Paladins as predators: Invasive waves in a spatial evolutionary adversarial game

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July 2014, 19(5): 1437-1457. doi: 10.3934/dcdsb.2014.19.1437

Paladins as predators: Invasive waves in a spatial evolutionary adversarial game

Scott G. McCalla ^1,

1.	Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095, United States

Received March 2012 Revised August 2012 Published April 2014

Abstract
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Invasive waves are numerically found in a variant of a reaction-diffusion system used to extend an evolutionary adversarial game into space wherein the influence of various strategies is allowed to diffuse. The diffusion of various strategies corresponds to peer-pressure. The invasive waves are driven by a nonlinear instability that enables an otherwise unstable state to travel through an initially uncooperative state leaving a cooperative state behind. The wave speed's dependence on the various diffusion parameters is examined in one- and two-dimensions through numerically solving the reaction-diffusion equations. Various other phenomena, such as pinning near a diffusive inhomogeneity, are also explored.

Keywords: evolutionary game theory, reaction-diffusion equations, spatial dynamics., social modeling, Traveling waves.

Mathematics Subject Classification: Primary: 35K57, 92D25; Secondary: 91A2.

Citation: 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

References:

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[9]	X. Hou, Y. Li and K. R. Meyer, Traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities, Discrete Contin. Dyn. Syst., 26 (2010), 265-290. doi: 10.3934/dcds.2010.26.265.
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[13]	J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8.
[14]	M. A. Lewis, B. Li and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233. doi: 10.1007/s002850200144.
[15]	B. Li, H. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98. doi: 10.1016/j.mbs.2005.03.008.
[16]	Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, Journal of Differential Equations, 131 (1996), 79-131, URL http://www.sciencedirect.com/science/article/pii/S0022039696901576. doi: 10.1006/jdeq.1996.0157.
[17]	S. McCalla, 2D invasion movie, http://www.math.ucla.edu/ mccalla/2DInvasion.mpg, 2012.
[18]	H. P. McKean, Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov, Comm. Pure Appl. Math., 28 (1975), 323-331. doi: 10.1002/cpa.3160280302.
[19]	H. P. McKean, Application of Brownian motion to the equation of Kolmogorov-Petrovskiĭ -Piskonov, Comm. Pure Appl. Math., 28 (1975), 323-331. doi: 10.1002/cpa.3160280302.
[20]	M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, Journal of Mathematical Biology, 9 (1980), 49-64. doi: 10.1007/BF00276035.
[21]	J. A. Sherratt, M. A. Lewis and A. C. Fowler, Ecological chaos in the wake of invasion, Proceedings of the National Academy of Sciences, 92 (1995), 2524-2528, URL http://www.pnas.org/content/92/7/2524.abstract. doi: 10.1073/pnas.92.7.2524.
[22]	J. A. Sherratt, On the evolution of periodic plane waves in reaction-diffusion systems of $\lambda $-$\omega$ type, SIAM Journal on Applied Mathematics, 54 (1994), 1374-1385, URL http://link.aip.org/link/?SMM/54/1374/1. doi: 10.1137/S0036139993243746.
[23]	N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, Journal of Theoretical Biology, 79 (1979), 83-99. doi: 10.1016/0022-5193(79)90258-3.
[24]	M. B. Short, P. J. Brantingham and M. R. D'Orsogna, Cooperation and punishment in an adversarial game: How defectors pave the way to a peaceful society, Phys. Rev. E, 82 (2010), 066114. doi: 10.1103/PhysRevE.82.066114.
[25]	G. Szabó and G. Fáth, Evolutionary games on graphs, Physics Reports, 446 (2007), 97-216, URL http://www.sciencedirect.com/science/article/pii/S0370157307001810. doi: 10.1016/j.physrep.2007.04.004.
[26]	G. Szabó and C. Hauert, Phase transitions and volunteering in spatial public goods games, Phys. Rev. Lett., 89 (2002), 118101. doi: 10.1103/PhysRevLett.89.118101.
[27]	W. van Saarloos, Front propagation into unstable states, Physics Reports, 386 (2003), 29-222, URL http://www.sciencedirect.com/science/article/pii/S0370157303003223.
[28]	X.-S. Wang, H. Wang and J. Wu, Traveling waves of diffusive predator-prey systems: Disease outbreak propagation, Discrete Contin. Dyn. Syst., 32 (2012), 3303-3324. doi: 10.3934/dcds.2012.32.3303.
[29]	H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396. doi: 10.1137/0513028.
[30]	H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218. doi: 10.1007/s002850200145.
[31]	H. F. Weinberger, M. A. Lewis and B. Li, Anomalous spreading speeds of cooperative recursion systems, J. Math. Biol., 55 (2007), 207-222. doi: 10.1007/s00285-007-0078-6.

show all references

References:

[1]	G. Dee and J. S. Langer, Propagating pattern selection, Phys. Rev. Lett., 50 (1983), 383-386. doi: 10.1103/PhysRevLett.50.383.
[2]	D. del Castillo-Negrete, B. Carreras and V. Lynch, Front propagation and segregation in a reaction-diffusion model with cross-diffusion, Physica D: Nonlinear Phenomena, 168/169 (2002), 45-60, URL http://www.sciencedirect.com/science/article/pii/S0167278902004943, VII Latin American Workshop on Nonlinear Phenomena. doi: 10.1016/S0167-2789(02)00494-3.
[3]	M. R. D'Orsogna, R. Kendall, M. McBride and M. B. Short, Criminal defectors lead to the emergence of cooperation in an experimental, adversarial game, PloS one, 8 (2013), e61458.
[4]	S. R. Dunbar, Travelling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32. doi: 10.1007/BF00276112.
[5]	S. R. Dunbar, Traveling wave solutions of diffusive lotka-volterra equations: A heteroclinic connection in r4, Transactions of the American Mathematical Society, 286 (1984), 557-594. doi: 10.2307/1999810.
[6]	R. A. Fisher, The wave of advance of advantageous genes, Annals of Human Genetics, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.
[7]	K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations, Journal of Mathematical Biology, 2 (1975), 251-263. doi: 10.1007/BF00277154.
[8]	M. Holzer and A. Scheel, A slow pushed front in a Lotka-Volterra competition model, Nonlinearity, 25 (2012), 2151-2179. doi: 10.1088/0951-7715/25/7/2151.
[9]	X. Hou, Y. Li and K. R. Meyer, Traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities, Discrete Contin. Dyn. Syst., 26 (2010), 265-290. doi: 10.3934/dcds.2010.26.265.
[10]	A. Kolmogorov, I. Petrovskii and N. Piscounov, Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application a un problème biologique, Mosc. Univ. Bull. Math., 1 (1937), 1-25.
[11]	J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. I. Le cas stationnaire, C. R. Math. Acad. Sci. Paris, 343 (2006), 619-625. doi: 10.1016/j.crma.2006.09.019.
[12]	J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal, C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684. doi: 10.1016/j.crma.2006.09.018.
[13]	J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8.
[14]	M. A. Lewis, B. Li and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233. doi: 10.1007/s002850200144.
[15]	B. Li, H. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98. doi: 10.1016/j.mbs.2005.03.008.
[16]	Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, Journal of Differential Equations, 131 (1996), 79-131, URL http://www.sciencedirect.com/science/article/pii/S0022039696901576. doi: 10.1006/jdeq.1996.0157.
[17]	S. McCalla, 2D invasion movie, http://www.math.ucla.edu/ mccalla/2DInvasion.mpg, 2012.
[18]	H. P. McKean, Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov, Comm. Pure Appl. Math., 28 (1975), 323-331. doi: 10.1002/cpa.3160280302.
[19]	H. P. McKean, Application of Brownian motion to the equation of Kolmogorov-Petrovskiĭ -Piskonov, Comm. Pure Appl. Math., 28 (1975), 323-331. doi: 10.1002/cpa.3160280302.
[20]	M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, Journal of Mathematical Biology, 9 (1980), 49-64. doi: 10.1007/BF00276035.
[21]	J. A. Sherratt, M. A. Lewis and A. C. Fowler, Ecological chaos in the wake of invasion, Proceedings of the National Academy of Sciences, 92 (1995), 2524-2528, URL http://www.pnas.org/content/92/7/2524.abstract. doi: 10.1073/pnas.92.7.2524.
[22]	J. A. Sherratt, On the evolution of periodic plane waves in reaction-diffusion systems of $\lambda $-$\omega$ type, SIAM Journal on Applied Mathematics, 54 (1994), 1374-1385, URL http://link.aip.org/link/?SMM/54/1374/1. doi: 10.1137/S0036139993243746.
[23]	N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, Journal of Theoretical Biology, 79 (1979), 83-99. doi: 10.1016/0022-5193(79)90258-3.
[24]	M. B. Short, P. J. Brantingham and M. R. D'Orsogna, Cooperation and punishment in an adversarial game: How defectors pave the way to a peaceful society, Phys. Rev. E, 82 (2010), 066114. doi: 10.1103/PhysRevE.82.066114.
[25]	G. Szabó and G. Fáth, Evolutionary games on graphs, Physics Reports, 446 (2007), 97-216, URL http://www.sciencedirect.com/science/article/pii/S0370157307001810. doi: 10.1016/j.physrep.2007.04.004.
[26]	G. Szabó and C. Hauert, Phase transitions and volunteering in spatial public goods games, Phys. Rev. Lett., 89 (2002), 118101. doi: 10.1103/PhysRevLett.89.118101.
[27]	W. van Saarloos, Front propagation into unstable states, Physics Reports, 386 (2003), 29-222, URL http://www.sciencedirect.com/science/article/pii/S0370157303003223.
[28]	X.-S. Wang, H. Wang and J. Wu, Traveling waves of diffusive predator-prey systems: Disease outbreak propagation, Discrete Contin. Dyn. Syst., 32 (2012), 3303-3324. doi: 10.3934/dcds.2012.32.3303.
[29]	H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396. doi: 10.1137/0513028.
[30]	H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218. doi: 10.1007/s002850200145.
[31]	H. F. Weinberger, M. A. Lewis and B. Li, Anomalous spreading speeds of cooperative recursion systems, J. Math. Biol., 55 (2007), 207-222. doi: 10.1007/s00285-007-0078-6.

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