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).