## Spatially explicit models on succession and disturbance in grasslands

Disturbance can play a major role in structuring grasslands by producing a spatiotemporal mosaic of patches by locally resetting the successional clock after a disturbance. The spatiotemporal distribution of species within the resulting mosaic depends upon an interaction between species' life-history traits and the spatial and temporal structure of the ecological processes controlling species' distributions. The availability of powerful computers and the advent of spatially explicit simulation models revitalized the conceptual work done by Watts in the 1940s and allowed to elucidate process from patterns. In this line, several spatially explicit plant dynamics models investigate questions related to succession and disturbance.

The Steppe model. D. B. Coffin and W. Lauenroth developed at the end of the 1980s a spatially explicit gap dynamics simulation model (STEPPE) to evaluate the effects of disturbances at the scale of a landscape for a semiarid grassland in north-central Colorado, USA. The approach goes back to spatially implicit gap models of forest succession (see Forest Models). These models simulate succession in a gap left after the death of a large tree. Succession dynamics are then estimated by the average behavior of 50-100 plots of the size of a single large tree. The exact location of each individual is not used to compute competition in these models. However, gap models can be made spatially explicit by linking several individual plots together.

The STEPPE model is a gap model and simulates the establishment, growth, and death of individual grass plants on a small plot (0.12 m ) through time at an annual time step. Landscapes were simulated either as a collection of independent plots or as a collection of interacting plots. In the 1990s, extending their first approach, W. Lauenroth and colleagues coupled compartment models of nutrient cycling and soil water-plant relation with the STEPPE model. The aim was to understand the interactions between vegetation structure and ecosystem processes across ecosystems. However, the approach of coupling several models was quickly abandoned, probably because of problems with increased model complexity.

A gap model similar to STEPPE was developed at the beginning of the 2000s by D. B. Peters (formerly Coffin). This individual-based, gap dynamics model (ECOTONE) was used to predict the effects of climatic fluctuations on regional patterns of vegetation dynamics, and the effects of disturbance on vegetation dynamics at an arid-semiarid ecotone between shortgrass steppe grassland and a Chihuahuan desert community in central New Mexico, USA.

The Jasper model. K. A. Moloney and S. A. Levin developed in the 1990s a spatially explicit simulation model of a serpentine grassland, focusing primarily on the role of disturbance. The model is hierarchical in design and population dynamics were modeled as occurring within local sites, which were then arranged to form a landscape and interact primarily through seed dispersal. Several components of the disturbance architecture were varied systematically among model runs to determine their impact on population dynamics at the scale of the landscape. Results suggested that predicting the impact of disturbance on ecological communities will require an explicit understanding of at least some aspects of the spatial and temporal architecture of the disturbance regime.

Neural networks. Where long-term site data on species composition and environmental factors are available, an approach using artificial neural networks may be feasible to predict grassland succession. The method takes advantage of the ability of neural networks to learn, recognize, and generalize from patterns contained in the ecological data. In the 1990s, S. S. Tan applied this approach to a 30-year data set on vegetation changes and climatic factors in grassland communities in Kansas (USA). Their model predicted future community composition using input data on present conditions that have not been used to develop the model. The model performs well for 1-4 year predictions and performance significantly deteriorated from the fifth year on. One disadvantage of the neural networks is that it is purely descriptive and gives little insight into underlying causes and mechanisms of grassland succession. On the other hand, if predictions are required for highly complex systems as they occur in natural rangeland communities, they are an appropriate technique for qualitative analysis and statistical forecasting.

Matrix models. Matrix models of succession are mathematically and conceptually the most straightforward among the succession models. Matrix models are constructed by determining the probability that the vegetation on a local plot will transform to some other vegetation state after a given time interval. To construct the model, the vegetation must be classified into identifiable states. The model consists of a vector representing the state variables of the system and a matrix containing the (usually) constant probabilities of possible state transitions. A question of interest is how a population growth rate depends on factors such as rainfall or grazing. Matrix models provide the means for calculating population growth rates, thereby allowing for an assessment of the influence of different population processes such as seed production and growth and mortality. For example, T. G. O'Connor used in the 1990s stage-structured matrix models to assess the influence of rainfall and grazing on the demography of some African savanna grasses. Results showed that the population growth rates of most species were positively correlated, indicating that an extrinsic force, presumably rainfall, had the greatest effect on population growth.

Optimal life-history strategies, competition, coexistence, and biodiversity

A considerable number of models on grassland dynamics and processes are motivated from general theory in ecology and are often put into a spatially explicit context. These models, developing since the mid-1990s, are concerned with tradeoffs in optimal reproduction strategies, and investigate how optimal strategies depend on disturbance and other spatially explicit factors such as a patchy habitat. One early example of this approach is a spatially explicit, two-species simulation model by S. Lavorel to examine the interaction between dispersal, dormancy, and small-scale disturbances on coexistence of two annual plant species in landscapes with varying degree of patchi-ness. Coexistence patterns depended on the degree of suitability and the patchiness of the landscape, mostly in relation to the interactions between landscape structure and mean dispersal distance.

A model of grassland community dynamics developed in the 2000s by Y. G. Matsinos and A. Y Troumbis aimed to quantify the role of competition and dispersal under disturbance, and investigated the resilience of the communities with respect to gap-creating disturbances that were imposed at different spatial extent. Model results showed that plants with longer seed dispersal distances may have a competitive advantage in their colonization success as compared to the better competitors, especially in the cases of a disturbance-mediated creation of gaps in the landscape. An increase of the species number led to more stable end communities and a higher vegetation cover in the landscape.

A series of papers by S. Schwinning and A. J. Parsons investigated, in the 1990s, coexistence mechanisms for grasses and legumes in grazing systems. This is an important question since legumes fix nitrogen which in turn is beneficial for grasses.

Other questions investigated with spatially explicit models attempt to gain an understanding of clonal growth and ramification of grass tillers in grasslands. For example, E. Winkler and colleagues developed a series of grid-based models of grassland communities to investigate long-term control of species abundances and reproduction strategies in patchy landscapes undergoing disturbances. One example is a study with M. Fischer investigating tradeoffs between sexual and vegetative reproduction of clonal plants. Depending on spatial habitat structure and disturbances, different reproduction strategies lead to different long-term fitness. The model simulated plant population dynamics on a two-dimensional cellular grid consisting of 70 x 70 square cells. In an extension of this approach, J. Stocklin and E. Winkler used a spatially explicit, individual-based metapopulation model of Hieracium pilosella to examine the consequences of tradeoffs between vegetative and sexual reproduction and between short and far-distance dispersal of seeds. They found that in a spatially heterogeneous landscape, sexual seed production in a clonal plant is advantageous even at the expense of local vegetative growth.

While most of the questions from this section were analyzed by means of grid-based models, advances in analytical modeling made by B. M. Bolker, S. W. Pacala, and others at the end of the 1990s suggested that spatial interactions may be approximated by means of moment equations that describe changes in the mean densities and spatial patterns (covariances) of competing species. The formalism of moment equations is borrowed from physics where it was used to describe phase transitions and is an elegant way to approximate simple spatial dynamics by tracking the dynamics of the first moments (mean densities) and second moments (variances and covariances or spatial covariances) of the spatial distributions of populations. For example, to understand at a general level how plants coexist in communities, Bolker and Pacala studied three different strategies to compete for resources in a spatially variable environment: colonizing new areas, exploiting resources in those areas quickly before other plants arrive, or tolerating competition once other plants arrive. However when adding more realism the moment equations become quickly lengthy, simulations are often required to find an appropriate moment closure.

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