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Wodarz D, Komarova NL. Mutant fixation in the presence of a natural enemy. Nat Commun 2023; 14:6642. [PMID: 37863909 PMCID: PMC10589345 DOI: 10.1038/s41467-023-41787-5] [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: 03/02/2023] [Accepted: 09/19/2023] [Indexed: 10/22/2023] Open
Abstract
The literature about mutant invasion and fixation typically assumes populations to exist in isolation from their ecosystem. Yet, populations are part of ecological communities, and enemy-victim (e.g. predator-prey or pathogen-host) interactions are particularly common. We use spatially explicit, computational pathogen-host models (with wild-type and mutant hosts) to re-visit the established theory about mutant fixation, where the pathogen equally attacks both wild-type and mutant individuals. Mutant fitness is assumed to be unrelated to infection. We find that pathogen presence substantially weakens selection, increasing the fixation probability of disadvantageous mutants and decreasing it for advantageous mutants. The magnitude of the effect rises with the infection rate. This occurs because infection induces spatial structures, where mutant and wild-type individuals are mostly spatially separated. Thus, instead of mutant and wild-type individuals competing with each other, it is mutant and wild-type "patches" that compete, resulting in smaller fitness differences and weakened selection. This implies that the deleterious mutant burden in natural populations might be higher than expected from traditional theory.
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Affiliation(s)
- Dominik Wodarz
- Department of Population Health and Disease Prevention, University of California, Irvine, CA, 92697, US.
- Department of Mathematics, University of California, Irvine, CA, 92697, US.
- School of Biological Sciences, Ecology, Behavior & Evolution Department, University of California San Diego, 9500 Gilman Drive, La Jolla, CA, 92093, USA.
| | - Natalia L Komarova
- Department of Mathematics, University of California, Irvine, CA, 92697, US
- Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA, 92093, USA
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2
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Mazzolini A, Grilli J. Universality of evolutionary trajectories under arbitrary forms of self-limitation and competition. Phys Rev E 2023; 108:034406. [PMID: 37849158 DOI: 10.1103/physreve.108.034406] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/20/2023] [Accepted: 08/25/2023] [Indexed: 10/19/2023]
Abstract
The assumption of constant population size is central in population genetics. It led to a large body of results that is robust to modeling choices and that has proven successful to understand evolutionary dynamics. In reality, allele frequencies and population size are both determined by the interaction between a population and the environment. Relaxing the constant-population assumption has two big drawbacks. It increases the technical difficulty of the analysis, and it requires specifying a mechanism for the saturation of the population size, possibly making the results contingent on model details. Here we develop a framework that encompasses a great variety of systems with an arbitrary mechanism for population growth limitation. By using techniques based on scale separation for stochastic processes, we are able to calculate analytically properties of evolutionary trajectories, such as the fixation probability. Remarkably, these properties assume a universal form with respect to our framework, which depends on only three parameters related to the intergeneration timescale, the invasion fitness, and the carrying capacity of the strains. In other words, different systems, such as Lotka-Volterra or a chemostat model (contained in our framework), share the same evolutionary outcomes after a proper remapping of their parameters. An important and surprising consequence of our results is that the direction of selection can be inverted, with a population evolving to reach lower values of invasion fitness.
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Affiliation(s)
- Andrea Mazzolini
- Laboratoire de physique de l'École normale supérieure (PSL University), CNRS, Sorbonne Université, and Université de Paris, 75005 Paris, France
| | - Jacopo Grilli
- Quantitative Life Sciences, The Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste 34151, Italy
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3
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Czuppon P, Day T, Débarre F, Blanquart F. A stochastic analysis of the interplay between antibiotic dose, mode of action, and bacterial competition in the evolution of antibiotic resistance. PLoS Comput Biol 2023; 19:e1011364. [PMID: 37578976 PMCID: PMC10449190 DOI: 10.1371/journal.pcbi.1011364] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/22/2023] [Revised: 08/24/2023] [Accepted: 07/17/2023] [Indexed: 08/16/2023] Open
Abstract
The use of an antibiotic may lead to the emergence and spread of bacterial strains resistant to this antibiotic. Experimental and theoretical studies have investigated the drug dose that minimizes the risk of resistance evolution over the course of treatment of an individual, showing that the optimal dose will either be the highest or the lowest drug concentration possible to administer; however, no analytical results exist that help decide between these two extremes. To address this gap, we develop a stochastic mathematical model of bacterial dynamics under antibiotic treatment. We explore various scenarios of density regulation (bacterial density affects cell birth or death rates), and antibiotic modes of action (biostatic or biocidal). We derive analytical results for the survival probability of the resistant subpopulation until the end of treatment, the size of the resistant subpopulation at the end of treatment, the carriage time of the resistant subpopulation until it is replaced by a sensitive one after treatment, and we verify these results with stochastic simulations. We find that the scenario of density regulation and the drug mode of action are important determinants of the survival of a resistant subpopulation. Resistant cells survive best when bacterial competition reduces cell birth and under biocidal antibiotics. Compared to an analogous deterministic model, the population size reached by the resistant type is larger and carriage time is slightly reduced by stochastic loss of resistant cells. Moreover, we obtain an analytical prediction of the antibiotic concentration that maximizes the survival of resistant cells, which may help to decide which drug dosage (not) to administer. Our results are amenable to experimental tests and help link the within and between host scales in epidemiological models.
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Affiliation(s)
- Peter Czuppon
- Institute for Evolution and Biodiversity, University of Münster, Münster, Germany
- Institute of Ecology and Environmental Sciences of Paris, Sorbonne Université, UPEC, CNRS, IRD, INRA, Paris, France
- Center for Interdisciplinary Research in Biology, CNRS, Collège de France, PSL Research University, Paris, France
| | - Troy Day
- Department of Mathematics and Statistics, Department of Biology, Queen’s University, Kingston, Canada
| | - Florence Débarre
- Institute of Ecology and Environmental Sciences of Paris, Sorbonne Université, UPEC, CNRS, IRD, INRA, Paris, France
| | - François Blanquart
- Center for Interdisciplinary Research in Biology, CNRS, Collège de France, PSL Research University, Paris, France
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4
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Wang G, Su Q, Wang L, Plotkin JB. Reproductive variance can drive behavioral dynamics. Proc Natl Acad Sci U S A 2023; 120:e2216218120. [PMID: 36927152 PMCID: PMC10041125 DOI: 10.1073/pnas.2216218120] [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: 09/22/2022] [Accepted: 02/10/2023] [Indexed: 03/18/2023] Open
Abstract
The concept of fitness is central to evolution, but it quantifies only the expected number of offspring an individual will produce. The actual number of offspring is also subject to demographic stochasticity-that is, randomness associated with birth and death processes. In nature, individuals who are more fecund tend to have greater variance in their offspring number. Here, we develop a model for the evolution of two types competing in a population of nonconstant size. The fitness of each type is determined by pairwise interactions in a prisoner's dilemma game, and the variance in offspring number depends upon its mean. Although defectors are preferred by natural selection in classical population models, since they always have greater fitness than cooperators, we show that sufficiently large offspring variance can reverse the direction of evolution and favor cooperation. Large offspring variance produces qualitatively new dynamics for other types of social interactions, as well, which cannot arise in populations with a fixed size or with a Poisson offspring distribution.
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Affiliation(s)
- Guocheng Wang
- Center for Systems and Control, College of Engineering, Peking University, Beijing100871, China
| | - Qi Su
- Center for Mathematical Biology, University of Pennsylvania, Philadelphia, PA19104
- Department of Mathematics, University of Pennsylvania, Philadelphia, PA19104
- Department of Biology, University of Pennsylvania, Philadelphia, PA19104
| | - Long Wang
- Center for Systems and Control, College of Engineering, Peking University, Beijing100871, China
- Center for Multi-Agent Research, Institute for Artificial Intelligence, Peking University, Beijing100871, China
| | - Joshua B. Plotkin
- Center for Mathematical Biology, University of Pennsylvania, Philadelphia, PA19104
- Department of Mathematics, University of Pennsylvania, Philadelphia, PA19104
- Department of Biology, University of Pennsylvania, Philadelphia, PA19104
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5
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Fixation in the stochastic Lotka-Volterra model with small fitness trade-offs. J Math Biol 2022; 85:8. [PMID: 35819503 DOI: 10.1007/s00285-022-01774-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/24/2020] [Revised: 12/01/2021] [Accepted: 04/05/2022] [Indexed: 10/17/2022]
Abstract
We study the probability of fixation in a stochastic two-species competition model. By identifying a naturally occurring fast timescale, we derive an approximation to the associated backward Kolmogorov equation that allows us to obtain an explicit closed form solution for the probability of fixation of either species. We use our result to study fitness tradeoff strategies and show that, despite some tradeoffs having nearly negligible effects on the corresponding deterministic dynamics, they can have large implications for the outcome of the stochastic system.
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6
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Balasekaran M, Johanis M, Rychtář J, Taylor D, Zhu J. Quasi-neutral evolution in populations under small demographic fluctuations. J Theor Biol 2022; 538:111040. [DOI: 10.1016/j.jtbi.2022.111040] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/02/2021] [Revised: 01/11/2022] [Accepted: 01/19/2022] [Indexed: 10/19/2022]
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7
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Park JI, Kim BJ, Park HJ. Stochastic resonance of abundance fluctuations and mean time to extinction in an ecological community. Phys Rev E 2021; 104:024133. [PMID: 34525626 DOI: 10.1103/physreve.104.024133] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/04/2021] [Accepted: 08/11/2021] [Indexed: 11/07/2022]
Abstract
Periodic environmental changes are commonly observed in nature from the amount of daylight to seasonal temperature. These changes usually affect individuals' death or birth rates, dragging the system from its previous stable states. When the fluctuation of abundance is amplified due to such changes, extinction of species may be accelerated. To see this effect, we examine how the abundance and the mean time to extinction respond to the periodic environmental changes. We consider a population wherein two species coexist together implemented by three rules-birth, spontaneous death, and death from competitions. As the interspecific interaction strength is varied, we observe the resonance behavior in both fluctuations of abundances and the mean time to extinction. Our result suggests that neither too high nor too low competition rates make the system more susceptible to environmental changes.
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Affiliation(s)
- Jong Il Park
- Department of Physics, Sungkyunkwan University, Suwon 16419, Republic of Korea
| | - Beom Jun Kim
- Department of Physics, Sungkyunkwan University, Suwon 16419, Republic of Korea
| | - Hye Jin Park
- Asia Pacific Center for Theoretical Physics, Pohang 37673, Republic of Korea.,Department of Physics, POSTECH, Pohang 37673, Republic of Korea
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8
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Czuppon P, Traulsen A. Understanding evolutionary and ecological dynamics using a continuum limit. Ecol Evol 2021; 11:5857-5873. [PMID: 34141189 PMCID: PMC8207364 DOI: 10.1002/ece3.7205] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/06/2020] [Revised: 12/08/2020] [Accepted: 12/23/2020] [Indexed: 01/08/2023] Open
Abstract
Continuum limits in the form of stochastic differential equations are typically used in theoretical population genetics to account for genetic drift or more generally, inherent randomness of the model. In evolutionary game theory and theoretical ecology, however, this method is used less frequently to study demographic stochasticity. Here, we review the use of continuum limits in ecology and evolution. Starting with an individual-based model, we derive a large population size limit, a (stochastic) differential equation which is called continuum limit. By example of the Wright-Fisher diffusion, we outline how to compute the stationary distribution, the fixation probability of a certain type, and the mean extinction time using the continuum limit. In the context of the logistic growth equation, we approximate the quasi-stationary distribution in a finite population.
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Affiliation(s)
- Peter Czuppon
- Institute of Ecology and Environmental Sciences ParisUPECCNRSIRDINRASorbonne UniversitéParisFrance
- Centre Interdisciplinaire de Recherche en BiologieCNRSCollège de FrancePSL Research UniversityParisFrance
- Department of Evolutionary TheoryMax Planck Institute for Evolutionary BiologyPlönGermany
| | - Arne Traulsen
- Department of Evolutionary TheoryMax Planck Institute for Evolutionary BiologyPlönGermany
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9
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Overton CE, Sharkey KJ. Evolutionary bet-hedging in structured populations. J Math Biol 2021; 82:43. [PMID: 33796960 PMCID: PMC8016807 DOI: 10.1007/s00285-021-01597-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/15/2020] [Revised: 03/08/2021] [Accepted: 03/16/2021] [Indexed: 11/21/2022]
Abstract
As ecosystems evolve, species can become extinct due to fluctuations in the environment. This leads to the evolutionary adaption known as bet-hedging, where species hedge against these fluctuations to reduce their likelihood of extinction. Environmental variation can be either within or between generations. Previous work has shown that selection for bet-hedging against within-generational variation should not occur in large populations. However, this work has been limited by assumptions of well-mixed populations, whereas real populations usually have some degree of structure. Using the framework of evolutionary graph theory, we show that through adding competition structure to the population, within-generational variation can have a significant impact on the evolutionary process for any population size. This complements research using subdivided populations, which suggests that within-generational variation is important when local population sizes are small. Together, these conclusions provide evidence to support observations by some ecologists that are contrary to the widely held view that only between-generational environmental variation has an impact on natural selection. This provides theoretical justification for further empirical study into this largely unexplored area.
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10
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Murray R, Young G. Neutral competition in a deterministically changing environment: Revisiting continuum approaches. J Theor Biol 2020; 486:110104. [PMID: 31809716 DOI: 10.1016/j.jtbi.2019.110104] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/17/2019] [Revised: 11/27/2019] [Accepted: 12/02/2019] [Indexed: 11/17/2022]
Abstract
Environmental variation can play an important role in ecological competition by influencing the relative advantage between competing species. Here, we consider such effects by extending a classical, competitive Moran model to incorporate an environment that fluctuates periodically in time. We adapt methods from work on these classical models to investigate the effects of the magnitude and frequency of environmental fluctuations on two important population statistics: the probability of fixation and the mean time to fixation. In particular, we find that for small frequencies, the system behaves similar to a system with a constant fitness difference between the two species, and for large frequencies, the system behaves similar to a neutrally competitive model. Most interestingly, the system exhibits nontrivial behavior for intermediate frequencies. We conclude by showing that our results agree quite well with recent theoretical work on competitive models with a stochastically changing environment, and discuss how the methods we develop ease the mathematical analysis required to study such models.
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Affiliation(s)
- Ryan Murray
- Department of Mathematics, North Carolina State University, Raleigh, NC 27695, United States
| | - Glenn Young
- Department of Mathematics, Kennesaw State University, Marietta, GA 30060, United States.
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11
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Badali M, Zilman A. Effects of niche overlap on coexistence, fixation and invasion in a population of two interacting species. ROYAL SOCIETY OPEN SCIENCE 2020; 7:192181. [PMID: 32257357 PMCID: PMC7062080 DOI: 10.1098/rsos.192181] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/19/2019] [Accepted: 01/21/2020] [Indexed: 06/11/2023]
Abstract
Synergistic and antagonistic interactions in multi-species populations-such as resource sharing and competition-result in remarkably diverse behaviours in populations of interacting cells, such as in soil or human microbiomes, or clonal competition in cancer. The degree of inter- and intra-specific interaction can often be quantified through the notion of an ecological 'niche'. Typically, weakly interacting species that occupy largely distinct niches result in stable mixed populations, while strong interactions and competition for the same niche result in rapid extinctions of some species and fixations of others. We investigate the transition of a deterministically stable mixed population to a stochasticity-induced fixation as a function of the niche overlap between the two species. We also investigate the effect of the niche overlap on the population stability with respect to external invasions. Our results have important implications for a number of experimental systems.
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Affiliation(s)
- Matthew Badali
- Department of Physics, University of Toronto, 60 St George St., Toronto, CanadaM5S 1A7
| | - Anton Zilman
- Department of Physics, University of Toronto, 60 St George St., Toronto, CanadaM5S 1A7
- Institute for Biomaterials and Biomedical Engineering, University of Toronto, Toronto, Canada
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12
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Park HJ, Pichugin Y, Huang W, Traulsen A. Population size changes and extinction risk of populations driven by mutant interactors. Phys Rev E 2019; 99:022305. [PMID: 30934279 DOI: 10.1103/physreve.99.022305] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/17/2018] [Indexed: 11/07/2022]
Abstract
Spontaneous random mutations are an important source of variation in populations. Many evolutionary models consider mutants with a fixed fitness, chosen from a fitness distribution without considering microscopic interactions among the residents and mutants. Here, we go beyond this and consider "mutant interactors," which lead to new interactions between the residents and invading mutants that can affect the population size and the extinction risk of populations. We model microscopic interactions between individuals by using a dynamic interaction matrix, the dimension of which increases with the emergence of a new mutant and decreases with extinction. The new interaction parameters of the mutant follow a probability distribution around the payoff entries of its ancestor. These new interactions can drive the population away from the previous equilibrium and lead to changes in the population size. Thus, the population size is an evolving property rather than an externally controlled variable. We calculate the average population size of our stochastic system over time and quantify the extinction risk of the population by the mean time to extinction.
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Affiliation(s)
- Hye Jin Park
- Department of Evolutionary Theory, Max Planck Institute for Evolutionary Biology, 24306 Plön, Germany
| | - Yuriy Pichugin
- Department of Evolutionary Theory, Max Planck Institute for Evolutionary Biology, 24306 Plön, Germany
| | - Weini Huang
- Complex Systems and Networks Research Group, School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom.,Group of Theoretical Biology, The State Key Laboratory of Biocontrol, School of Life Science, Sun Yat-sen University, Guangzhou 510060, China
| | - Arne Traulsen
- Department of Evolutionary Theory, Max Planck Institute for Evolutionary Biology, 24306 Plön, Germany
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13
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Chalub FACC, Souza MO. From Fixation Probabilities to d-player Games: An Inverse Problem in Evolutionary Dynamics. Bull Math Biol 2019; 81:4625-4642. [PMID: 30635836 DOI: 10.1007/s11538-018-00566-w] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/14/2017] [Accepted: 12/27/2018] [Indexed: 10/27/2022]
Abstract
The probability that the frequency of a particular trait will eventually become unity, the so-called fixation probability, is a central issue in the study of population evolution. Its computation, once we are given a stochastic finite population model without mutations and a (possibly frequency dependent) fitness function, is straightforward and it can be done in several ways. Nevertheless, despite the fact that the fixation probability is an important macroscopic property of the population, its precise knowledge does not give any clear information about the interaction patterns among individuals in the population. Here we address the inverse problem: from a given fixation pattern and population size, we want to infer what is the game being played by the population. This is done by first exploiting the framework developed in Chalub and Souza (J Math Biol 75:1735-1774, 2017), which yields a fitness function that realises this fixation pattern in the Wright-Fisher model. This fitness function always exists, but it is not necessarily unique. Subsequently, we show that any such fitness function can be approximated, with arbitrary precision, using d-player game theory, provided d is large enough. The pay-off matrix that emerges naturally from the approximating game will provide useful information about the individual interaction structure that is not itself apparent in the fixation pattern. We present extensive numerical support for our conclusions.
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Affiliation(s)
- Fabio A C C Chalub
- Departamento de Matemática and Centro de Matemática e Apliçoes, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829-516, Caparica, Portugal
| | - Max O Souza
- Instituto de Matemática e Estatística, Universidade Federal Fluminense, R. Prof. Marcos Waldemar de Freitas Reis, s/n, Niterói, RJ, 24210-201, Brasil.
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14
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Disentangling eco-evolutionary effects on trait fixation. Theor Popul Biol 2018; 124:93-107. [PMID: 30359662 DOI: 10.1016/j.tpb.2018.10.002] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/17/2018] [Revised: 10/08/2018] [Accepted: 10/11/2018] [Indexed: 11/23/2022]
Abstract
In population genetics, fixation of traits in a demographically changing population under frequency-independent selection has been extensively analysed. In evolutionary game theory, models of fixation have typically focused on fixed population sizes and frequency-dependent selection. A combination of demographic fluctuations with frequency-dependent interactions such as Lotka-Volterra dynamics has received comparatively little attention. We consider a stochastic, competitive Lotka-Volterra model with higher order interactions between two traits. The emerging individual-based model allows for stochastic fluctuations in the frequencies of the two traits and the total population size. We calculate the fixation probability of a trait under differing competition coefficients. This fixation probability resembles, qualitatively, the deterministic evolutionary dynamics. Furthermore, we partially disentangle the selection effects into their ecological and evolutionary components. We find that changing the evolutionary selection strength also changes the population dynamics and vice versa. Thus, a clean separation of the ecological and evolutionary effects is not possible. Instead, our results imply a nested interaction of the evolutionary and ecological effects. The entangled eco-evolutionary processes thus cannot be ignored when determining fixation properties in a co-evolutionary system.
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15
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McAvoy A, Fraiman N, Hauert C, Wakeley J, Nowak MA. Public goods games in populations with fluctuating size. Theor Popul Biol 2018; 121:72-84. [PMID: 29408219 DOI: 10.1016/j.tpb.2018.01.004] [Citation(s) in RCA: 25] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/10/2017] [Revised: 01/16/2018] [Accepted: 01/24/2018] [Indexed: 11/18/2022]
Abstract
Many mathematical frameworks of evolutionary game dynamics assume that the total population size is constant and that selection affects only the relative frequency of strategies. Here, we consider evolutionary game dynamics in an extended Wright-Fisher process with variable population size. In such a scenario, it is possible that the entire population becomes extinct. Survival of the population may depend on which strategy prevails in the game dynamics. Studying cooperative dilemmas, it is a natural feature of such a model that cooperators enable survival, while defectors drive extinction. Although defectors are favored for any mixed population, random drift could lead to their elimination and the resulting pure-cooperator population could survive. On the other hand, if the defectors remain, then the population will quickly go extinct because the frequency of cooperators steadily declines and defectors alone cannot survive. In a mutation-selection model, we find that (i) a steady supply of cooperators can enable long-term population survival, provided selection is sufficiently strong, and (ii) selection can increase the abundance of cooperators but reduce their relative frequency. Thus, evolutionary game dynamics in populations with variable size generate a multifaceted notion of what constitutes a trait's long-term success.
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Affiliation(s)
- Alex McAvoy
- Program for Evolutionary Dynamics, Harvard University, Cambridge, MA 02138, United States.
| | - Nicolas Fraiman
- Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, United States
| | - Christoph Hauert
- Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, Canada V6T 1Z2
| | - John Wakeley
- Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138, United States
| | - Martin A Nowak
- Program for Evolutionary Dynamics, Harvard University, Cambridge, MA 02138, United States; Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138, United States; Department of Mathematics, Harvard University, Cambridge, MA 02138, United States
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