Problem 29 - 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 real, distinct, and nonzero. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a sink, a source, or a saddle point. $$ A=\left[\begin{array}{rr} 2 & -1 \\ 0 & 3 \end{array}\right] $$

Step-by-Step Solution

Answer

The equilibrium \((0,0)\) is an unstable source.

Step 1: Determine the Eigenvalues

To analyze the stability of the equilibrium, we first need to calculate the eigenvalues of the matrix \( A \). The eigenvalues are found by solving the characteristic equation \( \det(A - \lambda I) = 0 \). For matrix \( A = \begin{bmatrix} 2 & -1 \ 0 & 3 \end{bmatrix} \), the characteristic equation is calculated as follows:\[\det \begin{bmatrix} 2 - \lambda & -1 \ 0 & 3 - \lambda \end{bmatrix} = (2 - \lambda)(3 - \lambda) - (0)(-1) = 0\]From the equation \( (2 - \lambda)(3 - \lambda) = 0 \), we find the eigenvalues: \( \lambda_1 = 2 \) and \( \lambda_2 = 3 \). Both eigenvalues are real, distinct, and nonzero.

Step 2: Analyze Stability

The stability of the equilibrium at \((0,0)\) depends on the sign of the eigenvalues. If all eigenvalues are positive, \((0,0)\) is an unstable source. If all eigenvalues are negative, it is a stable sink. If eigenvalues have mixed signs, it is a saddle point.In our case, both \( \lambda_1 = 2 \) and \( \lambda_2 = 3 \) are positive. This indicates that the equilibrium \((0,0)\) is an unstable equilibrium point.

Step 3: Classify the Equilibrium

Since both eigenvalues are positive, \((0,0)\) is classified as a source. This means that trajectories starting near \((0,0)\) will move away from the equilibrium point as time \( t \) increases.

Key Concepts

Eigenvalues

Eigenvalues play a crucial role in the stability analysis of differential equations, predominantly when dealing with systems of the form \(\frac{d \mathbf{x}}{d t} = A \mathbf{x}(t)\). The matrix \(A\) can be considered the heart of this system, impacting how solutions evolve over time.
One common way to find eigenvalues is by solving the characteristic equation \(\det(A - \lambda I) = 0\), where \(I\) is the identity matrix.
For instance, in our solution, the matrix \(A\) is given by:

• \(A = \begin{bmatrix} 2 & -1 \ 0 & 3 \end{bmatrix} \)

Solving the characteristic equation for this matrix means you evaluate \((2 - \lambda)(3 - \lambda)\), which gives eigenvalues \(\lambda_1 = 2\) and \(\lambda_2 = 3\). These eigenvalues, being real and distinct, suggest important insights about the behavior of solutions in a two-dimensional phase space.
Knowing eigenvalues provides a window into whether solutions grow, decay, or move in other ways relative to the system's equilibrium point.

Stability Analysis

Understanding the stability of an equilibrium point is fundamental when analyzing differential equations. In general, stability is assessed by examining the eigenvalues of the matrix \(A\).
If all eigenvalues are negative, the equilibrium

• It indicates a stable sink, meaning nearby trajectories converge to the equilibrium.

On the other hand, if all are positive, like in our example with \(\lambda_1 = 2\) and \(\lambda_2 = 3\), the equilibrium point becomes an unstable source.
This is because solutions will diverge from

• ('move away from') the point as time increases.

It's crucial to also consider mixed eigenvalue signs, which imply a saddle point.
Thus, through eigenvalues, one can predict not only the immediate reaction of a trajectory but also the long-term system's behavior.

Equilibrium Classification

Classifying equilibria involves not only determining stability but also understanding the nature of the equilibrium, typically categorized as a sink, source, or saddle point. With reference to a system

• Where \(A = \begin{bmatrix} 2 & -1 \ 0 & 3 \end{bmatrix}\), both eigenvalues are positive.

This classifies

• The equilibrium point \((0,0)\) as a source, meaning trajectories in its vicinity will move away as time progresses, aligning with the concept of an unstable source.

Contrast this with the case of negative eigenvalues, which would suggest a sink, with all paths leading towards the equilibrium.
In scenarios of mixed-signed eigenvalues,

• Systems usually exhibit a saddle point behavior, resulting in some trajectories approaching the equilibrium, while others recede.

Such classifications help in visualizing the overall flow and tendencies of the system, making it easier to predict potential outcomes.