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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 |
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. |
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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