)-a-b)$.

Can someone verify that $p(t)$ is a constant in the long run?

Is it critical here how fast $c(t)$ converges or not?