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

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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.