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.\(C=0\)

Short answer

The equilibrium at \( P = 0 \) is stable, and the directional field shows \( P \) decreasing towards zero.

Step-by-step solution

Identify the Logistic Equation

The logistic equation is given as \( P^{\prime} = C P - P^2 \). In this exercise, we are considering when \( C = 0 \). Thus, the logistic equation simplifies to \( P^{\prime} = -P^2 \).

Determine the Equilibrium Points

Equilibrium points occur where \( P^{\prime} = 0 \). For the equation \( P^{\prime} = -P^2 \), setting \( -P^2 = 0 \) gives the equilibrium point \( P = 0 \). Thus, the equilibrium point is \( P = 0 \).

Analyze Stability of Equilibrium

To analyze stability, consider a small perturbation around the equilibrium. The derivative \( P^{\prime} = -P^2 \) implies that if \( P \) is positive or negative, \( P^2 \) is positive, making \( P^{\prime} \) negative. This results in \( P \) decreasing towards zero. Therefore, the equilibrium at \( P = 0 \) is a stable equilibrium.

Draw the Directional Field

For the equation \( P^{\prime} = -P^2 \), the directional field has arrows pointing downwards for all \( P e 0 \), indicating that \( P \) tends to decrease towards zero for both positive and negative values. At \( P = 0 \), the field remains constant, showing stability at that point. The directional field, therefore, consists of downward-pointing arrows except at the equilibrium.

Key concepts

Equilibrium Points

In the context of the logistic equation, an equilibrium point occurs where the rate of change is zero. It means that at this point, the system is in balance and the variable does not change over time.
For the logistic equation given in this problem, which is simplified as \(P^{\prime} = -P^2\) when \(C = 0\), we find equilibrium points by setting \(P^{\prime} = 0\).

• This translates to \(-P^2 = 0\), which implies that the equilibrium point is at \(P = 0\).
• Equilibrium points act as critical markers, revealing where a system might stabilize over time.

Understanding equilibrium points offers insight into how a system behaves in the long run, essential for analyzing systems modeled by differential equations.

Stability Analysis

Stability analysis tells us how the system behaves when slightly disturbed from an equilibrium point. In simpler words, it answers the question: "Will things return to normal if they're slightly off?"
For the equation \(P^{\prime} = -P^2\), let's consider minor fluctuations around the point \(P = 0\).

• The term \(-P^2\) ensures that \(P^{\prime}\) remains negative whether \(P\) is slightly positive or slightly negative.
• This continual negative slope indicates that, regardless of small changes, \(P\) will always trend back towards zero.
• Therefore, the equilibrium at \(P = 0\) is stable, as small perturbations will decay back to this point.

In this analysis, a stable equilibrium means small deviations from \(P = 0\) will naturally correct themselves over time.

Directional Field

A directional field provides a visual representation of the behavior of differential equations, serving as a guide to understanding how solutions evolve over time.
For the equation \(P^{\prime} = -P^2\), the directional field can be visualized by plotting arrows that represent the direction of change in \(P\).

• For every point where \(P eq 0\), the arrows point downwards, indicating that \(P\) will decrease.
• At the equilibrium point \(P = 0\), the arrows are flat, showing no direction of change, representing stability.
• This means all trajectories lead back towards \(P = 0\), regardless of the initial position of \(P\).

The directional field is a powerful tool that enables us to visualize dynamic systems and predict their future behavior by simply interpreting arrow directions.