Environmental Engineering Reference
In-Depth Information
Fig. 19.2 Example of a
foodweb model; for part of
the Lake Michigan aquatic
ecosystem
Lake Trout
Forage Fish
Bloater
Chub
Rainbow
Smelt
Sculpin
Alewife
Bythotrephes
Benthos
diporeia
Herbivorous
Zooplankton
Nysis
Zooplankton
Detritus
Phytoplankton
19.2 Competing Species
More specifically we are interested in the development of two species that are
competing for exactly the same resources. For very low populations the growth
rates are given by the specified values
r
1
and
r
2
. For increased populations the
growth rate is reduced by a term which takes into account the reduced foodstock of
both species. If the foodstock is reduced by
Dh
, the following system of two
differential equations can be used to describe the temporal development:
9
=
@
c
1
@t
¼ c
1
r
1
a
1
Dh
ð
Þ
with
Dh ¼ h
1
c
1
þ h
2
c
2
(19.3)
@
c
2
@t
¼ c
2
r
2
a
2
Dh
;
ð
Þ
The system of (
19.3
) can be re-written in analogy to formulation (
19.1
):
with
8
<
k
10
c
1
k
1
r
1
c
1
1
k
1
¼
¼
þ lc
2
=c
1
1
@
@t
c
1
c
2
(19.4)
with
k
20
:
c
2
k
2
r
2
c
2
1
k
2
¼
þ l
1
c
1
=c
2
1
with capacities
k
i
0
¼ r
i
=a
i
h
i
for single species cases and the dimensionless system
h
2
r
2
r
1
a
1
a
2
k
10
parameter
l ¼
h
1
¼
(compare: Richter
1985
). When the ratios
r
i
=a
i
are
k
20
equal, the capacities are related by the formula:
k
10
¼ lk
20
. If species 1 is more
efficient than species 2 concerning resource consumption, it holds:
l<k
10
=k
20
;
while the opposite inequality holds if species 2 is more efficient.
