Dynamics of Populations in Extended Systems - Cellular Automata - page 196

Hence z j ∈ [0 , 1]. Survival probability defined by (3) increases with the indi-
vidual's fitness but goes down with increasing age and the selection pressure.
Since the exact form of the survival probability is not known, exponential or
gaussian dependence on the selection is quite often used by biologists (see
e.g. [15]). It should be noted that the results do not depend quantitatively
on the form of the survival probability.
The notion of the optimal phenotype is known since a long time [14] and it
represents here a vector of 32 components taking values equal either 0 or 1.
3. A random number r j is chosen. If r j ≤ p the individual lives, otherwise it is
removed from the system and the procedure goes back to 1.
4. If the individual's age is greater than 1, it may breed. To do so it must go
through the steps listed below
5. Find an empty place in the von Neumann neighborhood [8]. On a square
lattice used here, it means in the four principal directions - N,E,S,W . Only
one such search is made (blind ant rule).
6. After moving to the new place, find a partner in the von Neumann neigh-
borhood of the new position.
7. Thetwoproduce,atmost4offspring.Eachofthemreceivesitsowngenotype
constructed through recombination and formation of two gametes from each
parent (genetic shu K ing). The process may be described as follows. The two
strings of a parent's genotype are cut at a random place. The resulting four
piecesaregluedacross,formingtwogametes.Thesameisdoneforthesecond
parent. From each parent one gamete is chosen randomly, thus forming two
strings for the genotype of the offspring.
8. On each site of the chosen gametes a harmful mutation may take place with
probability p mut . This means that if at a given site (locus in biological terms)
the value agrees with the value at the corresponding site of the optimal
phenotype, it may be changed with that probability. If the value is different
fromthatoftheoptimalphenotype,noactionistaken.Eachprogenyreceives
its genotype independently. Only harmful mutations are considered because
their frequency is much larger than the one for positive mutations.
9. From the genotype a phenotype is produced along the lines described above.
10. ForeachoffspringasearchforanemptyplaceismadeintheMooreneighbor-
hood [8] of the first parent. On a square lattice it means 8 sites - N,NE,E ,
SE,S,SW,W,NW . If the place is found, the offspring is put there, if not,
it lost its chance and is not born.
11. After choosing as many first parents as there are individuals in the popula-
tion at that time, one Monte Carlo step (MCS) has been made and the age
of all individuals is increased by one.
At the beginning of simulations the optimal phenotype is the same in all
three regions and is the most favorable, i.e. it is a string of zeros only (zero is a
dominant allele !). The initial population is random, i.e. it has random genotypes
and the spatial distribution is random too, but restricted to the first region only,
with a given initial concentration. We have checked that if that concentration
is above a certain level ( about 0.05) its precise value is unimportant, since the