Complexity and coexistence in a simple spatial model for arid savanna ecosystems

Pattern Formation in Mesic Savannas

We analyze a spatially extended version of a well-known model of forest-savanna dynamics, which presents as a system of nonlinear partial integro-differential equations, and study necessary conditions for pattern-forming bifurcations. Homogeneous solutions dominate the dynamics of the standard forest-savanna model, regardless of the length scales of the various spatial processes considered. However, several different pattern-forming scenarios are possible upon including spatial resource limitation, such as competition for water, soil nutrients, or herbivory effects. Using numerical simulations and continuation, we study the nature of the resulting patterns as a function of system parameters and length scales, uncovering subcritical pattern-forming bifurcations and observing significant regions of multistability for realistic parameter regimes. Finally, we discuss our results in the context of extant savanna-forest modeling efforts and highlight ongoing challenges in building a unifying mathematical model for savannas across different rainfall levels.

Evasion of tipping in complex systems through spatial pattern formation

[Figure: see text].

Self-organised pattern formation increases local diversity in metacommunities

Self-organised formation of spatial patterns is known from a variety of different ecosystems, yet little is known about how these patterns affect the diversity of communities. Here, we use a food chain model in which autotroph diversity is described by a continuous distribution of a trait that affects both growth and defence against heterotrophs. On isolated patches, diversity is always lost over time due to stabilising selection, and the local communities settle on one of two alternative stable community states that are characterised by a dominance of either defended or undefended species. In a metacommunity context, dispersal can destabilise these states and complex spatio-temporal patterns in the species' abundances emerge. The resulting biomass-trait feedback increases local diversity by an order of magnitude compared to scenarios without self-organised pattern formation, thereby maintaining the ability of communities to adapt to potential future changes in biotic or abiotic environmental conditions.

An integrodifference model for vegetation patterns in semi-arid environments with seasonality

Vegetation patterns are a characteristic feature of semi-deserts occurring on all continents except Antarctica. In some semi-arid regions, the climate is characterised by seasonality, which yields a synchronisation of seed dispersal with the dry season or the beginning of the wet season. We reformulate the Klausmeier model, a reaction–advection–diffusion system that describes the plant–water dynamics in semi-arid environments, as an integrodifference model to account for the temporal separation of plant growth processes during the wet season and seed dispersal processes during the dry season. The model further accounts for nonlocal processes involved in the dispersal of seeds. Our analysis focusses on the onset of spatial patterns. The Klausmeier partial differential equations (PDE) model is linked to the integrodifference model in an appropriate limit, which yields a control parameter for the temporal separation of seed dispersal events. We find that the conditions for pattern onset in the integrodifference model are equivalent to those for the continuous PDE model and hence independent of the time between seed dispersal events. We thus conclude that in the context of seed dispersal, a PDE model provides a sufficiently accurate description, even if the environment is seasonal. This emphasises the validity of results that have previously been obtained for the PDE model. Further, we numerically investigate the effects of changes to seed dispersal behaviour on the onset of patterns. We find that long-range seed dispersal inhibits the formation of spatial patterns and that the seed dispersal kernel’s decay at infinity is a significant regulator of patterning.

Metastability as a Coexistence Mechanism in a Model for Dryland Vegetation Patterns

Vegetation patterns are a ubiquitous feature of water-deprived ecosystems. Despite the competition for the same limiting resource, coexistence of several plant species is commonly observed. We propose a two-species reaction-diffusion model based on the single-species Klausmeier model, to analytically investigate the existence of states in which both species coexist. Ecologically, the study finds that coexistence is supported if there is a small difference in the plant species' average fitness, measured by the ratio of a species' capabilities to convert water into new biomass to its mortality rate. Mathematically, coexistence is not a stable solution of the system, but both spatially uniform and patterned coexistence states occur as metastable states. In this context, a metastable solution in which both species coexist corresponds to a long transient (exceeding [Formula: see text] years in dimensional parameters) to a stable one-species state. This behaviour is characterised by the small size of a positive eigenvalue which has the same order of magnitude as the average fitness difference between the two species. Two mechanisms causing the occurrence of metastable solutions are established: a spatially uniform unstable equilibrium and a stable one-species pattern which is unstable to the introduction of a competitor. We further discuss effects of asymmetric interspecific competition (e.g. shading) on the metastability property.

Prediction and scale in savanna ecosystems

Contents Summary 52 I. Introduction 52 II. Determinants of savanna vegetation structure 53 III. Are trees in savannas really more heterogeneous? 53 IV. Are trees in savannas a 'slow' variable? 54 V. Are trees in savannas spatially patterned? 55 VI. Conclusions 55 Acknowledgements 55 References 56 SUMMARY: Savannas are highly variable systems, and predicting variation, especially in the tree layer, represents a major unresolved challenge for forecasting biosphere responses to global change. Prediction to date has focused on disentangling interactions between resource limitation and chronic disturbances to identify what determines local savanna vegetation heterogeneity. By focusing at too fine a scale, this approach overlooks: sample size limitation arising from sparse tree distributions; stochasticity in demographic and environmental processes that is preserved as heterogeneity among tree populations with slow dynamics; and spatial self-organization. Renewed focus on large (1-50 ha) permanent plots and on spatial patterns of tree-layer variability at even larger landscape spatial scales (≥1000s of ha) promises to resolve these limitations, consistent with the goal of predicting large-scale biosphere responses to global change.

Analysis of a model for banded vegetation patterns in semi-arid environments with nonlocal dispersal

Vegetation patterns are a characteristic feature of semi-arid regions. On hillsides these patterns occur as stripes running parallel to the contours. The Klausmeier model, a coupled reaction-advection-diffusion system, is a deliberately simple model describing the phenomenon. In this paper, we replace the diffusion term describing plant dispersal by a more realistic nonlocal convolution integral to account for the possibility of long-range dispersal of seeds. Our analysis focuses on the rainfall level at which there is a transition between uniform vegetation and pattern formation. We obtain results, valid to leading order in the large parameter comparing the rate of water flow downhill to the rate of plant dispersal, for a negative exponential dispersal kernel. Our results indicate that both a wider dispersal of seeds and an increase in dispersal rate inhibit the formation of patterns. Assuming an evolutionary trade-off between these two quantities, mathematically motivated by the limiting behaviour of the convolution term, allows us to make comparisons to existing results for the original reaction-advection-diffusion system. These comparisons show that the nonlocal model always predicts a larger parameter region supporting pattern formation. We then numerically extend the results to other dispersal kernels, showing that the tendency to form patterns depends on the type of decay of the kernel.

Formation of banded vegetation patterns resulted from interactions between sediment deposition and vegetation growth

This research investigates the formation of banded vegetation patterns on hillslopes affected by interactions between sediment deposition and vegetation growth. The following two perspectives in the formation of these patterns are taken into consideration: (a) increased sediment deposition from plant interception, and (b) reduced plant biomass caused by sediment accumulation. A spatial model is proposed to describe how the interactions between sediment deposition and vegetation growth promote self-organization of banded vegetation patterns. Based on theoretical and numerical analyses of the proposed spatial model, vegetation bands can result from a Turing instability mechanism. The banded vegetation patterns obtained in this research resemble patterns reported in the literature. Moreover, measured by sediment dynamics, the variation of hillslope landform can be described. The model predicts how treads on hillslopes evolve with the banded patterns. Thus, we provide a quantitative interpretation for coevolution of vegetation patterns and landforms under effects of sediment redistribution.

An impulsive modelling framework of fire occurrence in a size-structured model of tree-grass interactions for savanna ecosystems

Fires and mean annual rainfall are major factors that regulate woody and grassy biomasses in savanna ecosystems. Within the savanna biome, conditions of long-lasting coexistence of trees and grasses have been often studied using continuous-time modelling of tree-grass competition. In these studies, fire is a time-continuous forcing while the relationship between woody plant size and fire-sensitivity is not systematically considered. In this paper, we propose a new mathematical framework to model tree-grass interactions that takes into account both the impulsive nature of fire occurrence and size-dependent fire sensitivity (via two classes of woody plants). We carry out a qualitative analysis that highlights ecological thresholds and bifurcation parameters that shape the dynamics of the savanna-like systems within the main ecological zones. Through a qualitative analysis, we show that the impulsive modelling of fire occurrences leads to more diverse behaviors including cases of grassland, savanna and forest tristability and a more realistic array of solutions than the analogous time-continuous fire models. Numerical simulations are carried out with respect to the three main ecological contexts (moist, mesic, semi-arid) to illustrate the theoretical results and to support a discussion about the bifurcation parameters and the advantages of the model.

When does colonisation of a semi-arid hillslope generate vegetation patterns?

Patterned vegetation occurs in many semi-arid regions of the world. Most previous studies have assumed that patterns form from a starting point of uniform vegetation, for example as a response to a decrease in mean annual rainfall. However an alternative possibility is that patterns are generated when bare ground is colonised. This paper investigates the conditions under which colonisation leads to patterning on sloping ground. The slope gradient plays an important role because of the downhill flow of rainwater. One long-established consequence of this is that patterns are organised into stripes running parallel to the contours; such patterns are known as banded vegetation or tiger bush. This paper shows that the slope also has an important effect on colonisation, since the uphill and downhill edges of an isolated vegetation patch have different dynamics. For the much-used Klausmeier model for semi-arid vegetation, the author shows that without a term representing water diffusion, colonisation always generates uniform vegetation rather than a pattern. However the combination of a sufficiently large water diffusion term and a sufficiently low slope gradient does lead to colonisation-induced patterning. The author goes on to consider colonisation in the Rietkerk model, which is also in widespread use: the same conclusions apply for this model provided that a small threshold is imposed on vegetation biomass, below which plant growth is set to zero. Since the two models are quite different mathematically, this suggests that the predictions are a consequence of the basic underlying assumption of water redistribution as the pattern generation mechanism.

Pattern selection and hysteresis in the Rietkerk model for banded vegetation in semi-arid environments

Banded vegetation is a characteristic feature of semi-arid environments. It occurs on gentle slopes, with alternating stripes of vegetation and bare ground running parallel to the contours. A number of mathematical models have been proposed to investigate the mechanisms underlying these patterns, and how they might be affected by changes in environmental conditions. One of the most widely used models is due to Rietkerk and co-workers, and is based on a water redistribution hypothesis, with the key feedback being that the rate of rainwater infiltration into the soil is an increasing function of plant biomass. Here, for the first time, we present a detailed study of the existence and stability of pattern solutions of the Rietkerk model on slopes, using the software package wavetrain (www.ma.hw.ac.uk/wavetrain). Specifically, we calculate the region of the rainfall-migration speed parameter plane in which patterns exist, and the sub-region in which these patterns are stable as solutions of the model partial differential equations. We then perform a detailed simulation-based study of the way in which patterns evolve when the rainfall parameter is slowly varied. This reveals complex behaviour, with sudden jumps in pattern wavelength, and hysteresis; we show that these jumps occur when the contours of constant pattern wavelength leave the parameter region giving stable patterns. Finally, we extend our results to the case in which a diffusion term for surface water is added to the model equations. The parameter regions for pattern existence and stability are relatively insensitive to small or moderate levels of surface water diffusion, but larger diffusion coefficients significantly change the subdivision into stable and unstable patterns.

Predictability of the terrestrial carbon cycle

Terrestrial ecosystems sequester roughly 30% of anthropogenic carbon emission. However this estimate has not been directly deduced from studies of terrestrial ecosystems themselves, but inferred from atmospheric and oceanic data. This raises a question: to what extent is the terrestrial carbon cycle intrinsically predictable? In this paper, we investigated fundamental properties of the terrestrial carbon cycle, examined its intrinsic predictability, and proposed a suite of future research directions to improve empirical understanding and model predictive ability. Specifically, we isolated endogenous internal processes of the terrestrial carbon cycle from exogenous forcing variables. The internal processes share five fundamental properties (i.e., compartmentalization, carbon input through photosynthesis, partitioning among pools, donor pool-dominant transfers, and the first-order decay) among all types of ecosystems on the Earth. The five properties together result in an emergent constraint on predictability of various carbon cycle components in response to five classes of exogenous forcing. Future observational and experimental research should be focused on those less predictive components while modeling research needs to improve model predictive ability for those highly predictive components. We argue that an understanding of predictability should provide guidance on future observational, experimental and modeling research.

Using wavelength and slope to infer the historical origin of semiarid vegetation bands

Landscape-scale patterns of vegetation occur worldwide at interfaces between semiarid and arid climates. They are important as potential indicators of climate change and imminent regime shifts and are widely thought to arise from positive feedback between vegetation and infiltration of rainwater. On gentle slopes the typical pattern form is bands (stripes), oriented parallel to the contours, and their wavelength is probably the most accessible statistic for vegetation patterns. Recent field studies have found an inverse correlation between pattern wavelength and slope, in apparent contradiction with the predictions of mathematical models. Here I show that this "contradiction" is based on a flawed approach to calculating the wavelength in models. When pattern generation is considered in detail, the theory is fully consistent with empirical results. For realistic parameters, degradation of uniform vegetation generates patterns whose wavelength increases with slope, whereas colonization of bare ground gives the opposite trend. Therefore, the empirical finding of an inverse relationship can be used, in conjunction with climate records, to infer the historical origin of the patterns. Specifically, for the African Sahel my results suggest that banded vegetation originated by the colonization of bare ground during circa 1760-1790 or since circa 1850.

Inferring resilience to fragmentation-induced changes in plant communities in a semi-arid Mediterranean ecosystem

Predicting the capacity of ecosystems to absorb impacts from disturbance events (resilience), including land-use intensification and landscape fragmentation, is challenging in the face of global change. Little is known about the impacts of fragmentation on ecosystem functioning from a multi-dimensional perspective (multiple traits). This study used 58 500-m linear transects to quantify changes in the functional composition and resilience of vascular plant communities in response to an increase in landscape fragmentation in 18 natural scrubland fragments embedded within a matrix of abandoned crop fields in Cabo de Gata-Níjar Natural Park, Almería, Spain. Changes in functional community composition were measured using functional diversity indices (functional richness and functional dispersion) that were based on 12 plant traits. Resilience was evaluated using the functional redundancy and response diversity from the perspective of plant dispersal, which is important, particularly, in fragmented landscapes. Scrubland fragmentation was measured using the Integral Index of Connectivity (IIC). The functional richness of the plant communities was higher in the most fragmented scrubland. Conversely, the functional dispersion (i.e., spread) of trait values among species in the functional trait space was lower at the most fragmented sites; consequently, the ecological tolerance of the vegetation to scrubland fragmentation decreased. Classifying the plant species into four functional groups indicated that fragmentation favoured an increase in functional redundancy in the ‘short basal annual forbs and perennial forbs’ group, most of which are species adapted to degraded soils. An assessment based on the traits associated with plant dispersal indicated that the resilience of ‘woody plants’, an important component in the Mediterranean scrubland, and habitat fragmentation were negatively correlated; however, the correlation was positive in the ‘short basal annual forbs and perennial forbs’ and the ‘grasses’ groups.