Problem 28 - Calculus for Biology and Medicine

Question

Suppose that the size of a population at time \(t\) is given by $$N(t)=\frac{50}{1+6 e^{-2 t}}, \quad t \geq 0$$ (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

As \(t \rightarrow \infty\), the population size approaches 50, matching the graph's asymptote at 50.

Step 1: Understand the Function

The function provided is a logistic function: \( N(t)=\frac{50}{1+6 e^{-2 t}} \). It models population growth that is constrained by a carrying capacity, in this case, 50. As \(t\) increases, the exponential term \(e^{-2t}\) approaches zero.

Step 2: Sketch the Graph

Use a graphing calculator (like Desmos) to plot the function \( N(t)=\frac{50}{1+6 e^{-2 t}} \). You should observe that the graph starts at a small value when \(t = 0\) and rises to approach 50 as \(t\) increases. The curve approaches a horizontal asymptote at \(N = 50\).

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

To find \( \lim_{t \to \infty} N(t) \), note as \(t\) approaches infinity, \(e^{-2t} \rightarrow 0\). Thus, the expression becomes \( N(t)=\frac{50}{1+6 \times 0} = 50 \).

Step 4: Compare with the Graph

Reflecting on the graph sketched earlier, observe that the population size indeed levels off towards 50 as \(t\) increases indefinitely, confirming the calculated limit.

Key Concepts

Understanding Population Dynamics

Population dynamics is the study of how and why populations change over time. This includes how the size, structure, and distribution of populations are impacted by different factors.
In the context of logistic growth, population dynamics is used to model how populations grow rapidly at first when they are small, and then slow down as they near a limit or maximum size. This limit is known as the carrying capacity.

• Initial rapid growth due to low competition and abundant resources.
• Rate of growth slows as the population reaches environmental limits.
• Eventually, the population stabilizes around the carrying capacity.

Such models are integral for understanding ecosystems, conservation efforts, and managing animal populations effectively.

Carrying Capacity in Logistic Growth

Carrying capacity is a fundamental concept in environmental science and ecology. It refers to the maximum number of individuals of a species that an environment can sustainably support.
In the logistic growth model described by the equation \( N(t)=\frac{50}{1+6 e^{-2 t}} \), the carrying capacity is represented by the number 50.

• Acts as a ceiling on population size.
• Causes population growth to decelerate as it approaches this threshold.
• Influences the shape of the logistic growth curve, leading to an "S" shape that flattens out.

Keep in mind that carrying capacity can change with environmental conditions, such as availability of resources or space, making it a dynamic concept.

Exploring Limits in Calculus

Limits are a crucial concept in calculus that describe the behavior of functions as inputs approach a certain point. Calculating a limit can show where a function is heading as its variables get exceedingly large.
In the given exercise, we calculate the limit of the logistic function as time \( t \) approaches infinity. This method helps us understand the ultimate growth level of the population over a long period:

• When \( t \to \infty \), \( e^{-2t} \to 0 \).
• Substituting into the function gives \( N(t)=\frac{50}{1+6 \times 0} = 50 \).
• This limit indicates that the population will stabilize at 50.

Limits help verify theoretical predictions about growth and provide validity to such mathematical models by comparing them with real-world behavior.