- The horizontal asymptote of the logistic model is \(\boxed{y = 250}\).
- The point of inflection for the function occurs at \(\boxed{(168.40, 125)}\).
- The limiting value of 250 represents the \(\boxed{\text{maximum possible number of salmon that will be tagged}}\).
Explanation
- Analysis of the Logistic Model Parameters
The number of salmon tagged is modeled by the logistic function \(y = \frac{250}{1 + 2499 e^{-0.04646x}}\). In a logistic model of the form \(y = \frac{L}{1 + Ae^{-kx}}\), the parameters \(L\), \(A\), and \(k\) determine the behavior of the function. Here, \(L = 250\), \(A = 2499\), and \(k = 0.04646\). The variable \(x\) represents the number of days since the beginning of the migration.
- Determining the Horizontal Asymptote
As the number of days \(x\) approaches infinity, the term \(e^{-0.04646x}\) approaches 0. Therefore, the value of the function approaches:
\[ \lim_{x \to \infty} \frac{250}{1 + 2499e^{-0.04646x}} = \frac{250}{1 + 2499(0)} = 250 \]
Thus, the horizontal asymptote is at \(y = 250\).
- Finding the Point of Inflection
The point of inflection for a logistic function \(y = \frac{L}{1 + Ae^{-kx}}\) occurs when \(y = \frac{L}{2}\).
Substituting \(L = 250\), we get:
\[ y = \frac{250}{2} = 125 \]
To find the x-coordinate, we solve the equation:
\[ \frac{250}{1 + 2499e^{-0.04646x}} = 125 \]
\[ 1 + 2499e^{-0.04646x} = 2 \]
\[ 2499e^{-0.04646x} = 1 \]
\[ e^{-0.04646x} = \frac{1}{2499} \]
\[ -0.04646x = \ln\left(\frac{1}{2499}\right) \]
\[ x = \frac{-\ln(2499)}{-0.04646} = \frac{\ln(2499)}{0.04646} \approx 168.395 \]
Rounding to two decimal places, the point of inflection is approximately \((168.40, 125)\).
- Interpretation of the Limiting Value
The horizontal asymptote of this model is 250, which represents the carrying capacity of the tagged salmon population. In the context of the wildlife program, the limiting value represents the maximum possible number of salmon that will be tagged as the migration period progresses indefinitely.
- Final Summary
The function has an asymptote at \(y = 250\). The point of inflection is \((168.40, 125)\). The limiting value represents the maximum possible number of salmon that can be tagged during the migration.
Examples
Logistic growth models are widely used by ecologists to predict population trends. For instance, when monitoring a fish migration, researchers use these models to estimate not just the current number of fish tagged, but the overall capacity of the ecosystem or the peak of the migration event. This helps in conservation efforts by providing insight into when migration activity will likely plateau, allowing for better resource allocation and scheduling for wildlife biologists.