The 2D Boussinesq system is potentially relevant to the study of atmospheric and oceanographic turbulence, as well as other astrophysical situations where rotation and stratification play a dominant role. In fluid mechanics, the 2D Boussinesq system is commonly used in the field of buoyancy-driven flow. It describes the motion of incompressible inhomogeneous viscous fluid subject to convective heat transfer under the influence of gravitational force. It is well-known that the 2D Boussinesq equations are closely related to 3D Euler or Navier-Stokes equations for incompressible flow, and it shares a similar vortex stretching effect as that in the 3D incompressible flow. In fact, in vortex formulation, the 2D inviscid Boussinesq equations are formally identical to the 3D incompressible Euler equations for axisymmetric swirling flow. Therefore, the qualitative behaviors of the solutions to the two systems are expected to be identical. Better understanding of the 2D Boussinesq system will undoubtedly shed light on the understanding of 3D flows. In this talk, I will discuss some recent results concerning global existence, uniqueness and asymptotic behavior of classical solutions to initial boundary value problems for 2D Boussinesq equations with partial viscosity terms on bounded domains for large initial data.

The transcription factor NF-kB is critical to the control of responses to cellular stress, inter- and intracellular signaling, cell growth, survival and apoptosis. At rest, NF-kB is sequestered by its inhibitor IkB in the cytoplasm. Upon stimulation, such as tumor necrosis factor $\alpha$ (TNF$\alpha$), NF-kB gets released from IkB and translocates to the nucleus and regulates genes transcription, including regulating transcription of gene IkB. Then the newly synthesized IkB, on the other hand, removes NF-kB from the nucleus. Hence, NF-kB and IkB form a negative feedback loop. Negative feedback loop is often associated to oscillations. Indeed, oscillations of the concentration of nuclear NF-kB has been observed both at population and single cell levels by Hoffmann et al. and Nelson et al. respectively. Ashall et al. recently reported that different frequencies of the oscillations leads to different gene expression. It has been reported in many works that NF-kB signaling pathway may interact with many other signaling pathways, including P53 signaling pathway. So it is important to understand that how the frequencies of NF-kB oscillations may be influenced by its interacting signals. However, the existence and mechanism of those potential interactions are not clear. In this talk, I study this issue by considering the pathway subjected to two types of putative signals: sinusoid and pulsate signals. A rich variety of nonlinear dynamics can be observed. In addition, we consider possible cell-cell communication by secretion of TNF$\alpha$.

While alternans in a single cardiac cell appears through a simple period-doubling bifurcation, in extended tissue the exact nature of the bifurcation is unclear. In particular, the phase of alternans can exhibit wave-like spatial dependence, either stationary or travelling, which is known as discordant alternans. We study these phenomena in simple cardiac models through a modulation equation proposed by Echebarria-Karma. We perform a bifurcation analysis for their modulation equation. We also find that for some extreme range of parameters, there are chaotic solutions. Chaotic waves in recent years have been regarded to be closely related to dreadful cardiac arrhythmia. Proceeding work illustrates some chaotic phenomena in two- or three-dimensional space, for instance spiral and scroll waves. We show the existence of chaotic waves in one dimension, which may provide a different mechanism accounting for the instabilities in cardiac dynamics.

Thyroid hormone regulation is a classic example of biological feedback control, and thyroid disorders such as hypothyroidism affect more than 300 million people worldwide. We developed a physiologically based, ordinary differential equation model of the human hypothalamic-pituitary-thyroid axis, in order to address several clinical applications. The model is broken into two major components-- the thyroid and brain submodels, each quantified from human clinical data. We combined these two submodels to form a complete closed loop model, which we validated using additional independent clinical data. Using the closed-loop model, we address several applications in replacement thyroid hormone (L-T4) bioequivalence (equivalence between different brands/preparations of L-T4), circadian rhythms, and thyroid cancer.

The goal of this talk is to describe the analysis of a specific aspect of transient dynamics not covered by previous theory. The question addressed is whether one component of a perturbed solution to a system of differential equations can overtake the corresponding component of a reference solution as both converge to a stable node at the origin, given that the perturbed solution was initially farther away and that both solutions are nonnegative for all time. We call this phenomenon tolerance, for its relation to a biological effect.

Using geometric arguments it is shown that tolerance will exist in generic linear systems with a complete set of eigenvectors and in excitable nonlinear systems. A notion of inhibition is also defined that may constrain the regions in phase space where the possibility of tolerance arises in general systems. However, these general existence theorems do not yield an assessment of tolerance for specific initial conditions. To address that issue, some analytical tools were developed to determine if particular perturbed and reference solution initial conditions will exhibit tolerance.

Spatial segregation among life cycle stages has been observed in many stage-structured species, both in homogeneous and heterogeneous environments. We investigate density dependent dispersal of life cycle stages as a mechanism responsible for this separation by using stage-structured, integrodifference equation (IDE) models that incorporate density dependent dispersal kernels. After investigating mechanisms that can lead to spatial patterns in two dimensional Juvenile-Adult IDE models, we construct spatial models to describe the population dynamics of the flour beetle species T. castaneum, T. confusum and T. brevicornis and use them to assess density dependent dispersal mechanisms that are able to explain spatial formations observed in these species.

Under certain conditions, predation acts as a selective pressure that drives prey adaptation. In response, the predator can evolve counter-defenses to increase the likelihood of successful attack. Investment in such traits is often costly, so that there is a trade-off between trait investment and reproductive ability. There is some evidence that cost, at least for the prey, can vary with changes in the environment such as low resource availability. For our investigation, we assume that competition for resources is most likely to occur at high prey densities. The result is that as prey density increases, so does the cost for prey defense. Quantitative trait models are employed to examine the stability and dynamics of the system. We find that variable cost of prey defense tends to stabilize the system when the rate of prey evolution is either very fast or very slow.

Phylogenetics is the area of research concerned with finding the genetic relationship between species. The relationship can be represented by a phylogenetic tree, which is a simple, connected, acyclic graph equipped with some statistical information. This furnishes a certain polynomial map and we are interested in polynomials, called phylogenetic invariants, which vanish for every choice of model parameters. The set of phylogenetic invariants forms a certain algebraic object and we want to compute this object explicitly. One of the reasons that we want an explicit description of these polynomials is because it is claimed by Casanellas and Fernandez-Sanchez that using the entire set of phylogenetic invariants is an efficient phylogenetic reconstruction method. More importantly, phylogenetic invariants were used by Allman and Rhodes to study the problem of identifiability of tree topology for a number of phylogenetic models. In other words, given a distribution of observations that a certain model predicts, is it possible to uniquely determine all the parameters of the model? It is an important question since, if a tree is not uniquely determined by an expected joint distribution, then we cannot use that model for inference.

This presentation will explore in some detail group-based models and their invariants. (The content is drawn from the joint work with Sonja Petrović, UIC , petrovic@math.uic.edu)