Problem 7

Question

The Tojolobal Mayan Indian community in Southern Mexico has available a fixed amount of land." The proportion, $P,$ of land in use for farming $t$ years after 1935 is modeled with the logistic function $$P=\frac{1}{1+3 e^{-0.0275 t}}$$ (a) What proportion of the land was in use for farming in $1935 ?$ (b) What is the long-run prediction of this model? (c) When was half the land in use for farming? (d) When is the proportion of land used for farming increasing most rapidly?

Step-by-Step Solution

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Answer

1Step 1: Calculate Proportion in 1935

To find the proportion of land in use for farming in 1935, we need to substitute $t = 0$ into the logistic function, since 1935 is the base year. Thus, substitute $t = 0$ in $P(t) = \frac{1}{1 + 3e^{-0.0275t}}$. The equation becomes: $P(0) = \frac{1}{1 + 3e^{0}}$. Simplifying this gives $P(0) = \frac{1}{1 + 3} = \frac{1}{4} = 0.25$. This means that in 1935, 25% of the land was in use for farming.

2Step 2: Determine Long-run Prediction

The long-run behavior of the logistic function can be determined as $t$ approaches infinity. As $t\rightarrow \,\infty$, the term $3e^{-0.0275t}$ approaches zero, so $P(t)$ approximates $\frac{1}{1 + 0} = 1$. This indicates that in the long run, 100% of the land is predicted to be used for farming.

3Step 3: Solve for Half Used Land

We need to find $t$ such that $P(t) = 0.5$. Substitute 0.5 for $P(t)$ in the equation: $0.5 = \frac{1}{1 + 3e^{-0.0275t}}$. Solving gives $1 + 3e^{-0.0275t} = 2$. Rearranging, we have $3e^{-0.0275t} = 1$. Taking the logarithm of both sides, $-0.0275t = \ln(\frac{1}{3})$. Solving for $t$, we get $t = \frac{\ln(3)}{0.0275} \approx 39.8409$. Thus, the year is approximately 1935 + 40 = 1975.

4Step 4: Determine Maximum Rate of Increase

In a logistic model, the rate of increase is maximum when $P(t) = 0.5$, which is when half the land is in use as calculated before. From step 3, we know this happens when $t \approx 39.8409$, corresponding to the year 1975. Thus, the most rapid increase in land use occurs around 1975.

Key Concepts

land use modelinglogistic functiontime series analysis

land use modeling

Land use modeling is a powerful tool that helps us understand how land is utilized over time. It involves using mathematical and computational techniques to predict how land resources might change under different conditions. In the context of the Tojolobal Mayan Indian community, this type of modeling is useful to determine how much land is being cultivated for farming.
The process involves:

Making predictions about future land use based on current trends and data.
Assessing the impact of various factors like socio-economic changes and environmental conditions.
Using data-driven models, like the logistic function, to simulate potential future scenarios.

These models are crucial for planning and decision-making, as they can inform policy makers about potential future demands on land resources and the sustainability of land use practices.

logistic function

The logistic function is a mathematical concept often used to model growth processes. It is particularly adept at describing scenarios where growth starts slowly, accelerates, and then levels off as it reaches a maximum limit. This characteristic makes the logistic function ideal for modeling land use, where resources such as land are limited.
In our land use example, the logistic function is expressed as:\[ P = \frac{1}{1 + 3e^{-0.0275t}} \]where $P$ represents the proportion of land in use, and $t$ is time in years since the base year, 1935.
Key points to understand about logistic functions:

They have an S-shaped curve, known as a sigmoid curve, reflecting the stages of growth.
The function approaches a maximum value asymptotically, which represents the carrying capacity or the maximum resource utilization.
In this case, it predicts that the land use will eventually reach a saturation point of 100%.

Understanding logistic functions helps us interpret how land use can evolve over time and when it will stabilize, aiding effective resource management.

time series analysis

Time series analysis is a method to analyze data points collected over time to identify patterns or trends. It is used to understand how variables, like land use, change over a specified period. In the Tojolobal community example, a time series is formed from the proportion of land used annually.
Time series analysis involves several key steps:

Collecting data over consistent time intervals.
Analyzing the series for trends, seasonal patterns, or cyclical behavior.
Using models, such as the logistic model in our example, to predict future outcomes.

By employing time series analysis, we can forecast the dynamics of land use over decades, making it a vital tool for planners and researchers. It helps answer questions such as:
- When will half of the available land be in use?
- When does the land use growth experience its rapid increase?
These insights allow communities and stakeholders to prepare adequately for future scenarios and ensure sustainable resource allocation.

Problem 7

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