Problem 29 - Calculus for Biology and Medicine

Question

Suppose that the size of a population at time \(t\) is given by $$N(t)=\frac{50}{1+3 e^{-t}}$$ for \(t \geq 0\). (a) Use a graphing calculator to sketch the graph of \(N(t)\). (b) Determine the size of the population as \(t \rightarrow \infty\), using the basic rules for limits. Compare your answer with the graph that you sketched in (a).

Step-by-Step Solution

Answer

The population approaches 50 as \(t \rightarrow \infty\). The graph should confirm this by leveling off at 50.

Step 1: Interpret the function

We are given the function \(N(t) = \frac{50}{1 + 3e^{-t}}\), which describes the size of the population at time \(t\). This is a form of the logistic function, which typically models population growth.

Step 2: Graph the function

Use a graphing calculator or any graph plotting tool to sketch the function \(N(t) = \frac{50}{1 + 3e^{-t}}\). Input the function and observe how the population size changes with time \(t\). Note the behavior of the function as \(t\) increases.

Step 3: Evaluate the limit as \(t \rightarrow \infty\)

To find \(\lim_{t \to \infty} N(t)\), observe that as \(t\) increases, \(e^{-t}\) approaches zero. Therefore, the equation simplifies to \(N(t) = \frac{50}{1 + 0} = 50\). Thus, the population approaches a size of 50 as \(t\) tends to infinity.

Step 4: Compare limit result with the graph

Compare the horizontal asymptote of the graph with the limit found in Step 3. The graph should approach a horizontal line at \(N(t) = 50\) as \(t\) increases and confirm that the population stabilizes at 50.

Key Concepts

Population Models

Population models are mathematical representations used to describe how populations change over time. One fascinating model is the logistic growth model, which reflects more realistic conditions compared to exponential growth models. In nature, resources such as food and space are usually limited, and populations can't grow indefinitely.
The function given in the exercise, \(N(t) = \frac{50}{1 + 3e^{-t}}\), is a logistic model. It represents how a population grows rapidly at first, then slows as it approaches its carrying capacity, a maximum sustainable population size, due to limited resources.
Key characteristics of logistic growth models include:

• Initial growth phase: The population grows almost exponentially.
• Deceleration phase: As resources begin to dwindle, growth slows down.
• Equilibrium phase: The population settles at the carrying capacity, where birth rates and death rates are balanced.

These models are used in ecology, wildlife management, and conservation biology to predict and plan for changes in population sizes over time.

Limits in Calculus

Limits play a crucial role in understanding the behavior of functions as values approach certain points. In the context of the exercise, evaluating the limit of the function \(N(t) = \frac{50}{1 + 3e^{-t}}\) as \(t\) approaches infinity helps us determine the long-term behavior of the population.
When we say a function tends towards a limit, we mean that as the input, in this case, time \(t\), increases indefinitely, the function's output approaches a specific value. Limits are particularly useful when assessing horizontal asymptotes in graphing - lines that a graph gets closer to but never actually reaches. This is what happens in the context of a carrying capacity in logistic growth.
In our exercise, by calculating \(\lim_{t \to \infty} N(t)\), we find that the population approaches a limit of 50. Here's why:

• As \(t\) increases, \(e^{-t}\) tends towards zero.
• This makes the denominator, \(1 + 3e^{-t}\), approach 1.
• Hence, \(N(t)\) approaches \(\frac{50}{1} = 50\).

Understanding limits help us see beyond the immediate values and predict behavior as variables become large or small.

Exponential Functions

Exponential functions are a fundamental type of mathematical function where a constant base is raised to a variable exponent. They are written in the form \(a^x\), where \(a\) is a positive real number. In this exercise, the function includes \(e^{-t}\), which is an example of an exponential decay.
Exponential decay refers to the process of reducing an amount by a consistent percentage over a time period. As \(t\) increases, \(e^{-t}\) rapidly approaches zero. This attribute is critical in population models like the logistic function \(N(t) = \frac{50}{1 + 3e^{-t}}\), where it describes the slowing growth rate as a population reaches its carrying capacity.
Key aspects of exponential functions used in logistic models include:

• Growth or decay rate: Dictated by the exponent; negative for decay (as in the exercise), which compresses the value quickly towards zero.
• Impact over time: Seen through changes in the exponent. As time passes, the effect of the exponential part diminishes, slowing growth.

Overall, recognizing how exponential functions work helps us understand and predict trends in scientific and financial domains, ensuring better analysis and decision-making.