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.

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.

New conservation laws bifurcating from the classical form of conservation laws are constructed to the nonlinear Boussinesq model describing internal Kelvin waves propagating in a cylindrical wave field of an uniformly stratified water affected by the earth’s rotation. The obtained conservation laws are different from the well known energy conservation law for internal waves and they are associated with symmetries of the Boussinesq model. Particularly, it is shown that application of Lie group analysis provide three infinite sets of nontrivial integral conservation laws depending on two arbitrary functions, namely a(t, θ), b(t, r) and an arbitrary function c(t, θ, r) which is given implicitly as a nontrivial solution of a partial differential equation involving a(t, θ) and b(t, r).

This paper is concerned with the existence and stability of travelling front solutionsfor more general autocatalytic chemical reaction systems ut = duxx − uf(v), vt = vxx + uf(v)with d > 0 and d ≠ 1, wheref(v) has super-linear or linear degeneracy atv = 0. By applying Lyapunov-Schmidt decomposition method in someappropriate exponentially weighted spaces, we obtain the existence and continuousdependence of wave fronts with some critical speeds and with exponential spatial decay ford near 1. By applying special phase plane analysis and approximatecenter manifold theorem, the existence of traveling waves with algebraic spatial decay orwith some lower exponential decay is also obtained for d > 0. Further,by spectral estimates and Evans function method, the wave fronts with exponential spatialdecay are proved to be spectrally or linearly stable in some suitable exponentiallyweighted spaces. Finally, by adopting the main idea of proof in [12] and some similar arguments as in [21], the waves with critical speeds or with non-critical speeds are proved to belocally exponentially stable in some exponentially weighted spaces and Lyapunov stable inCunif(ℝ) space, if the initial perturbation of the waves issmall in both the weighted and unweighted norms; the perturbation of the waves also stayssmall in L1(ℝ) norm and decays algebraically inCunif(ℝ) norm, if the initial perturbation is in additionsmall in L1 norm.

Propagation of nonlinear baroclinic Kelvin waves in a rotating column of uniformlystratified fluid under the Boussinesq approximation is investigated. The model isconstrained by the Kelvin’s conjecture saying that the velocity component normal to theinterface between rotating fluid and surrounding medium (e.g. a seashore) is possibly zeroeverywhere in the domain of fluid motion, not only at the boundary. Three classes ofdistinctly different exact solutions for the nonlinear model are obtained. The obtainedsolutions are associated with symmetries of the Boussinesq model. It is shown that oneclass of the obtained solutions can be visualized as rotating whirlpools along which thepressure deviation from the mean state is zero, is positive inside and negative outside ofthe whirlpools. The angular velocity is zero at the center of the whirlpools and it ismonotonically increasing function of radius of the whirlpools.

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

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 construct the heat kernels of the sub-Laplacian and the Laplacian on theh-Heisenberg group and compute the limits as h → 0 ofthe heat kernels.

The bifurcation to one-dimensional weakly subcritical periodic patterns is described bythe cubic-quintic Ginzburg-Landau equation

       At = µA + Axx + i(a1|A|2Ax + a2A2Ax*) + b|A|2A - |A|4A.

These periodic patterns may in turn become unstable through one of two differentmechanisms, an Eckhaus instability or an oscillatory instability. We study the dynamicsnear the instability threshold in each of these cases using the corresponding modulationequations and compare the results with those obtained from direct numerical simulation ofthe equation. We also study the stability properties and dynamical evolution of differenttypes of fronts present in the protosnaking region of this equation. The results providenew predictions for the dynamical properties of generic systems in the weakly subcriticalregime.

An overview of recently obtained authors’ results on traveling wave solutions of someclasses of PDEs is presented. The main aim is to describe all possible travelling wavesolutions of the equations. The analysis was conducted using the methods of qualitativeand bifurcation analysis in order to study the phase-parameter space of the correspondingwave systems of ODEs. In the first part we analyze the wave dynamic modes of populationsdescribed by the “growth - taxis - diffusion" polynomial models. It is shown that“suitable" nonlinear taxis can affect the wave front sets and generate non-monotone waves,such as trains and pulses, which represent the exact solutions of the model system.Parametric critical points whose neighborhood displays the full spectrum of possible modelwave regimes are identified; the wave mode systematization is given in the form ofbifurcation diagrams. In the second part we study a modified version of theFitzHugh-Nagumo equations, which model the spatial propagation of neuron firing. We assumethat this propagation is (at least, partially) caused by the cross-diffusion connectionbetween the potential and recovery variables. We show that the cross-diffusion version ofthe model, besides giving rise to the typical fast travelling wave solution exhibited inthe original “diffusion" FitzHugh-Nagumo equations, additionally gives rise to a slowtraveling wave solution. We analyze all possible traveling wave solutions of the model andshow that there exists a threshold of the cross-diffusion coefficient (for a given speedof propagation), which bounds the area where “normal" impulse propagation is possible. Inthe third part we describe all possible wave solutions for a class of PDEs withcross-diffusion, which fall in a general class of the classical Keller-Segel modelsdescribing chemotaxis. Conditions for existence of front-impulse, impulse-front, andfront-front traveling wave solutions are formulated. In particular, we show that anon-isolated singular point in the ODE wave system implies existence of free-boundaryfronts.

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.

The Generalized Elastic Model is a linear stochastic model which accounts for the behaviour of many physical systems in nature, ranging from polymeric chains to single-file systems. If an external perturbation is exerted only on a single point x⋆ (tagged probe), it propagates throughout the entire system. Within the fractional Langevin equation framework, we study the effect of such a perturbation, in cases of a constant force applied. We report most of the results arising from our previous analysis and, in the present work, we show that the Fox H-functions formalism provides a compact, elegant and useful tool for the study of the scaling properties of any observable. In particular we show how the generalized Kubo fluctuation relations can be expressed in terms of H-functions.

This work deals with the consequences on structural stability of Gause type predator-preymodels, when are considered three standard functional responses and the prey growth rateis subject to an Allee effect.

An important consequence of this ecological phenomenon is the existence of a separatrixcurve dividing the behavior of trajectories in the phase plane. The origin is an attractorfor any set of parameters and the existence of heteroclinic curves can be also shown.

Conditions on the parameter values are established to ensure the existence of a uniquepositive equilibrium, which can be either an attractor or a repellor surrounded by one ormore limit cycles.

The influence of the Allee effect on the number of limit cycles is analyzed and theresults are compared with analogous models without this phenomenon, and which mainfeatures have been given in various above works. Ecological interpretations of theseresults are also given.

We study origin, parameter optimization, and thermodynamic efficiency of isothermal rocking ratchets based on fractional subdiffusion within a generalized non-Markovian Langevin equation approach. A corresponding multi-dimensional Markovian embedding dynamics is realized using a set of auxiliary Brownian particles elastically coupled to the central Brownian particle (see video on the journal web site). We show that anomalous subdiffusive transport emerges due to an interplay of nonlinear response and viscoelastic effects for fractional Brownian motion in periodic potentials with broken space-inversion symmetry and driven by a time-periodic field. The anomalous transport becomes optimal for a subthreshold driving when the driving period matches a characteristic time scale of interwell transitions. It can also be optimized by varying temperature, amplitude of periodic potential and driving strength. The useful work done against a load shows a parabolic dependence on the load strength. It grows sublinearly with time and the corresponding thermodynamic efficiency decays algebraically in time because the energy supplied by the driving field scales with time linearly. However, it compares well with the efficiency of normal diffusion rocking ratchets on an appreciably long time scale.

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 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.

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).

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 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.

We review recent stability and separation results in volume comparison problems and usethem to prove several hyperplane inequalities for intersection and projection bodies.