We consider the discrete-time threshold-$theta ge 2$ contact process on a random r-regular graph on n vertices. In this process, a vertex with at least theta occupied neighbors at time t will be occupied at time t+1 with probability p, and vacant otherwise. We show that if $theta ge 2$ and $r ge theta+2$, $epsilon_1$ is small and p is at least $p_1(epsilon_1)$, then starting from all vertices occupied the fraction of occupied vertices stays above $1-2epsilon_1$ up to time $exp(gamma_1(r)n)$ with probability at least $1 - exp(-gamma_1(r)n)$. In the other direction, we show that for $p_2 < 1$ there is an $epsilon_2(p_2)>0$ so that if $p le p_2$ and the number of occupied vertices in the initial configuration is at most $epsilon_2(p_2)n$, then with high probability all vertices are vacant at time $C_2(p_2) log(n)$. These two conclusions imply that on the random r-regular graph there cannot be a quasi-stationary distribution with density of occupied vertices between 0 and $epsilon_2(p_1)$, and allow us to conclude that the process on the r-tree has a first order phase transition.
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