论文标题
随机逻辑图的长期平均值
Long-term Averages of the Stochastic Logistic Map
论文作者
论文摘要
逻辑图是文献中对非线性差异方程式进行了很好的研究，用于模拟某些人群中的自限值。众所周知，在某些规律性条件下，随机逻辑图，该参数根据指定的分布而变化，具有独特的不变分布。在这些情况下，我们可以将随机系统的长期行为与以平均参数值进行评估的确定性系统的长期行为进行比较。在这里，我们研究了随机逻辑方程的平均值与在参数的期望值下确定性逻辑方程的轨道平均值之间的关系。我们正式证明，在某些情况下，增加噪声对人群有益，从某种意义上说，它会增加平均值，而对于其他参数范围，这是有害的。最后提出了基于数值证据的猜想。
The logistic map is a nonlinear difference equation well studied in the literature, used to model self-limiting growth in certain populations. It is known that, under certain regularity conditions, the stochastic logistic map, where the parameter is varied according to a specified distribution, has a unique invariant distribution. In these cases we can compare the long-term behavior of the stochastic system with that of the deterministic system evaluated at the average parameter value. Here we examine the relationship between the mean of the stochastic logistic equation and the mean of orbits of the deterministic logistic equation at the expected value of the parameter. We formally prove that, in some cases, the addition of noise is beneficial to the populations, in the sense that it increases the mean, while for other ranges of parameters it is detrimental. A conjecture based on numerical evidence is presented at the end.
