Environmental Engineering Reference
In-Depth Information
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.