# American Institute of Mathematical Sciences

January  2016, 3(1): 51-74. doi: 10.3934/jdg.2016003

## Similarity solutions of a multidimensional replicator dynamics integrodifferential equation

 1 Department of Mathematics, National Technical University of Athens, Zografou Campus, 157 80 Athens, Greece, Greece

Received  November 2015 Revised  January 2016 Published  March 2016

We consider a nonlinear degenerate parabolic equation containing a nonlocal term, where the spatial variable $x$ belongs to $\mathbb{R}^d$, $d \geq 2$. The equation serves as a replicator dynamics model where the set of strategies is $\mathbb{R}^d$ (hence a continuum). In our model the payoff operator (which is the continuous analog of the payoff matrix) is nonsymmetric and, also, evolves with time. We are interested in solutions $u(t, x)$ of our equation which are positive and their integral (with respect to $x$) over the whole space $\mathbb{R}^d$ is $1$, for any $t > 0$. These solutions, being probability densities, can serve as time-evolving mixed strategies of a player. We show that for our model there is an one-parameter family of self-similar such solutions $u(t, x)$, all approaching the Dirac delta function $\delta(x)$ as $t \to 0^+$. The present work extends our earlier work [11] which dealt with the case $d=1$.
Citation: Vassilis G. Papanicolaou, Kyriaki Vasilakopoulou. Similarity solutions of a multidimensional replicator dynamics integrodifferential equation. Journal of Dynamics & Games, 2016, 3 (1) : 51-74. doi: 10.3934/jdg.2016003
##### References:
 [1] I. Bomze, Dynamical Aspects of Evolutionary Stability,, Monaish. Mathematik, 110 (1990), 189. doi: 10.1007/BF01301675. [2] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations,, Robert E. Krieger Publishing Company, (1987). [3] J. Hofbauer and K. Sigmund, Evolutionary game dynamics,, Bulletin (New Series) of the American Mathematical Society, 40 (2003), 479. doi: 10.1090/S0273-0979-03-00988-1. [4] L. A. Imhof, The long-run behavior of the stochastic replicator dynamics,, Ann. Appl. Probab., 15 (2005), 1019. doi: 10.1214/105051604000000837. [5] N. I. Kavallaris, J. Lankeit and M. Winkler, On a Degenerate Non-Local Parabolic Problem Describing Infinite Dimensional Replicator Dynamics,, 2015, (). [6] D. Kravvaritis, V. G. Papanicolaou and A. Yannacopoulos, Similarity solutions for a replicator dynamics equation,, Indiana Univ. Math. Journal, 57 (2008), 1929. doi: 10.1512/iumj.2008.57.3297. [7] J. Lankeit, Equilibration of Unit Mass Solutions to a Degenerate Parabolic Equation with a Nonlocal Gradient Nonlinearity,, 2015, (). [8] J. Oechssler and F. Riedel, Evolutionary dynamics on infinite strategy spaces,, Economic Theory, 17 (2001), 141. doi: 10.1007/PL00004092. [9] J. Oechssler and F. Riedel, On the dynamic foundation of evolutionary stability in continuous models,, Journal of Economic Theory, 107 (2002), 223. doi: 10.1006/jeth.2001.2950. [10] V. G. Papanicolaou and G. Smyrlis, Similarity solutions for a multidimensional replicator dynamics equation,, Nonlinear Analysis, 71 (2009), 3185. doi: 10.1016/j.na.2009.01.227. [11] V. G. Papanicolaou and K. Vasilakopoulou, Similarity solutions for a replicator dynamics equation associated to a continuum of pure strategies,, Electronic Journal of Differential Equations, 2015 (2015), 1. [12] J. Smith, Maynard, Evolution and the Theory of Games,, Cambridge University Press, (1982). [13] P. D. Taylor and L. B. Jonker, Evolutionary stable strategies and game dynamics,, Mathematical Biosciences, 40 (1978), 145. doi: 10.1016/0025-5564(78)90077-9.

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##### References:
 [1] I. Bomze, Dynamical Aspects of Evolutionary Stability,, Monaish. Mathematik, 110 (1990), 189. doi: 10.1007/BF01301675. [2] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations,, Robert E. Krieger Publishing Company, (1987). [3] J. Hofbauer and K. Sigmund, Evolutionary game dynamics,, Bulletin (New Series) of the American Mathematical Society, 40 (2003), 479. doi: 10.1090/S0273-0979-03-00988-1. [4] L. A. Imhof, The long-run behavior of the stochastic replicator dynamics,, Ann. Appl. Probab., 15 (2005), 1019. doi: 10.1214/105051604000000837. [5] N. I. Kavallaris, J. Lankeit and M. Winkler, On a Degenerate Non-Local Parabolic Problem Describing Infinite Dimensional Replicator Dynamics,, 2015, (). [6] D. Kravvaritis, V. G. Papanicolaou and A. Yannacopoulos, Similarity solutions for a replicator dynamics equation,, Indiana Univ. Math. Journal, 57 (2008), 1929. doi: 10.1512/iumj.2008.57.3297. [7] J. Lankeit, Equilibration of Unit Mass Solutions to a Degenerate Parabolic Equation with a Nonlocal Gradient Nonlinearity,, 2015, (). [8] J. Oechssler and F. Riedel, Evolutionary dynamics on infinite strategy spaces,, Economic Theory, 17 (2001), 141. doi: 10.1007/PL00004092. [9] J. Oechssler and F. Riedel, On the dynamic foundation of evolutionary stability in continuous models,, Journal of Economic Theory, 107 (2002), 223. doi: 10.1006/jeth.2001.2950. [10] V. G. Papanicolaou and G. Smyrlis, Similarity solutions for a multidimensional replicator dynamics equation,, Nonlinear Analysis, 71 (2009), 3185. doi: 10.1016/j.na.2009.01.227. [11] V. G. Papanicolaou and K. Vasilakopoulou, Similarity solutions for a replicator dynamics equation associated to a continuum of pure strategies,, Electronic Journal of Differential Equations, 2015 (2015), 1. [12] J. Smith, Maynard, Evolution and the Theory of Games,, Cambridge University Press, (1982). [13] P. D. Taylor and L. B. Jonker, Evolutionary stable strategies and game dynamics,, Mathematical Biosciences, 40 (1978), 145. doi: 10.1016/0025-5564(78)90077-9.
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