We consider a cell population structured by a positive real number, describing the number of P-glycoproteins carried by the cell. We are interested in the effect of those proteins on the growth of the population: those proteins are indeed involve in the resistance of cancer cells to chemotherapy drugs. To describe this dynamics, we introduce a kinetic model. We then introduce a rigorous hydrodynamic limit, showing that if the exchanges are frequent, then the dynamics of the model can be described by a system of two coupled differential equations. Finally, we also show that the kinetic model converges to a unique limit in large times. The main idea of this analysis is to use Wasserstein distance estimates to describe the effect of the kinetic operator, combined to more classical estimates on the macroscopic quantities.
@cite_14: The theory of quantitative genetics bridges the gap between observable statistical properties of a character and the genetic and environmental factors which are thought to determine its expression. This book covers a wide range of genetic topics from Mendelian genetics to selection theory in breeding. It looks at the decomposition of the genotypic value into components representing various types of gene action; estimation of the components of variance and other quantities generated by this decomposition, and stochastic effects of finite population size.
@cite_4: Eusociality, in which some individuals reduce their own lifetime reproductive potential to raise the offspring of others, underlies the most advanced forms of social organization and the ecologically dominant role of social insects and humans. For the past four decades kin selection theory, based on the concept of inclusive fitness, has been the major theoretical attempt to explain the evolution of eusociality. Here we show the limitations of this approach. We argue that standard natural selection theory in the context of precise models of population structure represents a simpler and superior approach, allows the evaluation of multiple competing hypotheses, and provides an exact framework for interpreting empirical observations.
@cite_1: Inclusive fitness theory, summarised in Hamilton's rule, is a dominant explanation for the evolution of social behaviour. A parallel thread of evolutionary theory holds that selection between groups is also a candidate explanation for social evolution. The mathematical equivalence of these two approaches has long been known. Several recent papers, however, have objected that inclusive fitness theory is unable to deal with strong selection or with non-additive fitness effects, and concluded that the group selection framework is more general, or even that the two are not equivalent after all. Yet, these same problems have already been identified and resolved in the literature. Here, I survey these contemporary objections, and examine them in the light of current understanding of inclusive fitness theory.
@cite_3: Population genetics is concerned with the study of the genetic, ecological, and evolutionary factors that influence and change the genetic composition of populations. The emphasis here is on models that have a direct bearing on evolutionary quantitative genetics. Applications concerning the maintenance of genetic variation in quantitative traits and their dynamics under selection are treated in detail.
@cite_13: 
@cite_17: We consider populations structured by a phenotypic trait and a space variable, in a non-homogeneous environment. In the case of sexual populations, we are able to derive models close to existing models in theoretical biology, from a structured population model. We then analyze the dynamics of the population using a simplified model, where the population either propagates through the whole space or it survives but remains confined in a limited range. For asexual populations, we show that the dynamics are simpler. In this case, the population cannot remain confined in a limited range, i. e. the population, if it does not get extinct, propagates through the whole space.
Finally, let us mention two biological problems linked to the present article. The first class of biological problematic is cooperation in biological populations. Cooperation is a widespread behaviour: it exists at all levels of life, and appearance and stability of such phenomena is a challenging problem, and we refer to @cite_1 @cite_4 @cite_13 . One of the mechanisms of cooperation is the exchange of goods. These exchanges could be modelled with an operator similar to the one we derived in . The second class of problems is linked to sexual reproduction, where the formation of gametes and recombinations implies that the DNA of the offspring is a combination of the genomes of both parents. A model used in biology to describe the effect of those events on a given phenotype is the so-called infinitesimal model, see @cite_14 @cite_3 . The properties of the operator appearing in this model are similar to the exchange operator described in .We believe the analysis methods developed here could also be useful in those two other contexts. In particular, they could be an interesting step towards a rigorous mathematical description of the hydrodynamic limits introduced in @cite_17 .