Hal L. Smith - US grants

This project will study various classes of differential equations which generate dynamical systems with some monotonicity properties which limit the complexity of their asymptotic behavior. Included are certain classes of ordinary differential equations, delay differential equations and reaction-diffusion systems with time delays. The goal of the research is to describe global qualitative features of the flow. For finite dimensional systems we focus on existence and nonexistence of nontrivial periodic orbits and associated invariant manifolds. For infinite dimensional systems, the principal investigator will focus on convergence of solutions to equilibrium, invariance and comparison type results. Systems of the type considered here occur frequently in the applied literature and particularly in mathematical models in biology. Results with significant applications are expected.

9300974 Smith Ordinary differential equations, equations with time delays, reaction-diffusion systems containing time delays, and hyperbolic partial differential equations with nonlocal boundary conditions arise naturally in population biology, epidemiology, and in neural modeling. They are commonly used in the biological modeling of populations. The focus of this project is on means of determining the long-term behavior of solutions of this wide variety of differential equations. From the dynamical systems point of view, the investigator aims to find and exploit common features of the dynamics exhibited by this wide class of systems in order to determine asymptotic behavior. Monotonicity, the preservation of a partial order relation on the space of initial states by the dynamics, is one such feature shared by many systems that model competing populations of organisms, neurons, etc. More realistic models often take the form of complex hyperbolic systems modeling the time-evolution of densities of populations structured by size, age, stored nutrient, etc. Often, in their original form, these systems are mathematically intractable and means must be found to transform them to simpler, more well-understood, dynamical systems. Finding transformations of these complicated systems to simpler ones is an aim of the present study. Mathematical modeling of interacting populations of organisms, neurons, etc., is an important area of current interest. The goal is to try to explain observations of complex behavior in real populations by ignoring all but the "essential" features of the interactions and constructing simplified models of these systems that are amenable to mathematical analysis. If too much biology is ignored in the modeling then the models do not mirror the interactions of real populations; if too little is ignored, the models become so complicated that present knowledge does not permit successful analysis. Progress in applying the rapidly d eveloping field of dynamical systems to biological systems has steadily pushed this fine line between biologically irrelevant models and mathematically intractable ones forward, so that ever more biology can be incorporated into the models, making them more realistic to biologists, better mirrors of real populations, while yet remaining tractable. Essentially, the aim of this project is to push further in this direction. ***

Smith
0107160
The investigator develops mathematical methods that exploit
the special properties inherent in dynamical systems arising in
the biological sciences and applies them to specific problems.
The main focus is on the long-term behavior of the dynamics after
transient effects have disappeared. The work on biofilms focuses
on systems of advection-diffusion equations coupled to ordinary
differential equations with coupled nonlinear boundary conditions
in piecewise-smooth domains. Aside from well-posedness and the
existence of multiple steady states, which are nontrivial
questions, the important biological information comes from key
eigenvalues for associated nonstandard eigenvalue problems and
their dependence on parameters. Both mathematical analysis and
numerical simulations are required for an understanding. The work
on the paradox of the plankton focuses on the classical model of
n-species exploitative competition for k essential
(non-substitutable) nutrients. If n is larger than k there can be
no equilibrium coexistence of all n species so one must seek
oscillatory coexistence. The investigator exploits certain
heteroclinic structures (cycles of equilibria) that arise
naturally in such systems in order to seek periodic coexistence
solutions. The theory of competitive and cooperative systems can
play a significant role. Abstract dynamical systems theory comes
into play in the work on robust persistence. Specifically,
understanding chain recurrent sets and Morse decompositions of
the boundary dynamics (where one or more species are absent) are
key to showing that persistence for a dynamical system is robust
to perturbation of that system or of its parameters.
Dynamical systems that arise naturally in many areas in the
biological sciences often have special features not shared by
systems arising in the physical sciences. Broadly speaking, the
investigator develops mathematical methods that exploit these
special properties and applies them to specific problems. The
main focus is on the long-term behavior of the dynamics after
transient effects have disappeared. Progress in this area can
help to answer important biological questions that are subject to
mathematical modeling, such as whether an infectious disease
becomes endemic in a population or becomes extinct, or which
species in an ecosystem can survive and which will become
extinct. A substantial effort is devoted to understanding the
dynamics of mathematical models in three areas of the
biosciences. The first is microbial growth and competition in
environmental settings where biofilms may form on surfaces.
Biofilms are of great importance in the health sciences and food
industry where their formation typically results in negative
outcomes. They are responsible for food and water contamination,
dental caries and periodontal disease, and the contamination of
medical implants. The investigator studies under what conditions
biofilms may form and what bacterial densities can be expected.
The second is a fundamental issue in population biology, the
so-called "paradox of the plankton": why can so many plankton
species (and, more generally, species of other taxa) be supported
by so few limiting resources? The investigator provides
mathematically rigorous results for the existence of oscillatory
coexistence states of the relevant mathematical models when the
number of species exceeds the number of resources, as is typical
in natural environments. The third is a more robust theory of
persistence (also called permanence) for dynamical systems
arising in population dynamics. In its simplest form, this theory
seeks not to determine the global behavior of solutions of a
dynamical system representing interacting populations but rather
to answer the more basic question: What species are present at
the end of the day? While past research in this area has
primarily ignored the fact that population models are only
approximately correct, the investigator seeks to provide answers
to the basic question that hold true for all small perturbations
of the model equations.

An interdisciplinary team of investigators carry out an
undergraduate training initiative at Arizona State University.
The training plan intimately combines new cross-disciplinary
courses and summer research programs. The former are constructed
to allow maximal participation among undergraduate cadres, and
facilitate life science majors to achieve a minor in mathematics,
and, likewise, mathematics majors to enrich their education with
a minor in bioscience. The summer research program is a
competitive enterprise involving at least eight ASU faculty
members from life sciences, mathematics, and biophysics.
Research projects span modeling of ecological and evolutionary
processes through the new lens of stoichiometric constraints,
bio-economics, chemostat theory, and modeling of visual
perception.

This project has potential to make broad impact in both
local and global education environs. Regarding the former, the
ASU UBM team is truly interdisciplinary, with members in
mathematics, biology and biophysics, exceptionally well suited
for interdisciplinary training for undergraduates in biological
and mathematical sciences. Its collaborative efforts can provide
undergraduate and graduate students of diverse ethnic/racial
backgrounds with first-hand educational experience in
cross-disciplinary communication and exploration. As for global
impact, the proposed holistic approach (involving mathematical
biology courses at various levels and summer research projects)
in mathematical biology training can vertically integrate all the
components in the ASU education system. It is therefore expected
that this proposed program may yield many invaluable lessons to
serve mathematical bioscience education and research nationwide,
enriching the experience for the next generation of students in
this integrative and interdisciplinary scientific endeavor.

The project has both a theoretical component and mathematical modeling component. On the theoretical side, it will contribute to the mathematical theory of persistence, an important tool for the analysis of biological models as it provides a mathematically rigorous method to establish the long term persistence or viability of a population. Presently, the theory is difficult to apply to discrete-time dynamical systems arising in epidemiology and population dynamics because of the complexity of (boundary) attractors for discrete-time systems (e.g., the standard quadratic map). Recently, the theory of Lyapunov exponents has been applied to detect when a boundary invariant set, representing a "resident regime" is normally repelling, meaning it is invadable by a non-resident. One aspect of the project is to further develop this approach and to develop methods of computing/estimating the key normal Lyapunov exponents and their dependence on system parameters. The newly developed tools will be applied to mathematical models of a fungal disease model in amphibian populations. The project aims to improve current state-of-the-art mathematical models of in-vivo bacterial infections by including more realistic immune system responses, antibiotic treatment and the development of resistance to antibiotic, and examining appropriate antibiotic dosing strategies. A parallel effort will aim to improve mathematical models of virus disease by exploring the effects of multiple viral infections of host target cells.

Many important questions in the life sciences center on whether or not a biological species or set of species can survive indefinitely in given environment or whether they will become extinct. Issues of biological diversity revolve around the question of which species can or cannot survive in an (possibly human-altered) environment. If a disease or parasite is introduced into a population, can it survive and become endemic in the host or will it die out or become rare? Will it drive the host population to extinction as fungal diseases appear to be doing to amphibian populations? Similar questions arise when an inoculum of bacteria or virus gains entry into a host animal or plant: can it exploit its host and initiate a chronic disease or will it be removed by a host immune response? Currently, there are few guiding principles to decide these questions but mathematical modeling and analysis can and has played a role. Motivated by these important biological questions, a mathematical theory (called persistence theory) has been developing in recent decades to provide answers. The project will contribute to this theory by sharpening the available mathematical tools and applying them to important biological problems. On the practical side, the new tools should help to predict long term persistence or extinction in particular models. A focus of the project is to apply newly developed tools to study mathematical models of disease within human and animal populations. Models of the fungal disease Chytridiomycosis, an infectious disease of amphibians caused by the chytrid fungus, implicated in the mass die-offs and species extinction of frogs in many areas of the world, will be studied. Another arena of important applications are disease processes within a given animal or plant. Mathematical modeling of in vivo disease processes is a relatively new field holding great promise due to the difficulty of real-time monitoring of these processes. Prior work aimed at creating more realistic mathematical models of generic bacterial infections of mammalian tissues including host immune response and antibiotic therapy will be extended to include known pathogen heterogeneity in virulence and susceptibility to antibiotic treatment, and alternative antimicrobial dosing strategies. The proposed extensions should lead to better understanding of the infection and treatment processes, allowing computer simulation of processes which are difficult to monitor in vivo. Mathematical modeling of virus infections have also proved to be useful, most notably for understanding HIV replication within the human host. Mathematical modeling can predict and confirm features of virus-host cell dynamics that are difficult to monitor in animal models. The investigator will examine effects of multiply infected host cells on disease dynamics.