We consider a size structured cell population model where a mother cell gives birth totwo daughter cells. We know that the asymptotic behavior of the density of cells is given by thesolution to an eigenproblem. The eigenvector gives the asymptotic shape and the eigenvalue givesthe exponential growth rate and so the Maltusian parameter. The Maltusian parameter depends onthe division rule for the mother cell, i.e., symmetric (the two daughter cells have the same size) orasymmetric. We use a min-max principle and a differentiation principle to find the variation of thefirst eigenvalue with respect to a parameter of asymmetry of the cell division. We prove that thesymmetrical division is not always the best fitted division, i.e., the Maltusian parameter may be notoptimal.

We study the coexistence of multiple periodic solutions for an analogue of theintegrate-and-fire neuron model of two-neuron recurrent inhibitory loops with delayedfeedback, which incorporates the firing process and absolute refractory period. Uponreceiving an excitatory signal from the excitatory neuron, the inhibitory neuron emits aspike with a pattern-related delay, in addition to the synaptic delay. We present atheoretical framework to view the inhibitory signal from the inhibitory neuron as aself-feedback of the excitatory neuron with this additional delay. Our analysis shows thatthe inhibitory feedbacks with firing and the absolute refractory period can generate fourbasic types of oscillations, and the complicated interaction among these basicoscillations leads to a large class of periodic patterns and the occurrence ofmultistability in the recurrent inhibitory loop. We also introduce the average time ofconvergence to a periodic pattern to determine which periodic patterns have the potentialto be used for neural information transmission and cognition processing in the nervoussystem.

We study a class of bistable reaction-diffusion systems used to model two competingspecies. Systems in this class possess two uniform stable steady states representingsemi-trivial solutions. Principally, we are interested in the case where the ratio of thediffusion coefficients is small, i.e. in thenear-degenerate case. First, limiting arguments are presented to relatesolutions to such systems to those of the degenerate case where one species is assumed notto diffuse. We then consider travelling wave solutions that connect the two stablesemi-trivial states of the non-degenerate system. Next, a general energy function for thefull system is introduced. Using this and the limiting arguments, we are able to determinethe wave direction for small diffusion coefficient ratios. The results obtained onlyrequire knowledge of the system kinetics.

Plants, algae, and fungi are essential for nearly all life on earth. Throughphotosynthesis, plants and algae convert solar energy to chemical energy in the form oforganic compounds that sustains essentially all life on earth. In addition, plants andalgae convert the carbon dioxide produced by respiring organisms to oxygen that is neededfor respiration. Fungi decompose complex organic compounds produced by respiring organismsso that molecules can be recycled in photosynthesis and respiration. Plants, algae, andfungi have one important feature in common, their cells have walls. Expansive growth andits regulation are central to the life and development of plant, algal, and fungal cells,i.e. cells with walls. In recent decades there has been an explosion of informationrelevant to expansive growth of cells with walls. Mathematical models have beenconstructed in an attempt to organize and evaluate this information, to gain insight, toevaluate hypotheses, and to assist in the selection and development of new experimentalstudies. In this article some of the mathematical models constructed to study expansivegrowth of cells with walls are reviewed. It is nearly impossible to review all relevantresearch conducted in this area. Instead, the review focuses on the development ofmathematical equations that have been used to model expansive growth, morphogenesis, andgrowth rate regulation of cells with walls. Also, relevant experimental findings arereviewed, conceptual models are presented, and suggestions for future research areproposed. The authors have attempted to provide an overview that is accessible toresearchers that are not working in this field.

The paper is devoted to mathematical modelling of erythropoiesis, production of red blood cells in the bone marrow. We discuss intra-cellular regulatory networks which determine self-renewal and differentiation of erythroid progenitors. In the case of excessive self-renewal, immature cells can fill the bone marrow resulting in the development of leukemia. We introduce a parameter characterizing the strength of mutation. Depending on its value, leukemia will or will not develop. The simplest model of treatment of acute myeloid leukemia with chemotherapy allows us to determine the conditions of successful treatment or of its failure. We show that insufficient treatment can worsen the situation. In some cases curing may not be possible even without resistance to treatment. Modelling presented in this work is based on ordinary differential equations, reaction-diffusion systems and individual based approach.

We present an overview of recent results for the classic problem of the survival probability of an immobile target in the presence of a single mobile trap or of a collection of uncorrelated mobile traps. The diffusion exponent of the traps is taken to be either γ = 1, associated with normal diffusive motion, or 0 < γ < 1, corresponding to subdiffusive motion. We consider traps that can only die upon interaction with the target and, alternatively, traps that may die due to an additional evanescence process even before hitting the target. The evanescence reaction is found to completely modify the survival probability of the target. Such evanescence processes are important in systems where the addition of scavenger molecules may result in the removal of the majority species, or ones where the mobile traps have a finite intrinsic lifetime.

The Susceptible-Infected-Recovered (SIR) model for the spread of an infectious disease ina population of constant size is considered. In order to control the spread of infection,we propose the model with four bounded controls which describe vaccination of newborns,vaccination of the susceptible, treatment of infected, and indirect strategies aimed at areduction of the incidence rate (e. g. quarantine). The optimal control problem ofminimizing the total number of the infected individuals on a given time interval is statedand solved. The optimal solutions are obtained with the use of the Pontryagin MaximumPrinciple and investigated analytically. Numerical results are presented to illustrate theoptimal solutions.

A nonlinear system of two delay differential equations is proposed to modelhematopoietic stem cell dynamics. Each equation describes the evolution of asub-population, either proliferating or nonproliferating. The nonlinearityaccounting for introduction of nonproliferating cells in the proliferating phaseis assumed to depend upon the total number of cells. Existence and stabilityof steady states are investigated. A Lyapunov functional is built to obtain theglobal asymptotic stability of the trivial steady state. The study ofeigenvalues of a second degree exponential polynomial characteristic equationallows to conclude to the existence of stability switches for the uniquepositive steady state. A numerical analysis of the role of each parameter on theappearance of stability switches completes this analysis.

RNA viruses replicate as complex and dynamic mutant distributions. They are termed viralquasispecies, in recognition of the fundamental contribution of quasispecies theory in ourunderstanding of error-prone replicative entities. Viral quasispecies have launched afertile field of transdiciplinary research, both experimental and theoretical. Here wereview the origin and some implications of the quasispecies concept, with emphasis oninternal interactions among components of the same mutant virus ensemble, a critical factto design new antiviral strategies. We make the distinction between “intrinsic” and“extrinsic” properties of mutant distributions, and emphasize that there are severallevels of complexity that can influence viral quasispecies behavior.

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.

This paper is devoted to the introduction of a new variant of the extendedfinite element method (Xfem) for the approximation of elastostatic fractureproblems. This variant consists in a reduced basis strategy for the definitionof the crack tip enrichment. It is particularly adapted when the asymptoticcrack-tip displacement is complex or even unknown. We give a mathematical resultof quasi-optimal a priori error estimate and some computational tests includinga comparison with some other strategies.

A mathematical model of infiltrative tumour growth is investigated taking into account transitions between two possible states of malignant cells: proliferation and migration. These transitions are considered to depend on oxygen level in a threshold manner where high oxygen concentration allows cell proliferation, while concentration below a certain critical value induces cell migration. The infiltrative tumour spreading rate dependence on model parameters is obtained. It is shown that the tumour growth rate depends on the tissue oxygen level in a threshold manner.

Continuum mechanics (e.g., hydrodynamics, elasticity theory) is based on the assumptionthat a small set of fields provides a closed description on large space and time scales.Conditions governing the choice for these fields are discussed in the context of granularfluids and multi-component fluids. In the first case, the relevance of temperature orenergy as a hydrodynamic field is justified. For mixtures, the use of a total temperatureand single flow velocity is compared with the use of multiple species temperatures andvelocities.

Contact behavior plays an important role in influenza transmission. In the progression ofinfluenza spread, human population reduces mobility to decrease infection risks. In thispaper, a mathematical model is proposed to include adaptive mobility. It is shown that themobility response does not affect the basic reproduction number that characterizes theinvasion threshold, but reduces dramatically infection peaks, or removes the peaks.Numerical calculations indicate that the mobility response can provide a very goodprotection to susceptible individuals, and a combination of mobility response andtreatment is an effective way to control influenza outbreak.

We study ensembles of similar systemsunder load of environmental factors. The phenomenon of adaptationhas similar properties for systems of different nature. Typically,when the load increases above some threshold, then the adaptingsystems become more different (variance increases), but thecorrelation increases too. If the stress continues to increasethen the second threshold appears: the correlation achievesmaximal value, and start to decrease, but the variance continue toincrease. In many applications this second threshold is a signalof approaching of fatal outcome.
This effect is supported by many experiments and observation ofgroups of humans, mice, trees, grassy plants, and on financialtime series. A general approach to explanation of the effectthrough dynamics of adaptation is developed. H. Selye introduced“adaptation energy" for explanation of adaptation phenomena. Weformalize this approach in factors – resource models anddevelop hierarchy of models of adaptation. Different organizationof interaction between factors (Liebig's versus synergisticsystems) lead to different adaptation dynamics. This gives anexplanation to qualitatively different dynamics of correlationunder different types of load and to some deviation from thetypical reaction to stress.
In addition to the “quasistatic" optimization factor – resourcemodels, dynamical models of adaptation are developed, and asimple model (three variables) for adaptation to one factor loadis formulated explicitly.

We investigate the structure oftravelling waves for a model of a fungal disease propagating overa vineyard. This model is based on a set of ODEs of the SIR-typecoupled with two reaction-diffusion equations describing thedispersal of the spores produced by the fungus inside and over thevineyard. An estimate of the biological parameters in the modelsuggests to use a singular perturbation analysis. It allows us tocompute the speed and the profile of the travelling waves. Theanalytical results are compared with numerical simulations.

The aim of this paper is to study the effect of vibrations on convective instability ofreaction fronts in porous media. The model contains reaction-diffusion equations coupledwith the Darcy equation. Linear stability analysis is carried out and the convectiveinstability boundary is found. The results are compared with direct numericalsimulations.

Pollen tubes are tip growing plant cells that display oscillatory growth behavior. It hasbeen demonstrated experimentally that the reduction of the average pollen tube growth ratethrough elevated extracellular calcium or borate concentrations coincides with a greateramplitude of the growth rate oscillation and a lower oscillation frequency. We present asimple numerical model of pollen tube growth that reproduces these results, as well asanalytical calculations that suggest an underlying mechanism. These data show that thepollen tube oscillator is non-isochronous, and is different from harmonic oscillation.

This paper is related to the spectral stability of traveling wave solutions of partialdifferential equations. In the first part of the paper we use the Gohberg-Rouche Theoremto prove equality of the algebraic multiplicity of an isolated eigenvalue of an abstractoperator on a Hilbert space, and the algebraic multiplicity of the eigenvalue of thecorresponding Birman-Schwinger type operator pencil. In the second part of the paper weapply this result to discuss three particular classes of problems: the Schrödingeroperator, the operator obtained by linearizing a degenerate system of reaction diffusionequations about a pulse, and a general high order differential operator. We studyrelations between the algebraic multiplicity of an isolated eigenvalue for the respectiveoperators, and the order of the eigenvalue as the zero of the Evans function for thecorresponding first order system.

Blood clotting system (BCS) modelling is an important issue with a plenty of applications in medicine and biophysics. The BCS main function is to form a localized clot at the site of injury preventing blood loss. Mutual influence of fibrin clot consisting mainly of fibrin polymer gel and blood flow is an important factor for BCS to function properly. The process of fibrin polymer mesh formation has not adequately been described by current mathematical models. That is why it is not possible to define the borders of growing clot and model its interaction with a blood flow. This paper main goal is to propose physically well-founded mathematical model of fibrin polymerization and gelation. The proposed model defines the total length of fibrin polymer fibers in the unit volume, determines a position of the border between gel and liquid and allows to evaluate the permeability of growing gel. Without significant structural changes the proposed model could be modified to include the blood shear rate influence on the fibrin polymerization and gelation.