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 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 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 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.

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.

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.

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.

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.