Problem 27 - Calculus for Biology and Medicine

Question

Suppose that the size of a population at time \(t\) is \(N(t)\) and the growth rate of the population is given by the logistic growth function $$ \frac{d N}{d t}=r N\left(1-\frac{N}{K}\right), \quad t \geq 0 $$ where \(r\) and \(K\) are positive constants. The per capita growth rate is defined by $$ g(N)=\frac{1}{N} \frac{d N}{d t}=r\left(1-\frac{N}{K}\right) $$ (a) Graph \(g(N)\) as a function of \(N\) for \(N \geq 0\) when \(r=3\) and \(K=10\) (b) The function \(g(N)=r(1-N / K), N \geq 0\), is differentiable for \(N>0\). Compute \(g^{\prime}(N)\), and determine where the function \(g(N)\) is increasing and where it is decreasing.

Step-by-Step Solution

Answer

The function is decreasing for all \( N > 0 \), as \( g'(N) = -\frac{3}{10} \).

Step 1: Understand the Per Capita Growth Function

The given per capita growth function is \( g(N) = r\left(1 - \frac{N}{K}\right) \). For the specific problem, \( r = 3 \) and \( K = 10 \). Hence, the function becomes \( g(N) = 3\left(1 - \frac{N}{10}\right) \).

Step 2: Graph the Per Capita Growth Function

Consider the function \( g(N) = 3\left(1 - \frac{N}{10}\right) \). This is a linear function with a negative slope. At \( N = 0 \), \( g(0) = 3 \), and at \( N = 10 \), \( g(10) = 0 \). The function decreases linearly from 3 to 0 as \( N \) increases from 0 to 10.

Step 3: Find the Derivative of the Function

To find where \( g(N) \) is increasing or decreasing, compute the derivative: \( g'(N) = \frac{d}{dN}3\left(1 - \frac{N}{10}\right) = 3(-\frac{1}{10}) = -\frac{3}{10} \). The derivative is constant, indicating a linear decrease.

Step 4: Determine Where the Function is Increasing or Decreasing

Since the derivative \( g'(N) = -\frac{3}{10} \) is negative, \( g(N) \) is decreasing for all \( N > 0 \). The function is not increasing at any point because the slope is always negative.

Key Concepts

Population Dynamics

Population dynamics refers to how populations change over time regarding their size and structure. The logistic growth model is a common tool for studying these changes. It describes how a population can expand rapidly when numbers are small and resources are plentiful, but moderates to a steady state as it approaches the carrying capacity, represented by \(K\).
This model acknowledges that unlimited growth is unrealistic in natural settings because resources such as food, space, or mates are not infinite.

• In the logistic growth function \(\frac{d N}{d t} = r N\left(1-\frac{N}{K}\right)\), \(dN/dt\) represents the rate of population change over time.
• The constant \(r\) is the intrinsic growth rate, indicating how quickly the population can grow under ideal conditions.
• The term \(1 - \frac{N}{K}\) introduces the concept of resource limitation, slowing growth as \(N\) approaches \(K\).

This balance between growth forces and limiting factors allows for a sustainable population level, portraying real-world biological systems effectively.

Per Capita Growth Rate

The per capita growth rate is vital for understanding how individual members of a population contribute to total population growth. It is defined by the function \(g(N) = \frac{1}{N} \frac{dN}{dt}\), showing the average contribution to growth by each individual.
In our logistic model example, \(g(N) = r\left(1-\frac{N}{K}\right)\) further simplifies our understanding.

• In this case, with \(r = 3\) and \(K = 10\), the formula becomes \(g(N) = 3\left(1-\frac{N}{10}\right)\).
• Graphing \(g(N)\) against \(N\) shows a linear decline as \(N\) increases from 0 to 10, starting at 3 when \(N = 0\) and reaching 0 at \(N = K\).

This negative slope indicates each individual's growth contribution diminishes as population size approaches carrying capacity.

Differentiation

Differentiation helps us to find how functions change over input values, which in this case is crucial for analyzing \(g(N) = r\left(1-\frac{N}{K}\right)\).
Applying differentiation to \(g(N)\) reveals how and where the function changes.

• By computing the derivative \(g'(N)\), we find \(g'(N) = -\frac{3}{10}\).
• The constant negative value of the derivative indicates that the function is decreasing linearly for all \(N \gt 0\).

This means, as the population size increases, the per capita growth rate decreases steadily, emphasizing the impact of resource limitation as populations near their environmental carrying capacity.