******* CANCELED ****************** In recent years it has become increasingly clear that stochasticity plays an important role in many biological processes. Examples include bistable genetic switches, noise enhanced robustness of oscillations, and fluctuation enhanced sensitivity or “stochastic focusing".. Numerous cellular systems rely on spatial stochastic noise for robust performance. We examine the need for stochastic models, report on the state of the art of algorithms and software for modeling and simulation of stochastic biochemical systems, and identify some computational challenges.

This talk will discuss previous work on quadrupedal gaits and recent work on a generalized model for binocular rivalry proposed by Hugh Wilson. Both applications show how rigid phase-shift synchrony in periodic solutions of coupled systems of differential equations can help understand high level collective behavior in the nervous system.

The interface between the mathematical sciences and the biosciences is two-way and may be summarized as "math -> bio" and "bio -> math." This talk will have two parts. First, I will describe previous work on gaits of four-legged animals (based on distinguishing gaits, such as walk, trot, and pace, by their spatio-temporal symmetries). Second, I will discuss how the application to gaits has led to results about phase-shift synchrony in periodic solutions of coupled systems of differential equations.

We use results of Cox, Durrett, and Perkins for voter model perturbations to study spatial evolutionary games on $\ZZ^d$, $d\ge 3$ when the interaction kernel is finite range, symmetric, and has covariance matrix $\sigma^2I$. The games we consider have matrices of the form ${\bf 1} + wG$ where ${\bf 1}$ is matrix of 1's and $w$ is small and positive. We prove that the effect of space is equivalent to (i) changing the entries of the game matrix and (ii) replacing the replicator ODE by a related PDE. The first idea is due to Ohtsuki and Nowak (for the pair approximation) while the second is well known in the theory of stochastic spatial processes. A remarkable aspect of our result is that the limiting PDE depends on the interaction kernel only through the values of two simple noncoalescence probabilities. Due to results of Aronson and Weinberger, and Fife and McLeod, we can analyze any 2x2 game. However, when there are three strategies the limiting object is a system of reaction diffusion equations, so we only have results for special cases.

Living systems are characterized by variability; they are subject to constant evolution through the three processes of population growth, selection and mutations, a principle established by C. Darwin. In a very simple, general and idealized description, their environment can be considered as a nutrient shared by all the population. This allows certain individuals, characterized by a 'phenotypical trait', to expand faster because they are better adapted to use the environment. This leads to select the'fittest trait' in the population (singular point of the system). On the other hand, the new-born individuals undergo small variations of the trait under the effect of genetic mutations. In these circumstances, is it possible to describe the dynamical evolution of the current trait? An area of population biology that aims at describing mathematically these processes is born in the 1980's under the name of 'adaptive dynamics' and, compared to population genetics, considers usually asexual reproduction, a continuous phenotypical trait and population growth. We will give a self-contained mathematical model of such dynamics, based on parabolic equations, and show that an asymptotic method allows us to formalize precisely the concepts of monomorphic or polymorphic population. Then, we can describe the evolution of the 'fittest trait' and eventually to compute various forms of branching points which represent the cohabitation of two different populations. The concepts are based on the asymptotic analysis of the above mentioned parabolic equations once appropriately rescaled. This leads to concentrations of the solutions and the difficulty is to evaluate the weight and position of the moving Dirac masses that describe the population. We will show that a new type of Hamilton-Jacobi equation, with constraints, naturally describes this asymptotic. Some additional theoretical questions as uniqueness for the limiting H.-J. equation will also be addressed. Several other modeling methods have been proposed, stochastic individual based models, evolutionary game theory, dynamical systems. Connections will be presented too. This talk is based on collaborations with G. Barles, J. Carrillo, S. Cuadrado, O. Diekmann, M. Gauduchon, S. Genieys, P.-E. Jabin, S. Mirahimmi, S. Mischler and P. E. Souganidis.

Biology is a tremendously diverse field covering systems operating at vastly different scales, with differing levels of interaction between these. Much of the effort in mathematical biology has been driven by the desire to develop the general principles by which biological systems are organized and operate. Yet at the same time there are demands for answers to quite specific questions to better manage natural systems, to enhance human health, and to plan for the future impacts of human actions. I will give a variety of examples of projects in mathematical ecology that lie at the interface between theory and practice, providing some indication of the utility of quantitative methods to elucidate general patterns of natural system response to management actions. These will include applications of optimal control methods to problems in wildlife management and disease ecology as well as a discussion of individual-based models. An objective of these approaches is to develop hypothetical "best" methods to manage a system, and use this as a template to compare and contrast management scenarios arising from the differing view points of diverse stakeholders in a relative assessment framework.

Sleep and wake states are regulated by the interactions among a number of
brainstem and hypothalamic neuronal populations and the expression of
their neurotransmitters. Based on experimental studies, several different
structures have been proposed for this sleep-wake regulatory network with
particular debate over components involved in rapid-eye movement (REM)
sleep regulation. We have developed a mathematical modeling framework
that is uniquely suited for investigating the structure and dynamics of
proposed sleep-wake regulatory networks. Using this framework, we are
analyzing the competing proposed network structures for the regulation of
REM sleep to determine how the structure of the sleep-wake regulatory
network determines sleep-wake behavior and the dynamics of behavioral
state transitions.