# Population Model

## Homework Statement

The differential equation $$\frac{dP}{dt} = P(a - bP)$$ is a well-known population model. Suppose the DE is changed to:

$$\frac{dP}{dt} = P(aP - b)$$

Where a and b are positive constants. Discuss what happens to the population P as time t increases.

## Homework Equations

$$\frac{dP}{dt} = P(aP - b)$$

## The Attempt at a Solution

Well I thought the population would increase because in the original equation on the intervals:

critical points
$$P(t) = 0$$
$$P(t) = a/b$$

$$-\infty < P < 0$$ it is decreasing
$$0 < P < a/b$$ it is increasing
$$a/b < P < \infty$$ it is decreasing

so in the changed equation:

critical points
$$P(t) = 0$$
$$P(t) = b/a$$

$$-\infty < P < 0$$ it is increasing
$$0 < P < b/a$$ it is increasing
$$b/a < P < \infty$$ it is increasing

HallsofIvy