Insect outbreaks, where population sizes can change over multiple orders of magnitude, can be detrimental to natural and agricultural systems, yet the mechanisms driving these changes are not well understood. Proposed bottom-up mechanisms include plant responses to herbivory that affect the quantity and quality of the edible plant biomass. Plant quality responses, such as resistance, change the quality of the edible biomass and subsequently cause a decrease in the fecundity or survival of the herbivore. Plant quantity responses occur when plants regrow biomass lost to herbivory, thus decreasing the impact of herbivory on the plant but also producing more food for the herbivore. In order to understand the impact of plant responses to herbivory on both herbivores and plants, I analyzed mathematical models of plants and their herbivores that included these plant quality and quantity responses. To expand my analysis beyond stability analysis, I also computed the quasi-potential with the QPot package in R to measure the stability of equilibria and visualize complex dynamics, such as long transients.
Results/Conclusions
In mathematical models that focus on herbivore population dynamics, plant resistance and plant tolerance, where plants regrow some but not all biomass that was consumed by herbivores, lead to stable herbivore populations. Overcompensation by plants, where plants regrow more biomass than was consumed by the herbivore, can cause herbivore populations to exhibit outbreaks as bounded fluctuations. When inter-annual plant dynamics are included in the model, tolerance and overcompensation expands the range of parameter values that lead to stable population densities. When fluctuations do occur, the cycles can be bounded fluctuations or outbreaks with long transients, where the herbivore density remains at low levels for long periods of time, but then quickly increases in number, consumes much of the plant biomass, and declines back to the low level. Understanding these effects of plant responses on herbivore and plant population dynamics may help us recognize plant-driven insect outbreaks and facilitate the use of plant responses to herbivory in sustainable agriculture. The quasi-potential and the associated QPot package provides a useful tool for analyzing mathematical models, quantifying the stability of equilibria, and visualizing these complex dynamics.