We study the non-local reaction–diffusion system with Neumann boundary conditions $$\begin{aligned}& u_{t}(x,t)= \int_{\Omega }J(x-y) \bigl(u(y,t)-u(x,t)\bigr)\,dy+v^{p}(x,t), \quad (x,t)\in \Omega \times (0,T), \\& v_{t}(x,t)= \int_{\Omega }J(x-y) \bigl(v(y,t)-v(x,t)\bigr)\,dy+u^{q}(x,t), \quad (x,t)\in \Omega \times (0,T), \\& u(x,0)=u_{0}(x),\quad\quad v(x,0)=v_{0}(x),\quad x\in \Omega , \end{aligned}$$ where $p, q>0$ , $u_{0}(x), v_{0}(x)\in C(\overline{\Omega })$ are nonnegative and nontrivial functions, $\Omega \in \mathbb{R}^{N}$ a bounded connected and smooth domain. We determine the existence and uniqueness of the solution. The blow-up phenomenon is considered and the blow-up rates are obtained.