Multifractal analysis is known as a useful tool in signal analysis. However, themethods are often used without methodological validation. In this study, wepresent multidimensional models in order to validate multifractal analysismethods.

The inner knowledge of volumes from imagesis an ancient problem. This question becomes complicated when itconcerns quantization, as the case of any measurement and inparticular the calculation of fractal dimensions. Trabecular bonetissues have, like many natural elements, an architecture whichshows a fractal aspect. Many studies have already been developedaccording to this approach. The question which arises however is toknow to which extent it is possible to get an exact determination ofthe fractal dimension of a given volume only from the fractalmeasurement made on the projections or slice images given by medicalimaging. This paper gives general results about the Minkowskidimensions and contents of projections and sections of a set. Wealso show with examples that they depend essentially on thedirections of the planes and so there is - in general case - norelation between 3D and 2D fractal dimensions. This consideration isthen illustrated with examples from synthetic models and from CTscan images of wrists. In conclusion, this study reveals that thequantitative characterization of an organic volume (in particularosseous) requires taking into account the whole volume, and not onlysome of its slices or projections.

The challenges to live in the open water and the diversity of habitats in the marine environmentsprompts phytoplankton to devise strategies which often involve production of toxinsby Harmful Algal Bloom (HAB) and rapid production of metabolites from non-toxic precursor.The functional response of the predator is described by Holling type IV. We investigate wave phenomenaand non-linear non-equilibrium pattern formation in a phytoplankton-zooplankton systemwith Holling type IV functional response. Type IV functional response yields to type II responsein the event of large immunity from or tolerance of prey. It has been found that the Wave of Chaos(WoC) is still effective mechanism for the propagation of chaotic dynamics in predation and competitivesystems. Fish predation has a significant role in the temporal evolution of spatial patternsof phytoplankton-zooplankton system. From a field ecologist's perspective, it is not only importantto know the stable stationary patterns of an evolving phytoplankton-zooplankton system, butalso to know at what point of time these observations are carried out. This has implications forecological theory.

Proteus mirabilis are bacteria that make strikingly regular spatial-temporal patterns on agar surfaces.In this paper we investigate a mathematical model that has beenshown to display these structures when solved numerically. The model consists of an ordinarydifferential equation coupled with a partial differential equation involving a first-orderhyperbolic aging term together with nonlinear degenerate diffusion. The system is shown toadmit global weak solutions.

We study the role of interactions between habitats in rotifer dynamics. For this purpose we use a modified version of the Consensus model. The Consensus model has been shown to be realistic enough to reproduce distinguishing features of the rotifer species dynamics. Being uncoupled, intrinsically bistable rotifer populations, which inhabit the regions under different environmental conditions, do not impact each other. We show that migration of the rotifers between the habitats leads to the transformation of bistability to oscillatory dynamics. The coupling-induced oscillations of the rotifer biomass are shown to be either regular or irregular depending on the value of the model parameters, which describe environmental conditions and the biomass exchange between the habitats. We show that the irregular oscillations are characterized by positive values of the Lyapunov exponent, which means that they are chaotic.

Propagation of chemical waves at very low temperatures, observedexperimentally [V.V. Barelko et al., Advances in Chem. Phys. 74 (1988), 339-384.] at velocities of order  10 cm/s, is due toa very non- standard physical mechanism. The energy liberated bythe chemical reaction induces destruction of the material, therebyfacilitating the reaction, a process very different from standardcombustion. In this work we present recent experimental results and develop anew mathematical model which takes into account mechanical and chemicalprocesses. We carry out numerical simulations and suggest a simplified model inorder to obtain an explicit expression for the wave speed. This modelling allowsus to give a possible explanation of the existence of two modes of propagationobserved experimentally and of the dependence of the speed of propagation on theradiation dose.

Hill functions follow from the equilibrium state of the reaction in which n ligands simultaneouslybind a single receptor. This result if often employed to interpret the Hill coefficient asthe number of ligand binding sites in all kinds of reaction schemes. Here, we study the equilibriumstates of the reactions in which n ligand bind a receptor sequentially, both non-cooperatively andin a cooperative fashion. The main outcomes of such analysis are that: n is not a good estimate,but only an upper bound, for the Hill coefficient; while the Hill coefficient depends quite stronglyon the cooperativity level among ligands. We finally use these results to discuss the feasibilityand constrains of using Hill functions to model the regulatory functions in mathematical models ofgene regulatory networks.

The stability and evolution of very thin, single component, metallic-melt films isstudied by analysis of coupled strongly nonlinear equations for gas-melt (GM) and crystal-melt (CM) interfaces, derived using the lubrication approximation. The crystal-melt interface is deformable by freezing and melting, and there is a thermal gradient applied across thefilm. Linear analysis reveals that there is a maximum applied far-field temperature in thegas, beyond which there is no film instability. Instabilities observed in the absence of CMsurface energy are oscillatory for all marginally stable states. The effect of the CM surfaceenergy is to expand the parameter range over which a film is unstable. The new range ofinstabilities are of longer wavelength and are stationary, compared to the range found inthe absence of CM surface energy. Numerical analysis illustrates how perturbations grow torupture by standing waves. With CM surface energy, an initially longer (stationary) wavelength perturbation has a relatively slow growth rate, but it can trigger the appearance ofmuch faster growing shorter wavelength (oscillatory) instabilities, leading to an acceleratedfilm rupture process.

The shape and velocity of a sliding droplet are computed by solving the Navier-Stokes equation with free interface boundary conditions. The Galerkin finite element methodis implemented in a 2D computation domain discretized using an unstructured mesh withtriangular elements. The mesh is refined recursively at the corners (contact points). Thestationary sliding velocity is found to be strongly dependent on grid refinement, which isa consequence of the contact line singularity resolved through the effective slip across thefinite elements adjacent to the contact point. For small droplets, this dependence is wellapproximated by a theoretical estimates obtained using multiscale expansion and matchingtechnique in lubrication approximation, where the corner element size is used as a microscaleparameter. For larger droplets, the shape is also dependent on grid refinement. This questions the validity of numerous computations of flows with moving contact line where gridsare invariably much more coarse than molecular scales on which the singularity is resolved. Itis suggested that extrapolation to molecular scales should be used to obtain realistic results.

Impairment of left ventricular contractility due tocardiovascular diseases is reflected in left ventricle (LV) motionpatterns. An abnormal change of torsion or long axis shortening LVvalues can help with the diagnosis and follow-up of LVdysfunction. Tagged Magnetic Resonance (TMR) is a widely spreadmedical imaging modality that allows estimation of the myocardialtissue local deformation. In this work, we introduce a novelvariational framework for extracting the left ventricle dynamicsfrom TMR sequences. A bi-dimensional representation space of TMRimages given by Gabor filter banks is defined. Tracking of thephases of the Gabor response is combined using a variationalframework which regularizes the deformation field just at areaswhere the Gabor amplitude drops, while restoring the underlyingmotion otherwise. The clinical applicability of the proposedmethod is illustrated by extracting normality models of theventricular torsion from 19 healthy subjects.

Nowadays there has been a growing consciousness among the tea industry to reducethe use of the chemical pesticides for pest control. Predators are beneficial insects that feed onharmful insects and mites, which incur considerable loss of production of tea. In this paper wehave considered a tritrophic model consisting of tea plant, pest and predator to analyze differentfield observations. The effect of discrete time-delay on the tritrophic model is studied critically.The dynamical behaviours are studied both analytically and numerically by computer simulation.Our results suggest some theoretical measures which may be implemented for successful biologicalcontrol.

Recently, adult stem cells have become a focus of intensive biomedical research, but thecomplex regulation that allows a small population of stem cells to replenish depleted tissues is stillunknown. It has been suggested that specific tissue structures delimit the spaces where stem cellsundergo unlimited proliferation (stem cell niche). In contrast, mathematical analysis suggests thata feedback control of stem cells on their own proliferation and differentiation (denoted QuorumSensing) suffices for preserving homeostasis in developing tissues. In the present work we examinethis discrepancy by analyzing a mathematical model of signalling pathways in a mammary stemcell. Our work shows that the model has a unique equilibrium depending on the environmentalconditions, which is stable for appropriate parameter values. We show how this dependence onthe micro-environment can control the cell decision to either proliferate or differentiate. Our workfurther suggests that this description is sufficient for the creation of niche conditions, with no needof additional structures. The analysis also pinpoints the pathways that may serve as the QuorumSensing mechanism.

It is well established that resource variability generated by spatial patchiness and turbulence is an importantinfluence on the growth and recruitment of planktonic fish larvae. Empirical data show fractal-like prey distributions, and simulationsindicate that scale-invariant foraging strategies may be optimal. Here we show how larval growth and recruitment in a turbulent environment canbe formulated as a hitting time problem for a jump-diffusion process. We present two theoretical results. Firstly, if jumps are of a fixed sizeand occur as a Poisson process (embedded within a drift-diffusion), recruitment is effectively described by a diffusion process alone.Secondly, in the absence of diffusion, and for “patchy” jumps (of negative binomial size with Pareto inter-arrivals), the encounter processbecomes superdiffusive. To synthesise these results we conduct a strategic simulation study where “patchy” jumps are embedded in adrift-diffusion process. We conclude that increasingly Lévy-like predator foraging strategies can have a significantly positive effect onrecruitment at the population level.

The self-organization of porous nanostructures in anodic metal oxideis considered. A mathematical model which incorporates the chemicalreactions at the metal-oxide and oxide-electrolyte interfaces andelastic stress caused by the electrostrictive effects isdeveloped. It is shown through linear stability analysis, that ashort-wave instability exists in certain parameter regimes which canlead to the formation of hexagonally ordered pores observed inanodized aluminum oxide.

Parameterization of zooplankton functional response is crucial for constructing plankton models. Theoreticalstudies predict enhancing of system stability in case the response is of sigmoid type. Experiments on feeding in laboratories tell us infavor of non-sigmoid types for most herbivorous zooplankton species. However, recent field observations show that the overall functionalresponse of zooplankton in the whole euphotic zone can exhibit a sigmoid behavior even when the response for the same species in laboratorymesocosms is non-sigmoid. Here we propose a simple model explaining the observed alterations of functional response. We divide the euphoticzone into a number of layers and take into account the food-dependent migration of zooplankton. In each layer, the functional response (localresponse) is suggested to be non-sigmoid. We show that the overall response of zooplankton exhibits different behavior compared to thepatterns of the local response. In particular, the model predicts emergence of a sigmoid type as a result of zooplankton accumulation andfeeding in layers with high phytoplankton density. We show the importance of light attenuation by phytoplankton on the alteration offunctional response. The modelling results allow us to hypothesize that the sigmoid functional response in real communities should emergemore often than it was suggested earlier based only on experimental studies on zooplankton feeding.

In this paper we explore a new model of field carcinogenesis, inspired by lungcancer precursor lesions, which includes dynamics of a spatially distributed population ofpre-cancerous cells c(t, x), constantly supplied by an influx μ of mutated normal cells. Cellproliferation is controlled by growth factor molecules bound to cells, b(t, x). Free growthfactor molecules g(t, x) are produced by precancerous cells and may diffuse before they becomebound to other cells. The purpose of modelling is to investigate the existence of solutions,which correspond to formation of multiple spatially isolated lesions of pre-cancerous cellsor, mathematically, to stable spike solutions. These multiple lesions are consistent with thefield theory of carcinogenesis. In a previous model published by these authors, the influxof mutated cells was equal to zero, μ = 0, which corresponded to a single pre-malignantcolony of cells. In that model, stable patterns appeared only if some of the growth factorwas supplied from outside, arguably, a biologically tenuous hypothesis. In the present model,when μ > 0, that hypothesis is no more required, which makes this model more realistic.We present a range of results, both mathematical and computational, which taken togetherallow understanding the dynamics of this model. The equilibrium solutions in the currentmodel result from the balance between new premalignant colonies being initiated and theold ones dying out.

Relaxation oscillations are limit cycles with two clearly differenttime scales. In this article the spatio-temporal dynamics of astandard prey-predator system in the parameter region of relaxationoscillation is investigated. Both prey and predator population aredistributed irregularly at a relatively high average level between amaximal and a minimal value. However, the slowly developing complexpattern exhibits a feature of “inverse excitability”: Bothpopulations show collapses which occur erratically both in space andin time. The nature of these collapses is analysed statistically andit is shown that the model behaviour can be interpreted as aresolution of the paradox of enrichment.

Detailed descriptions of cardiacgeometry and architecture are necessary for examining andunderstanding structural changes to the myocardium that are theresult of pathologies, for interpreting the results ofexperimental studies of propagation, and for use as athree-dimensional orthotropically anisotropic model for thecomputational reconstruction of propagation during arrhythmias.Diffusion tensor imaging (DTI) provides a means to reconstructfibre and sheet orientation throughout the ventricles. Wereconstruct and quantify canine cardiac architecture in selectedregions of the left and right ventricular free walls and theinter-ventricular septum. Fibre inclination angle rotates smoothlythrough the wall in all regions, from positive in the endocardiumto negative in the epicardium. However, fibre transverse and sheetangles show large variability in basal regions. Additionally,regions where two populations (positive and negative) of sheetstructure merge are identified. From these data, we conclude thata single DTI-derived atlas model of ventricular architectureshould be applicable to modelling propagation in wedges from theequatorial and apical left ventricle, and allow comparisons toexperimental studies carried out in wedge preparations. However,due to inter-individual variability in basal regions, individual(rather than atlas) DTI models of basal wedges or of the wholeventricles will be required.

Composition gradients in miscible liquids can create volume forces resulting in variousinterfacial phenomena. Experimental observations of these phenomena are related to some difficultiesbecause they are transient, sufficiently weak and can be hidden by gravity driven flows.As a consequence, the question about their existence and about adequate mathematical models isnot yet completely elucidated. In this work we present some experimental evidences of interfacialphenomena in miscible liquids and numerical simulations of miscible drops and diffuse interfaces.