The weight w(t) of an individual fish at time t is governed by the differential equation.
$ \frac{dw}{dt} = \alpha w^{2/3}-\beta w $
with initial condition $ w(0) = w_{0} $ ( the weight at birth) and where $ \alpha$ and $ \beta$ are positive parameters depending on the fish species. Without solving the differential equation, determine
$$ \lim_{t\to \infty} w(t)$$
I am not sure how to analyse this without solving? Maybe graph it, but I am not even sure how to start that, any help help appreciated.
Best Answer
You have $\frac{dw}{dt} = w^{2/3}(\alpha - \beta w^{1/3}),$ so if $w^{1/3} = \alpha/\beta$, then $\frac{dw}{dt} = 0$ for all $t$, so the graph of the solution with $w(0) = (\alpha/\beta)^3$ will be a horizontal line.
If $w(0) > (\alpha/\beta)^3$ then $\frac{dw}{dt} <0$ by looking at the first equation I wrote. So the graph of that solution is decreasing.
Likewise, if $w(0) < (\alpha/\beta)^3$ then $\frac{dw}{dt} >0$, and the graph is increasing. So no matter what the initial condition, the graph of the solution will be asymptotic to the horizontal line above. That makes your limit $(\alpha/\beta)^3.$