A diploid population model for copy number variation of genetic elements. (English) Zbl 1519.92148

Summary: We study the following model for a diploid population of constant size \(N\): Every individual carries a random number of (genetic) elements. Upon a reproduction event each of the two parents passes each element independently with probability \(\frac{1}{2}\) on to the offspring. We study the process \(X^N=(X^N(1),X^N(2),\ldots)\), where \({X_t^N}(k)\) is the frequency of individuals at time \(t\) that carry \(k\) elements, and prove convergence (in some weak sense) of \({X^N}\) jointly with its empirical first moment \({Z^N}\) to the “slow-fast” system \((Z,X)\), where \({X_t}=\text{Poi}({Z_t})\) and \(Z\) evolves according to a critical Feller branching process. We discuss heuristics explaining this finding and some extensions and limitations.


92D15 Problems related to evolution
60J85 Applications of branching processes
60G57 Random measures


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