Progress report on "QEIB: Spatially-distributed population models with external forcing and spatial control (DMS-0110920)" Louis Gross and Suzanne Lenhart University of Tennessee March 5, 2003 This project involves several sub-projects all of which have as a central theme the problem of spatial control for natural systems: what to do, where to do it, how to do it and how to assess whether or not the control is successful. The project involves components from very theoretical ones, requiring the development of new mathematics, to ones that intimately connect modeling methods to field data in order to address concerns of those involved in day-to-day management of natural systems. The techniques range from analytical mathematical methods for ordinary and partial differential equations and numerical methods for control of integro- difference equations, to complex computer models that project the behavior of thousands of individuals within a population across space and time. The main project components are: 1. Optimal control for integro-difference equations. These equations have been developed over the past decade as appropriate models for populations in which there are discrete, non-overlapping generations (appropriate for many insect species) which disperse in a continuous space. They are an alternative to patch-models in which space is not continuous but broken into "patches" and are more appropriate for wind-dispersed species and as well as those with some active transport mechansism (flight in insects). There is essentially no body of mathematics available for optimal control of these equations and we have begun to develop this theory. This involves finding the optimal spatial pattern of control (e.g. harvesting, pesticide application, etc.) in order to maximize or minimize some criteria (e.g. population size). We have focused initially on problems related to bioeconomics in which there is an explicit monetary cost associated with the control as well as a benefit associated with the population distribution in space. 2. Spatial control of plant/pathogen systems via intercropping. These equations link ordinary-differential equation models for plant growth to reaction-diffusion partial differential equations for a pathogen which disperses in space and infects the plants, reducing their growth rate. This is the intercropping problem in agricultural systems, to which we have added the control problem of choosing the appropriate spatial pattern for planting different varieties of crops which have differential resistances to a pathogen. We have been developing the mathematical and numerical methods needed to solve the optimization problem when there is a trade-off in plant yield associated with enhanced resistance to the pathogen. 3. Spatial control to manage the spread of antibiotic resistance. A wide variety of human bacterial diseases have the ability to evolve resistance to commonly used antibiotics, thereby reducing the efficacy of these drugs to cure the infection and requiring the continuing development of new antibiotics in order to cure the infections. A suggested method to reduce the need for new antibiotics is drug rotation, in which several different antibiotics are rotated in application. There is no theory available for the potential benefits of such a rotation strategy, in part because of the difficulty in accounting for spatial movement of individuals carrying resistant bacteria. We have developed a modeling approach for this and applied it to tuberculosis in spatial regions associated with countries in Europe. We have shown for this case that although there are indeed benefits to carrying out a spatially-explicit pattern of drug rotation, the magnitude of these benefits is small relative to that obtainable through an optimal spatially-uniform rotation strategy. The solution of this problem required extensive use of parallel numerical computation. 4. Spatial control in an individual-based model for black-bears. Individual-based models track the behavior of a population through time and space by following the movement, growth, behavior and reproduction of the individuals which make up the population. Through close collaboration with several field biologists with extensive knowledge of black-bear biology in the southern Appalachians, we have developed an analytic model (in a meta-population formulation in which there are a few discrete types of patches available to bears) which allows us to apply optimal control theory to analyze the impacts of different harvesting strategies on the bear population. In order to analyze in more detail the potential for human-bear interactions, we have also developed an individual-based model that accounts for the explicit location of the various bear preserves (in which no hunting is allowed) throughout the southern Appalachians. We have been applying this model to determine the relative impacts of alternative spatial patterning of the preserves, with an objective of reducing the potential for harmful human-bear interactions. 5. Spatial control of a harmful invasive plant species. Numerous non-native plant species have devastating effects on natural systems by reducing biodiversity and altering the natural species composition of ecological communities. One of these is Old World Climbing Fern which has recently been invading the tree islands which exist within the matrix of fresh-water marshes in South Florida. These tree islands have unique flora and fauna and the invading Fern greatly reduces the species diversity on these tree islands. We have been developing models which would project the potential benefits of active management of the hydrologic conditions in the Everglades for reducing the spread of this invasive species. This is being developed in close collaboration with biologists who are collecting basic field data for this species, as well as managers involved in attempting to actively control the spread of the invasive through cutting and herbicide applications. A wide variety of presentations have been given at professional meetings about the above projects. Additionally, several papers are either in submission for publication to scientific journals, or are currently being written associated with the above. The key personnel associated with these projects include Drs. Gross and Lenhart (Principal Investigators), Drs. Jon Cline, Holly Gaff and Hem Raj Joshi (Post-doctoral researchers), and Scott Duke-Sylvester and Rene' Salinas (Graduate students in Ecology and Evolutionary Biology and Mathematics. Questions about any aspect of this project may be directed to Dr. Louis Gross, Departments of Ecology and Evolutionary Biology and Mathematics, University of Tennessee, Knoxville, TN 37996-1610, gross@tiem.utk.edu, 865-974-4295 or 865-974-3065 (Secretary).