Mata May Anne, Tyson Rebecca C, Greenwood Priscilla
Department of Math, Physics, and Computer Science, University of the Philippines Mindanao, Davao City, Philippines.
CMPS Department (Mathematics), Barber School of Arts and Sciences, University of British Columbia Okanagan, Kelowna, BC, Canada.
J Math Biol. 2019 Dec;79(6-7):2133-2155. doi: 10.1007/s00285-019-01423-7. Epub 2019 Sep 13.
Many real populations exhibit stochastic behaviour that appears to have some periodicity. In terms of populations, these time series can occur as limit cycles that arise through seasonal variation of parameters such as, e.g., disease transmission rate. The general mathematical context is that of a stochastic differential system with periodic parametric forcing whose solution is a stochastically perturbed limit cycle. Earlier work identified the power spectral density (PSD) features of these fluctuations by computation of the autocorrelation function of the stochastic process and its transform. Here, we present an alternative analysis which shows that the structure of the fluctuations around the limit cycle is analogous to that of fluctuations about a fixed point. Furthermore, we show that these fluctuations can be expressed, approximately, as a factorization which reveals the combined frequencies of the limit cycle and the stochastic perturbation. This result, based on a new limit theorem near a Hopf point, yields an understanding of the previously found features of the PSD. Further insights are obtained from the corresponding stochastic equations for phase and amplitude.
许多实际种群表现出具有一定周期性的随机行为。就种群而言，这些时间序列可以作为极限环出现，极限环是通过诸如疾病传播率等参数的季节性变化而产生的。一般的数学背景是一个具有周期性参数强迫的随机微分系统，其解是一个随机扰动的极限环。早期的工作通过计算随机过程的自相关函数及其变换来确定这些波动的功率谱密度（PSD）特征。在这里，我们提出一种替代分析方法，该方法表明极限环周围波动的结构类似于围绕固定点的波动结构。此外，我们表明这些波动可以近似表示为一种因式分解，该因式分解揭示了极限环和随机扰动的组合频率。基于霍普夫点附近的一个新的极限定理，这一结果使我们对先前发现的PSD特征有了理解。从相应的相位和振幅随机方程中可以获得进一步的见解。
