If you want to examine the behaviour of a mathematical function, you study not only the function itself, but also its derivative. The derivative tells you something about the rising or falling of the function and helps determine local maxima and minima.
It is therefore not surprising, that when modelling a dynamical process, special attention is paid to the change in quantity. Often, mathematical modelling of a dynamic process starts with making an equation in which both the quantity to be modelled and its derivative must meet certain requirements. Such an equation is called a differential equation. Mathematical modelling of a dynamic process often begins with the drafting of a differential equation. After this, one tries to determine the behaviour of general solutions of the differential equation and where possible to find explicit formulas for solutions. You can then also perform calculations with these formulas to further explore the modelled phenomenon and example to make predictions.
In this chapter, we look at three types of models for unlimited growth:
- linear growth
- quadratic growth
- exponential growth
Of course unlimited growth is not very realistic: in practice a growing system will always be limited by its resources. However, these are the easiest examples. After these examples, we are prepared (in terms of mathematics) for more complex models. Examples of application areas are microbial growth, chemical reaction kinetics, and pharmacokinetics.