Differential Equations Problem, logistic models

For part c), you can use the fact that as t approaches infinity, e^(-AP1 t) goes to 0, so that should give you a clue on how to find the limit.In summary, the given conversation is about a population, P, modeled by a logistic model with initial population of 30 and a differential equation of dP/dt = .3 P (3.5 - P/40). The equations and attempt at solving the problem are also provided. The solution involves finding the values of k1 and k3 and applying them to the given equation to solve for the population after 2.5 months and the limit of the population as t approaches infinity.
  • #1
beccajd
2
0

Homework Statement



Given that a population, P, after t months, can be modeled by the logistic model
dP/dt = .3 P (3.5 - P/40).
P(0) = 30

a) Solve the diff eq

b) Find the population after 2.5 months

c) Find lim P(t) as t -> infinity

Homework Equations



P(t) = P0 P1 /(P0 + (P1 - P0 )e^(-AP1 t))

A = k3 /2
P1 = (2k1 / k3 ) +1

The Attempt at a Solution


 
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  • #2
hi, beccajd
You have pretty much done part a) already. You have written down the correct solution. So now you can work out what k1 and k3 should be, by looking at the numbers in the equation given to you. I think this is everything they expect from part a). Try doing part b), it shouldn't be too difficult, since you have got the equation for it.
 

1. What is a differential equation?

A differential equation is a mathematical equation that relates a function to its derivatives. It is used to model relationships between different variables and their rates of change.

2. How are differential equations used in logistic models?

Differential equations are used in logistic models to describe the growth or decline of a population over time. They help us understand how the population size changes in response to various factors such as limited resources or competition.

3. What is the logistic equation?

The logistic equation is a type of differential equation that is used to model population growth or decline. It takes into account a carrying capacity, which is the maximum population size that an environment can sustain.

4. What is the difference between a logistic model and an exponential model?

A logistic model takes into account a limiting factor, such as a carrying capacity, while an exponential model does not. This means that a logistic model predicts a population size that will eventually level off, while an exponential model predicts continuous growth.

5. How can we solve a logistic differential equation?

There are different methods for solving a logistic differential equation, including separation of variables, substitution, and using the logistic function. The chosen method depends on the specific equation and its initial conditions.

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