$u < v$ if $u$ is an ancestor of $v$ So for example $\varnothing < \varnothing3 < \varnothing32$. We shall write $|u|$ for the number of ancestors of the particle $u$. So for example $|\varnothing32| = 2$.

We denote the set of all particles in the system at time $t$ by $N_t$ and for every particle $u \in N_t$ we let $X^u_t$ denote its spatial position at time $t$ and $(X^u_s)_{s \in [0,t]}$ its histrorical path up to time $t$ with $L_t^u$ the local time at $0$ of $(X^u_s)_{s \in [0,t]}$. We also define $$R_t := \sup \{X^u_t \ : \ u \in N_t\}$$ to be the position of the rightmost particle at time $t$.

We let $m_0 = \sum_{n \geq 1}n q_n$ be the mean of the offspring distribution due to catalytic branching and $m = \sum_{n \geq 1}n p_n$ the mean of the offspring distribution due to homogeneous branching. For convenience we also define effective branching rates $$\hat{\beta} := \beta(m-1)$$ and $$\hat{\beta}_0 := \beta_0(m_0 - 1) \text{.}$$ Finally, we let $(\mathcal{F}_t)_{t \geq 0}$ be the natural filtration of the branching process and $P$ the associated probability measure with the corresponding expectation $E$.

Motivation
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In this subsection we 