Environmental Engineering Reference
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Net increase
positive
0
0
Net increase
negative
N
N
Fig. 6.3 Rate level graph of a one-dimensional system [(6.9) and (6.10)] with one trivial equilib-
rium at N ¼ 0. The other equilibrium is stable ( left side ) or unstable ( right side )
slightly disturbed (i.e. is shifted to the vicinity of the equilibrium), it will move
back. The variable increases in the case that N is smaller than the equilibrium and
decreases for N larger than the equilibrium. In the opposite case we have an
unstable equilibrium . Then the variable decreases for N smaller than the equilib-
rium and increases for N larger than the equilibrium - it successively moves away
from the equilibrium - regardless how close it is to the equilibrium point, as long as
a difference exists [( 6.9 ), ( 6.10 ), Fig. 6.3 ].
dN
dt ¼
N 2 with stable equilibrium at C 1
C 1
N
C 2
=
C 2
(6.9)
dN
dt ¼
N 2 with unstable equilibrium at C 2
C 1
N
þ
C 2
=
C 1
(6.10)
It is important to note that in a rate level graph only the rate of change for
different N is shown, not the change over time. Rate over stock is different from
stock over time. Figure 6.4a, b show the respective examples with N plotted over
time.
Using this kind of functional approach, the change of animal-, plant- or micro-
bial populations can be approximated by treating the population sizes as pools.
With the concept outlined so far, we can now look at a frequently considered
starting point for quantitative population ecology, the Lotka Volterra model.
6.4 Lotka-Volterra Equations as a Starting Point
for Ecological Modelling
The Lotka-Volterra model is the simplest way to describe the interaction of a
predator population and a prey population. It was proposed independently by Alfred
Lotka (1925) and Vito Volterra (1926). It is extremely simplified and thus not very
realistic; however, this simplicity is what makes it interesting. Frequently, models
are started with a by far too simple approach and then refined in a step-wise process.
 
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