Diaz-Ruelas Alvaro, Jeldtoft Jensen Henrik, Piovani Duccio, Robledo Alberto
Instituto de Física, Universidad Nacional Autónoma de México, Ciudad Universitaria, Ciudad de México 04510, Mexico.
Centre for Complexity Science and Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom.
Chaos. 2016 Dec;26(12):123105. doi: 10.1063/1.4968207.
It is well known that low-dimensional nonlinear deterministic maps close to a tangent bifurcation exhibit intermittency and this circumstance has been exploited, e.g., by Procaccia and Schuster [Phys. Rev. A 28, 1210 (1983)], to develop a general theory of 1/f spectra. This suggests it is interesting to study the extent to which the behavior of a high-dimensional stochastic system can be described by such tangent maps. The Tangled Nature (TaNa) Model of evolutionary ecology is an ideal candidate for such a study, a significant model as it is capable of reproducing a broad range of the phenomenology of macroevolution and ecosystems. The TaNa model exhibits strong intermittency reminiscent of punctuated equilibrium and, like the fossil record of mass extinction, the intermittency in the model is found to be non-stationary, a feature typical of many complex systems. We derive a mean-field version for the evolution of the likelihood function controlling the reproduction of species and find a local map close to tangency. This mean-field map, by our own local approximation, is able to describe qualitatively only one episode of the intermittent dynamics of the full TaNa model. To complement this result, we construct a complete nonlinear dynamical system model consisting of successive tangent bifurcations that generates time evolution patterns resembling those of the full TaNa model in macroscopic scales. The switch from one tangent bifurcation to the next in the sequences produced in this model is stochastic in nature, based on criteria obtained from the local mean-field approximation, and capable of imitating the changing set of types of species and total population in the TaNa model. The model combines full deterministic dynamics with instantaneous parameter random jumps at stochastically drawn times. In spite of the limitations of our approach, which entails a drastic collapse of degrees of freedom, the description of a high-dimensional model system in terms of a low-dimensional one appears to be illuminating.
众所周知，接近切分岔的低维非线性确定性映射表现出间歇性，例如，普罗卡西亚和舒斯特[《物理评论A》28, 1210 (1983)]利用这种情况发展了1/f谱的一般理论。这表明研究高维随机系统的行为在多大程度上可以用这种切分映射来描述是很有趣的。进化生态学的缠结本质（TaNa）模型是进行此类研究的理想候选者，它是一个重要的模型，因为它能够再现宏观进化和生态系统的广泛现象学。TaNa模型表现出强烈的间歇性，让人联想到间断平衡，并且与大规模灭绝的化石记录一样，该模型中的间歇性是非平稳的，这是许多复杂系统的典型特征。我们推导了控制物种繁殖的似然函数演化的平均场版本，并找到了一个接近相切的局部映射。通过我们自己的局部近似，这个平均场映射只能定性地描述完整TaNa模型间歇性动力学的一个阶段。为了补充这一结果，我们构建了一个由连续切分岔组成的完整非线性动力系统模型，该模型在宏观尺度上产生类似于完整TaNa模型的时间演化模式。在这个模型中产生的序列中，从一个切分岔到下一个切分岔的切换本质上是随机的，基于从局部平均场近似中获得的标准，并且能够模仿TaNa模型中物种类型和总人口的变化集。该模型将完全确定性动力学与在随机抽取的时间的瞬时参数随机跳跃相结合。尽管我们的方法存在局限性，即导致自由度的急剧减少，但用低维模型系统来描述高维模型系统似乎很有启发性。
