Differential equation finding limit without solving explicitly.

Question

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.

Answer 1

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 $(\alpha/\beta)^3.$

Answer 2

Hint Find the critical points of this ODE and then draw the Phase Line. Then, classify the critical points as attractors, repellors or shunts so that you can see the behaviour qualitatively.