A Fish Migration Model - Cellular Automata - page 213

i.e. fish,we will choose a fish,locate its position and decide whether it will
migrate in contrast to choosing a cell to be iterated. So the rules read:

1. Choose a fish. Let x i be the cell where it is.
2. Choose a neighbouring cell of x i ,let this be x j .
3. With probability p j the fish migrates from x i to x j ,otherwise it stays at x i .

These rules imply asynchronous dynamics. The first rule has to be specified
more detailed,i.e. we have to select a method to choose the 'next' fish. The most
satisfying approximation of continuous time from theoretical point of view is to
assign exponentially distributed waiting times to each fish. For many applications
the easier choice with uniform distributed probability will be su @ cient. If we
want to formulate the first rule such that we choose a cel l instead of choosing
fish,then the probability a certain cell is selected should depend on the state
of the cell. The more fish it contains,the more likely it should be selected.
Otherwise fish in crowded cells would migrate slower than fish in cells with low
density. In the second rule the simplest possibility is to chose the neighbour cell
with equal probability from all neighbours of the cell. Certainly neighbours can
also be chosen with different probabilities q i ,with i q i = 1. In many cases a
cell will anyway have only one neighbour.
Our model fits in the framework of dimer automata [12]. If we want to formu-
late this model as a cellular automaton,the grid will be G as before and also the
states of the cells will be in E =0 , 1 , ..., n . The problem is to enable migration
of fish. To ensure mass conservation,i.e. a constant total fish number,we have
to change the states of two cells. For this we have to introduce some kind of
handshake between the cell the fish migrates from and the cell it migrates to.
Also at confluences we have to decide to which neighbour cell the fish moves.
This means that here stochasticity has to be involved. Therefore this automaton
can not be formulated as a classical deterministic cellular automaton. Also the
different values of p i and the structure of the grid gives a spatial inhomogeneous
model.
This model can be also formulated as a flow on a directed graph where every
vertex represents the segments between two dams. The parameter p i will then
be assigned to every edge - an example is shown in Figure 1c. Such a graph
is a subset of a binary tree (if we ignore river 'crossings' and circuits). This
formulation shows that fish migration can be seen as a percolation problem.
So far we only described how fish migrate on a river with given dam per-
meabilities. Our problem however is to invest our resources and change these
permeabilities such that most fish will arrive in suitable spawning habitats for
example. Therefore we may not specify p i but the amount of resources r i used on
a certain dam. To every dam (or cell) we have to give a function that gives p i for
any value of r i . Usually this function will be nonlinear,i.e. if we want to double
the permeability,the resources used at this dam have to be more than doubled.
Let p i 0

be the actual estimated permeability of dam i . Then p i = p i 0 + f i ( r i )

where f i is the cost function of dam i .