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Király B, Varga T, Szabó G, Garay J. Evolutionarily stable payoff matrix in hawk-dove games. BMC Ecol Evol 2024; 24:65. [PMID: 38769504 PMCID: PMC11107024 DOI: 10.1186/s12862-024-02257-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/05/2024] [Accepted: 05/15/2024] [Indexed: 05/22/2024] Open
Abstract
BACKGROUND Classical matrix game models aim to find the endpoint of behavioural evolution for a set of fixed possible interaction outcomes. Here, we introduce an evolutionary model in which not only the players' strategies but also the payoff matrix evolves according to natural selection. RESULTS We start out from the hawk-dove matrix game and, in a way that is consistent with the monomorphic model setup of Maynard Smith and Price, introduce an evolving phenotypic trait that quantifies fighting ability and determines the probability of winning and the cost of losing escalated hawk-hawk fights. We define evolutionarily stable phenotypes as consisting of an evolutionarily stable strategy and an evolutionarily stable trait, which in turn describes a corresponding evolutionarily stable payoff matrix. CONCLUSIONS We find that the maximal possible cost of escalating fights remains constant during evolution assuming a separation in the time scales of fast behavioural and slow trait selection, despite the fact that the final evolutionarily stable phenotype maximizes the payoff of hawk-hawk fights. Our results mirror the dual nature of Darwinian evolution whereby the criteria of evolutionary success, as well as the successful phenotypes themselves, are a product of natural selection.
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Affiliation(s)
- Balázs Király
- Institute of Technical Physics and Materials Science, HUN-REN Centre for Energy Research, Konkoly-Thege Miklós út 29-33., Budapest, H-1121, Hungary.
| | - Tamás Varga
- Bolyai Institute, University of Szeged, Aradi vértanúk tere 1., Szeged, H-6720, Hungary
| | - György Szabó
- Institute of Technical Physics and Materials Science, HUN-REN Centre for Energy Research, Konkoly-Thege Miklós út 29-33., Budapest, H-1121, Hungary
- Institute of Evolution, HUN-REN Centre for Ecological Research, Konkoly-Thege Miklós út 29-33., Budapest, H-1121, Hungary
| | - József Garay
- Institute of Evolution, HUN-REN Centre for Ecological Research, Konkoly-Thege Miklós út 29-33., Budapest, H-1121, Hungary
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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] [Abstract] [Key Words] [MESH Headings] [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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Křivan V, Cressman R. The asymmetric Hawk-Dove game with costs measured as time lost. J Theor Biol 2022; 547:111162. [DOI: 10.1016/j.jtbi.2022.111162] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/17/2022] [Revised: 05/06/2022] [Accepted: 05/10/2022] [Indexed: 10/18/2022]
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4
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Kurokawa S. For whom is it more beneficial to stop interactions with defectors: Cooperators or defectors? ECOLOGICAL COMPLEXITY 2021. [DOI: 10.1016/j.ecocom.2021.100968] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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5
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Miękisz J, Bodnar M. Evolution of populations with strategy-dependent time delays. Phys Rev E 2021; 103:012414. [PMID: 33601581 DOI: 10.1103/physreve.103.012414] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/16/2020] [Accepted: 01/04/2021] [Indexed: 11/07/2022]
Abstract
We study the effects of strategy-dependent time delays on the equilibria of evolving populations. It is well known that time delays may cause oscillations in dynamical systems. Here we report a novel behavior. We show that microscopic models of evolutionary games with strategy-dependent time delays lead to a new type of replicator dynamics. It describes the time evolution of fractions of the population playing given strategies and the size of the population. Unlike in all previous models, the stationary states of such dynamics depend continuously on time delays. We show that in games with an interior stationary state (a globally asymptotically stable equilibrium in the standard replicator dynamics), at certain time delays it may disappear or there may appear another interior stationary state. In the Prisoner's Dilemma game, for time delays of cooperation smaller than time delays of defection, there appears an unstable interior equilibrium, and therefore for some initial conditions the population converges to the homogeneous state with just cooperators.
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Affiliation(s)
- Jacek Miękisz
- Institute of Applied Mathematics and Mechanics, University of Warsaw, Warsaw, Poland
| | - Marek Bodnar
- Institute of Applied Mathematics and Mechanics, University of Warsaw, Warsaw, Poland
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6
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Broom M, Křivan V. Two-strategy games with time constraints on regular graphs. J Theor Biol 2020; 506:110426. [PMID: 32777217 DOI: 10.1016/j.jtbi.2020.110426] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/10/2019] [Revised: 07/24/2020] [Accepted: 07/27/2020] [Indexed: 11/19/2022]
Abstract
Evolutionary game theory is a powerful method for modelling animal conflicts. The original evolutionary game models were used to explain specific biological features of interest, such as the existence of ritualised contests, and were necessarily simple models that ignored many properties of real populations, including the duration of events and spatial and related structural effects. Both of these areas have subsequently received much attention. Spatial and structural effects have been considered in evolutionary graph theory, and a significant body of literature has been built up to deal with situations where the population is not homogeneous. More recently a theory of time constraints has been developed to take account of the fact that different events can take different times, and that interaction times can explicitly depend upon selected strategies, which can, in turn, influence the distribution of different opponent types within the population. Here, for the first time, we build a model of time constraint games which explicitly considers a spatial population, by considering a population evolving on an underlying graph, using two graph dynamics, birth-death and death-birth. We consider one short time scale along which frequencies of pairs and singles change as individuals interact with their neighbours, and another, evolutionary time scale, along which frequencies of strategies change in the population. We show that for graphs with large degree, both dynamics reproduce recent results from well-mixed time constraint models, including two ESSs being common in Hawk-Dove and Prisoner's Dilemma games, but for low degree there can be marked differences. For birth-death processes the effect of the graph degree is small, whereas for death-birth dynamics there is a large effect. The general prediction for both Hawk-Dove and Prisoner's dilemma games is that as the graph degree decreases, i.e., as the number of neighbours decreases, mixed ESS do appear. In particular, for the Prisoner's dilemma game this means that cooperation is easier to establish in situations where individuals have low number of neighbours. We thus see that solutions depend non-trivially on the combination of graph degree, dynamics and game.
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Affiliation(s)
- Mark Broom
- Department of Mathematics, City, University of London, London, UK.
| | - Vlastimil Křivan
- Centre for Mathematical Biology, Department of Mathematics, Faculty of Science, University of South Bohemia, Branišovská 1760, 370 05 České Budějovice, Czech Republic; Czech Academy of Sciences, Biology Centre, Institute of Entomology, Branišovská 31, 370 05 České Budějovice, Czech Republic.
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7
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Argasinski K, Rudnicki R. Replicator dynamics for the game theoretic selection models based on state. J Theor Biol 2020; 526:110540. [PMID: 33221278 DOI: 10.1016/j.jtbi.2020.110540] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/13/2019] [Revised: 11/06/2020] [Accepted: 11/10/2020] [Indexed: 11/16/2022]
Abstract
The paper presents an attempt to integrate the classical evolutionary game theory based on replicator dynamics and the state-based approach of Houston and McNamara. In the new approach, individuals have different heritable strategies; however, individuals carrying the same strategy can differ in terms of state, role or the situation in which they act. Thus, the classical replicator dynamics is completed by the additional subsystem of differential equations describing the dynamics of transitions between different states. In effect, the interactions described by game structure, in addition to the demographic payoffs (constituted by births and deaths), can lead to the change in state of the competing individuals. Special cases of reversible and irreversible incremental stage-structured models, where the state changes can describeenergy accumulation, developmental steps or aging, are derived for discrete and continuous versions. The new approach is illustrated using the example of the Owner-Intruder game with explicit dynamics of the role changes. The new model presents a generalization of the demographic version of the Hawk-Dove game,with the difference being that the opponents in the game are drawn from two separate subpopulations consisting of Owners and Intruders. Here, the Intruders check random nest sites and play the Hawk-Dove game with the Owner if they are occupied. Meanwhile, the Owners produce newborns that become Intruders, since they must find a free nest site to reproduce. An interesting feedback mechanism is produced via the fluxes of individuals between the different subpopulations. In addition, the population growth suppression mechanism resulting from the fixation Bourgeois strategy is analyzed.
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Affiliation(s)
- Krzysztof Argasinski
- Institute of Mathematics of Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa, Poland.
| | - Ryszard Rudnicki
- Institute of Mathematics of Polish Academy of Sciences, Bankowa 14, 40-007 Katowice, Poland.
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8
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Defectors' intolerance of others promotes cooperation in the repeated public goods game with opting out. Sci Rep 2020; 10:19511. [PMID: 33177552 PMCID: PMC7659018 DOI: 10.1038/s41598-020-76506-3] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/03/2020] [Accepted: 10/22/2020] [Indexed: 12/21/2022] Open
Abstract
The theoretical and experimental research on opting out (also called conditional dissociation) in social dilemmas has concentrated on the effect this behavior has on the level of cooperation when used against defectors. The intuition behind this emphasis is based on the common property of social dilemmas that individuals are worse off the more their opponents defect. However, this article shows clearly that other opting out mechanisms are better at increasing cooperative behavior. In fact, by analyzing the stable Nash equilibria for the repeated multi-player public goods game with opting out, our results provide a strong argument that the best opting out rule is one whereby the only groups that voluntarily stay together between rounds are those that are homogeneous (i.e., those groups that are either all cooperators or all defectors), when these groups stay together for enough rounds. This outcome emerges when defectors are completely intolerant of individuals who cooperate (e.g., defectors exhibit xenophobic behavior toward cooperators) and so opt out whenever their group has a cooperator in it. The strong preference by defectors to be with like-minded individuals causes all heterogeneous groups to disband after one round.
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9
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Garay J, Cressman R, Xu F, Broom M, Csiszár V, Móri TF. When optimal foragers meet in a game theoretical conflict: A model of kleptoparasitism. J Theor Biol 2020; 502:110306. [DOI: 10.1016/j.jtbi.2020.110306] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/04/2019] [Revised: 03/26/2020] [Accepted: 04/27/2020] [Indexed: 10/24/2022]
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10
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Reducing courtship time promotes marital bliss: The Battle of the Sexes game revisited with costs measured as time lost. J Theor Biol 2020; 503:110382. [DOI: 10.1016/j.jtbi.2020.110382] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/09/2020] [Accepted: 06/15/2020] [Indexed: 11/18/2022]
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11
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A temporal model of territorial defence with antagonistic interactions. Theor Popul Biol 2020; 134:15-35. [DOI: 10.1016/j.tpb.2020.03.005] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/05/2019] [Revised: 03/26/2020] [Accepted: 03/27/2020] [Indexed: 11/18/2022]
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12
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Broom M, Cressman R, Křivan V. Revisiting the "fallacy of averages" in ecology: Expected gain per unit time equals expected gain divided by expected time. J Theor Biol 2019; 483:109993. [PMID: 31491495 DOI: 10.1016/j.jtbi.2019.109993] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/18/2019] [Revised: 07/26/2019] [Accepted: 09/02/2019] [Indexed: 11/16/2022]
Abstract
Fitness is often defined as the average payoff an animal obtains when it is engaged in several activities, each taking some time. We point out that the average can be calculated with respect to either the time distribution, or to the event distribution of these activities. We show that these two averages lead to the same fitness function. We illustrate this result through two examples from foraging theory, Holling II functional response and the diet choice model, and one game-theoretic example of Hamilton's rule applied to the time-constrained Prisoner's dilemma (PD). In particular, we show that in these models, fitness defined as expected gain per unit time equals fitness defined as expected gain divided by expected time. We also show how these fitnesses predict the optimal outcome for diet choice and the prevalence of cooperation in the repeated PD game.
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Affiliation(s)
- Mark Broom
- Department of Mathematics, City, University of London, London, UK.
| | - Ross Cressman
- Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada.
| | - Vlastimil Křivan
- Department of Mathematics, Faculty of Science, University of South Bohemia, Branišovská 1760, České Budějovice 370 05, Czech Republic; Czech Academy of Sciences, Biology Centre, Institute of Entomology, Branišovská 31, České Budějovice 370 05, Czech Republic.
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13
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The ESS for evolutionary matrix games under time constraints and its relationship with the asymptotically stable rest point of the replicator dynamics. J Math Biol 2019; 80:743-774. [DOI: 10.1007/s00285-019-01440-6] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/10/2019] [Revised: 09/23/2019] [Indexed: 10/25/2022]
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14
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Sellinger T, Müller J, Hösel V, Tellier A. Are the better cooperators dormant or quiescent? Math Biosci 2019; 318:108272. [PMID: 31647933 DOI: 10.1016/j.mbs.2019.108272] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/01/2019] [Revised: 10/14/2019] [Accepted: 10/14/2019] [Indexed: 01/01/2023]
Abstract
Despite the wealth of empirical and theoretical studies, the origin and maintenance of cooperation is still an evolutionary riddle. In this context, ecological life-history traits which affect the efficiency of selection may play a role despite being often ignored. We consider here species such as bacteria, fungi, invertebrates and plants which exhibit resting stages in the form of a quiescent state or a seed bank. When quiescent, individuals are inactive and reproduce upon activation, while under seed bank parents produce offspring remaining dormant for different amount of time. We assume weak frequency-dependent selection modeled using game-theory and the prisoner's dilemma (cooperation/defect) as payoff matrix. The cooperators and defectors are allowed to evolve different quiescence or dormancy times. By means of singular perturbation theory we reduce the model to a one-dimensional equation resembling the well known replicator equation, in which the gain functions are scaled with lumped parameters reflecting the time scale of the resting state of the cooperators and defectors. If both time scales are identical cooperation cannot persist in a homogeneous population. If, however, the time scale of the cooperator is distinctively different from that of the defector, cooperation may become a locally asymptotically stable strategy. Interestingly enough, in the seed bank case the cooperator needs to become active faster than the defector, while in the quiescent case the cooperator has to be slower. We use adaptive dynamics to identify situations where cooperation may evolve and form a convergent stable ESS. We conclude by highlighting the relevance of these results for many non-model species and the maintenance of cooperation in microbial, invertebrate or plant populations.
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Affiliation(s)
- Thibaut Sellinger
- Section of Population Genetics, Center of Life and Food Sciences Weihenstephan, Technische Universität München, Freising 85354, Germany
| | - Johannes Müller
- Center for Mathematics, Technische Universität München, Garching 85748, Germany; Institute for Computational Biology, Helmholtz Center Munich, Neuherberg 85764, Germany.
| | - Volker Hösel
- Center for Mathematics, Technische Universität München, Garching 85748, Germany
| | - Aurélien Tellier
- Section of Population Genetics, Center of Life and Food Sciences Weihenstephan, Technische Universität München, Freising 85354, Germany
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15
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Premo LS, Brown JR. The opportunity cost of walking away in the spatial iterated prisoner's dilemma. Theor Popul Biol 2019; 127:40-48. [PMID: 30946861 DOI: 10.1016/j.tpb.2019.03.004] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/02/2018] [Revised: 01/29/2019] [Accepted: 03/27/2019] [Indexed: 11/30/2022]
Abstract
Previous work with the spatial iterated prisoner's dilemma has shown that the ability to respond to a partner's defection by simply "walking away" allows so-called walk away cooperators to outcompete defectors as well as cooperators that do not respond to defection. These findings are important because they suggest a relatively simple route by which cooperation can evolve. But it remains to be seen just how robust the walk away strategy is to ecologically important variables such as population density, strategic error, and offspring dispersal. The results of our simulation experiments show that the evolutionary success of walk away cooperators decreases with decreasing population density and/or with increasing error. This relationship is best explained by the ways in which population density and error jointly affect the opportunity cost of walking away. This opportunity cost also explains why naive cooperators regularly outcompete walk away cooperators in pair-wise competition, something not observed in previous studies. Our results further show that local offspring dispersal can inhibit the evolution of cooperation by negating the protection low population density affords the most vulnerable cooperators. Our research identifies socio-ecological conditions in which forgiveness trumps flight in the spatial iterated prisoner's dilemma.
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Affiliation(s)
- L S Premo
- Department of Anthropology, Washington State University, Pullman, WA 99164-4910, USA; Department of Human Evolution, Max Planck Institute for Evolutionary Anthropology, Deutscher Platz 6, 04103 Leipzig, Germany.
| | - Justin R Brown
- Department of Anthropology, Washington State University, Pullman, WA 99164-4910, USA
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16
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Garay J. Technical review on derivation methods for behavior dependent functional responses. COMMUNITY ECOL 2019. [DOI: 10.1556/168.2019.20.1.4] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
- J. Garay
- MTA Centre for Ecological Research, Evolutionary Systems Research Group, Klebelsberg Kunó u. 3, H-8237 Tihany, Hungary and
- MTA-ELTE Research Group in Theoretical Biology and Evolutionary Ecology and Department of Plant Systematics, Ecology and Theoretical Biology, ELTE Eötvös Loránd University, Pázmány Péter sétány1/c, H-1117 Budapest, Hungary. Phone: , Fax:
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17
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Rose D, Hawkes K, Kim PS. Adult sex ratio as an index for male strategy in primates. Theor Popul Biol 2019; 126:40-50. [PMID: 30771361 DOI: 10.1016/j.tpb.2019.02.002] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/20/2018] [Revised: 01/18/2019] [Accepted: 02/04/2019] [Indexed: 11/17/2022]
Abstract
The adult sex ratio (ASR) is defined as the number of fertile males divided by the number of fertile females in a population. We build an ODE model with minimal age structure, in which males compete for paternities using either a multiple-mating or searching-then-guarding strategy, to investigate the value of ASR as an index for predicting which strategy males will adopt, with a focus in our investigation on the differences of strategy choice between chimpanzees (Pan troglodytes) and human hunter-gatherers (Homo sapiens). Parameters in the model characterise aspects of life history and behaviour, and determine both dominant strategy and the ASR when the population is at or near equilibrium. Sensitivity analysis on the model parameters informs us that ASR is strongly influenced by parameters characterising life history, while dominant strategy is affected most strongly by the effectiveness of guarding (average length of time a guarded pair persists, and resistance to paternity theft) and moderately by some life history traits. For fixed effectiveness of guarding and other parameters, dominant strategy tends to change from multiple mating to guarding along a curve that aligns well with a contour of constant ASR, under variation of parameters such as longevity and age female fertility ends. This confirms the hypothesis that ASR may be a useful index for predicting the optimal male mating strategy, provided we have some limited information about ecology and behaviour.
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Affiliation(s)
- Danya Rose
- School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia.
| | - Kristen Hawkes
- Department of Anthropology, University of Utah, Salt Lake City, UT 84112, USA
| | - Peter S Kim
- School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia
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18
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Cressman R, Křivan V. Bimatrix games that include interaction times alter the evolutionary outcome: The Owner–Intruder game. J Theor Biol 2019; 460:262-273. [DOI: 10.1016/j.jtbi.2018.10.033] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/03/2018] [Revised: 10/02/2018] [Accepted: 10/12/2018] [Indexed: 11/16/2022]
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19
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Křivan V, Galanthay TE, Cressman R. Beyond replicator dynamics: From frequency to density dependent models of evolutionary games. J Theor Biol 2018; 455:232-248. [PMID: 29990466 DOI: 10.1016/j.jtbi.2018.07.003] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
Game theoretic models of evolution such as the Hawk-Dove game assume that individuals gain fitness (which is a proxy of the per capita population growth rate) in pair-wise contests only. These models assume that the equilibrium distribution of phenotypes involved (e.g., Hawks and Doves) in the population is given by the Hardy-Weinberg law, which is based on instantaneous, random pair formation. On the other hand, models of population dynamics do not consider pairs, newborns are produced by singles, and interactions between phenotypes or species are described by the mass action principle. This article links game theoretic and population approaches. It shows that combining distribution dynamics with population dynamics can lead to stable coexistence of Hawk and Dove population numbers in models that do not assume a priori that fitness is negative density dependent. Our analysis shows clearly that the interior Nash equilibrium of the Hawk and Dove model depends both on population size and on interaction times between different phenotypes in the population. This raises the question of the applicability of classic evolutionary game theory that requires all interactions take the same amount of time and that all single individuals have the same payoff per unit of time, to real populations. Furthermore, by separating individual fitness into birth and death effects on singles and pairs, it is shown that stable coexistence in these models depends on the time-scale of the distribution dynamics relative to the population dynamics. When explicit density-dependent fitness is included through competition over a limited resource, the combined dynamics of the Hawk-Dove model often lead to Dove extinction no matter how costly fighting is for Hawk pairs.
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Affiliation(s)
- Vlastimil Křivan
- Czech Academy of Sciences, Biology Centre, Institute of Entomology, Branišovská 31, České Budějovice 370 05, Czech Republic; Department of Mathematics, Faculty of Sciences, University of South Bohemia, Branišovská 1760, České Budějovice 370 05, Czech Republic.
| | | | - Ross Cressman
- Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada.
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Garay J, Cressman R, Móri TF, Varga T. The ESS and replicator equation in matrix games under time constraints. J Math Biol 2018; 76:1951-1973. [DOI: 10.1007/s00285-018-1207-0] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/27/2017] [Revised: 12/02/2017] [Indexed: 11/28/2022]
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