Autonomous equations

Let $x'=(x-1)(x-2)x^2$.

1. Sketch the phase diagram and find critical points.
2. Classify the critical points.
3. If $x(0)=0.5$, then find $\displaystyle \lim_{t\to\infty} x(t)$.

Let $x'=e^{-x}$.

1. Find and classify all critical points.
2. Find $\displaystyle \lim_{t\to\infty} x(t)$ given any initial condition.

Assume that a population of fish in a lake satisfies $\frac{dx}{dt} = kx(M-x)$. Now suppose that fish are continually added at $A$ fish per unit of time.

1. Find the differential equation for $x$.
2. What is the new limiting population?

Suppose $\frac{dx}{dt} = (x-\alpha)(x-\beta)$ for two numbers $\alpha < \beta$.

1. Find the critical points, and classify them.

For b), c), d), find $\displaystyle \lim_{t\to\infty} x(t)$ based on the phase diagram.

1. $x(0) < \alpha$,
2. $\alpha < x(0) < \beta$,
3. $\beta < x(0)$.