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An exact steady state solution of Fisher’s geometric model and other models. (English) Zbl 1210.92030

Summary: Because nearly neutral substitutions are thought to contribute substantially to molecular evolution, and much of our insight about the workings of nearly neutral evolution relies on theory, solvable models of this process are of particular interest. I present an analytical method for solving models of nearly neutral evolution at steady state. The steady state solution applies to any constant fitness landscape under a dynamic of successive fixations, each of which occurs on the background of the population’s most recent common ancestor. Because this dynamic neglects the effects of polymorphism in the population beyond the mutant allele under consideration, the steady state solution provides a decent approximation of evolutionary dynamics when the population mutation rate is low (\(Nu \ll 1\)). To demonstrate the method, I apply it to two examples: R. A. Fisher’s geometric model (FGM) [The genetical theory of natural selection. Revised reprint of the 1930 original. (1999; Zbl 1033.92022); Oxford Univ. Press (1930; JFM 56.1106.13)], and a simple model of molecular evolution. Since recent papers have studied the steady state behavior of FGM under this dynamic, I analyze its behavior in detail and compare the results with previous work.

MSC:

92D15 Problems related to evolution
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
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