Electron. J. Diff. Equ.,
Vol. 2015 (2015), No. 231, pp. 116.
Similarity solutions of a replicator dynamics equation associated with a
continuum of pure strategies
Vassilis G. Papanicolaou, Kyriaki Vasilakopoulou
Abstract:
We introduce a nonlinear degenerate parabolic equation containing a nonlocal term.
The equation serves as a replicator dynamics model where the set of strategies is
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 is 1, for any t > 0.
These solutions, being probability densities, can serve as timeevolving mixed
strategies of a player. We show that for our model there is an oneparameter
family of selfsimilar such solutions
,
all approaching the Dirac delta
function
as
.
Submitted August 24, 2015. Published September 10, 2015.
Math Subject Classifications: 35C06, 91A22, 91B52.
Key Words: Replicator dynamics model; nonlinear degenerate parabolic PDE;
nonlocal term; probability densities evolving in time;
selfsimilar solutions.
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Vassilis G. Papanicolaou
Department of Mathematics
National Technical University of Athens
Zografou Campus
157 80 Athens, Greece
email: papanico@math.ntua.gr


Kyriaki Vasilakopoulou
Department of Mathematics
National Technical University of Athens
Zografou Campus
157 80 Athens, Greece
email: kiriakivasilakopoulou@gmail.com

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