Treating cancer patients with metastatic disease remains an ultimate challenge inclinical oncology. Because invasive cancer precludes or limits the use of surgery,metastatic setting is often associated with (poor) survival, rather than sustainedremission, in patients with common cancers like lung, digestive or breast carcinomas.Mathematical modeling may help us better identify non detectable metastatic status to inturn optimize treatment for patients with metastatic disease. In this paper we present afamily of models for the metastatic growth. They are based on four principles : to be assimple as possible, involving the least possible number of parameters, the maininformations are obtained from the primary tumor and being able to recover the variety ofphenomena observed by the clinicians. Several simulations of therapeutic strategies arepresented illustrating possible applications of modeling to the clinic.

A simple explicit numerical scheme is proposed for the solution of the Gardner–Ostrovskyequation (ut + cux + α uux + α1u2ux + βuxxx)x = γuwhich is also known as the extended rotation-modified Korteweg–de Vries(KdV) equation. This equation is used for the description of internal oceanic wavesaffected by Earth’ rotation. Particular versions of this equation with zero some ofcoefficients, α, α1, β, orγ are also known in numerous applications. The proposed numericalscheme is a further development of the well-known finite-difference scheme earlier usedfor the solution of the KdV equation. The scheme is of the second order accuracy both ontemporal and spatial variables. The stability analysis of the scheme is presented forinfinitesimal perturbations. The conditions for the calculations with the appropriateaccuracy have been found. Examples of calculations with the periodic boundary conditionsare presented to illustrate the robustness of the proposed scheme.

The circulatory system is one of thefirst to function during development. The earliest event in thesystem's development is vasculogenesis, whereby vascularprogeniter cells form clusters called blood islands, which laterfuse to form capillary networks. There exists a very goodin vitro system that mimics this process. When HUVECs(Human Umbilical Vein Endothelial Cells) are cultured on Matrigel,they spontaneously form a capillary network structure. Twotheoretical models have been proposed to explain the patternformation of this in vitro system. Both models utilizechemotaxis to generate spatial instability, and one modelspecifies VEGF as the chemoattractant. However, there are severalunknown factors concerning the experimental model. First, thepattern formation process occurs at the interface between theliquid medium and Matrigel, and it is unclear whether diffusion inthe liquid or gel is critical. Second, the diffusion coefficientof VEGF, which determines the spatial scale of the capillarystructure, has not been properly measured. In the present study,we modified the experimental system to clarify the effect ofdiffusion in Matrigel, and experimentally measured the diffusioncoefficient of VEGF in this system. The relationship with thespatial scale of the pattern generated is discussed.

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.

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.

Reaction-diffusion equations with degenerate nonlinear diffusion are in widespread use asmodels of biological phenomena. This paper begins with a survey of applications toecology, cell biology and bacterial colony patterns. The author then reviews mathematicalresults on the existence of travelling wave front solutions of these equations, and theirgeneration from given initial data. A detailed study is then presented of the form ofsmooth-front waves with speeds close to that of the (unique) sharp-front solution, for theparticular equation ut =(uux)x + u(1− u). Using singular perturbation theory, the author derives anasymptotic approximation to the wave, which gives valuable information about the structureof smooth-front solutions. The approximation compares well with numerical results.

Bacillus subtilis swarms rapidly over the surface of a synthetic mediumcreating remarkable hyperbranched dendritic communities. Models to reproduce such effectshave been proposed under the form of parabolic Partial Differential Equations representingthe dynamics of the active cells (both motile and multiplying), the passive cells(non-motile and non-growing) and nutrient concentration. We test the numerical behavior ofsuch models and compare them to relevant experimental data together with a criticalanalysis of the validity of the models based on recent observations of the swarmingbacteria which show that nutrients are not limitating but distinct subpopulations growingat different rates are likely present.

In this paper, with the assumptions that an infectious disease has a fixedlatent period in a population and the latent individuals of the population maydisperse, we reformulate an SIR model for the population living in two patches(cities, towns, or countries etc.), which is a generalization of the classicKermack-McKendrick SIR model. The model is given by a system of delaydifferential equations with a fixed delay accounting for the latency andnon-local terms caused by the mobility of the individuals during the latentperiod. We analytically show that the model preserves some properties that theclassic Kermack-McKendrick SIR model possesses: the disease always dies out,leaving a certain portion of the susceptible population untouched (calledfinal sizes). Although we can not determine the two final sizes, we are able toshow that the ratio of the final sizes in the two patches is totally determinedby the ratio of the dispersion rates of the susceptible individuals between thetwo patches. We also explore numerically the patterns by which the disease diesout, and find that the new model may have very rich patterns for the diseaseto die out. In particular, it allows multiple outbreaks of the disease before itgoes to extinction, strongly contrasting to the classic Kermack-McKendrick SIRmodel.

This paper is devoted to the study of a predator-prey model in a patchy environment.The model represents the interactions between phytoplankton and zooplankton in the water column.Two patches are considered with respect to light availability: one patch is associated to thesurface layer, the second patch describes the bottom layer. We show that this spatial heterogeneitymay destabilize the predator-prey system, even in oligotrophic system where the nutrient is lowenough to avoid ”paradox-enrichment” phenomenon. Indeed, in this case, an heterogeneity indexcan be used as a bifurcation parameter, leading to a Hopf bifurcation. Moreover, we assume that individualscan be dispersed in both patches via hydrodynamism processes, like in a mixed layer. Theeffect of mixing intensity is analysed as well as interactions between dispersion and enrichment.We also show that, in some cases, spatial heterogeneity has a stabilizing effect. These contrastedresults are examined by considering the non linear interaction between heterogeneity, dispersal andenrichment and some mechanisms leading to stabilization/destabilization are exhibited.

The notion of inside dynamics of traveling waves has been introduced in the recent paper[14]. Assuming that a traveling waveu(t,x) = U(x − c   t)is made of several components υi ≥ 0(i ∈ I ⊂ N), the inside dynamics of the wave is thengiven by the spatio-temporal evolution of the densities of the componentsυi. For reaction-diffusion equations of theform∂tu(t,x) = ∂xxu(t,x) + f(u(t,x)),where f is of monostable or bistable type, the results in [14] show that traveling waves can be classified intotwo main classes: pulled waves and pushed waves. Using the same framework, we study thepulled/pushed nature of the traveling wave solutions of delay equations

            ∂tu(t,x) = ∂xxu(t,x) + F(u(t −τ,x),u(t,x))

We begin with areview of the latest results on the existence of traveling wave solutions of suchequations, for several classical reaction terms. Then, we give analytical and numericalresults which describe the inside dynamics of these waves. From a point of view ofpopulation ecology, our study shows that the existence of a non-reproductive andmotionless juvenile stage can slightly enhance the genetic diversity of a speciescolonizing an empty environment.

We propose and analyze a mathematical model of hematopoietic stem cell dynamics. This model takesinto account a finite number of stages in blood production, characterized by cell maturity levels,which enhance the difference, in the hematopoiesis process, between dividing cells thatdifferentiate (by going to the next stage) and dividing cells that keep the same maturity level (bystaying in the same stage). It is described by a system of n nonlinear differential equationswith n delays. We study some fundamental properties of the solutions, such as boundedness andpositivity, and we investigate the existence of steady states. We determine some conditions for thelocal asymptotic stability of the trivial steady state, and obtain a sufficient condition for itsglobal asymptotic stability by using a Lyapunov functional. Then we prove the instability of axialsteady states. We study the asymptotic behavior of the unique positive steady state and obtain theexistence of a stability area depending on all the time delays. We give a numerical illustration ofthis result for a system of four equations.

The present paper introduces a tumormodel with two time scales, the time t during which the tumorgrows and the cycle time of individual cells. The model alsoincludes the effects of gene mutations on the population densityof the tumor cells. The model is formulated as a free boundaryproblem for a coupled system of elliptic, parabolic and hyperbolicequations within the tumor region, with nonlinear and nonlocalterms. Existence and uniqueness theorems are proved, andproperties of the free boundary are established.

Modification of behaviour in response to changes in the environment or ambientconditions, based on memory, is typical of the human and, possibly, many animalspecies.One obvious example of such adaptivity is, for instance, switching to a saferbehaviour when in danger, from either a predator or an infectious disease. In humansociety such switching to safe behaviour is particularly apparent during epidemics.Mathematically, such changes of behaviour in response to changes in the ambient conditionscan be described by models involving switching. In most cases, this switching is assumedto depend on the system state, and thus it disregards the history and, therefore, memory.Memory can be introduced into a mathematical model using a phenomenon known as hysteresis.We illustrate this idea using a simple SIR compartmental model that is applicable inepidemiology. Our goal is to show why and how hysteresis can arise in such a model, andhow it may be applied to describe a variety of memory effects. Our other objective is tointroduce a unified paradigm for mathematical modelling with memory effects inepidemiology and ecology. Our approach treats changing behaviour as an irreversible flowrelated to large ensembles of elementary exchange operations that recently has beensuccessfully applied in a number of other areas, such as terrestrial hydrology, andmacroeconomics. For the purposes of illustrating these ideas in an application to biology,we consider a rather simple case study and develop a model from first principles. Weaccompany the model with extensive numerical simulations which exhibit interestingqualitative effects.

This paper extends the volume filling chemotaxis model [18, 26] by taking into account the cell population interactions. Theextended chemotaxis models have nonlinear diffusion and chemotactic sensitivity dependingon cell population density, which is a modification of the classical Keller-Segel model inwhich the diffusion and chemotactic sensitivity are constants (linear). The existence andboundedness of global solutions of these models are discussed and the numerical patternformations are shown. The further improvement is proposed in the end.

We present a unified mathematical approach to epidemiological models with parametricheterogeneity, i.e., to the models that describe individuals in the population as havingspecific parameter (trait) values that vary from one individuals to another. This is anatural framework to model, e.g., heterogeneity in susceptibility or infectivity ofindividuals. We review, along with the necessary theory, the results obtained using thediscussed approach. In particular, we formulate and analyze an SIR model with distributedsusceptibility and infectivity, showing that the epidemiological models for closedpopulations are well suited to the suggested framework. A number of known results from theliterature is derived, including the final epidemic size equation for an SIR model withdistributed susceptibility. It is proved that the bottom up approach of the theory ofheterogeneous populations with parametric heterogeneity allows to infer the populationlevel description, which was previously used without a firm mechanistic basis; inparticular, the power law transmission function is shown to be a consequence of theinitial gamma distributed susceptibility and infectivity. We discuss how the generaltheory can be applied to the modeling goals to include the heterogeneous contactpopulation structure and provide analysis of an SI model with heterogeneous contacts. Weconclude with a number of open questions and promising directions, where the theory ofheterogeneous populations can lead to important simplifications and generalizations.

Mathematical models of tumour spheroids, proposed since the early seventies, have beengenerally formulated in terms of a single diffusive nutrient which is critical for cell replicationand cell viability. Only recently, attempts have been made to incorporate in the models the cellenergy metabolism, by considering the interplay between glucose, oxygen and lactate (or pH).By assuming glucose and lactate as the only fuel substrates, we propose a simple model for thecell ATP production which takes into account the main reactions that occur in the glycolytic andthe oxidative pathway. Under the assumption that cell death occurs when ATP production fallsbelow a critical level, we have studied the free boundary problem for the concentration of glucose,lactate and oxygen inside the spheroid viable rim. We show that the existence of a necrotic core isguaranteed for a sufficiently large size of the spheroid.

We obtain solvability conditions in H6(ℝ3) for asixth order partial differential equation which is the linearized Cahn-Hilliard problemusing the results derived for a Schrödinger type operator without Fredholm property in ourpreceding article [18].

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.

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.

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.