0.3 Exponential growth

Since it is the dynamic nature of a system that we want to model and understand, the simplest form will be considered. This will involve onestate variable and will give rise to so-called "exponential" growth.

Two examples

First consider a mathematical model of a bank savings account. Assume that there is an initial deposit but after that, no deposits orwithdrawals. The bank has an interest rate $r_i$ and service charge rate $r_s$ that are used to calculate the interest and service charge once each time period. If the net income (interest less service charge)is re-invested each time, and each time period is denoted by the integer $n$ , the future amount of money could be calculated from

A net growth rate $r$ is defined as the difference

and this is further combined to define $R$ by

The basic model in [link] simplifies to give

This equation is called a first-order difference equation , and the solution $M\left(n\right)$ is found in a fairly straightforward way. Consider the equation for the first fewvalues of $n=0,1,2,...$

The solution to [link] is a geometric sequence that has an initial value of $M\left(0\right)$ and increases as a function of $n$ if $R$ is greater than 1 $\left(r0\right)$ , and decreases toward zero as a function of $n$ if $R$ is less than 1 $\left(r0\right)$ . This makes intuitive sense. One's account grows rapidly with a high interest rate and low service charge rate, andwould decrease toward zero if the service charges exceeded the interest.

A second example involves the growth of a population that has no constraints. If we assume that the population is a continuous function of time $p\left(t\right)$ , and that the birth rate $r_b$ and death rate $r_d$ are constants ( $$ not functions of the population $p\left(t\right)$ or time $t$ ), then the rate of increase in population can be written

There are a number of assumptions behind this simple model, but we delay those considerations until later and examine the nature of the solution ofthis simple model. First, we define a net rate of growth

which gives

which is a first-order linear differential equation. If the value of the population at time equals zero is $p_o$ , then the solution of [link] is given by

The population grows exponentially if $r$ is positive (if $r_b{r}_d$ ) and decays exponentially if $r$ is negative ( $r_b{r}_d$ ). The fact that [link] is a solution of [link] is easily verified by substitution. Note that in order to calculate future values ofpopulation, the result of the past as given by $p\left(0\right)$ must be known. ( $p\left(t\right)$ is a state variable and only one is necessary.)

Exponential and geometric growth

It is worth spending a bit of time considering the nature of the solution of the difference [link] and the differential [link] . First, note that the solutions of both increase at the same "rate". If we samplethe population function $p\left(t\right)$ at intervals of $T$ time units, a geometric number sequence results. Let $p_n$ be the samples of $p\left(t\right)$ given by

This give for [link]

which is the same as [link] if

This means that one can calculate samples of the exponential solution of differential equations exactly by solving the difference [link] if $R$ is chosen by [link] . Since difference equations are easily implemented on a digital computer, this is an important result; unfortunately,however, it is exact only if the equations are linear. Note that if the time interval $T$ is small, then the first two terms of the Taylor's series give