1986 — 1988 
Smith, Hal 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Mathematical Sciences: Nonlinear Competitive and Cooperativedynamical Systems @ Arizona State University 
0.915 
1988 — 1993 
Smith, Hal 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Mathematical Sciences: Monotone Dynamical Systems @ Arizona State University
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 reactiondiffusion 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.

0.915 
1993 — 1998 
Smith, Hal 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Mathematical Sciences: Dynamical Systems in Biology @ Arizona State University
9300974 Smith Ordinary differential equations, equations with time delays, reactiondiffusion 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 longterm 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 timeevolution 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 wellunderstood, 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. ***

0.915 
1997 — 2008 
Smith, Hal 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Dynamical Systems in Biology @ Arizona State University
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 longterm behavior of the dynamics after transient effects have disappeared. The work on biofilms focuses on systems of advectiondiffusion equations coupled to ordinary differential equations with coupled nonlinear boundary conditions in piecewisesmooth domains. Aside from wellposedness 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 nspecies exploitative competition for k essential (nonsubstitutable) 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 longterm 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 socalled "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.

0.915 
2004 — 2011 
Kuang, Yang [⬀] Smith, Hal Elser, James (coPI) [⬀] Anderies, John (coPI) [⬀] CastilloChavez, Carlos (coPI) [⬀] 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Ubm: Interdisciplinary Training For Undergraduates in Biological and Mathematical Sciences At Asu @ Arizona State University
An interdisciplinary team of investigators carry out an undergraduate training initiative at Arizona State University. The training plan intimately combines new crossdisciplinary 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, bioeconomics, 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 firsthand educational experience in crossdisciplinary 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.

0.915 
2009 — 2014 
Smith, Hal 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Dynamical Systems in Biology: Persistence and Models of Bacterial and Viral Infection and Treatment @ Arizona State University
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 discretetime dynamical systems arising in epidemiology and population dynamics because of the complexity of (boundary) attractors for discretetime 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 nonresident. 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 stateoftheart mathematical models of invivo 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 humanaltered) 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 dieoffs 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 realtime 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 virushost cell dynamics that are difficult to monitor in animal models. The investigator will examine effects of multiply infected host cells on disease dynamics.

0.915 