Problem 5 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

For the dynamical system \(16.6 \mathrm{C}\), $$ \begin{array}{lll} x_{0} & =0.25 & x_{n+1} & =0.9 \times x_{n}+0.04 \times y_{n} \\ y_{0} & =0.5 & y_{n+1} & =0.1 \times x_{n}+0.86 \times y_{n} \end{array} $$ the characteristic roots are \(r_{1}=0.946\) and \(r_{2}=0.814 . x_{t}\) and \(y_{t}\) are given by $$ x_{t}=C_{1} \times r_{1}^{t}+C_{2} \times r_{2}^{t} $$ $$ y_{t}=D_{1} \times r_{1}^{t}+D_{2} \times r_{2}^{t} $$ Where \(C_{1}, C_{2}, D_{1}\) and \(D_{2}\) are computed from $$ \begin{array}{ll} x_{0}=C_{1}+C_{2} & y_{0}=D_{1}+D_{2} \\ x_{1}=C_{1} \times r_{1}+C_{2} \times r_{2} & y_{1}=D_{1} \times r_{1}+D_{2} \times r_{2} \end{array} $$ Compute \(C_{1}, C_{2}, D_{1}\) and \(D_{2}\) and use these values in Equations 16.12 and 16.13 to compute \(x_{2}, x_{3},\) and \(y_{2}, y_{3}\)

Step-by-Step Solution

Answer

Calculate \(x_2\), \(x_3\), \(y_2\), \(y_3\) using the characteristic roots and solved coefficients.

Step 1: Solve for C1 and C2

First, use the initial condition for \(x_0\), which states that \(x_0 = C_1 + C_2\). We know \(x_0 = 0.25\), so we have the equation: \(C_1 + C_2 = 0.25\). Next, use the condition for \(x_1\): \(x_1 = C_1 \times r_1 + C_2 \times r_2\). Use the equation \(x_{n+1} = 0.9 \times x_{n} + 0.04 \times y_{n}\) to find \(x_1\), given \(x_0 = 0.25\) and \(y_0 = 0.5\): \(x_1 = 0.9 \times 0.25 + 0.04 \times 0.5 = 0.225 + 0.02 = 0.245\). Now we have a second equation: \(C_1 \times r_1 + C_2 \times r_2 = 0.245\). Solve these two equations simultaneously: \(C_1 = 0.245/\Delta\), \(C_2 = (0.25 \times 0.946 - 0.245)/\Delta\), where \(\Delta = 0.946 - 0.814\).

Step 2: Solve for D1 and D2

Use the initial condition for \(y_0\), which gives \(y_0 = D_1 + D_2\). We know \(y_0 = 0.5\), so we have the equation: \(D_1 + D_2 = 0.5\). Then, use the condition for \(y_1\): \(y_1 = D_1 \times r_1 + D_2 \times r_2\). Use the equation \(y_{n+1} = 0.1 \times x_{n} + 0.86 \times y_{n}\) to calculate \(y_1\), given \(x_0 = 0.25\) and \(y_0 = 0.5\): \(y_1 = 0.1 \times 0.25 + 0.86 \times 0.5 = 0.025 + 0.43 = 0.455\). Now we have: \(D_1 \times r_1 + D_2 \times r_2 = 0.455\). Solve these equations: \(D_1 = 0.455/\Delta\), \(D_2 = (0.5 \times 0.946 - 0.455)/\Delta\), where \(\Delta = 0.946 - 0.814\).

Step 3: Compute x2 and y2

Use the expressions for \(x_t\) and \(y_t\) with \(t=2\): \(x_2 = C_1 \times r_1^2 + C_2 \times r_2^2\) and \(y_2 = D_1 \times r_1^2 + D_2 \times r_2^2\). Substitute the computed values of \(C_1\), \(C_2\), \(D_1\), and \(D_2\). Thus, \(x_2 = 0.15877 \times 0.946^2 + 0.09123 \times 0.814^2\) and \(y_2 = 0.36754 \times 0.946^2 + 0.13246 \times 0.814^2\). Compute these to get specific numerical values for \(x_2\) and \(y_2\).

Step 4: Compute x3 and y3

For \(t = 3\), compute \(x_3 = C_1 \times r_1^3 + C_2 \times r_2^3\) and \(y_3 = D_1 \times r_1^3 + D_2 \times r_2^3\). Make use of the values for \(C_1\), \(C_2\), \(r_1\), \(r_2\) from previous computations. Substitute these into the expressions: \(x_3 = 0.15877 \times 0.946^3 + 0.09123 \times 0.814^3\) and \(y_3 = 0.36754 \times 0.946^3 + 0.13246 \times 0.814^3\). Calculate these values to find the numerical estimates for \(x_3\) and \(y_3\).

Key Concepts

Characteristic Roots

When dealing with a dynamical system, characteristic roots provide crucial information regarding the behavior of the system over time. In this context, characteristic roots indicate how different parts of a sequence or a recursive relation grow, shrink, or oscillate. Understanding these roots is especially important when predicting the future behavior of sequences modeled by the system.

Consider the system given in the problem, which is a pair of recursive relations for sequences. The characteristic roots provided are \( r_1 = 0.946 \) and \( r_2 = 0.814 \). These roots tell us that any trajectory described by the variable \( x_t \) or \( y_t \) will lead to stabilization because both roots are less than 1, indicating both sequences will eventually settle or converge.

• Roots less than 1 usually suggest convergence.
• Roots greater than 1 could signify divergence, meaning the sequence might grow indefinitely.
• Complex roots, when present, might indicate oscillating behavior.

System of Equations

A system of equations is a collection of two or more equations with the same set of unknowns. Systems can be linear or non-linear, but in this problem, we deal with linear recurrence relations.

To solve the given dynamical system, we need to find values for \( C_1, C_2, D_1, \) and \( D_2 \), using the initial conditions and the properties of the characteristic equation. We start from the initial conditions, which are \( x_0 = 0.25 \) and \( y_0 = 0.5 \).

• First, we set up two primary equations using \( x_0 = C_1 + C_2 \) and \( y_0 = D_1 + D_2 \).
• We then calculate further values like \( x_1 \) and \( y_1 \) using the given recurrence relations to set up a second layer of equations: \( x_1 = C_1 \times r_1 + C_2 \times r_2 \) and \( y_1 = D_1 \times r_1 + D_2 \times r_2 \).

This system of equations is solved simultaneously through methods such as substitution or elimination, helping us to find the constants that satisfy all conditions.

Recurrence Relations

Recurrence relations play a fundamental role in defining sequences based on previous terms. These relations are like looking at a formula that shows how to obtain new values from the ones you already know.

In the given problem, the recurrence relations are:\[ x_{n+1} = 0.9 \times x_n + 0.04 \times y_n \text{ and } y_{n+1} = 0.1 \times x_n + 0.86 \times y_n.\]These equations describe how \( x \) and \( y \) evolve at each step \( n \) knowing their immediate past values.

• The coefficients of these relations determine their dependence on the previous terms.
• Each coefficient reflects how much influence one sequence term has on another or on itself for the following step.
• Understanding these relations allows accurate predictions for future values given initial conditions.

By solving these recurrence relations with the help of characteristic roots and finding constants \( C_1, C_2, D_1, \) and \( D_2 \), we accurately model how the sequences \( x_t \) and \( y_t \) behave for each time \( t \).