particle can produce also $n\neq 1$ offsprings or die (case with $n=0$) with the probability $$\label{E:phn}
p_{*}(h, n) = p(h,0,0)\cdot p(h,n)=b_nh + o(h),$$ or remain unchanged with the probability $$\label{E:ph1}
p_*(h, 1)=p(h,0,0)\cdot p(h,1) = 1+ a(0,0)h +b_{1}h + o(h).$$

As a result, each particle stays at the source for a random time, exponentially distributed with the parameter $-(a(0,0) + b_{1})$, after which it produces a random number of offsprings or jumps at some other point. Each of the particles evolves independently of others by the same law.

Denote as usual by $E$ the expectation of a random variable. In the sequel the main subject of our interest will be the integer moments of the numbers of particles $\eta_{t}(y)$, $\eta_{x,t}$, and $\eta_{x, t}(y) $ at the point $y$, respectively: $$M_{\infty, n}(t, y):=E\eta_{t}^{n}(y),\quad M_{n}(t, x):=E\eta_{x, t}^{n},\quad
M_{n}(t, x, y):=E\eta_{x, t}^{n}(y).$$

Introduce the operator $$\mathscr{H}:= \mathscr{A} + \beta\Delta_{0},$$ where $\Delta_{0}$ is the following operator on $l^{p}(\mathbb{Z}^{d})$ for every $p$, $1 \leq p \leq \infty$: $$(\Delta_{0}u)(x):=\delta_{0}(x)u(x)\equiv u(0)\delta_{0}(x), \quad u(\cdot)\in l^{p}(\mathbb{Z}^{d}),~x\in\mathbb{Z}^{d}.$$ Introduce also the Laplace generating functions for random variables $\eta_{t}(y) $, $\eta_{x,t}$, and $\eta_{x, t}(y) $, defined by the following equations $$F_{\infty}(z; t, y) := Ee^{-z\eta_{t}(y)},\quad F_{1}(z; t, x) := Ee^{-z\eta_{x, t}}, \quad
F_{1}(z; 