Environmental Engineering Reference
In-Depth Information
same: marginally stable oscillations. This holds even though the speed of
change can become so extremely rapid (or slow), that standard numeric approx-
imations fail. From what has been previously discussed, we can derive conclu-
sions for
phase transitions
. In nonlinear systems with multiple equilibria,
parameter changes can lead to the emergence of new alternative equilibria or
induce the destabilization of previously stable states. In particular, models with
more complex nonlinearities facilitate the occurrence of complex combinations.
A still relatively simple type of transition is the breaking up of a stable
equilibrium and the emergence of a limit cycle. This transition was named
after the Austrian mathematician Eberhard Hopf. For (6.18) we can observe
such a Hopf-bifurcation when changing the parameter
C
3. For small and large
parameter values the limit cycle vanishes and the system approaches a steady
state instead of a limit cycle (Figs.
6.9
and
6.15
).
To be sufficiently careful in the interpretation of a model, the modeller needs to
have an overview of the potential range of dynamic behaviour. Since ecological
models are always a simplification focussing on a limited set of interactions and
conditions (“
ceteris paribus
” - i.e. all other conditions remain the same), it is useful
to know about potential implications. This applies not only to the field work, where
the empirical background for structural and functional model specifications are
conceptualized - the number of variables, the quantitative relations and the param-
eter ranges. It is also necessary to have an intuition about the potential properties of
dynamic systems representations.
For a modeller it is not enough to be able to write down equations and let the
computer evaluate them numerically. An appropriate interpretation of simulation
results should also take into account that slight parameter changes could lead
to phase transitions, which would alter the overall dynamics. Without knowing
the potential effects of the phase transitions between alternative equilibria (which
are quite difficult to experimentally investigate in the field, if the system state
cannot be arbitrarily manipulated), adequate interpretation of the model results
might be difficult. This leaves much potential for uncertainty. However, other
difficulties can be managed by understanding the equations and protecting against
simulation artefacts. Some of these strategies to tackle “standard” uncertainties are
compiled at
http://www.mced-ecology.org
and in Chaps. 2 and 23.
6.7 Solving Differential Equations Analytically
and Numerically
For most ecological questions where dynamic models are developed, a mathe-
matical solution is not possible and a numerical evaluation of the equations is
required. Analytical solutions exist only in relatively simple cases; i.e. linear
equations and some simple nonlinear equations. These include the logistic growth
function, and the Lotka-Volterra model. More complex, nonlinear ecological
