Sadhu Susmita, Kuehn Christian
Department of Mathematics, Georgia College & State University, Milledgeville, Georgia 31061, USA.
Chaos. 2018 Mar;28(3):033606. doi: 10.1063/1.4994830.
The effect of demographic stochasticity, in the form of Gaussian white noise, in a predator-prey model with one fast and two slow variables is studied. We derive the stochastic differential equations (SDEs) from a discrete model. For suitable parameter values, the deterministic drift part of the model admits a folded node singularity and exhibits a singular Hopf bifurcation. We focus on the parameter regime near the Hopf bifurcation, where small amplitude oscillations exist as stable dynamics in the absence of noise. In this regime, the stochastic model admits noise-driven mixed-mode oscillations (MMOs), which capture the intermediate dynamics between two cycles of population outbreaks. We perform numerical simulations to calculate the distribution of the random number of small oscillations between successive spikes for varying noise intensities and distance to the Hopf bifurcation. We also study the effect of noise on a suitable Poincaré map. Finally, we prove that the stochastic model can be transformed into a normal form near the folded node, which can be linked to recent results on the interplay between deterministic and stochastic small amplitude oscillations. The normal form can also be used to study the parameter influence on the noise level near folded singularities.
研究了高斯白噪声形式的人口统计学随机性在具有一个快变量和两个慢变量的捕食者 - 猎物模型中的影响。我们从一个离散模型推导出随机微分方程（SDEs）。对于合适的参数值，模型的确定性漂移部分存在一个折叠节点奇点，并表现出奇异霍普夫分岔。我们关注霍普夫分岔附近的参数区域，在没有噪声的情况下，该区域存在小振幅振荡作为稳定动力学。在这个区域，随机模型允许噪声驱动的混合模式振荡（MMOs），它捕捉了种群爆发两个周期之间的中间动力学。我们进行数值模拟，以计算不同噪声强度和到霍普夫分岔距离下连续尖峰之间小振荡随机数的分布。我们还研究了噪声对合适的庞加莱映射的影响。最后，我们证明随机模型可以在折叠节点附近转化为一种范式，这可以与确定性和随机小振幅振荡相互作用的最新结果联系起来。该范式还可用于研究折叠奇点附近参数对噪声水平的影响。
