Problem 82 - Calculus for Biology and Medicine

Question

Find all equilibria, and, by calculating the eigenvalue of the differential equation, determine which equilibria are stable and which are unstable. $$ \frac{d N}{d t}=\frac{N-1}{N+1} \quad N \geq 0 $$

Step-by-Step Solution

Answer

The equilibrium is at \( N = 1 \), and it is unstable because the eigenvalue is positive.

Step 1: Set Differential Equation to Zero

To find the equilibria, set \( \frac{dN}{dt} \) to zero. The equation becomes: \[ \frac{N-1}{N+1} = 0 \]

Step 2: Solve for Equilibria

Solve the equation from Step 1, \( \frac{N-1}{N+1} = 0 \). This implies \( N - 1 = 0 \), so \( N = 1 \) is the equilibrium point.

Step 3: Linearize the System

To analyze stability, find the derivative of \( \frac{dN}{dt} \) with respect to \( N \). The derivative is: \[ \frac{d}{dN} \left( \frac{N-1}{N+1} \right) = \frac{2}{(N+1)^2} \] This will be used to evaluate stability at \( N = 1 \).

Step 4: Evaluate the Derivative at the Equilibrium Point

Substitute \( N = 1 \) into the derivative \( \frac{2}{(N+1)^2} \): \[ \frac{2}{(1+1)^2} = \frac{2}{4} = 0.5 \]

Step 5: Determine Stability from Eigenvalue

The eigenvalue, \( 0.5 \), is positive, indicating that the equilibrium \( N = 1 \) is unstable, because any positive eigenvalue means the system moves away from equilibrium.

Key Concepts

Understanding Differential Equations

A differential equation is like a mathematical equation that relates a function to its derivatives. It's a way of expressing how a quantity changes in relation to another. In our exercise, the equation \( \frac{dN}{dt}=\frac{N-1}{N+1} \) tells us how the quantity \( N \) changes over time, \( t \). This particular equation is a first-order differential equation because it relates \( N \) to its first derivative, \( \frac{dN}{dt} \).

Here's how it works in practice:

• The left side, \( \frac{dN}{dt} \), represents the rate of change of the variable \( N \).
• The right side, \( \frac{N-1}{N+1} \), defines how this change depends on \( N \) itself.

When we solve differential equations, we're looking to find the value or behavior of \( N \) over time. Specifically, finding equilibria involves determining when the rate of change is zero, meaning \( N \) doesn't change at all, leading to stable or unstable conditions.

Exploring Stability Analysis

Stability analysis helps us understand how small changes to our system affect its behavior over time. For the equilibrium found at \( N = 1 \), the main question is whether small deviations from this point grow or shrink as time passes.

In simpler terms, when we slightly disturb the system, will it return to equilibrium or continue to move away? To answer this, we look at changes in \( N \) using the derivative of our differential equation. By focusing on the value of the derivative evaluated at the equilibrium, we can predict how the system responds:

• If this derivative is positive, as with our example where it equals 0.5, it tells us disturbances will grow, implying instability.
• If it were negative, that would indicate stability, meaning disturbances would lessen over time, and the system would return to equilibrium.

Stability analysis is crucial in fields like engineering, physics, and biology, where systems are constantly adjusted for balance and control.

Deciphering Eigenvalues

Eigenvalues might sound complex, but they boil down to being vital for understanding system stability, particularly in linear systems. In our problem, once we linearized the differential equation, we ended up with an eigenvalue of 0.5.

Here's how that affects stability:

• Eigenvalues help quantify the impact of perturbations near equilibrium.
• A positive eigenvalue, like 0.5, suggests instability because any small change will lead the system to move away from the equilibrium.
• On the other hand, a negative eigenvalue would indicate that the system naturally moves back towards equilibrium, reflecting stability.

Understanding eigenvalues can often be intimidating. Yet, they play a central role in many theoretical and practical applications, serving as a measure of system robustness in contexts ranging from quantum mechanics to economic models.