with respect to $z$ (which is possible due to Remark \[Rem1\]) we get $$- \partial_{t}\partial_{z}F_{1}(z; t, x, y) = -(\mathscr{A}\partial_{z} F_{1}(z; t, \cdot, y))(x) + \delta_{0}(x)\partial_{z}f(F_{1}(z; t, x, y)).$$ Here, expressing the last term with the help of following formula for the derivative of the superposition $f(F_{1}(z; t, x, y))$ of the functions $f$ and $F_{1}(z; t, x, y)$: $$\partial_{z}f(F_{1}(z; t, x, y)) = f'(F_{1}(z; t, x, y))\cdot \partial_{z}F_{1}(z; t, x, y),$$ we obtain the equation: $$- \partial_{t}\partial_{z}F_{1}(z; t, x, y) = -(\mathscr{A}\partial_{z} F_{1}(z; t, \cdot, y))(x) + \delta_{0}(x)f'(F_{1}(z; t, x, y))\cdot \partial_{z}F_{1}(z; t, x, y).$$ From here, by passing to the limit with respect to $ z\rightarrow 0 $ we get equation .

\[Rem2\]Owing to Remark \[Rem1\] equations and can be interpreted as the Cauchy problems in the Banach spaces $l^{\infty}(\mathbb{Z}^d)$ and $l^{2}(\mathbb{Z}^d)$, and the solutions of these Cauchy problems are unique and belong to the Banach spaces $l^{\infty}(\mathbb{Z}^d)$ and $l^{2}(\mathbb{Z}^d)$.

\[L3\] For $ n\geq 0 $ and $ t\geq 0 $ the higher moments of subpopulations: $\eta_{x,t}$ and $\eta_{x,t}(y)$, at $y\in \mathbb{Z}^{d}$, whose initial position was at point $x$, satisfy the following backward Kolmogorov differential equations: $$\label{E1.8}
\partial_{t}M_{n}(t, x) = (\mathscr{H}M_{n}(t, \cdot))(x) + \delta_0(x)g_{n}(M_{1}(t,x),\ldots, M_{n-1}(t, x)),$$ with the 