Environmental Engineering Reference
In-Depth Information
Stationary States (Single or Multiple Equilibria)
The third alternative, that a one-dimensional system approaches a nontrivial equi-
librium (equilibrium at a finite value of
N
), requires in a minimum setting the
combination of linear increase and exponential decrease (
6.14
). The introduction of
higher nonlinearities can lead to multiple equilibria
dN
dt
¼
C
1
C
2
N
(6.14)
with
C
1 and
C
2
>
0. A stable dynamic equilibrium exists for
N
¼
C
1/
C
2. The
logistic equation [see (6.9)] has also a stable equilibrium.
6.5.2 Dynamic Properties in Two-Dimensional Systems
In two dimensional systems, the same types of dynamic behaviour can occur as in
one-dimensional systems. Oscillations are an additional type of dynamics that are
not found in 1D systems. There are different types of oscillations. The simplest
oscillator results from a positive and negative coupling of two variables, as we saw
in (6.8) and Fig.
6.2
.
In case of models that include nonlinearities, the oscillations can either
increase or decrease in amplitude over time. At the transition point between
both types there are so-called
marginal stable oscillations
, which neither increase
nor decrease in amplitude, but maintain the amplitude set by the initial condi-
tions. This is the dynamic behaviour of the Lotka-Volterra equations (Fig.
6.6
).
An additional type of oscillation occurs, if a dynamic behaviour which leads to
increasing oscillation in a certain part of the phase space is limited by a region of
the phase space where a decreasing amplitude prevails. Over the long term, a
specific cycle results, independent of the initial conditions. This is called a stable
limit cycle
. This dynamic type will be presented in a simple model example
below. An unstable limit cycle is also possible. It would be more difficult to
observe, since the slightest, infinitesimal deviation would induce a transit to either
one of two different alternative states, which could be explosion, collapse or
another stationary state.
Oscillations with Damped Amplitude (Approximating a Steady State)
The following system (6.15) oscillates with decreasing amplitude and approaches
an equilibrium point (stationary steady state). Over time, the oscillations decay.
To obtain this behaviour, the Lotka-Volterra equations (
6.11
) are used and it is
additionally assumed that the growth capacity of the prey is limited.
