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| R.A. Fisher argued that populations at a stationary state in the presence of natural selection must be subject to a "deterioration of the environment" in order to balance the increase in mean fitness due to the Fundamental Theorem of Natural Selection. Here it is shown that this "deterioration" is a universal feature of positive semigroups whose generators include operators of multiplication. Specifically, the non-multiplicative part of the generator must produce a net flow at equilibrium that is negative when weighted by the operator of multiplication. In biological applications, the operator of multiplication corresponds to fitness coefficients or heterogeneous growth rates, while the other parts of the generator correspond to diffusion, dispersal, mutation, recombination, sexual reproduction, or other transformations. As a consequence, the mean effect of these transformations at a stationary distribution is always to reduce fitness. This provides an intuitive explanation for the Reduction Principle for the evolution of reduced rates of transformation. |
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