We develop the qualitative theory of thesolutions of the McKendrick partial differential equation ofpopulation dynamics. We calculate explicitly the weak solutionsof the McKendrick equation and of the Lotka renewal integralequation with time and age dependent birth rate. Mortality modulusis considered age dependent. We show the existence of demographycycles. For a population with only one reproductive age class,independently of the stability of the weak solutions and after atransient time, the temporal evolution of the number ofindividuals of a population is always modulated by a time periodicfunction. The periodicity of the cycles is equal to the age ofthe reproductive age class, and a population retains the memoryfrom the initial data through the amplitude of oscillations. For apopulation with a continuous distribution of reproductive ageclasses, the amplitude of oscillation is damped. The periodicityof the damped cycles is associated with the age of the firstreproductive age class. Damping increases as the dispersion of thefertility function around the age class with maximal fertilityincreases. In general, the period of the demography cycles isassociated with the time that a species takes to reach thereproductive maturity.

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.

We propose a matrix population modellingapproach in order to describe the dynamics of a grayling(Thymallus thymallus, L. 1758) population living in the Ainriver (France). We built a Leslie like model, which integrates theclimate changes in terms of temperature and discharge. First, weshow how temperature and discharge can be related to life historytraits like survival and reproduction. Second, we show how to usethe population model to precisely examine the life cycle ofgrayling : estimated numbers of individuals at the beginning ofthe reproduction season, estimated number of individuals insummer, comparison between observed and simulated numbers ofindividuals. Finally, we discuss our results in the framework ofthe assessment of impact of possible climate changes on graylingpopulations.

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 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.

We review the basic pathology of cyclical neutropenia in both humansand the grey collie, and examine the role that mathematical modelingof hematopoietic cell production has played in our understanding ofthe origins of this fascinating dynamical disease.

We study a reaction-diffusion equationwith an integral term describing nonlocal consumption of resourcesin population dynamics. We show that a homogeneous equilibrium canlose its stability resulting in appearance of stationary spatialstructures. They can be related to the emergence of biologicalspecies due to the intra-specific competition and randommutations. Various types of travelling waves are observed.

Neutropenia is a significant dose-limiting toxicity of cancerchemotherapy, especially in dose-intensified regimens. It iswidely treated by injections of Granulocyte Colony-StimulatingFactor (G-CSF). However, optimal schedules of G-CSF administrationare still not determined. In order to aid in identifying moreefficacious and less neutropenic treatment protocols, we studied adetailed physiologically-based computer model of granulopoiesis,as affected by different treatment schedules of doxorubicin and/orGranulocyte Colony-Stimulating Factor (G-CSF). We validated themodel as evident from accurate prediction of clinical data onhuman granulopoiesis in healthy individuals and indoxorubicin-treated cancer patients, with or without G-CSFsupport. Based on our model, we suggest new G-CSF administrationregimens. These regimens include reduced G-CSF doses, optimallytimed post-chemotherapy. Application of these regimens can lead tominimization of G-CSF side effects, as well as more cost-effectiveand less myelotoxic protocols. Currently clinical trials are beingdesigned in order to test these new treatment regimens.

We present models of the dynamics ofphytoplankton aggregates. We start with an individual-based modelin which aggregates can grow, divide, joint and move randomly.Passing to infinity with the number of individuals, we obtain amodel which describes the space-size distribution of aggregates.The density distribution function satisfies a non-linear transport equation, which contains terms responsible for thegrowth of phytoplankton aggregates, their fragmentation,coagulation, and diffusion.

In this work, we introduce a new software created to study hematopoiesis at the cellpopulation level with the individually based approach. It can be used as an interface between theoreticalworks on population dynamics and experimental observations. We show that this softwarecan be useful to study some features of normal hematopoiesis as well as some blood diseases suchas myelogenous leukemia. It is also possible to simulate cell communication and the formation ofcell colonies in the bone marrow.

A large variety of complexspatio-temporal patterns emerge from the processes occurring inbiological systems, one of them being the result of propagatingphenomena. This wave-like structurescan be modelled via reaction-diffusion equations. If a solution ofa reaction-diffusion equation represents a travelling wave, theshape of the solution will be the same at all time and the speedof propagation of this shape will be a constant. Travelling wavesolutions of reaction-diffusion systems have been extensivelystudied by several authors from experimental, numerical andanalytical points-of-view.In this paper we focus on two reaction-diffusion modelsfor the dynamics of the travelling waves appearing during theprocess of the cells aggregation. Using singular perturbationmethods to study the structure of solutions, we can deriveanalytic formulae (like for the wave speed, for example) in termsof the different biochemical constants that appear in the models.The goal is to point out if the models can describe inquantitative manner the experimental observations.

Patients with acute myeloblastic leukemia (AML) are divided according to the FrenchAmerican British (FAB) classification into eight subgroups (M0 to M7) on the basis of their degreeof maturation/differentiation. However, even if immunophenotypical characterization by flow cytometryis routinely used to distinguish between AML and acute lymphoblastic leukemia (ALL), itis not yet well established for the identification within the AML subgroups. Here we show that certainsubgroups of AML can be identified with a good accuracy (80% of the patients we analyzed)by means of the expression of specific markers.

An optimal control problem is studiedfor a Lotka-Volterra system of three differential equations. Itmodels an ecosystem of three species which coexist. The speciesare supposed to be separated from each others. Mathematically,this is modeled with the aid of two control variables. Somenecessary conditions of optimality are found in order to maximizethe total number of individuals at the end of a given timeinterval.

Spatial dynamics of fibrin clot formation in non-stirred system activated by glasssurface was studied as a function of FIX activity. Haemophilia B plasma was obtained fromuntreated patients with different levels of FIX deficiency and from severe haemophilia B patienttreated with FIX concentrate (Ahemphil B) during its clearance with half-life t1/2=12 hours. Asreported previously (Ataullakhanov et al. Biochim Biophys Acta 1998; 1425: 453-468), clotgrowth in space showed two distinct phases: activation and propagation. The activation phase ischaracterized by the time required to start clot growth from the activator, while the characteristicparameter of the propagation phase is the clot elongation rate. This rate reaches steady state inapproximately ten minutes after the beginning of growth. In haemophilia B plasma, clotformation is substantially impaired: clot starts to grow from the activating surface later than inhealthy donor plasma, and its propagation rate is considerably lower. The most significantabnormalities in clot growth kinetics are observed at FIX activity below 10% of normal.Simulation of these experiments was performed theoretically using a detailed biochemical model(Panteleev et al. Biophys J 2006; 90: 1489-1500) adapted for experimental conditions used.Suitability of the assumptions used to describe triggering contact activation was verified.

We consider an ecosystem in whichspiders may be transported by the wind from vineyards into thesurrounding woods and vice versa. The model takes into accountthis tranport phenomenon without building space explicitly intothe governing equations. The equilibria of the dynamical systemare analyzed together with their stability, showing thatbifurcations may occur. Then the effects of indiscriminatedspraying to keep pests under control is also investigated viasuitable simulations.