Two problems in population dynamics are addressed in a slow or rapid periodic environment. We first obtain a Taylor expansion for the probability of non-extinction of a supercriticial linear birth-and-death process with periodic coefficients when the period is large or small. If the birth rate is lower than the mortality for part of the period and the period tends to infinity, then the probability of non-extinction tends to a discontinuous limit related to a "canard" in a slow-fast system. Secondly, a nonlinear S-I-R epidemic model is studied when the contact rate fluctuates rapidly. The final size of the epidemic is close to that obtained by replacing the contact rate with its average. An approximation of the correction can be calculated analytically when the basic reproduction number of the epidemic is close to 1. The correction term, which can be either positive or negative, is proportional to both the period of oscillations and the initial fraction of infected people.

中文翻译：

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环境友好的人口与环境发展的双重标准。

在缓慢或快速的周期性环境中解决了人口动态中的两个问题。我们首先获得泰勒展开式，用于当周期较大或较小时具有周期系数的超临界线性生灭过程没有灭绝的概率。如果部分时期的出生率低于死亡率，并且该时期趋于无穷大，则不灭绝的可能性倾向于达到与慢速系统中“鸭嘴”有关的不连续极限。其次，研究了接触率快速波动时的非线性SIR传染病模型。该流行病的最终规模接近通过用平均接触率代替接触率获得的规模。当流行病的基本繁殖次数接近1时，可以通过分析计算出修正的近似值。修正项，