Ch01.10 Linear Models in Business, Science, and Engineering

01.10.02 : Difference Equation

• In many fields, a dynamic system that changes over a time should be mathematically modeled.
• Several features of the system are each measured at discrete time intervals, producing a sequence of vectors, $\textbf{x}_0, \textbf{x}_1, \textbf{x}_2, \cdots,$.
• The entries in $\textbf{x}_k$ provide information about the state of system at the time of the $k$th measurement.
• If there is a matrix $A$ such that $\textbf{x}_1=A\textbf{x}_0, \textbf{x}_2=A\textbf{x}_1$, and , in general $$\textbf{x}_{k+1}=A\textbf{x}_k \text{ for }k=0,1,2,\cdots$$
• then it is called a linear difference equation (or recurrence relation)
• Given such an equation, one can compute $\textbf{x}_1,\textbf{x}_2$, and so on, provided $\textbf{x}_0$ is known.
• The discussion below illustrates how a difference equation might arise.
• A subject of interest to demographers is the movement of populations or groups of people from one region to another. The simple model here considers the changes in the population of a certain city and its surrounding suburbs over a period of years.
• Fix an initial year-say,2019- and denote the populations of the city and suburbs that year by $r_0$ and $s_0$, respectively. Let $\textbf{x}_0$ be the population vector $$\textbf{x}_0 = \begin{bmatrix}r_0 \\ s_0 \end{bmatrix} \begin{matrix} \text{City population, 2019} \\ \text{Suburban population, 2019} \end{matrix}$$
• For 2020 and subsequent years, denote the populations of the city and suburbs by vectors $$\textbf{x}_1 = \begin{bmatrix}r_1 \\ s_1 \end{bmatrix}, \textbf{x}_2 = \begin{bmatrix}r_2 \\ s_2 \end{bmatrix}, \textbf{x}_3 = \begin{bmatrix}r_3 \\ s_3 \end{bmatrix}$$
• Our goal is to describe mathematically how these vectors might be related.
• Suppose demographic studies show that each year about 5% of the city's population moves to the suburbs (and 95% remains in the city), while 3% of the suburban population moves to the city (and 97% remains in the suburs). See Fig.2 below:
• After 1 year, the original $r_0$ persons in the city are now distributed between city and suburbs as $$\begin{bmatrix}.95 r_0 \\.05 r_0\end{bmatrix} = r_0 \begin{bmatrix}.95 \\ .05 \end{bmatrix} \begin{matrix} \text{Remain in city} \\ \text{Move to suburbs} \end{matrix} \tag{6}$$
• The so persons in the suburbs in 2019 are distributed 1 year later as $$s_0\begin{bmatrix}.03 \\.97 \end{bmatrix} \begin{matrix} \text{Move to city} \\ \text{Remain in suburbs} \end{matrix} \tag{7}$$
• The vectors in (6) and (7) account for all of the populations in 2020. Thus $$\begin{bmatrix} r_1 \\ s_1 \end{bmatrix} =r_0\begin{bmatrix} .95 \\ .05 \end{bmatrix} +s_0\begin{bmatrix} .03 \\ .97 \end{bmatrix} =\begin{bmatrix} .95 &.03 \\ .05 & .97 \end{bmatrix} \begin{bmatrix} r_0 \\ s_0 \end{bmatrix}$$
• That is, $$\textbf{x}_1 = M \textbf{x}_0 \tag{8}$$
• Where $M$ is the migration matrix determined by the following table: $$M=\begin{bmatrix} .95 &.03 \\ .05 & .97 \end{bmatrix}$$
• Equation (8) describes how the population changes from 2019 to 2020. If the migration percentages remain constant, then the change from 2020 to 2021 is given by $$\textbf{x}_2 = M \textbf{x}_1$$
• And similarly for 2021 to 2022 and subsequent years. In general, $$\textbf{x}_{k+1} = M\textbf{x}_k \text{ for }k=0,1,2,\cdots \tag{9}$$
• The sequence of vectors $\left\{ \textbf{x}_0, \textbf{x}_1, \textbf{x}_2, \cdots \right\}$ describes the population of the city/suburban region over a period of years.