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.
dp/dt = A P
p(t) = P0 ekt
dp/dt = A P (P1 – P)
The Attempt at a Solution
P1 = 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) = P0 ekt 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.