Similarity solutions for a multi-dimensional replicator dynamics equation

Similarity solutions of a multidimensional replicator dynamics integrodifferential equation

“…In order to make some progress in this direction, the works [6] and [10] initiated the study of the case where S is the set R d , d ≥ 1, and the payoff operator A is the Laplacian operator ∆. Then the evolution law (1) becomes…”

“…References [6] and [10] deal only with the special problem of constructing an oneparameter family of self-similar solutions for (2), namely, solutions u of the form u(t, x) = t −κ g d rt −λ , where r :…”

Equilibration of unit mass solutions to a degenerate parabolic equation with a nonlocal gradient nonlinearity

“…In this setting, anyhow, Ω is a set of "strategies" and, in the manner of a probability density, u gives the relative frequency with which they are pursued in the population of "players" considered -and accordingly the case of highest interest from an applications viewpoint is that of Ω u 0 = 1. Let us mention some results concerning explicit examples of solutions to (1) and solutions of a particular form: In [11] and [17], self-similar solutions to (1) were constructed in Ω = R and Ω = R n , respectively, and [16] investigates self-similar solutions in Ω = R for a related model, where the Laplacian is perturbed by a time-dependent term involving first derivatives as well. In [10,9], stationary solutions of (1) were studied; however, it remained open if those accurately capture the long-term behaviour of solutions to (1).…”

On a class of operator equations arising in infinite dimensional replicator dynamics

“…[1,16,17]) and applications. However, most of the work on replicator dynamics, with the notable exception of [2][3][4][5][6], has focused on games that have a finite strategy space, thus leading to a dynamical system for the frequencies of the population which is finite dimensional. According to [2][3][4], interesting applications may arise (either in biology or economics) in cases where the strategy space is not finite or, even, not discrete.…”