生态系统中随机性引起的混合模振荡和复发性爆发的分布
Stochasticity induced mixed-mode oscillations and distribution of recurrent outbreaks in an ecosystem
摘要：
研究了具有两个不同时标的捕食-食饵模型中高斯白噪声形式的随机性效应。一个超临界奇异的Hopf分岔产生的确定性模型中的II型兴奋性。我们探讨的随机性的影响，在兴奋的制度，导致其确定性的对应物没有预期的动态。随机模型允许几种噪声驱动的混合模式振荡，捕捉两个周期之间的人口爆发的中间动态。根据噪声的强度，猎物种群表现出中等到高振幅的波动（分别与中度或重度爆发有关）。我们将这些波动分类为孤立的或间歇性的或集群取决于他们的复发。我们研究了随机变量$N$的分布，代表连续尖峰之间的小振荡的数量，作为噪声强度和距离的函数的Hopf分叉。$N$的分布是“渐近几何”的，相应的参数与次随机马尔可夫链的主特征值有关。最后，随机模型转化为它的“正常形式”，这是用来获得重复爆发的概率估计。
英文摘要：
The effect of stochasticity, in the form of Gaussian white noise, in a predator-prey model with two distinct time-scales is presented. A supercritical singular Hopf bifurcation yields a Type II excitability in the deterministic model. We explore the effect of stochasticity in the excitable regime, leading to dynamics that are not anticipated by its deterministic counterpart. The stochastic model admits several kinds of noise-driven mixed-mode oscillations which capture the intermediate dynamics between two cycles of population outbreaks. Depending on the strength of noise, the prey population exhibits intermediate to high-amplitude fluctuations (related to moderate or severe outbreaks respectively). We classify these fluctuations as isolated or intermittent or as clusters depending on their recurrences. We study the distribution of the random variable $N$, representing the number of small oscillations between successive spikes, as a function of the noise intensity and the distance to the Hopf bifurcation. The distribution of $N$ is "asymptotically geometric" with the corresponding parameter related to the principal eigenvalue of a substochastic Markov chain. Finally, the stochastic model is transformed into its "normal form" which is used to obtain an estimate of the probability of repeated outbreaks.
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