Evolutionary Games in Space

N. Kronik; Y. Cohen

Mathematical Modelling of Natural Phenomena (2009)

  • Volume: 4, Issue: 6, page 54-90
  • ISSN: 0973-5348

Abstract

top
The G-function formalism has been widely used in the context of evolutionary games for identifying evolutionarily stable strategies (ESS). This formalism was developed for and applied to point processes. Here, we examine the G-function formalism in the settings of spatial evolutionary games and strategy dynamics, based on reaction-diffusion models. We start by extending the point process maximum principle to reaction-diffusion models with homogeneous, locally stable surfaces. We then develop the strategy dynamics for such surfaces. When the surfaces are locally stable, but not homogenous, the standard definitions of ESS and the maximum principle fall apart. Yet, we show by examples that strategy dynamics leads to convergent stable inhomogeneous strategies that are possibly ESS, in the sense that for many scenarios which we simulated, invaders could not coexist with the exisiting strategies.

How to cite

top

Kronik, N., and Cohen, Y.. "Evolutionary Games in Space." Mathematical Modelling of Natural Phenomena 4.6 (2009): 54-90. <http://eudml.org/doc/222357>.

@article{Kronik2009,
abstract = { The G-function formalism has been widely used in the context of evolutionary games for identifying evolutionarily stable strategies (ESS). This formalism was developed for and applied to point processes. Here, we examine the G-function formalism in the settings of spatial evolutionary games and strategy dynamics, based on reaction-diffusion models. We start by extending the point process maximum principle to reaction-diffusion models with homogeneous, locally stable surfaces. We then develop the strategy dynamics for such surfaces. When the surfaces are locally stable, but not homogenous, the standard definitions of ESS and the maximum principle fall apart. Yet, we show by examples that strategy dynamics leads to convergent stable inhomogeneous strategies that are possibly ESS, in the sense that for many scenarios which we simulated, invaders could not coexist with the exisiting strategies. },
author = {Kronik, N., Cohen, Y.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {mathematical modeling; game theory; reaction-diffusion equation; G-function; evolutionary ecology; -function; evolutionarily stable strategies},
language = {eng},
month = {11},
number = {6},
pages = {54-90},
publisher = {EDP Sciences},
title = {Evolutionary Games in Space},
url = {http://eudml.org/doc/222357},
volume = {4},
year = {2009},
}

TY - JOUR
AU - Kronik, N.
AU - Cohen, Y.
TI - Evolutionary Games in Space
JO - Mathematical Modelling of Natural Phenomena
DA - 2009/11//
PB - EDP Sciences
VL - 4
IS - 6
SP - 54
EP - 90
AB - The G-function formalism has been widely used in the context of evolutionary games for identifying evolutionarily stable strategies (ESS). This formalism was developed for and applied to point processes. Here, we examine the G-function formalism in the settings of spatial evolutionary games and strategy dynamics, based on reaction-diffusion models. We start by extending the point process maximum principle to reaction-diffusion models with homogeneous, locally stable surfaces. We then develop the strategy dynamics for such surfaces. When the surfaces are locally stable, but not homogenous, the standard definitions of ESS and the maximum principle fall apart. Yet, we show by examples that strategy dynamics leads to convergent stable inhomogeneous strategies that are possibly ESS, in the sense that for many scenarios which we simulated, invaders could not coexist with the exisiting strategies.
LA - eng
KW - mathematical modeling; game theory; reaction-diffusion equation; G-function; evolutionary ecology; -function; evolutionarily stable strategies
UR - http://eudml.org/doc/222357
ER -

References

top
  1. P.A. Abrams. Adaptive dynamics: Neither F nor G. Evol. Ecol. Res., 3 (2001), 369–373.  
  2. P.A. Abrams, H. Matsuda. Evolutionarily unstable fitness maxima and stable fitness minima of continuous traits. Evol. Ecol., 7 (1993), 465–487.  
  3. D. Alonso, F. Bartumeus, J. Catalan. Mutual interference between predators can give rise to Turing spatial patterns. Ecology, 83 (2002), 28–34.  
  4. H. Anton.and C. Rorres. Elementary linear algebra: applications version. 8th Edition. John Wiley & Sons, New York, 2000.  Zbl1286.15001
  5. J. Apaloo. Revisting strategic models of evolution: The concept of neighborhood invader strategies. Theor. Pop. Biol.,52 (1997), 71–77.  Zbl0889.92019
  6. N.F. Britton. Reaction-diffusion equations and their applications to biology. Academic Press, New York, 1986.  Zbl0602.92001
  7. J.S. Brown, N.B. Pavlovic. Evolution in heterogeneous environments - effects of migtation on habitat specialization. Evol. Ecol. 6 (1992),360–382.  
  8. J.S. Brown, T.L. Vincent. A theory for the evolutionary game. Theor. Pop. Biol., 31 (1987), 140–166.  Zbl0618.92014
  9. J.S. Brown, T.L. Vincent. Organiztion of predator-prey communities as an evolutionary game. Evolution, 46 (1992), 1269–1283.  
  10. R.G. Casten, C.J. Holland. Stability properties of solutions of systems of reaction-diffusion equations. SIAM J. Appl. Math. 33 (1977), 353–364.  Zbl0372.35044
  11. Y. Cohen, J. Pastor, T.L. Vincent. Evolutionary strategies and nutrient cycling in ecosystems. Evol. Ecol. Res., 2 (2000), 719–743.  
  12. Y. Cohen, T.L. Vincent, J.S. Brown. A G-function approach to fitness minima, fitness maxima, evolutionarily stable strategies and adaptive landscapes. Evol. Ecol. Res., 1 (1999), 923–942.  
  13. R. Cressman, G.T. Vickers. Spatial and density effects in evolutionary game theory. Math. Biol., 184 (1997), 359–369.  
  14. U. Dieckmann, R. Law. The dynamical theory of coevolution: A derivation from stochastic ecological processes. J. Math. Biol., 34 (1996), 579–612.  Zbl0845.92013
  15. R. Durrett, S. Levin. The importance of being discrete (and spatial). Theor. Pop. Biol., 46 (1994), 363–394.  Zbl0846.92027
  16. R. Durrett, S. Levin. Allelopathy in spatially distributed populations. J. Theor. Biol., 185 (1997), 165–171.  
  17. I. Eshel. Evolutionary and continuous stability. J. Theor. Biol. 108 (1983), 99–111.  
  18. I. Eshel. On the changing concept of evolutionary population stability as a reflection of a changing point of view in the quantitative theory of evolution. J. Math. Biol. 34 (1996), 485–510.  Zbl0851.92011
  19. I. Eshel, U. Motro. Kin selection and strong evolutionary stability of mutual help. Theor. Pop. Biol. 19 (1981), 420–433.  Zbl0473.92014
  20. G. Gause. The struggle for existence. Williams and Wilkins, Baltimore, 1934.  
  21. S.A.H. Geritz, M. Gyllenberg, F.J.A. Jacobs, K. Parvinen. Invasion dynamics and attractor inheritance. J. Math. Biol., 44 (2002), 548–560.  Zbl0990.92029
  22. S.A.H. Geritz, S.A.H. Kisdi, G. Meszéna, J.A.J. Metz. Evolutionary singular strategies and the adaptive growth and branching of the evolutionary tree. Evol. Ecol., 12 (1998), 35–57.  
  23. S.A.H. Geritz, J.A.J. Metz. É. Kisdi, G. Meszéna. Dynamics of adaptation and evolutionary branching. Physical Review Letters78, 2024–2027.  
  24. A. Gorban. Selection theorem for systems with inheritance. Math. Model. Nat. Phenom., 2 (2007), 1–45.  Zbl06543021
  25. P. Grindrod. The theory and applications of reaction-diffusion equations: patterns and waves. 2nd Edition. Clarendon press, Oxford, 1996.  Zbl0867.35001
  26. M. Gyllenberg, J.A. Metz. On fitness in structured metapopulations. J. Math. Biol., 43 (2001), 545–560.  Zbl0995.92034
  27. K.P. Hadeler. Diffusion in Fisher's population model. Rocky Mountain J. Math., 11 (1981), 39–45.  Zbl0445.35061
  28. J. Haldane. The causes of evolution. Princeton University Press, 1932.  
  29. W.G.S. Hines. Evolutionary stable strategies: A review of basic theory. Theor. Pop. Biol., 31 (1987), 195–272.  Zbl0608.92008
  30. V. C.L. Hutson, G.T. Vickers. Travelling waves and dominance of ESS's. J. Math. Biol., 30 (1992), 457–471.  Zbl0763.92007
  31. N. Kalev-Kronik. Evolutionary games in space. Ph.D. Thesis, University of Minneosta, 2006.  
  32. W. Kaplan. Advanced calculus. Addison-Wesley, Reading, 1952.  Zbl0047.28308
  33. C.L. Lehman, D. Tilman. Spatial Ecology : The Role of Space in Population Dynamics and Interspecific Interactions,chapter: Competition in Spatial Habitats.. Princeton University Press, Princeton, 1997.  
  34. J.L. Lions. Equations differentielles operationelles. Springer-Verlag, New-York, 1961.  
  35. S. Lipschutz. Linear algebra. McGraw-Hill, New York, 1991.  Zbl0699.15001
  36. J. Maynard-Smith. Evolution and the theory of games. Cambridge University Press, Cambridge, 1982.  Zbl0526.90102
  37. J. Maynard-Smith, G. Price. The logic of animal conflict. Nature, 246 (1973), 15–18.  
  38. J.A.J. Metz, M. Gyllenberg. How should we define fitness in structured metapopulation models?. Proc. Royal Soc. London B, 268 (2001), 499–508.  
  39. J. Murray. Mathematical biology, 2nd Edition, Springer-Verlag, Berlin, 1993.  Zbl0779.92001
  40. C. Neuhauser. Habitat destruction and competitive coexistence in spatially explicit models with local interactions. J. Theor. Biol., 193 (1998), 445–463.  
  41. C. Neuhauser, S.W. Pacala. An explicit spatial version of the lotka-volterra model with interspecific competition. Ann. Appl. Probab., 9 (1999), 1226–1259.  Zbl0948.92022
  42. H.G. Othmer, L.E. Scriven. Interactions of reaction and diffusion in open systems. Ind. Eng. Chem. Fund., 8 (1969), 302–313.  
  43. K. Parvinen. Evolution of migration in a metapopulation. Bul. Math. Biol., 61 (1999), 531–550.  Zbl1323.92183
  44. H. Qian, J. Murray. A simple method of parameter space determination for diffusion-driven instability with three species. Appl. Math. Let., 9 (2001), 405–411.  Zbl0980.35062
  45. A. Sasaki, I. Kawaguchi, A. Yoshimori. Spatial mosaic and interfacial dynamics in a Müllerian mimicry system. Theor. Pop. Biol., 61 (2002), 49–71.  Zbl1048.92024
  46. L.E. Segel, J.L. Jackson. Dissipative structure: An explanation and an ecological example. J. Theor. Biol., 37 (1972), 545–559.  
  47. J. Smoller. Shock waves and reaction-diffusion equations. Springer-Verlag, New York, 1983.  Zbl0508.35002
  48. T. Takada, J. Kigami. The dynamical attainability of ESS in evolutionary games. J. Math. Biol., 29 (1991), 513–529.  Zbl0734.92021
  49. P.D. Taylor. Evolutionary stability in one-parameter models under weak selection. Theor. Pop. Biol., 36 (1989), 125–143.  Zbl0684.92014
  50. D. Tilman, P. Kareiva eds. Spatial ecology : the role of space in population dynamics and interspecific interactions. Princeton University Press, Princeton, 1997.  
  51. A.M. Turing. On the chemical basis of morphogenesis. Phil. Trans. B., 237 (1952), 37–37.  
  52. G.T. Vickers, Spatial patterns and ESS's. J. Theo. Biol., 140 (1989), 129–135.  
  53. G.T. Vickers,V.C.L. Hutson, C.J. Budd. Spatial patterns in population conflicts. J. Math. Biol., 31 (1993), 411–430.  Zbl0773.92014
  54. T. Vincent, Evolutionary games. J. Optim. Theor. Appl., 46 (1985), 605–612.  Zbl0548.90110
  55. T. Vincent, J. Brown. Evolution under nonequilibrium dynamics. Math. Model., 8 (1987), 766–771.  Zbl0632.92017
  56. T.L. Vincent, J. Brown. Evolutionary game theory, natural selection, and Darwinian dynamics. Cambridge University Press, Cambridge, 2005.  Zbl1140.91015
  57. Vincent, T. Evolutionary stable strategies in differential and difference equation models. Evol. Ecol., 2, (1988), 321–337.  
  58. T.L. Vincent, Y. Cohen, J.S. Brown. Evolution via strategy dynamics. Theor. Pop. Biol., 44 (1993), 149–176.  Zbl0788.92018
  59. T.L. Vincent, M.V. Van, G.S. Goh. Ecological stability, evolutionary stability, and the ESS Maximum Principle. Evol. Ecol., 10 (1996), 567–591.  
  60. V. Zakharov, V.S. L'vov, S.S. Starobinets. Spin-wave turbulence beyond the parametric excitation threshold. Soviet Physics Uspekhii, 17 (1975), 896–919.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.