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## Homework Statement

Population Dynamics: Logistic model. Suppose the environmental carrying capacity of the population is 100000 and the growth rate a t=0 is 5%. . If the population starts at 10000, how long does it take for the population to reach half the carrying capacity?

*dp/dt*= A

*P*(P1 –

*P*), where P1 = 100 using 1000 as the unit of population. Here

*P*'/

*P*= 0.05 at t = 0.

Use P'/P = 0.05 and the value of P0 given above in the ODE

*P*' = A

*P*(P1 –

*P*) to find A.

You have the solution of the ODE… use it to answer the question.

## Homework Equations

*dp/dt =*A P

p(t) = P

_{0}e

^{kt}

*dp/dt*= A

*P*(P1 –

*P*)

## The Attempt at a Solution

P

_{1}= 100

P(0) = 10

From my understanding you want to find P(t) = 50 and t = ?

I'm just having a hard time connecting dp/dt = AP and

*dp/dt*= A

*P*(P1 –

*P*)

do i find what AP is and then set it equal to A

*P*(P1 –

*P*)?

or do i use p(t) = P

_{0}e

^{kt}in some way.

It says "

*you have the solution of the ODE, use it to answer the question*"

I am also not understanding what P'/P represents, as it is equal to .05, but thats the answer when i divide 50 by 1000.

Any help is appreciated.