# Grass, Trees, and Fire: Elements of a Savanna Lifecycle

Imagine a savanna — an undulating plain of long grass stretching towards a horizon dotted with scrubby trees. Unobscured by foliage, the sky is expansive. Winds ripple the golden grass while distant smoke curls from a recent fire, still smoldering.

The team defines a savanna as a combination of grasses, saplings, and adult savanna trees, with fire as the driving element. Grasses fuel fires and sparse tree covering, which encourages grass growth and instigates more fires. Fires damage saplings and prevent tree maturity. This loop sustains savannas. Adult trees, however, withstand fires and shade the ground. Robbed of sunlight, shaded grasses wither and fire decreases, thus allowing continued tree propagation. Such conditions foster woodlands.

Touboul, Staver, and Levin codify these relationships in units of aerial cover where variables \(G\), \(S\), and \(T\) represent the fractional cover of grass, saplings, and savanna trees. This yields a three-dimensional system in which

\[G'= \mu S + \nu T - \beta GT\]

\[S' = \beta GT - (\omega (G) + \mu)S \]

\[T' = \omega (G)S - \nu T,\]

where \(\beta\) is the savanna sapling birth rate and \(\mu\) and \(\nu\) are respectively the rates at which savanna saplings or adult trees die. Saplings grow into trees at a rate \(\omega\), a nonlinear decreasing function of grass cover that accounts for the ecological system’s nonlinear response to fires. Fires readily spread in systems with sufficient grass biomass but are quickly limited when tree cover exceeds a certain threshold. Therefore, this function has a sigmoidal shape that approximates a steep drop at a crucial percolation threshold associated with fire spread probability. Because \(G\), \(S\), and \(T\) represent fractional cover, they are nonnegative and can be expressed as \(G + S + T = 1\); this reduces the above equations to a two-dimensional system.

**Figure 1.**Grasslands and woodlands in the absence of forests yield a bifurcation diagram with respect to the savanna tree birth rate. In ecosystems with low savanna tree birth rate, grass covers the landscape and prevents woodlands from emerging; at high rates, woodlands dominate and grasslands cannot establish themselves. For intermediate birth rates, both woodlands and savannas may emerge and remain stable over time. Figure courtesy of [1].

This behavior conflicts with an underlying tenet of ecology. “In the traditional view in ecology, if you know something about the environment you basically know what kind of vegetation you’re going to have,” Staver said. “What’s different about this is that…even in a very rainy place that ought to be able to be a forest, if it ends up being a savanna we should expect it to continue to be a savanna.”

But high densities of hardy scrub trees do not signify a real forest. Forest trees are more sensitive to fires than savanna trees; they also create denser shade. To better approximate these different landscapes, Touboul, Staver, and Levin add a variable \(F\), a forest tree with birth rate \(\alpha\), and a mortality that depends on fire’s ability to spread. To incorporate this element, the researchers model the mortality rate as an increasing function \(\phi (G + \gamma (S + T))\), which accounts for the nonlinear dependence of fire spread on grass and tree cover and refines this dependence by considering the level at which savanna and woodland tree foliage propagate fires (parameter \(\gamma\)).

Using their model, the team studies this element’s possible effect on landscape dynamics and identifies the system’s complex co-dimensional two and three bifurcations that organize intricate multistability or oscillatory dynamics. They also account for differences between savanna and woodland tree foliage by varying the degree to which they exclude grasses. Savannas, scrub woodlands, and true forests can coexist in this more realistic model, described by the set

\[G'= \mu S + \nu T - \beta GT + \phi (G + \gamma (S + T)) F - \alpha GF\]

\[S' = \beta GT -(\omega (G + \gamma (S + T)) + \mu)S- \: \alpha SF\]

\[T' = \omega (G + \gamma (S + T))S -\: \nu T -\: \alpha TF\]

\[F' = \big(\alpha (1 - F) - \phi (G + \gamma (S + T))\big) F,\]

where \(\alpha\) and \(\phi\) represent forest tree birth and death rates respectively. This system is reduced by \(G + S + T + F = 1\) to

\[G' = \mu(1 - G - T - F) + \:\nu T - \beta GT + \phi (G + \gamma(1 - G - F))F - \alpha GF\]

\[T' = \omega \big(G + \gamma (1 - G - F)\big) (1 - G - T - F) - \nu T - \alpha TF\]

\[F' = \big( \alpha(1 - F) - \phi (G + \gamma (1 - G - F))\big) F.\]

The result is a complex system that is quite sensitive to the parameters (see Figure 2). A variety of equilibria appear, and families of periodic orbits undergo complex transformations.

**Figure 2.**Complex dynamics—including periodic orbits—emerge in the presence of a forest tree subtype.

**2a.**Highly intricate dependences in the birth rates of forest and savanna trees organize these dynamics.

**2b.**For a fixed savanna birth rate, increasing the forest tree birth rate drives the system from savannas (left trajectory) to oscillatory behaviors with the emergence of forest trees (right trajectory), and from bistable forest-grassland regimes to forests at high birth rates. Figure adapted from [1].

Levin and Staver were surprised to find heteroclinic orbits—trajectories that connect and cycle between multiple states—during which the system flipped from one state to another. At the saddle node bifurcation, the orbit expands and approaches three coexisting unstable equilibria, which results in sharp, slow oscillations (see Figure 3).

These oscillations describe what the researchers call a “winnerless competition.” Hardy savanna trees invade grasslands, shade the ground, and reduce grass growth and fire risk. Diminished fires promote tree growth, and soon forests dominate the landscape. But forest trees are more sensitive to fires than savanna trees, and they give way to grassy plains as fire again sweeps across the land.

Strangely, these patterns were not well-behaved. “They didn’t have many of the features that we expected of heteroclinic orbits,” Levin said. “Namely, as they approach the equilibrium they ought to be slowing down. That’s what we didn’t understand.”

**Figure 3.**Heteroclinic loop and nearby periodic orbits.

**3a.**The heteroclinic loop (orange) joins a grassland, forest, and woodland equilibrium — all of which are unstable for those parameters.

**3b.**For smaller values of the forest tree birth rate, periodic orbits emerge and progressively deviate from this cycle. The forest equilibrium is stabilized for larger values. Figure courtesy of [1].

Levin and Staver consulted Touboul, who was familiar with multiple timescales thanks to his neuroscience experience. However, unlike in the brain, all savanna species evolve at comparable timescales. “It’s the dynamics and the nonlinearity itself that create these heteroclinic cycles,” Touboul realized. “The cycles may have branches that are very fast and branches that are very slow.” Contrary to normal ecological assumptions, the model demonstrated that the landscape could change independently of any external process.

In this case, change—even rapid change—might just be a cycle from one state to the next rather than an indication that something or someone is pushing it. If something does drive it, like climate change, the corresponding effects would manifest in even the simplest model — as increased fire risk from less precipitation or faster tree growth due to warmer temperatures, for example.

However, a real savanna is more than simplified trees and grass with constant rates of initiation and growth. Fires are sporadic. Tree growth depends on water, nutrients, sunlight, and the voracity of predators. Researchers often assume that this extra “noise” averages out, but Touboul, Staver, and Levin realized that this was not the case upon examining these stochastic effects to test the model’s relevance to real systems.

Rather than averaging out, relatively small noise perturbations exacerbated bifurcations and caused the model to veer off. For noise perturbations away from bifurcations, the system deviated only slightly from deterministic trajectories. But for ecological systems near the heteroclinic cycle, these perturbations triggered large amplitude periodic oscillations between grassland, savanna woodland, and forest where the noiseless system stabilized on a fixed landscape.

In the vicinity of another transition called fold of limit cycles, these perturbations had the opposite effect. They dampened the emergence of oscillations in the model where such stochastic effects are ignored. “When I applied noise to the system, I was very surprised to see that noise could actually act both ways, either in creating regular oscillations or destroying existing oscillations in the noiseless system,” Touboul said. Figure 4 illustrates these novel results.

**Figure 4.**Complex responses to noise in various parameter regimes.

**4a.**Near the heteroclinic loop, an optimal level of noise triggers highly periodic responses, where the noiseless system stabilizes in a forest (stochastic resonance).

**4b.**An optimal level of noise cancels the oscillations in the noiseless system (inverse stochastic resonance) near the fold of limit cycles.

**4c.**Noise induces irregular switches between the different attractors in the bistable regime. Figure courtesy of Jonathan Touboul, adapted from [1].

The team asserts that one should not view this model as predictive; an ecologist cannot simply tabulate all parameters, feed them into the model, and expect to know the landscape’s vegetation type in coming centuries. Furthermore, parameters are difficult to estimate and check. “I can say with some confidence that when the system is pushed far enough, you will obtain these transitions,” Levin said. “But in ecological systems, one must be very cautious about using models for prediction.”

Touboul, Staver, and Levin are currently using paleoecological data to test the model’s ability to describe past ecosystems. Ancient pollen grains embedded in layers of lake sediment provide clues about vegetation types that existed as early as 20,000 years ago. “It’s complicated because the vegetation dynamics are often on the same timescale as climate change,” Staver said. “So it’s really difficult to determine if those cycles are occurring independently.”

The team is also working to refine the model to account for spatial dynamics. The current model assumes the system is well-mixed, but in reality, one patch of vegetation influences another even if they are identical.

The researchers’ work demonstrates an elegant and practical example of how simple starting conditions can produce highly complex non-equilibrium phenomena. It is also a cautionary tale against ignoring the sometimes-substantial effects of ecological noise and ascribing change to external drivers. Their model provides scientists with an improved understanding of this complex ecological system — with expandable techniques to help understand global vegetation patterns and potential changes due to climate change. And it is not limited to savannas. Other multi-layered systems—such as coral reefs, brain activity, and financial markets—would benefit from similar modeling.

**References**

[1] Touboul, J.D., Staver, A.C., & Levin, S.A. (2018). On the complex dynamics of savanna landscapes. *PNAS, 115*(7), E1336-E1345.