We prove convergence of the spectra of certain class of nonlocal bounded operators to the spectrum of a Laplace operator as a parameter tends to zero. The operator under consideration takes the form L_e u(x) = int_Omega J((x-y)/e)/e^d[u(y)-u(x)]/e^² dy, where Omega is a smoothly bounded domain in R^d.. One application of this result is (nonlocal) diffusion-driven instability in systems with large differences in the diffusion coefficients, as in a Turing system. This is joint work with Guangyu Zhao.

We present and overview of the dynamics that are present on the bifurcation diagram including Hopf bifurcations and double-Hopf bifurcations, fold bifurcations and tori bifurcations. We also perform a numerical exploration of the stable tori emerging from the double Hopf bifurcations. Our numerical findings include Arnold tongues, computation of the rotation number, 1D manifolds on the Poincar'e section and torus break-up. This is joint work with A.R. Humphries and B. Krauskopf.

In this talk, traveling waves for nonlocal evolution systems with bistable nonlinearity are discussed. The spectrum of the operator obtained by linearizing about such a traveling wave are also examined.

We propose a new fast algorithm for finding a global shortest path connecting two points while avoiding obstacles in a region by solving a finite set of initial value problems for ordinary differential equations under random (white noise) perturbations. The idea is based on the fact that every shortest path possesses a simple and the same geometric structure. This enables us to restrict our search to a set of feasible paths that share a common structure. Each of the search set is based on a union of finite dimensional compact manifolds representing the obstacle boundaries. Comparing to the existing methods, such as combinatorial methods or partial differential equations methods, our algorithm seems to be faster and easier to implement. We can also handle cases in which obstacle shapes are arbitrary and/or the dimension of the base space is three or higher. In addition, we can also consider similar problems with obstacles that are not necessary stationary.

This is joint work with Jun Lu and Haomin Zhou.

The dispersal of organisms has many significant ecological effects, and hence the evolution of dispersal has been a subject of considerable interest in evolutionary ecology. An important problem in the study of the evolution of dispersal is determining what kinds of dispersal strategies are evolutionary stable in the sense that populations using them cannot be invaded by ecologically similar populations using other strategies. A class of strategies that have been shown to be evolutionarily stable in various contexts are those that produce an ideal free distribution of the population, that is, a spatial distribution where no individual can increase its fitness by moving to another location. This talk will present results on the evolutionary stability of ideal free dispersal strategies in the context of continuous time nonlocal dispersal models. These results partially extend some recent work on the evolutionary stability of ideal free dispersal for reaction-advection-diffusion equations and discrete diffusion models to nonlocal dispersal models. They also include an extension of an inequality from matrix theory to the case of nonlocal dispersal operators, which may be of independent interest.

I will present some recent results on the convergence to the unique equilibrium in some mutation selection model. I will first start by exposing some results concerning a Lotka-Volterra competition system with mutation. Then I will present some extension to a PDE version of the Lotka-Volterra competition system with mutation and point some consequence on the dynamics.

Schloegl's 2nd model on a lattice (for spontaneous particle annihilation and autocatalytic creation) exhibits bistable steady states. In the mean-field approximation, discrete reaction-diffusion equations describe spatially heterogenous states of this system: propagation of planar interfaces between the states, and evolution of droplets of one state embedded in the other. We find a remarkably rich variety of stationary droplet solutions in contrast to the typical unique (critical) solution.

References: Liu et al. Phys Rev Lett 98, 050601 (2007); Guo et al J Chem Phys 130, 074106 (2009); Wang et al Phys Rev E 85, 041109 (2012).

Bistable space-time discrete systems commonly possess a large variety of stable stationary solutions with periodic profile. In this context, it is natural to ask about the fate of trajectories composed of interfaces between steady configurations with periodic pattern and in particular, to study their propagation as traveling fronts. In this talk, I will consider such fronts in piecewise affine bistable recursions on the one-dimensional lattice. By introducing a definition inspired by symbolic dynamics, I will present results on the existence of front solutions and the uniqueness of their velocity, upon existence of their ground patterns. Moreover, the velocity dependence on parameters and the co-existence of several fronts with distinct ground patterns will also be described. Finally, robustness of the results to small C1-perturbations of the piecewise affine map will be argued by mean of continuation arguments.

In this talk, we shall survey some recent results on the wave propagation in the two species competition systems with Lotka-Volterra type nonlinearity. This includes systems with continuous and discrete diffusion. Both monostable and bistable cases shall be discussed. Questions on minimal wave speed for the monostable case, propagation failure in the bistable case, monotonicity of wave profiles, and uniqueness of wave speed in the bistable case shall be addressed.

Random attractor is an important concept to describe long term behavior of solutions for a given stochastic system. In this talk we will first provide sufficient conditions for the existence of a global compact random attractors for general dynamical systems in weighted space of infinite sequences. We then apply the result to show the existence of a unique global compact random attractor for first order, second order and partly dissipative stochastic lattice differential equations with random coupled coefficients and multiplicative/additive white noise in weighted spaces.

In this talk we study a class of nonlocal and non-autonomous Fokker-Planck equations that has recently been introduced in order to describe the hysteretic behaviour of many-particle systems with dynamical control. Relying on methods from asymptotic analysis we identify several parameter regimes and derive reduced evolution equations for certain macroscopic quantities. In particular, we discuss the fast reaction regime, which can be understood by adapting Kramers formula for large deviation, and the slow reaction regime, in which the dynamics is governed by a subtle interplay of parabolic and hyperbolic effects.

This is joint work with Barbara Niethammer and Juan Velazquez.

We study traveling waves for bistable reaction diffusion equations on the spatially discrete domain rectangular lattice in two spatial dimensions. Pinning or propagation failure refers to the existence of a stationary planar front at parameter values for which the two spatially homogeneous stable equilibria are energetically distinct. This front blocks the invasion by the energetically favorable stable equilibrium of the spatial domain occupied by the less energetically favorable stable equilibrium. Crystallographic pinning refers to roughness in the strength of the pinning regarded as a function of the direction in the two-dimensional lattice which the stationary front faces. We give a generic condition under which crystallographic pinning is guaranteed to hold in the lattice directions. The proof is based on dynamical systems. This is joint work with J. Mallet-Paret.

In the early 20th century, S. Bose and A. Einstein predicted the existence of a state of matter composed of weakly interacting bosons (integer spin particles). Today, this is known as the Bose-Einstein condensate. The BEC was first experimentally realized in 1995 by E. Cornell and C. Wieman (U. of Colorado at Boulder) and W. Ketterle (MIT). These experiments have generated a plethora of research concerning both theory and experiment in this area. The work presented in this talk focuses on understanding solitary waves in a spinor BEC lattice system. This system is motivated by the spinor BEC which can be described by a quasione dimensional model. Here, we discuss two- and three-component dynamical lattice which contains a mean field nonlinearity. Our analysis of solitary waves involves (i) an examination of the anti-continuum limit for our model of interest, (ii) the existence and stability of these solitary waves via a perturbative approach and (iii) understanding the structure of these waves in excited sites of the lattice.

We consider general reaction diffusion systems posed on rectangular lattices in two or more spatial dimensions. We show that travelling wave solutions to such systems that propagate in rational directions are nonlinearly stable under small perturbations. We employ recently developed techniques involving point-wise Green's functions estimates for functional differential equations of mixed type (MFDEs), allowing our results to be applied even in situations where comparison principles are not available.

[This project is joint work with Erik van Vleck and Aaron Hoffman.]

In this talk, we 'll start by reviewing some of the developments on nonlinear dynamical lattices of the discrete nonlinear Schrodinger type. We will explore ideas of continuation from the so-called anti-continuum limit, in order to identify discrete solitons and their stability in 1d lattices, as well as discrete vortices and more complex entities (such as vortex cubes) in two-dimensional and three dimensional case examples. Time-permitting we will present some extensions of these nearest-neighbor lattices to longer range interaction examples and how similar ideas carry over to the setting of Klein-Gordon lattices. More importantly, we 'll venture to go beyond the Hamiltonian realm to a setting which has recently gained significant momentum in the physical community but has very slightly been touched upon in the mathematical literature, namely that of the PT-symmetric lattices. The latter are, in a sense, a very special case example that stands between the Hamiltonian and the dissipative case. We will attempt to illustrate via some prototypical case examples how what we know from the Hamiltonian case is drastically modified in this PT-symmetric setting, highlighting some of the emerging mathematical challenges in this field.

It is well known that reaction rates can be strongly affected by the ambient fluid flow. The phenomenon where reaction rates in biology can be influenced by chemical attraction of species appears to be much less studied. I will consider a simple model involving diffusion, advection, chemotaxis, and absorbing reaction, motivated by the modelling of coral life cycle. We prove that in the framework of this model, chemotaxis plays a crucial role in enhancing reaction rates.

In this talk, we shall consider some aspects of nonlocal dispersal equations. First, we will present some relations between local (random) and nonlocal dispersal problems and then report our recent results on traveling waves and entire solutions of nonlocal dispersal equations. This talk is based on some joint works with Yu-Juan Sun, Zhi-Cheng Wang and Guo-Bao Zhang.

In this talk we will study the existence of pulsating fronts for a nonlocal dispersal equation with KPP type nonlinearity, which is spatially inhomogeneous but periodic in space. The nonlocal dispersal is accounted by a convolution term, with a compactly supported kernel. The existence of such fronts will be proved using a vanishing viscosity method, which relies in a priori estimates for the solutions. This is joint work with J. Covilla (INRA, Avignon) and J. Dávila (U. de Chile).

We investigate grain boundary motion in polycrystalline material layers (solids) with columnar structure, particularly in thin films, where the microstructure is given by a network of grains and grain boundaries. Our work focuses on the numerical simulation and analysis of grain boundary migration in an idealized system of three grains embedded in a thin film. The aim of the numerical simulations is to get insight into the coupled motion, and to understand certain known phenomena such as grain grooving, break-up of thin films, pitting at quadruple junctions, jerky motion, which influence the robustness and stability of polycrystalline materials.

We review work with Guy Gilboa on the use of nonlocal operators to define new types of functionals for image processing and elsewhere. This gives an advantage in handling textures and repetitive structures. then we will discuss new joint work with Hayden Schaeffer, Russle Caflisch and Cory Hauck on sparse solvers for multiscale PDE. we seem to automatically very efficiently represent the dynamics of multiscale PDE's including Navier-Stokes equation, with a very simple sparsification idea.

We study bifurcations of periodic travelling waves in diatomic granular chains from the anti-continuum limit, when the mass ratio between the light and heavy beads is zero. We show that every limiting periodic wave is uniquely continued with respect to the mass ratio parameter and the periodic waves with the wavelength larger than a certain critical value are spectrally stable.

Α model of one-dimensional metamaterial formed by a discrete array of nonlinear resonators is considered. The existence and uniqueness results of periodic and asymptotic travelling waves of the system are presented. The existence and the stability of asymptotic waves are also computed and discussed numerically.

While the theory of nonlinear waves in partial differential equations is very well developed, understanding travelling waves in systems posed on lattices is challenging, and many basic questions remain open. Indeed, travelling waves on lattices can be found only by solving functional differential equations of mixed mode, which are ill-posed as initial-value problems. In addition, propagation failure or pinning occurs frequently for waves with small speeds, which makes it hard to find such waves using perturbation arguments. In this talk, I will outline work on the existence and stability of travelling waves for the discrete FitzHugh-Nagumo system using geometric singular perturbation theory and, if time permits, for weak shocks in semidiscrete systems of conservation laws.

The current talk is concerned with the spectral theory, in particular, the principal eigenvalue theory, of nonlocal dispersal operators with time periodic dependence, and its applications. Nonlocal and random dispersal operators are widely used to model diffusion systems in applied sciences and share many properties. There are also some essential differences between nonlocal and random dispersal operators, for example, a random dispersal operator always has a principal eigenvalue, but a nonlocal dispersal operator may not have a principal eigenvalue. In this talk, I will present criteria for the existence of principal eigenvalues of nonlocal dispersal operators with time periodic dependence and consider the applications of the criteria to monostable equations with nonlocal dispersal.

I will describe several works dealing with the construction of different solutions on lattices. I will first investigate the problem of pulsating trvaelling waves known as travelling breathers. I will secondly deal with qusi-periodic motions on the lattice via a KAM theorem. I will construct finite dimensional and infinite dimensional invariant tori.

We consider traveling wave solutions of bistable lattice differential equations with repelling first neighbor and/or second neighbor interactions. Such equations arise as prototypical discrete models of phase transitions. Traveling wave solutions in this case correspond to heteroclinic connections between spatially periodic solutions and in some cases results can be obtained by rewriting as an appropriate vector equation. We present some recent results when there are both repelling first and second nearest neighbor interactions and for repelling first neighbor interactions in higher space dimensions.

This talk represents joint work with Maila Brucal-Hallare, Hermen Jan Hupkes, Anna Vainchtein, and Aijun Zhang.

We discuss the asymptotic behavior of a class of non-autonomous stochastic lattice systems driven by multiplicative white noise. We first prove the existence and uniqueness of tempered random attractors in a weighted space containing all bounded sequences, and then establish the upper semi-continuity of these attractors as the intensity of noise approaches zero. We also prove the existence of maximal and minimal tempered random complete solutions which bound the attractors from above and below, respectively. When deterministic external terms are periodic in time, we show the random attractors are pathwise periodic. Finally, we discuss a stochastic system which possesses an infinite-dimensional tempered random attractor.

This is joint work with Peter W. Bates and Kening Lu.

Spatial epidemic models on a square, cubic, or hypercubic lattice (d=2, 3 or more dimensions) involves: (i) spontaneous recovery of sick individual at lattice sites with rate p; and (ii) infection of healthy individual at a rate proportional to the number of diagonal sick neighbor pairs.

This model provides a prototype for nonequilibrium discontinuous phase transitions. However, it also exhibits a non-trivial generic two-phase coexistence: Stable infected and all-healthy states coexist for a finite range, pf(d) < p < pe(d), spanned by the orientation-dependent stationary points for planar interfaces. Our interface dynamic analysis from kinetic Monte Carlo simulation and from discrete reaction-diffusion equations (dRDEs) obtained from truncations of the exact master equation, reveals that pe(f) ∼ 0.2113765 + ce(f)/d as d →∞. The dRDEs display artificial propagation failure absent due to fluctuations in the stochastic model, and the propagation failure regimes are amplified for increasing d.

Non-mean-field behavior of catalytic conversion reactions in narrow pores is controlled by interplay between fluctuations in adsorption-desorption at pore openings, restricted diffusion, and reaction. Behavior is captured by generalized hydrodynamic formulation of the reaction-diffusion equations (RDE). These incorporate an appropriate description of chemical diffusion in mixed-component quasi-single-file systems, which is based on a refined picture of tracer diffusion. The RDE elucidate the non-exponential decay of the steady-state reactant concentration into the pore and anomalous scaling of the reactant penetration depth.

We study the interaction of small amplitude, long wavelength solitary waves in the Fermi-Pasta-Ulam model with general nearest-neighbor interaction potential. We establish global-in-time existence and stability of counter-propagating solitary wave solutions. These solutions are close to the linear superposition of two solitary waves for large positive and negative values of time; for intermediate values of time these solutions describe the interaction of two counter-propagating pulses. These solutions are stable with respect to perturbations in the space of square integrable sequences and asymptotically stable with respect to perturbations which decay exponentially at spatial infinity. This is joint work with Aaron Hoffman from Olin College

The discrete nonlinear Schroedinger equation (dNLS) breaks Galilean invariance. Are there discrete solitary traveling waves of dNLS? Numerical simulations and formal analyses go back to the work of M. Peyrard and M.D. Kruskal. A localized state propagating through the lattice excites radiation modes (lattice phonons). The propagating structure slows as it radiates some of its energy away. The structure then stops advancing and is eventually pinned to a fixed lattice site, where it converges to a discrete solitary standing wave.

I will describe recent joint work with Michael Jenkinson (Columbia University), where we construct on-site and off-site solitary waves solitary standing waves by bifurcation methods. These are related to the ``Peireles-Nabbaro barrier'', believed to play an important role in the above phenomena.

We will present results on the asymptotic behavior of solutions to a non-local diffusion equation, u_t=J*u-u:=Lu, in an exterior domain, which excludes one or several holes, and with zero Dirichlet data on its complement. When the space dimension is three or more this behavior is given by a multiple of the fundamental solution of the heat equation away from the holes. On the other hand, if the solution is scaled according to its decay factor, close to the holes it behaves like a function that is L-harmonic, Lu=0, in the exterior domain and vanishes in its complement. The height of such a function at infinity is determined through a matching procedure with the multiple of the fundamental solution of the heat equation representing the outer behavior. The inner and the outer behavior can be presented in a unified way through a suitable global approximation.

The study involves a thorough understanding of the stationary solutions of the Dirichlet problem in the exterior domain and a conservation law for the evolution problem that gives the nontrivial final mass.

If time allows, we will comment on the differences in the case of 1 dimension where the local decay factor differs from the global one making the study more involved.

This is joint work with C. Cortazar and M. Elgueta from PUC-Chile and F. Quiros from UAM, Spain.

This talk is concerned with the approximations of random dispersal operators/equations by nonlocal dispersal operators/equations. It rst proves that the solutions of properly rescaled nonlocal dispersal initial-boundary value problems converge to the solutions of the corresponding random dispersal initial-boundary value problems. Next, it proves that the principal spectrum points of nonlocal dispersal operators with properly rescaled kernels converge to the principal eigenvalues of the corresponding random dispersal operators. Finally, it proves that the unique positive stationary solutions of nonlocal dispersal KPP equations with properly rescaled kernels converge to the unique positive stationary solutions of the corresponding random dispersal KPP equations.

We discuss the existence, uniqueness, asymptotic behavior and other qualitative properties of traveling wave solutions of Allen-Cahn equation with fractional Laplacians where the double well potential has unequal depths. The existence is proved by using a continuity argument, where a key ingredient is the estimate of the speed of the traveling wave in terms of the potential in order to get the uniform estimates of the solutions. The upper bound of the speed is given explicitly in terms of potential. The method can be applied to other elliptic operators such as the standard Laplacian and the convolution-form operators, etc.

This is a joint work with Professor Changfeng Gui.

We study the Cauchy problems for bistable nonlinearity reaction-diffusion equations by using energy argument. We prove the one to one relation between long time behavior of solution and the time limit of energy. Moreover, for a suitable monotone one-parameter family of initial data, there exists a sharp threshold between extinction and propagation.