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).