
-
Previous Article
On bounding exact models of epidemic spread on networks
- DCDS-B Home
- This Issue
-
Next Article
Dynamical systems associated with adjacency matrices
Evolving multiplayer networks: Modelling the evolution of cooperation in a mobile population
1. | Department of Mathematics, City, University of London, 10 Northampton Square, London, EC1V 0HB, UK |
2. | Department of Mathematics and Statistics, The University of North Carolina at Greensboro, Greensboro, NC 27412, USA |
We consider a finite population of individuals that can move through a structured environment using our previously developed flexible evolutionary framework. In the current paper the behaviour of the individuals follows a Markov movement model where decisions about whether they should stay or leave depends upon the group of individuals they are with at present. The interaction between individuals is modelled using a public goods game. We demonstrate that cooperation can evolve when there is a cost associated with movement. Combining the movement cost with a larger population size has a positive effect on the evolution of cooperation. Moreover, increasing the exploration time, which is the amount of time an individual is allowed to explore its environment, also has a positive effect. Unusually, we find that the evolutionary dynamics used does not have a significant effect on these results.
References:
[1] |
C. Aktipis,
Know when to walk away: Contingent movement and the evolution of cooperation, Journal of Theoretical Biology, 231 (2004), 249-260.
doi: 10.1016/j.jtbi.2004.06.020. |
[2] |
C. Aktipis,
Is cooperation viable in mobile organisms? Simple walk away rule favors the evolution of cooperation in groups, Evolution and Human Behavior, 32 (2011), 263-276.
doi: 10.1016/j.evolhumbehav.2011.01.002. |
[3] |
B. Allen and M. Nowak,
Games on graphs, EMS Surveys in Mathematical Sciences, 1 (2014), 113-151.
doi: 10.4171/EMSS/3. |
[4] |
B. Allen and C. Tarnita,
Measures of success in a class of evolutionary models with fixed population size and structure, Journal of Mathematical Biology, 68 (2014), 109-143.
doi: 10.1007/s00285-012-0622-x. |
[5] |
T. Antal and I. Scheuring,
Fixation of strategies for an evolutionary game in finite populations, Bulletin of Mathematical Biology, 68 (2006), 1923-1944.
doi: 10.1007/s11538-006-9061-4. |
[6] |
M. Archetti and I. Scheuring,
Coexistence of cooperation and defection in public goods games, Evolution, 65 (2011), 1140-1148.
doi: 10.1111/j.1558-5646.2010.01185.x. |
[7] |
M. Archetti and I. Scheuring,
Review: Game theory of public goods in one-shot social dilemmas without assortment, Journal of Theoretical Biology, 299 (2012), 9-20.
doi: 10.1016/j.jtbi.2011.06.018. |
[8] |
M. Broom, C. Cannings and G. Vickers,
Multi-player matrix games, Bulletin of Mathematical Biology, 59 (1997), 931-952.
doi: 10.1007/BF02460000. |
[9] |
M. Broom, C. Lafaye, K. Pattni and J. Rychtář,
A study of the dynamics of multi-player games on small networks using territorial interactions, Journal of Mathematical Biology, 71 (2015), 1551-1574.
doi: 10.1007/s00285-015-0868-1. |
[10] |
M. Broom and J. Rychtář,
An analysis of the fixation probability of a mutant on special classes of non-directed graphs, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 464 (2008), 2609-2627.
doi: 10.1098/rspa.2008.0058. |
[11] |
M. Broom and J. Rychtář,
A general framework for analysing multiplayer games in networks using territorial interactions as a case study, Journal of Theoretical Biology, 302 (2012), 70-80.
doi: 10.1016/j.jtbi.2012.02.025. |
[12] |
M. Broom and J. Rychtář,
Ideal cost-free distributions in structured populations for general payoff functions, Dynamic Games and Applications, 8 (2018), 79-92.
doi: 10.1007/s13235-016-0204-4. |
[13] |
M. Bruni, M. Broom and J. Rychtář,
Analysing territorial models on graphs, Involve, a Journal of Mathematics, 7 (2014), 129-149.
doi: 10.2140/involve.2014.7.129. |
[14] |
M. Bukowski and J. Miekisz,
Evolutionary and asymptotic stability in symmetric multi-player games, International Journal of Game Theory, 33 (2004), 41-54.
doi: 10.1007/s001820400183. |
[15] |
M. Cavaliere, S. Sedwards, C. E. Tarnita, M. A. Nowak and A. Csikász-Nagy,
Prosperity is associated with instability in dynamical networks, Journal of Theoretical Biology, 299 (2012), 126-138.
doi: 10.1016/j.jtbi.2011.09.005. |
[16] |
X. Chen, A. Szolnoki and M. Perc, Risk-driven migration and the collective-risk social dilemma Physical Review E, 86 (2012), 036101.
doi: 10.1103/PhysRevE.86.036101. |
[17] |
R. Cong, B. Wu, Y. Qiu and L. Wang, Evolution of cooperation driven by reputation-based migration PLoS One, 7 (2012), e35776.
doi: 10.1371/journal.pone.0035776. |
[18] |
G. W. Constable and A. J. McKane,
Population genetics on islands connected by an arbitrary network: An analytic approach, Journal of Theoretical Biology, 358 (2014), 149-165.
doi: 10.1016/j.jtbi.2014.05.033. |
[19] |
P. Domenici, R. Batty, T. Similä and E. Ogam,
Killer whales (orcinus orca) feeding on schooling herring (clupea harengus) using underwater tail-slaps: Kinematic analyses of field observations, Journal of Experimental Biology, 203 (2000), 283-294.
|
[20] |
M. Enquist and O. Leimar,
The evolution of cooperation in mobile organisms, Animal Behaviour, 45 (1993), 747-757.
doi: 10.1006/anbe.1993.1089. |
[21] |
I. Erovenko and J. Rychtář,
The evolution of cooperation in one-dimensional mobile populations, Far East Journal of Applied Mathematics, 95 (2016), 63-88.
|
[22] |
J. A. Fletcher and M. Doebeli,
A simple and general explanation for the evolution of altruism, Proceedings of the Royal Society of London B: Biological Sciences, 276 (2009), 13-19.
doi: 10.1098/rspb.2008.0829. |
[23] |
F. Fu, C. Hauert, M. A. Nowak and L. ~Wang, Reputation-based partner choice promotes cooperation in social networks, Physical Review E, 78 (2008), 026117. |
[24] |
C. Gokhale and A. Traulsen,
Evolutionary games in the multiverse, Proceedings of the National Academy of Sciences, 107 (2010), 5500-5504.
doi: 10.1073/pnas.0912214107. |
[25] |
C. Gokhale and A. Traulsen,
Evolutionary multiplayer games, Dynamic Games and Applications, 4 (2014), 468-488.
doi: 10.1007/s13235-014-0106-2. |
[26] |
W. Hamilton,
Extraordinary sex ratios, Science, 156 (1967), 477-488.
doi: 10.1126/science.156.3774.477. |
[27] |
R. Ibsen-Jensen, K. Chatterjee and M. A. Nowak,
Computational complexity of ecological and evolutionary spatial dynamics, Proceedings of the National Academy of Sciences, 112 (2015), 15636-15641.
doi: 10.1073/pnas.1511366112. |
[28] |
S. Karlin and H. Taylor, A First Course in Stochastic Processes, London, Academic Press, 1975. |
[29] |
A. Li, M. Broom, J. Du and L. Wang, Evolutionary dynamics of general group interactions in structured populations, Physical Review E, 93 (2016), 022407, 7pp.
doi: 10.1103/PhysRevE.93.022407. |
[30] |
A. Li, B. Wu and L. Wang,
Cooperation with both synergistic and local interactions can be worse than each alone, Scientific Reports, 4 (2014), 1-6.
doi: 10.1038/srep05536. |
[31] |
E. Lieberman, C. Hauert and M. Nowak,
Evolutionary dynamics on graphs, Nature, 433 (2005), 312-316.
doi: 10.1038/nature03204. |
[32] |
W. Maciejewski and G. Puleo,
Environmental evolutionary graph theory, Journal of Theoretical Biology, 360 (2014), 117-128.
doi: 10.1016/j.jtbi.2014.06.040. |
[33] |
N. Masuda,
Directionality of contact networks suppresses selection pressure in evolutionary dynamics, Journal of Theoretical Biology, 258 (2009), 323-334.
doi: 10.1016/j.jtbi.2009.01.025. |
[34] |
J. Maynard Smith,
The theory of games and the evolution of animal conflicts, Journal of Theoretical Biology, 47 (1974), 209-221.
doi: 10.1016/0022-5193(74)90110-6. |
[35] |
J. Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, 1982. |
[36] |
J. Maynard Smith and G. R. Price,
The logic of animal conflict, Nature, 246 (1973), 15-18.
doi: 10.1038/246015a0. |
[37] |
P. Moran,
Random processes in genetics, in Mathematical Proceedings of the Cambridge Philosophical Society, vol. 54, Cambridge Univ Press, (1958), 60-71.
doi: 10.1017/S0305004100033193. |
[38] |
P. Moran, The Statistical Processes of Evolutionary Theory, Clarendon Press, Oxford, 1962. |
[39] |
M. Nowak, Evolutionary Dynamics, Exploring the Equations of Life Harward University Press, Cambridge, Mass, 2006. |
[40] |
H. Ohtsuki, C. Hauert, E. Lieberman and M. Nowak,
A simple rule for the evolution of cooperation on graphs and social networks, Nature, 441 (2006), 502-505.
doi: 10.1038/nature04605. |
[41] |
H. Ohtsuki, M. Nowak and J. Pacheco, Breaking the symmetry between interaction and replacement in evolutionary dynamics on graphs, Physical Review Letters, 98 (2007), 108106. |
[42] |
J. M. Pacheco, A. Traulsen and M. A. Nowak,
Active linking in evolutionary games, Journal of Theoretical Biology, 243 (2006), 437-443.
doi: 10.1016/j.jtbi.2006.06.027. |
[43] |
J. M. Pacheco, A. Traulsen and M. A. Nowak, Coevolution of strategy and structure in complex networks with dynamical linking, Physical Review Letters, 97 (2006), 258103.
doi: 10.1103/PhysRevLett.97.258103. |
[44] |
G. Palm,
Evolutionary stable strategies and game dynamics for n-person games, Journal of Mathematical Biology, 19 (1984), 329-334.
doi: 10.1007/BF00277103. |
[45] |
K. Pattni, M. Broom, J. Rychtář and L. J. Silvers, Evolutionary graph theory revisited: When is an evolutionary process equivalent to the moran process? in Proc. R. Soc. A, The Royal Society, 471 (2015), 20150334, 19 pp.
doi: 10.1098/rspa.2015.0334. |
[46] |
M. Perc, J. Gómez-Gardeñes, A. Szolnoki, L. M. Floría and Y. Moreno, Evolutionary dynamics of group interactions on structured populations: A review, Journal of The Royal Society Interface, 10 (2013), 20120997.
doi: 10.1098/rsif.2012.0997. |
[47] |
M. Perc and A. Szolnoki,
Coevolutionary games-a mini review, BioSystems, 99 (2010), 109-125.
doi: 10.1016/j.biosystems.2009.10.003. |
[48] |
P. Shakarian and P. Roos,
Fast and Deterministic Computation of Fixation Probability in Evolutionary Graphs, Technical report, DTIC Document, 2012. |
[49] |
P. Shakarian, P. Roos and A. Johnson,
A review of evolutionary graph theory with applications to game theory, Biosystems, 107 (2012), 66-80.
doi: 10.1016/j.biosystems.2011.09.006. |
[50] |
T. Similä,
Sonar observations of killer whales (orcinus orca) feeding on herring schools, Aquatic Mammals, 23 (1997), 119-126.
|
[51] |
G. Szabó and G. Fath,
Evolutionary games on graphs, Physics Reports, 446 (2007), 97-216.
doi: 10.1016/j.physrep.2007.04.004. |
[52] |
M. van Veelen and M. Nowak,
Multi-player games on the cycle, Journal of Theoretical Biology, 292 (2012), 116-128.
doi: 10.1016/j.jtbi.2011.08.031. |
[53] |
B. Voorhees and A. Murray, Fixation probabilities for simple digraphs in Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., The Royal Society, 469 (2013), 20120676, 18pp.
doi: 10.1098/rspa.2012.0676. |
[54] |
J. Wang, B. Wu, D. Ho and L. Wang, Evolution of cooperation in multilevel public goods games with community structures, EPL (Europhysics Letters), 93 (2011), 58001. |
[55] |
B. Wu, A. Traulsen and C. S. Gokhale,
Dynamic properties of evolutionary multi-player games in finite populations, Games, 4 (2013), 182-199.
doi: 10.3390/g4020182. |
[56] |
B. Wu, J. Arranz, J. Du, D. Zhou and A. Traulsen, Evolving synergetic interactions Journal of The Royal Society Interface 13 (2016), 20160282.
doi: 10.1098/rsif.2016.0282. |
[57] |
T. Wu, F. Fu, Y. Zhang and L. Wang, Expectation-driven migration promotes cooperation by group interactions, Physical Review E, 85 (2012), 066104.
doi: 10.1103/PhysRevE.85.066104. |
[58] |
L. Zhou, A. Li and L. Wang, Evolution of cooperation on complex networks with synergistic and discounted group interactions, EPL (Europhysics Letters) 110 (2015), 60006.
doi: 10. 1209/0295-5075/110/60006. |
show all references
References:
[1] |
C. Aktipis,
Know when to walk away: Contingent movement and the evolution of cooperation, Journal of Theoretical Biology, 231 (2004), 249-260.
doi: 10.1016/j.jtbi.2004.06.020. |
[2] |
C. Aktipis,
Is cooperation viable in mobile organisms? Simple walk away rule favors the evolution of cooperation in groups, Evolution and Human Behavior, 32 (2011), 263-276.
doi: 10.1016/j.evolhumbehav.2011.01.002. |
[3] |
B. Allen and M. Nowak,
Games on graphs, EMS Surveys in Mathematical Sciences, 1 (2014), 113-151.
doi: 10.4171/EMSS/3. |
[4] |
B. Allen and C. Tarnita,
Measures of success in a class of evolutionary models with fixed population size and structure, Journal of Mathematical Biology, 68 (2014), 109-143.
doi: 10.1007/s00285-012-0622-x. |
[5] |
T. Antal and I. Scheuring,
Fixation of strategies for an evolutionary game in finite populations, Bulletin of Mathematical Biology, 68 (2006), 1923-1944.
doi: 10.1007/s11538-006-9061-4. |
[6] |
M. Archetti and I. Scheuring,
Coexistence of cooperation and defection in public goods games, Evolution, 65 (2011), 1140-1148.
doi: 10.1111/j.1558-5646.2010.01185.x. |
[7] |
M. Archetti and I. Scheuring,
Review: Game theory of public goods in one-shot social dilemmas without assortment, Journal of Theoretical Biology, 299 (2012), 9-20.
doi: 10.1016/j.jtbi.2011.06.018. |
[8] |
M. Broom, C. Cannings and G. Vickers,
Multi-player matrix games, Bulletin of Mathematical Biology, 59 (1997), 931-952.
doi: 10.1007/BF02460000. |
[9] |
M. Broom, C. Lafaye, K. Pattni and J. Rychtář,
A study of the dynamics of multi-player games on small networks using territorial interactions, Journal of Mathematical Biology, 71 (2015), 1551-1574.
doi: 10.1007/s00285-015-0868-1. |
[10] |
M. Broom and J. Rychtář,
An analysis of the fixation probability of a mutant on special classes of non-directed graphs, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 464 (2008), 2609-2627.
doi: 10.1098/rspa.2008.0058. |
[11] |
M. Broom and J. Rychtář,
A general framework for analysing multiplayer games in networks using territorial interactions as a case study, Journal of Theoretical Biology, 302 (2012), 70-80.
doi: 10.1016/j.jtbi.2012.02.025. |
[12] |
M. Broom and J. Rychtář,
Ideal cost-free distributions in structured populations for general payoff functions, Dynamic Games and Applications, 8 (2018), 79-92.
doi: 10.1007/s13235-016-0204-4. |
[13] |
M. Bruni, M. Broom and J. Rychtář,
Analysing territorial models on graphs, Involve, a Journal of Mathematics, 7 (2014), 129-149.
doi: 10.2140/involve.2014.7.129. |
[14] |
M. Bukowski and J. Miekisz,
Evolutionary and asymptotic stability in symmetric multi-player games, International Journal of Game Theory, 33 (2004), 41-54.
doi: 10.1007/s001820400183. |
[15] |
M. Cavaliere, S. Sedwards, C. E. Tarnita, M. A. Nowak and A. Csikász-Nagy,
Prosperity is associated with instability in dynamical networks, Journal of Theoretical Biology, 299 (2012), 126-138.
doi: 10.1016/j.jtbi.2011.09.005. |
[16] |
X. Chen, A. Szolnoki and M. Perc, Risk-driven migration and the collective-risk social dilemma Physical Review E, 86 (2012), 036101.
doi: 10.1103/PhysRevE.86.036101. |
[17] |
R. Cong, B. Wu, Y. Qiu and L. Wang, Evolution of cooperation driven by reputation-based migration PLoS One, 7 (2012), e35776.
doi: 10.1371/journal.pone.0035776. |
[18] |
G. W. Constable and A. J. McKane,
Population genetics on islands connected by an arbitrary network: An analytic approach, Journal of Theoretical Biology, 358 (2014), 149-165.
doi: 10.1016/j.jtbi.2014.05.033. |
[19] |
P. Domenici, R. Batty, T. Similä and E. Ogam,
Killer whales (orcinus orca) feeding on schooling herring (clupea harengus) using underwater tail-slaps: Kinematic analyses of field observations, Journal of Experimental Biology, 203 (2000), 283-294.
|
[20] |
M. Enquist and O. Leimar,
The evolution of cooperation in mobile organisms, Animal Behaviour, 45 (1993), 747-757.
doi: 10.1006/anbe.1993.1089. |
[21] |
I. Erovenko and J. Rychtář,
The evolution of cooperation in one-dimensional mobile populations, Far East Journal of Applied Mathematics, 95 (2016), 63-88.
|
[22] |
J. A. Fletcher and M. Doebeli,
A simple and general explanation for the evolution of altruism, Proceedings of the Royal Society of London B: Biological Sciences, 276 (2009), 13-19.
doi: 10.1098/rspb.2008.0829. |
[23] |
F. Fu, C. Hauert, M. A. Nowak and L. ~Wang, Reputation-based partner choice promotes cooperation in social networks, Physical Review E, 78 (2008), 026117. |
[24] |
C. Gokhale and A. Traulsen,
Evolutionary games in the multiverse, Proceedings of the National Academy of Sciences, 107 (2010), 5500-5504.
doi: 10.1073/pnas.0912214107. |
[25] |
C. Gokhale and A. Traulsen,
Evolutionary multiplayer games, Dynamic Games and Applications, 4 (2014), 468-488.
doi: 10.1007/s13235-014-0106-2. |
[26] |
W. Hamilton,
Extraordinary sex ratios, Science, 156 (1967), 477-488.
doi: 10.1126/science.156.3774.477. |
[27] |
R. Ibsen-Jensen, K. Chatterjee and M. A. Nowak,
Computational complexity of ecological and evolutionary spatial dynamics, Proceedings of the National Academy of Sciences, 112 (2015), 15636-15641.
doi: 10.1073/pnas.1511366112. |
[28] |
S. Karlin and H. Taylor, A First Course in Stochastic Processes, London, Academic Press, 1975. |
[29] |
A. Li, M. Broom, J. Du and L. Wang, Evolutionary dynamics of general group interactions in structured populations, Physical Review E, 93 (2016), 022407, 7pp.
doi: 10.1103/PhysRevE.93.022407. |
[30] |
A. Li, B. Wu and L. Wang,
Cooperation with both synergistic and local interactions can be worse than each alone, Scientific Reports, 4 (2014), 1-6.
doi: 10.1038/srep05536. |
[31] |
E. Lieberman, C. Hauert and M. Nowak,
Evolutionary dynamics on graphs, Nature, 433 (2005), 312-316.
doi: 10.1038/nature03204. |
[32] |
W. Maciejewski and G. Puleo,
Environmental evolutionary graph theory, Journal of Theoretical Biology, 360 (2014), 117-128.
doi: 10.1016/j.jtbi.2014.06.040. |
[33] |
N. Masuda,
Directionality of contact networks suppresses selection pressure in evolutionary dynamics, Journal of Theoretical Biology, 258 (2009), 323-334.
doi: 10.1016/j.jtbi.2009.01.025. |
[34] |
J. Maynard Smith,
The theory of games and the evolution of animal conflicts, Journal of Theoretical Biology, 47 (1974), 209-221.
doi: 10.1016/0022-5193(74)90110-6. |
[35] |
J. Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, 1982. |
[36] |
J. Maynard Smith and G. R. Price,
The logic of animal conflict, Nature, 246 (1973), 15-18.
doi: 10.1038/246015a0. |
[37] |
P. Moran,
Random processes in genetics, in Mathematical Proceedings of the Cambridge Philosophical Society, vol. 54, Cambridge Univ Press, (1958), 60-71.
doi: 10.1017/S0305004100033193. |
[38] |
P. Moran, The Statistical Processes of Evolutionary Theory, Clarendon Press, Oxford, 1962. |
[39] |
M. Nowak, Evolutionary Dynamics, Exploring the Equations of Life Harward University Press, Cambridge, Mass, 2006. |
[40] |
H. Ohtsuki, C. Hauert, E. Lieberman and M. Nowak,
A simple rule for the evolution of cooperation on graphs and social networks, Nature, 441 (2006), 502-505.
doi: 10.1038/nature04605. |
[41] |
H. Ohtsuki, M. Nowak and J. Pacheco, Breaking the symmetry between interaction and replacement in evolutionary dynamics on graphs, Physical Review Letters, 98 (2007), 108106. |
[42] |
J. M. Pacheco, A. Traulsen and M. A. Nowak,
Active linking in evolutionary games, Journal of Theoretical Biology, 243 (2006), 437-443.
doi: 10.1016/j.jtbi.2006.06.027. |
[43] |
J. M. Pacheco, A. Traulsen and M. A. Nowak, Coevolution of strategy and structure in complex networks with dynamical linking, Physical Review Letters, 97 (2006), 258103.
doi: 10.1103/PhysRevLett.97.258103. |
[44] |
G. Palm,
Evolutionary stable strategies and game dynamics for n-person games, Journal of Mathematical Biology, 19 (1984), 329-334.
doi: 10.1007/BF00277103. |
[45] |
K. Pattni, M. Broom, J. Rychtář and L. J. Silvers, Evolutionary graph theory revisited: When is an evolutionary process equivalent to the moran process? in Proc. R. Soc. A, The Royal Society, 471 (2015), 20150334, 19 pp.
doi: 10.1098/rspa.2015.0334. |
[46] |
M. Perc, J. Gómez-Gardeñes, A. Szolnoki, L. M. Floría and Y. Moreno, Evolutionary dynamics of group interactions on structured populations: A review, Journal of The Royal Society Interface, 10 (2013), 20120997.
doi: 10.1098/rsif.2012.0997. |
[47] |
M. Perc and A. Szolnoki,
Coevolutionary games-a mini review, BioSystems, 99 (2010), 109-125.
doi: 10.1016/j.biosystems.2009.10.003. |
[48] |
P. Shakarian and P. Roos,
Fast and Deterministic Computation of Fixation Probability in Evolutionary Graphs, Technical report, DTIC Document, 2012. |
[49] |
P. Shakarian, P. Roos and A. Johnson,
A review of evolutionary graph theory with applications to game theory, Biosystems, 107 (2012), 66-80.
doi: 10.1016/j.biosystems.2011.09.006. |
[50] |
T. Similä,
Sonar observations of killer whales (orcinus orca) feeding on herring schools, Aquatic Mammals, 23 (1997), 119-126.
|
[51] |
G. Szabó and G. Fath,
Evolutionary games on graphs, Physics Reports, 446 (2007), 97-216.
doi: 10.1016/j.physrep.2007.04.004. |
[52] |
M. van Veelen and M. Nowak,
Multi-player games on the cycle, Journal of Theoretical Biology, 292 (2012), 116-128.
doi: 10.1016/j.jtbi.2011.08.031. |
[53] |
B. Voorhees and A. Murray, Fixation probabilities for simple digraphs in Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., The Royal Society, 469 (2013), 20120676, 18pp.
doi: 10.1098/rspa.2012.0676. |
[54] |
J. Wang, B. Wu, D. Ho and L. Wang, Evolution of cooperation in multilevel public goods games with community structures, EPL (Europhysics Letters), 93 (2011), 58001. |
[55] |
B. Wu, A. Traulsen and C. S. Gokhale,
Dynamic properties of evolutionary multi-player games in finite populations, Games, 4 (2013), 182-199.
doi: 10.3390/g4020182. |
[56] |
B. Wu, J. Arranz, J. Du, D. Zhou and A. Traulsen, Evolving synergetic interactions Journal of The Royal Society Interface 13 (2016), 20160282.
doi: 10.1098/rsif.2016.0282. |
[57] |
T. Wu, F. Fu, Y. Zhang and L. Wang, Expectation-driven migration promotes cooperation by group interactions, Physical Review E, 85 (2012), 066104.
doi: 10.1103/PhysRevE.85.066104. |
[58] |
L. Zhou, A. Li and L. Wang, Evolution of cooperation on complex networks with synergistic and discounted group interactions, EPL (Europhysics Letters) 110 (2015), 60006.
doi: 10. 1209/0295-5075/110/60006. |














Table of Notation | ||
Notation | Definition | Description |
| | Population size. |
| | Number of places. |
| Individual | |
| Place | |
| Place where | |
| | Population distribution at time |
| Population distribution history. | |
| Probability population has distribution | |
| | Population distribution probability function (PDPF). |
| | Probability that population has history |
| | Individual distribution probability function (IDPF). |
| Fitness contribution of | |
| | Fitness of |
| | Direct group: group that |
| | Replacement weight that |
| | Weighted adjacency matrix of evolutionary graph. |
| Replacement weight contribution that | |
| Two types of individuals in the population. | |
| | Population state, |
| | State consisting of all type |
| | Probability of transitioning from |
| | Fixation probability of type |
| | Probability that |
| | Probability that |
| | Staying propensity: probability that individual |
| Cooperator and defector interactive strategy. | |
| | Benefit of being with a cooperator (defector). |
| | Sensitivity shown to group members. |
| | Reward as a multiple of background fitness. |
| | Cost as a multiple of background fitness. |
| | Payoff to |
| | Movement cost. |
| | Exploration time. |
| Cooperator (defector) with staying propensity | |
| | Nash equilibrium staying propensity of cooperator (defector). |
Table of Notation | ||
Notation | Definition | Description |
| | Population size. |
| | Number of places. |
| Individual | |
| Place | |
| Place where | |
| | Population distribution at time |
| Population distribution history. | |
| Probability population has distribution | |
| | Population distribution probability function (PDPF). |
| | Probability that population has history |
| | Individual distribution probability function (IDPF). |
| Fitness contribution of | |
| | Fitness of |
| | Direct group: group that |
| | Replacement weight that |
| | Weighted adjacency matrix of evolutionary graph. |
| Replacement weight contribution that | |
| Two types of individuals in the population. | |
| | Population state, |
| | State consisting of all type |
| | Probability of transitioning from |
| | Fixation probability of type |
| | Probability that |
| | Probability that |
| | Staying propensity: probability that individual |
| Cooperator and defector interactive strategy. | |
| | Benefit of being with a cooperator (defector). |
| | Sensitivity shown to group members. |
| | Reward as a multiple of background fitness. |
| | Cost as a multiple of background fitness. |
| | Payoff to |
| | Movement cost. |
| | Exploration time. |
| Cooperator (defector) with staying propensity | |
| | Nash equilibrium staying propensity of cooperator (defector). |
Dynamics | |||
BDB | | BDD | |
DBD | | DBB | |
LB | | LD | |
Dynamics | |||
BDB | | BDD | |
DBD | | DBB | |
LB | | LD | |
Parameter Set | 1 | 2 | 3 | 4 | 5 | 6 |
10 | 10 | 10 | 20 | 10 | 10 | |
| 10 | 5 | 25 | 10 | 10 | 10 |
| Variable | Variable | Variable | Variable | 0.20 | 0.20 |
| 0.04 | 0.04 | 0.04 | 0.04 | 0.04 | 0.09 |
| 0.40 | 0.40 | 0.40 | 0.4 | Variable | Variable |
Parameter Set | 1 | 2 | 3 | 4 | 5 | 6 |
10 | 10 | 10 | 20 | 10 | 10 | |
| 10 | 5 | 25 | 10 | 10 | 10 |
| Variable | Variable | Variable | Variable | 0.20 | 0.20 |
| 0.04 | 0.04 | 0.04 | 0.04 | 0.04 | 0.09 |
| 0.40 | 0.40 | 0.40 | 0.4 | Variable | Variable |
[1] |
Astridh Boccabella, Roberto Natalini, Lorenzo Pareschi. On a continuous mixed strategies model for evolutionary game theory. Kinetic and Related Models, 2011, 4 (1) : 187-213. doi: 10.3934/krm.2011.4.187 |
[2] |
Anna Lisa Amadori, Astridh Boccabella, Roberto Natalini. A hyperbolic model of spatial evolutionary game theory. Communications on Pure and Applied Analysis, 2012, 11 (3) : 981-1002. doi: 10.3934/cpaa.2012.11.981 |
[3] |
Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic and Related Models, 2021, 14 (1) : 115-148. doi: 10.3934/krm.2020051 |
[4] |
King-Yeung Lam. Dirac-concentrations in an integro-pde model from evolutionary game theory. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 737-754. doi: 10.3934/dcdsb.2018205 |
[5] |
Scott G. McCalla. Paladins as predators: Invasive waves in a spatial evolutionary adversarial game. Discrete and Continuous Dynamical Systems - B, 2014, 19 (5) : 1437-1457. doi: 10.3934/dcdsb.2014.19.1437 |
[6] |
William H. Sandholm. Local stability of strict equilibria under evolutionary game dynamics. Journal of Dynamics and Games, 2014, 1 (3) : 485-495. doi: 10.3934/jdg.2014.1.485 |
[7] |
John Cleveland. Basic stage structure measure valued evolutionary game model. Mathematical Biosciences & Engineering, 2015, 12 (2) : 291-310. doi: 10.3934/mbe.2015.12.291 |
[8] |
Hassan Najafi Alishah, Pedro Duarte. Hamiltonian evolutionary games. Journal of Dynamics and Games, 2015, 2 (1) : 33-49. doi: 10.3934/jdg.2015.2.33 |
[9] |
Shui-Nee Chow, Kening Lu, Yun-Qiu Shen. Normal forms for quasiperiodic evolutionary equations. Discrete and Continuous Dynamical Systems, 1996, 2 (1) : 65-94. doi: 10.3934/dcds.1996.2.65 |
[10] |
Alexey Cheskidov, Landon Kavlie. Pullback attractors for generalized evolutionary systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 749-779. doi: 10.3934/dcdsb.2015.20.749 |
[11] |
Andrzej Swierniak, Michal Krzeslak. Application of evolutionary games to modeling carcinogenesis. Mathematical Biosciences & Engineering, 2013, 10 (3) : 873-911. doi: 10.3934/mbe.2013.10.873 |
[12] |
Stamatios Katsikas, Vassilli Kolokoltsov. Evolutionary, mean-field and pressure-resistance game modelling of networks security. Journal of Dynamics and Games, 2019, 6 (4) : 315-335. doi: 10.3934/jdg.2019021 |
[13] |
Jiahui Chen, Rundong Zhao, Yiying Tong, Guo-Wei Wei. Evolutionary de Rham-Hodge method. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3785-3821. doi: 10.3934/dcdsb.2020257 |
[14] |
Jim M. Cushing. The evolutionary dynamics of a population model with a strong Allee effect. Mathematical Biosciences & Engineering, 2015, 12 (4) : 643-660. doi: 10.3934/mbe.2015.12.643 |
[15] |
Minette Herrera, Aaron Miller, Joel Nishimura. Altruistic aging: The evolutionary dynamics balancing longevity and evolvability. Mathematical Biosciences & Engineering, 2017, 14 (2) : 455-465. doi: 10.3934/mbe.2017028 |
[16] |
Jeremias Epperlein, Vladimír Švígler. On arbitrarily long periodic orbits of evolutionary games on graphs. Discrete and Continuous Dynamical Systems - B, 2018, 23 (5) : 1895-1915. doi: 10.3934/dcdsb.2018187 |
[17] |
Amy Veprauskas, J. M. Cushing. Evolutionary dynamics of a multi-trait semelparous model. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 655-676. doi: 10.3934/dcdsb.2016.21.655 |
[18] |
Jinyuan Zhang, Aimin Zhou, Guixu Zhang, Hu Zhang. A clustering based mate selection for evolutionary optimization. Big Data & Information Analytics, 2017, 2 (1) : 77-85. doi: 10.3934/bdia.2017010 |
[19] |
Siegfried Carl. Comparison results for a class of quasilinear evolutionary hemivariational inequalities. Conference Publications, 2007, 2007 (Special) : 221-229. doi: 10.3934/proc.2007.2007.221 |
[20] |
Caichun Chai, Tiaojun Xiao, Eilin Francis. Is social responsibility for firms competing on quantity evolutionary stable?. Journal of Industrial and Management Optimization, 2018, 14 (1) : 325-347. doi: 10.3934/jimo.2017049 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]