In this work we review the aggregation of variables method for discrete dynamicalsystems. These methods consist of describing the asymptotic behaviour of a complex systeminvolving many coupled variables through the asymptotic behaviour of a reduced systemformulated in terms of a few global variables. We consider population dynamics modelsincluding two processes acting at different time scales. Each process has associated a mapdescribing its effect along its specific time unit. The discrete system encompassing bothprocesses is expressed in the slow time scale composing the map associated to the slow oneand the k-th iterate of the map associated to the fast one. In the linear case a result isstated showing the relationship between the corresponding asymptotic elements of bothsystems, initial and reduced. In the nonlinear case, the reduction result establishes theexistence, stability and basins of attraction of steady states and periodic solutions ofthe original system with the help of the same elements of the corresponding reducedsystem. Several models looking over the main applications of the method to populationsdynamics are collected to illustrate the general results.

We establish sharp semiclassical upper bounds for the moments of some negative powers forthe eigenvalues of the Dirichlet Laplacian. When a constant magnetic field is incorporatedin the problem, we obtain sharp lower bounds for the moments of positive powers notexceeding one for such eigenvalues. When considering a Schrödinger operator with therelativistic kinetic energy and a smooth, nonnegative, unbounded potential, we prove thesharp Lieb-Thirring estimate for the moments of some negative powers of itseigenvalues.

We perform a computationl study of front speeds of G-equation models in time dependentcellular flows. The G-equations arise in premixed turbulent combustion, and areHamilton-Jacobi type level set partial differential equations (PDEs). The curvature-strainG-equations are also non-convex with degenerate diffusion. The computation is based onmonotone finite difference discretization and weighted essentially nonoscillatory (WENO)methods. We found that the large time front speeds lock into the frequency of timeperiodic cellular flows in curvature-strain G-equations similar to what occurs in thebasic inviscid G-equation. However, such frequency locking phenomenon disappears inviscous G-equation, and in the inviscid G-equation if time periodic oscillation of thecellular flow is replaced by time stochastic oscillation.

Directional multiscale representations such as shearlets and curvelets have gainedincreasing recognition in recent years as superior methods for the sparse representationof data. Thanks to their ability to sparsely encode images and other multidimensionaldata, transform-domain denoising algorithms based on these representations are among thebest performing methods currently available. As already observed in the literature, theperformance of many sparsity-based data processing methods can be further improved byusing appropriate combinations of dictionaries. In this paper, we consider the problem of3D data denoising and introduce a denoising algorithm which uses combined sparsedictionaries. Our numerical demonstrations show that the realization of the algorithmwhich combines 3D shearlets and local Fourier bases provides highly competitive results ascompared to other 3D sparsity-based denosing algorithms based on both single and combineddictionaries.

Spatial heterogeneity greatly affects the population spread. Although the theory forbiological invasion in heterogeneous spatially continuous habitats have receivedconsiderable attention, spatially discrete models have remained outside of the mainstream.In this study, we formulate and analyze a Coupled Map Lattice model for a single speciespopulation invading a two dimensional heterogeneous environment. The population growthrate and dispersal coefficient depend on the site quality. We first find an analyticalcriterium for the spread success in terms of the population growth rate and the dispersalcoefficient in unfavorable regions. We then implemented our model for two distinct spatialconfigurations: periodical stripe-like and randomized environments. The spread rate iscomputed numerically and it shows a decrease with an increase of the fraction of thehostile sites. However, we observed that invasion success does not depend on the fractionof favorable sites but crucially depends on the connectivity of favorable regions.

Deformable cell model is developed to study pattern formation and to simulate planttissue growth. Each cell represents a polygon with a number of vertices connected bysprings. Some cells in the tissue can grow and divide, other cells are differentiated anddo not grow or divide but remain deformable. The model is used to investigate formation ofself-similar structures which reproduce the same cell organization during their growth. Innumerical experiments we observed that self-similar solutions can exist for a ratherprecise choice of plant structure and mechanical properties of cell walls. We test themodel for simulation of apical meristems functioning which represent self-similar cellstructures in plants. At the next stage of modelling, auxin distribution is introduced bymeans of diffusion and polar transport mechanisms. The existence of steady auxindistribution in a growing root is investigated. Single as well as multiple auxin maximahave been observed in model solutions.

We introduce new classes of modulation spaces over phase space. By means of theKohn-Nirenberg correspondence, these spaces induce norms on pseudo-differential operatorsthat bound their operator norms on Lp–spaces,Sobolev spaces, and modulation spaces.

Assuming the negative part of the potential is uniformly locallyL1, we prove a pointwiseLp estimate on derivatives ofeigenfunctions of one-dimensional Schrödinger operators. In particular, if aneigenfunction is in Lp, then so is itsderivative, for 1 ≤ p ≤ ∞.

We are interested in the stability of the localized stationary solutions of a three-component reaction-diffusion system with one activator and two inhibitors. We show that depending on control parameters, solutions in form of moving and breathing localized structures can be observed in the vicinity of the codimension-two bifurcation point. We analyze this situation performing multiple scale perturbation expansion in the vicinity of the bifurcation point and derive a set of order parameter equations, explicitly describing the dynamics of the single localized structure. Numerical simulations are carried out, showing good agreement with the analytical predictions.

Let {pn}∞n=0 be a sequence of orthogonal polynomials. We brieflyreview properties of pn that have been usedto derive upper and lower bounds for the largest and smallest zero ofpn. Bounds for theextreme zeros of Laguerre, Jacobi and Gegenbauer polynomials that have been obtained usingdifferent approaches are numerically compared and new bounds for extreme zeros ofq-Laguerre and little q-Jacobi polynomials are proved.

We carried out a computational study of propagation speeds ofreaction-diffusion-advection fronts in three dimensional (3D) cellular andArnold-Beltrami-Childress (ABC) flows with Kolmogorov-Petrovsky-Piskunov(KPP)nonlinearity. The variational principle of front speeds reduces the problem to a principaleigenvalue calculation. An adaptive streamline diffusion finite element method is used inthe advection dominated regime. Numerical results showed that the front speeds areenhanced in cellular flows according to sublinear power lawO(δp),p ≈ 0.13, δ the flow intensity. In ABC flows however,the enhancement is O(δ) which can be attributed to thepresence of principal vortex tubes in the streamlines. Poincaré sections are used tovisualize and quantify the chaotic fractions of ABC flows in the phase space. The effectof chaotic streamlines of ABC flows on front speeds is studied by varying the threeparameters (a,b,c) of the ABC flows. Speed enhancement alongx direction is reduced as b (the parameter controlingthe flow variation along x) increases at fixed(a,c) > 0, more rapidly as the corresponding ABC streamlines becomemore chaotic.

The paper is devoted to a reaction-diffusion equation in an infinite two-dimensionalstrip with nonlinear boundary conditions. The existence of travelling waves is proved inthe bistable case by the Leray-Schauder method. It is based on a topological degree forelliptic problems in unbounded domains and on a priori estimates of solutions.

We review recent results on stability of traveling waves in partly parabolicreaction-diffusion systems with stable or marginally stable equilibria. We explain howattention to what are apparently mathematical technicalities has led to theorems thatallow one to convert spectral calculations, which are used in the sciences and engineeringto study stability of a wave, into detailed, theoretically-based information about thebehavior of perturbations of the wave.

The inverse stable subordinator provides a probability model for time-fractionaldifferential equations, and leads to explicit solution formulae. This paper reviewsproperties of the inverse stable subordinator, and applications to a variety of problemsin mathematics and physics. Several different governing equations for the inverse stablesubordinator have been proposed in the literature. This paper also shows how theseequations can be reconciled.

We study dynamics and bifurcations of three-dimensional diffeomorphisms with nontransversal heteroclinic cycles. We show that bifurcations under consideration lead to the birth of Lorenz-like attractors. They can be viewed as attractors in the Poincare map for periodically perturbed classical Lorenz attractors and hence they can allow for the existence of homoclinic tangencies and wild hyperbolic sets.

The paper is devoted to optimization of resonances in a 1-D open optical cavity. Thecavity’s structure is represented by its dielectric permittivity functionε(s). It is assumed thatε(s) takes values in the range1 ≤ ε1 ≤ ε(s) ≤ ε2.The problem is to design, for a given (real) frequency α, a cavity havinga resonance with the minimal possible decay rate. Restricting ourselves to resonances of agiven frequency α, we define cavities and resonant modes with locallyextremal decay rate, and then study their properties. We show that such locally extremalcavities are 1-D photonic crystals consisting of alternating layers of two materials withextreme allowed dielectric permittivities ε1 andε2. To find thicknesses of these layers, a nonlineareigenvalue problem for locally extremal resonant modes is derived. It occurs thatcoordinates of interface planes between the layers can be expressed via arg-function ofcorresponding modes. As a result, the question of minimization of the decay rate isreduced to a four-dimensional problem of finding the zeroes of a function of twovariables.

We develop a framework for analysing the outcome of resource competition based onbifurcation theory. We elaborate our methodology by readdressing the problem ofcompetition of two species for two resources in a chemostat environment. In the case ofperfect-essential resources it has been extensively discussed using Tilman’srepresentation of resource quarter plane plots. Our mathematically rigorous analysisyields bifurcation diagrams with a striking similarity to Tilman’s method including theinterpretation of the consumption vector and the resource supply vector. However, ourapproach is not restricted to a particular class of models but also works with othertrophic interaction formulations. This is illustrated by the analysis of a modelconsidering interactively-essential or complementary resources instead ofprefect-essential resources. Additionally, our approach can also be used for otherecosystem compositions: multiple resources–multiple species communities with equilibriumor oscillatory dynamics. Hence, it gives not only a new interpretation of Tilman’sgraphical approach, but it constitutes an extension of competition analyses to communitieswith many species as well as non-equilibrium dynamics.

This paper is concerned with a non-homogeneous in space and non-local in time random walkmodel for anomalous subdiffusive transport of cells. Starting with a Markov modelinvolving a structured probability density function, we derive the non-local in timemaster equation and fractional equation for the probability of cell position. We derivethe fractional Fokker-Planck equation for the density of cells and apply this equation tothe anomalous chemotaxis problem. We show the structural instability of fractionalsubdiffusive equation with respect to the partial variations of anomalous exponent. Wefind the criteria under which the anomalous aggregation of cells takes place in thesemi-infinite domain.

In various biological systems and small scale technological applications particlestransiently bind to a cylindrical surface. Upon unbinding the particles diffuse in thevicinal bulk before rebinding to the surface. Such bulk-mediated excursions give rise toan effective surface translation, for which we here derive and discuss the dynamicequations, including additional surface diffusion. We discuss the time evolution of thenumber of surface-bound particles, the effective surface mean squared displacement, andthe surface propagator. In particular, we observe sub- and superdiffusive regimes. Aplateau of the surface mean-squared displacement reflects a stalling of the surfacediffusion at longer times. Finally, the corresponding first passage problem for thecylindrical geometry is analysed.

Traveling waves for the nonlocal Fisher Equation can exhibit much more complex behaviourthan for the usual Fisher equation. A striking numerical observation is that a travelingwave with minimal speed can connect a dynamically unstable steady state 0 to a Turingunstable steady state 1, see [12]. This is provedin [1, 6] inthe case where the speed is far from minimal, where we expect the wave to be monotone.

Here we introduce a simplified nonlocal Fisher equation for which we can build simpleanalytical traveling wave solutions that exhibit various behaviours. These travelingwaves, with minimal speed or not, can (i) connect monotonically 0 and 1, (ii) connectthese two states non-monotonically, and (iii) connect 0 to a wavetrain around 1. Thelatter exist in a regime where time dynamics converges to another object observed in[3, 8]: awave that connects 0 to a pulsating wave around 1.