Impulsive Fire Disturbance in a Savanna Model: Tree–Grass Coexistence States, Multiple Stable System States, and Resilience

Abstract

Savanna ecosystems are shaped by the frequency and intensity of regular fires. We model savannas via an ordinary differential equation (ODE) encoding a one-sided inhibitory Lotka–Volterra interaction between trees and grass. By applying fire as a discrete disturbance, we create an impulsive dynamical system that allows us to identify the impact of variation in fire frequency and intensity. The model exhibits three different bistability regimes: between savanna and grassland; two savanna states; and savanna and woodland. The impulsive model reveals rich bifurcation structures in response to changes in fire intensity and frequency—structures that are largely invisible to analogous ODE models with continuous fire. In addition, by using the amount of grass as an example of a socially valued function of the system state, we examine the resilience of the social value to different disturbance regimes. We find that large transitions (“tipping”) in the valued quantity can be triggered by small changes in disturbance regime.

Emergent structure and dynamics of tropical forest-grassland landscapes

“… 15 ), inclusion of other tree types (such as in savanna dynamics: refs. 10 , 12 , 15 , 17 , and 33 – 38 ), or vegetation-rainfall feedback ( 39 ). This may result in additional relevant quantities in Eq.…”

Is resilience a unifying concept for the biological sciences?

“…One interesting assessment of flow-kick systems studied at the population and ecosystem levels is that resilience is overestimated by traditional approaches that consider the consequences of each disturbance separately. 51 This has been observed in ecological systems, such as fire-disturbance in savannahs, 61 and has been proposed for the physiological vulnerability of individuals facing repeated stressors that cause wear and tear. 62 These dynamics might even be generated through within-system feedback, as demonstrated by a recent model that predicts complex transient immune protection for a population due to flow-kick dynamics from the epidemiological behavior of disease spread.…”

Pattern Formation in Mesic Savannas

“…Table 1 summarizes the parameters of the SL model, along with their ecological interpretations and default numerical values. In addition to the four-functional-type model described above, numerous other related models with the similar underlying interaction rules, some including explicit spatial extent, have been studied in the literature (Durrett and Zhang 2015 ; Durrett and Ma 2018 ; Hoyer-Leitzel and Iams 2021 ; Li et al. 2019 ; Schertzer et al.…”

“…Typically, these models involve interactions between one or more species of trees, shrubs and grasses, with each functional type of vegetation having different tolerances or reactions to fire and herbivory (Staver and Hempson 2020 ). Some of these models of savannas are spatially implicit and describe only proportions of a landscape occupied by each cover type (Hoyer-Leitzel and Iams 2021 ; Staver and Levin 2012 ). These models are accurate for describing well-mixed regions of vegetation, which is typically appropriate at small length scales.…”