Max Rietkerk

Early-warning signals for critical transitions

Complex dynamical systems, ranging from ecosystems to financial markets and the climate, can have tipping points at which a sudden shift to a contrasting dynamical regime may occur. Although predicting such critical points before they are reached is extremely difficult, work in different scientific fields is now suggesting the existence of generic early-warning signals that may indicate for a wide class of systems if a critical threshold is approaching.

Spatial vegetation patterns and imminent desertification in Mediterranean arid ecosystems

Humans and climate affect ecosystems and their services, which may involve continuous and discontinuous transitions from one stable state to another. Discontinuous transitions are abrupt, irreversible and among the most catastrophic changes of ecosystems identified. For terrestrial ecosystems, it has been hypothesized that vegetation patchiness could be used as a signature of imminent transitions. Here, we analyse how vegetation patchiness changes in arid ecosystems with different grazing pressures, using both field data and a modelling approach. In the modelling approach, we extrapolated our analysis to even higher grazing pressures to investigate the vegetation patchiness when desertification is imminent. In three arid Mediterranean ecosystems in Spain, Greece and Morocco, we found that the patch-size distribution of the vegetation follows a power law. Using a stochastic cellular automaton model, we show that local positive interactions among plants can explain such power-law distributions. Furthermore, with increasing grazing pressure, the field data revealed consistent deviations from power laws. Increased grazing pressure leads to similar deviations in the model. When grazing was further increased in the model, we found that these deviations always and only occurred close to transition to desert, independent of the type of transition, and regardless of the vegetation cover. Therefore, we propose that patch-size distributions may be a warning signal for the onset of desertification.

Effects of fire and herbivory on the stability of savanna ecosystems

Savanna ecosystems are characterized by the co-occurrence of trees and grass- es. In this paper, we argue that the balance between trees and grasses is, to a large extent, determined by the indirect interactive effects of herbivory and fire. These effects are based on the positive feedback between fuel load (grass biomass) and fire intensity. An increase in the level of grazing leads to reduced fuel load, which makes fire less intense and, thus, less damaging to trees and, consequently, results in an increase in woody vegetation. The system then switches from a state with trees and grasses to a state with solely trees. Similarly, browsers may enhance the effect of fire on trees because they reduce woody biomass, thus indirectly stimulating grass growth. This consequent increase in fuel load results in more intense fire and increased decline of biomass. The system then switches from a state with solely trees to a state with trees and grasses. We maintain that the interaction between fire and herbivory provides a mechanistic explanation for observed discontinuous changes in woody and grass biomass. This is an alternative for the soil degradation mechanism, in which there is a positive feedback between the amount of grass biomass and the amount of water that infiltrates into the soil. The soil degradation mechanism predicts no discontinuous chang- es, such as bush encroachment, on sandy soils. Such changes, however, are frequently ob- served. Therefore, the interactive effects of fire and herbivory provide a more plausible explanation for the occurrence of discontinuous changes in savanna ecosystems.

Self-organization of vegetation in arid ecosystems

Scientists are still searching for possible unifying mechanisms to explain this range of spatial patterns (Tongway and Ludwig 2001), and an important question of this research is whether this range is the result of preexisting environmental heterogeneity, the result of spatial selforganization, or both (Klausmeier 1999; Couteron and Lejeune 2001; HilleRisLambers et al. 2001; Von Hardenberg et al. 2001). Here, we contribute to the ongoing debate about vegetation pattern formation in arid ecosystems by presenting novel, spatially explicit model analyses and results, extending on the work of HilleRisLambers et al. (2001). Our results show that these different vegetation patterns observed in arid ecosystems might all be the result of spatial self-organization, caused by one single mechanism: water infiltrates faster into vegetated ground than into bare soil, leading to net displacement of surface water to vegetated patches. This model differs from earlier model results (Klausmeier 1999; Couteron and Lejeune 2001; HilleRisLambers et al. 2001; Von Hardenberg et al. 2001) primarily in two ways: it is fully mechanistic, and it treats the lateral flow of water above and below the soil as separate, not independent, variables. Although the current model greatly simplifies the biophysics of arid systems, it can reproduce the whole range of distinctive vegetation patterns as observed in arid ecosystems, indicating that the proposed mechanism might be generally applicable. We further show that self-organized vegetation patterns can persist far into regions of high aridity, where plants would become extinct if homogeneously distributed, pointing to the importance of this mechanism for maintaining productivity of arid ecosystems (Noy-Meir 1973). Our analyses are based on the model first developed in HilleRisLambers et al. (2001)

Regular pattern formation in real ecosystems

Localized ecological interactions can generate striking large-scale spatial patterns in ecosystems through spatial self-organization. Possible mechanisms include oscillating consumer-resource interactions, localized disturbance-recovery processes and scale-dependent feedback. Despite abundant theoretical literature, studies revealing spatial self-organization in real ecosystems are limited. Recently, however, many examples of regular pattern formation have been discovered, supporting the importance of scale-dependent feedback. Here, we review these studies, showing regular pattern formation to be a general phenomenon rather than a peculiarity. We provide a conceptual framework explaining how scale-dependent feedback determines regular pattern formation in ecosystems. More empirical studies are needed to better understand regular pattern formation in ecosystems, and how this affects the response of ecosystems to global environmental change.

VEGETATION PATTERN FORMATION IN SEMI-ARID GRAZING SYSTEMS

Hypotheses about the origin of vegetation pattern formation in semi-arid areas around the world almost all include a common feature of semi-arid areas: the presence of a positive feedback between plant density and water infiltration. We investigate whether this positive feedback and the spatial redistribution of runoff water are sufficient to explain vegetation pattern formation. For this purpose, we analyze a spatially explicit model con- sisting of partial differential equations using a method for demonstrating pattern formation (Turing analysis). Our analysis reveals that pattern formation can occur in semi-arid areas given only the positive feedback between plant density and local water infiltration coupled with the spatial redistribution of runoff water. Thus, slope and underlying heterogeneity are not essential conditions. Other factors in the model, such as herbivory, plant dispersal, rainfall, and drought tolerance of plants, appear to determine under what conditions pattern formation is likely but are not the primary factors that generate the patterns. The model is in agreement with field observations and indicates the conditions for which vegetation pattern formation can be expected in arid and semi-arid grazing systems.

Catastrophic vegetation shifts and soil degradation in terrestrial grazing systems.

It has long been recognized that alternative vegetation states may occur in terrestrial grazing systems. This phenomenon may be of great importance as small environmental fluctuations may lead to relatively sudden and irreversible jumps between vegetation states. Early theoretical studies emphasized saturation of herbivore feeding to explain multiple stable states and catastrophic behaviour. Recent studies on semi-arid grasslands and arctic salt marshes, however, relate catastrophic events in these systems to plant-soil interactions.

Alternate stable states and threshold effects in semi-arid grazing systems

Models that explain the discontinuous behaviour of semi-arid grazing systems usually emphasize herbivore feeding characteristics or plant competition as possible mechanisms. Field studies indicate, however, that plant-soil relations could be more important. We show by means of a graphical model that the interactions between water infiltration or nutrient retention and plant density potentially give rise to the existence of alternate stable vegetation states and threshold effects in semi-arid grazing systems, even without the effect of a non-linear herbivore functional response or plant competition. These interactions may trigger a positive feedback between reduced plant density and reduced resource availability, and lead to a collapse of the system. The model results are in line with well-documented observations of spatial and temporal patterns such as two-phase mosaics and stably degraded grasslands.

Scale-Dependent Feedback and Regular Spatial Patterns in Young Mussel Beds

In the past decade, theoretical ecologists have emphasized that local interactions between predators and prey may invoke emergent spatial patterning at larger spatial scales. However, empirical evidence for the occurrence of emergent spatial patterning is scarce, which questions the relevance of the proposed mechanisms to ecological theory. We report on regular spatial patterns in young mussel beds on soft sediments in the Wadden Sea. We propose that scale‐dependent feedback, resulting from short‐range facilitation by mutual protection from waves and currents and long‐range competition for algae, induces spatial self‐organization, thereby providing a possible explanation for the observed patterning. The emergent self‐organization affects the functioning of mussel bed ecosystems by enhancing productivity and resilience against disturbance. Moreover, self‐organization allows mussels to persist at algal concentrations that would not permit survival of mussels in a homogeneous bed. Our results emphasize the importance of self‐organization in affecting the emergent properties of natural systems at larger spatial scales.

Experimental Evidence for Spatial Self-Organization and Its Emergent Effects in Mussel Bed Ecosystems

Spatial self-organization is the main theoretical explanation for the global occurrence of regular or otherwise coherent spatial patterns in ecosystems. Using mussel beds as a model ecosystem, we provide an experimental demonstration of spatial self-organization. Under homogeneous laboratory conditions, mussels developed regular patterns, similar to those in the field. An individual-based model derived from our experiments showed that interactions between individuals explained the observed patterns. Furthermore, a field study showed that pattern formation affected ecosystem-level processes in terms of improved growth and resistance to wave action. Our results imply that spatial self-organization is an important determinant of the structure and functioning of ecosystems, and it needs to be considered in their conservation.