Rate-induced tipping in savanna–forest ecosystems: effects of fast-varying fitness parameters in the framework of compactification

Abstract Savannas are grass-dominated ecosystems with scattered shrubs and savanna trees, often coexisting with tropical forests. They host a diverse range of plants and animals, act as a major carbon sink, regulate the global climate and support local economies. However, rapid environmental change is degrading these ecosystems, ultimately leading to a decline in biodiversity. Here, we consider a mathematical model of savanna–forest ecosystems to investigate rate-induced tipping (R-tipping) between distinct states under rapid variations in fitness parameters of different functional types (i.e. grass, savanna saplings and adult trees and forest trees) over time. To establish the existence of R-tipping, we determine basin instability (BI) in the corresponding frozen system with fixed-in-time inputs. Since time-dependent parameters make the model non-autonomous, it lacks compact invariant sets, which preclude the use of classical bifurcation theory. To address this, we reformulate the non-autonomous system with bi-asymptotically constant inputs as an autonomous one using compactification with an additional bounded variable. Furthermore, compactification enables us to identify the R-tipping threshold, which is crossed when tipping occurs. Overall, our analysis reveals distinct rate-induced transitions, including shifts to dominant states of grass, adult savanna trees and oscillatory dynamics.