Nature of Flagger/Coward Distribution

A linear function was chosen to describe the number of flaggers in a group, such that the total number of flaggers across all the groups was equal to the total number of cowards.

f(x) = a + x (n-2a)/g , where n is the total number of rabbits in each group, g is the number of groups and a is the number of flaggers in the 1st group. This integrates to ng/2 i.e. half the rabbits. The gradient of the function is given by grad=(n-2a)/g.
Using this as f(x) the expression Gc – Gf (from before) was calculated. This gives the average predator pressure experienced by flaggers relative to cowards for populations of heterogenous groups with varying numbers of flaggers and cowards as described by ‘a’.
Gf – Gc =


Substituting the expression a = (n-g*grad)/2 into the expression enables a clearer expression of heterogeneity.

Results
p=0.5, n=10, g=10

Once the gradient is steeper then 60% then flaggers experience rapidly decreasing predator pressure. This is a distribution with group one having 8 flaggers and 2 cowards, and group 10 having 2 flaggers and 8 cowards a range of {2,8}
For 50 groups (g=50) the graph has the same shape but the gradient only needs to be 12% before flaggers benefit more than cowards. The general rule is that the cut-off is proportional to 6/g.
p=0.5, n=20, g=10


For groups of large size there is little extra benefit to cowards until the groups have a larger range than {5,15}. After which there is considerable advantage to flaggers.
p=0.1, n=10, g=10

If the probability of not seeing the predator is very low (10%) there is range of distributions which benefit cowards very much. Suddenly when the range reaches {0.5,9.5} is benefits flaggers. 0.5 isn’t possible so there are no distributions which favour flaggers in small groups.
p=0.9, n=10, g=10

A simple curve where ranges {2.5,7.5} and steeper benefit flaggers. Flagging strategy is an anti-predator strategy and obviously only benefits where predators are skillful and not easily seen by individuals.
ConclusionFlagging as a strategy depends upon strong heterogeneity in flagger/coward ratios. Low dispersal of offspring i.e. children remaining in the warren will create such a heterogeneity. Sexual selection for flaggers is vulnerable to the evolution of a “slag” gene but may provide a mechanism for heterogeneity also.
Another avenue of investigation is in free markets. The local marginal strategy is always to defect and become a coward. However under certain situations strategies that benefit the group will lead to selection for groups of individuals who do not do this, i.e.e those who abandon local marginal thinking.