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site but distributed over several sites. This implies that the goal function will
not be as simple as getting as much individuals as possible to one location. Also
upstream migrating salmon for example will more and more exhausted and per-
haps die before reaching suitable gravel beds for reproduction. However in at
least some of these settings analytic solutions can be found.
The structure of optimal fish migration is related to problems from quite
different fields,for example to optimal vaccination strategy problems [10],and
results may be transferred to our question. An individual based model approach
however offers a very flexible method to test different strategies and include
various parameters. Also the model is simple and transparent for the users.
A fish passage model enables to test different strategies of fish passage facilities
extensions,however it does not solve the optimisation problem directly. One idea
is to add a part that forecasts permeabilities on the base of resources invested in
a certain dam - a cost-effect function has to specified for every barrier. The best
way to distribute resources can be found with stochastic hill-climbing algorithms
for example,the goal function being the number of successfully migrating fish.
2 Fish Passage Model
For simplicity we will first describe an
upstream
migration model for one fish
species. The fragmentation of rivers by dams suggests discrete units in space.
Every cell will represent an impoundment which is the segment between two
successive dams. A river without tributary rivers for example can be seen as a
one-dimensional grid
G ⊂
Z
. Since we look only at upstream migration,in this
simple case every cell will have only one neighbour cell,except the last cell at the
river's source. Please note that cells are not their own neighbours. To describe
more complex river systems,like for example the river Moselle with its tributary
system,we have to include branching. Then some cells will have two neighbours
or even more - although every cell is only neighbour of one other cell. This gives
us a grid
G
of cells
x
with 'branches' embedded in two-dimensional space.
Every cell can take states from the set
E
=0
,
1
, ..., n
where
n
is the total
number of simulated individuals. The state of a cell is interpreted as the number
of fish present. The state of the system is then a function
z
:
G → E
,
z
:
x → z
(
x
).
We first explain the dynamics in an individual based formulation. To every
cell a number
p
i
∈
[0
,
1] is assigned. We interpret it as probability that one fish
successfully migrates
into
this impoundment in one try. For upstream migration
it therefore describes the permeability of the dam located downstream of the cell.
At confluences,were tributary rivers join,
p
i
can be seen also as the preference
for fish to migrate into tributary rivers versus following the main stream. At
these branching points cells have more than one neighbour. Then we have to be
more careful with the interpretation of
p
i
. Consider for example a cell with two
neighbours with permeabilities
p
1
and
p
2
. Suppose that fish try to migrate in
one of the neighbours with equal probability. Then the probability that a fish
successfully migrates into the first or second neighbouring cell is
p
1
/
2 respectively
p
2
/
2. Since in the individual based formulation we take the view of individuals,
