Petrovskii Sergei, Li Bai-Lian, Malchow Horst
Shirshov Institute of Oceanology, Russian Academy of Science, Nakhimovsky Prospekt 36, Moscow 117218, Russia.
Bull Math Biol. 2003 May;65(3):425-46. doi: 10.1016/S0092-8240(03)00004-1.
The need to study spatio-temporal chaos in a spatially extended dynamical system which exhibits not only irregular, initial-value sensitive temporal behavior but also the formation of irregular spatial patterns, has increasingly been recognized in biological science. While the temporal aspect of chaotic dynamics is usually characterized by the dominant Lyapunov exponent, the spatial aspect can be quantified by the correlation length. In this paper, using the diffusion-reaction model of population dynamics and considering the conditions of the system stability with respect to small heterogeneous perturbations, we derive an analytical formula for an 'intrinsic length' which appears to be in a very good agreement with the value of the correlation length of the system. Using this formula and numerical simulations, we analyze the dependence of the correlation length on the system parameters. We show that our findings may lead to a new understanding of some well-known experimental and field data as well as affect the choice of an adequate model of chaotic dynamics in biological and chemical systems.
在不仅展现出不规则的、对初始值敏感的时间行为，还会形成不规则空间模式的空间扩展动力系统中，研究时空混沌的必要性在生物科学中已日益得到认可。虽然混沌动力学的时间方面通常由主导李雅普诺夫指数来表征，但空间方面可通过关联长度来量化。在本文中，我们利用种群动力学的扩散 - 反应模型，并考虑系统相对于小的非均匀扰动的稳定性条件，推导出了一个“固有长度”的解析公式，该公式与系统关联长度的值似乎非常吻合。利用这个公式和数值模拟，我们分析了关联长度对系统参数的依赖性。我们表明，我们的发现可能会引发对一些著名实验和现场数据的新理解，并影响生物和化学系统中混沌动力学适当模型的选择。
