Many modellers suggest that as a preliminary to writing down a mathematical model, that is, a mathematical equation, the modeller should always try to write down a word equation. This is probably a good idea since modellers should always be able to communicate their ideas both to fellow modellers, often mathematicians, and also to non-mathematicians who may need to use the model predictions ultimately obtained.
A word equation describing the model that we now have could be:
Over a unit time period, the fish population exhibits the following relationship:
Population at the end of the time period | = | the population at the start of the time period | + | the number of fish resulting from births | - | the number of fish resulting from natural deaths. |
If we now take the time period to be one year we can write the difference equation
This equation can be presented as
where k=b-d.
If we apply the same ideas to a time period of six months, the basic equation is
and so
In a similar fashion, we can obtain the equation
corresponding to a monthly application, and
corresponding to a daily application.
You should be able to deduce the answer corresponding to an infinitely small time interval from the above information.