Problem 84 - Calculus for Biology and Medicine

Question

Suppose that you follow a population over time. When you plot your data on a semilog plot, a straight line with slope \(0.03\) results. Furthermore, assume that the population size at time 0 was 20 . If \(N(t)\) denotes the population size at time \(t\), what function best describes the population size at time \(t\) ?

Step-by-Step Solution

Answer

The population size at time \( t \) is modeled by \( N(t) = 20 e^{0.03t} \).

Step 1: Understand the semilog plot

In a semilog plot, the y-axis (population size, in this case, denoted by \( N(t) \)) is logarithmic, while the x-axis (time, \( t \)) is linear. A straight line in a semilog plot implies that \( N(t) \) follows an exponential growth model. The slope in a semilog plot represents the growth rate of the exponential function.

Step 2: Interpret the slope

Given that the slope of the line is \( 0.03 \), this represents the growth constant \( r \) in the exponential growth equation \( N(t) = N_0 e^{rt} \). Here, \( r = 0.03 \).

Step 3: Use initial condition

The initial population size \( N_0 \) at time \( t = 0 \) is given as 20. This means \( N(0) = 20 \).

Step 4: Write the population function

Using the information from steps 2 and 3, the exponential growth function can be written as:\[N(t) = 20 e^{0.03t}\]

Step 5: Final verification

Check that the function \( N(t) = 20 e^{0.03t} \) satisfies both the initial condition and the slope of the line on the semilog plot. This confirms that this function correctly models the given scenario.

Key Concepts

Semilog Plot

A semilogarithmic plot, or semilog plot, is a particular type of graph that is useful when dealing with exponential growth. In this plot, the y-axis is scaled logarithmically, while the x-axis remains linear. This unique setup allows exponential relationships to appear as straight lines. Here’s why and how it works:

• Exponential relationships: When data follows an exponential curve, a semilog plot offers a way to simplify the visualization by transforming the curve into a straight line. This makes identifying patterns and rates much easier.
• Slope significance: The slope of the line on a semilog plot directly corresponds to the rate of growth or decay in the system being analyzed. For our exercise, the slope of 0.03 implies that the population grows at this constant exponential rate.

When you plot data in this manner, it becomes much simpler to interpret and derive important mathematical properties like the growth rate or initial quantity. In the context of population dynamics, this tool is invaluable for understanding how quickly populations grow.

Population Dynamics

Population dynamics is a field that studies how and why the number of individuals in a population changes over time. It considers various factors such as birth rates, death rates, immigration, and emigration that affect population size. In the context of exponential growth, the focus is on how populations expand when resources are abundant and conditions are optimal.

• Exponential growth phase: When populations grow exponentially, each individual contributes to the growth, leading to a rapid increase in numbers. This occurs when resources are unlimited, and environmental conditions are favorable.
• Modeling growth: Population dynamics modeling often involves using mathematical equations such as the exponential growth function to predict future sizes and behaviors of populations. For this model, the formula used is the exponential model, which incorporates the initial population size and the growth rate.

Understanding population dynamics allows researchers and scientists to predict trends in wildlife populations, foresee challenges in managing natural resources, and address issues in ecology and conservation.

Exponential Function

The exponential function is a mathematical expression that models a wide array of natural phenomena, where change occurs at constant relative rates. When applied to population dynamics, this function helps illustrate how populations increase over time.

• Formula: The general form of the exponential growth model is given by the equation \( N(t) = N_0 e^{rt} \), where:
  • \( N(t) \) is the population size at time \( t \)
  • \( N_0 \) is the initial population size
  • \( r \) is the growth rate
  • \( t \) is the time elapsed
• Characteristics: Exponential functions are characterized by their rapid increase after each time period, an aspect that makes them unique when describing phenomena like population growth, which accelerates as the population gets larger.

In the exercise, we learned to formulate an exponential growth model for a population starting with 20 individuals, growing at a rate of 0.03, represented by the function \( N(t) = 20 e^{0.03t} \). This function captured both the initial population and the consistent growth rate afforded by favorable conditions.