Population Model: Increasing P as t Increases

[KillerZ]

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]

The derivative is positive (and so the function is increasing) when P(aP- b)> 0. Now, if P< 0, both P and aP-b are negative so P(aP- b)> 0. For 0< P < b/a, P is positive but aP- b is still negative so P(aP-b)< 0 and P is decreasing. Was that a typo?

From that, it seems to me that what happens to the population "as time t increases" depends upon the initial value. What happens if P(0) is less than b/a? What happens if P(0) is larger than b/a? What happens if P(0)= b/a?

 

[KillerZ]

Ok I got it thanks.