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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.
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 and service charge rate 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 , the future amount of money could be calculated from
A net growth rate is defined as the difference
and this is further combined to define by
The basic model in [link] simplifies to give
This equation is called a first-order difference equation , and the solution is found in a fairly straightforward way. Consider the equation for the first fewvalues of
The solution to [link] is a geometric sequence that has an initial value of and increases as a function of if is greater than 1 , and decreases toward zero as a function of if is less than 1 . 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 , and that the birth rate and death rate are constants ( not functions of the population or time ), 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 , then the solution of [link] is given by
The population grows exponentially if is positive (if ) and decays exponentially if is negative ( ). 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 must be known. ( is a state variable and only one is necessary.)
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 at intervals of time units, a geometric number sequence results. Let be the samples of 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 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 is small, then the first two terms of the Taylor's series give
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