Introduction: When Variables Interact

In our study of calculus so far, we have mostly looked at how one variable changes in relation to time. But in the real world, things rarely act alone. Think about a predator-prey relationship: the number of rabbits depends on how many foxes there are, and the number of foxes depends on how many rabbits they can eat. This interaction is called a coupled system.

In this chapter, we will learn how to model these interactions using pairs of differential equations and how to "see" the long-term behavior of these systems using phase portraits. Don't worry if it sounds complex—it’s essentially just looking at the "flow" of a system on a graph!

1. Euler’s Method for Coupled Systems

Sometimes, differential equations are too messy to solve exactly by hand. When this happens, we use Euler's Method to estimate the values of \(x\) and \(y\) step-by-step. In a coupled system, we have two equations happening at the same time:

\(\frac{dx}{dt} = f_1(x, y, t)\)

\(\frac{dy}{dt} = f_2(x, y, t)\)

How the "Step" Works

To find the next values (\(x_{n+1}\) and \(y_{n+1}\)) based on the current values (\(x_n\) and \(y_n\)), we use a small time step, \(h\):

\(x_{n+1} = x_n + h \cdot \left(\frac{dx}{dt}\right)_n\)

\(y_{n+1} = y_n + h \cdot \left(\frac{dy}{dt}\right)_n\)

Step-by-Step Process:

  1. Start with your initial conditions: \(t_0\), \(x_0\), and \(y_0\).
  2. Calculate the rates of change \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) using the initial values.
  3. Multiply these rates by the step size \(h\).
  4. Add these values to your current \(x\) and \(y\) to get the new values.
  5. Repeat the process for the next time interval.

Quick Tip: Always use the same values of \(x_n\) and \(y_n\) to calculate both new values. Don't use the "new" \(x\) you just found to calculate the "new" \(y\) in the same step!

2. Linear Coupled Systems

In this course, we focus heavily on linear systems that look like this:

\(\frac{dx}{dt} = ax + by\)

\(\frac{dy}{dt} = cx + dy\)

We can write this in matrix form (which connects back to Topic 1):

\(\begin{pmatrix} \frac{dx}{dt} \\ \frac{dy}{dt} \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}\)

The matrix \(\mathbf{M} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\) holds the "DNA" of the system. By finding its eigenvalues (\(\lambda\)), we can predict exactly how the system will behave without ever solving the equations!

3. Phase Portraits: The "Map" of the System

A phase portrait is a graph where the x-axis represents \(x\) and the y-axis represents \(y\). We draw trajectories (paths) that show how the state of the system moves over time.

Key Terms:

  • Equilibrium Point: A point where \(\frac{dx}{dt} = 0\) and \(\frac{dy}{dt} = 0\). For the systems we study, this is usually at the origin \((0,0)\). At this point, the system is "at rest."
  • Trajectory: A curve representing the path a system takes as time passes.
  • Stable: If trajectories move toward the equilibrium point.
  • Unstable: If trajectories move away from the equilibrium point.

4. Qualitative Analysis using Eigenvalues

The behavior of the phase portrait depends on the eigenvalues (\(\lambda_1\) and \(\lambda_2\)) of the matrix. Here is how to interpret them:

Case A: Distinct Real Eigenvalues

  1. Both Positive (\(\lambda_1, \lambda_2 > 0\)): An unstable node (source). All paths move away from the origin.
  2. Both Negative (\(\lambda_1, \lambda_2 < 0\)): A stable node (sink). All paths move toward the origin. Think of this as a marble settling at the bottom of a bowl.
  3. Opposite Signs (\(\lambda_1 > 0, \lambda_2 < 0\)): A saddle point. Trajectories move toward the origin along one direction but get pushed away in another. It's always unstable.

Case B: Complex Eigenvalues (\(\lambda = \alpha \pm \beta i\))

  1. Real part is positive (\(\alpha > 0\)): An unstable spiral. The system spirals outward.
  2. Real part is negative (\(\alpha < 0\)): A stable spiral. The system spirals inward toward the origin.

Case C: Purely Imaginary Eigenvalues (\(\lambda = \pm \beta i\))

  • The trajectories are ellipses or circles around the origin. This is called a center. The system orbits forever, like a perfect pendulum with no friction.

Did you know? In biology, "centers" are used to model predator-prey cycles where the populations rise and fall in a repeating, periodic pattern.

5. Stable Populations and Eigenvectors

When dealing with stable populations in a real-world context, we look at the eigenvectors. If the system has a dominant positive eigenvalue, the population ratio will eventually settle to align with the direction of the corresponding eigenvector.

Example: If the eigenvector for the largest eigenvalue is \(\begin{pmatrix} 2 \\ 1 \end{pmatrix}\), it means that in the long run, there will be 2 of species \(x\) for every 1 of species \(y\).

Common Mistakes to Avoid

  • Mixing up \(x\) and \(t\): Remember that in a phase portrait, time (\(t\)) is not an axis. The axes are the two variables (usually \(x\) and \(y\)).
  • Eigenvalue Signs: Double-check your signs! A single negative sign can turn a "source" into a "saddle point."
  • Euler's Step Size: Using a step size (\(h\)) that is too large will make your Euler's Method estimation very inaccurate. Smaller \(h\) is usually better but takes more steps.

Quick Review Box

1. Saddle Point: One positive, one negative real eigenvalue.
2. Stable Sink: Both eigenvalues have negative real parts.
3. Unstable Source: Both eigenvalues have positive real parts.
4. Center: Eigenvalues are purely imaginary (no real part).
5. Euler's for Coupled Systems: Update both variables simultaneously using the current values.

Note: For more on how to calculate eigenvalues and eigenvectors, refer to the chapter on "Matrices (AHL)". For more on basic differential equations, see "Differential equations, slope fields and Euler's method (HL)".