Problem 29 - Calculus for Biology and Medicine

Question

Suppose that the growth rate of a population is given by $$ f(N)=N\left(1-\left(\frac{N}{K}\right)^{\theta}\right) $$ where \(N\) is the size of the population, \(K\) is a positive constant denoting the carrying capacity, and \(\theta\) is a parameter greater than 1\. Find \(f^{\prime}(N)\), and determine where the growth rate is increasing and where it is decreasing.

Step-by-Step Solution

Answer

The growth rate is increasing when \(N < \left(\frac{1}{1 + \theta}\right)^{\frac{1}{\theta - 1}} K\) and decreasing otherwise.

Step 1: Differentiate the function

We need to find the derivative of the given function with respect to \(N\). Start by differentiating the function \(f(N) = N \left(1 - \left(\frac{N}{K}\right)^\theta\right)\) using the product rule. Let \(u = N\) and \(v = 1 - \left(\frac{N}{K}\right)^\theta\). Then, \(f'(N) = u'v + uv'\).

Step 2: Derive each part separately

The derivative of \(u = N\) with respect to \(N\) is \(u' = 1\).The derivative of \(v = 1 - \left(\frac{N}{K}\right)^\theta\) with respect to \(N\) is \(v' = 0 - \theta \left(\frac{N}{K}\right)^{\theta - 1}\cdot \frac{1}{K}\).

Step 3: Apply the product rule

Using the results from Step 2, substitute into the product rule:\[ f'(N) = 1 \cdot \left(1 - \left(\frac{N}{K}\right)^\theta\right) + N \cdot \left(-\theta \frac{1}{K} \left(\frac{N}{K}\right)^{\theta-1}\right) \].Simplify it to obtain:\[ f'(N) = 1 - \left(\frac{N}{K}\right)^\theta - \frac{\theta N}{K} \left(\frac{N}{K}\right)^{\theta-1} \].

Step 4: Simplify further

Factor out \(\left(\frac{N}{K}\right)^{\theta - 1}\) from the last two terms:\[ f'(N) = 1 - \left(\frac{N}{K}\right)^{\theta - 1} \left( \frac{N}{K} + \theta N \right) \].This gives:\[ f'(N) = 1 - \left(\frac{N}{K}\right)^{\theta - 1} N \left( \frac{1}{K} + \theta \right) \].Thus, it can be *simplified to \[ 1 - N \left( 1 + \theta \right) \left( \frac{N}{K^2} \right)^{\theta - 1} \].*

Step 5: Find increasing and decreasing intervals

Set \(f'(N) = 0\) to find critical points:\[ 1 - N \left( 1 + \theta \right) \left( \frac{N}{K^2} \right)^{\theta - 1} = 0 \].Solve for \(N\):\[ N = \left(\frac{1}{1 + \theta}\right)^{\frac{1}{\theta - 1}} \cdot K \].Determine intervals of increase or decrease by testing values around this critical point in \(f'(N)\).

Key Concepts

Differentiation

Differentiation is a powerful mathematical tool used to determine the rate at which a quantity changes with respect to another. In the context of population growth, differentiation helps us understand how the growth rate of a population changes as the population size itself changes.

When we differentiate the function \( f(N) = N \left(1 - \left(\frac{N}{K}\right)^\theta\right) \), we are seeking the expression for \( f'(N) \), which tells us how the function \( f(N) \) changes in response to changes in \( N \). This involves using various rules of differentiation, such as the product rule, to handle multiple components within the function.

• Identify each part of the function as a product of two functions.
• Use the product rule to differentiate each component accordingly.
• Simplify the resulting derivative expression to gain insights about the original function's behavior.

Understanding differentiation in this context provides insights into the population's tendencies, such as where it might grow faster or slower as it changes.

Carrying Capacity

Carrying capacity, denoted as \( K \), is a vital concept in ecology and population dynamics. It represents the maximum population size that an environment can support sustainably, based on the available resources.

In a mathematical model of population growth, the carrying capacity keeps the population from growing indefinitely. It acts like an upper barrier, limiting the growth rate as the population approaches this threshold.

• At low population sizes, the growth rate is usually higher because resources are abundant.
• As the population size approaches the carrying capacity, the growth slows down.
• This concept is represented in growth functions, ensuring they reflect the limitations imposed by the environment.

By incorporating carrying capacity into the growth models, we can more accurately simulate realistic population behavior and predict potential outcomes over time.

Critical Points

Critical points of a function are specific values where the function's derivative is zero or undefined. These points are significant because they can indicate potential maxima, minima, or points of inflection in the function.

To identify critical points in our population growth model, we set the derivative \( f'(N) \) to zero and solve for \( N \). This process tells us where the growth rate transitions from increasing to decreasing or vice versa.

• Derivatives equal to zero identify potential turning points in the population growth rate.
• By evaluating these points, we can determine whether they correspond to local maximum or minimum growth rates.
• Testing intervals around the critical point helps us understand the overall behavior of the growth function.

Understanding critical points ensures we know where important shifts in population dynamics occur, allowing better resource management and planning.

Product Rule

The product rule is a fundamental concept in calculus that deals with differentiating products of two functions. It states that the derivative of a product is the derivative of the first function multiplied by the second function, plus the first function multiplied by the derivative of the second function.

In our given population growth function \( f(N) = N \left(1 - \left(\frac{N}{K}\right)^\theta\right) \), applying the product rule is essential for correctly finding \( f'(N) \).

• Identify \( u = N \) and \( v = 1 - \left(\frac{N}{K}\right)^\theta \).
• Differentiate each part: \( u'(N) = 1 \) and for \( v(N) \), use the chain rule to obtain \( v' = -\theta \left(\frac{N}{K}\right)^{\theta - 1} \cdot \frac{1}{K} \).
• Substitute these derivatives back into the product rule formula.
• Simplify to obtain the final expression of \( f'(N) \).

Using the product rule accurately ensures that we capture all interactions within the function and provides a clearer view of its behavior.