Problem 5 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Find a formula for a population that grows exponentially and a. Has an initial population of 50 and a doubling time of 10 years. b. Has an initial population of 1000 and a doubling time of 50 years. c. Has in initial population of 1000 and a doubling time of 100 years.

Step-by-Step Solution

Answer

Exponential growth formulas for each scenario: (a) \( P(t) = 50 \cdot 2^{t/10} \), (b) \( P(t) = 1000 \cdot 2^{t/50} \), (c) \( P(t) = 1000 \cdot 2^{t/100} \).

Step 1: Understand the Exponential Growth Formula

The general formula for exponential population growth is \( P(t) = P_0 \cdot 2^{t/T} \), where \( P(t) \) is the population at time \( t \), \( P_0 \) is the initial population, and \( T \) is the doubling time.

Step 2: Substitute Initial Population for Part (a)

For part (a), the initial population \( P_0 \) is 50, and the doubling time \( T \) is 10 years. Substitute these values into the formula: \( P(t) = 50 \cdot 2^{t/10} \).

Step 3: Verify Part (a) Form

Ensure the formula \( P(t) = 50 \cdot 2^{t/10} \) correctly represents an initial population of 50 and doubles every 10 years.

Step 4: Substitute Initial Population for Part (b)

For part (b), use \( P_0 = 1000 \) and \( T = 50 \) years. Substitute into the formula: \( P(t) = 1000 \cdot 2^{t/50} \).

Step 5: Verify Part (b) Form

Check the formula \( P(t) = 1000 \cdot 2^{t/50} \) to ensure it aligns with the given values of an initial population of 1000 and a doubling time of 50 years.

Step 6: Substitute Initial Population for Part (c)

For part (c), the initial population \( P_0 = 1000 \) and the doubling time \( T = 100 \) years. Plug these values into the formula: \( P(t) = 1000 \cdot 2^{t/100} \).

Step 7: Verify Part (c) Form

Ensure that \( P(t) = 1000 \cdot 2^{t/100} \) reflects the conditions of an initial population of 1000 and a doubling time of 100 years.

Key Concepts

Doubling Time

Doubling time is a simple yet powerful concept in understanding exponential growth. It tells us how long it takes for a population to double in size. Consider this: if you start with a small group, doubling time gives you a clear view of how quickly it will grow.

• It provides a straightforward way to visualize growth, making complex calculations easier to understand.
• Knowing the doubling time, you can forecast future population sizes without needing intricate calculations at each step.

In a formula, doubling time appears in the exponent's denominator. For instance, with a doubling time of 10 years, the formula can be written as \( 2^{t/10} \) where \( t \) is the time in years. This means every 10 years the population size doubles, making it a crucial factor in population studies.

Population Growth Formula

The population growth formula is at the heart of understanding exponential growth. It is represented by:\[ P(t) = P_0 \cdot 2^{t/T} \]This formula breaks down as follows:

• \( P(t) \): the population at a given time \( t \).
• \( P_0 \): the initial population, or the starting point.
• \( T \): the doubling time, or the time needed for the population to double.
• \( t \): the time elapsed.

The use of base 2 in the exponent \( 2^{t/T} \) is crucial because it signifies doubling behavior. As time advances, this formula predicts how populations evolve over time, showing clear patterns of rapid growth as predicted by their doubling time.It’s particularly useful for modeling situations where populations grow at a constant percentage rate.

Initial Population

The initial population, denoted as \( P_0 \), is the starting number of individuals before growth begins. Understanding this concept is vital because it acts as the baseline from which all calculations stem.An increase or decrease in the initial population can significantly impact growth predictions.

• A larger initial population leads to greater overall growth, even with the same doubling time.
• If the initial population is small, it may take time for noticeable growth to occur.

Imagine you have an initial population of 50 with a doubling time of 10 years. Using the exponential growth formula, you start with that amount before applying any time-based growth. With each passing period equivalent to the doubling time, the population doubles. This explains why this value plays a critical role in any projections and simulations about population dynamics.