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.