General solution(s) to Logistic model with harvesting?

In summary, the conversation revolves around finding the limiting population for a model that includes harvesting (or immigration). The person has attempted to solve for this using separation of variables and partial fractions, but is struggling due to not having a value for the harvesting term. After some more time and effort, they finally solved it by substituting variables for the equilibrium values and taking the limit as time approaches infinity.
  • #1
clarkie_49
6
0

Homework Statement


Hi guys! I am trying to show the limiting population for this model:

Homework Equations


P'(t) = P(S - P) + H

The harvesting, is actually immigration, so its a positive unknown. I have shown this before (without harvesting) by solving for P(t) using separation of variables. Then taking the limit t -> infinity. However i can't work this out, without an actual value for H. Problem is there isn't one!

The Attempt at a Solution



So far i attempted to change the formula to use separation and method of partial fractions:

P'(t) = [ (S + sqrt{S^2 + 4H} - 2P)/2 ] [ (2P- S - sqrt{S^2 + 4H} )/2 ]

then (2AP - AS - Asqrt{S^2 + 4H} + BS + Bsqrt{S^2 + 4H} - 2BP)/2

is the numerator in my attempt at a partial fraction. Here's about as far is i get. Assuming S^2 + 4H is positive i can't seem to group to find a partial fraction. All i can obviously see is P(A - B) = 0

Any help would be greatly appreciated.
 
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  • #2
I managed to solve it, after a few more hours :)

I just subbed in variables for the equilibria, rather than leave them in square root form. Taking the limit of t -> infinity showed population approaching the equilibrium value which made

P''(equilibrium_value) < 0
 

Related to General solution(s) to Logistic model with harvesting?

1. What is the Logistic model with harvesting?

The Logistic model with harvesting is a mathematical model used in population dynamics to describe the growth and decline of a population over time. It takes into account both the natural growth rate of the population and the impact of harvesting or harvesting restrictions on the population.

2. How is the Logistic model with harvesting different from the basic Logistic model?

The basic Logistic model only takes into account the natural growth rate of a population, while the Logistic model with harvesting also includes the impact of harvesting on the population. This allows for a more accurate representation of real-life population dynamics where human intervention, such as harvesting, can affect the growth and decline of a population.

3. What is the general solution to the Logistic model with harvesting?

The general solution to the Logistic model with harvesting is a formula that describes the population size at any given time, taking into account the initial population size, natural growth rate, harvesting rate, and carrying capacity. It is expressed as P(t) = K / [1 + A * e^(-rt)], where P(t) is the population size at time t, K is the carrying capacity, r is the growth rate, and A is a constant determined by the harvesting rate.

4. How can the general solution to the Logistic model with harvesting be used?

The general solution to the Logistic model with harvesting can be used to predict the future population size of a species under different scenarios. By changing the values of the parameters, such as the growth rate or harvesting rate, we can see how these factors will affect the population over time. This information can be useful in making management and conservation decisions for the species.

5. What are some limitations of the Logistic model with harvesting?

Like any mathematical model, the Logistic model with harvesting has its limitations. It assumes a constant carrying capacity and growth rate, which may not always hold true in real-life populations. It also does not take into account environmental factors or other external factors that may affect the population. Therefore, it should be used as a tool to understand general population dynamics rather than as an exact prediction of population size.

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