We discuss how distributed delays arise in biological models and review theliterature on such models. We indicate why it is important to keep thedistributions in a model as general as possible. We then demonstrate, throughthe analysis of a particular example, what kind of information can be gainedwith only minimal information about the exact distribution of delays.In particular we show that a distribution independent stability region maybe obtained in a similar way that delay independent results are obtained forsystems with discrete delays. Further, we show how approximations to theboundary of the stability region of an equilibrium point may be obtained withknowledge of one, two or three moments of the distribution. We compare theapproximations with the true boundary for the case of uniform and gammadistributions and show that the approximations improve as more moments are used.

While drawing a link between the papers contained in this issue and those present in a previous one (Vol. 2, Issue 3), this introductory article aims at putting in evidence some trends and challenges on cancer modelling, especially related to the development of multiphase and multiscale models.

We consider mixed and hybrid variational formulations to the linearizedelasticity system in domains with cracks. Inequality type conditions areprescribed at the crack faces which results in unilateral contact problems. Thevariational formulations are extended to the whole domain including the crackswhich yields, for each problem, a smooth domain formulation. Mixedfinite element methods such as PEERS or BDM methods are designed to avoidlocking for nearly incompressible materials in plane elasticity. We study andimplement discretizations based on such mixed finite element methods for thesmooth domain formulations to the unilateral crack problems. We obtainconvergence rates and optimal error estimates and we present some numericalexperiments in agreement with the theoretical results.

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.

The contact between two membranes can be described by a system of variationalinequalities, where the unknowns are the displacements of the membranes and theaction of a membrane on the other one. We first perform the analysis of thissystem. We then propose a discretization, where the displacements areapproximated by standard finite elements and the action by alocal postprocessing. Such a discretization admits an equivalent mixedreformulation. We prove the well-posedness of the discrete problem and establishoptimal a priori error estimates.

Polysystem methodology elaborated for comprehensive analysis of geographicalobjectsconsiders them as interrelated systems of different types. Each systematicinterpretation of aterritorial object is formed as a theory describing this object with a speciallanguage used for constructionof a certain type of models. This paper proposes new methods to developgeographicalmodels and describes several types of systematic models constructed by thesemethods.

This review aims at presenting asynoptic, if not exhaustive, point of view on some of the problemsencountered by biologists and physicians who deal with naturalcell proliferation and disruptions of its physiological control incancer disease. It also aims at suggesting how mathematicians arenaturally challenged by these questions and how they might help,not only biologists to deal theoretically with biologicalcomplexity, but also physicians to optimise therapeutics, on whichlast point the focus will be set here. To this purpose,mathematical modelling should represent proliferating cellpopulation dynamics with natural built-in control targets (whichimplies modelling the cell division cycle), together with thedistribution of drugs in the organism and their molecular actionson different targets at the cell level on proliferation, i.e.,molecular pharmacokinetics-pharmacodynamics of antiproliferativedrugs. This should make possible optimal control of drug deliverywith constraints to be determined according to the mainpharmacological issues encountered in the clinic: unwanted toxicside-effects, occurrence of drug resistance. Mathematicalmodelling should also take into account physiological determinantsof cell and tissue proliferation, such as intervention of theimmune system, circadian control on cell cycle checkpointproteins, and activity of intracellular drug processing enzymestogether with individual variations in the activities of theseproteins (genetic polymorphism). Taking these points into accountwill add to the rich scenery of normal or disrupted cell andtissue regulations, and their corrections by drugs, a naturalenvironmental, whole body physiological, frame. It is necessaryindeed to consider such a framework if one wants to eventually beactually helpful to clinicians who routinely treat by combinationsof drugs living Humans with their complex whole body regulations,often dependent on genotypic variations, and not isolated cells ortissues.

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.

We discuss theoretical and experimental approaches to three distinctdevelopmental systems that illustrate how theory can influence experimental workand vice-versa. The chosen systems – Drosophila melanogaster,bacterial pattern formation, and pigmentation patterns – illustrate thefundamental physical processes of signaling, growth and cell division, and cellmovement involved in pattern formation and development. These systems exemplifythe current state of theoretical and experimental understanding of how theseprocesses produce the observed patterns, and illustrate how theoretical andexperimental approaches can interact to lead to a better understanding ofdevelopment. As John Bonner said long ago
`We have arrived at the stage where models are useful to suggestexperiments, and the facts of the experiments in turn lead to new and improvedmodels that suggest new experiments. By this rocking back and forth between thereality of experimental facts and the dream world of hypotheses, we can moveslowly toward a satisfactory solution of the major problems of developmentalbiology.'

The G-function formalism has beenwidely used in the context of evolutionary games for identifyingevolutionarily stable strategies (ESS). This formalism wasdeveloped for and applied to point processes. Here, we examine the G-functionformalism in the settings of spatial evolutionarygames and strategy dynamics, based on reaction-diffusion models. We startbyextending the point process maximum principle to reaction-diffusion modelswith homogeneous, locally stable surfaces. We then develop the strategy dynamics forsuch surfaces. When the surfaces are locally stable, but nothomogenous, the standard definitions of ESS and the maximumprinciple fall apart. Yet, we show by examples that strategy dynamics leads toconvergent stable inhomogeneous strategies that are possibly ESS, in the sensethat for many scenarios which we simulated,invaders could not coexist with the exisiting strategies.

Many tumours undergo periods in whichthey apparently do not grow but remain at a roughly constant sizefor extended periods. This is termed tumour dormancy. Themechanisms responsible for dormancy include failure to develop aninternal blood supply, individual tumour cells exiting the cellcycle and a balance between the tumour and the immune response toit. Tumour dormancy is of considerable importance in the naturalhistory of cancer. In many cancers, and in particular in breastcancer, recurrence can occur many years after surgery to removethe primary tumour, following a long period of occult disease.Mathematical modelling suggested that continuous growth of tumourswas incompatible with data of the times of recurrence in breastcancer, suggesting that tumour dormancy was a common phenomenon.Modelling has also been applied to understanding the mechanismsresponsible for dormancy, how they can be manipulated and theimplications for cancer therapy. Here, the literature onmathematical modelling of tumour dormancy is reviewed. Inconclusion, promising future directions for research arediscussed.

This paper suggests models of the landscape complexes based on the theory ofcomplexsystems and category-functorial approach. With the use of the space Landsatimages the existenceof the Earth's landscape field is proved and criteria of its parametercharacterization is suggested.The unity of spatiotemporal changes of the landscape units is substantiated.

Wilkins' method has been successfully used since early 60s for numericalsimulation of high velocity contact elastic-plastic flows. The present workproposes some effective modifications of this method including moresophisticated material model including the Baushinger effect; modification ofthe algorithm based on correction of the initial configuration of a solid; acontact algorithm based on the idea of a mild contact.

Many models in biology and ecology can be described by reaction-diffusionequations wit time delay. One of important solutions for these type of equationsis the traveling wave solution that shows the phenomenon of wave propagation.The existence of traveling wave fronts has been proved for large class ofequations, in particular, the monotone systems, such as the cooperativesystems and some competition systems. However, the problem on the uniqueness oftraveling wave (for a fixed wave speed) remains unsolved. In this paper, we showthat, for a class of monotone diffusion systems with time delayed reaction term,the mono-stable traveling wave font is unique whenever it exists.

Since Rosenzweig showed the destabilisation of exploited ecosystems, the so called Paradox of enrichment, severalmechanisms have been proposed to resolve this paradox. In this paper we will show that a feeding threshold in the functional response for predators feeding on a prey population stabilizes the system and that there exists a minimumthreshold value above which the predator-prey system is unconditionally stable with respect to enrichment. Two models areanalysed, the first being the classical Rosenzweig-MacArthur (RM) model with an adapted Holling type-II functional response to include a feeding threshold. This mathematical model can be studied using analytical tools, which gives insight into the mathematical properties of the two dimensional ode system and reveals underlying stabilisation mechanisms. The second model is a mass-balance (MB) model for a predator-prey-nutrient system with complete recycling ofthe nutrient in a closed environment. In this model a feeding threshold is also taken into account for the predator-prey trophic interaction. Numerical bifurcation analysis is performed on both models. Analysis results are compared between models and are discussed in relation to the analytical analysis of the classical RM model. Experimental data from the literature of a closed system with ciliates, algae and a limiting nutrient are used to estimate parameters for the MB model. This microbial system forms the bottom trophic levels of aquatic ecosystems and therefore a complete overview of its dynamics is essential for understanding aquatic ecosystem dynamics.

During vertebrate development cells acquire different fates depending largely ontheir location in the embryo. The definition of a cell's developmentalfate relies on extensive intercellular communication that produces positionalinformation and ultimately generates an appropriately proportioned anatomy. Herewe place reaction-diffusion mechanisms in the context of general conceptsregarding the generation of positional information during development and thenfocus on these mechanisms as parsimonious systems for positioning anatomicalstructures relative to one another. In particular, we discuss the evidence forreaction-diffusion systems operating in the developing skin to yield theperiodic arrangements of hairs and feathers and discuss how best tobring together experimental molecular biology and numerical simulations to yielda more complete understanding of the mechanisms of development and naturalvariation.

The maintenance of a stable stem cellpopulation in the epidermis is important for robust regenerationof the stratified epithelium. The population size is usuallyregulated by cell secreted extracellular signalling molecules aswell as intracellular molecules. In this paper, a simple modelincorporating both levels of regulation is developed to examinethe balance between growth and differentiation for the stem cellpopulation. In particular, the dynamics of a known differentiationregulator c-Myc, its threshold dependent differentiation, andfeedback regulation on maintaining a stable stem cell populationare investigated.

We construct a stochastic model ofbacteriophage parasitism of a host bacteria that accounts fordemographic stochasticity of host and parasite and allows formultiple bacteriophage adsorption to host. We analyze the associateddeterministic model, identifying the basic reproductive number forphage proliferation, showing that host and phage persist when itexceeds unity, and establishing that the distribution of adsorbedphage on a host is binomial with slowly evolving mean. Notsurprisingly, extinction of the parasite or both host and parasitecan occur for the stochastic model.