Dynamics of Populations in Extended Systems - Cellular Automata - page 199

all three regions with similar densities. When the mutation rate increases to
p mut =0 . 003, although the population also survives, the mutation rate is so
high, that life in the I region became di ? cult. Hence the population moves to
more friendly regions II and III, with only very few individuals remaining in
the region I. Both cases correspond to the active state. When p mut =0 . 004, the
population becomes, after some time, extinct. This is the absorbing state. The
fitness of the individuals in the three regions raises very quickly to about 0.8
and remains at that level.
The cases in which the mutation rate is kept constant and the selection
pressure is varied have also been studied. If the living conditions in the region
I are not too di ? cult, the population will recolonize it and the density grows
there to about 0.5. This is the case illustrated in Figures 1, where the density
dependence on time and some ”snapshots”, showing the spatial location of the
population, are presented.
We have performed a series of simulations keeping either the selection
pressure or mutation rate fixed and changing the other parameter. We have
found that in each case, for some values of the varying parameter, the pop-
ulation arrived at an active state, while for others it became extinct. Far
away from the critical point, i.e. the lowest value of the parameter which
was changed (selection or mutation rate) and for which the populations
died, average density stabilized pretty soon. The closer the system was to
the critical point, the more the density fluctuated. The fact is well known to
biologistsasademographicstochasticity,affectingpopulationsofsmallsizes[16].
Examining the final, quasi-stationary, states obtained from the time depen-
dencies of the density of the populations, we can construct a phase diagram in
the (selection, mutation rate) plane, presented in Figure 2. The diagram shows
a critical line separating the alive (active) and extinct (absorbing) phases. The
line is not symmetric, since the role played by the two parameters is not equal.
Selection acts in the same way in all three regions, while mutation rate affects
individuals differently. First of all, if a mutation (there are only harmful muta-
tions) occurred for an individual in the region II, then if it immigrates either to
the region I or III, the mutation may turn beneficial, because of different optimal
phenotypes there. Next, a mutation changes only one allele in the genotype, and
for having an effect on the phenotype, also the second allele at the same locus
(site on the genotype) has to be 'wrong'. In biological terms it means that if a
harmful mutation is to affect the phenotype it has to occur in a heterozygote
(differentallelesinthesamesite)atthatlocus.Third,thereisageneticshu K ing,
meaning that progeny receives genotypes of the parents changed in the process
of recombination.
In many biological studies, see e.g. [17,16,18], the average time to extinction
is considered. In most cases the parameters are the reproduction rate (number
of offspring) and the size of the carrying capacity of the habitat (maximum
number of individuals who can live there at the same time). Here we have found