The dynamics of interacting consumer and resource populations is one of the most thoroughly studied problems of theoretical population biology. Among the key results from the study of simple mathematical models of interacting populations is that the Holling Type 2 functional response tends to be unstable for a wide range of realistic parameters. Functional responses such as Holling Type 3, which might be thought of as implicitly incorporating the existence of consumer refuges, are more stable than the Type 2. We studied consumer—resource models with these different functional responses on a landscape level by modeling grazers that can disperse across a space of patchily distributed grass resources. For certain assumptions concerning the movement of grazers on the landscape, the effect of these functional responses on stability is reversed; the Holling Type 2 functional response confers greater stability. The reason for this apparently paradoxical result is that the Holling Type 2 functional response allows grazers to graze individual grass patches to lower levels than Type 3, as the energy balance remains favorable for grazing at lower grass biomasses. However, this local overexploitation leads the grazers to be slower in reaching areas of the landscape where resource densities are higher. It decreases the likelihood that the resource will be overexploited over the whole landscape simultaneously, which results in a stronger tendency towards system stability. It appears, then, that consumer overexploitation of resources locally may contribute to lower stability.