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.
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 .
@cite_14 @cite_4 @cite_1 @cite_3 @cite_13 @cite_17
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Cooperation is a difficult proposition in the face of Darwinian selection. Those that defect have an evolutionary advantage over cooperators who should therefore die out. However, spatial structure enables cooperators to survive through the formation of homogeneous clusters, which is the hallmark of network reciprocity. Here we go beyond this traditional setup and study the spatiotemporal dynamics of cooperation in a population of populations. We use the prisoner's dilemma game as the mathematical model and show that considering several populations simultaneously gives rise to fascinating spatiotemporal dynamics and pattern formation. Even the simplest assumption that strategies between different populations are payoff-neutral with one another results in the spontaneous emergence of cyclic dominance, where defectors of one population become prey of cooperators in the other population, and vice versa. Moreover, if social interactions within different populations are characterized by significantly different temptations to defect, we observe that defectors in the population with the largest temptation counterintuitively vanish the fastest, while cooperators that hang on eventually take over the whole available space. Our results reveal that considering the simultaneous presence of different populations significantly expands the complexity of evolutionary dynamics in structured populations, and it allows us to understand the stability of cooperation under adverse conditions that could never be bridged by network reciprocity alone. How do human groups maintain a high level of cooperation despite a low level of genetic relatedness among group members? We suggest that many humans have a predisposition to punish those who violate group-beneficial norms, even when this imposes a fitness cost on the punisher. Such altruistic punishment is widely observed to sustain high levels of cooperation in behavioral experiments and in natural settings. We offer a model of cooperation and punishment that we call strong reciprocity: where members of a group benefit from mutual adherence to a social norm, strong reciprocators obey the norm and punish its violators, even though as a result they receive lower payoffs than other group members, such as selfish agents who violate the norm and do not punish, and pure cooperators who adhere to the norm but free-ride by never punishing. Our agent-based simulations show that, under assumptions approximating likely human environments over the 100,000 years prior to the domestication of animals and plants, the proliferation of strong reciprocators when initially rare is highly likely, and that substantial frequencies of all three behavioral types can be sustained in a population. As a result, high levels of cooperation are sustained. Our results do not require that group members be related or that group extinctions occur. A wide variety of practical problems related to the interaction of agents can be examined using biological metaphors. This paper applies the theory of G-networks to agent systems by considering a biological metaphor based on three types of entities: normal cells C, cancerous or bad cells B, and immune defense agents A which are used to destroy the bad cells B, but which sometimes have the effect of being able to destroy the good cells C as well (autoimmune response). Cells of type C can mutate into cells of Type B, and vice-versa. In the presence of probabilities of correct detection and false alarm on the part of agents of Type A, we examine how the dose of agent A will influence the desired outcome which is that most bad cells B are destroyed while the damage to cells C is limited to an acceptable level. In a second part of the paper we illustrate how a similar model can be used to represent a mixture of agents with the ability to cooperate as well as to compete. Cooperation in organisms, whether bacteria or primates, has been a difficulty for evolutionary theory since Darwin. On the assumption that interactions between pairs of individuals occur on a probabilistic basis, a model is developed based on the concept of an evolutionarily stable strategy in the context of the Prisoner's Dilemma game. Deductions from the model, and the results of a computer tournament show how cooperation based on reciprocity can get started in an asocial world, can thrive while interacting with a wide range of other strategies, and can resist invasion once fully established. Potential applications include specific aspects of territoriality, mating, and disease.