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

Equilibria at \( P=0 \) (unstable), \( P=3 \) (stable). Solutions approach \( P=3 \).

Step-by-step solution

Identify the Logistic Equation

First, let's identify the given logistic differential equation. The equation provided is in the form \( P' = C P - P^2 \), where we are given \( C = 3 \). Substitute \( C \) into the equation to obtain: \( P' = 3P - P^2 \).

Find Equilibria

To find the equilibria of the equation, set \( P' = 0 \). This gives \( 3P - P^2 = 0 \). Factoring out \( P \), we have \( P(3 - P) = 0 \). The solutions are \( P = 0 \) and \( P = 3 \). These are the equilibrium points.

Analyze Stability of Equilibria

To determine stability, we can use the derivative of \( P' \) with respect to \( P \) evaluated at each equilibrium. The derivative \( \frac{d}{dP}(3P - P^2) = 3 - 2P \). Evaluate at \( P = 0 \): \( 3 - 2(0) = 3 \) (positive, so \( P = 0 \) is unstable). Evaluate at \( P = 3 \): \( 3 - 2(3) = -3 \) (negative, so \( P = 3 \) is stable).

Draw the Directional Field

Sketch a directional field by plotting values of \( P' \) for various \( P \) values. At \( P = 0 \) and \( P = 3 \), draw horizontal lines. For \( 0 < P < 3 \), \( P' > 0 \), indicating increasing values (arrows up). For \( P > 3 \), \( P' < 0 \), indicating decreasing values (arrows down). This shows \( P = 0 \) as a repeller and \( P = 3 \) as an attractor.

Conclusion

The directional field and stability analysis show that \( P = 0 \) is an unstable equilibrium and \( P = 3 \) is a stable equilibrium, indicating that solutions head towards \( P = 3 \) starting from initial values near \( P = 0 \).

Key concepts

Equilibrium Stability

In the context of differential equations, equilibrium points are solutions where the system doesn't change over time. For the logistic equation, these are found by setting the derivative of the population function, \( P' \), to zero. This means solving the equation \( 3P - P^2 = 0 \). From the solution, we determine the equilibrium points: \( P = 0 \) and \( P = 3 \).

Equilibrium stability tells us whether small disturbances or perturbations will decay back to the equilibrium (stable) or grow away from it (unstable). This can be determined by substituting the equilibrium points into the derivative of \( P' \), which is \( 3 - 2P \). If this derivative is negative at an equilibrium point, it is stable, as disturbances tend to die out. If the derivative is positive, the point is unstable since disturbances will magnify.

• For \( P = 0 \), the derivative is \( 3 \), indicating instability.
• For \( P = 3 \), the derivative is \( -3 \), suggesting stability.

Therefore, small changes around \( P = 3 \) will return to that point, while any changes around \( P = 0 \) will move away.

Directional Field

A directional field for a differential equation like the logistic equation visualizes how solutions evolve over time. It is a graph where each point represents the direction in which the system moves at that point. On the plot of our equation \( P' = 3P - P^2 \), you would draw small arrows indicating the sign and magnitude of \( P' \) at different values of \( P \). This helps illustrate how different initial conditions influence the behavior of the solution over time.

In our case, we note that:

• At the equilibria \( P = 0 \) and \( P = 3 \), the arrows are horizontal, signifying no change.
• For \( 0 < P < 3 \), the arrows point upwards, indicating that the population is increasing.
• For \( P > 3 \), the arrows point downwards, showing that the population is decreasing.

This visualization supports the idea that \( P = 3 \) acts as an attractor (stable equilibrium), and \( P = 0 \) acts as a repeller (unstable equilibrium).

Differential Equations

Differential equations are mathematical expressions that describe how a quantity changes with respect to another. They are vital in modeling real-world phenomena where rates of change are involved, like population dynamics using the logistic equation. The logistic equation \( P' = 3P - P^2 \) is a first-order nonlinear ordinary differential equation. It models how a population grows rapidly initially, then slows as it reaches a carrying capacity (here, \( P = 3 \)).

Understanding differential equations involves:

• Identifying the type of differential equation (e.g., ordinary vs. partial; linear vs. nonlinear).
• Finding equilibrium points by setting the derivative \( P' = 0 \), which identifies constant solutions.
• Examining the stability of these points through derivatives, predicting how or if the system returns to equilibrium after perturbations.

Learning to solve and analyze these equations allows students to predict long-term behaviors, an essential skill in fields like biology, economics, and engineering.