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Lessard, Sabin

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Full Professor

Faculty of Arts and Science - Department of Mathematics and Statistics

André-Aisenstadt Office 5233

514 343-6818

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Affiliations

  • Membre Centre de recherches mathématiques
  • Membre CRM — Centre de recherches mathématiques

Research area

Student supervision Expand all Collapse all

Evolution of cooperation in evolutionary games with the opting-out strategy and under random environmental noise Theses and supervised dissertations / 2020-07
Li, Cong
Abstract
In this thesis, we study the effects of a stochastic environment and the use of an opting-out strategy on the evolution of cooperation in evolutionary games. The thesis contains 8 articles, among which 6 are already published in peer-reviewed journals. Apart from the introduction, the thesis is divided into two parts, Part 1 made with 5 articles and Part 2 with 3 articles. Part 1 studies randomized payoffs in evolutionary games. Article 1 introduces stability concepts for 2x2 matrix games in infinite populations undergoing discrete, non-overlapping generations in a stochastic environment and gives conditions for an equilibrium, either on the boundary or in the interior of the simplex of all strategy frequencies, to be stochastically locally stable or unstable. Article 2 extends the results of Article 1 to the case where fitness is an exponential function of expected payoff in random pairwise interactions and shows that, unexpectedly, environmental random noise can break a periodic cycle and promote stability of an interior equilibrium. Article 3 discusses the effects of weak selection. While stability conditions in a random environment return to conditions in the deterministic case as selection intensity diminishes, random fluctuations in payoffs can accelerate the speed of convergence toward a stable equilibrium under weaker selection. Article 4 applies stochastic evolutionary stability theory to a randomized Prisoner's dilemma game and shows that increasing the variance in payoffs for defection is conducive to the evolution of cooperation. Article 5 studies randomized matrix games in finite populations and gives conditions for selection to favor the evolution of cooperation in the context of a randomized Prisoner's dilemma. Part 2 considers a repeated Prisoner's dilemma game with an opting-out behavior adopted by every player in pairwise interactions. Article 6 studies the evolutionary dynamics of cooperation and defection in this context and shows possible long-term coexistence, assuming an infinite population and fast (actually, instantaneous) equilibrium in the pair frequencies. Article 7 reports experimental results with 264 university students using the opting-out strategy that support the theoretical prediction of a long-term coexistence of cooperation and defection. Article 8 extends the analysis of the model with the opting-out strategy to the case of a finite population and provides a rigorous proof of the two-time scales for the frequencies of cooperation and defection on one hand and the frequencies of strategy pairs on the other.

Évolution dans des populations structurées en classes Theses and supervised dissertations / 2019-05
Soares, Cíntia Dalila
Abstract
Firstly, we consider an age-structured asexual haploid population with constant age-specific birth and death (or survival) parameters for each type. In this simplified framework, the rate of change in mean fitness in continuous time is expected to be exactly equal to the genetic variance in fitness (or to the genetic variance in fitness divided by the mean fitness in discrete time), which can be seen as a generalized growth-rate theorem. Fisher’s (1930) Fundamental Theorem of Natural Selection (FTNS), and in particular the development of an explicit age-structured version of the theorem, is shown to hold with the Malthusian parameter used as the definition of fitness. Then, we consider class-structured populations, in discrete time in the limit of weak selection and with the inverse of the intensity of selection as unit of time. The aim is to establish a continuous model that approximates the discrete model. More precisely, we study frequency-dependent growth in an infinite haploid population structured into a finite number of classes such that individuals in each class contribute to a given subset of classes from one time step to the next. These contributions take the form of generalized fecundity parameters with perturbations of order 1/N that depends on the class frequencies of each type and the type frequencies. The main result is that the type frequencies as N → +∞ obey the replicator equation with instantaneous growth rates for the different types that depend only on instantaneous equilibrium class frequencies and reproductive values. An application to evolutionary game theory complemented by simulation results is presented. We ascertain the validity of a diffusion approximation for the frequencies of different types under recurrent mutation and frequency-dependent viability and fertility selection in a haploid population with a fixed age structure in the limit of a large population size. The approximation is used to study, and explain in terms of selection coefficients, reproductive values and population-age-structure coefficients, the differences in the effects of viability versus fertility selection on the fixation probability of an advantageous mutant. In the same context, we deduce the first-order effect of frequency-dependent viability and fertility selection on the probability of fixation of a mutant in a large finite haploid population with a fixed age structure by applying a direct small perturbation method to the neutral two-timescale genealogical process. This effect is expressed in terms of fixation-fitness coefficients times ancestry coefficients that are related to the effective population size. In the case of constant selection, the fixation-fitness coefficients are functions of the coefficients of viability and fertility selection weighted by reproductive values and population-structure coefficients for the different age classes. This explains the difference between the effects of viability selection and fertility selection on fixation probability in age-structured populations. With frequency-dependent selection in the form of a public goods game, the fixation-fitness coefficients depend also on the public good allocation strategy of the population and the resource allocation strategies of the individuals. In this case, the results show that weak selection may favor the fixation of cooperation if the cooperators allocate more resources to reproduction versus survival than the defectors.

Étude de quelques populations structurées : processus de coalescence et abondance d'une stratégie Theses and supervised dissertations / 2016-03
Kroumi, Dhaker
Abstract
In this thesis, we study some examples of structured populations. In particular, we analyze the evolution of cooperation in the sense of determining conditions that favor it. We study the evolution of cooperation in a phenotype space of any size. We study also the evolution of cooperation in a finite population subdivided into hierarchical groups of any size. Finally, we study the effect of aspiration on the evolutionary process in a finite population distributed on a circle with only a local interaction by pairwise.

Applications du processus ancestral avec recombinaison et conversion en génétique statistique Theses and supervised dissertations / 2013-12
Saidi, Lamiae
Abstract
The ancestral process is applied to investigate the amount of DNA variation and the amount of linkage disequilibrium ; it is also applied to make statistical inference about the multiple factors responsible for this variation. Considering genetic drift, mutation, and recombination events, the exact solutions for linkage disequilibrium between two loci are obtained. Furthermore, the association measure between two loci is obtained by using an approximation of the exact process, SMC (sequentially Markov chain), and correcting a covariance formula. After introducing intrachromosomal gene conversion under the Moran model, the expected amounts of variation within and between two loci are obtained using expected times spent in the ancestral states. Furthermore, the expectation of linkage disequilibrium is obtained and it is shown to decrease as the recombination rate is increased. Using these theoretical results, a method for estimating the mutation, recombination and gene conversion parameters is presented.

Probabilité et temps de fixation à l'aide de processus ancestraux Theses and supervised dissertations / 2013-11
Elgbeili, Guillaume
Abstract
The expected time for fixation given its occurrence, and the probability of fixa- tion of a new mutant allele in populations subject to various biological phe- nomena are analyzed using the approach of the ancestral process. First, the paper of Tajima (1990) is analyzed, and the missing or incomplete proofs are fully worked out in this Master thesis in order to familiarize ourselves with calculations of fixation times. Our study of Tajima’s paper helps to show the importance of the fixation time in some biological phenomena. Thereafter, we extend the work of Tajima (1990) by introducing the effect of natural selec- tion in the model. Using a diffusion approximation, the work of Mano (2009) provides an interesting result about the expected time of fixation given its oc- currence. We derived an alternative method that uses an ancestral process that approximates well Mani’s result. Simulations are made to verify the accuracy ofthenewapproach.Finally,onemodelsubjecttogeneconversionisanalyzed, since this phenomenon, in the presence of bias, has a similar effect as selection. We deduce an analytical result for the probability of fixation of a new mutant in the population. Finally, simulations are made to determine the probability of fixation and the time of fixation given its occurrence when rates are too large to be calculated analytically.

Mesures d'apparentement pour des modèles de sélection avec interactions dans une population structurée en groupes Theses and supervised dissertations / 2009
Martin, Géraldine
Abstract
Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal.

Probabilité de fixation dans des modèles génétiques de populations à plusieurs allèles Theses and supervised dissertations / 2008
Lahaie, Philippe
Abstract
Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal.

Processus de coalescence dans une population subdivisée avec possibilité de coalescences multiples Theses and supervised dissertations / 2008
Lasalle Ialongo, David
Abstract
Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal.

Effet de l'échantillonnage non proportionnel de cas et de témoins sur une méthode de vraisemblance maximale pour l'estimation de la position d'une mutation sous sélection Theses and supervised dissertations / 2008
Villandré, Luc
Abstract
Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal.

Probabilité de fixation dans des modèles de sélection avec consanguinité Theses and supervised dissertations / 2006
Langevin, Samuel
Abstract
Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Cartographie génétique fine par le graphe de recombinaison ancestral Theses and supervised dissertations / 2003
Larribe, Fabrice
Abstract
Thèse diffusée initialement dans le cadre d'un projet pilote des Presses de l'Université de Montréal/Centre d'édition numérique UdeM (1997-2008) avec l'autorisation de l'auteur.

Effets de la consanguinité dans des modèles de sélection pour des populations structurées en familles Theses and supervised dissertations / 2002
Rocheleau, Ghislain
Abstract
Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal.

Research projects Expand all Collapse all

Supplément COVID-19 CRSNG_Mathematical population genetics and evolutionary games CRSNG/Conseil de recherches en sciences naturelles et génie du Canada (CRSNG) / 2020 - 2021

Mathematical population genetics and evolutionary games CRSNG/Conseil de recherches en sciences naturelles et génie du Canada (CRSNG) / 2016 - 2024

Congrès canadien des étudiants en mathématiques FRQNT/Fonds de recherche du Québec - Nature et technologies (FQRNT) / 2016 - 2018

CENTRE DE RECHERCHES MATHEMATIQUES (CRM) FRQNT/Fonds de recherche du Québec - Nature et technologies (FQRNT) / 2015 - 2023

COMPUTATIONAL RESOURCES FOR RESEARCH IN MATHEMATICS AND STATISTICS CRSNG/Conseil de recherches en sciences naturelles et génie du Canada (CRSNG) / 2013 - 2015

MATHEMATICAL POPULATION GENETICS AND EVOLUTIONARY GAMES / 2011 - 2015

CENTRE DE RECHERCHES MATHEMATIQUES (CRM) FRQNT/Fonds de recherche du Québec - Nature et technologies (FQRNT) / 2008 - 2016

MATHEMATICAL POPULATION GENETICS AND EVOLUTIONARY GAMES CRSNG/Conseil de recherches en sciences naturelles et génie du Canada (CRSNG) / 1994 - 2017

Selected publications Expand all Collapse all

Evolution of cooperation in a multidimensional phenotype space

KROUMI Dhaker et LESSARD Sabin, Evolution of cooperation in a multidimensional phenotype space , (2015), , Theoretical Population Biology

Processus stochastiques

Lessard, Sabin, Processus stochastiques , 285 (2014), , Ellipses

Conditions for cooperation to be more abundant than defection in a hierarchically structured population

KROUMI Dhaker et LESSARD Sabin, Conditions for cooperation to be more abundant than defection in a hierarchically structured population , (2014), , Dynamic Games and Applications

Fixation probability in a two-locus model by the ancestral recombination-selection graph

Lessard, S. et Kermany, A. R. , Fixation probability in a two-locus model by the ancestral recombination-selection graph , 691-707 (2012), , Genetics 190,

Effect of epistasis and linkage on fixation probability in three-locus models: An ancestral recombination-selection graph approach

KERMANY Amir R. et LESSARD Sabin, Effect of epistasis and linkage on fixation probability in three-locus models: An ancestral recombination-selection graph approach 82, 131-145 (2012), , Theoretical Population Biology

Effective game matrix and inclusive payoff in group-structured populations

Lessard, Sabin, Effective game matrix and inclusive payoff in group-structured populations 1, 301--318 (2011), , Dyn. Games Appl.

On the robustness of the extension of the one-third law of evolution to the multi-player game

Lessard, Sabin, On the robustness of the extension of the one-third law of evolution to the multi-player game 1, 408--418 (2011), , Dyn. Games Appl.

Recurrence equations for the probability distribution of sample configurations in exact population genetics models

Lessard, Sabin, Recurrence equations for the probability distribution of sample configurations in exact population genetics models 47, 732--751 (2010), , J. Appl. Probab.

Diffusion approximations for one-locus multi-allele kin selection, mutation and random drift in group-structured populations: a unifying approach to selection models in population genetics

Lessard, Sabin, Diffusion approximations for one-locus multi-allele kin selection, mutation and random drift in group-structured populations: a unifying approach to selection models in population genetics 59, 659--696 (2009), , J. Math. Biol.

A composite-conditional-likelihood approach for gene mapping based on linkage disequilibrium in windows of marker loci

Larribe, Fabrice et Lessard, Sabin, A composite-conditional-likelihood approach for gene mapping based on linkage disequilibrium in windows of marker loci 7, Art. (2008), , Stat. Appl. Genet. Mol. Biol.

Evolutionary game dynamics in a finite asymmetric two-deme population and emergence of cooperation

Ladret, Véronique et Lessard, Sabin, Evolutionary game dynamics in a finite asymmetric two-deme population and emergence of cooperation 255, 137--151 (2008), , J. Theoret. Biol.

The probability of fixation of a single mutant in an exchangeable selection model

Lessard, Sabin et Ladret, Véronique, The probability of fixation of a single mutant in an exchangeable selection model 54, 721--744 (2007), , J. Math. Biol.

Sex ratio evolution through group selection using diffusion approximation

Courteau, Josiane et Lessard, Sabin, Sex ratio evolution through group selection using diffusion approximation 48, 340--356 (2004), , J. Math. Biol.

The two-locus ancestral graph in a subdivided population: convergence as the number of demes grows in the island model

Lessard, Sabin et Wakeley, John, The two-locus ancestral graph in a subdivided population: convergence as the number of demes grows in the island model 48, 275--292 (2004), , J. Math. Biol.

Change in frequency of a rare mutant allele: a general formula and applications to partial inbreeding models

Lessard, Sabin et Rocheleau, Ghislain, Change in frequency of a rare mutant allele: a general formula and applications to partial inbreeding models 46, 71--94 (2003), , J. Math. Biol.

Frequency-dependent selection in sexual family-structured populations

Tao, Yi et Lessard, Sabin, Frequency-dependent selection in sexual family-structured populations 217, 525--534 (2002), , J. Theoret. Biol.

Equilibrium structure and stability in a frequency-dependent, two-population diploid model

Lemire, Mathieu et Lessard, Sabin, Equilibrium structure and stability in a frequency-dependent, two-population diploid model 43, 1--21 (2001), , J. Math. Biol.

Stability analysis of the partial selfing selection model

Rocheleau, Ghislain et Lessard, Sabin, Stability analysis of the partial selfing selection model 40, 541--574 (2000), , J. Math. Biol.

On the non-existence of an optimal migration rate

Lemire, Mathieu et Lessard, Sabin, On the non-existence of an optimal migration rate 35, 657--682 (1997), , J. Math. Biol.

Convergence of covariance structures in additive Gaussian polygenic models

Mahdi, Smail et Lessard, Sabin, Convergence of covariance structures in additive Gaussian polygenic models 52, 833--845 (1996), , Biometrics

La sélection de parentèle (kin selection), In: Dictionnaire du Darwinisme et de l'Evolution (Patrick Tort, Ed.)

LESSARD Sabin, La sélection de parentèle (kin selection), In: Dictionnaire du Darwinisme et de l'Evolution (Patrick Tort, Ed.) , (1996), , Presses Universitaires de France

Statistique : Concepts et méthodes

LESSARD Sabin et MONGA, Statistique : Concepts et méthodes , 421 (1993), , Les Presses de l'Université de Montréal et Masson Éditeur

The mathematics of resource allocation. II. Intraspecific competition

LESSARD Sabin, The mathematics of resource allocation. II. Intraspecific competition , 16 (1992), , Département de mathématiques et de statistique, Université de Montréal

Statistical models for sex ratio evolution, In: Statistical Methods in Biological and Medical Sciences

LESSARD Sabin, Statistical models for sex ratio evolution, In: Statistical Methods in Biological and Medical Sciences 8, 347-372 (1991), , Handbook of Statistics, North-Holland, New York

Compte rendu sur ?Histoire d'un génome: Population et génétique dans l'est du Québec

LESSARD Sabin, Compte rendu sur ?Histoire d'un génome: Population et génétique dans l'est du Québec , 607 (1991), , Presses de l'Université du Québec

Mathematical and Statistical Developments of Evolutionary Theory

LESSARD Sabin, Mathematical and Statistical Developments of Evolutionary Theory 299, 363 (1990), , Kluwer Academic Publishers

Evolution du rapport numérique des sexes et modèles dynamiques connexes, In: Mathematical and Statistical Developments of Evolutionary Theory

LESSARD Sabin, Evolution du rapport numérique des sexes et modèles dynamiques connexes, In: Mathematical and Statistical Developments of Evolutionary Theory 299, 269-325 (1990), , NATO ASI Series C: Mathematical and Physical Sciences

Evolutionary stability: one concept, several meanings

Lessard, Sabin, Evolutionary stability: one concept, several meanings 37, 159--170 (1990), , Theoret. Population Biol.

The mathematics of resource allocation in population biology. I. Interspecific competition

Lessard, Sabin, The mathematics of resource allocation in population biology. I. Interspecific competition 32, 207--226 (1989), , Appl. Math. Comput.

Population genetics of sex allocation, In: Some Mathematical Questions in Biology, Sex Allocation and Sex changes: Experiments and Models

LESSARD Sabin, Population genetics of sex allocation, In: Some Mathematical Questions in Biology, Sex Allocation and Sex changes: Experiments and Models 22, 109-126 (1989), , Mathematics in the Life Sciences

The role of recombination and selection in the modifier theory of sex-ratio distortion

Lessard, Sabin, The role of recombination and selection in the modifier theory of sex-ratio distortion 31, 339--358 (1987), , Theoret. Population Biol.

Evolutionary principles for general frequency-dependent two-phenotype models in sexual populations

Lessard, Sabin, Evolutionary principles for general frequency-dependent two-phenotype models in sexual populations 119, 329--344 (1986), , J. Theoret. Biol.

The role of recombination in the modifier theory of autosomal segregation distortion

Lessard, Sabin, The role of recombination in the modifier theory of autosomal segregation distortion 28, 133--149 (1985), , Theoret. Population Biol.

On the optimal sex-ratio: a stability analysis based on a characterization for one-locus multiallele viability models

Karlin, Samuel et Lessard, Sabin, On the optimal sex-ratio: a stability analysis based on a characterization for one-locus multiallele viability models 20, 15--38 (1984), , J. Math. Biol.

On the optimal sex ratio

Karlin, Samuel et Lessard, Sabin, On the optimal sex ratio 80, 5931--5935 (1983), , Proc. Nat. Acad. Sci. U.S.A.

A criterion for stability-instability at fixation states involving an eigenvalue one with applications in population genetics

Lessard, Sabin et Karlin, Samuel, A criterion for stability-instability at fixation states involving an eigenvalue one with applications in population genetics 22, 108--126 (1982), , Theoret. Population Biol.

La loi forte des grands nombres pour certaines classes de variables aléatoires faiblement dépendantes

Lessard, Sabin, La loi forte des grands nombres pour certaines classes de variables aléatoires faiblement dépendantes 26, 1355--1366 (1981), , Rev. Roumaine Math. Pures Appl.

Is the between-population variance negligible in the total variance of heterozygosity? Case of a finite number of loci subject to the infinite-allele model in f

Lessard, S., Is the between-population variance negligible in the total variance of heterozygosity? Case of a finite number of loci subject to the infinite-allele model in f 20, 394--410 (1981), , Theoret. Population Biol.

Une double généralisation du théorème de Fejér-Lebesgue

Lessard, Sabin, Une double généralisation du théorème de Fejér-Lebesgue 23, 29--35 (1980), , Canad. Math. Bull.