Mutation and Selection on
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So far we have stipulated that movement rates are fixed, and moreover, are the same for all agents regardless of the strategy they play. It seems, however, that Cooperators and Defectors might each have own optimum movement rates, there being different considerations that apply. Cooperators need to stay in one place long enough to reap the gains of reciprocity with their kin. They need to move enough for dispersal adequate to insure survival of the strategy when the patch is inevitably invaded by Defectors. Defectors need to move fast enough to find cooperator patches before "accident" befalls them, slow enough to stay in those patches for a while. High movment rates may also take Defectors too quickly into the unihabitable outer region. Moreover, it would be reasonable to suspect that the cooperator populations in the boundary region can persist if Defectors are allowed their optimal movment rate.
What we have done here is to start our population as in Step 2 (mixed initial population, no mutation on strategies) with the exception that each individual carries its own movment parameter. Moreover, movement values of offspring may differ from that of their parent. To be precise, there is a 5% chance each offspring's movement rate being greater than that of its parent by 0.001, and a 5% chance of the offspring's movement rate being 0.001 less. These differences are heritbale and subject to natural selection, with the more successful rates for each strategy gradually coming to dominate the population.
The graph below shows what happened in a run of over one million cycles (20,000 baseline generations). Cooperator movment rates seem to oscillate around the original 1% value, with a mean value of 0.009987. Recall that this is a rate of approximately one move each two generations. Defector movement rates climb to about 4.5% by about cycle 200,000, and oscillate around that level thereafter, albeit at a rather low frequency. Note the initial decline in average cooperator levels as Defector movement rate increases. The important point is that this trend seems to stop, leaving a relatively stable cooperator population of between 150 and 600. Notice, however, that there are a number of near extinctions, where the number of Cooperators dropped as low as 69. Presumably, a larger boundary region would mitigate this instability.
In contrast, if instead of individual mortatiliy
risks, whole patches are visited by extinction events, selection favors higher movment
rates for cooperators as well. Higher movement rates combined with the greater
stochasticity of patch exinction patterns undermines the apparent long term stabiliy of
boundary region cooperator populations with evolving movement rates. The graph to the
right shows what happened in a trial with these characteristics. Note that in the cases we
have looked at in steps 2 through 4, patch extinction patterns allow the survival of
cooperation almost as well as individual mortality risks. It may be that larger grids
would smooth out the random fluctuations enough to stabilize cooperation with patch
extinction and evolving mutation rates as well.
Copyright © William Harms
1999. (Author, designer, and programmer.)
Evolving Artificial Moral Ecologies Project
Centre for Applied Ethics, UBC