In this paper, we are concerned with the long time behavior and the spreading speed of a general system of species modeled by a heterogeneous reaction–diffusion system with nonlocal dispersals as follows : \(\begin{aligned} \left\{ \begin{array}{l} \partial _t u_1(t, x)=d_1 \mathcal {N}_1[u_1](t, x)+f_1\left( x-c t, u_1, u_2\right) \\ \partial _t u_2(t, x)=d_2 \mathcal {N}_2[u_2](t, x)+f_2\left( x-c t, u_1, u_2\right) \end{array}\right. \end{aligned}\) where \(\mathcal {N}_1[u](x, t)\) and \(\mathcal {N}_2[v](x, t)\) stand for the spatial nonlocal dispersal of individuals and \(c>0\) is the shifting speed. Under a subhomogeneity and some asymptotic conditions \(f(\pm \infty ,u_1,u_2),\partial _{u_i}f_i(\pm \infty ,u_1,u_2)\) , we first established the existence of stationary solution via Schauder fixed point theory and its uniqueness by an improvement of the sliding method for all \(c>0\) . Later, we utilize the abstract dynamical system theory of Weinberger et al. (J Math Biol 45:183–218, 2002) and Yi and Zhao (J Funct Anal 279:108722, 2020) to prove the long-time dynamics for all \(c>0\) and show the existence of spreading speed \(c^*:=\max \left\{ 2 \sqrt{d_1 \cdot \partial _{u_1} f_1(+\infty , 0)}, 2 \sqrt{d_2 \cdot \partial _{u_2} f_2(+\infty , 0)}\right\} \) such that the extinction holds as \(c<c^*\) while propagation holds under a subhomogeneity condition. Due to the different structure of the traveling wave, our results strikingly contrast with the interesting result obtained by Fang et al. (J Math Pures Appl 147:1–28, 2021), who based on PDE approach to prove that the existence of unique forced wave holds if and only if \(c<c^*\) defined in a similar manner formula for single equation. Finally, some numerical experiments have been made to illustrate our theoretical results.
