Question

For the following problems, consider the logistic equation in the form \(P^{\prime}=C P-P^{2} .\) Draw the directional field and find the stability of the equilibria.[T] Two monkeys are placed on an island. After 5 years, there are 8 monkeys, and the estimated carrying capacity is 25 monkeys. When does the population of monkeys reach 16 monkeys?

Short answer

The population reaches 16 monkeys after approximately 7.24 years.

Step-by-step solution

Understand the Logistic Equation

The logistic equation is given as \( P^{\prime}=C P-P^{2} \), where \( P \) is the population and \( C \) is the initial per-capita growth rate. The term \( -P^2 \) represents the limiting factor due to the carrying capacity.

Find Equilibria and Stability

Set \( P' = 0 \) to find equilibria: \( CP - P^2 = 0 \Rightarrow P(C - P) = 0 \). Solutions are \( P = 0 \) and \( P = C \). To determine stability, linearize around these points or use phase line analysis. Generally, \( P=0 \) is unstable and \( P=C \) is stable for \( C \gt 0 \).

Determine the Constant C

Given the carrying capacity \( K = 25 \), we use the relationship \( C = K \) because at \( P=K \), the growth stops. Hence, \( C = 25 \). The modified equation becomes \( P' = 25P - P^2 \).

Setup and Solve the Logistic Differential Equation

The equation \( P' = 25P - P^2 \) is separable. Rewrite as \( \frac{dP}{P(25-P)} = dt \). Use partial fraction decomposition to integrate: \( \int \left( \frac{1}{P} + \frac{1}{25-P} \right) dP = \int dt \).

Integrate and Simplify

Integrate: \( \ln|P| - \ln|25-P| = 25t + C_1 \) leads to \( \ln\left(\frac{P}{25-P}\right) = 25t + C_1 \). Solve for \( P \): \( \frac{P}{25-P} = e^{25t + C_1} \).

Use Initial Condition to Find Solution Constant

When \( t = 0 \), \( P = 2 \), so \( \frac{2}{25-2} = e^{C_1} \Rightarrow e^{C_1} = \frac{2}{23} \). This implies \( \frac{P}{25-P} = \frac{2}{23}e^{25t} \).

Solve for When Population is 16

Set \( P = 16 \): \( \frac{16}{9} = \frac{2}{23}e^{25t} \). Solve for \( t \) by rearranging: \( e^{25t} = \frac{16 \times 23}{2 \times 9} \). Calculate \( t \) using the natural logarithm: \( t = \frac{1}{25} \ln\left( \frac{368}{18} \right) \).

Calculate Time to Reach 16 Monkeys

Evaluate \( t = \frac{1}{25} \ln\left( \frac{368}{18} \right) \) to find \( t \) approx 2.24 years. Therefore, the population will reach 16 monkeys in about 2.24 years from \( t=0 \).

Key concepts

Equilibria Stability

When studying differential equations like the logistic equation, understanding equilibria and their stability is critical. Equilibria are the points where the system doesn't change, meaning the derivative or rate of change is zero. In the logistic equation \( P^{\prime}=C P-P^{2} \), setting \( P' = 0 \) gives us the equilibria points. Solving this gives \( P = 0 \) and \( P = C \).
To investigate stability:

• The point \( P = 0 \) is usually unstable, meaning any small increase in population will lead it to grow.
• The point \( P = C \) is stable if \( C > 0 \). This means if the population reaches the carrying capacity, any small disturbance will balance out, keeping the population around that level.

Understanding stability helps in predicting long-term behavior of populations in a given ecosystem. Stabile equilibria act as attractors; the population will tend to stabilize around these points over time.

Carrying Capacity

The concept of carrying capacity is central to the logistic equation. It represents the maximum population that the environment can sustainably support. In mathematical terms, this is where the growth of the population ceases, which can be seen in the logistic equation as the point where the population stabilizes: \( P = C \).
For instance, in our problem, the carrying capacity \( K \) is 25 monkeys. This means the island can support up to 25 monkeys, and if the population goes beyond this, resources will not be sufficient to sustain them. In the differential equation \( P' = 25P - P^2 \), the term \(-P^2\) ensures that as \( P \) approaches 25, the growth slows and stops, reflecting the limiting effects of carrying capacity. By understanding carrying capacity, scientists and ecologists can manage wildlife preserves and predict how populations will respond to changes, ensuring sustainable resource usage.

Differential Equation

A differential equation is an equation involving an unknown function and its derivatives. They are powerful tools in modeling real-world phenomena because they describe how things change over time. The logistic equation, \( P^{\prime}=C P-P^{2} \), is a fundamental example of a differential equation.
This equation models population growth by considering two significant factors:

• The growth rate, represented by \( CP \). It describes how the population increases when resources are ample.
• The limiting factor, \(-P^2\), reflects the reduction in growth as the population reaches the carrying capacity due to limited resources.

By solving differential equations like this, we gain insights into the dynamics of population growth. Such solutions can show us how quickly populations can grow, stabilize, or decline, depending on initial conditions and environmental constraints. The power of differential equations lies in their ability to model change and predict future states of dynamic systems.