# Poster Presentations

Workshop on Nonlinear Differential Equations, Dynamical Systems and Applications

## Poster Presentations

**Jianing Chen**, New Mexico Institute of Mining and Technology*Boundary Layer Effects on Ionic Flows via Poisson-Nernst-Planck Models with Local Hard-Sphere Potentials*

We study a quasi-one-dimensional steady-state Poisson-Nernst-Planck model with a local hard-sphere potential for ionic flows of two oppositely charged ion species through a membrane channel. Of particular interest is to study qualitative properties of ionic flows in terms of individual fluxes without assuming electro-neutrality boundary conditions (indicating the existence of boundary layers). Compared with our previous results with electro-neutrality boundary conditions, new phenomena are observed, which further indicates the rich and complicated dynamics of the flow properties of interest in the study of ion channel problem.

**Hung Le**, University of Missouri*On the Existence and Instability of Solitary Water Waves with a Finite Dipole*

We investigate the existence and instability of two-dimensional traveling capillary- gravity water waves with a finite dipole. In particular, we consider the case where the fluid has infinite depth, and the vorticity is a sum of two weighted d-functions. Using the implicit function theorem, we can construct a family of solitary waves for the finite dipole problem. Our main result is that this family is orbitally unstable. This is proved using a modification of the Grillakis-Shatah-Strauss method recently introduced by Varholm, Wahlen, and Walsh.

**Hamidreza Mofidi**, University of Kansas*Effects of Permanent Charges and Ion Sizes on Ionic Fluxes*

In this talk, we will report our ongoing project concerning combined effects of (small) permanent charges and ionic sizes on ionic fluxes via the Poisson-Nernst-Planck models with a hard-sphere component for ion channel problems. For the three- constant-piece permanent charge with one nonzero region, we are able to derive the expansion of the fluxes in the small nonzero permanent charge and ionic diameter, up to quadratic order terms.

Based on the leading order terms, we determine several critical potentials in terms of the other key physical parameters that identify boundaries of different effects on fluxes rom the interaction between the permanent charge and ionic sizes. Other effects on fluxes due to high order terms will be reported too.

**Mayowa Michael Ojo**, University of Kansas*Modeling the Effect of Vaccination on Neisseria Meningitidis*

An impulsive differential equation model for assessing the impact of vaccine on the dynamics of meningitis is developed and analyzed. The impulsive reproduction number T0 and the maximum period of vaccination at T0=1 were obtained. The susceptible population is globally asymptotically stable at any given fixed point. Sensitivity analysis of the continuous model reveals that the transmission rate (β), recovery rate of carrier and vaccine efficacy are the parameters with most influence on the transmission of the disease. Numerical result of the study shows possibilities of effectively controlling the disease in the population using an imperfect vaccine with an efficacy of >75% and high vaccine coverage rate of at least 75%.

**Wesley Perkins**, University of Kansas*Modulational Instability of Ascending Periodic Magma Waves*

The Whitham modulation equations are widely used to describe the behavior of modulated periodic waves on large space and time scales, and hence are expected to give insight into the stability of spatially periodic structures. However, the derivation of these equations is based on formal asymptotic (WKB) methods, thus removing a layer of rigor that would otherwise support their predictions. In this study, we aim at rigorously verifying the predictions of the Whitham modulation equations in the context of the so-called conduit equations, a nonlinear dispersive PDE governing the evolution of the circular interface separating a light, viscous fluid rising buoyantly through a heavy, more viscous, miscible fluid at small Reynolds numbers. In particular, using rigorous spectral perturbation theory, we connect the predictions of the Whitham modulation equations to the rigorous spectral (in particular, modulational) stability of the underlying wave trains. This makes rigorous recent formal results on the conduit equation obtained by Maiden and Hoefer. This is joint work with Mat Johnson.

**Antoine Remond-Tiedrez**, Carnegie Mellon University*The Viscous Surface Wave Problem with Generalized Surface Energies*

We study a three-dimensional incompressible viscous fluid in a horizontally periodic domain with finite depth whose free boundary is the graph of a function. The fluid is subject to gravity and generalized forces arising from a surface energy. The surface energy incorporates both bending and surface tension effects. We prove that for initial conditions sufficiently close to equilibrium the problem is globally well-posed and solutions decay to equilibrium exponentially fast, in an appropriate norm. Our proof is centered around a nonlinear energy method that is coupled to careful estimates of the fully nonlinear surface energy.

**Rebekah Wagner**, University of Kansas*A Mechanistic Model of Plant-Symbiont Interactions*

The influence of microbial symbionts on the phenotype of their plant host is not well understood despite their known importance. Microbial symbionts, such as mycorrhizal fungi and rhizobia bacteria, reside within plant roots and exchange resources with their plant host. These relationships can be classified as mutualistic or non-mutualistic depending on environmental conditions and life history characteristics of plants and symbionts. Mycorrhizal fungal mutualists supply their host with phosphorus in exchange for allotted carbon produced through photosynthesis by the host plant. Rhizobia bacteria also deliver fixed nitrogen in exchange for allotted carbon. The resource complimentary of these relationships can facilitate increased host growth and can generate synergisms. A mechanistic model of plant-symbiont interactions was developed and shows when a pool resource is available to the host plant (i.e. soil fertility), the relative abundance of each symbiont decreases. The cost of preferentially allocated carbon to the symbionts decreases when the resource pool of phosphorus or nitrogen is available. Productivity of the host plant increases as a measure of total plant biomass increases over time before reaching the saturating equilibrium when interacting with either symbiont.

**Zhenshu Wen**, New Mexico Institute of Mining and Technology*Boundary Layer Effects on Ionic Flows via Classical Poisson-Nernst-Planck Models*

We study a quasi-one-dimensional steady-state Poisson-Nernst-Planck model for ionic flows of multiple charged ion species through a membrane channel. Of particular interest is to study boundary layer effects on ionic flows in terms of individual fluxes. More precisely, we study the flow properties of interest without assuming electro- neutrality boundary. Two cases are carefully analyzed, case one involves two ion species, one negatively charged and one positively charged, while case two involves two positively charged ion species and one negatively charged ion species. In both cases, new phenomena are observed and much more rich dynamics of ionic flows are obtained.