Problem 61 - Calculus Single Variable

Question

Let \(K\) and \(M\) be positive constants. Suppose that as population \(P(t)\) increases from \(M / 10\) to \(M,\) it satisfies the equation $$ P^{\prime}(t)=k \cdot P(t) \cdot(M-P(t)) $$ Show that the instantaneous rate of change of the population is greatest when the population size is \(M / 2\).

Step-by-Step Solution

Answer

The population size for the greatest rate of change is when \( P = M/2 \).

Step 1: Understanding the Rate of Change

The differential equation given is \( P'(t) = k \cdot P(t) \cdot (M - P(t)) \). This represents a logistic growth model, where \( P(t) \) is the population at time \( t \), and \( k \) is a constant. Our job is to find when this rate of change is maximized.

Step 2: Determine Critical Points

Set up the derivative of the rate of change \( P'(t) \) with respect to \( P(t) \) and find the critical points. This means we calculate the derivative \( \frac{d}{dP}[k \cdot P(t) \cdot (M - P(t))] \) and set it to zero.

Step 3: Compute the Derivative

Compute the derivative of \( P'(t) = k \cdot P(t) \cdot (M - P(t)) \) with respect to \( P(t) \):\[\frac{d}{dP}[k \cdot P \cdot (M - P)] = k \cdot (M - 2P(t))\] Set this equal to zero to find the critical points: \[ k \cdot (M - 2P) = 0 \] which gives \( M - 2P = 0 \).

Step 4: Solve for Critical Population Size

Solve the equation \( M - 2P = 0 \) for \( P \): \[ M = 2P \] Thus, \( P = \frac{M}{2} \). This means the population size for the maximum instantaneous rate of change is \( \frac{M}{2} \).

Step 5: Verify Maximum using Second Derivative

To confirm that this critical point is indeed a maximum, compute the second derivative of \( P'(t) \) with respect to \( P \):\[\frac{d^2}{dP^2}[k \cdot P \cdot (M - P)] = -2k\]since \( -2k \) is negative, this confirms that the critical point corresponding to \( P = \frac{M}{2} \) is a maximum.

Key Concepts

Differential Equations in Logistic Growth

Differential equations are mathematical tools used to describe various rates of change. In the context of population growth, they help us understand how a population changes over time. The logistic growth model is a classic example, described by the differential equation \( P'(t) = k \cdot P(t) \cdot (M - P(t)) \), where \( P(t) \) is the population at time \( t \), \( k \) is a constant, and \( M \) is the carrying capacity—the maximum population size the environment can sustain.
The equation is a product of three components:

• \( k \): The growth rate constant, which influences how quickly the population reaches the carrying capacity.
• \( P(t) \): The current population, showing that growth is proportional to the population size.
• \( M - P(t) \): The remaining capacity for growth, which accounts for the slowing growth as the population nears \( M \).

Understanding this equation is crucial for predicting how populations behave over time, especially when resources are limited.

Critical Points and Their Significance

Critical points in calculus are where the first derivative of a function is zero or undefined, indicating potential maximum or minimum points. In our context, determining the critical points involves finding where the derivative of the logistic growth rate itself, \( P'(t) \), is zero. This helps us identify where the population growth rate changes most rapidly.
To find the critical points of \( P'(t) = k \cdot P(t) \cdot (M - P(t)) \), we take its derivative concerning \( P(t) \) and set it to zero:

• \( \frac{d}{dP}[k \cdot P \cdot (M - P)] = k \cdot (M - 2P) \)
• Setting \( k \cdot (M - 2P) = 0 \) yields \( M - 2P = 0 \), giving us \( P = \frac{M}{2} \).

At \( P = \frac{M}{2} \), the population growth rate is at one of its critical points. Analyzing these points is essential to understanding where and how the population changes most effectively, providing insight into resource allocation and management strategies.

Understanding Maximum Rate of Change

The maximum rate of change indicates the point at which a process like population growth is fastest. In the logistic growth model, this is when the derivative of the rate of growth, \( P'(t) \), is at a maximum with respect to the population size \( P(t) \). Finding this point helps us understand under what conditions a population grows most efficiently.
As outlined, the first derivative condition \( M - 2P = 0 \) provides our critical point at \( P = \frac{M}{2} \). To confirm that this is indeed a maximum, one examines the second derivative:

• \( \frac{d^2}{dP^2}[k \cdot P \cdot (M - P)] = -2k \)
• Since \(-2k\) is negative, it suggests a concave down curve, confirming a local maximum.

This analysis indicates that logistic population models predict the most rapid increase in population at half the carrying capacity. Such insight is vital for understanding population dynamics and implementing appropriate ecological or resource management practices.