Okabe Takuya, Yoshimura Jin
Graduate School of Integrated Science and Technology, Shizuoka University, 3-5-1 Johoku, Hamamatsu, 432-8561, Japan.
Department of International Health and Medical Anthropology, Institute of Tropical Medicine, Nagasaki University, Nagasaki, 852-8523, Japan.
Bull Math Biol. 2022 Jan 4;84(2):25. doi: 10.1007/s11538-021-00984-3.
Temporal variations in population size under unpredictable environments are of primary concern in evolutionary ecology, where time scale enters as an important factor while setting up an optimization problem. Thus, short-term optimization with traditional (arithmetic) mean fitness may give a different result from long-term optimization. In the long-term optimization, the concept of geometric mean fitness has been received well by researchers and applied to various problems in ecology and evolution. However, the limit of applicability of geometric mean has not been addressed so far. Here we investigate this problem by analyzing numerically the probability distribution of a random variable obeying stochastic multiplicative growth. According to the law of large number, the expected value (i.e., arithmetic mean) manifests itself as a proper measure of optimization as the number of random processes increases to infinity. We show that the finiteness of this number plays a crucial role in arguing for the relevance of geometric mean. The geometric mean provides a satisfactory picture of the random variation in a long term above a crossover time scale that is determined by this number and the standard deviation of the randomly varying growth rates. We thus derive the applicability condition under which the geometric mean fitness is valid. We explore this condition in some examples of risk-spreading behavior.
在不可预测的环境中，种群规模的时间变化是进化生态学的主要关注点，在建立优化问题时，时间尺度是一个重要因素。因此，用传统（算术）平均适应度进行短期优化可能会得到与长期优化不同的结果。在长期优化中，几何平均适应度的概念已被研究人员广泛接受，并应用于生态学和进化中的各种问题。然而，几何平均的适用范围至今尚未得到探讨。在这里，我们通过数值分析服从随机乘法增长的随机变量的概率分布来研究这个问题。根据大数定律，随着随机过程的数量增加到无穷大，期望值（即算术平均值）表现为优化的适当度量。我们表明，这个数量的有限性在论证几何平均的相关性方面起着关键作用。在由这个数量和随机变化增长率的标准差决定的交叉时间尺度以上，几何平均提供了长期随机变化的令人满意的图景。因此，我们推导出几何平均适应度有效的适用条件。我们在一些风险分散行为的例子中探讨了这个条件。
