We study the large time behavior of a class of diffusive predator–prey systems posed on the whole
Euclidean space. By studying a family of similar problems with all possible spatial translations, we first
prove the asymptotic persistence of the prey for the spatially heterogeneous case under certain assumptions on the coefficients. Then, applying this persistence theorem, we prove the convergence of the solution to the unique positive equilibrium for the spatially homogeneous case, under certain restrictions on the space dimension and the predation coefficient.