  A continuous time mixed state branching process is constructed as the scaling
limits of two-type Galton-Watson processes. The process can also be obtained by
the pathwise unique solution to a stochastic equation system. From the
stochastic equation system we derive the distribution of local jumps and the
exponential ergodicity in Wasserstein-type distances of the transition
semigroup is given. Meanwhile, we study immigration structures associated with
the process and prove the existence of the stationary distribution of the
process with immigration.
