Population Dynamics: Logistic Model (Differential Equations)

In summary: Maybe it was given in a previous conversation and we're just supposed to use it to answer the question. In that case, we don't actually need to solve the ODE. We just need to use the given solution to find the value of A.In summary, the question asks how long it will take for a population to reach half of its carrying capacity, given the growth rate and starting population. The solution to the logistic equation, dp/dt = AP(P1 - P), is needed to answer this question. The value of A can be found by using the given solution, P(t) = P0e^kt, and setting P(0) = 10000 and P(t) = 50000. P'/
  • #1
KTiaam
53
1

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 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 that's the answer when i divide 50 by 1000.

Any help is appreciated.
 
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  • #2
KTiaam said:
dp/dt = A P
p(t) = P0 ekt
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.
 
  • #3
haruspex said:
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?
 
  • #4
KTiaam said:
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.
 
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  • #5
Mark44 said:
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.
 
  • #6
haruspex said:
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.
 

Related to Population Dynamics: Logistic Model (Differential Equations)

What is the logistic model in population dynamics?

The logistic model is a mathematical model used to describe the growth of a population over time. It takes into account factors such as birth rate, death rate, and carrying capacity to predict the size of a population at a given time.

What is the differential equation used in the logistic model?

The differential equation used in the logistic model is known as the logistic growth equation. It is a second-order differential equation that takes into account both the rate of change of the population and the current population size.

What is carrying capacity in the logistic model?

Carrying capacity is the maximum number of individuals that an environment can support without causing significant strain on its resources. In the logistic model, it is often represented by the letter K and is a crucial factor in determining the growth of a population.

What is the difference between exponential and logistic growth?

Exponential growth occurs when a population grows at a constant rate, while logistic growth takes into account the limiting factors of carrying capacity and slows down the growth rate as the population approaches the maximum size. In other words, exponential growth is unsustainable, while logistic growth is more realistic.

How is the logistic model used to study real-world populations?

The logistic model is used to analyze and predict the growth of populations in various fields such as ecology, epidemiology, and economics. By incorporating factors such as birth and death rates, it can provide insight into how a population will change over time and help inform decision-making and resource management.

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