work we focus on the long time behavior of this multi-type growth model. However, differently from the above mentioned authors, we choose a first-passage percolation set-up on a random Delaunay triangulation [@VW90] whose spherical symmetry (isotropy) ensures that the asymptotic shape of the corresponding growth process is an euclidean ball. This choice allows us to prove various statements concerning minimizing paths, such as $\bP\big(K(\Gamma)=\infty\big)=1$, who could mostly only be conjectured by Newman in the standard model. In this setting, the main results we will prove are the following:

-   If a type of infection survives then the region it conquers is (asymptotically) a cone with a random angle (Theorem \[t1\], Remark \[rand+strai\]);

-   If the $k$ initial sites form a regular polygon centered at the origin with radius $r$, then the probability that $k$ coexistence occurs tends to $1$ when $r$ tends to infinity. Moreover, for all $\epsilon>0$, the probability that for all $j\in\{1,\dots,k\}$ the region conquered by infection $j$ contains (asymptotically) the cone with axis through $\0$ and $\x_j$ and angle $\frac{\pi}{k}-\epsilon$ also tends to $1$ (Theorem \[t2\]).

The main idea to prove our results is to explore the relation between this multi-type growth model and the asymptotic behavior 