Environmental Engineering Reference
In-Depth Information
and top-down elements can be combined, it will facilitate a more directed develop-
ment. Especially for two-dimensional systems, the isoclines (see Sect.
6.4
)areof
particular importance in model construction. They allow one to see when small
changes in the system, e.g. the size of a single parameter, can give rise to basic
changes of the overall system behaviour. Intersections of the isoclines are the points
where the temporal changes in the system are zero. If such an intersection emerges or
disappears as a function of changes in parameter values, this represents an important
change in system behaviour. The possibilities of isoclines intersecting are limited if
isoclines are straight lines. Nonlinearities in the system can give rise to curved
isoclines. This can bring various kinds of interesting dynamic properties.
6.6.1 Logistic Growth
An interesting nonlinearity consists in the introduction of a negative quadratic term.
As we saw above, when this is added to a simple exponential equation, it yielded the
logistic curve (
6.9
). Simulating the equation, we saw, that starting with very small
N
, a rapid increase occurred that transitioned towards a stationary state. Starting
with a very large
N
, we observe a declining trend, which stabilizes to the same
steady state (Fig.
6.3
, logistic growth). Logistic terms are frequently used in
ecological models to simulate limited carrying capacities.
6.6.2 Multiple Equilibria States and Hysteresis
The mathematical term “catastrophe” refers to a rapid transition of a system
equilibrium point from one state to another. It can occur, if the change of a
condition causes the system to leave a stable domain beyond a critical point, after
which a
bifurcation
occurs. Then, the system shifts towards another alternative
stable state. Such a situation is shown in Fig.
6.12
. Here the upper and the lower
branch of the graph (solid lines) are the stable state regions of the system. Once the
induced change in conditions forces the system to cross a bifurcation point (either
of the two black dots in the graph), the system shifts towards another alternative
state. The dashed line in the graph represents the unstable region between the two
alternative system states. One consequence is that such a system can reach multiple
equilibrium points, only depending on the impact of the driving forces (see
Fig.
6.12
). These dynamics can result in
hysteresis
behaviour, and they have been
intensively studied in lake ecosystems. Scheffer et al. (1993) found such concurrent
alternative equilibrium states exist in shallow lakes, which tend to be either algae or
macrophyte dominated. The transitions between these alternative states can occur
through changes in the nutrient load. Scheffer et al. found, also, that the stability of
a given stable state (see the upper and lower branches in Fig.
6.12
) prevents an
