  In this work we study the properties of segregation processes modeled by a
family of equations $$ L(u_i) (x) = u_i(x)\: F_i (u_1, \ldots, u_K)(x)\qquad
i=1,\ldots, K $$ where $F_i (u_1, \ldots, u_K)(x)$ is a non-local factor that
takes into consideration the values of the functions $u_j$'s in a full
neighborhood of $x.$ We consider as a model problem $$\Delta u_i^\ep (x) =
\frac1{\ep^2} u_i^\ep (x)\sum_{i\neq j} H(u_j^\ep)(x)$$ where $\ep$ is a small
parameter and $H(u_j^\ep)(x)$ is for instance $$H(u_j^\ep)(x)=
\int_{\mathcal{B}_1 (x)} u_j^\ep (y)\, \text{d}y$$ or $$H(u_j^\ep)(x)=
\sup_{y\in \mathcal{B}_1(x)} u_j^\ep (y).$$ Here the set $\mathcal{B}_1(x)$ is
the unit ball centered at $x$ with respect to a smooth, uniformly convex norm
$\rho$ of $\real^n$. Heuristically, this will force the populations to stay at
$\rho$-distance 1, one from each other, as $\ep\to0$.
