On holiday while looking at rabbits was wondering how genes for particular sexual attractions spread in populations. What if there was a gene that made brave individuals attractive. Decided to look at the old area I did at college that of tail flagging in rabbits to warn of predators (assume it’s a warning signal rather than predator visual confusion). If this is in doubt then alarm calls will do as another model. Let there be two genes F and C which cause their bearer to F=flag (or give an alarm call) or C=just run away. Let there be two other genes that cause their bearer to find this behaviour attractive Lf and Lc. I’ll ignore a ‘Slag’ gene that just breeds with anything for now (for good reason since it messes this whole argument up, unless there is another gene which doesn’t like Slags etc). I assume a simple haploid genetics to make like easy.
Cowards and Braves
F = brave flagger
C= coward who saves himself
Lf= find flagger attractive
Lc=find coward attractive
These parents: flag+likes cowards x coward+likes flaggers give the following children:
FLc x CLf –> FLc,FLf,CLc,CLf (in equal proportion)
It can be seen that only 1/4 of FLc and CLf matings are like the parent and 1/4 are FLf and CLc.
FLf and CLc parents however only produce themselves.
The other combinations won’t happen since the very nature of the Lf and Lc genes is to make the bearer look to mate with only flaggers and cowards respectively. So if you are Lc you will only look for cowards to mate with, and if you are a flagger yourself only Lf will mate with you so FLc must mate with only CLf.
As a result the population of FLc and CLf both drop by 75% each generation (assuming that all individuals mate freely). If low density means that it is harder to find a mate then this will be larger.
By contrast FLf and CLc populations replace themselves each generation and get added to by the 25% of the FLc and CLf populations.
CONCLUSION: (without the Slag gene) flagging genes rapidly become associated with genes that mate with flaggers at the exclusion of cowards, and vice versa.
The Population Models
Let p=the probability that an individual does not see a predator. This is the probability the individual will be eaten in some time frame.
Take a group n numbering f flaggers and c cowards (n=f+c). The group comes under attack from an ambush predator looking for 1 individual. To succeed the predator needs only catch the prey unawares, after which it is a certain kill. The probability of being taken P is:
P = 1 / Number of unaware individuals
However the flaggers alert the group so they all need to be unaware at once. If all the flaggers are unaware (prob = p^f) then the total number of unaware individuals is f (since all the flaggers are unaware) + pc (the unaware cowards). The probability of being taken as a flagger is then:
Pf = p^f / (f + pc)
The cowards benefit from the flaggers flagging but also the probability that they are looking too or conversely the probability they are taken is p^(f+1). The probability of being taken as a coward is thus:
Pc = p^(f +1) / (f + pc)
CONCLUSION: Thus as a coward your probability of being taken is always p less than being a flagger. Flaggers because of their relative generosity will decline in number relative to cowards in a given population.
However note that Pc has a maximum value when c==n and a minimum value when c==0.
CONCLUSION: Thus having flaggers around instead of cowards is beneficial to cowards. It is better for cowards to move towards areas with higher proportions of flaggers. However as will be seen this leads to mating problems if there are genes that don’t like cowards.
Model 1
Let the population be spread out in g groups. The predation pressure on the population as a whole can be estimated by looking at the individual average, that is the sum of f*Pf for each group divided by the total number . As a continuous function we therefore need (given that f+c=n so c=n-f),
If f[x] is a constant f (as before) then the equations decompose to the equations before i.e. Gf = Pf and Gc = Pc
If f[x]=x i.e. it linearly increases through all the groups so that groups have increasing numbers of flaggers and let there be n groups so that flaggers range from 0 to n, then the integrals to calculate are (where g=n):
Which numerically integrated for p=1, n=10 gives
Gf = 0.1
Gc = 0.1
As expected since the average predation in any number of groups of 10 individuals if they are blind (unable to detect the predator) will be 1/10. i.e. Gf/Gc = 1. For the following probabilities where group size is n=10, Gf/Gc gives:
p=0.25 0.264687
p=0.5 0.295375
p=0.75 0.496825
That is the average predation of flaggers is around 30% that of cowards for lower probabilities and rises relatively as both groups become less vigilant. For larger groups (and a larger range of groups) the effect is ever more apparent so that with groups of size n=30 predation of flaggers is 10% that of cowards.
CONCLUSION
When the ratio of flaggers to cowards varies linearly across all the groups it remains beneficial to be a coward as an individual in any one group. However the flaggers in mostly cooperative groups do better than the cheats in mostly cheating groups so the average globally benefits the flaggers. So while cowards will locally do better than their neighbouring flaggers, globally their population will drop significantly faster.
Model 2
In all groups the rate of reduction of cowards is p less than that of flaggers. That is cowards will therefore come to dominate all static groups and if the group size is limited (as by limiting resources) then eventually flaggers will become extinct in these groups. However it is beneficial for cowards to move to areas of high flagging density.
In reality offspring, likely to carry the parents flagging or coward gene, will disperse around the parents creating pockets of high density. Infiltrating individuals may be excluded by territorial behaviour. It would be in the interests of the cowards to let flaggers into their territory, but not vice-versa.
Sexual selection may also provide a barrier to migration as described above.
2Do
Model that looks at changing rate of populations in cells (with genetics) dy/dx = ry(genetic factors) – pred(y)
Model that looks at changing rate of populations globally (with genetics)
Model that looks at changing rate of populations in a cell (with steady influx of individuals) i.e. like mandelbrot… look to see if there are critical influxes that lead to changes in the cell ratios… then we know any cell with neighb\ours that cause that level of migration will turn themselves into such a cell. i.e. if their outflux goes to the level that needed to promote them.
Examine the shape of distribution of ratios across groups. Population heterogeneity can create incongruent local versus global properties! Apply to markets where marginal action (i.e. to become a coward) is offset by global benefits of being a flagger. Relevance of markets? Conditions where “Invisible Hand of Markets” no longer operates… consider information dispersal through a population as one means to create heterogeneity.
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