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.  Google Scholar

[2]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations,, Robert E. Krieger Publishing Company, (1987).   Google Scholar

[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.  Google Scholar

[4]

L. A. Imhof, The long-run behavior of the stochastic replicator dynamics,, Ann. Appl. Probab., 15 (2005), 1019.  doi: 10.1214/105051604000000837.  Google Scholar

[5]

N. I. Kavallaris, J. Lankeit and M. Winkler, On a Degenerate Non-Local Parabolic Problem Describing Infinite Dimensional Replicator Dynamics,, 2015, ().   Google Scholar

[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.  Google Scholar

[7]

J. Lankeit, Equilibration of Unit Mass Solutions to a Degenerate Parabolic Equation with a Nonlocal Gradient Nonlinearity,, 2015, ().   Google Scholar

[8]

J. Oechssler and F. Riedel, Evolutionary dynamics on infinite strategy spaces,, Economic Theory, 17 (2001), 141.  doi: 10.1007/PL00004092.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[12]

J. Smith, Maynard, Evolution and the Theory of Games,, Cambridge University Press, (1982).   Google Scholar

[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.  Google Scholar

show all references

References:
[1]

I. Bomze, Dynamical Aspects of Evolutionary Stability,, Monaish. Mathematik, 110 (1990), 189.  doi: 10.1007/BF01301675.  Google Scholar

[2]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations,, Robert E. Krieger Publishing Company, (1987).   Google Scholar

[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.  Google Scholar

[4]

L. A. Imhof, The long-run behavior of the stochastic replicator dynamics,, Ann. Appl. Probab., 15 (2005), 1019.  doi: 10.1214/105051604000000837.  Google Scholar

[5]

N. I. Kavallaris, J. Lankeit and M. Winkler, On a Degenerate Non-Local Parabolic Problem Describing Infinite Dimensional Replicator Dynamics,, 2015, ().   Google Scholar

[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.  Google Scholar

[7]

J. Lankeit, Equilibration of Unit Mass Solutions to a Degenerate Parabolic Equation with a Nonlocal Gradient Nonlinearity,, 2015, ().   Google Scholar

[8]

J. Oechssler and F. Riedel, Evolutionary dynamics on infinite strategy spaces,, Economic Theory, 17 (2001), 141.  doi: 10.1007/PL00004092.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[12]

J. Smith, Maynard, Evolution and the Theory of Games,, Cambridge University Press, (1982).   Google Scholar

[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.  Google Scholar

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