Outbreaks of herbivorous insects are detrimental to natural and agricultural systems, but the mechanisms driving outbreaks are not well understood. Plant responses to herbivory have the potential to produce outbreaks, but long-term effects of plant responses on herbivore dynamics are understudied. To quantify these effects, we analyze mathematical models of univoltine herbivores consuming annual plants with two responses: (1) compensatory regrowth, which affects herbivore survival in food-limited situations by increasing the amount of food available to the herbivore; and (2) induced resistance, which reduces herbivore survival proportional to the strength of the response. Compensatory regrowth includes tolerance, where plants replace some or all of the consumed biomass, and overcompensation, where plants produce more biomass than was consumed. We found that overcompensation can cause bounded fluctuations in the herbivore density (called outbreaks here) by itself, whereas neither tolerance nor induced resistance can cause an outbreak on its own. Food limitation and induced resistance can also drive outbreaks when they act simultaneously. Tolerance damps these outbreaks, but overcompensation, by contrast, qualitatively changes the conditions under which the outbreaks occur. Not properly accounting for these interactions may explain why it has been difficult to document plant-driven insect outbreaks and could undermine efforts to control herbivore populations in agricultural systems.