Biophysics and Evolutionary Dynamics Group, Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany.
Proc Natl Acad Sci U S A. 2011 Feb 1;108(5):1783-7. doi: 10.1073/pnas.1013529108. Epub 2010 Dec 27.
Traveling waves are ubiquitous in nature and control the speed of many important dynamical processes, including chemical reactions, epidemic outbreaks, and biological evolution. Despite their fundamental role in complex systems, traveling waves remain elusive because they are often dominated by rare fluctuations in the wave tip, which have defied any rigorous analysis so far. Here, we show that by adjusting nonlinear model details, noisy traveling waves can be solved exactly. The moment equations of these tuned models are closed and have a simple analytical structure resembling the deterministic approximation supplemented by a nonlocal cutoff term. The peculiar form of the cutoff shapes the noisy edge of traveling waves and is critical for the correct prediction of the wave speed and its fluctuations. Our approach is illustrated and benchmarked using the example of fitness waves arising in simple models of microbial evolution, which are highly sensitive to number fluctuations. We demonstrate explicitly how these models can be tuned to account for finite population sizes and determine how quickly populations adapt as a function of population size and mutation rates. More generally, our method is shown to apply to a broad class of models, in which number fluctuations are generated by branching processes. Because of this versatility, the method of model tuning may serve as a promising route toward unraveling universal properties of complex discrete particle systems.
行波在自然界中无处不在，控制着许多重要动力学过程的速度，包括化学反应、传染病爆发和生物进化。尽管它们在复杂系统中起着基础性的作用，但行波仍然难以捉摸，因为它们通常受到波峰处罕见波动的控制，而这些波动迄今为止一直难以进行任何严格的分析。在这里，我们表明通过调整非线性模型的细节，可以精确地求解出噪声行波。这些调整后的模型的矩方程是封闭的，具有类似于确定性近似的简单解析结构，并补充了一个非局部截止项。截止项的特殊形式塑造了行波的噪声边缘，对于正确预测波速及其波动至关重要。我们使用简单微生物进化模型中出现的适应性波的例子来说明和基准测试了我们的方法，这些模型对数量波动非常敏感。我们明确地展示了如何调整这些模型来考虑有限的种群大小，并确定种群随着种群大小和突变率的变化而适应的速度。更一般地说，我们的方法被证明适用于广泛的模型类，其中数量波动是由分支过程产生的。由于这种多功能性，模型调整方法可能成为揭示复杂离散粒子系统普遍性质的有前途的途径。
