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Fontanari JF, Santos M. The dynamics of casual groups can keep free-riders at bay. Math Biosci 2024; 372:109188. [PMID: 38570010 DOI: 10.1016/j.mbs.2024.109188] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/28/2024] [Revised: 03/07/2024] [Accepted: 03/27/2024] [Indexed: 04/05/2024]
Abstract
Understanding the conditions for maintaining cooperation in groups of unrelated individuals despite the presence of non-cooperative members is a major research topic in contemporary biological, sociological, and economic theory. The N-person snowdrift game models the type of social dilemma where cooperative actions are costly, but there is a reward for performing them. We study this game in a scenario where players move between play groups following the casual group dynamics, where groups grow by recruiting isolates and shrink by losing individuals who then become isolates. This describes the size distribution of spontaneous human groups and also the formation of sleeping groups in monkeys. We consider three scenarios according to the probability of isolates joining a group. We find that for appropriate choices of the cost-benefit ratio of cooperation and the aggregation-disaggregation ratio in the formation of casual groups, free-riders can be completely eliminated from the population. If individuals are more attracted to large groups, we find that cooperators persist in the population even when the mean group size diverges. We also point out the remarkable similarity between the replicator equation approach to public goods games and the trait group formulation of structured demes.
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Affiliation(s)
- José F Fontanari
- Instituto de Física de São Carlos, Universidade de São Paulo, 13560-970 São Carlos, São Paulo, Brazil.
| | - Mauro Santos
- Departament de Genètica i de Microbiologia, Grup de Genòmica, Bioinformàtica i Biologia Evolutiva (GBBE), Universitat Autònoma de Barcelona, Spain; cE3c - Centre for Ecology, Evolution and Environmental Changes & CHANGE - Global Change and Sustainability Institute, Lisboa, Portugal.
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2
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John S, Müller J. Age structure, replicator equation, and the prisoner's dilemma. Math Biosci 2023; 365:109076. [PMID: 37716407 DOI: 10.1016/j.mbs.2023.109076] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/15/2023] [Revised: 08/25/2023] [Accepted: 09/09/2023] [Indexed: 09/18/2023]
Abstract
We investigate the evolutionary dynamics of an age-structured population subject to weak frequency-dependent selection. It turns out that the weak selection is affected in a non-trivial way by the life-history trait. We disentangle the dynamics, based on the appearance of different time scales. These time scales, which seem to form a universal structure in the interplay of weak selection and life-history traits, allow us to reduce the infinite dimensional model to a one-dimensional modified replicator equation. The modified replicator equation is then used to investigate cooperation (the prisoner's dilemma) by means of adaptive dynamics. We identify conditions under which age structure is able to promote cooperation. At the end we discuss the relevance of our findings.
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Affiliation(s)
- Sona John
- School of Computation, Information and Technology, Department of Mathematics, Technical University of Munich, 85748 Garching, Germany; Comprehensive Pneumology Center (CPC)/Institute of Lung Health and Immunity (LHI), Helmholtz Center Munich, 81377 Munich, Germany
| | - Johannes Müller
- School of Computation, Information and Technology, Department of Mathematics, Technical University of Munich, 85748 Garching, Germany; Institute for Computational Biology, Helmholtz Center Munich, 85764 Neuherberg, Germany.
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3
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Le TMT, Gjini E, Madec S. Quasi-neutral dynamics in a coinfection system with N strains and asymmetries along multiple traits. J Math Biol 2023; 87:48. [PMID: 37640832 DOI: 10.1007/s00285-023-01977-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/05/2021] [Revised: 07/21/2023] [Accepted: 07/27/2023] [Indexed: 08/31/2023]
Abstract
Understanding the interplay of different traits in a co-infection system with multiple strains has many applications in ecology and epidemiology. Because of high dimensionality and complex feedback between traits manifested in infection and co-infection, the study of such systems remains a challenge. In the case where strains are similar (quasi-neutrality assumption), we can model trait variation as perturbations in parameters, which simplifies analysis. Here, we apply singular perturbation theory to many strain parameters simultaneously and advance analytically to obtain their explicit collective dynamics. We consider and study such a quasi-neutral model of susceptible-infected-susceptible (SIS) dynamics among N strains, which vary in 5 fitness dimensions: transmissibility, clearance rate of single- and co-infection, transmission probability from mixed coinfection, and co-colonization vulnerability factors encompassing cooperation and competition. This quasi-neutral system is analyzed with a singular perturbation method through an appropriate slow-fast decomposition. The fast dynamics correspond to the embedded neutral system, while the slow dynamics are governed by an N-dimensional replicator equation, describing the time evolution of strain frequencies. The coefficients of this replicator system are pairwise invasion fitnesses between strains, which, in our model, are an explicit weighted sum of pairwise asymmetries along all trait dimensions. Remarkably these weights depend only on the parameters of the neutral system. Such model reduction highlights the centrality of the neutral system for dynamics at the edge of neutrality and exposes critical features for the maintenance of diversity.
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Affiliation(s)
- Thi Minh Thao Le
- Department of Mathematics and Statistics, Masaryk University, Brno, Czech Republic
| | - Erida Gjini
- Center for Computational and Stochastic Mathematics, Instituto Superior Técnico, Lisbon, Portugal
| | - Sten Madec
- Laboratory of Mathematics, Institut Denis Poisson, University of Tours, Tours, France.
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Müller J, Tellier A. Life-History traits and the replicator equation. Math Biosci 2022; 349:108826. [PMID: 35489522 DOI: 10.1016/j.mbs.2022.108826] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/16/2021] [Revised: 03/10/2022] [Accepted: 04/12/2022] [Indexed: 10/18/2022]
Abstract
Due to the relevance for conservation biology, there is an increasing interest to extend evolutionary genomics models to plant, animal or microbial species. However, this requires to understand the effect of life-history traits absent in humans on genomic evolution. In this context, it is fundamentally of interest to generalize the replicator equation, which is at the heart of most population genomics models. However, as the inclusion of life-history traits generates models with a large state space, the analysis becomes involving. We focus, here, on quiescence and seed banks, two features common to many plant, invertebrate and microbial species. We develop a method to obtain a low-dimensional replicator equation in the context of evolutionary game theory, based on two assumptions: (1) the life-history traits are per se neutral, and (2) frequency-dependent selection is weak. We use the results to investigate the evolution and maintenance of cooperation based on the Prisoner's dilemma and the snowdrift game. We first consider the generalized replicator equation, and then refine the investigation using adaptive dynamics. It turns out that, depending on the structure and timing of the quiescence/dormancy life-history trait, cooperation in a homogeneous population can be stabilized. We finally discuss and highlight the relevance of these results for plant, invertebrate and microbial communities.
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Affiliation(s)
- Johannes Müller
- Center for Mathematics, Technische Universität München, 85748 Garching, Germany; Institute for Computational Biology, Helmholtz Center Munich, 85764 Neuherberg, Germany.
| | - Aurélien Tellier
- Professorship for Population Genetics, Department of Life Science Systems, School of Life Sciences, Technische Universität München, 85354 Freising, Germany
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Ezoe H. Adaptive partner recruitment can help maintain an intra-guild diversity in mutualistic systems. J Theor Biol 2019; 478:40-7. [PMID: 31220467 DOI: 10.1016/j.jtbi.2019.06.017] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/19/2019] [Revised: 06/13/2019] [Accepted: 06/17/2019] [Indexed: 11/20/2022]
Abstract
Mutualisms between assemblages of multiple species or strains (guilds) are considered unstable because of positive feedback between the guilds. Previous studies suggest that negative inter-guild feedback due to asymmetry in the exchange of benefits between the guilds can stabilize them, but preferential association for more beneficial partners may reduce such asymmetry and strengthen the positive inter-guild feedback. Here I develop a replicator dynamics model for mutualistic systems between two host and two symbiont strains to investigate conditions that stabilize mutualisms when feedback between host-symbiont guilds is positive. I assume that one symbiont strain is mutualistic for one host strain but parasitic for the other, whereas the other symbiont strain is the opposite. Hosts recruit their symbionts from the environment and discriminately offer them resources (partner preference), and only mutualistic symbionts return benefits to their hosts. I show that the two host and symbiont strains can coexist under strong partner preference by hosts if they adaptively adjust the number of associating symbionts, even when the intra-host strain competition is not so strong. Interestingly, there can be a stable coexistence equilibrium also under weak partner preference, but it disappears under intermediate levels of partner preference.
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Abstract
We describe an iterated game between two players, in which the payoff is to survive a number of steps. Expected payoffs are probabilities of survival. A key feature of the game is that individuals have to survive on their own if their partner dies. We consider individuals with hardwired, unconditional behaviors or strategies. When both players are present, each step is a symmetric two-player game. The overall survival of the two individuals forms a Markov chain. As the number of iterations tends to infinity, all probabilities of survival decrease to zero. We obtain general, analytical results for n-step payoffs and use these to describe how the game changes as n increases. In order to predict changes in the frequency of a cooperative strategy over time, we embed the survival game in three different models of a large, well-mixed population. Two of these models are deterministic and one is stochastic. Offspring receive their parent's type without modification and fitnesses are determined by the game. Increasing the number of iterations changes the prospects for cooperation. All models become neutral in the limit (n→∞). Further, if pairs of cooperative individuals survive together with high probability, specifically higher than for any other pair and for either type when it is alone, then cooperation becomes favored if the number of iterations is large enough. This holds regardless of the structure of pairwise interactions in a single step. Even if the single-step interaction is a Prisoner's Dilemma, the cooperative type becomes favored. Enhanced survival is crucial in these iterated evolutionary games: if players in pairs start the game with a fitness deficit relative to lone individuals, the prospects for cooperation can become even worse than in the case of a single-step game.
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Affiliation(s)
- John Wakeley
- Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA, 02138, USA.
| | - Martin Nowak
- Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA, 02138, USA; Program for Evolutionary Dynamics, Harvard University, Cambridge, MA 02138, USA; Department of Mathematics, Harvard University, Cambridge, MA 02138, USA
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Cassese D. Replicator equation on networks with degree regular communities. Appl Netw Sci 2018; 3:29. [PMID: 30839808 PMCID: PMC6214305 DOI: 10.1007/s41109-018-0083-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 03/08/2018] [Accepted: 07/13/2018] [Indexed: 06/09/2023]
Abstract
The replicator equation is one of the fundamental tools to study evolutionary dynamics in well-mixed populations. This paper contributes to the literature on evolutionary graph theory, providing a version of the replicator equation for a family of connected networks with communities, where nodes in the same community have the same degree. This replicator equation is applied to the study of different classes of games, exploring the impact of the graph structure on the equilibria of the evolutionary dynamics.
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Affiliation(s)
- Daniele Cassese
- Department of Mathematics, University of Namur, NaXys, Rempart de la Vierge 8, Namur, Belgium
- ICTEAM, University of Louvain, Av Georges Lemaître, Louvain-la-Neuve, Belgium
- Oxford Mathematical Institute, Woodstock Road, Oxford, OX2 6GG UK
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8
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Lessard S, Soares CD. Frequency-dependent growth in class-structured populations: continuous dynamics in the limit of weak selection. J Math Biol 2018; 77:229-59. [PMID: 29236142 DOI: 10.1007/s00285-017-1195-5] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/02/2017] [Revised: 11/16/2017] [Indexed: 10/18/2022]
Abstract
In this paper 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. Moreover, they satisfy some mild conditions that ensure mixing in the long run. The dynamics in the limit as [Formula: see text] with N time steps as unit of time is considered first in the case of a single type, and second in the case of multiple types. The main result is that the type frequencies as [Formula: see text] 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.
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Morsky B, Cressman R, Bauch CT. Homophilic replicator equations. J Math Biol 2016; 75:309-325. [PMID: 27995300 DOI: 10.1007/s00285-016-1083-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/03/2015] [Revised: 06/22/2016] [Indexed: 10/20/2022]
Abstract
Tags are conspicuous attributes of organisms that affect the behaviour of other organisms toward the holder, and have previously been used to explore group formation and altruism. Homophilic imitation, a form of tag-based selection, occurs when organisms imitate those with similar tags. Here we further explore the use of tag-based selection by developing homophilic replicator equations to model homophilic imitation dynamics. We assume that replicators have both tags (sometimes called traits) and strategies. Fitnesses are determined by the strategy profile of the population, and imitation is based upon the strategy profile, fitness differences, and similarity in tag space. We show the characteristics of resulting fixed manifolds and conditions for stability. We discuss the phenomenon of coat-tailing (where tags associated with successful strategies increase in abundance, even though the tags are not inherently beneficial) and its implications for population diversity. We extend our model to incorporate recurrent mutations and invasions to explore their implications upon tag and strategy diversity. We find that homophilic imitation based upon tags significantly affects the diversity of the population, although not the ESS. We classify two different types of invasion scenarios by the strategy and tag compositions of the invaders and invaded. In one scenario, we find that novel tags introduced by invaders become more readily established with homophilic imitation than without it. In the other, diversity decreases. Lastly, we find a negative correlation between homophily and the rate of convergence.
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Affiliation(s)
- Bryce Morsky
- Department of Mathematics and Statistics, University of Guelph, Guelph, Canada.
| | - Ross Cressman
- Department of Mathematics, Wilfrid Laurier University, Waterloo, Canada
| | - C T Bauch
- Department of Applied Mathematics, University of Waterloo, Waterloo, Canada
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Vessman B, Gerlee P, Lundh T. Estimating the probability of coexistence in cross-feeding communities. J Theor Biol 2016; 408:13-21. [PMID: 27484301 DOI: 10.1016/j.jtbi.2016.07.043] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/09/2015] [Revised: 07/22/2016] [Accepted: 07/28/2016] [Indexed: 11/15/2022]
Abstract
The dynamics of many microbial ecosystems are driven by cross-feeding interactions, in which metabolites excreted by some species are metabolised further by others. The population dynamics of such ecosystems are governed by frequency-dependent selection, which allows for stable coexistence of two or more species. We have analysed a model of cross-feeding based on the replicator equation, with the aim of establishing criteria for coexistence in ecosystems containing three species, given the information of the three species' ability to coexist in their three separate pairs, i.e. the long term dynamics in the three two-species component systems. The triple-system is studied statistically and the probability of coexistence in the species triplet is computed for two models of species interactions. The interaction parameters are modelled either as stochastically independent or organised in a hierarchy where any derived metabolite carries less energy than previous nutrients in the metabolic chain. We differentiate between different modes of coexistence with respect to the pair-wise dynamics of the species, and find that the probability of coexistence is close to 12 for triplet systems with three pair-wise coexistent pairs and for the so-called intransitive systems. Systems with two and one pair-wise coexistent pairs are more likely to exist for random interaction parameters, but are on the other hand much less likely to exhibit triplet coexistence. Hence we conclude that certain species triplets are, from a statistical point of view, rare, but if allowed to interact are likely to coexist. This knowledge might be helpful when constructing synthetic microbial communities for industrial purposes.
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Affiliation(s)
- Björn Vessman
- Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, 412 96 Göteborg, Sweden.
| | - Philip Gerlee
- Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, 412 96 Göteborg, Sweden
| | - Torbjörn Lundh
- Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, 412 96 Göteborg, Sweden
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Iacobelli G, Madeo D, Mocenni C. Lumping evolutionary game dynamics on networks. J Theor Biol 2016; 407:328-38. [PMID: 27475842 DOI: 10.1016/j.jtbi.2016.07.037] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/22/2016] [Revised: 07/21/2016] [Accepted: 07/22/2016] [Indexed: 11/24/2022]
Abstract
We study evolutionary game dynamics on networks (EGN), where players reside in the vertices of a graph, and games are played between neighboring vertices. The model is described by a system of ordinary differential equations which depends on players payoff functions, as well as on the adjacency matrix of the underlying graph. Since the number of differential equations increases with the number of vertices in the graph, the analysis of EGN becomes hard for large graphs. Building on the notion of lumpability for Markov chains, we identify conditions on the network structure allowing to reduce the original graph. In particular, we identify a partition of the vertex set of the graph and show that players in the same block of a lumpable partition have equivalent dynamical behaviors, whenever their payoff functions and initial conditions are equivalent. Therefore, vertices belonging to the same partition block can be merged into a single vertex, giving rise to a reduced graph and consequently to a simplified system of equations. We also introduce a tighter condition, called strong lumpability, which can be used to identify dynamical symmetries in EGN which are related to the interchangeability of players in the system.
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Kebir A, Fefferman NH, Ben Miled S. Understanding hermaphrodite species through game theory. J Math Biol 2015; 71:1505-24. [PMID: 25749650 DOI: 10.1007/s00285-015-0866-3] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/30/2014] [Revised: 01/15/2015] [Indexed: 10/23/2022]
Abstract
We investigate the existence and stability of sexual strategies (sequential hermaphrodite, successive hermaphrodite or gonochore) at a proximate level. To accomplish this, we constructed and analyzed a general dynamical game model structured by size and sex. Our main objective is to study how costs of changing sex and of sexual competition should shape the sexual behavior of a hermaphrodite. We prove that, at the proximate level, size alone is insufficient to explain the tendency for a pair of prospective copulants to elect the male sexual role by virtue of the disparity in the energetic costs of eggs and sperm. In fact, we show that the stability of sequential vs. simultaneous hermaphrodite depends on sex change costs, while the stability of protandrous vs. protogynous strategies depends on competition cost.
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Abstract
The basic idea of evolutionary game theory is that payoff determines reproductive rate. Successful individuals have a higher payoff and produce more offspring. But in evolutionary and ecological situations there is not only reproductive rate but also carrying capacity. Individuals may differ in their exposure to density limiting effects. Here we explore an alternative approach to evolutionary game theory by assuming that the payoff from the game determines the carrying capacity of individual phenotypes. Successful strategies are less affected by density limitation (crowding) and reach higher equilibrium abundance. We demonstrate similarities and differences between our framework and the standard replicator equation. Our equation is defined on the positive orthant, instead of the simplex, but has the same equilibrium points as the replicator equation. Linear stability analysis produces the classical conditions for asymptotic stability of pure strategies, but the stability properties of internal equilibria can differ in the two frameworks. For example, in a two-strategy game with an internal equilibrium that is always stable under the replicator equation, the corresponding equilibrium can be unstable in the new framework resulting in a limit cycle.
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