A quasispecies continuous contact model in a subcritical regime

We study a non-equilibrium dynamical model: a marked continuous contact model in $d$-dimensional space, $d \ge 1$. In contrast with the continuous contact model in a critical regime, see previous work by Kondratiev, Kutoviy, Pirogov, and Zhizhina, the model under consideration is in the subcritical regime and it contains an additional spontaneous spatially homogeneous birth from an external source. We prove that this system has an invariant measure. We prove also that the process starting from any initial distribution converges to this invariant measure.