Environmental Engineering Reference
In-Depth Information
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What should be done in addition is a consideration of the expected dynamics:
Does the model contain conditions under which critical phases occur - are there
extreme changes in very short time intervals? Are there higher order nonlinea-
rities which are important during particular phases of the simulated time? If this
is the case or it could occur under particular boundary conditions this represents
a challenge for numerical correctness, and all indicated phases should be inves-
tigated in detail.
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For critical phases the model behaviour in extreme (but still plausible) situations
should be looked at, e.g. by running the model with a higher resolution for the
critical conditions. The time span when a population outbreak or collapse occurs
can be an example for a critical phase of model development.
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Are there hints for the occurrence of chaotic dynamics (e.g. the occurrence of
continuing endogenous changes of amplitude and frequency without an imple-
mented stochasticity)? For chaotic dynamics infinitely small changes are suc-
cessively amplified. These models usually are not suitable for long-term
projections other than statistical interpretations.
In ecology, most of the dynamic processes we deal with do not involve numeric
extremes and run relatively smoothly, however, relevant differences of a correct
and a numerically approximated result can even be observed in quite simple cases.
The Lotka-Volterra equations for a predator prey interaction (see 6.11) can be used
for a demonstration. The “true” solutions of the equations are accessible through
mathematical integration - unlike most of the other more complex cases. Therefore,
we know that the model result is a closed trajectory with constant amplitude. Using
the Euler integration routine for simulating the equations, errors accumulate and
yield an oscillatory pattern with successively increasing amplitude. Reducing the
integration step width improves the situation only gradually. Even for extremely
small intervals the effect is still observable. Changing to the Runge Kutta 4th order
integration, the results are considerably better. For beginners it can be an interesting
and instructive exercise to start such a simulation experiment and to recognize that
numeric artefacts are not only a myth and model results must be analyzed carefully
before drawing conclusions.
23.3.1 Model Structure and Parametrization: The Issue
of Aggregated Parameters
Model parameters do not necessarily derive from direct measurements. In particular,
in equations which describe population dynamics (e.g. Chaps. 6, 7, 9), parameters
usually represent aggregated averages of a larger set of phenomena, e.g. the overall
lifespan, mean population increase or specific tolerance ranges. Sometimes, they can
be determined empirically for a specific set of individuals but rarely for the whole
population. Regardless of their origin, either from a field campaign or even from
further assumptions (“educated guess”), such parameters for aggregated phenomena,
