Introduction

Bush encroachment has been a global trend in savanna systems all over the world (Stevens et al. 2017). It describes the increase in woody cover and the spread of shrubs and trees into areas that were previously dominated by grasses (Scholes and Archer 1997). Bush encroachment potentially implies adverse effects on various ecosystem functions and services, such as soil carbon storage and soil nitrogen (Berthrong et al. 2012), infiltration of water into the soil (Tietjen et al. 2009) and the biodiversity of herbaceous species (Siraj and Abdella 2018) and mammals (Soto-Shoender et al. 2018). Additionally, there are negative economic consequences, due to a decreased availability of fodder for grazing livestock (Angassa and Baars 2000) and reduced attractiveness of savannas to tourists (Gray and Bond 2013). For the development of reasonable management and restoration strategies it is therefore crucial to understand the exact nature and drivers of bush encroachment.

Unsustainable grazing management with high livestock densities is seen as one of the main causes of bush encroachment (Scholes and Archer 1997, Van Auken 2009, Kgosikoma and Mogotsi 2013). Savannas have diverse herbivore communities (Olff et al. 2002) that consist of browsers which feed mostly on trees, grazers that feed on grasses, as well as mixed feeders (Shorrocks and Bates 2015). Historically, high levels of both grazing and browsing from ungulate herbivores stabilized savanna vegetation, but many indigenous animal species have now been replaced with domestic livestock (De Klerk 2004, Smit 2004). The increased density of grazers is assumed to lead to an overgrazing of grasses, causing a dominance of trees and shrubs. Several observational studies confirm that encroachment is associated with high levels of grazing (Van Vegten 1984, Skarpe 1990a, b1990b, Roques et al. 2001) but also show that herbivore impact varies from site to site, demonstrating that the exact mechanisms and dynamics associated with herbivory are still unclear. Moreover, savanna vegetation structure is not only influenced by herbivory, but also by a variety of other factors, including intraspecific competition, soil water availability, recurrent bush fires, climate change (Scholes and Archer 1997, Sankaran et al. 2004) and an increased atmospheric CO₂ level (Bond and Midgley 2012). Due to the multitude of potential effects, the causal factors are difficult to disentangle from each other.

Bush encroachment is assumed to be an example of a critical transition between one system state that is dominated by grasses to another state dominated by shrubs (Walker et al. 1981, Hirota et al. 2011, Aleman et al. 2020). The concept of critical transitions is known from dynamical systems theory. These transitions explain abrupt and irreversible shifts, where a system ‘tips over' from one stable state to another (Noy-Meir 1975, May 1977, Scheffer et al. 2001). Important in our understanding of critical transitions (within savanna systems and elsewhere) is the threshold between the two stable states. In mathematical terms, this threshold corresponds to a third, unstable system state which is characterised by positive, self-reinforcing feedback mechanisms (Levins 1974, Scheffer et al. 2001, van Nes et al. 2016). Because of this amplifying effect, a system that is pushed beyond this threshold by an external perturbation, e.g. an extreme weather event or human interference, will consequently be pushed even further until it reaches the alternate stable state.

A comprehensive understanding of the drivers of bush encroachment therefore requires insights into the key positive feedback loops at play (Marzloff et al. 2011, Briske 2017). Increased grazing pressure is probably an important element of these feedback loops but its direct negative impact on grass abundance alone is not sufficient to explain the existence of alternative stable states, because it does not create positive feedback. While several possible positive feedback mechanisms arising for example from interactions between vegetation and fire or between vegetation and soil erosion have already been identified (Van Langevelde et al. 2003, D'Odorico et al. 2012), such feedback loops are usually studied in isolation, and only from a conceptual point of view. This is problematic since real systems contain many feedback loops, both positive (self-reinforcing) and negative (self-dampening), that together determine the system's response to perturbations (Dambacher et al. 2003, Van de Leemput et al. 2016). To gain a full understanding of the forces that enable bush encroachment, it would be necessary to address all feedback loops that act in savanna systems in concert. In principle, this can be done by describing the system as an interaction network. Looking at circuits of effects within this network then allows us to not only quantify specific feedback loops, but also to analyse their combined effect on system dynamics (Levins 1974, Puccia and Levins 1991).

We aim at taking a first step into this direction by focusing on several feedback mechanisms that arise from biotic interactions between grasses and shrubs and their herbivorous consumers. It is well known that the exclusion of browsers (Augustine and McNaughton 2004, Staver et al. 2009) or grazers (Riginos and Young 2007, Goheen et al. 2010) can have severe impacts on savanna vegetation. At the same time, the population of herbivores is obviously influenced by the availability of grasses and trees as food source (Van De Koppel and Rietkerk 2000), creating consumer–resource feedback loops. Grazing cattle are additionally influenced by management decisions, meaning that grazing pressure is in fact an emergent property that arises from the interplay between ecological dynamics and management decisions. It is very challenging to unravel all these dependencies using field experiments (Staver and Bond 2014) so that based on empirical data we cannot predict how exactly herbivore populations respond to changes in the vegetation structure and vice versa. On the other hand, most theoretical models that aim at explaining bistability of savanna systems focus on feedback loops that consider the role of fire or soil moisture availability but neglect the dynamic role of herbivores (Holdo et al. 2013). In cases where herbivory is included, it is rather seen as a constant rate being applied from outside the system, instead of being a dynamic system element (Anderies et al. 2002, Van Langevelde et al. 2003). However, vegetation–herbivore interactions together with competitive interactions between plants form a complex interaction network that contains several self-reinforcing and self-dampening feedback loops. Their combined effect might play an important role in explaining why savannas exhibit bistability and also how savanna systems react to external disturbances, such as droughts.

We therefore introduce a model that includes herbivores as dynamic system elements to explore the impacts of vegetation–herbivore interactions on the dynamics of savanna systems. The model consists of four interacting populations: grasses and shrubs as two contrasting plant functional types as well as two groups of herbivores that differ in their feeding preference: browsers and grazers. We consider direct competition between the two plant types for space and other resources, as well as indirect competition between the two herbivore types, as we account for a moderate level of cross-feeding. This describes grazers that occasionally also feed on shrub seedlings, as well as browsers (such as goats or elephants) that feed on both grasses and shrubs. Our model thus includes several feedback mechanisms whose effects have so far not been explored. In addition, we analyse the possible impact of management on the interaction network by assuming that grazers, who mainly represent livestock, receive a certain level of care from farmers, which reduces their net loss rates. Grazing pressure is thus not a static model parameter but a dynamic model component that emerges from the interplay between farmer support and biotic interactions. The goal of our analysis is to explore possible system dynamics in a general way, with a focus on potential management impact on system stability.

We expect that an increase in farmer support directly translates into increased grazer densities compared to a natural, grass dominated system without management influence. We assume that this results in 1) a strong negative impact on grass biomass via herbivory, 2) a weak negative impact on shrub biomass, also via herbivory and 3) a negative impact on browser density via resource competition among herbivores. Mechanism 1) and 3) should result in an indirect positive effect on shrub biomass, as competition from grasses as well as herbivory by browsers are decreased. We hypothesize that this combined indirect positive effect outweighs the weak impact of mechanism 2) if farmer support becomes sufficiently strong. We furthermore hypothesize that a shift in the relative importance of these interactions increases the strength of already existing positive feedback loops in the system, allowing for the emergence of a second system state in which shrubs dominate over grasses.

With our interaction network model, we test these hypotheses by investigating how different levels of farmer support affect the existence of two alternative stable states. Furthermore, we include external disturbances that mimic drought or other drastic events that reduce plant abundances, in particular of the grasses. We analyse under which conditions such disturbances trigger critical transitions towards an encroached state and how farmer support affects the ability of the system to recover from disturbances. Finally, we quantitatively analyse the underlying feedback loops that (de-)stabilise the observed system states and analyze their relative importance in shaping system dynamics. We conclude with a discussion of the potential of explicit feedback loop analysis to increase system understanding.

Models and methods

A dynamical system for bush encroachment

We consider a simple food web module containing two plant functional types (PFTs), namely grasses (P_H) and shrubs (P_S). We assume that both PFTs compete for available resources, such as water and space, so that our system represents a semi-arid, rather than an arid savanna, where facilitative effects may dominate the interactions between grasses and shrubs (Synodinos et al. 2015). Our model furthermore contains two types of herbivores, namely browsers (C_B) and grazers (C_G). It has become increasingly clear that a simple distinction between grazers and browsers is insufficient as grazers are often mixed grazers/browsers and vice versa (Bodmer 1990, Codron et al. 2007). We therefore assume that both herbivores consume both PFTs. They differ in their preferences for the resources, with grazers preferring grasses and browsers preferring shrubs, as illustrated in Fig. 1.

The resulting change in biomass of each population is given via the following set of differential equations:

(1)

(2)

(3)

(4)

Both PFTs follow a logistic growth with growth rate r_x and are limited by intra- and interspecific competition for resources leading to a total PFT-specific carrying capacity (K_x). Own standing biomass and the weighted standing biomass of the respective other PFT (weighting factor c) are evaluated relative to K_x. The amount of biomass consumption of resource i by consumer j is given via a Holling type II functional response, as at high plant densities, consumers need little time to find food and spend more time handling it:

(5)

Consumed plant biomass leads to linear increase of herbivore biomass with a conversion efficiency e. Losses of browsers occur constantly with background death rate m_b and additionally density dependent with loss rate m_d, e.g. as a result of increased levels of pathogens or interference competition in dense populations. The same applies for grazers but their loss rates are reduced by factors f_b and f_d, respectively. These parameters describe a certain level of care by farmers who might provide fresh water, shelter, protection against predators, medical treatment and additional food in times of food scarcity. Farmers might also buy new animals to replace the deceased. An increase in farmer support thus translates into a reduced net mortality loss rate (m_b(1 − f_b)) and also into a reduced net density dependent loss rate (m_d(1 − f_b)C_G) in Eq. 4. All parameter values are summarised in Table 1.

Table 1. A summary of all model parameters, including the standard parameter values used in our analysis

Symbol	Meaning	Value
rH	Intrinsic growth grate of grasses	1
rS	Intrinsic growth grate of shrubs	0.5
KH	Carrying capacity of grasses	2
KS	Carrying capacity of browsers	3
c	Strength of interference competition between plants	0.3
e	Conversion efficiency	0.45
a	Attack rate of consumers	1
h	Handling time of consumers	3
pHB	Feeding preference of browsers for grasses	0.3
pSB	Feeding preference of browsers for shrubs	1 − p11
pHG	Feeding preference of grazers for grasses	0.7
pSG	Feeding preference of grazers for shrubs	1 − p12
mb	Consumer background mortality loss rate	0.15
md	Consumer density dependent loss rate	0.05
fb	Farmer support to reduce mortality losses	[0, 0.8]
fd	Farmer support to reduce density dependent losses	[0, 0.8]

Specific model assumptions

Our model investigation is based on several key assumptions. First, we assume that grasses grow faster than shrubs (r_H > r_S), while shrubs have a higher carrying capacity (K_S > K_H), as total shrub biomass is normally higher than grass biomass if they cover the same space. For the given set of parameters and in the absence of herbivores (C_B = C_G = 0), this translates into a competitive advantage for the shrubs. More precisely, the model then reduces into a simple Lotka–Volterra competition system with equilibrium densities

and

Second, we assume that the key difference between both herbivore populations is their feeding behaviour. We therefore choose identical parameter values for both types of herbivores, except for the feeding preferences p_ij. More precisely, we assume that grazers feed mostly on grasses (p_HG > p_HB), while browsers feed mostly on shrubs (p_HB < p_HS). Each herbivore is thus specialised on a different type of resource but resource competition is still at play.

Third, we are primarily interested in large scale patterns. Our simulations are therefore not spatially explicit so that spatiotemporal processes that take place locally or at a short time scale (e.g. ‘islands of fertility' due to uneven water distribution, or a patchy distribution of vegetation due to selective feeding) are consequently not taken into account. More precisely, we assume homogeneous soil conditions, homogeneous management and an even distribution of plants and herbivores. Modelled processes are described at an annual time scale without explicit periods of drought or rain e.g. triggering events such as mass recruitment or vegetation die-back.

Numerical investigations

The source code to reproduce our data is available online, see the Supporting information. If not stated otherwise, we initialise the model with random population densities and let it run for 1000 time steps using the parameter values summarised in Table 1. Population densities that fall below the extinction threshold ϵ = 10⁻⁵ are removed from the system. We neglect all transient dynamics at the beginning of each simulation and focus only on the stationary part of the time series, that is on the population densities P_H(t), P_S(t), C_B(t) and C_G(t) with t ∈ [700, 1000]. Therefore, the specific length of a time step is not of importance. From this data, we calculate minimum, maximum and mean population densities, as well as the ratio of shrubs in the total plant population,

and the ratio of browsers in the total herbivore population,

These measures allow us to distinguish between different model outcomes: the system typically reaches either a stable equilibrium (minimum and maximum densities are identical) or a limit cycle (the system oscillates, minimum and maximum densities differ). Furthermore, they also allow us to identify the parameter ranges for which the systems shows alternative states. In this case, the shrub and browser ratio depend on the initial population densities and therefore differ between simulations that are run with identical parameter values, as it is the case for the exemplary time series shown in Fig. 1.

We start our analysis by varying the level of farmer support f_b and f_d from 0 to 0.8 to analyse how this support impacts the measures explained above. Values larger than 0.8 are intentionally not considered because they translate into extremely low grazer loss rates. As a result, model outcomes become unrealistic and show oscillations with very high amplitudes or unbounded consumer growth. In a second step, we then zoom into the parameter space of interest and analyse the possible system states, as well as the transitions between them, using bifurcation diagrams and additional time series.

Some of these transitions occur automatically, e.g. whenever the farmer support reaches a critical value, while others are triggered only via external disturbances. During a disturbance, d% of the grasses and % of the shrubs are assumed to die and are therefore removed from the system. In reality, such a reduction in biomass could for example be caused by a drought event. We corroborate our analysis with a sensitivity analysis by asking which level of drought severity d the system can tolerate before tipping over into the encroached state, and how this in turn depends on the level of farmer support.

Results

Increasing levels of farmer support trigger bistability

The model outcome strongly depended on the level of farmer support (Fig. 2). First, in the absence of farmer support, we found that the system approached a stable state with all four populations being present (see bottom left corner in Fig. 2a–c). The ratio of shrubs within the plant population and also the ratio of browsers within the animal population was approximately 0.5. A minor increase in farmer support resulted in a minor shift towards relatively more grazers and fewer shrubs but it did not affect the qualitative outcome of the model. However, increasing the level of farmer support beyond a certain threshold enabled bistability. Depending on the composition of the initial population, the system then either ran into the state described above or into an alternative state, characterised by shrubs dominating over grasses (Fig. 2a) and browsers close to extinction (Fig. 2b). This transition from one to two possible stable states occurred for a critical level of farmer support f_b,crit ≈ 0.32 if f_d = 0. The value of f_b,crit decreased for increasing values of f_d. Both vegetation types and both herbivore types, i.e. all four populations, remained viable in both states, as long as the farmer support was still moderate. Extinctions only occurred for even higher levels of farmer support (Fig. 2c).

Figure 2d summarises all possible population densities of the four populations for increasing farmer support to reduce grazer mortality rate (increasing values of f_b) but without any effect on density dependent losses (f_d = 0). This data thus corresponds to the x-axis in the panels above. At f_b,crit ≈ 0.32, the system shifted from having only one to three equilibria via a saddle-node bifurcation. However, one of the equilibria that emerged at the critical threshold is unstable and therefore difficult to detect using numerical simulations. The bifurcation diagrams consequently show only two branches. The first branch, which is present for f_b > 0.7, represents the ‘grass dominated' state, while the second branch, which is present for f_b > 0.32, represents the ‘encroached' state. Note that very intense farmer support always led to system degradation via additional transcritical bifurcations. This is obviously true if the system was in the encroached state, where browsers already disappeared if f_b > 0.4 and grasses disappeared if f_b > 0.55, but also for the grass dominated state, where browsers disappeared if f_b > 0.55 and shrubs disappeared if f_b > 0.7.

The described patterns were also found when farmer support affected density dependent losses in addition to net mortality, albeit with slightly different critical parameter values, as summarised in Fig. 2e. However, a decrease in density dependent losses due to increased farmer support was associated with a reduced dampening of consumer–resource oscillations, which explained the occurrence of additional Hopf-bifurcations at f_b = f_d ≈ 0.66 in the grass dominated state and at f_b = f_d ≈ 0.28, 0.39 and 0.75 in the encroached state.

External disturbances push the system towards bush encroachment

A gradual increase in the level of farmer support was not sufficient to push the system towards encroachment. Without external disturbances, the system simply stayed in the grass dominated state with a relatively stable density of grasses and a gradually decreasing density of shrubs, always in a quasi equilibrium, until it finally degraded (Fig. 2d). However, real ecosystems always experience a certain degree of variation in environmental conditions, such as fluctuations in temperature or rainfall. Population densities consequently fluctuate as well, in response to ever changing environmental conditions, and cannot be considered to be at equilibrium.

To study system dynamics away from equilibrium, we therefore applied artificial disturbances, which drastically reduced plant biomass, as shown in the exemplary time series in Fig. 3. At t = 0, both time series were initialised with random population densities and quickly approached a stable state. The system was very resilient, meaning that it quickly returned to the grass dominated equilibrium state after the disturbance occurring at t = 1000. The level of farmer support was then increased at t = 2000. The system stayed in the grass dominated state but approached different equilibrium densities and lost stability. The latter became apparent from the system's response to a second disturbance, which took place at t = 3000 and forced the system into the alternative stable state dominated by shrubs. Whether or not browsers survived the second disturbance depended on how much their population had already declined in response to increasing farmer support. Browser survival in turn determined whether a simple reduction of farmer support was sufficient to push the system back into the grass dominated state (Fig. 3a) or whether shrubs continued to dominate even if f_x = 0 (Fig. 3b).

Resistance to external disturbances declines for increasing levels of farmer support

Whether or not a disturbance pushed the system towards the encroached state obviously depended on the severity of the drought, but also on the level of farmer support, as illustrated in Fig. 4. We found that disturbances that reduced grass biomass by less than 70% were in general not severe enough to enable bush encroachment (Fig. 4a). We furthermore found that the positive impact of increasing farmer support on livestock abundances (Fig. 4c) came at a cost, because even for moderate levels of farmer support, the system already started to lose resistance. More precisely, the maximum severity of a disturbance that the system could tolerate without tipping into the encroached state drastically decreased for increasing values of farmer support (Fig. 4a with 0.32 < f_x < 0.55), while equilibrium grazer densities were still increasing (Fig. 4c).

Even higher levels of farmer support then led to reversed trends, that is an apparent increase in resistance (Fig. 4a) and a decrease in livestock abundance (Fig. 4c). However, this interpretation of the results is misleading, since the system was already degraded at that point and did no longer contain all four populations. In fact, in consistency with our previous results, we found that browsers had no chance to survive once the farmer support became sufficiently high. They either gradually went extinct, in direct response to increasing levels of farmer support, or abruptly, in response to a disturbance (Fig. 4b).

Discussion

We showed that our model exhibits two alternative stable states in response to increasing levels of farmer support. While most studies explain the bistability of savannas as a result of abiotic factors, e.g. due to non-linear responses of vegetation to gradients of rainfall (Hirota et al. 2011, Aleman et al. 2020) or fire intensity (Staver et al. 2011, Dantas et al. 2016), our results indicate that bistability could also be a direct consequence of management influence on vegetation–herbivore interactions. Bush encroachment, the critical transition from a grass-dominated to an encroached state, was only triggered by a high farmer support in combination with an external disturbance, such as drought. A gradual increase in farmer support alone, without disturbance, did not cause the system to tip over. However, we found that the system's resilience towards these disturbances declined as farmer support was increased. Our findings have important consequences for predicting critical transitions in savanna systems, since the type of tipping point that we describe here cannot be predicted by resilience-based indicators (Scheffer et al. 2009).

To assess potential management impacts on system stability, we focused our analysis on the farmer support parameter, which translates into a reduced grazer mortality. Note that the grazer abundance and hence grazing pressure can be seen as an emergent property of the system that is determined both by management, via farmer support, but also by the state of the environment, that is the system's ability to support the grazers. We hypothesised that an increase in farmer support would lead to an increase in grazing pressure, and hence indirectly to a release of competitive and herbivorous pressure on shrubs, which in turn results in a dominance of woody vegetation. This expectation was mostly confirmed, given that the system indeed showed critical transitions between a grass dominated and a shrub dominated system state. As long as the savanna remained in the grass-dominated state, farmer support indeed caused an increase in grazer densities while grass densities remained relatively constant, indicating that growth was fast enough to compensate for removal by grazing. The expectation that higher levels of farmer support always lead to increasing grazer biomass was however not confirmed after the system transitioned into the encroached state. Grazer densities then declined significantly, even if farmer support was further increased. Such a decline in the cattle number in encroached savannas is consistent with empirical literature (Lange et al. 1998) and suggests that encroached savannas have a reduced carrying capacity and can thus no longer support the same amount of herbivores as before (Scholes and Archer 1997, Smit 2004, Ward 2005).

We furthermore found that the level of farmer support during the critical transition also determined whether the grass-dominated state could be restored by reducing farmer support. This was only possible for moderate levels of farmer support. More precisely, we found that the possibility of reversing was critically linked to the survival of the browser population, which was close to extinction under high levels of farmer support. However, after we reintroduced the browser population, the system could always return to the grass-dominated state. This is in line with empirical studies which have shown that native browsers play an important role in maintaining open savannas and in mitigating bush encroachment (Augustine and McNaughton 2004, Staver and Bond 2014, Stevens et al. 2016).

Even though our model is very simple and rather abstract, the results are qualitatively in line with many observational studies reporting an increase in woody cover in response to overgrazing (Van Vegten 1984, Skarpe 1990a, b1990b, Roques et al. 2001). This is remarkable, given the fact that we deliberately did not include other factors that are known to play a role in bush encroachment. Due to its simplicity, and because parameters have not been calibrated based on empirical data, our model is not well suited to forecast specific tipping points quantitatively. However, its abstractness allowed us to focus on certain key mechanisms in a generalising way, which is often impossible using more complex models or empirical methods.

One important goal of our study was to identify which processes cause the transition from a grass-dominated to an encroached state. By quantifying all positive and negative feedback loops within the system, we showed that the longer feedback loops describing the combined effect of interactions between plants and between plants and herbivores were responsible for destabilising the unstable system state. This is somewhat surprising, as the direct influence of herbivores on vegetation via consumer–resource interactions was actually a negative feedback loop, which added a direct stabilising effect on the system (Briske et al. 2006). Herbivore populations that are regulated by vegetation have thus been assumed to stabilise savannas (Van De Koppel and Rietkerk 2000). Accordingly, in a similar model by Holdo et al. (2013), the bistable parameter region became smaller when browsers and grazers were included. However, in contrast to the model presented here, their model assumed that both groups of herbivores feed exclusively on one group of plants, so that herbivores are neither affected by each other nor by management. This indicates that the inclusion of cross-feeding between the two herbivore guilds has important consequences on the model outcome. We also found that the extent of cross-feeding, which is controlled by the preference parameters, altered the level of farmer support at which the critical transition appeared in our model (Supplementary information).

In summary, our results point to an urgent need to not only recognize feedback mechanisms at a conceptual level, but also to quantify them and to investigate how they interact in shaping system stability. The apparent contradiction between our own results compared with (Holdo et al. 2013) clearly shows that even small differences in the configuration of the underlying interaction network can affect how exactly feedback loops are formed and thus lead to completely different model outcomes. To better understand how exactly these small changes can lead to completely different model outcomes, similar studies are needed that apply feedback loop analysis to various interaction networks. Furthermore, our analysis shows that the stability of a given system is not determined simply by the presence of positive feedback loops but by the balance of all positive and negative loops within that system. While this has long been known in theory (Levins 1974) only very few studies (Marzloff et al. 2011, Van de Leemput et al. 2016) have so far applied this knowledge to real systems. In our specific case, we found that instability only occurred when an increase in farmer support caused the combined effect of all positive loops to exceed the combined effect of the all negative 2-link loops between plants and herbivores, as well as the negative feedback from self-regulation.

In order to keep our equation simple and to be able to focus on interactions at the system level, our model greatly simplifies certain processes. This includes the interactions between the two plant-functional types, which we represent as a direct negative link without considering competition for explicit resources like water or light. In reality, grasses and trees interact in an extremely complex fashion (Scholes and Archer 1997, Sankaran et al. 2004, Riginos and Young 2007, Synodinos et al. 2015) that varies between life stages as well as with environmental conditions. Similarly, our representation of human management impact as the ‘farmer support' parameter is only one possibility. We focused on a positive effect on grazers, such as farmers supporting their cattle, and thereby neglected scenarios where farmer management might also positively affect browsers. Such potential impacts include the support of browsing livestock, such as goats, or the installations of water points, which benefits both grazing and browser animals. Furthermore, instead of only affecting mortality, management could also influence the population's growth terms, e.g. via the provision of fodder to livestock and wild live in dry periods. Finally, all of the processes mentioned above could additionally be influenced by spatial heterogeneity, like uneven distributions of vegetation or herbivores. All these aspects could, in principle, affect the emergence of bistability and critical transitions by modulating the strength of existing feedback loops or by adding new ones.

Models that take such complexity into account typically aim at improving management strategies at the cost of being too complex to unravel the relative importance of the different processes at play (Tietjen and Jeltsch 2007). Simpler models based on equilibrium concepts, on the other hand, have long been assumed to be unsuitable to capture tree–grass interactions and have therefore traditionally received little attention (Jeltsch et al. 2000). In line with related models based on dynamical systems theory (Anderies et al. 2002, Van Langevelde et al. 2003, Accatino et al. 2010, Staver et al. 2011, Holdo et al. 2013, Touboul et al. 2018) we challenge this view by demonstrating that simple models can indeed contribute to an improved understanding of the forces that push savanna systems towards encroachment. We consider our model a proof of concept. It demonstrates that interaction networks and provide a powerful tool-set because they allow us to study the relative importance of the underlying feedback loops. In line with even simpler, very early plant–herbivore models that contain only one plant and one herbivore (Noy-Meir 1975, Walker et al. 1981), we show that bistability can arise already from biotic interactions, without considering other known self-reinforcing feedback that involves rainfall or fire. However, both abiotic and biotic factors obviously affect each other, and hence the dynamics of real savanna systems. We therefore argue that future modelling approaches should address them in concert, and allowing for feedback between both, to further improve our understanding of tipping points within savanna systems.