Problem 4 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

Find the roots to the characteristic equation, \(r^{2}-p r+q=0\) and write the solutions for \(A_{t}\) in the following systems. a. \(\begin{array}{rlrl}A_{0} & =10 & A_{t+1} & =0.8 A_{t}+0.2 B_{t} \\ B_{0} & =0 & B_{t+1} & =0.1 A_{t}+0.7 B_{t}\end{array}\) b. \(\begin{array}{rlrl}A_{0} & =0 & A_{t+1} & =0.6 A_{t}+0.3 B_{t} \\ B_{0} & =5 & B_{t+1} & =0.2 A_{t}+0.7 B_{t}\end{array}\) c. \(\begin{array}{rlrl}A_{0} & =1 & A_{t+1} & =0.26 A_{t}+0.04 B_{t} \\ B_{0} & =1 & B_{t+1} & =0.06 A_{t}+0.24 B_{t}\end{array}\) d. \(\begin{aligned} A_{0} &=2 & & A_{t+1}=& 1.04 A_{t}+0.16 B_{t} \\ B_{0} &=3 & & B_{t+1} &=0.24 A_{t}+0.96 B_{t} \end{aligned}\) e. \(\begin{array}{rlrl}A_{0} & =20 & A_{t+1} & =0.86 A_{t}+0.04 B_{t} \\\ B_{0} & =10 & B_{t+1} & =0.06 A_{t}+0.84 B_{t}\end{array}\)

Step-by-Step Solution

Answer

Solve the characteristic equation for eigenvalues, use them to express \( A_t \), fit initial conditions to finalize solutions for each system.

Step 1: Identify System Matrix

For each system, define the matrix \( M \) based on the transition equations given. The general form is \( \begin{bmatrix} A_{t+1} \ B_{t+1} \end{bmatrix} = M \begin{bmatrix} A_{t} \ B_{t} \end{bmatrix} \), where \( M = \begin{bmatrix} a & b \ c & d \end{bmatrix} \).

Step 2: Calculate Eigenvalues

The characteristic equation of \( M \) is found by setting \( \det(M - rI) = 0 \). Calculate the determinant \( r^2 - (a+d)r + (ad-bc) \). Solve for \( r \), which are the eigenvalues.

Step 3: Solve Characteristic Equation

The roots of the characteristic equation are found using the quadratic formula \( r = \frac{-(a+d) \pm \sqrt{(a+d)^2 - 4(ad-bc)}}{2} \). Use these roots to understand the behavior of the system over time.

Step 4: Construct Solution for A_t

Using the eigenvalues \( r_1 \) and \( r_2 \), the general solution format for \( A_t \) is \( A_t = C_1 r_1^t + C_2 r_2^t \), where \( C_1 \) and \( C_2 \) are constants determined by initial conditions.

Step 5: Apply Initial Conditions for Each System

Substitute the initial conditions to solve for constants \( C_1 \) and \( C_2 \) for each system. Use \( A_0 \) and \( B_0 \) to find these constants.

Step 6: Complete Solution for Each System

For each subsystem (e.g., a, b, c, d, e), finalize \( A_t \) by plugging the determined constants back into \( A_t = C_1 r_1^t + C_2 r_2^t \). This equation provides the behavior of \( A_t \) over time.

Key Concepts

Characteristic Equation

In systems of equations, especially those involving matrices, understanding the characteristic equation is crucial for analyzing system stability and behavior over time. The characteristic equation helps us determine the eigenvalues of a matrix, which are key to predicting how a system evolves.

For a system matrix, the characteristic equation is derived from the equation \( \det(M - rI) = 0 \). Here, \( M \) is the system matrix, \( r \) represents potential eigenvalues, and \( I \) is the identity matrix of the same dimension as \( M \). Calculating the determinant of \( M - rI \) provides a polynomial, typically a quadratic in simpler systems.

• The roots of this polynomial are the eigenvalues of the matrix.
• The form \( r^2 - (a+d)r + (ad-bc) = 0 \) is common when dealing with 2x2 matrices.

Understanding the eigenvalues obtained from the characteristic equation can indicate whether a system will grow, shrink, or oscillate over time, based on their real or complex nature.

System Matrix

A system matrix is a key component in analyzing linear systems represented by sets of equations. This matrix provides a compact way to express how variables like \( A_t \) and \( B_t \) interact and transform over each time step.

In the context of discrete dynamical systems, the system matrix \( M \) is defined as:
\[M = \begin{bmatrix} a & b \ c & d \end{bmatrix}\]
and is used in relation:\[\begin{bmatrix} A_{t+1} \ B_{t+1} \end{bmatrix} = M \begin{bmatrix} A_{t} \ B_{t} \end{bmatrix}\]

• The elements of the matrix \( a, b, c, \) and \( d \) usually come from coefficients in the given update equations for a system.
• By analyzing this matrix, particularly its eigenvalues and eigenvectors, we can understand the long-term behavior of the system.

Formulating such matrices from a system allows for convenient use of linear algebra techniques to solve and predict system behavior.

Quadratic Formula

The quadratic formula is an essential tool for solving the characteristic equation, which appears as a quadratic polynomial when analyzing systems of equations.

This formula is given by:
\[r = \frac{-(a+d) \pm \sqrt{(a+d)^2 - 4(ad-bc)}}{2}\]
where \( a, b, c, \) and \( d \) are elements of the system matrix, and \( (a+d)^2 - 4(ad-bc) \) is called the discriminant.

• If the discriminant is positive, the equation has two distinct real roots. These may signify two different growth or decay rates in a dynamical system.
• A zero discriminant indicates a single repeated root, which could lead to a stable system if the eigenvalue is less than 1.
• A negative discriminant suggests complex roots, often leading to oscillatory systems.

This powerful formula offers an exact solution to quadratic equations and is invaluable in assessing system stability through eigenvalues.

Initial Conditions

In solving the system of equations, initial conditions are instrumental in determining the specific trajectory of system states over time. They allow us to find the constants needed to fully express the system's behavior.

When we have derived the general solution for \( A_t \) using the eigenvalues, we end up with:
\[A_t = C_1 r_1^t + C_2 r_2^t\]
where \( r_1 \) and \( r_2 \) are the eigenvalues, and \( C_1 \) and \( C_2 \) are constants dependent on the initial conditions.

• Initial conditions, like \( A_0 \) and \( B_0 \), are specific values given at \( t = 0 \) for the variables in the system.
• Substituting these initial values into the general solution allows determination of \( C_1 \) and \( C_2 \).

By knowing the initial state of a system, we can tailor the general solution to fit the particular dynamics and evolution expected upon its evolution.