1. We discuss a simple implicit-space model for the competition of trees and grasses in an idealized savanna environment. The model represents patch occupancy dynamics within the habitat and introduces life stage structure in the tree population, namely adults and seedlings. A tree can be out-competed by grasses only as long as it is a seedling.
2. The model is able to predict grassland, forest, savanna and bistability between forest and grassland, depending on the different characteristics of the ecosystem, represented by the model’s parameters.
3. The inclusion of stochastic fire disturbances significantly widens the parameter range where coexistence of trees and grasses is found. At the same time, grass-fire feedback can induce bistability between forest and grassland.
4.Synthesis. These results suggest that tree–grass coexistence in savannas can be either deterministically stable or stabilized by random disturbances, depending on prevailing environmental conditions and on the types of plant species present in the ecosystem.
Tree–grass coexistence is another important case, common in many ecosystems covering a large fraction of the Earth’s surface (Scholes 2003). One of the most widely studied examples is tropical savanna (e.g. Scholes & Walker 1993; Scholes & Archer 1997; Sankaran, Ratnam & Hanan 2004; Sankaran et al. 2005; Scheiter & Higgins 2007). Trees and grasses use practically overlapping resources (mainly soil water) and the outcome of their competition is varied. In some areas, competition results in the exclusion of one of the two life forms, leading to the formation of grasslands or forests, while in other regions of the world tree–grass coexistence is observed.
Species coexistence can either be stable or unstable (Chesson 2000). It is stable if species densities do not show long-term trends and are able to recover if they get low. Theoretically, this type of coexistence corresponds to a stable equilibrium where both species have non-zero densities. Unstable coexistence can be generated by a continuous disturbance acting on the system that produces the stabilization of an otherwise unstable coexistence equilibrium. This is similar to the stabilization of the unstable upright position of a physical pendulum, which can be achieved using appropriate forcing conditions (Kapitza 1951).
To theoretically address some of these issues, in this work we analysed the influence of life stage structure on savanna dynamics, considering also the effect of stochastic disturbances. Following the approach of Levins (1969), Hastings (1980) and Tilman (1994), we introduced a hierarchical model for competition, where space is considered implicitly. We inserted a new demographic element in the classical model for patch occupancy dynamics: two stages in tree life are considered, namely adults and seedlings, to include demographic structure, which Sankaran, Ratnam & Hanan (2004) suggested to be relevant in savanna models. Tree seedlings compete for resources with grasses, which can limit the growth of tree seedlings in different ways (Knoop & Walker 1985; Harrington 1991; Scholes & Archer 1997; Davis et al. 1998; Ball et al. 2002; Riginos & Young 2007). In our model we also make the assumption that adult trees cannot be directly displaced by herbaceous vegetation.
The model shows that, depending on parameter values, the system can display different types of behaviour. In some regions of parameter space, bistability between grassland and forest is observed. For other parameter values, stable coexistence of grasses and trees is possible, even in the absence of explicit niche separation or environmental heterogeneity. In addition, the role of stochastic disturbances (e.g. fires) is analysed, showing that disturbance can broaden the parameter range where trees and grasses coexist. If grass–fire feedbacks are introduced, bistability can emerge, with either forest or grassland as possible alternative solutions.
In the following, we provide a description of the model and we discuss its behaviour in different parameter ranges, both for a constant environment and in the presence of stochastic fire disturbances.
The model adopted here describes the dynamics of adult trees, grasses and tree seedlings. These three types of life forms are represented by the fraction of space they occupy in the habitat, and thus the model represents spatial dynamics, although implicitly. The total dynamics is the sum of all the individual processes of colonization of new sites, local extinction and replacement (Tilman 1994). We introduce three ordinary differential equations (ODEs) for the variables representing the fraction of space occupied respectively by adult trees (b1), grasses (b2) and tree seedlings (b3), without overlapping, i.e. the sum of the areas occupied by trees, grasses and tree seedlings cannot exceed unity (0 ≤ b1 +b2 +b3 ≤1):
( eqn 1)
( eqn 2)
( eqn 3)
In eqn 1, adult trees are obtained from the growth of the seedlings (g1b3), with g1 representing the rate of recruitment, or the rate at which tree seedlings become adult. Trees are subject to linear local mortality, last term on the right hand side (r.h.s.).
In eqn 2, grasses colonize new space proportionally to their actual coverage, b2, but also proportionally to the space which is not occupied by trees or grass, 1 − b1 − b2. The space where tree seedlings are settled is considered to be free for colonization, as grass can displace seedlings. Grass coverage decreases with linear local mortality, μ2b2.
Eqn 3 represents the dynamics of tree seedlings. They are produced by adult trees, and thus they occupy new areas proportionally to adult coverage (b1), and to the fraction of space free from vegetation (1 − b1 − b2 − b3). They are subject to linear local mortality (second term on the r.h.s.) and the growth into adulthood is linear with a growth rate g1, see also eqn 1. The last term on the r.h.s. represents the actual displacement of tree seedlings by grasses, which occurs proportionally to both grass and tree seedling coverage (b2 and b3).
In our model, grasses are competitively superior to tree seedlings and can displace them; conversely, a tree seedling cannot displace grass. Eqns 2 and 3 are similar to the classical hierarchical implicit space models (e.g. Levins 1969; Hastings 1980; Tilman 1994). The main idea underlying these models is that competitors are ranked by their ability to use resources. The best competitor can consume resources to such low levels that the other competitors cannot survive in its vicinity and are displaced. At the same time, since higher allocation to roots is often associated with lower allocation to reproduction and reduced dispersal (see e.g. Gleeson & Tilman 1990), inferior competitors can survive in a spatially extended environment thanks to their higher colonization ability.
The colonization rates of herbaceous and woody vegetation, c2 and c3, lump together multiple processes associated with fecundity, dispersal and seed establishment. The classical implicit space models consider a fixed species ranking, where superior competitors displace all of the inferior ones. Here, both an asymmetry in the hierarchy and some degree of circularity are introduced (see also Discussion): tree seedlings can be displaced by grasses but, when they survive, they grow into adult trees (eqn 1, first term on the r.h.s.). Then, trees cannot be out-competed by grass and are responsible for the production of tree seedlings (eqn 3, first term on the r.h.s., c3b1). Trees constitute a spatial limit to grass expansion, although adult trees are not strictly better competitors than grasses, because they do not actively displace them.
There are six parameters in the model and thus five of them are free (the sixth can be used to scale time). In the following, we fixed the time scale by assuming g1=0.2 year−1, which corresponds to a tree seedling becoming adult when it is 5 years old (van Wijk & Rodriguez-Iturbe 2002).
We also explored what happens to the tree–grass system when the assumption that tree seedlings can be out-competed by grasses is relaxed. If we assume that grasses and young trees cannot displace each other, we replace eqns 2 and 3 with:
( eqn 4)
( eqn 5)
where (1 − b1 − b2 − b3) represents the empty space and there is no displacement term.
In the next paragraph we describe the behaviour of the two systems (eqns 1–3 and 4 and 5) when the model parameters are varied. We also discuss the effects of stochastic disturbances, such as fire, on the dynamics of the systems.
First, we calculated the equilibrium states (also called fixed points) of the system and their stability. When the system is in an equilibrium state, it cannot leave it. Mathematically, equilibrium occurs when , i.e. the fraction of space occupied by the three vegetation compartments does not vary in time. When the fixed point is stable, the system returns to it after a small perturbation. Vice versa, when the equilibrium is unstable, as soon as a small perturbation occurs, the system leaves the equilibrium state. It is possible to calculate analytically the stability of the equilibrium points in a small neighbourhood with the method of ‘linear stability analysis’. When two different fixed points are simultaneously stable in the same range of parameter values (bistability), the system will in the long run reach one of the two equilibria depending on the initial values of vegetation cover. The presence of two stable attracting states may also lead to “catastrophic shifts” between the two equilibria (see e.g. Rietkerk et al. 2004).
For the analytic calculation of the value and linear stability of the equilibrium points, a matlab 7.4 code was implemented. The numerical integration of the system including the stochastic disturbances was performed implementing a Fortran 95 code.
Owing to its simplicity, the model of eqns 1–3 allows the analytical calculation of the fixed points and of their linear stability. The system has four different equilibrium points, corresponding to bare soil (b1=b2=b3=0), only trees (forest, b1,b3>0 and b2=0), only grasses (grassland, b1=b3=0 and b2>0) and tree–grass coexistence (savanna, b1,b2,b3 >0). The existence and stability of the different fixed points depend on the parameter values.
Figure 1 illustrates the behaviour of the system for different values of the colonization rates of grasses and tree seedlings, c2 and c3, with the other parameters kept fixed. In Fig. 1, we varied c2 and c3 between 0 and 10 year−1 (see e.g. Fernandez-Illescas & Rodriguez-Iturbe 2003). The highest value of 10 year−1 means that at very low covers, in one year vegetation can potentially colonize up to ten times its initial cover. Five regions appeared in parameter space. Bare soil with no trees or grasses was observed when the colonization rates were very low (the tiny white area close to the origin of the axes). For intermediate values of c2 and c3, either coexistence or bistability were possible. When c2 was smaller than c3, a region of coexistence was present (marked in black in Fig. 1). In the bistability region (marked in grey), found for c2>c3, the system converged either to a completely herbaceous state (grassland) or to a woody equilibrium (forest), depending on the initial value of tree cover. In the region of bistability, an unstable coexistence fixed point existed as well. The two lighter grey areas correspond to only grass, at the top of the panel, and only trees, at the bottom. When the grass colonization rate became high, the system shifted towards grassland; conversely, when tree seedlings had high ability to occupy new space, a forest was formed. For larger values of the grass extinction rate, m2, trees became favoured and the coexistence region was found at much higher values of the colonization rates (outside the region depicted in Fig. 1). Increasing the mortality of tree seedlings, m3, decreased the extension of the bistability region. A similar effect was obtained by increasing adult tree mortality, m1. The diagonal line, c2=c3, always separated the regions of bistability and coexistence.
The effect of changing mortalities is illustrated in Fig. 2, which shows the types of stable fixed points in the parameter plane of grass and tree seedling mortalities, m2 and m3, for fixed colonization rates. The qualitative behaviour of the system depended on whether c2>c3 or c2<c3, but it did not depend on the precise values of these parameters. If grass had a larger colonization rate than tree seedlings, as in Fig. 2, in a region of parameter space bistability was present, but coexistence was never stable. Conversely, when c3>c2 there was a stable coexistence range and bistability was never present (not shown).
To explore the effect of disturbances such as fire, we introduced stochastic perturbations in the system. We assumed fire events to occur randomly, with an exponential distribution of return times (van Kampen 1992; Li, Corns & Yang 1999; D’Odorico, Laio & Ridolfi 2006). Fires instantly burn a fraction of grass and tree seedling covers, reducing them to a value ɛ << 1. By contrast, adult trees are more resistant to fire, and we assumed that only a fraction f of the area they occupy survives the fire. Due to stochastic disturbances, the values of vegetation cover are forced to oscillate: in our formulation, this type of perturbation instantaneously disturbs the state of the system, without affecting the parameter values and forcing the system to stay in a transient state. The bottom panel of Fig. 3 shows three example time series from a stochastically disturbed situation. Grass and tree seedling covers (b2 and b3) dramatically decreased when a fire perturbation occurred (marked by arrows in Fig. 3b), and then tended to return rapidly to the equilibrium value. Adult tree cover was less affected by the occurrence of fire.
Taking the average of the different vegetation covers over a sufficiently long time, we obtained a figure similar to Fig. 1. Figure 3a shows that the system displayed four different states: bare soil, grassland, forest and tree–grass coexistence, according to the colonization rates of tree seedlings and grasses. The coexistence region was wider than in the undisturbed case: with stochastic perturbations, coexistence was possible even above the diagonal (c2 >c3). With this type of fire disturbances, no bistability was observed in the system (i.e. there was no dependence on initial conditions). The bare soil region was wider, because fire increases the chance of vegetation death. The details of Fig. 3 depended on the choice of the parameters that define the fire disturbance: ɛ, f and the fire frequency, but in general we observed that tree–grass coexistence was favoured. In Fig. 3, 80% of the tree cover survived a fire event (f =0.8). If f was assumed to be larger, coexistence occurred to grasses’ disadvantage, and the black region was displaced upwards. Conversely, when the fraction of area covered by trees that survive fire became lower, the coexistence region extended in the region where, in the constant case, only trees were found. In this case, the coexistence region also became smaller, and, for f about 50%, it was not larger than in the constant case. The level at which the fires reduce grasses and tree seedlings was set by ɛ, assumed here to be small (ɛ = 0.01 in Fig. 3). Increasing this parameter, according to the hypothesis that tree seedlings and grasses are affected by fires less strongly than trees (ɛ < f), the coexistence region scrolled down, to the grasses’ advantage, but it did not qualitatively affect the results. The system did not show a strong dependence on fire frequency, when it was varied in a realistic range from 1 to 10 year−1 (see e.g. Sankaran et al. 2005).
Since grass is the main fire fuel in savannas, and fire frequency has been observed to increase with grass biomass (e.g. van Wilgen et al. 2003), we also considered a case where fire frequency depends on grass coverage. As above, we assumed the presence of stochastic fire events with an exponential distribution of return times. However, the average fire frequency was assumed to increase linearly with b2, varying from 20 year−1 when no grass is present (b2=0) to 2 year−1 when only grass is present (D’Odorico, Laio & Ridolfi 2006). In addition to bare soil, only grass, only trees and coexistence of grasses and trees, bistability between grasses and trees was also present in a limited region of parameter space with c2>c3 and c2 ≤ 4 year−1 (not shown). In this region, if grasses were not damaged too strongly (e.g. ɛ ≥ 0.05 when f =0.8), either forest or grassland could be observed, depending on the initial conditions of tree cover. Thus, the feedback of grass cover on fire frequency allowed for the emergence of bistability also under stochastically forced conditions. Coexistence was still possibile, but it corresponded to larger values of grass and tree seedling colonization rate, c2 and c3. When the dependence of average fire frequency on b2 became less marked, the bistability region became smaller, until only coexistence was observed when fire frequency was constant (as in Fig. 3). Note, also, that in the transition region from tree–grass coexistence to forest, the presence of stochastic fire disturbances depending on grass cover can induce a form of intermittency characterized by long periods during which grass can stay at extremely low levels (b < 10−20). Albeit interesting mathematically, this behaviour is of limited ecological relevance. Completely analogous results are in general obtained if tree seedlings and grasses burn proportionally to their coverage, instead of being reduced to a threshold value.
We also briefly analysed the case when tree seedlings and grasses cannot outcompete each other. In this case (not shown), as expected from pre-emptive model theory, the only solutions were bare soil (no vegetation), forest and grassland. Neither coexistence nor tree–grass bistability were observed. Nevertheless, coexistence was observed when stochastic fire disturbances were introduced.
In the past, tree–grass coexistence in savannas has been associated either with a stable equilibrium state or with an unstable state that is stabilized by spatial or temporal random disturbances. Both explanations are valid in view of the results reported by Sankaran et al. (2005), who showed that coexistence is likely to be stable in the driest environments and unstable in the wettest.
Analogously, our modelling effort suggests that there is no conflict between the rationalizations of tree–grass coexistence as stable or unstable (but stabilized by the variations in the environmental conditions). The model discussed here is able to generate deterministic coexistence of trees and grasses. In addition, stochastic fire perturbations can expand the region of coexistence in parameter space. Tree–grass coexistence was obtained also for parameter values where, without disturbance, a stable forest would be found. As the fraction of tree cover destroyed by fire decreased, the coexistence region widened in the parameter space towards the region where grassland would be stable without disturbance. Thus, in our simple model, both stable and unstable coexistence can be realized, depending on the values of the colonization and extinction rates, which are in turn related to the type of environment and of vegetation species.
As observed by D’Odorico, Laio & Ridolfi (2006), grass–fire feedback increases the instability of tree–grass coexistence. Consistent with those findings, in our model the dependence of fire frequency on grass cover lead to the presence of bistability between forest and grassland in a portion of parameter space. This is an important element to consider for fire management purposes: if a strong grass–fire feedback exists, depending on the various vegetation characteristics, a variation in fire occurrence may lead to savanna thickening into forest, or thinning into a grassland.
The model does not implement explicit root depth separation. Nevertheless, the competitive hierarchy introduced here may also be interpreted in terms of different strategies of competition for water: adult trees out-compete grasses owing to their deeper roots and grasses out-compete tree seedlings as they can extend their roots earlier into the shared soil layer.
When we assumed no hierarchy between tree seedlings and grasses, as in eqns 4 and 5, coexistence via competition–colonization trade-off is no longer realized. ‘Pre-emptive’ models maintain that in reality it may be difficult, if not impossible, for a seedling to colonize a site already taken up by another species. Analogously, the ‘pre-emptive’ effect assumes that the superior competitor cannot colonize every site without distinguishing empty ones from those occupied by an inferior competitor, unless exogenous heterogeneity is present (Yu & Wilson 2001). On the other hand, if disturbances, simulated as explained in the Results, are introduced, the coexistence of trees and grasses becomes possible.
In our model, we assumed the environment to be homogeneous. The only sources of heterogeneity were ‘endogenous’, i.e. internal (Pacala & Levin 1997). In the domain of the superior competitor (grasses), the inferior one (trees) digs ‘holes’ (Murell & Law 2003), because when a tree seedling survives and becomes an adult, grass can no longer displace it, creating a region of the domain that cannot be occupied by the superior competitor and thus allowing coexistence. This feature is enhanced by the slower dynamical time scale of trees with respect to grasses and it explains why, in part of the parameter space, our system is particularly sensitive to the initial conditions on adult tree cover.
The adult tree–grass–tree seedling structure introduced here bears some similarity to intransitive competition models, also called ‘rock–scissors–paper’ from the children’s game, where a competitive cycle is present (Buss & Jackson 1979; Paquin & Adams 1983; Sinervo & Lively 1996). In a three-species case, species A out-competes species B, B out-competes C and C out-competes A (A>B>C>A), while the classical hierarchical competition would require A>B>C. The asymmetries help prevent competitive exclusion (Tainaka 1995; Durrett & Levin 1998; Frean & Abraham 2001; Kerr et al. 2002; Laird & Schamp 2006). These models differ from our case because A (adult tree) does not out-compete B (grass), although adult trees cannot be out-competed. Besides, tree seedlings (C) grow into adult trees (A) rather than out-competing them.
Clearly, simple conceptual models such as the one introduced here have a number of limitations. A critical element is the colonization rate, which lumps into one term several different processes such as seed production, dispersal and establishment. These models cannot describe other trade-offs, e.g. between fecundity and dispersal (Yu & Wilson 2001). Another possible drawback is the lack of ‘successional niche’ behaviour, i.e. a trade-off between competitive ability and rapid resource exploitation (Pacala & Rees 1998; Bolker & Pacala 1999). Further limitations of our model arise from the simplicity of the representation of woody and herbaceous vegetation. In general, the implicit treatment of spatial structure has been shown to be unfavourable to coexistence (Hurtt & Pacala 1995; Higgins, Mastrandrea & Schneider 2002), since the incorporation of spatially limited dispersal could further enhance coexistence (Calcagno et al. 2006). At the same time, considering space only implicitly disregards the important phenomenon of vegetation self-organization (von Hardenberg et al. 2001; Rietkerk et al. 2002). The model does not represent the complex network of relationships between vegetation and its environment. Resource (water, nutrients) dynamics are not included (Scholes & Archer 1997; van Wijk & Rodriguez-Iturbe 2002; Sankaran et al. 2005); fire mortality is extremely simplified, e.g. because the whole plant is hit without distinguishing between roots and shoots (Scheiter & Higgins 2007); animal effects, such as those due to grazers, browsers or facilitators, are neglected (Scholes & Archer 1997; van Langevelde et al. 2003; Bond 2003). Notwithstanding these limitations, the simple implicit-space model discussed in this work underlines the importance of the demographic structure in savanna environments, and it addresses fire effects as a stochastic disturbance, providing some insight into the dynamics of tree–grass coexistence and a stimulus for further theoretical and empirical research.
The authors are grateful to David Tilman, Paolo D’Odorico and Max Rietkerk for interesting discussions and advice. This work is funded by the program of Italian-French cooperation Galileo 2007/08, by the Italian Ministry of Education and Research (MIUR), PRIN project 2007 “Experimental measurements of the atmosphere-vegetation-soil interaction processes and their response to climate change” and by a research grant in the framework of the CMCC-CNR collaboration on climate modelling.