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which is somewhat similar to [link] . Another view of the relation can be seen by approximating [link] by
which gives
having a solution
This implies [link] also, and the method is known as Euler's method for numerically solving a differential equation.
These approximations are used often in modeling. For population models a differential evuation is often used, even though it is obvious thatbirths and deaths occur at random discrete times and populations can take on only integer values. The approximation makes sense only if we uselarge aggregates of individuals. We end up modeling a process that occurs at random discrete points in time by a continuous time mode, which is thenapproximated by a uniformly-spaced discrete time difference equation for solution on a digital computer!
The rapidity of increase of an exponential is usually surprising and it is this fact that makes understanding it important. There are several ways to describe the rate of growth.
This states that $r$ is the rate of growth per unit of $x\phantom{\rule{0.166667em}{0ex}}$ . For example, the growth rate for the U.S. is about 0.014 per year, or anincrease of 14 people per thousand people each year.
Another measure of the rate is the time for the variable to double in value. This doubling time, ${T}_{d}\phantom{\rule{4pt}{0ex}}$ , is constant and can easily be shown to be given by
For example, doubling times for several rates are given by
$r$ | ${T}_{d}$ |
.01 | 70 |
.02 | 35 |
.03 | 23 |
.04 | 17 |
.05 | 14 |
.06 | 12 |
The present world population is about three billion, and the growth rate is 2.1% per year. This gives
with $p\left(t\right)$ measured in billions of people and $t$ in years. This gives a doubling time of 33 years. While it is easy to talk of growthrate and doubling times, these have real predictive meaning only if the growth is exponential.
There are two rather different approaches that can be used when describing some physical phenomenon by exponential growth. It can be viewed as anempirical description of how some variables tend to evolve in time. This is a data-fitting view that is pragmatic and flexible, but does not givemuch insight or direction on how to conduct experiments or what other things might be implied.
The second approach primarily considers the underlying differential equation as a "law" of growth that results in exponential behavior. Thislaw has various assumptions and implications that can be examined for reasonableness or verified by independent experiment. While perhaps notso important for the first-order linear equation here, this approach becomes necessary for the more complicated models later.
These approaches must often be mixed. The data will imply a model or equation which will give direction as to what data should be taken, whichwill in turn imply modifications, etc. The process where structure is chosen and the parameters are chosen so that the model solution agreeswith observed data is a form of parameter identification. That was how [link] was determined.
When examining data that has been plotted in fashion, it is often hard to say much about its basic nature. For example if a time series is plotted on linear coordinates as follows
it would not be obvious if it were samples of an exponential, a parabola, or some other function. Straight lines, on the other hand, are easy to identify and so we will seek a method of displaying data that will use straight lines.If $x\left(t\right)$ is an exponential, then
Taking logarithms of base $e$ for both sides of [link] gives
If, rather than plotting $x$ versus $t\phantom{\rule{0.166667em}{0ex}}$ , we plot the log of $x$ versus $t\phantom{\rule{0.166667em}{0ex}}$ , then we have a straight line with a slope of $r$ and an intercept of log $k\phantom{\rule{4pt}{0ex}}$ . It would look like
Actually using the logarithm of a variable is awkward so the variable itself can be plotted on logarithmic coordinates to give the same result. Graph paper with logarithmic spacing along one coordinate and uniform spacing along the other is called semi-log paper.
Consider the plot of the U.S. population displayed on semi-log paper in Figure 3. Note that there were two distinct periods of exponentialgrowth, one from 1600 to 1650, and another from 1650 to 1870. To calculate the growth rate over the 1650 – 1870 period, we can calculatethe slope.
During that period, there was a 3% per year growth rate or, in other terms, a 23-year doubling time.
An alternative is to measure the doubling time and calculate $r$ from [link] . Still another approach is to measure the time necessary for the populationto increase by $e=2.72$ ... The growth rate is the reciprocal of that time interval. Derive and check these for yourself.
The data displayed in Figures 1, 2 and 3 illustrates exponential growth and the use of semi-log paper.The books [link] [link] give interesting discussions of growth.
In some cases, there is a choice between an analytical solution in the form of an equation and a numerical solution in the form of a sequence ofnumbers. A real advantage of an analytical form is the ability to easily see the effects of various parameters. For example, the exponentialsolution of [link] given in [link] directly shows the relation of the growth rate $r$ in equation [link] to the exponent in solution [link] . If the equation were numerically solved, say on a digital computer or calculatorusing Euler's method given in [link] , it would take numerous experimental runs to establish the same relations.
On the other hand, for complicated equations there are no known analytical expressions for the solutions, and numerical solutions are the onlyalternatives. It is still worth studying the analytical solution of simple equations to gain insight into the nature of the numerical solutions of complex equations.
The linear first-order differential equation model that is implied by exponential growth has many assumptions that are worth noting here.First, the growth rate is constant, independent of crowding, food, availability, etc. It also assumes that age distribution within thepopulation is constant, and that an average birth and death rate makes sense. There are many factors one will want to include effects ofcrowding and resource availability, time delays in reproduction, different birth and death rates for different age groups, and many more. In thenext section we will add one complication the effects of a limit to growth.
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