Environmental Engineering Reference
In-Depth Information
Chapter 19
Nonlinear Systems
Linear systems, as examined in the previous chapter, represent the simplest type of
models. But linear models are often too simplistic from the process aspect. The set-
up of a linear model is often motivated by the fact that few characteristics,
parameters or variables, of the system have been observed and that few data are
available to check the model approach, whatever that may be. This chapter
describes models slightly more complex than the linear ones. It is demonstrated
that even simple nonlinear terms in the differential equation open the door to
a much greater variety of phenomena than experienced by the work with linear
systems.
There are several mathematical textbooks on nonlinear systems of ordinary
differential equations. Jordan and Smith (
1977
) provide a wide range of examples,
not only for environmental systems. Hale and Ko¸ak (
1991
) focus on bifurcations in
nonlinear systems, but with hardly a connection to environmental sciences. For
MATLAB
users the topic of Polking (
2004
) is highly recommended, because the
accompanying software is extremely user-friendly and can be obtained via internet.
®
19.1 Logistic Growth
The linear differential equation
@c=@t ¼ rc
for a single species describes exponen-
tial growth, for
r>
0. With reference to discussions on earth's population this is
sometimes referred to as
Malthus
1
ian growth
. However, there is no environmental
system in which any species can grow infinitely. The model, described by the
simple linear equation above, can thus be valid for a limited range of parameter
or variable values only. If the model is to be valid for an extended parameter range,
1
Thomas Robert Malthus (1766-1834), English demographer and political economist.
