Problem 45 - Calculus for Biology and Medicine

Question

You should treat \(h\) as a constant. For what values of \(h\) (if any) does each equation have equilibria? Use a graphical argument to show which of the equilibria (if any) are stable. $$ \frac{d x}{d t}=x^{2}-h x $$

Step-by-Step Solution

Answer

The equilibrium points are at \(x = 0\) (stable) and \(x = h\) (unstable) for all values of \(h\).

Step 1: Set the Derivative to Zero for Equilibrium

To find points of equilibrium, we set \( \frac{d x}{d t} = 0 \). Therefore, we solve the equation: \[ x^2 - h x = 0 \].

Step 2: Factor the Equation

The equation \( x^2 - h x = 0 \) can be factored as follows:\[ x(x - h) = 0 \].

Step 3: Solve for Equilibrium Points

From the factored equation \( x(x - h) = 0 \), we can identify the equilibrium points by setting each factor to zero. Thus, the solutions are \( x = 0 \) and \( x = h \).

Step 4: Determine Stability Using a Graphical Argument

To determine stability, let's consider the sign of \( \frac{d x}{d t} \) around the equilibrium points.- For \( x = 0 \), just to the right (e.g., \( x > 0 \)), \( x^2 - hx > 0\), thus \( \frac{d x}{d t} > 0 \), and just to the left (e.g., \( x < 0 \)), \( x^2 - hx < 0 \), so \( \frac{d x}{d t} < 0 \). Therefore, \( x = 0 \) is a stable equilibrium.- For \( x = h \), as \( x \) increases beyond \( h \), \( x^2 - hx > 0 \) and \( \frac{d x}{d t} > 0 \), but as \( x \) decreases below \( h \), \( x^2 - hx < 0 \) and \( \frac{d x}{d t} < 0 \). Therefore, \( x = h \) is an unstable equilibrium.

Key Concepts

Equilibrium Points

To find equilibrium points, we need to determine where the rate of change, or the derivative, is zero. Given the differential equation \( \frac{d x}{d t} = x^2 - h x \), equilibrium points occur when \( x^2 - h x = 0 \).
Setting the equation to zero helps us identify which values of \( x \) cause the entire expression to equal zero, meaning there's no change at that moment. This makes these points stable. We can simplify this by factoring:

• The equation becomes \( x(x - h) = 0 \).
• This tells us the possible equilibria are \( x = 0 \) and \( x = h \).

By identifying these values, we know that these are the points where the system does not change — they "stay put." This is crucial for understanding what happens in the system over time.

Stability Analysis

Once the equilibrium points \( x = 0 \) and \( x = h \) are identified, it's important to determine their stability. A stable equilibrium means that if the system is slightly disturbed, it will return to equilibrium. If it's unstable, the system will move away from these points.
To analyze stability, we check the sign of \( \frac{d x}{d t} \) nearby the equilibrium points:

• For \( x = 0 \):
  • Just right of 0 (\( x > 0 \)), \( x^2 - hx > 0 \), so \( \frac{d x}{d t} > 0 \).
  • Just left of 0 (\( x < 0 \)), \( x^2 - hx < 0 \), so \( \frac{d x}{d t} < 0 \).
  
  This indicates that \( x = 0 \) is a stable point.


• For \( x = h \):
  • If \( x > h \), \( x^2 - hx > 0 \), so \( \frac{d x}{d t} > 0 \).
  • If \( x < h \), \( x^2 - hx < 0 \), so \( \frac{d x}{d t} < 0 \).
  
  This means \( x = h \) is an unstable point.

By observing the behavior around these points, you can predict whether small disturbances will settle or amplify.

Graphical Arguments

Graphical analysis is a powerful tool to visually interpret stability, especially if equations are complex. By sketching the derivative \( \frac{d x}{d t} = x^2 - h x \), the behavior around equilibrium can be appreciated more intuitively. Think of the graph as showing how \( x \) changes over time.
Here's how to use graphical arguments:

• Plot the graph of \( x^2 - hx \). The \( x \)-axis intersections where the graph touches zero are the equilibrium points \( x = 0 \) and \( x = h \).
• Look at the areas where the graph is above the \( x \)-axis. This indicates \( \frac{d x}{d t} > 0 \), meaning \( x \) is increasing.
• Check where the graph is below the \( x \)-axis. Here \( \frac{d x}{d t} < 0 \) entails \( x \) is decreasing.

This visual evidence supports our earlier algebraic conclusions. The graphical approach confirms that at \( x = 0 \), disturbances return to the state, reinforcing stability. Conversely, at \( x = h \), disturbances escalate, underscoring instability. This way, you get a geometric insight into differential equations, facilitating a deeper understanding.