Mathematical Sciences: Nonlinear Self-focussing as a Deterministic Mechanism for Generating Spatial Complexity in Ecosystems

研究人员和他的同事们研究种群的空间动态，开发和分析结合生物数据的模型，并调查种群个体聚集和分散的方式。聚合通过反馈导致空间复杂性。当个体用于聚合的线索依赖于个体数量时，反馈是非线性的。在允许移动的人口模型中，聚合反馈导致非线性自聚焦，即在小尺度上组织或“聚焦”人口群体。空间相关尺度的减小使扩散格局更加复杂，导致空间结构的复杂化。在数学上，模式形成的问题与入侵波的结构或生态状态之间的“前线”有关。当结构已知时，对前缘后面或前缘沿线状态的稳定性分析决定了复杂性是如何产生的。大量的数学工作集中在预测生物问题中入侵波的速度和形状上。不幸的是，已知的技术只解决低维系统（一个或两个因变量）的入侵问题，而不适合一般的生态应用。因此，在多变量系统中预测前传播的技术是研究的中心。前解或入侵波是入侵边界稳定性分析的基础，它预测了曲率不稳定性如何在入侵边界产生复杂性。这些数学努力的结果用于理解山松甲虫/松系统的复杂性生成。数学模型的结果与已知虫害的大小、形状和结构数据进行了比较。简而言之，这个想法是确定一个物种的集中或聚集在多大程度上决定了生态系统模式的自然形成。物质的聚集或聚集是自然发生的，并且可以用数学方法预测。研究人员研究了这种机制是如何在生态系统中创造复杂性的，并研究了如何确定有多少生态复杂性是自我产生的，而不是由环境的复杂性产生的。研究山松甲虫/洛奇波勒松生态系统是因为空间格局的元素（被攻击和被杀死的树木）是独特的，易于测量。此外，了解松甲虫侵染过程的可预测方面对于了解这种害虫在西部森林中所起的作用非常重要，包括甲虫杀死的松树与森林火灾之间的关系。

Powell The investigator and his colleagues study the spatial dynamics of populations, developing and analyzing models that incorporate biological data and investigating the way individuals of a population aggregate and disperse. Aggregation leads to spatial complexity through feedback. The feedback is nonlinear when the cues individuals use to aggregate depend on the number of individuals. In a population model that allows for movement, aggregative feedback causes nonlinear self-focussing, the organization or `focussing' of population groups on small scales. Decreases in the scale of spatial correlation make the dispersal pattern more complex, leading to complicated spatial structure. Mathematically, issues of pattern formation are related to the structure of invasion waves or `fronts' between ecological states. When the structure is known, stability analyses of states behind the front or along the front determine how complexity is generated. A body of mathematical work has focussed on predicting the speed and shape of invasion waves in biological problems. Unfortunately, known techniques only address the invasion problem in systems of low dimensionality (one or two dependent variables), and do not lend themselves to general ecological application. Techniques to predict front propagation in systems of many variables are therefore at the center of the research. Front solutions, or waves of invasion, are the basis for stability analyses at invasion boundaries, which predict how curvature instabilities produce complexity at the invasion boundaries. Results of these mathematical efforts are used to understand complexity generation in the mountain pine beetle/pine system. Results from the mathematical model are compared with data on size, shape, and structure of known infestations. In short, the idea is to determine how much focussing, or clumping, of a species determines the natural formation of patterns in ecosystems. Aggregation, or clumping, of spec ies occurs naturally, and can be predicted mathematically. The researchers examine how this mechanism creates complexity in ecosystems, and study how to determine how much of ecological complexity is self-generating, as opposed to being generated by the complexity of the environment. The Mountain Pine Beetle/Lodgepole Pine ecosystem is studied because elements of the spatial pattern (attacked and killed trees) are distinctive and easy to measure. Moreover, understanding the predictable aspects of the course of pine beetle infestations is important to understanding the role this pest plays in western forests, including the relationship between beetle-killed pines and forest fire.