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population very quickly (about 50 MCS) reaches the same value of about 0.6 and
then it follows its evolution, depending on the values of the parameters. Regions
II and III are empty. The values of the selection pressure,
s
, and mutation rate,
p
mut
, are fixed. We let the population adapt to the existing conditions and then
we change the optimal phenotype (climate) as follows. At the first change, after
200 MCS after the adaptation, the values at two, randomly chosen, sites in the
optimal phenotype in the first region are changed from 0 to 1. Since a zero in the
individual's phenotype can be obtained from three combinations of alleles in the
genotype, i.e., (00), (01) and (10), a change from 0 to 1 in the optimal phenotype
means that there are less sites in the genotypes that satisfy this condition. One
may therefore say that the living conditions turned to worse. At the second
change, which also occurred after 200 MCS, the optimal phenotype in the first
region is again changed in the same way, i.e. at two, randomly chosen sites,
zeros were switched for ones. Now however, also the optimal phenotype in the
second region has been changed. We simply adopted here the previous (changed)
optimum phenotype of the region I. One may see it as a gradual moving, from
the left, of a colder climate. Finally, once again after 200 MCS the climate is
changed. Two more sites in the optimal phenotype in the region I are switched
from 0 to 1, thus it contains now six sites with value one and twenty six sites
with value 0. In region II the optimal phenotype contains four sites with value
1, and in the region III it contains just two sites with values equal 1. Since the
average age of an individual in our simulations oscillates around 3.5 MCS, the
changes are made after about 100 generations. We consider here a system with
overlapping generations.
Although in our model there is no force pushing the individuals into the re-
gions II and III, it is not a simple colonization of empty territories. An individual
adapted to the climate in the first region finds, after the conditions changed, a
better chance of survival if it follows the climate, hence if it moves into the II or
III regions. In such a way the region I becomes depopulated. A simple invasion
process occurs in our model when the climatic changes are negligible. Then the
density in the first region remains approximately the same as in the other two.
For simulations we have used a lattice of the following dimensions -
L
x
=
150, hence each region was 50 lattice sites long,
L
y
= 1000. We have checked
that increasing the value of
L
x
did not change the results, while increasing
L
y
lead to very long simulations with only slightly better statistics. Typically we
let a population evolve for 5
×
10
4
to 10
5
MCS.
3.3 Properties of Model B
A detailed description of the properties of model B can be found in [12].
Accordingly, we shall only give here a summary of the results.
The time evolution of the densities in the three regions, for a given value of
the selection pressure and mutation rate, has been investigated.
There are three possible outcomes for a fixed value of
s
=0
.
065. When
the mutation rate is low (
p
mut
=0
.
001), the population survives and colonizes
