Summary: One can construct a sequence of rescaled perturbations of voter processes in dimension \(d=1\) whose approximate densities are tight. By combining both long-range models and fixed kernel models in the perturbations and considering the critical long-range case, results of J. T. Cox and E. A. Perkins [Ann. Probab. 33, No. 3, 904–947 (2005 Zbl 1078.60082)] are refined. As a special case, we are able to consider rescaled Lotka-Volterra models with long-range dispersal and short-range competition. In the case of long-range interactions only, the approximate densities converge to continuous space time densities which solve a class of SPDEs (stochastic partial differential equations), namely the heat equation with a class of drifts, driven by Fisher-Wright noise. If the initial condition of the limiting SPDE is integrable, weak uniqueness of the limits follows. The results obtained extend the results of C. Müller and R. Tribe [Probab. Theory Relat. Fields 102, No. 4, 519–545 (1995; Zbl 0827.60050)] for the voter model by including perturbations. In particular, spatial versions of the Lotka-Volterra model as introduced in C. Neuhauser and S. W. Pacala [Ann. Appl. Probab. 9, No. 4, 1226–1259 (1999; Zbl 0948.92022)] are covered for parameters approaching one. Their model incorporates a fecundity parameter and models both intra- and interspecific competition.