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## Homework Statement

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

[tex]\frac{dP}{dt} = P(aP - b)[/tex]

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

## Homework Equations

[tex]\frac{dP}{dt} = P(aP - b)[/tex]

## The Attempt at a Solution

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

critical points

[tex] P(t) = 0[/tex]

[tex] P(t) = a/b[/tex]

[tex]-\infty < P < 0[/tex] it is decreasing

[tex]0 < P < a/b[/tex] it is increasing

[tex]a/b < P < \infty[/tex] it is decreasing

so in the changed equation:

critical points

[tex] P(t) = 0[/tex]

[tex] P(t) = b/a[/tex]

[tex]-\infty < P < 0[/tex] it is increasing

[tex]0 < P < b/a[/tex] it is increasing

[tex]b/a < P < \infty[/tex] it is increasing