Abstract: Central questions in ecology seek explanation for distribution, abundance, and co-existence of species. It is well known that competitive exclusion can occur between species competing for a resource where one species may drive another species to extinction. However, the presence of a predator can force the mediation of a coexistent state between the species. We ask whether a repulsive prey defense mechanism can upset predator-mediated coexistence. We discuss one scenario involving two competing species with a common predator. All three populations are mobile via random dispersal within a bounded spatial domain. We will first define predator-mediated coexistence, then the notion of prey-taxis. This mechanism influences the predator's movement leading to various spatial distributions. We present an existence result, discuss existence of non-constant steady states through bifurcation, and through numerical simulation show the model captures a range of spatial patterns when small perturbations to the homogeneous steady state are made.

Inverse Problems for Neuronal Cables on Graphs

Abstract: We review motivation and analysis for recovering spatially varying conductance parameters for a linear cable equation on a compact tree graph. This extends previous work of recovering a single conductance parameter to an arbitrary number of conductance parameters. The Boundary Control Method is used.

A model of arterial plaque cap development and degradation

Abstract: Cardiovascular disease is a leading cause of death in the US, and a common form of cardiovascular disease is atherosclerosis. This is an inflammatory disease of the large and medium arteries due to plaques that develop in the arterial wall. In modeling development of a lesion in an artery wall, there are a number of chemotactic mechanisms going on within the wall layer that lead to an arterial plaque with fibrous cap. We introduce a model involving some of these dynamic processes, present some theoretical results, do some simulations, and examine the implications of the model results. Our main goal of the project is to isolate potential mechanisms that lead to plaque rupture through thinning of the fibrous cap.

Challenges in understanding atherosclerotic plaque rupture: a mathematical modeling strategy

Abstract: Cardiovascular disease is a leading cause of death in the US and many developed countries. Atherosclerosis is a major contributor to this disease profile. Atherosclerosis is an inflammatory disease of major arteries due to fatty lesions forming in arterial walls, causing stenosis (contracting blood flow) and thrombosis (blood clots, blockage). Certain lesions, called vulnerable plaques, are responsible for most deaths from atherosclerosis. The growth and degradation of these plaques is very dynamic, involving complex biochemical, hemodynamic, and mechanical interactions. But the present experimental means for studying arterial plaque development is limited, calling for augmenting such studies by mathematical modeling, analysis, and simulation. In this talk I will give a background to the biology and outline a strategy for model development, starting with an ODE model of principle chemical and cellular processes, and progressing to more complicated PDE models that include more mechanisms. At this stage little is proved, so the talk should be viewed as a possible roadmap for approaching a variety of questions. The major one for me is why do some plaques become unstable and rupture, while others do not.

Myotonia Modeling presentation

Abstract: Medical research has indicated that abnormalities of skeletal muscles, myotonia and periodic paralysis, are caused by alteration in the voltage-gated sodium channels. This assumption led to studies of channel behavior based on the dynamics of membrane potentials. Cannon, et al developed a two-compartment Hodgkin-Huxley type model that had a reformulation of the sodium current term and did some simulations to compare with experiment. Here we discuss a geometric perturbation analysis on the model system, reducing it from an eight-order system to a third-order system. The conditions on the system parameters under which the model exhibits dynamic behavior that resembles clinical observations are derived. We are able to detect slow-fast limit cycles which generate bursts of action potentials characteristic of the clinical case where active and non-active phases are observed to alternate in a pulsatile fashion, such as that in patients with Hyperkalemic periodic paralysis. Relying on the observation that the state variables possess drastically diversified dynamics, we explain the differences between the action potential dynamics of a normal subject and those of myotonia or periodic paralysis cases. The model seems to display mixed-mode oscillations that need further analysis.

Comments Concerning Models of Myelinated Fibers

Abstract: I will introduce three relatively simple models for myelinated neural fibers, and discuss what has, and has not, been done on developing and analyzing traveling wave solutions to such problems. Such solutions must satisfy nonlinear functional differential equations with both forward and backward delays that must be determined along with the wave solution.

Introduction to the Boundary Control Method and its Application to Inverse Problems

Abstract: The boundary control method is an approach to inverse problems based on the relationship between control and systems theory. I will first give some motivation for studying certain inverse problems, then reduce the problem to a "simple" case. Then I will develop aspects of the boundary control method in a way that leads to an algorithmic approach for estimating a certain distributed parameter. I will wrap up with comments about other problems I am, or would like to attack. This project is joint with S. Avdonin, U. Alaska, Fairbanks.

Neuronal Cable Theory on Dendritic Trees

Abstract: We are interested in the qualitative behavior of diffusion problems on metric tree graphs. In this talk we extend neuronal cable theory to tree graphs that represent (idealized) dendritic trees, and discuss analytical results concerning threshold conditions, traveling wave solutions, bounds on conduction speed, and conduction block. As time permits we will mention work on an (inverse) problem in linear cable theory on tree graphs of recovering a parameter, namely the conductance on each branch.

Persistence and Competition: A Review of these Ideas in various Environments

Abstract: In this presentation I will start with models of a single population, concentrating on historic models in a non-spatial setting. Next I move to addressing population dynamics when there is mobility via a diffusion mechanism. After presenting some solution behavior, we move on to an advection-driven setting, like a simple creek environment, then a branched environment (river network). After this I return to basics of adding a second, competitive, species, first discussing the competitive exclusion principle in a single compartment setting, then discussing how the picture changes in the presence of diffusion and advection. I will finish with presenting some problems worth pursuing. The presentation is designed to be reasonably accessible to students with some differential equations background, but should raise some interesting, but unresolved questions in dynamics of populations.

Predator-mediated coexistence with chemorepulsion

Abstract: We discuss analysis and simulations associated with a model system consisting of two competing populations and one common predator population; all populations are mobile (random dispersal), but the predator\222s movement is influenced by one prey's gradient representing a chemorepulsive effect on the predator population. There is no adaptive mechanism in the present model. We examine pattern formation through bifurcations with respect to the chemotactic sensitivity parameter, and the prey diffusivity parameter. We also mention existence and convergence to steady state results. This work is in collaboration with Evan Haskell (Nova Southeastern University, Ft. Lauderdale, FL).

Chemical and mechanical mechanisms making arterial plaques vulnerable to rupture: a mathematical modeling perspective

Abstract: While most arterial plaques are stable, a percentage of plaques become vulnerable to rupture, causing heart attacks, strokes, or organ damage, depending on their location. The main question is to pin down trigger mechanisms that destabilize a plaque. Biochemical, mechanical and hemodynamic mechanisms are involved. We model the cellular and chemical dynamics in a maturing plaque, where a fibrous cap is developing and chemotaxis plays a significant role. We explain cross-chemotaxis, presenting some theory and simulations. As time permits, we then briefly discuss the role of blood shear stress on the endothelial cell layer, and how to incorporate this mechanism into our plaque model.

Inverse Problems for Neuronal Cable Models on Graphs

Abstract: For a parabolic equation defined on a tree graph domain, a dynamic Neumann-to-Dirichlet map associated with the boundary vertices can be used to recover the topology of the graph, length of the edges, and unknown coefficients and source terms in the equation. The motivation for this investigation is that the parabolic equation comes from a (linear) neuronal cable equation defined on the dendritic tree of a neuron, and the inverse problem concerns parameter identification of k unknown distributed conductance parameters. The talk is based on joint work with Sergei Avdonin (University of Alaska, Fairbanks, AK)