In order to study the tree of infection, Hägggström and Pemantle [@hp98; @hp99] have introduced a multi-type growth model as follows. At time $0$ we start with $k$ different sites of $\bZ^2$, say $\x_1,\dots,\x_k$, each one representing a different type of infection. A site $\y\in\bZ^2$ is then infected at time $\min\big\{T(\x_1,\y),\dots,T(\x_k,\y)\big\}$ and it is acquired by the infection which first arrives there, i.e. by the unique type $j\in\{1,\dots,k\}$ such that $T(\x_j,\y)=\min\big\{T(\x_1,\y),\dots,T(\x_k,\y)\big\}$ (Figure \[f2\]) . It may happens that at some early stage one of the types of infection completely surrounds another one, which then is prevented to grow indefinitely. If this does not occur, or equivalently, if all types of infection grow unboundedly, we say that $k$-coexistence occurs.

![Growth and Competition[]{data-label="f2"}](tfgrow){width="50.00000%"}

Turning back to the question of topological ends of $\Gamma$, Häggström and Pemantle have noticed that if $k$-coexistence occurs with positive probability then $K(\Gamma)\geq k$ occurs with positive probability. They also have shown that, if one considers an exponential passage time distribution then $2$-coexistence occurs with positive probability, and thus $K(\Gamma)\geq 2$ occurs with positive probability. Later Garet and Marchand [@gm04] and Hoffman [@H05] have extended this last result for stationary and ergodic FPP models on $\bZ^d$.

In this 