Consider a function intended to represent the speed of a parachutist after the opening of the parachute where is the instantanous speed at time s. An appropriate function is
We will look at some of the properties and modelling implications of this function. Consider first the value of when :
This means the function predicts that the parachutist is moving at when the parachute opens. Consider next the value of when is arbitrarily large. For such a value of would be arbitrarily small, so would be very close to the value 12. The modelling interpretation of this is that eventually the speed becomes very close to a constant value, which will be maintained until the parachutist lands.
The steady speed which is approached by the parachutist (or anything else falling against air resistance) is called the terminal velocity . The parachute, of course, is designed to ensure that the terminal velocity is sufficiently low ( in the specific case we have looked at here) to give a reasonably gentle landing and avoid injury.
Now consider what happens as increases from near zero. When is near zero, the speed will be near . The amount being subtracted from 12, through the term 8 , is close to 8 because . As increases the value of 8 decreases fairly rapidly at first and then more gradually until is very nearly 12. This is sketched in Figure 10. In fact is never equal to 12 but gets imperceptibly close as anyone would like as increases. The value shown as a horizontal broken line in Figure 10 is called an asymptotic limit for .
Figure 10 :
The model concerned the approach of a parachutist’s velocity to terminal velocity but the kind of behaviour portrayed by the resulting function is useful generally in modelling any growth to a limit .
A general form of this type of growth-to-a-limit function is
where and are positive constants (parameters) and and represent values of the independent variable between which the function is valid. We will now check on the properties of this general function. When . As increases the exponential factor gets smaller, so will increase from the value but at an ever-decreasing rate. As becomes very small, , approaches the value . This value represents the limit, towards which grows. If a function of this general form was being used to create a model of population growth to a limit, then would represent the limiting population, and would represent the starting population.
There are three parameters, , and in the general form. Knowledge of the initial and limiting population only gives two pieces of information. A value for the population at some non-zero time is needed also to evaluate the third parameter .
As an example we will obtain a function to describe a food-limited bacterial culture that has 300 cells when first counted, has 600 cells after 30 minutes but seems to have approached a limit of 4000 cells after 18 hours.
We start by assuming the general form of growth-to-a-limit function for the bacteria population, with time measured in hours
When = 0 (the start of counting), = 300. Since the general form gives when = 0, this means that
The limit of as gets large, according to the general form , is , so = 4000. From this and the value of , we deduce that = 3700. Finally, we use the information that = 600 when (measuring time in hours) = 0.5. Substitution in the general form gives
Taking natural logs of both sides:
Note, as a check, that turns out to be positive as required for a growth-to-a-limit behaviour. Finally the required function may be written
As a check we should substitute = 18 in this equation. The result is = 3824 which is close to the required value of 4000.
Find a function that could be used to model the growth of a population that has a value of 3000 when counts start, reaches a value of 6000 after 1 year but approaches a limit of 12000 after a period of 10 years.
First find the modelling equation:
where is the number of members of the population at time years. The given data requires that is 12000 and that , so
The corresponding curve must pass through ( ) so
(using Rule 3b, Table 1, page 42)
So the population function is
Note that (10) according to this formula is approximately 11840, which is reasonably close to the required value of 12000.
Now sketch this function: