In this paper, an open problem in the multidimensional complex analysis is presented thatarises in the harmonic analysis related to the investigation of the regularity propertiesof Fourier integral operators and in the regularity theory for hyperbolic partialdifferential equations. The problem is discussed in a self-contained elementary way andsome results towards its resolution are presented. A conjecture concerning the structureof appearing affine fibrations is formulated.

The slow dynamics and linearized stability of a two-spike quasi-equilibrium solution to ageneral class of reaction-diffusion (RD) system with and without sub-diffusion isanalyzed. For both the case of regular and sub-diffusion, the method of matched asymptoticexpansions is used to derive an ODE characterizing the spike locations in the absence ofany 𝒪(1) time-scale instabilities of the two-spike quasi-equilibrium profile. These fastinstabilities result from unstable eigenvalues of a certain nonlocal eigenvalue problem(NLEP) that is derived by linearizing the RD system around the two-spike quasi-equilibriumsolution. For a particular sub-class of the reaction kinetics, it is shown that thediscrete spectrum of this NLEP is determined by the roots of some simple transcendentalequations. From a rigorous analysis of these transcendental equations, explicit sufficientconditions are given to predict the occurrence of either Hopf bifurcations or competitioninstabilities of the two-spike quasi-equilibrium solution. The theory is illustrated forseveral specific choices of the reaction kinetics.

The model we consider treats the cell as a viscoelastic medium filling one of two kindsof thin domains (“shapes” of cells): the thin slab being a caricature of a tissue and thethin circular cylinder mimicking a long cell. This enables us to simplify the system ofmechano-chemical equations. We construct abundant classes of explicit, but approximate,formulae for heteroclinic solutions to these equations.

A recently proposed model for the investigation of diffusivity in planktonic systemscontaining toxin-producing phytoplanktons is here reconsidered. We show the existence ofplanktonic travelling waves. Numerical simulations validate the analytical findings, toelucidate the sensitivity of the results in dependence of the diffusion coefficients.

The formation, persistence and movement of self-organised biological aggregations aremediated by signals (e.g., visual, acoustic or chemical) that organisms use to communicatewith each other. To investigate the effect that communication has on the movement ofbiological aggregations, we use a class of nonlocal hyperbolic models that incorporatesocial interactions and different communication mechanisms between group members. Weapproximate the maximum speed for left-moving and right-moving groups, and shownumerically that the travelling pulses exhibited by the nonlocal hyperbolic modelsactually travel at this maximum speed. Next, we use the formula for the speed of atravelling pulse to calculate the reversal time for the zigzagging behaviour, and showthat the communication mechanisms have an effect on these reversal times. Moreover, weshow that how animals communicate with each other affects also the density structure ofthe zigzags. These findings offer a new perspective on the complexity of the biologicalfactors behind the formation and movement of various aggregations.

This paper is devoted to the study of spreading speeds and traveling waves for a class ofreaction-diffusion equations with distributed delay. Such an equation describes growth anddiffusion in a population where the individuals enter a quiescent phase exponentially andstay quiescent for some arbitrary time that is given by a probability density function.The existence of the spreading speed and its coincidence with the minimum wave speed ofmonostable traveling waves are established via the finite-delay approximation approach. Wealso prove the existence of bistable traveling waves in the case where the associatedreaction system admits a bistable structure. Moreover, the global stability and uniquenessof the bistable waves are obtained in the case where the density function has zerotail

We study a nonlinear diffusion equationut = uxx + f(u)with Robin boundary condition at x = 0 and with a free boundary conditionat x = h(t), whereh(t) > 0 is a moving boundary representing theexpanding front in ecology models. For anyf ∈ C1 with f(0) = 0, weprove that every bounded positive solution of this problem converges to a stationary one.As applications, we use this convergence result to study diffusion equations withmonostable and combustion types of nonlinearities. We obtain dichotomy results and sharpthresholds for the asymptotic behavior of the solutions.

We present a mathematical model of a fishery on several sites with a variable price. Themodel takes into account the evolution during the time of the resource, fishes and boatsmovements between the different sites, fishing effort and price that varies with respectto supply and demand. We suppose that boats and fishes movements as well as pricesvariations occur at a fast time scale. We use methods of aggregation of variables in orderto reduce the number of variables and we derive a reduced model governing two globalvariables, respectively the biomass of the resource and the fishing effort of the wholefishery. We look for the existence of equilibria of the aggregated model. We show that theaggregated model can have 1, 2 or 3 non trivial equilibria. We show that a variation ofthe total number of sites can induce a switch from over-exploitation to sustainablefisheries.

We demonstrate that a piecewise linear slow-fast Hamiltonian system with an equilibriumof the saddle-center type can have a sequence of small parameter values for which aone-round homoclinic orbit to this equilibrium exists. This contrasts with the well-knownfindings by Amick and McLeod and others that solutions of such type do not exist inanalytic Hamiltonian systems, and that the separatrices are split by the exponentiallysmall quantity. We also discuss existence of homoclinic trajectories to small periodicorbits of the Lyapunov family as well as symmetric periodic orbits near the homoclinicconnection. Our further result, illustrated by simulations, concerns the complicatedstructure of orbits related to passage through a non-smooth bifurcation of a periodicorbit.

Mathematical models of plant growth are generally characterized by a large number ofinteracting processes, a large number of model parameters and costly experimental dataacquisition. Such complexities make model parameterization a difficult process. Moreover,there is a large variety of models that coexist in the literature with generally anabsence of benchmarking between the different approaches and insufficient modelevaluation. In this context, this paper aims at enhancing good modelling practices in theplant growth modelling community and at increasing model design efficiency. It gives anoverview of the different steps in modelling and specify them in the case of plant growthmodels specifically regarding their above mentioned characteristics.

Different methods allowing to perform these steps are implemented in a dedicated platformPYGMALION (Plant Growth Model Analysis, Identification and Optimization). Some of thesemethods are original. The C++ platform proposes a framework in which stochastic ordeterministic discrete dynamic models can be implemented, and several efficient methodsfor sensitivity analysis, uncertainty analysis, parameter estimation, model selection ordata assimilation can be used for model design, evaluation or application.

Finally, a new model, the LNAS model for sugar beet growth, is presented and serves toillustrate how the different methods in PYGMALION can be used for its parameterization,its evaluation and its application to yield prediction. The model is evaluated from realdata and is shown to have interesting predictive capacities when coupled with dataassimilation techniques.

We study a reaction diffusion system that models the dynamics of two species that displayinter-species competition and intra-species cooperation. We find that there are betweenthree and six different equilibrium states and a variety of possible travelling wavesolutions that can connect them. After examining the travelling waves that are generatedin three different ecologically-relevant initial value problems, we construct asymptoticsolutions in the limit λ ≪ 1 (fast diffusion, slow reaction for thesecond species relative to the first).

In this paper we study the complex dynamics of predator-prey systems with nonmonotonicfunctional response and harvesting. When the harvesting is constant-yield for prey, it isshown that various kinds of bifurcations, such as saddle-node bifurcation, degenerate Hopfbifurcation, and Bogdanov-Takens bifurcation, occur in the model as parameters vary. Theexistence of two limit cycles and a homoclinic loop is established by numericalsimulations. When the harvesting is seasonal for both species, sufficient conditions forthe existence of an asymptotically stable periodic solution and bifurcation of a stableperiodic orbit into a stable invariant torus of the model are given. Numerical simulationsare carried out to demonstrate the existence of bifurcation of a stable periodic orbitinto an invariant torus and transition from invariant tori to periodic solutions,respectively, as the amplitude of seasonal harvesting increases.

Building on earlier work, we have given in [29] aproof of existence of Abrikosov vortex lattices in the Ginzburg-Landau model ofsuperconductivity and shown that the triangular lattice gives the lowest energy perlattice cell. After [29] was published, we realizedthat it proves a stronger result than was stated there. This result is recorded in thepresent paper. The proofs remain the same as in [29], apart from some streamlining.

The Generalized Elastic Model is a linear stochastic model which accounts for thebehaviour of many physical systems in nature, ranging from polymeric chains to single-filesystems. If an external perturbation is exerted only on a single pointx⋆ (taggedprobe), it propagates throughout the entire system. Within the fractionalLangevin equation framework, we study the effect of such a perturbation, in cases of aconstant force applied. We report most of the results arising from our previous analysisand, in the present work, we show that the Fox H-functions formalismprovides a compact, elegant and useful tool for the study of the scaling properties of anyobservable. In particular we show how the generalized Kubo fluctuation relations can beexpressed in terms of H-functions.

Inverse scattering problem is discussed for the Maxwell’s equations. A reduction of theMaxwell’s system to a new Fredholm second-kind integral equation with a scalarweakly singular kernel is given for electromagnetic (EM) wave scattering. Thisequation allows one to derive a formula for the scattering amplitude in which only ascalar function is present. If this function is small (an assumption that validates aBorn-type approximation), then formulas for the solution to the inverse problem areobtained from the scattering data: the complex permittivityϵ′(x) in a bounded regionD ⊂ R3 is found from the scattering amplitudeA(β,α,k) known for a fixed k = ω √ϵ0μ0 >0 and allβ,α ∈ S2, whereS2 is the unit sphere in R3,ϵ0 and μ0 are constantpermittivity and magnetic permeability in the exterior regionD′ = R3\D. The novel pointsin this paper include: i) A reduction of the inverse problem for vectorEM waves to a vector integral equation with scalar kernelwithout any symmetry assumptions on the scatterer, ii) A derivation of thescalar integral equation of the first kind for solving the inversescattering problem, and iii) Presenting formulas for solving this scalar integralequation. The problem of solving this integral equation is an ill-posed one. A method fora stable solution of this problem is given.

This paper is concerned with entire solutions of a class of bistable delayed latticedifferential equations with nonlocal interaction. Here an entire solution is meant by asolution defined for all (n,t) ∈ ℤ × ℝ. Assuming that the equation has anincreasing traveling wave front with nonzero wave speed and using a comparison argument,we obtain a two-dimensional manifold of entire solutions. In particular,it is shown that the traveling wave fronts are on the boundary of the manifold.Furthermore, uniqueness and stability of such entire solutions are studied.

We study the destabilization mechanism in a unidirectional ring of identical oscillators,perturbed by the introduction of a long-range connection. It is known that for ahomogeneous, unidirectional ring of identical Stuart-Landau oscillators the trivialequilibrium undergoes a sequence of Hopf bifurcations eventually leading to thecoexistence of multiple stable periodic states resembling the Eckhaus scenario. We showthat this destabilization scenario persists under small non-local perturbations. In thiscase, the Eckhaus line is modulated according to certain resonance conditions. In the casewhen the shortcut is strong, we show that the coexisting periodic solutions split up intotwo groups. The first group consists of orbits which are unstable for all parametervalues, while the other one shows the classical Eckhaus behavior.

We study the behavior of linear nonstationary shallow water waves generated by aninstantaneous localized source as they propagate over and become trapped by elongatedunderwater banks or ridges. To find the solutions of the corresponding equations, we usean earlier-developed asymptotic approach based on a generalization of Maslov’s canonicaloperator, which provides a relatively simple and efficient analytic-numerical algorithmfor the wave field computation. An analysis of simple examples (where the bank and sourceshapes are given by certain elementary functions) shows that the appearance and dynamicsof trapped wave trains is closely related to a cascade of bifurcations of space-timecaustics, the bifurcation parameter being the bank length-to-width ratio.