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.\(C=-3\)

Short answer

Equilibrium at \( P=0 \) is unstable; \( P=-3 \) is stable.

Step-by-step solution

Identify the Logistic Equation

We are given the logistic equation in the form \( P' = C P - P^2 \). The problem specifies \( C = -3 \), so our equation becomes \( P' = -3P - P^2 \). This equation models the rate of change of a population \( P \) over time.

Determine Equilibria Points

Equilibria occur when \( P' = 0 \). This implies the equation \( -3P - P^2 = 0 \). Factoring out \( P \), we have \( P(-3 - P) = 0 \). Thus, the equilibrium points are \( P = 0 \) and \( P = -3 \).

Evaluate Stability of Equilibria

To assess stability, analyze the sign of \( P' \) around the equilibria points. For a simplified analysis, use a number line or test points:- For \( P < 0 \) (e.g., \( P = -4 \)), \( P' = -3(-4) - (-4)^2 = 12 - 16 = -4 \), so \( P' < 0 \). Thus, \( P = 0 \) is stable from the left.- For \( -3 < P < 0 \) (e.g., \( P = -1 \)), \( P' = -3(-1) - (-1)^2 = 3 - 1 = 2 \), so \( P' > 0 \). This implies \( P = 0 \) is unstable from the right and \( P = -3 \) is stable from the right.

Plot the Direction Field

For each point around the equilibrium values \( P = 0 \) and \( P = -3 \), plot the direction (i.e., arrows indicating \( P' \)) based on the calculated \( P' \) values. Negative values indicate arrows pointing downwards, and positive values indicate arrows pointing upwards.

Key concepts

Directional Field

In the context of differential equations, a directional field (also known as a slope field) is a visual representation that shows the behavior of solutions without solving the equation analytically. It helps us understand how the function evolves over time by drawing small arrows or line segments at various points on the graph. Each arrow points in the direction of the rate of change given by the differential equation at that point.

To create a directional field for the logistic equation given as \( P' = -3P - P^2 \), we compute the slope \( P' \) for several values of the population \( P \). After determining the slopes at several points, draw small arrows indicating whether the change is increasing or decreasing. For negative values of \( P' \), the arrows should point downwards, indicating a decrease. For positive values, the arrows point upwards, indicating an increase.

This visual tool provides a clear, intuitive understanding of how solutions behave, especially near equilibrium points, where changes can indicate stability or instability.

Stability of Equilibria

Stability of equilibria refers to whether small perturbations from an equilibrium position lead the system back to the equilibrium (stable) or away from it (unstable). In our logistic equation \( P' = -3P - P^2 \), we identify points where \( P' = 0 \), known as equilibrium points.

There are two kinds of stability analysis:

• Stable equilibrium: Small deviations from the equilibrium return to the equilibrium point.
• Unstable equilibrium: Small deviations result in moving away from the equilibrium point.

In our example, after calculating \( P' \) for values around each equilibrium (\( P = 0 \) and \( P = -3 \)), we find:

• \( P = 0 \) is stable from the left due to negative \( P' \), as disturbances result in a return to equilibrium.
• \( P = 0 \) is unstable from the right because disturbances grow, moving the system away.
• \( P = -3 \) is stable because any small deviation results in the system returning to \( P = -3 \).

This analysis is crucial for understanding the long-term behavior of solutions in many applied contexts, including population dynamics and other real-world scenarios.

Equilibrium Points

Equilibrium points occur where the rate of change \( P' \) equals zero, indicating that the population \( P \) remains constant over time. For the logistic equation \( P' = -3P - P^2 \), we determine these points by solving \( -3P - P^2 = 0 \).

Factoring this equation yields \( P(-3 - P) = 0 \), which gives us two equilibrium points: \( P = 0 \) and \( P = -3 \). These points signify populations at which there is no change over time under the current model. Depending on the stability analysis, knowing these points helps predict if a given initial population will restore to these points or move away, providing a crucial insight into the behavior of the system.

Rate of Change Analysis

Rate of change analysis involves studying how the quantity \( P \) changes over time as governed by the differential equation \( P' = -3P - P^2 \). The focus is on calculating \( P' \) at different values of \( P \) to understand if the population is increasing or decreasing.

By examining points around the equilibria, we obtain:

• For \( P < 0 \), such as \( P = -4 \), \( P' \) is negative, indicating a decrease in the population.
• For \( -3 < P < 0 \), like \( P = -1 \), \( P' \) is positive, indicating an increase in the population.

The rate of change helps anticipate future behavior of the system, aiding predictions of growth or decline towards equilibrium points. Understanding these dynamics is essential for applications like ecological modeling, where predicting population shifts is critical.