Problem 4 - Biology: The Unity and Diversity of Life

Question

A population of worms is growing exponentially in a compost heap. Thirty days ago there were 300 worms and now there are \(600 .\) How many worms will there be thirty days from now, assuming conditions remain constant? a. 1,200 b. 1,600 c. 3,200 d. 6,400

Step-by-Step Solution

Answer

There will be 1,200 worms in 30 more days. Option (a) 1,200.

Step 1: Understand Exponential Growth

Exponential growth means population increases by a constant percentage or factor over equal time periods. The formula to represent this growth is \( P(t) = P_0 \times r^t \), where \( P(t) \) is the population at time \( t \), \( P_0 \) is the initial population, \( r \) is the growth factor, and \( t \) is time.

Step 2: Calculate Growth Factor

Given that the worm population doubled from 300 to 600 in 30 days, we can find the growth factor \( r \). Substitute \( P(t=30) = 600 \) and \( P_0 = 300 \) into the formula: \( 600 = 300 \times r^{30} \). Solve for \( r \): \( r^{30} = 2 \), therefore \( r = 2^{1/30} \).

Step 3: Calculate Future Population

To find the worm population 30 days from now, which is 60 days from the initial observation, substitute \( t = 60 \) into the exponential growth formula: \( P(60) = 300 \times (2^{1/30})^{60} = 300 \times 2^{60/30} = 300 \times 2^2 = 300 \times 4 = 1200 \).

Step 4: Verify with Given Options

We calculated there will be 1,200 worms in 60 days. The correct option given is (a) 1,200.

Key Concepts

Population Dynamics

Population dynamics is a field in biology that examines how and why populations change over time. In the case of the worm population in the compost heap, we are interested in how their numbers grow.
This change in population size can be influenced by various factors such as birth rates, death rates, and migration. However, when we talk about exponential growth, as seen here, we're primarily focusing on reproduction rates with ideal conditions assumed.
Exponential growth occurs when the resources available to a population are unlimited, allowing the population to grow at a constant rate over time.

• This results in a rapid increase in population size.
• The size of each generation is larger than the one before.

Understanding this concept is crucial when predicting future population sizes, particularly in controlled environments such as our compost heap example.

Growth Factor

The growth factor is a critical component in understanding exponential growth. It tells us how much the population grows over a certain period. In our exercise, the worm population doubled in 30 days, indicating a clear process of exponential growth.
To mathematically calculate the growth factor, we use the formula for exponential growth: \[ P(t) = P_0 \times r^t \]

• Here, \( r \) represents the growth factor.
• When \( r > 1 \), the population is growing.
• If \( r < 1 \), the population is decreasing.

In the worm example, solving the equation \( r^{30} = 2 \), where the population doubled, gives us a growth factor of \( r = 2^{1/30} \). This growth factor is then used to predict future population sizes accurately.

Mathematical Modeling

Mathematical modeling is a powerful tool to predict population growth and other dynamic systems. It involves creating mathematical representations of real-world scenarios to anticipate future conditions, as seen with the worm population over time.
Using the exponential growth model, we can formulate equations that help us understand complex biological phenomena without requiring resource-intensive real-world trials.

• This provides a practical and efficient method to predict outcomes.
• We can also manipulate variables in models to observe how changes in conditions might affect predictions.

In our exercise, by applying the exponential growth formula \[ P(t) = P_0 \times r^t \]we determined the future worm population size by substituting known values and the calculated growth factor. This approach not only delivered precise predictions but also underscored how changes in growth factor might influence future studies.