Departamento de Ciências Exatas, Universidade Federal de Lavras, Lavras, Brazil.
Departamento de Matemática, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil.
J Math Biol. 2020 Jul;81(1):277-314. doi: 10.1007/s00285-020-01510-0. Epub 2020 May 31.
We study fixation probabilities for the Moran stochastic process for the evolution of a population with three or more types of individuals and frequency-dependent fitnesses. Contrary to the case of populations with two types of individuals, in which fixation probabilities may be calculated by an exact formula, here we must solve a large system of linear equations. We first show that this system always has a unique solution. Other results are upper and lower bounds for the fixation probabilities obtained by coupling the Moran process with three strategies with birth-death processes with only two strategies. We also apply our bounds to the problem of evolution of cooperation in a population with three types of individuals already studied in a deterministic setting by Núñez Rodríguez and Neves (J Math Biol 73:1665-1690, 2016). We argue that cooperators will be fixated in the population with probability arbitrarily close to 1 for a large region of initial conditions and large enough population sizes.
我们研究了具有三种或更多种个体类型和频率相关适合度的种群演化的 Moran 随机过程的固定概率。与具有两种个体类型的种群不同，在这种情况下，固定概率可以通过精确公式计算，而在这里，我们必须求解一个大型线性方程组。我们首先证明该系统始终有唯一解。其他结果是通过将 Moran 过程与具有三个策略的过程与仅具有两个策略的出生-死亡过程耦合而获得的固定概率的上下界。我们还将我们的界应用于 Núñez Rodríguez 和 Neves（2016 年，J Math Biol 73:1665-1690）已经在确定性环境中研究过的具有三种个体类型的种群中合作进化的问题。我们认为，对于较大的初始条件和足够大的种群规模区域，合作者在种群中被固定的概率将任意接近 1。
