An individual-based, stochastic and spatial model to simulate the ramification of grass tillers and their distribution in swards
Introduction
The vegetation dynamics of a grassland depends on plant competition [19] and on gap colonisation [6]. In a pasture, ingressions and competition are involved at spatial scales between a few square centimetres and a few square decimetres [43]. At such scales in perennial grass swards, the horizontal structure of the stand is very heterogeneous, with empty spaces adjoining very densely populated ones at changing places [26]. Grass patches are composed of grass sprigs, called tillers that are continually renewed by ramification, as the old ones die. In most species, these tillers live from a few months to one or a few years, but “plants” (that is sets of connected tillers) are perennial.
The density of all natural populations varies in space and time, according to density-dependent processes and environmental variability, which always interact [30]. Understanding the vegetation dynamics requires to break this interplay. Simulation of one of both interacting processes may be a powerful tool. The aim of our work is to generate fine-scale stand structures of pure grass swards from a spatial simulation of the demographic processes of a perennial grass species in given environmental conditions, in order to know the relative weights of the species-specific patterns and of the local environmental conditions in observed heterogeneities of local tiller densities. In a grass patch, demographic processes are the birth of new tillers, the death of the existing tillers and, when a spatial pattern is taken into account, the positioning of the tillers.
The few existing tiller-density simulators are sub-models in dry-mass growth models [38]. They simulate a mean density at a too coarse scale. Existing spatial demographic models for grasses focus on tussocks (for examples [45], [16]). We chose a model generating individual tillers from mother tillers and distributing them on the ground. In this way, we focused on clonal multiplication rather than on seedling recruitment, which is always possible but minor [41]. The biological side of this model was described and discussed in a companion paper [27]. In this study, we want to explain and discuss our modelling choices and to describe the computer model, its implementation and verification. At the end, we will show results and compare them with fine-scale observations in established swards.
Section snippets
Summarized analysis of the biological system
The local distribution of grass tillers is the result of:
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the emergence of new tillers,
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the death of existing tillers,
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the displacements of existing tillers.
In the absence of seedling recruitment, all new tillers come from “tillering”, that is the branching of existing tillers at ground level. The tillering and its regulation are well known. A tiller axis is built by a succession of discs, called nodes, each bearing a leaf (a sheath and a lamina) and a tiller bud. The buds are alternatively placed
Modelling choices
Cellular automata are the more common approach for spatial modelling of vegetation dynamics. They are based on rules of transition between successive states in spatial cells [2], which integrate environmental and plant processes. For our purpose, the environment must only be an external factor of the behaviour of individuals and thus, a spatial individual-based model [4] is more convenient.
The modelling of a biological system in which individual entities interact, such as the one presented in
The UML class diagram
SISTAL (“SImulation Spatialisée du TALlage”, that is “spatial simulation of the tillering”) is a stochastic, object-oriented [20], discrete event, “individual based” and bidimensional spatial simulator [13], [23], [4].
Each tiller is an active object of the simulation whose spatial location on the simulated site is given by the coordinates of the cell it occupies and whose height is integrated in the canopy characteristics. Such a spatial and stochastic model enables us to reproduce the
The displacement method
The phenomenon of displacement is due to the nearly horizontal position of the tiller bases just below ground (see Section 2). In the present version of our model, the speed of displacement is the same for all tillers (see Section 4.1). The displacement is done according to a direction given by an angle θ (the attribute “displacement angle”), and a step, s. It is a continuous phenomenon whereas SISTAL is a discrete event simulation model, so the displacement method is activated periodically
Inputs-Outputs
The inputs of the model are
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a “simulation file” giving all simulation parameters: coefficients of the equations, characteristics of the tillers, average values and amplitude of the random variations, size of the grid and of the cells, position of the initial tillers, etc., and the names of the environmental files to be used;
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the environmental files (daily temperatures, extinction of the light under the canopy, soil resources) required for the simulation. In the present stage of SISTAL, they are
The verification tool
A bulky and wordy diary is produced between selected simulation days, when it is requested at the configuration of a simulation.
For each day of the selected periods, the calendar date and the thermal time are given. All living tillers are listed in the order used to scan the grid. For each tiller, we give its individual number, its position in real and discrete coordinates, its status (reproductive or not), its age in days and in phyllochrons, the thermal time of the next displacement and the
Simulation choices and results
The sward observations and the simulations performed for comparisons were described and compared in detail in the companion paper [27]. Briefly: the simulated tillers were of an usual tall-fescue type (Festuca arundinacea L. Schreb) with a phyllochron of 230 ± 20 degree-days, a sheath diameter of 0.2 cm, a ramification step of 0.4 cm, a displacement speed of 0.15 cm/phyllochron and probabilities for the induction of the reproductive status and for the natural mortality inferred from the results by
Conclusions
In order to study the vegetation dynamics of pure grass swards we have given the focus to density-dependent processes. The purpose of this approach was to separate the role of the latter from the one of environmental variability. A spatial simulation of tiller interactions within given environmental conditions enabled to generate fine-scale stand structures of pure grass swards. This approach allows studying the relative weights of the species-specific patterns and of the local environmental
Acknowledgments
We acknowledge four former ISIMA students: Nicolas Prud’homme and Rémi Svahn who implemented the main parts of the simulation program in 1997 and 1998, respectively; Mathieu Brocchi and Cyrille Cormier who recently implemented an improved version of the mortality method.
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2012, Ecological ModellingCitation Excerpt :This structure affects their dynamics to an unknown degree; origin, maintenance and dynamical implications of the pronounced spatial structure is not fully understood (Bolker et al., 2003). There have been a number of models devoted to grassland simulation, ranging from fine-scale individual-based models at the level of ramets that attempt to scale up plant-level processes (Winkler and Klotz, 1997; Winkler et al., 1999; Mazel et al., 2005; Tomlinson et al., 2007) to large scale models of grassland ecosystems at the landscape level, often with agricultural relevance (Hutchings and Gordon, 2001; Seabloom and Richards, 2003; Lazzarotto et al., 2009). However, there have been a very few models that use the individual-based approach for several species and capture the whole community dynamics and spatial pattern in terms of individual-level processes (but see e.g. Seabloom et al., 2005).
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