1980 Preface * 1999 Preface * 1999 Acknowledgements * Introduction * 1 Circular Logic * 2 Phase Singularities (Screwy Results of Circular Logic) * 3 The Rules of the Ring * 4 Ring Populations * 5 Getting Off the Ring * 6 Attracting Cycles and Isochrons * 7 Measuring the Trajectories of a Circadian Clock * 8 Populations of Attractor Cycle Oscillators * 9 Excitable Kinetics and Excitable Media * 10 The Varieties of Phaseless Experience: In Which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways * 11 The Firefly Machine 12 Energy Metabolism in Cells * 13 The Malonic Acid Reagent ('Sodium Geometrate') * 14 Electrical Rhythmicity and Excitability in Cell Membranes * 15 The Aggregation of Slime Mold Amoebae * 16 Numerical Organizing Centers * 17 Electrical Singular Filaments in the Heart Wall * 18 Pattern Formation in the Fungi * 19 Circadian Rhythms in General * 20 The Circadian Clocks of Insect Eclosion * 21 The Flower of Kalanchoe * 22 The Cell Mitotic Cycle * 23 The Female Cycle * References * Index of Names * Index of Subjects

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/596B270197FE27DE5FABB598B6210622/S0025557200150892a.pdf/div-class-title-span-class-italic-the-geometry-of-biological-time-span-by-a-t-winfree-pp-544-dm68-corrected-second-printing-1990-isbn-3-540-52528-9-springer-div.pdf

The geometry of biological time , by A. T. Winfree. Pp 544. DM68. Corrected Second Printing 1990. ISBN 3-540-52528-9 (Springer)

Evolutionary games on graphs

Prevalence of cooperation within groups of selfish individuals is puzzling in that it contradicts with the basic premise of natural selection. Favoring players with higher fitness, the latter is key for understanding the challenges faced by cooperators when competing with defectors. Evolutionary game theory provides a competent theoretical framework for addressing the subtleties of cooperation in such situations, which are known as social dilemmas. Recent advances point towards the fact that the evolution of strategies alone may be insufficient to fully exploit the benefits offered by cooperative behavior. Indeed, while spatial structure and heterogeneity, for example, have been recognized as potent promoters of cooperation, coevolutionary rules can extend the potentials of such entities further, and even more importantly, lead to the understanding of their emergence. The introduction of coevolutionary rules to evolutionary games implies, that besides the evolution of strategies, another property may simultaneously be subject to evolution as well. Coevolutionary rules may affect the interaction network, the reproduction capability of players, their reputation, mobility or age. Here we review recent works on evolutionary games incorporating coevolutionary rules, as well as give a didactic description of potential pitfalls and misconceptions associated with the subject. In addition, we briefly outline directions for future research that we feel are promising, thereby particularly focusing on dynamical effects of coevolutionary rules on the evolution of cooperation, which are still widely open to research and thus hold promise of exciting new discoveries.

http://www.matjazperc.com/publications/BioSystems_99_109.pdf

Coevolutionary games--a mini review.

https://gisc.uc3m.es/~cuesta/PDFs/PLR06208.pdf

Evolutionary game theory: temporal and spatial effects beyond replicator dynamics

Evolutionary game dynamics is one of the most fruitful frameworks for studying evolution in different disciplines, from Biology to Economics. Within this context, the approach of choice for many researchers is the so-called replicator equation, that describes mathematically the idea that those individuals performing better have more offspring and thus their frequency in the population grows. While very many interesting results have been obtained with this equation in the three decades elapsed since it was first proposed, it is important to realize the limits of its applicability. One particularly relevant issue in this respect is that of non-mean-field effects, that may arise from temporal fluctuations or from spatial correlations, both neglected in the replicator equation. This review discusses these temporal and spatial effects focusing on the non-trivial modifications they induce when compared to the outcome of replicator dynamics. Alongside this question, the hypothesis of linearity and its relation to the choice of the rule for strategy update is also analyzed. The discussion is presented in terms of the emergence of cooperation, as one of the current key problems in Biology and in other disciplines.

/pdf/evolutionary-game-theory-temporal-and-spatial-effects-beyond-4hbhqjijrq.pdf

Evolutionary game theory: Temporal and spatial effects beyond replicator dynamics

The effect of heterogeneous influence of different individuals on the maintenance of co-operative behaviour is studied in an evolutionary Prisoner's Dilemma game with players located on the sites of regular small-world networks The players interacting with their neighbours can either co-operate or defect and update their states by choosing one of the neighbours and adopting its strategy with a probability depending on the payoff difference The selection of the neighbour obeys a preferential rule: the more influential a neighbour, the larger the probability it is picked It is found that this simple preferential selection rule can promote continuously the co-operation of the whole population with the strengthening of the disorder of the underlying network

Prisoner's Dilemma Game with Heterogeneous Influential Effect on Regular Small-World Networks

We investigate the dynamics of random walks on weighted networks. Assuming that the edge weight and the node strength are used as local information by a random walker. Two kinds of walks, weight-dependent walk and strength-dependent walk, are studied. Exact expressions for stationary distribution and average return time are derived and confirmed by computer simulations. The distribution of average return time and the mean-square displacement are calculated for two walks on the Barrat–Barthelemy–Vespignani (BBV) networks. It is found that a weight-dependent walker can arrive at a new territory more easily than a strength-dependent one.

https://iopscience.iop.org/article/10.1088/0256-307X/24/2/077/pdf

Walks on Weighted Networks

We obtained a class of approximate periodic solutions for the (2+1)-dimensional modified Zakharov-Kuznetsov equation by using the homotopy analysis method (HAM). The solutions we got agree with the exact solutions. The results indicate that the HAM is still valid for solving a class of higher dimensional evolution equations. We also made some efforts to extend the HAM to find the analytical solutions for more nonlinear evolution equations in an easier way.

Application of the homotopy analysis method to solving nonlinear evolution equations

The collective synchronization of a system of coupled logistic maps on random community networks is investigated. It is found that the synchronizability of the community network is affected by two factors when the size of the network and the number of connections are fixed. One is the number of communities denoted by the parameter m, and the other is the ratio σ of the connection probability p of each pair of nodes within each community to the connection probability q of each pair of nodes among different communities. Theoretical analysis and numerical results indicate that larger m and smaller σ are the key to the enhancement of network synchronizability. We also testify synchronous properties of the system by analysing the largest Lyapunov exponents of the system.

Synchronization of coupled logistic maps on random community networks

We investigate a spatial Prisoner's Dilemma game with nonlinear attractive effect on regular small-world networks. The players located on the sites of networks can either cooperate with their neighbours or defect. In every generation, each player updates its strategy by firstly choosing one of the neighbours with a probability proportional to α denoting the attractiveness of the neighbour, where is the collected payoff and α (≥0) is a free parameter characterizing the extent of nonlinear effect. Then each player adopts its strategy with a probability dependent on their payoff difference. Using Monte Carlo simulations, we investigate the density ρC of cooperators in the stationary state for various values of α and the rewiring probability q of the network. It is shown that the introduction of such attractive effect remarkably promotes the emergence and persistence of cooperation over a wide range of the temptation to defect for the same network structures. We also point out that long-range connections either enhance or inhibit the cooperation, which depends on the value of α and the payoff parameter b.

WU Zhi-Xi

Papers

Prisoner's Dilemma Game with Heterogeneous Influential Effect on Regular Small-World Networks

Walks on Weighted Networks

Application of the homotopy analysis method to solving nonlinear evolution equations

Synchronization of coupled logistic maps on random community networks

Prisoner's Dilemma Game with Nonlinear Attractive Effect on Regular Small-World Networks