Population Dynamics: Logistic Model (Differential Equations)

[KTiaam]

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) = P₀ e^kt
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 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 that's the answer when i divide 50 by 1000.

Any help is appreciated.

 

[haruspex]

dp/dt = A P
p(t) = P₀ e^kt
dp/dt = A P (P1 – P)

You have conlicting expressions for dP/dt. I believe the second is for the logistic model. The first is for a model that has no limit on carrying capacity.
You need to solve the logistic differential equation.

 

[KTiaam]

You have conlicting expressions for dP/dt. I believe the second is for the logistic model. The first is for a model that has no limit on carrying capacity.
You need to solve the logistic differential equation.

Could you explain a little bit more?
So use only p(t) = P0e^kt?

 

[Mark44]

Could you explain a little bit more?
So use only p(t) = P0e^kt?

No, haruspex was talking about the logistic equation -- dP/dt = AP(P₁ - P). This is the one you need to solve for P.

BTW, try to be consistent with the casing of your variables. Don't use dp/dt in one place and P on the other side. P represents population, so the equation should be as I wrote it above.

 

[haruspex]

No, haruspex was talking about the logistic equation -- dP/dt = AP(P₁ - P). This is the one you need to solve for P.
.

Yes, that's what I meant. But I just noticed in the OP it says "you have the solution to the equation". KTiaam, this implies you have already been given the solution to this differential equation, but you have not listed it. Instead, you have listed a simpler ODE and its solution.

 

[Mark44]

But I just noticed in the OP it says "you have the solution to the equation". KTiaam, this implies you have already been given the solution to this differential equation, but you have not listed it.

Yeah, I was wondering about that, and didn't see the solution listed anywhere.