Lotka-Volterra equation and replicator dynamics: A two-dimensional classification

Bomze, Immanuel M.

doi:10.1007/bf00318088

Cited by 185 publications

(226 citation statements)

References 11 publications

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“…The dynamic system (3) is analysed in the Mathematical Appendix using Bomze's (1983) classification for replicator equations. In the following section, we illustrate the basic features of dynamics generated by (3).…”

Section: The Evolutionary Game Modelmentioning

confidence: 99%

Online and Offline Social Participation and Social Poverty Traps: Can Social Networks Save Human Relations?

Antoci

Sabatini

Sodini

2015

The Journal of Mathematical Sociology

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In this study, we develop an evolutionary game model to analyse how human relations evolve in a context characterised by declining face-to-face interactions and growing online social participation. Our results suggest that online networks may constitute a coping response allowing individuals to "defend" their social life from increasing busyness and a reduction in the time available for leisure.Internet-mediated interaction can play a positive role in preventing the disruption of ties and the weakening of community life documented by empirical studies. In this scenario, the digital divide is likely to become an increasingly relevant factor of social exclusion, which may exacerbate inequalities in well-being and capabilities.

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Section: The Evolutionary Game Modelmentioning

confidence: 99%

Online and Offline Social Participation and Social Poverty Traps: Can Social Networks Save Human Relations?

Antoci

Sabatini

Sodini

2015

The Journal of Mathematical Sociology

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“…Using this notation Liebrand [6] discussed the social dilemmas. The introduction of replicator dynamics [7] initiated a different classification based on the evolutionarily stable strategies [8][9][10][11] and phase portrait [12,13]. Very recently the games have been analyzed by distinguishing intragroup and intergroup interactions within the framework of population dynamics [14].…”

Section: Introductionmentioning

confidence: 99%

Four classes of interactions for evolutionary games

et al. 2015

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The symmetric four-strategy games are decomposed into a linear combination of 16 basis games represented by orthogonal matrices. Among these basis games four classes can be distinguished as it is already found for the three-strategy games. The games with self-dependent (cross-dependent) payoffs are characterized by matrices consisting of uniform rows (columns). Six of 16 basis games describe coordination-type interactions among the strategy pairs and three basis games span the parameter space of the cyclic components that are analogous to the rock-paper-scissors games. In the absence of cyclic components the game is a potential game and the potential matrix is evaluated. The main features of the four classes of games are discussed separately and we illustrate some characteristic strategy distributions on a square lattice in the low noise limit if logit rule controls the strategy evolution. Analysis of the general properties indicates similar types of interactions at larger number of strategies for the symmetric matrix games.

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“…Bomze (15,17) lists more than 40 possibilities for the 3 × 3 case. Here, we do not consider games with unstable edge fixed points and only consider generic examples in which the six off-diagonal entries are nonzero and distinct, so we end up with 11 cases described in SI Appendix, Section S2.…”

Section: [6]mentioning

confidence: 99%

Spatial evolutionary games with weak selection

Nanda

Durrett

2017

Proc. Natl. Acad. Sci. U.S.A.

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Recently, a rigorous mathematical theory has been developed for spatial games with weak selection, i.e., when the payoff differences between strategies are small. The key to the analysis is that when space and time are suitably rescaled, the spatial model converges to the solution of a partial differential equation (PDE). This approach can be used to analyze all 2 × 2 games, but there are a number of 3 × 3 games for which the behavior of the limiting PDE is not known. In this paper, we give rules for determining the behavior of a large class of 3 × 3 games and check their validity using simulation. In words, the effect of space is equivalent to making changes in the payoff matrix, and once this is done, the behavior of the spatial game can be predicted from the behavior of the replicator equation for the modified game. We say predicted here because in some cases the behavior of the spatial game is different from that of the replicator equation for the modified game. For example, if a rock-paper-scissors game has a replicator equation that spirals out to the boundary, space stabilizes the system and produces an equilibrium.cancer modeling | public goods game | bone cancer | rock-paper-scissors E volutionary games are often studied by assuming that the population is homogeneously mixing, i.e., each individual interacts equally with all the others. In this case, the frequencies of strategies evolve according to the replicator equation. See, e.g., Hofbauer and Sigmund's book (1). If ui is the frequency of players using strategy i, thenwhere Fi = j Gi,j uj is the fitness of strategy i, Gi,j is the payoff for playing strategy i against an opponent who plays strategy j , andF = i ui Fi is the average fitness. The homogeneous mixing assumption is not satisfied for the evolutionary games that arise in ecology or modeling solid cancer tumors, so it is important to understand how spatial structure changes the outcome of games. The goal of this paper is to facilitate applications of spatial evolutionary games by giving rules to determine the limiting behavior of a large class of 3 × 3 games. Our spatial games will take place on the 3D integer lattice Z 3 . The theory (2, 3) has been developed under the assumption that the interactions between an individual and its neighbors are given by an irreducible probability kernel p(x ) on Z 3 with p(0) = 0, that is finite range, symmetric p(x ) = p(−x ), and has covariance matrix σ 2 I . Here we will restrict our attention to the nearest neighbor case, in which p(x ) = 1/6 for x = (1, 0, 0), (−1, 0, 0), . . . (0, 0, −1).To describe the dynamics we let ξt (x ) be the strategy used by the individual at x at time t and letbe the fitness of x at time t. In birth-death dynamics, site x gives birth at rate ψt (x ) and sends its offspring to replace the individual at y with probability p(y − x ). In death-birth dynamics, the individual at x dies at rate 1 and is replaced by a copy of the one at y with probability proportional to p(y − x )ψt (y). The theory developed in ref. 3 can be applied...

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Lotka-Volterra equation and replicator dynamics: A two-dimensional classification

Cited by 185 publications

References 11 publications

Online and Offline Social Participation and Social Poverty Traps: Can Social Networks Save Human Relations?

Online and Offline Social Participation and Social Poverty Traps: Can Social Networks Save Human Relations?

Four classes of interactions for evolutionary games

Spatial evolutionary games with weak selection

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