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\(F\)-quadratic stochastic operators. (English. Russian original) Zbl 1167.92023

Math. Notes 83, No. 4, 554-559 (2008); translation from Mat. Zametki 83, No. 4, 606-612 (2008).
Motivated by applications to mathematical genetics, the authors study a special class of quadratic transformations of the unit simplex \(S^m\subset {\mathbb{R}}^{m+1}\) into itself. Using elementary methods, they prove that for each function from this class and any \(x^{(0)} \in S^m\) the trajectory \(\{ x^{(n)}\}\) is exponentially convergent to the fixed point \((1,0,...,0)\).

MSC:

92D10 Genetics and epigenetics
47N60 Applications of operator theory in chemistry and life sciences
47H40 Random nonlinear operators
47H10 Fixed-point theorems
37N25 Dynamical systems in biology
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