Formulation and Analysis of Deterministic Models of Predation Among Acarine Populations

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

Gardner The investigator develops and analyzes models of persistent behavior of locally self-annihilating populations of predatory and herbivorous mites. The project is stimulated by Huffaker's classic experiment, which provided the first demonstration of the mediating role of spatial processes in certain ecological interactions. Huffaker noted that synchronies in the phases of population cycles of localized aggregates within host plants of herbivorous mites and of the mites that preyed upon them occasionally spread over an entire area, but that the synchrony can be broken by demographic and environmental stochasticity. He conjectured that the complementary regularizing mechanisms that generate synchrony, and the complementary stochastic mechanisms that generate asynchrony, together generate the complex waves and patterns seen in these predator-prey interactions in nature. Existing stochastic patch models have been formulated that successfully simulate complex, and evidently chaotic, spatio-temporal patterns that have typically accompanied persistent dynamics. Accordingly, such models have difficulty in providing reliable predictions of behavior due to the large variance seen in replicate data sets. This substantially limits their practical application to the biological control of crop pests. In previous work, the investigator formulated a family of deterministic models, including an idealized paradigm defining a class of reaction-diffusion systems. He studies two general issues relating to the analysis and practical applications of the reaction-diffusion system and related models. First, he investigates the presence of several distinct families of stable waves and spatial patterns of these equations, their instabilities and bifurcations, and the presence of exotic wave-like patterns. By combined analytical and approximate methods, he describes theoretical mechanisms by which such unstable waves can form the organizing center of spatio-temporal chaos, while in other regimes, complex but non-chaotic, stable spatio-temporal patterns appear spontaneously. A second topic concerns the derivation of the model, which is based on experiment. This addresses the manner in which the reaction-diffusion model simulates stochasticity through postulated low-density thresholds in the growth rates. The investigator studies hybrid numerical procedures addressing the crucial issue of parameter estimation, which can be problematic both for stochastic patch models and for the reaction-diffusion model, but at opposite spatio-temporal scales. For example, the low-density thresholds involve the micro-local dynamics of a small number of individuals, whereas large-scale emigration over large regions is difficult to measure in a laboratory. However, simple stable waves may suggest one instance in which this may be possible. The investigator examines this in numerical simulations with patch models, which may form the basis for new and simpler experimental designs that can detect such behavior. Together, these two approaches present a unified picture of how "dynamic stochasticity" driven by the presence of certain unstable waves and spatio-temporal patterns is theoretically linked to parameters of population change governing behavior at the smallest micro-local scales, and thus the generation of asynchrony through demographic and environmental stochasticity. Plant-eating mites, and the mites that eat them in turn, often show cycles of growth and decline in numbers that are closely related. Sometimes the populations decline rapidly to near-extinction levels. But in a classic experiment Huffaker and colleagues showed that these strongly coupled cycles can be moderated by small changes in demographic or geographic factors, leading to a long-term survival of both species that shows itself in patterns, in time and space, of the numbers of predator and prey mites. Of course, when the prey mite is eating a valuable crop, one wants both mite populations to go extinct. So issues of how mite populations vary together are of considerable practical importance in the biological control of crop pests. The economic impact of phytophageous species is considerable, and although the use of predatory mites as a biological control is widespread in certain local agricultures, many agricultural enterprises rely on chemical pest controls. Chemical controls provide a short-term solution, but have many damaging long-term effects. However, the risks and uncertainties of biological pest control are difficult to quantify due to their complex and unpredictable patterns of behavior. The investigator's numerical simulations indicate the presence of many different kinds of chaos, some of which achieve better levels of suppression of both herbivore populations and crop damage. The project explores new theoretical and computational tools that may find practical application in the biological control of crop pests and, in other contexts, in inoculative control of certain exotic species.