Spatial modeling of forest-savanna bistability: Impacts of fire dynamics and timescale separation

TL;DR

This work tackles forest–savanna bistability by developing a spatial stochastic FGBA model on a continuous domain, where sites switch among $F$, $G$, $B$, and $A$ under local seed dispersal, fire spread, and regrowth processes. The model couples spatial Markov jump dynamics with spreading kernels $W_F$, $W_G$, $W_B$ and percolation-like flammability terms $ ext{Φ}_F$, $ ext{Φ}_G$, yielding both a spatial IDE description and a nonspatial MF reduction in the $N\to\infty$ limit. Comparison between the spatial stochastic model and the mean-field approximation reveals that bistability persists but is profoundly shaped by transient, short-time dynamics and timescale separation, producing grass-dominated quasi-steady states and other behaviors absent in purely nonspatial analyses. The work provides bifurcation analyses, numerical simulations via the Gillespie algorithm, and a flexible framework to study fire–vegetation feedbacks under climate change, with potential extensions to seasonality and landscape heterogeneity.

Abstract

Forest-savanna bistability - the hypothesis that forests and savannas exist as alternative stable states in the tropics - and its implications are key challenges for mathematical modelers and ecologists in the context of ongoing climate change. To generate new insights into this problem, we present a spatial Markov jump process model of savanna forest fires that integrates key ecological processes, including seed dispersal, fire spread, and non-linear vegetation flammability. In contrast to many models of forest-savanna bistability, we explicitly model both fire dynamics and vegetation regrowth in a mathematically tractable framework. This approach bridges the gap between slow-timescale vegetation models and highly resolved fire dynamics, shedding light on the influence of short-term and transient processes on vegetation cover. In our spatial stochastic model, bistability arises from periodic fires that maintain low forest cover, whereas dense forest areas inhibit fire spread and preserve high tree density. The deterministic mean-field approximation of the model similarly predicts bistability, but deviates quantitatively from the fully spatial model, especially in terms of its transient dynamics. These results also underscore the critical role of timescale separation between fire and vegetation processes in shaping ecosystem structure and resilience.

Figures (16)

• Figure 1: State transition diagram of the spatial FGBA model. Transition arrows are labeled with the relevant parameters and/or flammability functions. Forest and grass/fire timescale transitions are shown in green and orange, respectively.
• Figure 2: FGBA model state transition diagram in the mean-field limit. Transition arrows are labeled by the transition rates. Forest and fire timescale transitions are shown in green and orange, respectively.
• Figure 3: A plot of the computed branch points governing the stable to unstable transition for the GBA steady state alongside the analytically predicted branch points in Eq. (\ref{['eqn:BP']}). Branch points are plotted as a function of $\beta_F$, the rate of fire spread over forest, and $\varphi$, the rate of forest spreading over grass and ash (assuming $\varphi = \varphi_A = \varphi_G$). Parameter values: $\beta_G = 50$, $\beta_F = 10$, $q = 30$, $\gamma = 10$, $\varphi = 0.1$, $f_0 = g_0 = 0.01$, $f_1 = 0.5$, $g_1 = 1$
• Figure 4: One parameter bifurcation diagrams for partially timescale separated parameter values. Red lines indicate stable equilibria, while black lines indicate unstable equilibria. LP indicates a saddle-node bifurcation point (limit point), BP indicates a transcritical bifurcation point (branch point), and N indicates a neutral saddle equilibrium with no stability change.
• Figure 5: Equilibrium land state proportions reached starting from a low forest state ($F=0.01$ initial condition) for partially timescale separated parameter values near branch points.

Theorems & Definitions (10)

• Proposition 1
• Proposition 2
• Theorem 1
• proof
• Proposition 3
• proof
• Proposition 4
• proof : Proof of Proposition (forward invariance of the GBA subspace)
• proof : Proof of Proposition \ref{['prop_2']} (existence and uniqueness of the GBA steady state)
• proof : Proof of Proposition \ref{['prop.fwd.FGBA']} (forward invariance for the nonspatial FGBA model)