Bernstein Polynomial Approximation of Fixation Probability in Finite Population Evolutionary Games

IF 1.8 4区 数学 Q2 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS
Jiyeon Park, Paul K. Newton
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引用次数: 0

Abstract

We use the Bernstein polynomials of degree d as the basis for constructing a uniform approximation to the rate of evolution (related to the fixation probability) of a species in a two-component finite-population, well-mixed, frequency-dependent evolutionary game setting. The approximation is valid over the full range $$0 \le w \le 1$$ , where w is the selection pressure parameter, and converges uniformly to the exact solution as $$d \rightarrow \infty $$ . We compare it to a widely used non-uniform approximation formula in the weak-selection limit ( $$w \sim 0$$ ) as well as numerically computed values of the exact solution. Because of a boundary layer that occurs in the weak-selection limit, the Bernstein polynomial method is more efficient at approximating the rate of evolution in the strong selection region ( $$w \sim 1$$ ) (requiring the use of fewer modes to obtain the same level of accuracy) than in the weak selection regime.
有限种群进化对策中固定概率的Bernstein多项式逼近
我们使用d度的Bernstein多项式作为在双组分有限种群、混合良好、频率依赖的进化博弈设置中构建物种进化速率(与固定概率相关)的统一近似的基础。该近似在整个范围内都是有效的$$0 \le w \le 1$$,其中w是选择压力参数,并均匀收敛到精确解$$d \rightarrow \infty $$。我们将其与在弱选择极限($$w \sim 0$$)中广泛使用的非均匀近似公式以及精确解的数值计算值进行比较。由于在弱选择极限中存在边界层,伯恩斯坦多项式方法在强选择区域($$w \sim 1$$)(需要使用更少的模式来获得相同的精度水平)中近似进化速率比在弱选择区域更有效。
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来源期刊
Dynamic Games and Applications
Dynamic Games and Applications MATHEMATICS, INTERDISCIPLINARY APPLICATIONS-
CiteScore
3.20
自引率
13.30%
发文量
67
期刊介绍: Dynamic Games and Applications is devoted to the development of all classes of dynamic games, namely, differential games, discrete-time dynamic games, evolutionary games, repeated and stochastic games, and their applications in all fields
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