We investigate the Hawkes processes on the positive real line exhibiting both self-excitation and inhibition. Each point of such a point process impacts its future intensity by the addition of a signed reproduction function. The case of a nonnegative reproduction function corresponds to self-excitation, and has been widely investigated in the literature. In particular, there exists a cluster representation of the Hawkes process which allows one to apply known results for Galton–Watson trees. We use renewal techniques to establish limit theorems for Hawkes processes that have reproduction functions which are signed and have bounded support. Notably, we prove exponential concentration inequalities, extending results of Reynaud-Bouret and Roy (2006) previously proven for nonnegative reproduction functions using a cluster representation no longer valid in our case. Importantly, we establish the existence of exponential moments for renewal times of M/G/$\infty$ queues which appear naturally in our problem. These results possess interest independent of the original problem.

Abstract. Mechanisms leading to speciation are a major focus in evolutionary biology. In this paper, we present and study a stochastic model of population where individuals, with type a or A, are equivalent from ecological, demographical and spatial points of view, and differ only by their mating preference: two individuals with the same genotype have a higher probability to mate and produce a viable offspring. The population is subdivided in several patches and individuals may migrate between them. We show that mating preferences by themselves, even if they are very small, are enough to entail reproductive isolation between patches, and we provide the time needed for this isolation to occur as a function of the population size. Our results rely on a fine study of the stochastic process and of its deterministic limit in large population, which is given by a system of coupled nonlinear differential equations. Besides, we propose several generalisations of our model, and prove that our findings are robust for those generalisations.

We are interested in the impact of natural selection in a prey-predator community. We introduce an individual-based model of the community that takes into account both prey and predator phenotypes. Our aim is to understand the phenotypic coevolution of prey and predators. The community evolves as a multi-type birth and death process with mutations. We first consider the infinite particle approximation of the process without mutation. In this limit, the process can be approximated by a system of differential equations. We prove the existence of a unique globally asymptotically stable equilibrium under specific conditions on the interaction among prey individuals. When mutations are rare, the community evolves on the mutational scale according to a Markovian jump process. This process describes the successive equilibria of the prey-predator community and extends the Polymorphic Evolutionary Sequence to a coevolutionary framework. We then assume that mutations have a small impact on phenotypes and consider the evolution of monomorphic prey and predator populations. The limit of small mutation steps leads to a system of two differential equations which is a version of the canonical equation of adaptive dynamics for the prey-predator coevolution. We illustrate these different limits with an example of prey-predator community that takes into account different prey defense mechanisms. We observe through simulations how these various prey strategies impact the community.

Birth and death processes with interactions, multitype branching processes, large population limits, mating preferences. The authors thank the CNRS for its financial support through its competitive funding programs on interdisciplinary research. This work was partially funded by the Chair "Modélisation Mathématique et Biodiversité" of VEOLIA-Ecole Polytechnique-MNHN-F.X. H.L. acknowledges support from CONACyT-MEXICO and the foundation Sofía Kovalevskaia of SMM. The authors are grateful to András Tóbiás and an anonymous referee for their careful reading of the paper and useful comments.

International audienceWe are interested in prey-predator communities where the predator population evolves much faster than the prey's (e.g. insect-tree communities). We introduce a piecewise deterministic model for these prey-predator communitiesthat arises as a limit of a microscopic model when the number of predators goes to infinity. Weprove that the process has a unique invariant probability measure and that it is exponentially ergodic. Further on, we rescale the predator dynamics in order to model predators of smaller size. This slow-fast system converges to a community process in which the prey dynamics is averaged on the predator equilibria. This averaged process admits an invariant probability measure which can be computed explicitly. We use numerical simulations to study the convergence of the invariant probability measures of the rescaled processes

CD8 + T cells are frontline defenders against cancer and primary targets of current immunotherapies. In CLL, specific functional alterations have been described in circulating CD8 + T cells, yet a global view of the CD8 + T cell compartment phenotype and of its real impact on disease progression is presently elusive. We developed a multidimensional statistical analysis of CD8 + T cell phenotypic marker expression based on whole blood multi-color flow-cytometry. The analysis comprises both unsupervised statistics (hClust and PCA) and supervised classification methods (Random forest, Adaboost algorithm, Decision tree learning and logistic regression) and allows to cluster patients by comparing multiple phenotypic markers expressed by CD8 + T cells. Our results reveal a global CD8 + T cell phenotypic signature in CLL patients that is significantly modified when compared to healthy donors. We also uncover a CD8 + T cell signature characteristic of patients evolving toward therapy within 6 months after phenotyping. The unbiased, not predetermined and multimodal approach highlights a prominent role of the memory compartment in the prognostic signature. The analysis also reveals that imbalance of the central/effector memory compartment in CD8 + T cells can occur irrespectively of the elapsed time after diagnosis. Taken together our results indicate that, in CLL patients, CD8 + T cell phenotype is imprinted by disease clinical progression and reveal that CD8 + T cell memory compartment alteration is not only a hallmark of CLL disease but also a signature of disease evolution toward the need for therapy.

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