An investigation is presented on the global boundedness for a system of nonlocal advection--diffusion equations which models a heterogeneous population over $R^d$, $d \\in N$. For each nonlocal advection of species $i$ according to the distribution of species $j$, the corresponding convolution kernel $K_{ij}$ is assumed to have its own regularity $\\nabla K_{ij} \\in L^{q_{ij}}(R^d)$, $1 < q_{ij} < \\infty$.

The first a global bound is established using a Nash-type inequality to bound an energy functional. Using a graph-theoretic arguement, it is then shown that the energy bound holds when the $q_{ij}$ values have geometric mean above $d$ along every index cycle of the form $i_1 \\to \\cdots i_n \\to i_1$ for any $1 \\leq n \\leq N$.

Alternatively, a sufficient smallness condition on the initial data is obtained by establishing an invairant set of the ratios between the $L^{p_i}$ norm and the mass of each species $i$ via a differential inequality.