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where j is the smallest index such that |
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(modified so that the actual solution is in integers). For example, let l0 = 2 with 2 alleles (attributes) at each locus, yielding schemata x1, x2, x3, x4 with Q1= 1, Q2 = 4, Q3 = 8, Q4 = 1. Then for M = 9 there will be 6 instances of x3, 3 instances of x2, and no instances of x1 or x4 in the stable distribution. |
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Here we have a simple example of speciation. If the population is restricted to M individuals (by factors other than the niche payoff rates), certain combinations of alleles appear in a stable competition while other combinations are proscribed by the same competition. The example can rapidly be made more realistic by letting the payoff to each schema x be a random variable with expected payoff |
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where Qx is the minimum of the renewal rates of resources characterizing the environmental niche associated with x, Mx (t) is the number of instances of x at time t, Q is the minimum of the renewal rates of resources required by all the schemata, and M(t) is the total population at time t. Now the schema x will increase its proportion at an intrinsic rate set by  until it reaches the "carrying capacity" of its niche, determined by Qx, or until the total population has increased to a point that the overall "carrying capacity," determined by Q, limits further expansion. (For the reader familiar with MacArthur and Wilson's [1967] work, the effect of Qx corresponds to a K selectioncrowded nicheeffect, whereas is the intrinsic rate of increase, possibly wasteful of resources, under classical r selection. Q sets an ultimate limit on the carrying capacity of the environment, no matter what the diversity or organization of the species.) With typical values for the {Qx} and Q, the population will once again develop into subpopulations characterized by certain combinations of alleles (schemata), with many combinations being proscribed. |
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The really interesting form of this theory would characterize niches (and hence the overall payoff function µ) in terms of the varieties of schemata that could exploit themdifferent schemata exploiting a given niche with differing efficiencies. The dynamics of speciation would then be determined by competition within and across niches. It is interesting that under these circumstances speciation |
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