Abstracts

ABSTRACTS

(alphabetically on last name)

Or click directly on the name of the speaker below

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Kevin Burrage [homepage] & Pamela Burrage [homepage], Queensland University of Technology, Australia

Time: Monday 18 July, 5:00pm - 5:30pm

Title: Solution and dynamical systems properties of fractional differential equations with applications

Abstract: TBA

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Chao-Nien Chen [homepage], National Tsing Hua University, Taiwan

Time: Wednesday 20 July, 10:45am - 11:45am

Title: Traveling waves for the FitzHugh-Nagumo system on an infinite channel

Abstract: In this talk we aim at the traveling wave solutions for the FitzHugh-Nagumo system. Based on a variational formulation in which a non-local term depends on a parameter, the speed of a traveling wave can be selected out. To show the existence of a traveling wave solution with such a speed, we seek a minimizer subject to a constraint. In particular, a truncation technique is used to obtain a minimizer located in a bounded region.

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Paige Davis, Queensland University of Technology, Australia

Time: Monday 18 July, 4:00pm - 4:15pm

Title: Absolute Instabilities in a Keller-Segel Model

Abstract: We investigate the spectral stability of travelling wave solutions supported by a Keller-Segel model of bacterial chemotaxis with a logarithmic chemosensitivity function and a constant, sublinear or linear consumption rate. We locate the essential and absolute spectrum. While all the travelling wave solutions have essential spectrum in the right half plane and are thus unstable, we also show that there exists a range of parameters such that the absolute spectrum is contained in the open left half plane. For these parameter values, the essential spectrum is also contained in the open left half plane in a particular range of weighted function spaces hinting at stability. For the constant and sublinear consumption rate models the absolute spectrum crosses into the right half plane (as a function of one of the system parameters) off of the real axis and we locate the critical parameter values where this crossing occurs.

This is joint work with P. van Heijster and R. Marangell.

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Gianne Derks [homepage], University of Surrey, UK

Time: Monday 18 July, 2:15pm - 3:15pm

Title: Existence and stability of fronts in inhomogeneous wave equations

Abstract: Models describing waves in anisotropic media or media with imperfections usually have inhomogeneous terms. Examples of such models can be found in many applications, for example in nonlinear optical waveguides, water waves moving over a bottom with topology, currents in nonuniform Josephson junctions, DNA-RNAP interactions etc. Homogeneous nonlinear wave equations are Hamiltonian partial differential equations with the homogeneity providing an extra symmetry in the form of the spatial translations. Inhomogeneities break the translational symmetry, though the Hamiltonian structure is still present. When the spatial translational symmetry is broken, travelling waves are no longer natural solutions. Instead, the travelling waves tend to interact with the inhomogeneity and get trapped, reflected, or slowed down.

In this talk, wave equations with finite length inhomogeneities will be considered, assuming that the spatial domain can be written as the union of disjoint intervals, such that on each interval the wave equation is homogeneous. The underlying Hamiltonian structure allows for a rich family of stationary front solutions and the values of the energy (Hamiltonian) in each intermediate interval provide natural parameters for the family of orbits. It will be shown that changes of stability can only occur at critical points of the length of the inhomogeneity as a function of the energy density inside the inhomogeneity and we give a necessary and sufficient criterion for the change of stability. These results will be illustrated with an example related to a Josephson junction system with a finite length inhomogeneity associated with variations in the Josephson tunneling critical current and an application to DNA-RNAP interactions. A discussion of open problems and questions will conclude this talk.

This is joint work with Arjen Doelman, Giuseppe Gaeta, Chris Knight and Hadi Susanto.

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Arjen Doelman [homepage], Leiden University, the Netherlands

Time: Thursday 21 July, 10:30am - 11:30am

Title: Pattern formation with a slowly varying parameter: desertification and pulse interactions

Abstract: Desertification is the ecological process in which an initially homogeneously vegetated area eventually collapses into a bare soil desert state -- typically as a consequence of a slow decrease in yearly rainfall or through an increasing grazing pressure. Vegetation patterns appear as intermediate structures, these patterns are quite resilient and are therefore playing a crucial role in the desertification process. In mathematical terms, desertification can be modeled by systems of singularly perturbed reaction-diffusion equations (that describe the interactions between water and vegetation). Vegetation patterns correspond to localized structures in semi-strong interaction. Pattern formation in reaction-diffusion equations is a well-studied subject, but the ecological point of view adds a novel aspect: what's the impact of a slowly varying parameter -- that for instance measures the yearly rainfall -- on the dynamics of the system? In this talk I will discuss the interactions of localized pulses under changing circumstances: what we do understand and what not? A typical and highly relevant ecological question I will consider is: When will a collapse into a desert state be catastrophic -- in the sense that in a large area all vegetation disappears simultaneously -- and when will it be more gradual -- by localized vegetation `pulses' disappearing one by one?

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Shin-Ichiro Ei [homepage], Hokkaido University, Japan

Time: Thursday 21 July, 9:00am - 10:00am

Title: Effect of boundaries on the motion of a spot solution in a two dimensional domain

Abstract: The motion of spot solution for reaction-diffusion systems in a bounded domain Ω ⊂ R2 with smooth boundaries is considered when the Neumann boundary condition is imposed. Supposing the existence of a stable stationary spot solution in the whole region R2, I am going to give a criterion to analyze the dynamics of a spot solution in the case that the spot exists in side of Ω apart from the boundary with sufficiently large distance.

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Cecilia González Tokman [homepage], University of Queensland, Australia

Time: Tuesday 19 July, 2:00pm - 2:30pm

Title: Multiplicative ergodic theorems and applications to the study of non-autonomous dynamical systems

Abstract: Multiplicative ergodic theorems (METs), first discovered by Oseledets in the 1960s, provide spectral-type decompositions for non-autonomous systems, with Lyapunov exponents and Oseledets spaces playing similar roles to eigenvalues and generalised eigenspaces, respectively. We will present recent developments on METs, including generalisations to semi-invertible and infinite-dimensional scenarios. We will discuss how, through the use of transfer operators, such progress has facilitated the use of METs as a powerful tool for the analysis of global transport properties of non-autonomous dynamical systems, including random invariant-measures and coherent structures.

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Hideo Ikeda [homepage], University of Toyama, Japan

Time: Wednesday 20 July, 4:00pm - 5:00pm

Title: Complex Dynamics of Bifurcating Front Solutions in a Three-Component Reaction-Diffusion System

Abstract: In the paper "Butterfly catastrophe for fronts in a three-component reaction-diffusion system (J. Nonlinear Science (2015)25,87-129)" by C.-Bruckner et al, several interesting front dynamics are studied. Based on this work, we want to discuss a bifurcation from a standing front solution of the above system. That is, we reduce the PDE dynamics to a finite-dimensional ODE system explicitly on a center manifold near a drift bifurcation point and analyze the dynamics of the reduced ODE system for several parameters. This indicates that the three-component system show a complex dynamics compared to the corresponding two-component system. Finally, we consider the type of criticality of the triple zero eigenvalue of the linearized eigenvalue problem.

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Zlatko Jovanoski [homepage], University of New South Wales (at the Australian Defence Force Academy), Australia

Time: Wednesday 20 July, 3:00pm - 3:30pm

Title: Population dynamics in a variable environment

Abstract: Many ecosystems are subject to external perturbations such as pollution, land clearing and sudden shocks to their environment. However, most current models used do not take the changing environment into consideration. In cases where the changes in the environment are taken into account this is usually done by specifying some time-dependent function for the carrying capacity that reflects the observed behaviour of the changing environment.

Recent models developed by our group directly couple the dynamics of one or two species with their environments. This is achieved by treating the carrying capacity, a proxy for the state of the environment, as a state variable in the governing equations of the model. Thereby, any changes to the environment can be naturally reflected in the survival, movement and competition of the species within the ecosystem.

Furthermore, a vast majority of models are deterministic and cannot adequately account for random external perturbation such as fires, drought, floods, contamination of water resources etc. By adding stochasticity (noise), it is possible to account for these anomalous impacts on population dynamics that deterministic models often ignore.

For this workshop a simple ecosystem consisting of a single species and its variable environment is proposed. Specifically, a logistic population model that incorporates a stochastic carrying capacity is investigated. The carrying capacity is treated as a state variable and is described by a stochastic differential equation. The statistical properties of the carrying capacity and the population are analysed. The probability distribution of the mean-time to extinction, the expected time evolution of the population and its variance are computed using the Monte Carlo method

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Bernd Krauskopf [homepage], the University of Auckland, New Zealand

Time: Tuesday 19 July, 9:00am - 10:00am

Title: Interacting isochron foliations

Abstract: The concept of isochrons was introduced by Winfree in the 1970s with the goal of understanding oscillations and syncronization in biological systems. For the important case of planar systems considered by Winfree and also here, each isochron is a curve of equal asymptotic phase in the basin of an attracting periodic orbit. From a dynamical systems point of view, isocrons are stabe manifolds of fixed points under the time-T map, where T is its period of the periodic orbit. They foliate the basin of attraction and, hence, encode geometrically how the system relaxes back to regular oscillation when a pertubation is applied. We present an efficient numerical methods — based on a boundary value problem setup — to compute isochrons as curves parameterized by arcength. We then introduce backward-time isochrones of unstable periodic orbits and spiraling equilibria, which foliate the basin of repulsion. This allows us to identify a cubic tangency between the two foliations as a mechanism for the onset of extreme phase sensitivity in the annulus where both sets of isochrons exist. As is illustrated with several examples, including the planar FitzHugh-Nagumo model, the cubic isochron foliation tangency emerges as a natural consequence of an increasing time-scale separation. We also discuss briefly a second source of phase sensitivity: the accumulation of isochrons on a nontrivial basin boundary.

This is joint work with Peter Langfield, James Hannam and Hinke Osinga, the University of Auckland.

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Dann Mallet [homepage], Queensland University of Technology, Australia

Time: Tuesday 19 July, 2:30pm - 3:00pm

Title: Some biologically and human inspired dynamical systems

Abstract: In this talk I will present a number of mathematical models that have arisen in my research covering problems in biological and human systems. Specifically, I will be presenting two models of the sexually transmitted infection, Chlamydia, and another model which relates to the assisted healing of a chronic wound. These models will move from a fairly simple ordinary differential equation model, through an optimally controlled differential equation model, to a reaction-diffusion style partial differential equation with a controllable boundary condition. While the construction of the models themselves and subsequent numerical simulation are at a fairly advanced stage, theoretical analysis is in its infancy. I will present a number of questions of interest to researchers in the fields of application that may benefit from collaboration with mathematicians focussed on analytical investigation of such systems.

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Robby Marangell [homepage], University of Sydney, Australia

Time: Monday 18 July, 4:30pm - 5:00pm

Title: Unstable Standing Waves in Inhomogeneous Schrödinger Equations

Abstract: In this talk I will discuss the stability of some solitons arising in models used to create a Bose-Einstein Condensate (BEC). Bose-Einstein Condensates are an exotic state of matter, first created in a laboratory in the 1990's. Mathematically, Bose-Einsten condensates are modelled by considering magnetic potential changes as defects in the standing wave equation of the Nonlinear Schrödinger (NLS) equation. The focus of the talk is on the instability of these defect-created states. I will outline how a topological tool, called the Maslov index, as well as composite phase portrait techniques can be used to establish the instability of standing waves to NLS equations with an inhomogeneous nonlinear term. The topological nature of the techniques used means that instability can be established by simple observations of the soliton's orbit in the phase plane.

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Jose Mujica [homepage], the University of Auckland, New Zealand

Time: Tuesday 19 July, 1:30pm - 1:45pm

Title: Canard orbits and mixed-mode oscillations near a singular Hopf bifurcation

Abstract: In slow-fast systems, such as those arising in the study of single neuron dynamics, one encounters the phenomenon referred to as a singular Hopf bifurcation: here a periodic orbit is born very close to a fold of the critical manifold, which implies that there is locally a mix between slow and fast directions. Near a singular Hopf bifurcation one finds a tangency between the repelling slow manifold and the two-dimensional unstable manifold of the bifurcating saddle-focus equilibrium. We show how this interaction has dramatic consequences for shaping the dynamics locally and globally.

This is joint work with Bernd Krauskopf and Hinke Osinga, the University of Auckland.

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Yasumasa Nishiura [homepage], Tohoku University, Japan

Time: Tuesday 19 July, 10:45am - 11:45am

Title: Open questions for the interplay between internal and external dynamics of localized patterns with oscillatory tails

Abstract: Spatially localized dissipative structures are observed in various fields, such as neural signaling, chemical reactions, discharge patterns, granular materials, vegetated landscapes and binary convection. These patterns are much simpler than single living cells, however they seem to inherit several characteristic “living state” feature, such as self-replication, self-healing and robustness as a whole system. Adaptive switching of dynamics can also be observed when these structures collide with each other, or when they encounter environmental changes in the media. These behaviors stem from an interplay between the intrinsic instability of each localized pattern and the strength of external signals. In this talk, I mainly focus on the the dynamics of pulses with oscillatory tails, in which the associated homoclinic orbits converge to the background state in an oscillatory manner. Even in a weak interaction regime, the behavior of them is different from those of monotone tail case. I will present a couple of preliminary results as well as many open issues and hopefully trigger future collaboration in this direction.

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Toshi Ogawa [homepage], Meji University, Japan

Time: Monday 18 July, 1:15pm - 2:15pm

Title: Drift bifurcation of traveling wave in reaction-diffusion system with 3 competing species

Abstract: There have been a lot of studies on the traveling wave solutions to a reaction-diffusion system with 2 competing species. Let us consider, here, the situation where we have third competing species adding to the original two competing species. Since this system has a trivial traveling solution which consists of the traveling wave solution of 2 species, the question is the stability of this solution in the full 3 component system. It turns out there is one critical point by taking the birth rate of the third species as a bifurcation parameter and we study the bifurcation structure around this critical point.

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Graeme Pettet [homepage], Queensland University of Technology, Australia

Time: Wednesday 20 July, 2:00pm - 2:30pm

Title: Travelling wave solutions for chemotactically-driven cell populations

Abstract: Continuum models of cell migration formulated as coupled advection-reaction-diffusion (ARD) systems in one spatial dimension have had a long and fruitful history in terms of providing insight and quantification of numerous biological processes.

Travelling wave solutions to ARD systems demonstrate the capacity of such models to support coherent structures and pattern development illustrative of complex behaviours that arise in cell populations. The existence and form of such travelling wave solutions depends strongly on the balance between the advective, reactive, and diffusive components of the models being considered.

In the context of dense cell populations in-vivo, as in embryologic, tumour or wound-repair growth and development, cell diffusion is arguably small in comparison to both the reactive and advective behaviours. Consequently, ARD models of these processes may develop travelling wave solutions with shock-like wavefronts, evocative of sharp interfaces between cell populations.

In this talk I will discuss recent work that examines some established ARD models with a view to characterising the conditions for existence of shock-like wavefronts. Results presented have required the development of a high resolution numerical scheme for reliably capturing the shock-fronts as they evolve, while proof of existence has led to the novel application of elements of geometric singular perturbation and canard theory.

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Marianito Rodrigo [homepage], University of Wollongong, Australia

Time: Tuesday 19 July, 3:00pm - 3:30pm

Title: A general reaction-diffusion model for acid-mediated tumour growth

Abstract: I will revisit the modelling of tumour invasion based on the acid-mediation hypothesis, i.e., the assumption that tumour progression is facilitated by acidification of the region around the tumour-host interface. The resulting destruction of the normal tissue environment promotes tumour growth. Gatenby and Gawlinski (1996) proposed a simplified reaction-diffusion system to model this hypothesis. Fasano, Herrero and Rodrigo (2009) used a nonstandard asymptotic analysis to study the properties of travelling waves that can be supported by the Gatenby-Gawlinksi model. Subsequently, Holder, Rodrigo and Herrero (2014) proposed an extension that incorporated a nonlinear acid production term. Another direction was given by McGillen, Gaffney, Martin and Maini (2014), where terms representing mutual competition between healthy and tumour cells, as well as acid-mediated tumour cell death, were added to the original Gatenby-Gawlinski model. In this talk I will consider a general reaction-diffusion model that includes the aforementioned models as special cases, with the aim of trying to determine under a quite broad framework the possibility of tumour progression that takes into account the acid-mediation hypothesis.

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Rashed Saifuddin [homepage], University of New South Wales (at the Australian Defence Force Academy), Australia

Time: Tuesday 19 July, 1:45pm - 2:00pm

Title: Application of a delayed logistic model with variable carrying capacity to a deer population

Abstract: A single-species population growth model with a variable carrying capacity is considered. The carrying capacity is treated as a state-variable, representing the availability of a non-renewable resource (the environment). We investigate models based on the logistic equation where the rate of decrease of the carrying capacity is proportional to the size of the population. We apply these models, with and without constant time-delays, to a deer population. The model that best fits the data includes a delay in the population and a delay in the term describing the carrying capacity. We provide a physical justification for these delays.

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Ayuki Sekisaka, Meji University, Japan

Time: Wednesday 20 July, 1:45pm - 2:00pm

Title: Gluing bifurcation problem in three-component reaction-diffusion systems

Abstract: We investigate the stability problem of traveling waves under the gluing bifurcation.A simplest gluing bifurcation is given by a homoclinic bifurcation from a heteroclitic cycle.There are many results about the stability property of such homoclinic pulse when the codimension of cycle equals two. In this talk, we show the homoclinic pulse from codimension-three heteroclinic cycle in some three-component reaction-diffusion systems which was discovered by the numerical simulation, and explain the stability problems and bifurcation structures with their difficulty.

This is joint work with Yasumasa Nishiura.

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Yoshitaro Tanaka, Hokkaido University, Japan

Time: Monday 18 July, 4:15pm - 4:30pm

Title: Instability caused by non-local interaction and its reaction-diffusion approximation

Abstract: Recently, various non-local interactions which influence globally in space arise in many fields, for example, in the spatial dispersals of the living things, the neural firing phenomenon in brain and the pigment cells in the skin of the zebra fish. To reveal the mechanisms of the pattern formations many mathematical models with convolutions were proposed. We found from nu- merical simulations that the destabilization of the solution was very sensitive for the shape of the convolution kernel. To specify this relationship, we ap- proximate the non-local interaction by reaction-diffusion systems (RDS) using the singular limit analysis. Through this approximation we reveal not only the relationship between the destabilization and the kernel shape but also that the destabilization is the diffusion driven instability. Finally, it is shown that any non-local interactions with the even convolution kernel can be realized by the RDS by increasing the components.

This is joint work with Hiroko Yamamoto and Hirokazu Ninomiya.

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Takashi Teramoto, Asahikawa Medical University, Japan

Time: Wednesday 20 July, 2:30pm - 3:00pm

Title: An action functional approach to stationary pulse solutions in a three-component FitzHugh-Nagumo model

Abstract: We combine geometrical singular perturbation techniques and an action functional approach to study the existence and stability of stationary localized patterns in a singularly perturbed three-component FitzHugh-Nagumo model. This new approach enables us to avoid technical computations of a Melnikov-type integral -- to derive existence conditions for the pulse solutions -- and of an Evans function -- to obtain critical information with regards to stability of the solution.

This is a joint work with Peter van Heijster, Chao-Nien Chen and Yasumasa Nishiura.

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Martin Wechselberger [homepage], University of Sydney, Australia

Time: Wednesday 20 July, 9:00am - 10:00am

Title: Neural excitability and singular bifurcations

Abstract: We discuss the notion of excitability in slow/fast neural models from a geometric singular perturbation theory point of view. In the first part of this presentation, we focus on the inherent singular nature of slow/fast neural models and define excitability via singular bifurcations. In particular, we show that type I excitability is associated with a singular Bogdanov-Takens/SNIC bifurcation while type II excitability is associated with a singular Andronov-Hopf bifurcation. In both cases, canards play an important role in the understanding of the unfolding of these singular bifurcation structures. We also explain the transition between the two excitability types and highlight all bifurcations involved, thus providing a complete analysis of excitability based on geometric singular perturbation theory.

In the second part, we focus on the inherent transient behaviour of type III excitability. Here, we show that canards play an essential role in defining firing threshold behaviour for these type of neurons, in particular, when exposed to (slow) dynamic forcing. The take-home message lies in the realisation that folded singularities and associated canards create local transient `attractor' states in multiple scales problems. This opens many opportunities for forging (more) connections between nonautonomous attractor theory and geometric singular perturbation theory.

This is joint work with Peter de Maesschalck (Hasselt), John Mitry (USyd) and John Rinzel (NYU).

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