Problem 13 - Biocalculus Calculus for the Life Sciences

Question

A Jacobian matrix and two equlibria are given. Determine if each is locally stable, unstable, or if the analysis is inconclusive. $$\begin{array}{l}{J=\left[ \begin{array}{cc}{\left(x_{1}-2\right) x_{2}+x_{1} x_{2}} & {x_{1}\left(x_{1}-2\right)} \\ {0} & {-1+2 x_{2}}\end{array}\right]} \\\ {\text { (i) } \hat{x}_{1}=0, \hat{x}_{2}=2} \\ {\text { (ii) } \hat{x}_{1}=2, \hat{x}_{2}=-1}\end{array}$$

Step-by-Step Solution

Answer

Equilibrium (i) is unstable, while equilibrium (ii) is locally stable.

Step 1: Evaluate the Jacobian at Equilibrium (i)

Substitute \( \hat{x}_1 = 0 \) and \( \hat{x}_2 = 2 \) into the Jacobian matrix. This results in: \[J = \begin{bmatrix} (0 - 2) \cdot 2 + 0 \cdot 2 & 0 \cdot (0 - 2) \ 0 & -1 + 2 \cdot 2 \end{bmatrix} = \begin{bmatrix} -4 & 0 \ 0 & 3 \end{bmatrix}.\]

Step 2: Determine Stability at Equilibrium (i)

To determine stability, evaluate the eigenvalues of the matrix from Step 1. The eigenvalues are the diagonal elements: \( \lambda_1 = -4 \) and \( \lambda_2 = 3 \). Since one eigenvalue is negative and the other is positive, this equilibrium point is a saddle point, indicating it's unstable.

Step 3: Evaluate the Jacobian at Equilibrium (ii)

Substitute \( \hat{x}_1 = 2 \) and \( \hat{x}_2 = -1 \) into the Jacobian matrix. This results in: \[J = \begin{bmatrix} (2 - 2)(-1) + 2(-1) & 2(2-2) \ 0 & -1 + 2(-1) \end{bmatrix} = \begin{bmatrix} -2 & 0 \ 0 & -3 \end{bmatrix}.\]

Step 4: Determine Stability at Equilibrium (ii)

For stability analysis, check the eigenvalues of the matrix from Step 3. Here, the eigenvalues are also the diagonal elements: \( \lambda_1 = -2 \) and \( \lambda_2 = -3 \). Both eigenvalues are negative, indicating that the system is locally stable (a stable node) at this equilibrium.

Key Concepts

Equilibrium Stability

In mathematical analysis, equilibrium stability refers to the behavior of a system at a point where it does not change over time. These points, called equilibrium points, occur when the system is in a state of balance. Understanding whether these points remain stable or not when subject to small perturbations is crucial in fields like economics, engineering, and biology.
An equilibrium point is stable if, when the system is slightly disturbed, it returns to its original state. Conversely, it is unstable if it moves away from its original state following such a disturbance.
Stability analysis often involves linearizing the system near equilibrium points and evaluating the Jacobian matrix. This matrix provides a linear approximation of the system dynamics at the equilibrium. An analysis of the eigenvalues of the Jacobian can then determine the nature of the equilibrium stability.

Eigenvalues

Eigenvalues are scalars associated with square matrices, which provide insight into the behavior of a system of linear equations. When dealing with the Jacobian matrix in stability analysis, eigenvalues indicate how the system evolves over time near equilibrium points.
The Jacobian matrix is calculated from a system of equations that describe how state variables change. By substituting an equilibrium point into this matrix, we can determine the eigenvalues.

• If all eigenvalues have negative real parts, the equilibrium is locally stable.
• If at least one eigenvalue has a positive real part, the equilibrium is unstable.
• If eigenvalues are zero, the analysis may be inconclusive without additional information.

Evaluating these eigenvalues can reveal whether a system will consistently maintain, diverge from, or oscillate around an equilibrium point.

Stability Analysis

Stability analysis involves examining how a system responds to small changes in its state near equilibrium points. The primary tool used for this analysis is the Jacobian matrix, which offers a snapshot of the dynamics of a system near these points. Once the Jacobian matrix is obtained, the stability of the system can be determined by calculating its eigenvalues.
If the eigenvalues suggest stability (i.e., they have negative real parts), disturbances will die out, allowing the system to return to equilibrium. If eigenvalues suggest instability, any disturbance will grow, causing the system to move away from equilibrium.
Thus, the first step in stability analysis is to find equilibrium points, then evaluate the Jacobian at those points, and finally analyze the eigenvalues to conclude the stability nature.

Saddle Point

A saddle point in the context of stability analysis is a type of equilibrium point characterized by having both stable and unstable directions. This occurs when the Jacobian matrix has eigenvalues with mixed signs—some are positive, while others are negative.
In a system with a saddle point, specific directions in the phase space will lead the trajectory away from the equilibrium, demonstrating instability. At the same time, in other directions, the system might seem stable as trajectories approach the point. However, overall, the presence of even one positive eigenvalue results in an unstable equilibrium.
Saddle points are critical to understanding since they indicate an inherently unstable equilibrium, regardless of the stability in certain directions.

Stable Node

A stable node is an equilibrium point where all trajectories in a system are attracted, indicating local stability. This condition arises when all the eigenvalues of the Jacobian matrix at this point have negative real parts.
For instance, consider a second equilibrium point where the Jacobian results in a matrix with entirely negative eigenvalues, like \(\lambda_1 = -2\) and \(\lambda_2 = -3\). Here, any perturbation or slight deviation from the equilibrium leads the system to return to this stable point.
Stable nodes are vital in modeling systems where predictable and consistent behavior is required. They ensure that amid small disturbances, the system naturally gravitates back to equilibrium, thus maintaining stability.