Similarity solutions of a multidimensional replicator dynamics integrodifferential equation

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 $d=1$.

1. Introduction -The replicator dynamics equation. The replicator dynamics models are popular in evolutionary game theory. They have significant applications in economics, population biology, as well as in other areas of science [3], [4], [12], [13].
Replicator dynamics has been studied extensively in the finite-dimensional case: Let A = (a ij ) be an m × m negative matrix. The typical replicator dynamics equation is [3] u (notice that if the above conditions are satisfied for t = 0, then they are satisfied for all t ≥ 0, under the flow (1)). The vector u represents the mixed strategy of one member of the population, i.e. one player against the rest of the population. The dependence of u in t allows the player to update his strategy, in order to increase his payoff. Infinite dimensional versions of this evolutionary strategy models have been proposed, e.g., in [1] and [8] (see also [9] and the survey [3]) in connection to certain economic and biological applications. For instance, there are situations where (pure) strategies correspond to geographical points, and hence it is natural to model the set of strategies by a continuum. However, the abstract form of the proposed equations does not allow one to obtain much insight, for example on the form of solutions.
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 where (· , ·) denotes the usual inner product of the (real) Hilbert space L 2 (R d ) of the square-integrable functions defined on R d , namely 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 := |x| = x 2 1 + · · · + x 2 d .
A peculiar feature of these solutions is that all of them are probability densities on R d , for all t > 0, and approach the Dirac delta function δ(x) as t → 0 + . It is worth mentioning that if we consider the equation (2) in a bounded domain Ω of R d with Dirichlet boundary conditions and an initial condition u(0, x) which is a probability density and does not vanish inside Ω, then it has been very recently established [5], [7] that u(t, x) exists for all t > 0 and approaches, as t → ∞ the equilibrium solution which is also a probability density (notice that there is exactly one such equilibrium solution). Of course, in the case where the domain is the whole R d such equilibrium solutions do not exist, and this is one reason why self-similar solutions become important. One criticism towards (2) is that the Laplacian operator ∆ is a symmetric operator and, also, time-independent. A payoff operator A which is symmetric with respect to the inner product (· , ·) corresponds to the case of a partnership game, where interests of both players coincide (see, e.g., [3]). These are unrealistic features for a payoff operator in a replicator dynamics model. For this reason, in our recent work [11], we considered a nonsymmetric and time-dependent payoff operator, namely

SIMILARITY SOLUTIONS OF A REPLICATOR DYNAMICS EQUATION 53
where α > 0 is a parameter. Then, in view of (5), the replicator dynamics equation (1) becomes with u = u(t, x). In [11] it was shown that (5) possesses an one-parameter family of self-similar solutions, all approaching the Dirac delta function δ(x) as t → 0 + . Viewed as functions of x, all those solutions are probability densities on R d , for each t > 0.
In the present work we study the d-dimensional version of (6), d ≥ 2, where the strategy space is S = R d , while the operator A is where α > 0 is an arbitrary but fixed constant, γ is a specific constant (as we will see γ = − 2 d+2 ), t > 0, x ∈ R d , while the operators ∆ and ∇ are acting on the x variable. In this case, the corresponding replicator dynamics problem (1) takes the form Incidentally, let us remark that (8), as well as (6) and (2), can be viewed as nonlinear (and nonlocal) Fokker-Planck equations.
The main result of this article is the construction of an one-parameter family of self-similar solutions for (8)-(9), i.e. solutions u of the form (4). As in the previous cases, all these solutions are probability densities on R d , for all t > 0, and approach the Dirac delta function δ(x) as t → 0 + . 2. Self similar solutions. Consider the problem (8)- (9). We will look for solutions u(t, x) of the form (4). We set s = rt −λ (hence r = st λ ). Then u(t, x) of (4) can be also written as u(t, x) = t −κ g d (s) (notice that 0 < s < ∞). It follows that Since u of (4) is radial in x, we have and From the above, we obtain By (7) we have
(15) Then, by (15), we have where S d−1 is the unit sphere in R d , σ is the surface measure on S d−1 (namely the measure on S d−1 induced by the d-dimensional Lebesgue measure), while σ d is the total measure of S d−1 . Recall that where Γ(·) is the Gamma function. Combining (16) with (4),(11), we obtain Finally, assuming that lim s→∞ g d (s) 2 s d = 0, we have Furthermore, using Green's first identity, we have where S d−1 (R) is the sphere of radius R in R d centered at the origin, and η is its outward unit normal vector. An easy calculation gives that for radial functions we have |∇u| 2 = u 2 r , thus We set and and by (13), (15), (11), operator A takes the form Substituting (4), (10), (22), (23) in (8), we have The only way that (24) is a meaningful equation is that it does not contain t. Therefore we are forced to take This gives Hence, by (25), (4) gives and the operator A becomes Finally, we notice that, under (26), (9) gives which is independent of t. Thus, if we set The following lemma summarizes what we have done so far.
is a probability density in x and satisfies (8), then we must have where and Conversely, if (29)-(34) hold, then u(t, x) given by (28) is a probability density in x and satisfies (8).
We need to show that there exist function(s) g d (s) satisfying (30), (31), (32), together with (33), (34). In order to do that we must first consider an auxiliary problem.
3. The auxiliary problem. Consider the problem is a natural number and µ is a real parameter satisfying We note that the above initial conditions (36) are interpreted in the sense of limits as s → 0 + . Equation (35) can be written in the form as long as q(s) = 0. By Proposition 1 of the Appendix we have that there is an > 0 such that (35) and (36) have a unique solution q(s) for s ∈ [0, ].
Lemma 3.1. The solution q(s) of (35), (36) exists for all s > 0 and it is a strictly positive function which is decreasing on (0, +∞). Also, Then, by a well-known theorem in the theory of ordinary differential equations [2], Claim 1. Suppose that the function q is positive on some interval (0, s 1 ) with 0 < s 1 < b. Then q must stay negative on (0, s 1 ).
Furthermore, we notice that Then, from (47) and (49), there is a sequence (s n ), such that s n → ∞, q attains a local minimum at s n for all n and lim n→∞ q (s n ) = −δ, for some δ > 0.
The proof of this key lemma is now complete.

Lemma 3.2. Let q(s) be the solution of the problem (35), (36). Then
Proof. First we notice that, since by Lemma 3.1 q is negative and lim s→∞ q (s) = 0, (iii) follows immediately from (i). We will use an inductive argument to show that for each n ∈ {1, 2, ...d} we have We will proof that (i ), (ii ) also hold for k = n. Multiplying both sides of (35) with s n−1 , integrating from 0 to s > 0 and using q (0) = 0, we obtain Since q(s) > 0 for all s ≥ 0 and q (s) < 0 for all s > 0, from (69) we have If n ≥ 3, our hypothesis enables us to apply (i ) for k = n − 2 and using the fact that q (s) < 0, for all s > 0 and q(s) ≤ A, for all s ≥ 0, we have Thus, For n = 2, the above inequality also holds. Indeed, for s > 0 we have We set Then, the inequality (70) can be written and this establishes (iv). If we suppose that N = 0, then the above limit tells us that N (s) := α(d + 2)s n−1 q(s) 2 +2s n−1 q(s), is asymptotic to 2(d+2)N s −1 , which is impossible because

By (69) we have
Since q(s) > 0, for all s ≥ 0 then and thus lim Then, consequently, which is (vi).
Using (78) together with the fact that q(0) = A, from the above inequality we obtain Thus, which is (80).