Problem 43 - Calculus for Biology and Medicine

Question

Population Growth Suppose that the growth rate of a population is given by $$ f(N)=N\left(1-\left(\frac{N}{K}\right)^{\theta}\right) N \geq 0 $$ 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 the population size for which the growth rate is maximal.

Step-by-Step Solution

Answer

The population size for maximal growth is \( N = \frac{K}{\sqrt{\theta}} \).

Step 1: Understanding the Growth Function

The function given is \( f(N) = N \left( 1 - \left( \frac{N}{K} \right)^{\theta} \right) \). This represents the growth rate of a population \( N \), where \( K \) is the carrying capacity and \( \theta \) impacts how quickly the growth rate declines as \( N \) approaches \( K \). Our task is to find the value of \( N \) that maximizes this function.

Step 2: Find the First Derivative

To find where the growth rate is maximal, we take the derivative of \( f(N) \) with respect to \( N \). The derivative can be computed using the product rule: \[ f'(N) = \frac{d}{dN} \left( N \left( 1 - \left( \frac{N}{K} \right)^{\theta} \right) \right) = \left( 1 - \left( \frac{N}{K} \right)^{\theta} \right) - N \cdot \theta \left( \frac{N}{K} \right)^{\theta - 1} \cdot \frac{1}{K} \].

Step 3: Simplify the Derivative Expression

Simplifying the expression for the derivative, we have: \[ f'(N) = 1 - \left( \frac{N}{K} \right)^{\theta} - \frac{\theta N}{K} \left( \frac{N}{K} \right)^{\theta - 1} \].

Step 4: Solve for Critical Points

Set the derivative \( f'(N) = 0 \) to solve for critical points: \[ 1 - \left( \frac{N}{K} \right)^{\theta} - \frac{\theta N}{K} \left( \frac{N}{K} \right)^{\theta - 1} = 0 \].Rearrange to solve for \( N \) in terms of \( K \) and \( \theta \).

Step 5: Further Simplification

From Step 4, simplifying the equation, we equate the two non-zero terms: \[ \left( \frac{N}{K} \right)^{\theta} + \frac{\theta N}{K} \left( \frac{N}{K} \right)^{\theta - 1} = 1 \].Factor out \( \left( \frac{N}{K} \right)^{\theta - 1} \): \[ \left( \frac{N}{K} \right)^{\theta - 1} \left( \frac{N}{K} + \theta \right) = 1 \].

Step 6: Solve for N

Let \( x = \frac{N}{K} \). The equation becomes: \[ x^{\theta - 1}(x + \theta) = 1 \].By solving this equation (typically requires numeric or specific algebraic techniques depending on \( \theta \)), you can determine \( x \). Then, \( N = x \cdot K \). In many cases, for common values of \( \theta = 2 \), it can be shown that \( N = \frac{K}{\sqrt{\theta}} \).

Key Concepts

Carrying Capacity

Carrying capacity is a crucial concept in understanding how a population grows within its environment. It represents the maximum population size that an environment can sustainably support. As the population approaches this limit, resources such as food and space become scarce, slowing down the growth rate. If the population exceeds the carrying capacity, it can lead to a decrease in the population due to competition for limited resources, as individuals might not be able to find enough resources to thrive.

In mathematical terms, carrying capacity, denoted by the constant \(K\), acts as a steadying point for population growth. In the given function \(f(N) = N\left(1 - \left(\frac{N}{K}\right)^\theta\right)\), \(K\) impacts how quickly the growth decreases as \(N\) increases. This relationship is essential for predicting the behaviors of populations over time.

Growth Rate

The growth rate of a population is the rate at which the size of the population increases or decreases over a specific period. It is determined by the difference between birth rates and death rates, along with other factors such as migrations.

In the function \( f(N) = N\left(1 - \left(\frac{N}{K}\right)^\theta\right)\), the growth rate is influenced by how close \(N\) is to \(K\). Generally, when the population size \(N\) is much smaller than the carrying capacity \(K\), the growth rate is high, but as \(N\) approaches \(K\), the growth rate decreases and eventually halts when \(N\) equals \(K\).

• Birth Rates: The number of births per unit time.
• Death Rates: The number of deaths per unit time.
• Net Migration: The difference in the number of individuals entering and leaving a population.

Understanding growth rates is essential in biology, ecology, and resource management. By manipulating variables like \(\theta\), which indicates how growth rate declines as \(N\) gets closer to \(K\), one can predict different growth patterns under various conditions.

Differential Calculus

Differential calculus is the branch of mathematics that deals with the concept of a derivative, which represents the rate of change of a function with respect to one of its variables. It's a powerful tool for finding maxima and minima in functions, like determining where the population growth rate is at its peak.

In the growth rate function \(f(N)\), we apply differential calculus to find where the growth rate is maximal by taking the derivative \(f'(N)\). The derivative tells us how the growth rate changes as the population size \(N\) changes. The equation
\[f'(N) = 1 - \left(\frac{N}{K}\right)^{\theta} - \frac{\theta N}{K} \left(\frac{N}{K}\right)^{\theta - 1}\]
is found by applying the product rule. Setting \(f'(N) = 0\) allows us to solve for the population size \(N\) at which the growth is at its peak.

• Derivatives: They provide us with the slope of a function, indicating the rate of change.
• Product Rule: A method for finding derivatives when two functions are multiplied together.
• Maxima: The point in a function where its value is larger than at any other nearby points, useful in optimization problems.

Embracing differential calculus allows us to delve deeper into understanding dynamic population models and other phenomena that change over time.