Keywords: **random** **size**-**structured** **population**, division kernel, nonparametric estima- tion, Goldenshluger-Lepski’s method, adaptive estimator, penalization, optimal rate.
1 Introduction
Models for populations of dividing cells possibly differentiated by covariates such as **size** have made the subject of an abundant literature in recent years (starting from Athreya and Ney [3], Harris [18], Jagers [28]...) Covariates termed as ‘**size**’ are variables that grow deterministically with time (such as volume, length, level of certain proteins, DNA content, etc.) Such models of **structured** populations provide descriptions for the evolution of the **size** distribution, which can be interesting for applications. For instance, in the spirit of Stewart et al. [33], we can imagine that each cell contains some toxicities whose quantity plays the role of the **size**. The asymmetric divisions of the cells, where one daughter contains more toxicity than the other, can lead under some conditions to the purge of the toxicity in the **population** by concentrating it into few lineages. These results are linked with the concept of aging for cell lineage. This concept has been tackled in many papers (e.g. Ackermann et al. [2], Aguilaniu et al. [1], C-Y. Lai et al. [22], Evans and Steinsaltz [15], Moseley [27]...).

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at location j of basin i. We estimated linear effects (β 1 , β 2 , β 3 , β 4 and β 5 ) of temperature, stream width, stream slope, month and year, implemented as numerical variables, to con- trol for changes in stream morphology, seasonal and yearly changes in the descriptors. We further modelled quadratic effects (γ 1 and γ 2 ) of temperature and month, to address nonlinearity in temperature effects and to control for non- linear seasonal variation in network structure that would not be explained by fluctuations in temperature. In addition, we also added Gaussian (mean of 0 and specific variances σ Ui , σ Vj(i) , σ Ll and σ Mm(l) ) **random** effects for station, nested in basin (1|basin) + (1|basin:station) and for technique, nested in method (1|method) + (1|technique:method), to account for changes sampling scheme and spatiotemporal pseudo- replication that may arise because of repeated sampling at the same stations and basins throughout the years. In this way the general effects of temperature that we recover from these models are robust to spatially and temporally repeated mea- surements in the same locations, and to seasonal and long- term changes in trophic structure that may not be accounted for by temperature. In total, we built 10 full-models, one for each topological descriptor, that we simplified following a step-wise AIC-based backward procedure. At each step, the most suitable term to delete was identified by considering the term associated with the smallest ∆AIC. Then the sig- nificance of the deletion of the identified term was assessed through log-likelihood ratio tests of the full model compared to the model without the term (χ 2 -test statistic as appropri- ate for large sample **size**). The simplification was performed first on **random** effects, and then on fixed effects (see the Supporting information for details). For a complete presen- tation of model outputs see the Supporting information.

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Perhaps the most delicate assumption is (b). The cell division process may result in daughters with unequal sizes. Equal division does not occur surely for all individuals of a microbiological **population**, but we may quantify how much they deviate from a perfect division by a statistical description, as explained in the work of Koch [7].
The bacteria E. Coli is known for its quite small variation in its subdivisions [7, 13, 14]. It is usually assumed (see [7]) that the fluctuations in the critical **size**-at-division of individual cells is **random** and not correlated with other cell cycle events in the current cell generation or in earlier generations. This justifies (c) for the bacteria E. Coli.

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1.2 The statistical approach
In this paper, we propose to overcome the limitation of the deterministic inverse problems approach by assuming that we have n data, each data being obtained from the measure- ment of an individual cell picked at **random**, after the system has evolved for a long time so that the approximation n(t, x) ≈ N(x)e λt is valid. This is actually what happens if one observes cell cultures in laboratory after a few hours, a typical situation for E. Coli cultures for instance, provided, of course, that the underlying aggregation-fragmentation equation is valid.

Definition 3. A subset Y of set X is a uniform sample of X of **size** k if Y is any possible subset of X of **size** k with equal probability.
This short note proves the two following theorems:
Theorem 4. Given an (ε, κ)-sample X in T d , a Bernoulli sample Y of probability α of X , and a point p, the expected number of faces of dimension i of the Delaunay triangulation of Y ∪ {p} that contain p is at most 72 di+O(d+i(log iκ)) . In particular this number is a constant with respect to n and α.

3. Stationary CB
In contrast to Wright-Fisher **population** models, CB models do not exhibit stationary distributions. However, by conditioning sub-critical CB to non-extinction (see [45], [20] and [31] for details), one get the so-called Q-process, which we denotes by Y ′′ . This process is also a CB process with immigration introduced in [28] and may have a stationary distribution. This process, as pointed out in [3] see also [19], has a heuristic interpretation by introducing a fixed ancestral lineage, namely it is an independent sum of the process Y and the **population** thrown off by an ”immortal individual” whose laws coincide with the law of a generic **population** Y . We introduce the process Y ′′ in Section 3.1 as well as its stationary version Z. Then we check in Section 3.2, that under (A1) the process Y ′′ is indeed the Q-process associated to

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randomized incremental construction of the Delaunay triangulation of an ε-net in general 51.. position takes time O(n log n) in any dimension.[r]

is only **structured** in age (see [39] for a study of the linear equation). This model has been
extended with other structuring variables as **size** for example (see [32, 37]) and then with more variables (representing DNA content, maturation, etc.) to illustrate biological phenomena, among many others, like cell division (see [23, 33]), proliferative and quiescent states of tumour cells (see [2, 26]). Space **structured** problems have also been extensively studied (see [28, 35, 36, 38]). The variable x can represent different biological quantities that evolve throughout the individual lifespan and that are not inherited at birth. These can be as diverse as, for example, the **size** of individuals, a physiological age, a parasite load and many others. Therefore we assume that the propgression speed A depends both on x and the trait y to keep the model (1) quite general. In the present paper, we refer to x as the age for simplicity. Studies in these contexts can be found in [13] about the existence of steady states for a selection-mutation model **structured** by physiological age and maturation age, which is considered as a phenotypical trait.

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In this article, we consider a species whose **population** density solves the steady diffusive logistic equation in a heterogeneous environment modeled with the help of a spatially non constant coefficient standing for a resources distribution. We address the issue of maximizing the total **population** **size** with respect to the resources distribution, considering some uniform pointwise bounds as well as prescribing the total amount of resources. By assuming the diffusion rate of the species large enough, we prove that any optimal configuration is bang- bang (in other words an extreme point of the admissible set) meaning that this problem can be recast as a shape optimization problem, the unknown domain standing for the resources location. In the one-dimensional case, this problem is deeply analyzed, and for large diffusion rates, all optimal configurations are exhibited. This study is completed by several numerical simulations in the one dimensional case.

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As in Rousset and Leblois (2007), Rousset and Leblois (2012) and Leblois et al. (2014) we can conduct the inference process as follow. We first define a set of parameter points through a stratified **random** sample within a range of the parameter space provided by the user. Then, at each parameter point, the multilocus likelihood is the product of the likelihoods for each locus, which are estimated through the SIS or SISR algorithm. The likelihoods inferred at the different parameter points are then smoothed by a Kriging scheme. After a first analysis of the smoothed likelihood surface, the algorithm can be repeated a second time to increase the density of parameter points in the neighborhood of the first MLE. The Kriging step removes part of the estimation error of likelihood in any given parameter point by assuming that the likelihood is a smooth function of the parameter. In this Section, we thus conducted numerical experiments to show that the gain of SISR over SIS in accuracy of likelihood estimates is retained through the Kriging step.

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A B S T R A C T
Changes in agricultural practices have lead to pollination deficits in entomophilous crops, leading to a growing interest in supplementing farmlands with managed colonies of honey bee, Apis mellifera. However, the metrics of a colony as a pollination unit is controversial due to the wide range of adult **population** sizes encountered in a colony, especially in relation with the time of year and beekeeping management. Correctly measuring the number of adult honey bees per hive is critical for farmers to adjust the number of colonies they need to meet crop pollination demand. We tested a simple non-invasive method to estimate the adult worker **population** **size** of colonies based on common beekeeping handlings. This method consisted in counting the number of inter-frames covered with adult bees (called IFB thereafter) from above the hive body. Based on the monitoring of 181 colonies, we investigated the nature of the relation between IFB and the adult bee **population** **size** and its context- dependence to the meterological conditions and hive type. We then evaluated the possible improvement of the method with additional IFB counted in the supers and from below the hive body. Finally, we analysed the robustness of the method by comparing estimates obtained from colonies observed by experimented and naive observers. We revealed a clear-cut logarithmic relation between the IFB and the adult **population** **size**, covering the effects of meteorological conditions and hive type. The counting of IFB from above the hive body were particularly sensitive to meteorological conditions, unlike those counted from below the hive body. Moreover, the counting of additional IFB from the supers slightly improved the estimates of adult **population** **size**. Inter- estingly, no difference of estimate was detected between experimented and naive observers, suggesting applied simplicity of the method. The IFB counting method thus provides a simple, non-invasive and robust indicator of the adult **population** **size** of a managed honey bee colony. The counting of IFB from below the hive body should be recommend due to the sensitivity to meteorological conditions of the counting of IFB from above the hive body. Beyond crop pollination, we also highlighted application perspectives of this method as an indicator of survival probability. This method can therefore be viewed as a standard for routine field monitoring (i) to help farmers to estimate rigorously the number of colonies they need to meet the crop pollination demand and (ii) to help beekeepers assessing the mortality risk of their colonies.

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Observe that if φ is the constant flow, then (X t ) t ≥0 is an homogeneous continuous time Markov chain on G with transition kernel p(x, d x ′ ) = q(x)r(x, d x ′ ).
Observe also that if the rate function q is constant along the flow of the dynamic system, that is q( φ t (x)) = q(x), then the construction of the sequence may be sim- plified as (if E (a) denotes the law of an exponential **random** variable of parameter

* nicolas.galtier@univ-montp2.fr
Abstract
The rate at which genomes adapt to environmental changes and the prevalence of adaptive processes in molecular evolution are two controversial issues in current evolutionary genet- ics. Previous attempts to quantify the genome-wide rate of adaptation through amino-acid substitution have revealed a surprising diversity of patterns, with some species (e.g. Dro- sophila) experiencing a very high adaptive rate, while other (e.g. humans) are dominated by nearly-neutral processes. It has been suggested that this discrepancy reflects between- species differences in effective **population** **size**. Published studies, however, were mainly focused on model organisms, and relied on disparate data sets and methodologies, so that an overview of the prevalence of adaptive protein evolution in nature is currently lacking. Here we extend existing estimators of the amino-acid adaptive rate by explicitly modelling the effect of favourable mutations on non-synonymous polymorphism patterns, and we apply these methods to a newly-built, homogeneous data set of 44 non-model animal spe- cies pairs. Data analysis uncovers a major contribution of adaptive evolution to the amino- acid substitution process across all major metazoan phyla —with the notable exception of humans and primates. The proportion of adaptive amino-acid substitution is found to be positively correlated to species effective **population** **size**. This relationship, however, appears to be primarily driven by a decreased rate of nearly-neutral amino-acid substitution because of more efficient purifying selection in large populations. Our results reveal that adaptive processes dominate the evolution of proteins in most animal species, but do not corroborate the hypothesis that adaptive substitutions accumulate at a faster rate in large populations. Implications regarding the factors influencing the rate of adaptive evolution and positive selection detection in humans vs. other organisms are discussed.

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Motivations. The Moea/d framework is based on the decomposition of the original MCOP into a set of smaller sub-problems that are mapped to a popu- lation of individuals. In its basic variant [22], Moea/d considers a set of single- objective sub-problems defined using a scalarizing function transforming a multi- dimensional objective vector into a scalar value w.r.t. one weight (or direction) vector in the objective space. The **population** is then typically **structured** by mapping one individual to one sub-problem targeting a different region of the objective space. Individuals from the **population** are evolved following a coopera- tive mechanism in order for each individual (i) to optimize its own sub-problem, and also (ii) to help solving its neighboring sub-problems.The **population** hence ends up having a good quality w.r.t. all sub-problems. Although being extremely simple and flexible, the computational flow of Moea/d is constantly redesigned to deal with different issues. Different Moea/d variants have been proposed so far in the literature, e.g., to study the impact of elitist replacements [19], of generational design [13], or of stable-matching based evolution [12], and other mechanisms [1]. In this paper, we are interested in the interdependence between the **population** **size**, which is implied by the number of sub-problems defined in the initial decomposition, and the internal evolution mechanisms of Moea/d.

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where denotes the brain volume for all 36 brain regions of the social atlas and denotes the (z- scored) age of the participants. Gaussian-distributed hyper-priors for beta coefficients underlying network-level volume variation jointly inform beta coefficients at the region level for each participant group . Variance that could be explained by the nuisance variables of body mass and head **size** was accounted for as potential confounds. The participant groups indicated stratification of our **population** sample into male and female with or without presence of a certain social trait. For the example regarding household **size**, the groups corresponded to [male lives alone], [male lives with others], [female lives alone], and [female lives with others]. This multi-group regression approach also capitalized on the fact that sex and age differences are among the by far largest sources of variability in MRI scans in general (Miller et al., 2016; Ritchie et al., 2018). In this way, we could get the most of our rich sample by borrowing statistical strength between clusters of individuals in our **population** through interlocking of their model coefficients. Parameters of the within-group regressions, placed at the bottom, were modeled themselves by the hyper-parameters of the across-group regression to pool information across batches of variance components. We could thus provide more coherent and detailed quantitative answers to questions about morphological differentiation of the social brain by a joint model estimation profiting from several sources of **population** variation.

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• deriving the 3D histogram from 2D sections, thanks to the so-called Saltykov method [ 25 ]; • fitting a lognormal distribution on the 3D histograms.
The Saltykov method (also known as Scheil-Schwartz-Saltykov method) is an iterative procedure, consisting in evaluating the probability of cutting each grain (considered as spherical) at a certain latitude, resulting in an reduced apparent **size**. This method is based on the finite histogram (finite number of classes) of apparent sizes. The fundamental assumption is that the upper class (i.e. larger apparent sizes) is given by the largest grains cut near their equatorial planes. When cutting a particle of radius R at a **random** latitude, the probability of finding an apparent radius r comprised in between r 1 and r 2 is given by:

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Effective population size and heterozygosity-fitness correlations in a population of the Mediterranean lagoon ecotype of long-snouted seahorse Hippocampus guttulatus... Population geneti[r]

maturation between mature male parr and anadromous males.
Similarly, this study is also the first to quantify the con- tribution of mature male parr to reducing relatedness among parents and inbreeding in fry in Atlantic salmon. While inbreeding was not significantly different between progeny fathered by mature male parr and anadromous males, relatedness was in average smaller between mature male parr and anadromous female than between anadro- mous partners. We thus propose that alternative sexual maturation strategies may result in reduced relatedness among breeders as well as long-term inbreeding depression at the **population** level. In salmonids, dispersal and active kin avoidance could play a role in inbreeding reduction. However, homing can be extremely accurate and gene flow low even between relatively close subpopulations of a single river-system (Dionne et al. 2008; Vaha et al. 2008). Simi- larly, while active mate choice based on MHC-related genes diversity has been suggested in salmonids (Landry et al. 2001), active kin avoidance among anadromous breeders has never been demonstrated. Two main mechanisms diminishing inbreeding in Atlantic salmon without imply- ing an active, and selected for, kin avoidance can thus be proposed: multiple mating and overlap among generations. In the absence of mature male parr and in semi-natural conditions, Garant et al. (2005) suggested that multiple mating increased individual reproductive success and the proportion of outbred progeny in female Atlantic salmon. Although generation overlap can be provided by differences in age structure in anadromous fish, we propose that mat- ing between anadromous females and asynchronously maturing male parr of an earlier cohort can be an effective way to decrease inbreeding. While mature male parr are most commonly 1+ or 2+ old (Dalley et al. 1983; Fig. 1), anadromous breeders are predominantly 4+, 5+, or 6+ (Palstra and Dionne 2011; Fig. 1), respectively. Therefore, while it is possible for anadromous breeders to breed with half-sibs or full sibs, it is virtually impossible for an anadro- mous female to mate with a mature male parr of same sex. Given the negative relationship between individuals’ inbreeding and fitness previously documented in salmonids (Wang et al. 2001; Houde et al. 2011; Naish et al. 2013), such a strategy minimizing inbreeding without imposing a reproductive cost through kin avoidance might increase individual fitness, especially in the case of small popula- tions where inbreeding is expected to rise. While the decrease of inbreeding associated with asynchronous matu- ration has been shown in subsocial spiders (Bukowski and Aviles 2002), this phenomenon has rarely been proposed in other animal taxa. Hence, future studies could further investigate the effect of asynchronous maturation in salmo- nids by linking parent age differences to their relatedness, their reproductive success and inbreeding in fry.

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/ La version de cette publication peut être l’une des suivantes : la version prépublication de l’auteur, la version acceptée du manuscrit ou la version de l’éditeur. Access and use of [r]

As the experimental data were performed in vitro in a liquid medium, and as cells were kept in the medium three days before the recordings in order not to be synchronised by a “serum shock” (meaning a steep increase in growth factors contained in FBS induced by sudden serum adjunction) from the beginning of recordings, we could consider that there was no synchronisation between cells, hence no time dependency of the control of the growth process at the cell **population** level. These facts were consistent with our assumption of stationary transition coefficients, i.e., the assumption under which transition rates from

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