Problem 30 - Calculus for Biology and Medicine

Question

For a population growing according to the logistic model we can calculate a per capita reproductive rate, which is defined to be equal to: $$ g(N)=\frac{f(N)}{N}=r\left(1-\frac{N}{K}\right), \quad N \geq 0 $$ (a) Plot the function \(g(N)\) for \(r=3\) and \(K=10\). (b) For the parameters \(r=3\) and \(K=10\), use calculus to find \(g^{\prime}(N)\), and determine where the function \(g(N)\) is increasing and where it is decreasing. (c) Now suppose that \(K=10\), but the value of \(r\) is not given to you (You may assume \(r>0 .\) ) Show that the reproduction rate \(g(N)\) is a decreasing function of \(N\) for all \(N>0\).

Step-by-Step Solution

Answer

(a) Plot is a line decreasing from 3 to 0; (b) g(N) decreases everywhere; (c) g(N) is always decreasing for N > 0 and r > 0.

Step 1: Plot the Function g(N)

To plot \( g(N) = 3 \left(1 - \frac{N}{10}\right) \), substitute \( r = 3 \) and \( K = 10 \). Thus, the equation simplifies to \( g(N) = 3 - \frac{3N}{10} \). Plotting this equation, we get a straight line with a slope of \(-0.3\) starting from \( g(0) = 3 \) and decreasing to zero at \( N = 10 \).

Step 2: Find the Derivative g'(N)

To find where \( g(N) \) is increasing or decreasing, calculate the derivative of \( g(N) = 3 - \frac{3N}{10} \). The derivative is \( g'(N) = -\frac{3}{10} \), which is a constant negative value.

Step 3: Determine Increasing/Decreasing Intervals

Since \( g'(N) = -\frac{3}{10} < 0 \) for all \( N \geq 0 \), \( g(N) \) is a decreasing function for all values of \( N \) greater than zero.

Step 4: Prove g(N) is Decreasing for N > 0 with General r

For the general case \( g(N) = r\left(1 - \frac{N}{10}\right) \), the derivative is \( g'(N) = -\frac{r}{10} \). Since \( r > 0 \), \(-\frac{r}{10} < 0 \), indicating \( g(N) \) is decreasing for all \( N > 0 \). Hence, the function consistently decreases regardless of the specific positive value of \( r \).

Key Concepts

Per Capita Reproductive Rate

In the logistic growth model, populations don't grow indefinitely because resources are limited. The per capita reproductive rate, denoted by \( g(N) \), helps us understand how the population grows relative to its size \( N \).
Simply put, it's the average rate of reproduction per individual member of the population. For our given model, \( g(N) = \frac{f(N)}{N} = r\left(1-\frac{N}{K}\right) \). This formula shows that the rate decreases as population \( N \) approaches the carrying capacity \( K \).

• \( r \) is the maximum per capita rate of increase.
• \( K \) is the carrying capacity, or the maximum sustainable population size.
• \( 1-\frac{N}{K} \) ensures the rate decreases as \( N \) grows.

Through this function, we can understand how effective population growth is as resources become scarce.

Derivative Calculation

To understand when the per capita reproductive rate is increasing or decreasing, we calculate its derivative, \( g'(N) \). Let's recall that the derivative indicates the slope of a function at any particular point, telling us whether it's increasing or decreasing.
Given \( g(N) = 3 - \frac{3N}{10} \), the derivative is a constant \( g'(N) = -\frac{3}{10} \). This negative constant shows that the rate \( g(N) \) declines steadily regardless of the value of \( N \).
More generally, with \( r \) unspecified but \( K = 10 \), derivative \( g'(N) = -\frac{r}{10} \) remains negative due to the positivity of \( r \).

• The derivative tells us there's a consistent downward trend.
• No matter the value of \( r \), maintaining \( r > 0 \), \( g(N) \) always decreases.
• This understanding helps verify predictions about growth limits.

Increasing and Decreasing Functions

Understanding when functions are increasing or decreasing is crucial, and it's all about analyzing the derivative. If a derivative is positive, the function increases; if negative, it decreases.
With our function, \( g(N) = 3 - \frac{3N}{10} \), the derivative \( g'(N) = -\frac{3}{10} \) shows a clear decrease for the entire range \( N \ge 0 \).
This means that each increase in \( N \) leads to a decrease in \( g(N) \) consistently. Similarly, if \( r \) instead of a fixed value is kept positive, the function \( g(N) = r(1-\frac{N}{10}) \) keeps its decreasing nature due to the constant negative slope \( -\frac{r}{10} \).

• The derivative gives insights into functional trends over the domain.
• Understanding this helps predict population behaviors in resource-limited ecosystems.
• It showcases how mathematics can explain real-world phenomena efficiently.

The analysis of this decreasing trend is critical in ecological modeling and understanding limits to population growth.