Problem 42 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 41-44, solve the logistic differential equation representing population growth with the given initial condition. Then use the solution to predict the population size at time \(t=3\). $$ y^{\prime}=\frac{1}{10} y(12-y), y(0)=2 $$

Step-by-Step Solution

Answer

At \( t = 3 \), the predicted population size is approximately 2.55.

Step 1: Identify the Logistic Differential Equation

The given logistic differential equation is \( y' = \frac{1}{10} y (12 - y) \) with the initial condition \( y(0) = 2 \).

Step 2: Determine the Logistic Growth Function

The general solution for a logistic differential equation \( y' = ry(1 - \frac{y}{K}) \) is \( y(t) = \frac{K}{1 + Ae^{-rt}} \), where \( K \) is the carrying capacity and \( A \) is a constant determined by initial conditions. For our equation, \( r = \frac{1}{10} \) and \( K = 12 \).

Step 3: Solve for the Constant A

Using the initial condition \( y(0) = 2 \), we substitute into the logistic equation to find \( A \). \( 2 = \frac{12}{1 + A} \).From this, solve for \( A \) :\[ A + 1 = \frac{12}{2} = 6 \rightarrow A = 5. \]

Step 4: Write the Specific Logistic Function

Substitute \( K = 12 \), \( A = 5 \) and \( r = \frac{1}{10} \) into the logistic growth function:\[ y(t) = \frac{12}{1 + 5e^{-\frac{1}{10}t}}. \]

Step 5: Predict Population Size at t = 3

To find the population at \( t = 3 \), substitute \( t = 3 \) into the logistic function:\[ y(3) = \frac{12}{1 + 5e^{-\frac{3}{10}}} \].

Step 6: Calculate and Interpret the Result

Calculate \( y(3) \) using the expression:\[ e^{-\frac{3}{10}} \approx 0.7408 \], so \[ y(3) = \frac{12}{1 + 5 \times 0.7408} \approx \frac{12}{4.704} \approx 2.55. \]Thus, the predicted population size at \( t = 3 \) is approximately 2.55.

Key Concepts

Population Growth

Understanding how populations grow is essential in many fields, from biology to economics. In the context of the logistic differential equation, population growth is modeled by considering resources' limitations. Unlike exponential growth, which assumes unlimited resources, logistic growth incorporates the fact that resources can run out. As such, the growth rate decreases as the population size approaches the environment's carrying capacity. This is reflected in the logistic differential equation:

• The term \( y^{\prime} = \frac{1}{10} y(12 - y) \) implies that growth is fastest at low population sizes.
• As the population increases, the term \( (12-y) \) decreases, slowing the growth rate.

This creates an S-shaped or sigmoid curve characteristic of logistic growth, depicting rapid growth at first, which stabilizes as it reaches its maximum sustainable size.

Initial Condition

An initial condition in differential equations determines the system's state at the beginning of a process. It is a crucial part of solving differential equations, as it allows us to solve for unknown constants. In our problem, the initial condition is given as \( y(0) = 2 \), meaning that the population size at time \( t = 0 \) is 2. Having this information is vital because:

• It allows us to find the specific solution tailored to this initial state.
• The constant \( A \) in the logistic growth function \( y(t) = \frac{K}{1 + Ae^{-rt}} \) is determined using this condition.

By substituting and solving, we find \( A = 5 \), leading us to a solution that accurately reflects initial conditions. Thus, the initial condition personalizes general solutions to fit specific real-world scenarios.

Carrying Capacity

Carrying capacity is a vital concept that indicates the maximum population size an environment can sustain indefinitely. It reflects the availability of resources such as food, shelter, and space. In the logistic differential equation, the carrying capacity is denoted as \( K \). In our case, \( K = 12 \).

• The function \( y(t) = \frac{12}{1 + 5e^{-\frac{1}{10}t}} \) demonstrates that as \( t \) increases, the population \( y(t) \) approaches 12.
• This asymptotic behavior highlights the environmental limitation imposed on population growth.

It is crucial to acknowledge that carrying capacity depends on the specific environment's conditions. Understanding this concept allows for more realistic models in population dynamics, where exceeding the carrying capacity could lead to resource depletion.

Predictive Modeling

Predictive modeling uses mathematical models to forecast future data points. In the context of logistic population growth, we use our logistic growth equation to predict future population sizes at different time intervals. This is achieved by:

• Substituting specific values of \( t \) in the logistic equation. In our case, predicting the population at \( t = 3 \).
• Conducting calculations such as finding \( e^{-\frac{3}{10}} \) to refine our predictions.

With the solution \( y(3) \approx 2.55 \), we can forecast that the population will increase to approximately 2.55 at \( t = 3 \). This process is integral to resource management, conservation planning, and understanding future trends in various disciplines, offering insights into population dynamics over time.