76 DS07 Abstracts CP1 exist in the neighbourhood of linear instability threshold A Note on the use of Lagrangian-Averaged Navier- for strongly non-Newtonian mud flows. Many bifurcation Stokes-Alpha Model for Wind-Driven Surface scenarios exhibited by the permanent wave equation ob- Waves tained at the second order approximation for film flows with moderate surface tension are identified, examined and The Lagrangian-averaged Navier-Stokes-α model was in- delineated in the parameter space and compared with the troduced in 1998. Since then, some developments have Newtonian results(R. Usha and B. Uma, Physics of Fluids, been made in mathematical and computational analysis of Vol 16, 2679-2696, 2004) the α-model. There, however, are few examples of the use of the model for real fluid problems. One of the obstacles R Usha is in the fact that the α-model is a system of fourth-order Professor, Department of Mathematics partial differential equations and needs additional bound- Indian Institute of Technology Madras, Chennai-36, India ary conditions for the well-posedness. We apply the α- [email protected] model to the generation of sea surface waves by winds and illustrate that such conditions might not be feasible, when I. Mohammed Rizwan Sadiq the regularizing parameter α is constant. We try to consol- Research Scholar, Department of Mathematics idate the Lagrangian-averaging modeling concept and look Indian Institute of Technology Madras, Chennai 36, India for possible alternatives. [email protected] Bong-Sik Kim Department of Mathematics and Statistics CP2 Arizona State University Drift-Diffusion Models for the Dynamics of Deci- [email protected] sion Making Behavioral and neural data from humans and animals at- CP1 tempting to identify randomly-presented stimuli can be de- Resonant Surface Waves scribed by a simple stochastic differential equation: the drift-diffusion (DD) process. In the two-alternative, forced- Interaction of resonant surface waves in an oscillating con- choice task the DD process describes how the logarithm of tainer is considered. Using the framework of Hierarchy of the likelihood ratio evolves as noisy incoming evidence ac- Bifurcations the averaged two-mode amplitude equations cumulates. DD and related Ornstein-Uhlenbeck processes are studied. The analysis explains globally the role of ini- emerge as reductions of multi-component neural networks tial profile properties vs forcing parameter magnitude. For on stochastic center manifolds, and also as continuum lim- several regimes of initial conditions it reconciles with the its of an optimal decision maker: the sequential probability Simonelli-Gollub experiment. Moreover, it proposes that ratio test. I will outline some background from cognitive several new types of solutions may appear. psychology and neuroscience, and explain how DD models with variable drift rates can represent ‘bottom-up’ informa- Vered Rom-Kedar tion on stimulus identity and reward magnitudes for correct The Weizmann Institute choices, can capture such ‘top-down’ phenomena as atten- Applied Math & Computer Sci tion and cognitive control, and can also describe changes [email protected] that occur during learning. This is joint work with Juan Gao, Philip Eckhoff, Sophie Liu, Angela Yu, Rafal Bogacz Eli Shlizerman and Jonathan Cohen. The Weizmann Institute of Science Department of Computer Science and Applied Philip Holmes Mathematics Princeton University [email protected] MAE Dept. [email protected] CP1 Dynamics of Waves in a Shallow Layer of Inelastic CP2 Non-Newtonian Viscous Fluid of Shear-Thinning Kuramoto-Sivashinsky Equation with Drift Type Flowing Down An Inclined Plane The Kuramoto–Sivashinsky equation is an important The nonlinear waves in a shallow layer of inelastic non- model for pattern formation in cases where the pattern Newtonian viscous fluid of shear-thinning type flowing forming instability has a preferred (non-zero) wavenumber. down an inclined plane are examined using dynamical sys- This equation has been studied extensively with periodic tems approach. A set of exact averaged equations from boundary conditions. Here we study the dynamics of the the complete Navier-Stokes equations for modified power- Kuramoto–Sivashinsky equation in a finite domain with re- law fluid flowing down an inclined plane is derived using flectional symmetry broken by the addition of a drift term. Energy Integral method. The linear stability of primary The results will be compared with those found in the pe- flow is investigated by the normal-mode formulation and riodic case where the effect of drift may be removed by the critical condition for the linear instability is obtained. changing to a moving frame. The permanent waves are investigated at the leading-order approximation in which the surface tension is absent and Steve Houghton therefore serves as a model for large-scale continuous bores University of Leeds, United Kingdom in mud flows. The analysis shows the existence of two [email protected] types of propagating bores. For weakly non-Newtonian mud flows, the retreating type exists only in the regime of linear-instability while the advancing type exists only in the regime of linear-stability. On the otherhand, both types DS07 Abstracts 77 CP2 Department of Mathematics Combat Modelling with Pdes [email protected] Limitations of Lanchester’s ODEs for modelling combat have long been recognised. We present work seeking to CP3 more realistically represent troop dynamics, enabling a Transport and Aggregation of Self-propelled Parti- deeper understanding of the nature of conflict. We extend cles in Fluid Flows Lanchesters ODEs, constructing a new PDE system and describe simulation results obtained by introducing spa- The distribution of swimming microorganisms represented tial force movement and troop interaction components as as self-propelled particles in a moving fluid medium is con- nonlocal terms. The spatial dynamics component takes ad- sidered. It is shown that the particles concentrate around vantage of swarming behaviour proposed by Mogilner et al, flow regions with chaotic trajectories. When the swimming producing cohesive realistic density profiles. velocity is larger than a treshold, dependent on the shape of the particles, all particles escape from regular elliptic Therese A. Keane regions and participate in global transport. For thin rod- Department of Mathematics like particles the threshold velocity vanishes and arbitrarily University of New South Wales weak swimming destroys all transport barriers. We derive [email protected] an expression for the swimming velocity required for escape based on a cicular flow approximation. James Franklin, Gary Froyland University of New South Wales Zoltan Neufeld, Colin Torney [email protected], [email protected] University College Dublin [email protected], [email protected] CP3 CP4 How Tadpoles Swims: Simple But Biologically Re- alistic Model Non-Linear Modelling of Cable Stayed Bridges A new model of tadpole swimming based on experimental Cable-stayed bridges frequently experience vibrations due studies of the spinal cord (Alan Roberts Lab, Bristol Uni- to a variety of mechanisms. Following on from previous versity, UK) is developed. We first consider a system of research at the University of Bristol, this paper studies two coupled Morris-Lecar neurons in the regime of post- nonlinear dynamics in a neighbourhood of multiple para- inhibitory rebound. Bifurcation analysis shows that this metric resonances. We examine a previously established simple model can generate robust anti-phase oscillations. cable-deck model, looking at the validity of the derivation A model of 2000 Morris-Lecar neurons of four different and compare the behaviour of the model to data obtained types is then developed. Experimental measurements and from parallel experimental work. realistic computer simulations of developmental processes Alan R. Champneys, Claire L. Massow, John Macdonald, in the spinal cord provide evidence for the connection ar- Veronica Vidal chitecture and parameter values of the model. Simulations University of Bristol show that the model can generate a metachronal wave re- [email protected], [email protected], sembling the tadpole swimming pattern. [email protected], [email protected] Roman M. Borisyuk University of Plymouth CP4 Centre for Theoretical and Computational Neuroscience [email protected] Driving a Chain from Stasis to Chaos We examine the dynamics of an inextensible hanging chain, Tom Cooke driven at one end. Although the physical system is quite University of Plymouth simple, the dynamics are rich, with solutions ranging from [email protected] rodlike motion to chaos, with swinging and whirling modes in between. We discuss the use of angular momentum in diagnosing symmetry breaking bifurcations and the role CP3 which different forms of dissipation have in determining The Iron Cycle and Thiobacillus Ferrooxidans Bac- the behavior of the system. teria Glenn Hollandsworth, Cavendish Q. McKay A non-spatial model for the iron cycle including pyrite as Marietta College waste rock is proposed. The biotic chemical reactions and [email protected], [email protected] reaction rates are based on experimental papers. The anal- ysis of the system indicates the possibility of bistability and the existence of a Hopf bifurcation indicates the presence of CP4 periodic orbits in ferric ion, bacteria and pH as suggested On the Stability of the Track