A Fish Migration Model - Cellular Automata - page 214

2.1 Example: Migration of (Atlantic) Salmon in the River Moselle
We give an example to illustrate our model concept. We simulate upstream
migrating adult salmon coming from river Rhine and passing river Moselle for
reaching spawning sites in its tributaries. The question is how to invest resources
for the improvement of fish passages to enable and optimise salmon migration.
In our simplification we consider a section of river Moselle with seven dams
and only two larger tributaries. This gives ten river fragments,i.e. ten cells
as shown in Figure 1b. To each of these cells a probability p i is assigned,a
measure for the permeability of the dam,respectively the preference of tributary
rivers. Figure 1c shows a more formal representation as a directed graph with
an example setting of p i values.
We look only at the number of fish that are able to migrate upstream to the
headwaters,i.e. to the most left cell (Fig. 1c). We start with a finite number
of fish at the rightmost cell,i.e. fish entering the River Moselle. The number
arriving at the headwaters will increase over time and finally,if all fish travelled
up,reach a constant level. It is shown in Figure 2 for two choices of p i : upstream
increasing permeability - these are the values drawn in Figure 1c - and constant
permeabilities (with the same mean) on the main stream.
Figure 3 shows also simulation snapshots. With constant p i fish arrive faster
at the headwater; this recovers the analytic result that equal permeabilities lead
to optimal migration in the case of a river without tributaries. In our example
with increasing permeabilities finally more individuals reach the headwater. The
p i values of the three leftmost cells are about 1.0,0.86 and 0.71 while with
constant permeabilities we have p i =0 . 57. The branches to the tributary rivers
both have p i =0 . 2,i.e. in the case of constant permeabilities more fish will swim
into the tributary and at the river's headwater finally less fish will arrive. We see
also that the variability between different realisations is a little bit larger with
constant permeabilities.
These results may be obtained also analytically. In more realistic settings
however,as described in the next section,this may be at least laborious.
2.2 Model Extentions
The model formulation allows to include almost arbitrary details of the fish
life cycle. Through upstream migration for example some fish will die by various
causes. This can be included by an additional process - once a fish is chosen it will
decided with certain probability whether it will migrate or die. This probability
may even depend on space,i.e. the position of the fish. It is also reasonable
that fish get tired the more often they try to migrate through a given dam.
We can give each fish a counter - the more times it makes a try,the smaller
the probability it succeeds. As well fish become more and more exhausted the
further they travel. This can be incorporated likewise giving each fish a 'fitness'
value (eventually according to a certain distribution) that decreases with time -
if it drops below a given threshold,then the fish will die.