Environmental Engineering Reference
In-Depth Information
Fish preservation by smoking over wood led to a decline in the stock of trees.
The spread of the infectious and often lethal bilharzia disease can also be related to
the introduction of the Nile perch (Murray
2002
).
19.3 Predator-Prey Models
Another type of model describes the development of populations of a predator and
its prey. The model covers two trophic levels. The simplest approach goes back to
publications in the 1920s of the last century. In 1926, Volterra
4
tried to explain the
observation of oscillatory fish catches in the Adriatic Sea with the set of two
ordinary differential equations; the first for prey population
c
1
, the second for the
predator concentration
c
2
:
@
c
1
@t
¼ c
1
r
1
a
1
c
2
ð
Þ
(19.12)
@
c
2
@t
¼ c
2
a
2
c
1
r
2
ð
Þ
with positive parameters
r
1
,
r
2
,
a
2
.
While Lotka
5
developed the same approach for a system of chemical species
(Lotka
1956
) Volterra was the first who applied the system to an ecological problem
(Murray
2002
). Often the term
Lotka-Volterra
-models is found. The assumptions
that lead to the system (
19.12
) are rather simplistic. Without the predator-prey
interaction the prey population would increase exponentially with rate
r
1
,
accompanied by the exponential decrease of the predator population with rate
r
2
.
However, it is assumed that for the overall behavior the interaction is crucially
relevant. Prey consumption and predator population growth, both increase or
decrease with
c
1
and
c
2
, which is expressed by the two terms including the product
c
1
c
2
. In the corresponding M-file the MATLAB
a
1
and
®
ode15s
solver is again utilized
for the calculation of the ordinary differential equations. The M-file in large parts
coincides with the previous examples, so that few comments are necessary only.
4
Vito Volterra (1860-1940), Italian mathematician and physicist.
5
Alfred James Lotka (1880-1949), Austrian-US-American mathematician, chemist and ecologist.
