Problem 5

Question

Investigate the effect of the parameter $C$ on the logistic curve $$ P=\frac{10}{1+C e^{-t}} $$ Substitute several values for $C$ and explain, with a graph and with words, the effect of $C$ on the graph.

Step-by-Step Solution

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Answer

Higher values of C result in a steeper and faster initial rise in the logistic curve.

1Step 1: Understanding the Logistic Curve

In the logistic curve equation $ P = \frac{10}{1 + C e^{-t}} $, $ P $ represents the population or quantity of interest over time $ t $, and $ C $ is a parameter that influences the shape of the curve. It affects how quickly $ P $ approaches its maximum value, which is 10 in this case. The function has a characteristic S-shaped curve.

2Step 2: Substituting Values for C

Let's substitute different values of $ C $ into the logistic curve equation and observe the changes. We'll consider $ C = 0.5, 1, 2, 5 $. This will help us understand how $ C $ affects the steepness and position of the curve.

3Step 3: Calculating the Curve for C = 0.5

For $ C = 0.5 $, the equation becomes $ P = \frac{10}{1 + 0.5 e^{-t}} $. This creates a curve that rises quickly and approaches its maximum at a relatively slow rate compared to higher values of $ C $.

4Step 4: Calculating the Curve for C = 1

With $ C = 1 $, the equation is $ P = \frac{10}{1 + e^{-t}} $. Here, the growth is moderate, and $ P $ approaches the maximum value of 10 at a typical rate without too much early steepness or slowness.

5Step 5: Calculating the Curve for C = 2

When $ C = 2 $, the formula becomes $ P = \frac{10}{1 + 2e^{-t}} $. The curve now rises more steeply early on and reaches closer to 10 more quickly than when $ C = 1 $. It indicates a faster initial growth rate.

6Step 6: Calculating the Curve for C = 5

For $ C = 5 $, the equation is $ P = \frac{10}{1 + 5e^{-t}} $. This results in a very steep rise after a slow start, quickly reaching close to the maximum value, which demonstrates the effect of a large $ C $ value on increasing early growth rates.

7Step 7: Analyzing Results

As $ C $ increases, the logistic curve's growth becomes faster initially, reaching close to the maximum value quicker. For smaller values of $ C $, the growth is slower, and the curve takes more time to approach 10. This shows $ C $ alters how sharply the curve turns (inflection point) and the initial growth rate.

Key Concepts

Logistic CurveParameter InfluenceGraph Interpretation

Logistic Curve

The logistic curve is a fundamental concept in mathematical modeling of population growth. It is described by the equation $ P = \frac{10}{1 + C e^{-t}} $. Here, $ P $ stands for the population size or any quantity of interest as a function of time $ t $. The curve is characterized by an S-shape, which mirrors how many populations grow.Initially, growth is slow. As time passes, it speeds up until it reaches a point of maximum growth rate called an inflection point. After that, growth slows and levels off as it gets closer to a maximum carrying capacity, which in this context is 10.The equation beautifully encapsulates the natural tendency of populations to experience rapid growth when resources are abundant, followed by a plateau as resources become limited.

Parameter Influence

The parameter $ C $ in the logistic curve equation has a profound impact on the curve's features. It dictates how quickly the curve approaches its maximum value.

When $ C $ is small, such as 0.5, the curve rises rapidly but slows as it approaches the maximum. The inflection point is lower and occurs later.
For moderate values like $ C = 1 $, the curve rises at a moderate pace throughout.
Higher values of $ C $ like 2 or 5 result in a steeper rise earlier in the timeline, followed by a quick leveling off.

Essentially, the larger the value of $ C $, the steeper the initial curve, indicating a faster initial growth rate and shifting the inflection point earlier in time.

Graph Interpretation

Understanding the graph of the logistic curve helps visualize how different values of $ C $ influence growth.Substituting values of $ C $ into the equation provides insight into these effects:

For $ C = 0.5 $, the curve gradually increases and takes longer to near the maximum value of 10.
With $ C = 1 $, the growth rate from start to finish is balanced, without any drastic increases or slowdowns.
When $ C = 2 $ or $ C = 5 $, the curves steeply incline early on, nearly touching the maximum much sooner.

In summary, altering $ C $ affects the steepness and timing with which the curve inches towards its plateau. The graphs clearly illustrate these variances, making the concept easier to comprehend.

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