2009-3-28, 2007-07-07

### Sozial systems

A species that form groups is social when the individual and the group are
elements of the selection and when the chance of success of the group depends
on the composition of his member. It is assumed that the chance of success of
an individual till the dividing of the group does not depend on the composition
of other groups. A individual is more social than an other if the ratio of
their actualy chance of success is greater than the ratio of their chance of
success if the group their belong does not duplicate or if the chance of
success of the group does not depend on his member.

As example is considered two groups with two individual assuming that the
number of groups and their size remain equal, the time of the duplication of an
individual is fixed, the number of generation of individual between a new
generation of the group are infinite and that they are twins in one of the
group.

g1={e1,e2} and g2={e,e}

After infinite generation of individual there is the probality:

w({e1,e1}=i1 w({e2,e2})=i2 w({e1,e2})=i0=0 and w({e,e})=1

After infinite generation of the group:

w({e1,e1})=u1 and w({e,e})=u01 or w({e2,e2})=u2 and w({e,e})=u02

The actualy probability of the reproduction rate are:

r(e1)=2*i1*u1 r(e2)=2*i2*u2

R=2*i1*u1/(2*i2*u2)

The probability of the reproduction rate assuming the group does not
reproduce:

i(e1)=2*i1 i(e2)=2*i2

I=2*i1/(2*i2)

The probability of the reproduction rate assuming that the reproduction rate of
the group does not depend on his member:

r(e1)=2*i2*u'1 and r(e2)=2*i2*u'2 with u'1=u'2

J=2*i1*u'1/(2*i2*u'2)=I

e1 ist more social than e2 if u1 > u2.

There is a value u1 so that r(e1)=r(e2).

i1*u1=i2*u2 <=> u1=i2*u2/i1

If i1 < i2 then e1 has the same chance of success than e2 even as long as
the group does not split it has a lower reproductive rate than e2.

In this example the composition of the reproducing group is the same as the
reproducted group. In general there is only needed that the reproducted group
are composed of a multiple of members of the reproducing group. A special form
for the generation step of a group is when the group split into two parts first
and then duplicate. In the example there would only be needed one individual
generation to get i0=0.