Spatial context in predator-prey systems has proven to have important dynamical consequences. Instabilities and spatial pattern formation driven by diffusion (Turing pattern formation) have been extensively observed and theorized on, but empirical examples of Turing pattern formation in ecological systems are few. In this study we construct and analyze a reaction-diffusion equation model of the aphid species Aphis helianthi under predation by two species of ladybugs: Coccinella septempunctata and Hippodamia convergens. The structure and parametrization of the model is entirely field derived and in analysis of model output it is compared extensively to field observations. This system fits the well known framework for diffusive instability and pattern formation: an activator-inhibitor system in which the inhibitor (predator) diffuses substantially faster than the activator (prey). Theory predicts that under these conditions the inhibitor will fail to strike a normal equilibrium with the activator; rather diffusing away from activator outbreaks too quickly to contain them, subsequently over-inhibiting the surrounding lower densities of activator (undermatching). This usually results in a patchy, bimodal distribution of prey resulting from cubic density dependence driven by undermatching. Aphid population distribution in the field is clearly bimodal and patchy. We looked for several indications of diffusive instability in field data; bimodality, cubic density dependence, and undermatching were all found. The focus of this paper is on a mathematical model we developed from field data to gain insight into the workings of the system. I found the model matched field data very well and corroborated the hypothesized functioning of a diffusive instability. I explored the role of self attraction (aggregation) among ladybugs. Aggregation is not considered a hallmark of diffusive instability but in this case it created some preytaxis in ladybugs (allowing aphids to act as an activator). Preytaxis by aggregation is slow though, which allowed some aphid populations to avoid detection long enough to reach the high attractor of cubic density dependence. Finally I considered the nature of space in our system. Although our model is constructed in Euclidean space it demonstrates some features of a network. Network structured systems manifest Turing patterns primarily as bimodal distributions. They also facilitate understanding of ladybug behavior and may increase efficiency of computer model execution.
Includes bibliographical references.