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

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

1. 

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

Received  March 2012 Revised  August 2012 Published  April 2014

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.
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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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.

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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.

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K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations, Journal of Mathematical Biology, 2 (1975), 251-263. doi: 10.1007/BF00277154.

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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.

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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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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.

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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.

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J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8.

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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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