  We consider recursion equations of the form $u_{n+1}(x)=Q[u_n](x),~n\ge
1,~x\in R$, with a non-local operator $Q[u](x)= g( u\ast q)$, where $g$ is a
polynomial, satisfying $g(0)=0$, $g(1)=1$, $g((0,1)) \subseteq (0,1)$, and $q$
is a (compactly supported) probability density with $\ast$ denoting
convolution. Motivated by a line of works for nonlinear PDEs initiated by
Etheridge, Freeman and Penington (2017), we show that for general $g$, a
probabilistic model based on branching random walk can be given to the solution
of the recursion, while in case $g$ is also strictly monotone, a probabilistic
threshold-based model can be given. In the latter case, we provide a
conditional tightness result. We analyze in detail the bistable case and prove
for it convergence of the solution shifted around a linear in $n$ centering.
