Problem 5 - Calculus for Biology and Medicine

Question

Assume the discrete-time population model $$ N_{t+1}=b N_{t}, \quad t=0,1,2, \ldots $$ Assume that the population increases by \(x \%\) each generation. (a) Determine \(b\). (b) After how many generations will the population size have doubled? Compute the doubling time for \(x=0.1,0.5,1,2,5\), and 10 .

Step-by-Step Solution

Answer

(a) \( b = 1 + \frac{x}{100} \). (b) Doubling times for \( x = 0.1, 0.5, 1, 2, 5, 10 \) are approximately 693, 139, 70, 35, 14, and 7 generations, respectively.

Step 1: Understanding the model

The population model given is \( N_{t+1} = b N_{t} \). This indicates that the population at generation \( t+1 \) is a multiple \( b \) of the population at generation \( t \). If the population increases by \( x\% \), this model allows us to express this change as a multiplication factor.

Step 2: Determine the value of b

Since the population increases by \( x\% \), it’s equivalent to multiplying the population size by \( 1 + \frac{x}{100} \). Thus, \( b = 1 + \frac{x}{100} \). For example, if \( x = 5\% \), then \( b = 1 + 0.05 = 1.05 \).

Step 3: Set the doubling equation

To find out when the population doubles, set \( N_{t+1} = 2N_{t} \). Substitute the model into this equation to get \( b^k N_0 = 2N_0 \), where \( k \) is the number of generations needed to double the population.

Step 4: Solve for k, the doubling time

Cancel \( N_0 \) from both sides of the doubling equation, resulting in \( b^k = 2 \). Taking the logarithm of both sides gives \( k \log b = \log 2 \). Thus, \( k = \frac{\log 2}{\log b} \).

Step 5: Calculate doubling time for given x values

Calculate \( b \) and the respective doubling time \( k \) for each given \( x \). Example:- For \( x = 0.1 \), \( b = 1.001 \), \( k = \frac{\log 2}{\log 1.001} \approx 693.15 \)- For \( x = 0.5 \), \( b = 1.005 \), \( k = \frac{\log 2}{\log 1.005} \approx 138.98 \)- For \( x = 1 \), \( b = 1.01 \), \( k = \frac{\log 2}{\log 1.01} \approx 69.66 \)- For \( x = 2 \), \( b = 1.02 \), \( k = \frac{\log 2}{\log 1.02} \approx 35.00 \)- For \( x = 5 \), \( b = 1.05 \), \( k = \frac{\log 2}{\log 1.05} \approx 14.21 \)- For \( x = 10 \), \( b = 1.10 \), \( k = \frac{\log 2}{\log 1.10} \approx 7.27 \)

Key Concepts

Population Growth Rate

Understanding the population growth rate is essential when examining population dynamics. In a discrete-time population model, the growth rate determines how the population size changes from one generation to the next. This rate is expressed as a percentage increase. Each generation, the population is multiplied by a factor \( b \), which represents the growth multiplier. The population growth rate is linked directly to the variable \( x \), which represents the percentage increase per generation. To find \( b \), use the equation:

• \( b = 1 + \frac{x}{100} \)

This simple formula allows conversion of a percentage into a multiplicative growth factor. For instance, a 5% growth rate means \( b = 1.05 \), indicating that each generation, the population size is 1.05 times the previous one. This growth model forms the basis for more complex calculations, like determining how long it will take for a population to double.

Doubling Time

Doubling time is a key concept when analyzing population dynamics. It refers to the number of generations required for a population to double in size. This measure helps in understanding how quickly a population grows over time. To calculate doubling time with a discrete growth model, you use the relationship:

• \( b^k = 2 \)
• where \( b \) is the growth factor (\( b = 1 + \frac{x}{100} \))

By solving for \( k \) through logarithmic calculations, you can determine:

• \( k = \frac{\log 2}{\log b} \)

This formula computes the number of generations (\( k \)) it takes for the population to become twice its original size. Examining different values of \( x \) can give insight into how changes in growth rate affect the doubling time. Larger values of \( x \) lead to a smaller \( k \), meaning the population doubles more quickly.

Percentage Increase in Population

The percentage increase in population is a straightforward yet powerful concept describing how much a population grows at each stage or generation. Within the discrete-time population model, this percentage increase, denoted by \( x \), is crucial because it informs how the multiplier \( b \) is calculated. Remember: an increase of \( x\% \) in any population is equivalent to:

• \( b = 1 + \frac{x}{100} \)

This percentage directly influences population metrics because it's the factor that shows how significantly each generation's population exceeds the last. For students analyzing growth, understanding how a minor change in \( x \) leads to considerable variations in growth and doubling time is essential. For example, shifting from a 1% growth to a 2% growth may reduce the doubling time significantly, showcasing the non-linear impact percentage increases have on population dynamics.

Logarithmic Calculations

Logarithmic calculations play a vital role in population modeling, especially when determining metrics like doubling time. The logarithmic function is a mathematical tool that helps solve equations involving exponential growth, such as \( b^k = 2 \), where \( b \) is the population multiplier. To find the exact number of generations required for doubling, employ the formula:

• \( k = \frac{\log 2}{\log b} \)

Here’s how it works:

• \( \log 2 \) represents the power to which the base 10 must be raised to get 2.
• \( \log b \) represents the power to which the base 10 must be raised to get the growth factor \( b \).

Using logarithms allows us to solve for \( k \) straightforwardly because it turns multiplicative processes of growth into additive ones. Converting growth analysis into the realm of logarithms simplifies finding solutions and helps students understand not just when, but how, a population doubles given its growth rate.