Pascual Mercedes, Roy Manojit, Laneri Karina
Department of Ecology and Evolutionary Biology, University of Michigan, Ann Arbor, MI USA ; HHMI, and Department of Ecology and Evolutionary, University of Michigan, Ann Arbor, MI 48109-1048 USA.
IC3, Institut Catala de Ciencies del Clima, Barcelona, Spain.
Theor Ecol. 2011;4(2):211-222. doi: 10.1007/s12080-011-0116-2. Epub 2011 Mar 11.
Simple temporal models that ignore the spatial nature of interactions and track only changes in mean quantities, such as global densities, are typically used under the unrealistic assumption that individuals are well mixed. These so-called mean-field models are often considered overly simplified, given the ample evidence for distributed interactions and spatial heterogeneity over broad ranges of scales. Here, we present one reason why such simple population models may work even when mass-action assumptions do not hold: spatial structure is present but it relates to global densities in a special way. With an individual-based predator-prey model that is spatial and stochastic, and whose mean-field counterpart is the classic Lotka-Volterra model, we show that the global densities and densities of pairs (or spatial covariances) establish a at the stationary state and also in their transient approach to this state. This relationship implies that the dynamics of global densities can be written simply as a function of those densities alone without invoking pairs (or higher order moments). The exponents of the bi-power law for the predation rate exhibit a remarkable robustness to changes in model parameters. Evidence is presented for a connection of our findings to the existence of a critical phase transition in the dynamics of the spatial system. We discuss the application of similar modified mean-field equations to other ecological systems for which similar transitions have been described, both in models and empirical data. The online version of this article (doi:10.1007/s12080-011-0116-2) contains supplementary material, which is available to authorized users.
简单的时间模型忽略了相互作用的空间性质，仅跟踪平均量的变化，如全局密度，通常是在个体充分混合这一不现实的假设下使用的。鉴于在广泛的尺度范围内存在分布式相互作用和空间异质性的充分证据，这些所谓的平均场模型通常被认为过于简化。在这里，我们给出一个原因，说明即使质量作用假设不成立，这种简单的种群模型仍可能起作用：存在空间结构，但它以一种特殊的方式与全局密度相关。通过一个基于个体的空间随机捕食者 - 猎物模型，其平均场对应模型是经典的洛特卡 - 沃尔泰拉模型，我们表明全局密度和成对密度（或空间协方差）在稳态以及它们向该状态的瞬态过程中建立了一种关系。这种关系意味着全局密度的动态可以简单地写成仅关于那些密度的函数，而无需涉及成对（或更高阶矩）。捕食率的双幂律指数对模型参数变化表现出显著的稳健性。给出了证据表明我们的发现与空间系统动态中存在临界相变有关。我们讨论了类似的修正平均场方程在其他生态系统中的应用，这些生态系统在模型和实证数据中都有类似的转变描述。本文的在线版本（doi:10.1007/s12080 - 011 - 0116 - 2）包含补充材料，授权用户可以获取。
