Logistic model/initial value problem

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Discussion Overview

The discussion revolves around solving an initial value problem using a logistic model, specifically focusing on the calculations involved in determining the value of \( e^{-k} \) based on population growth data.

Discussion Character

  • Technical explanation, Mathematical reasoning

Main Points Raised

  • One participant provides a link to a document containing solutions related to the logistic model.
  • Another participant presents the equation derived from the logistic model, indicating that the fish population tripled in the first year, leading to the equation \( P(1)=1200=\frac{10000}{1+24e^{-k}} \).
  • Subsequent calculations are shown to isolate \( e^{-k} \), resulting in \( e^{-k}=\frac{11}{36} \).
  • There is a clarification regarding the initial population, with participants confirming that the population was 400 before tripling to 1200 after one year.

Areas of Agreement / Disagreement

Participants generally agree on the calculations related to the tripling of the fish population and the resulting value of \( e^{-k} \). However, there is no explicit consensus on the overall approach or any additional implications of the logistic model.

Contextual Notes

The discussion does not address potential limitations or assumptions in the model, such as the conditions under which the logistic growth applies or the accuracy of the initial population estimate.

Who May Find This Useful

Individuals interested in mathematical modeling, particularly in population dynamics and logistic growth, may find this discussion relevant.

ineedhelpnow
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The number of fish tripled in the first year, hence:

$$P(1)=1200=\frac{10000}{1+24e^{-k}}$$

$$3=\frac{25}{1+24e^{-k}}$$

$$3\left(1+24e^{-k}\right)=25$$

$$3+72e^{-k}=25$$

$$72e^{-k}=22$$

$$36e^{-k}=11$$

$$e^{-k}=\frac{11}{36}$$
 
1200? that's because the number tripled right?
 
ineedhelpnow said:
1200? that's because the number tripled right?

Yes, the initial population was 400 so after one year there would be 1200 if it tripled during that time. :D
 

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