Accurate parametrization of functional terms in model equations is of great importance for reproducing the dynamics of real food webs. Constructing models over large spatial and temporal scales using mathematical expressions obtained based on microcosm experiments can be erroneous. Here, using a generic spatial predator-prey model, we show that scaling up the microscale functional response of a predator can result in qualitative alterations of functional response on macroscales. In particular, a global functional response of sigmoid type (Holling type III) can emerge as a result of non-linear averaging of non-sigmoid local responses (Holling type I or II). We demonstrate that alteration between the local and the global response in the model is a result of the interplay between density-dependent dispersal of the predator across the habitat and heterogeneity of the environment. Using the method of aggregation of variables, we analytically derive the mathematical formulation of the global functional response as a function of the total amount of prey in the system, and reveal the key parameters which control the emergence of a Holling type III global response. We argue that this mechanism by which a global Holling type III emerges from a local Holling type II response has not been reported in the literature yet: in particular, Holling type III can emerge in the case of a fixed gradient of resource distribution across the habitat, which would be impossible in priorly suggested mechanisms. As a case study, we consider the interaction between phytoplankton and zooplankton grazers in the water column; and we show that the emergence of a Holling type III global response can allow for the efficient top-down regulation of primary producers and stabilization of planktonic ecosystems under eutrophic conditions.