具有密度依赖演化博弈模型的随机稳定性分析  

Analysis of Stochastic Stability on Evolutionary Games Model with Density Dependence

作　　者：孙霞 呼文军 刘婕 SUN Xia;HU Wenjun;LIU Jie(School of Mathematics,North University of China,Taiyuan 030051,China;Department of Mathematics,Lyuliang University,Lüliang 033000,China)

机构地区：[1]中北大学数学学院,山西太原030051 [2]吕梁学院数学系,山西吕梁033000

出　　处：《中北大学学报(自然科学版)》2025年第1期64-75,共12页Journal of North University of China(Natural Science Edition)

基　　金：国家自然基金资助项目(12371494);山西省自然科学基金资助项目(2021L561)。

摘　　要：演化博弈提供了一种新的视角来审视和理解复杂系统中的互动行为,为多个领域的研究提供了有力的工具和方法。现有文献大部分集中于对确定型生态博弈模型动力学行为的研究,还未探索种群和环境的随机性对生态演化博弈动态的影响。基于此,本文引入环境随机噪声,构建了一个考虑个体出生率和死亡率的具有密度依赖的随机演化博弈动力学模型,并对所建模型进行动力学分析,以证实随机噪声对生态演化动态的影响。首先,通过分析具有密度依赖的确定型生态演化博弈动力学模型的均衡点及其稳定性,以及利用随机Lyapunov理论证明相应随机演化博弈模型的稳定性,以说明噪声对具有密度依赖的生态演化博弈模型的影响。其次,经研究得到了环境噪声的阈值,当环境噪声值大于阈值时,种群灭绝和表型消失是几乎必然指数稳定的,反之,当环境噪声值小于某一阈值时,出生率较高表型的频率将在时间均值意义下持续存在。最后,以猎鹿博弈模型为例进行理论验证及数值模拟,证实了理论的有效性和可行性。研究结果也为理解环境波动如何促进表型多样性的进化提供了一个新的思路。Evolutionary games provide a new perspective to examine and understand interactive behaviors in complex systems,and provide powerful tools and methods for research in several fields.Much of the existing literature focuses on the study of the dynamical behavior of deterministic ecological game models,and doesn’t explore the influence of population and environmental stochasticity on the dynamics of ecological evolutionary games.Thus,this paper considered environmental random noise,and constructed a stochastic evolutionary game dynamics model with density dependence considering individual birth and death rates.Dynamics of the constructed model was analyzed to confirm the effect of random noise on the trends of ecological evolution.Firstly,the effect of noise on ecological evolutionary game models with density dependence is shown by analyzing the equilibrium points and their stability of the dynamical models of ecological evolutionary games with density dependence indeed,and by proving the stability of the corresponding stochastic evolutionary game models using stochastic Lyapunov theory.Secondly,the threshold value of environmental noise was obtained from the study.When the environmental noise value is greater than the threshold,population extinction and phenotype disappearance are almost surely exponentially stable.Conversely,when the environmental noise value is less than a certain threshold,the frequency of phenotypes with higher birth rates is persistent in mean.Finally,the theoretical verification and numerical simulation were carried out with the deer hunting game model as an example to confirm the validity and feasibility of the theory.This result also provides a new idea for understanding how environmental fluctuations contribute to the evolution of phenotypic diversity.

关 键 词：演化博弈 密度依赖 几乎必然指数稳定 随机Lyapunov理论 稳定性 

分 类 号：O175.1[理学—数学]