Models & Qualitative Behavior · Superchallenge problem

An insect population $P$ (in thousands) grows logistically but is eaten by birds whose predation saturates: $P' = \dfrac{P}{2}\left(1 - \dfrac{P}{10}\right) - \dfrac{P^2}{1 + P^2}$. The population starts at $P(0) = 3$. What value does $P(t)$ approach as $t \to \infty$? Give an exact answer.