Problem 52 - Calculus for Biology and Medicine

Question

We consider differential equations of the form $$ \frac{d \mathbf{x}}{d t}=A \mathbf{x}(t) $$ where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of \(A\) will be complex conjugates. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a stable spiral, an unstable spiral, or a center. $$ A=\left[\begin{array}{rr} 3 & -2 \\ 1 & 3 \end{array}\right] $$

Step-by-Step Solution

Answer

The equilibrium \((0,0)\) is an unstable spiral because eigenvalues have positive real parts.

Step 1: Find the Characteristic Equation

To find eigenvalues, calculate the determinant of \( (A - \lambda I) \), where \( I \) is the identity matrix. For our matrix \( A \): \[ A - \lambda I = \begin{pmatrix} 3-\lambda & -2 \ 1 & 3-\lambda \end{pmatrix} \] The determinant is: \[ (3-\lambda)^2 + 2 = \lambda^2 - 6\lambda + 13 \] So, the characteristic equation is \( \lambda^2 - 6\lambda + 13 = 0 \).

Step 2: Solve the Characteristic Equation

Solve \( \lambda^2 - 6\lambda + 13 = 0 \) using the quadratic formula: \[ \lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 1, b = -6, c = 13 \). Substituting these gives: \[ \lambda = \frac{6 \pm \sqrt{(-6)^2 - 4 \times 1 \times 13}}{2} = \frac{6 \pm \sqrt{-4}}{2} = 3 \pm i \] The eigenvalues are \( \lambda = 3 + i \) and \( \lambda = 3 - i \).

Step 3: Analyze the Real part of Eigenvalues

The eigenvalues \( 3 \pm i \) have a real part of 3, which is positive. A positive real part indicates that the equilibrium point is unstable.

Step 4: Classify the Equilibrium

Since the real part of the eigenvalues is positive, the equilibrium \((0,0)\) is an unstable spiral (focus). For linear systems of the form \( \dot{\mathbf{x}} = A\mathbf{x}\), if \( A \) has complex eigenvalues with positive real parts, the origin is an unstable spiral.

Key Concepts

Complex Eigenvalues

In the realm of linear algebra and differential equations, eigenvalues are a critical concept. They represent special numbers that arise when analyzing matrices, like our matrix \( A \). When these eigenvalues are complex, it adds an intriguing layer to the story. Imagine a complex number, often noted as \( a + bi \) where \( i \) is the imaginary unit. In our case, the eigenvalues of the matrix are \( \lambda = 3 + i \) and \( \lambda = 3 - i \). These are complex conjugates, coming in pairs like a concisely coupled dance.

The real part of these eigenvalues, here represented by 3, tells us whether trajectories move away from or towards an equilibrium point as time progresses. Meanwhile, the imaginary part, \( i \), introduces oscillations or swirling motions around the equilibrium. Therefore, the presence of complex eigenvalues typically indicates spiral behaviors in the system's phase plane. To wrap up, recognizing complex eigenvalues aids in predicting dynamic behavior in systems, making it crucial for stability analysis.

Stability Analysis

Stability analysis involves determining whether solutions to a differential equation remain near an equilibrium point as time progresses. It's like figuring out if a spinning top will wobble to stop or continue spinning seamlessly. In the context of our differential equation system, the equilibrium point is \((0, 0)\).

To perform a stability analysis, we first find the eigenvalues of matrix \( A \). These eigenvalues offer insights into how solutions behave over time. For instance, if all eigenvalues have negative real parts, the system is stable, drawing solutions to the equilibrium. However, if there's an eigenvalue with a positive real part, as we have here, the equilibrium point is unstable, leading away from it.

Thus, because our eigenvalues \( 3 \pm i \) contain a positive real part (3), our equilibrium is classified as an unstable spiral. Picture it as a trajectory that spirals outwards, moving further from the origin over time.

Linear Algebra in Biology

Linear algebra, particularly concepts like eigenvalues and eigenvectors, plays a powerful role in biological modeling. Consider a population model where interaction among species or within a population is modeled by systems of differential equations similar to our exercise. Here, eigenvalues can inform us about the population's growth rate and stability, predicting phenomena such as whether populations will stabilize, oscillate, or collapse.

For instance, if a model predicts complex eigenvalues with a positive real part, biologists might forecast a population that oscillates while expanding—perhaps describing invasive species spread or disease propagation. Moreover, these mathematical tools allow for simulating ecosystems, understanding genetics through principal component analysis, and much more.

In summary, by utilizing linear algebra, we provide a structured approach to explore and solve intricate biological systems, enhancing our ability to anticipate and manage biological behaviors effectively.