Environmental Engineering Reference
In-Depth Information
Is the expectation of capturing and expressing the essence of ecological entities
in forms of simplicity a contradiction? Some aspects of life and technology can be
precisely forecast and operated. For instance, a computer would not work in the
case its states of operation were not fully determined. In other fields projections are
impossible, e.g., the efforts to forecast earthquakes precisely have only very limited
success. Comparatively, weather forecasts work sometimes but only within very
narrow time frames. Biological relationships are notoriously difficult to predict in
models. However, there are surprising applications of these relations. We can use
simple models to demonstrate the causes of complex dynamics. This will not tell us
which example out of the possible range we will actually meet next time in the field.
But it will help us to link different aspects of causal structures and consider joint
contexts of the ecological systems.
To create complex dynamics, we only need simple relations, slight modifica-
tions, and iterative repetition. Then we can show: Not all complex phenomena are
based on equally complex relations. Simple settings can generate complexity, and
hence complexity can be based on simple mechanisms. Here we start with a few of
the most simplistic examples. In the subsequent chapters we will then successively
turn to more sophisticated approaches and solutions, which will relate to advanced
ecological theory and application.
One Formal Step into the Kingdom of Chaos
To show how close simplicity and complexity are, we leave out all biological
realism for a moment. We only link numbers with each other and define a specific
predecessor and successor for each. Since numeric operations are frequently used in
ecological modelling, we are not too far away from our subject.
We define the successor (
y
) of each number (
x
) as the value we obtain if we
subtract the inverse. This is a simple mathematical operation:
y
Þ
Since the result is again a real number, we can apply the same operation again and
thus create chains of operations. These chains have interesting properties. They tend to
approach 0; however, when becoming smaller than
¼
x
ð
1
=
x
1)
the algebraic sign changes and they then alternate between positive and negative
values. Since we can use any starting point, the procedure successively intertwines the
chains like infinitely long and thin spaghetti. This becomes apparent if we follow the
fate of an interval that contains any possible starting points between two close numbers
that are far outside the interval between [
þ
1(andlargerthan
1] (see arrows in Fig
1.2
). It is apparent,
that the numbers will become subsequently smaller (when being positive - or larger
when being negative). The interval itself will increase in extent after each step.
This kind of movement, increasing the differences between originally close
values through continuous operations, can be observed in this simple equation.
It exhibits what is called deterministic chaos. In this case the chaos expands any
small interval by iteration successively until they become independent from each
other. A graphic representation of the process is shown in Fig.
1.2
.
1;
þ
