A minimalistic model of vegetation physiognomies in the savanna biome
Graphical abstract
Introduction
Savannas, as broadly defined as systems where tree and grass coexist (Scholes and Archer, 1997), occupy about 20% of the Earth land surface and are observed in a large range of Mean Annual Precipitation (MAP). In Africa, they particularly occur between 100 mm and 1500 mm (and sometimes more) of total mean annual precipitation (Lehmann et al., 2011, Baudena and Rietkerk, 2013), that is along a precipitation gradient leading from dense tropical forest to desert. There is widespread evidence that fire and water availabilities are variables which can exert determinant roles in mixed tree–grass systems (Scholes and Archer, 1997, Yatat Djeumen et al., 2018b and references therein). Empirical studies showed that vegetation properties such as biomass, leaf area, net primary production, maximal tree height and annual maximum standing crop of grasses vary along gradients of precipitation (Penning de Vries and Djitèye, 1982, Abbadie et al., 2006). It is widely accepted that water availability directly limits woody vegetation in the driest part of the rainfall gradient, see e.g. Sankaran et al. (2005). However. in the mesic and humid parts of this gradient, rainfall is known to influence indirectly the fire regime through what can be referred to as the grass-fire feedback (Yatat Djeumen et al., 2018b, Scholes, 2003 and references therein): grass biomass that grows during rainfall periods is fuel for fires occurring in the dry months. Sufficiently frequent and intense fires are known to prevent or at least delay the development of woody vegetation (Yatat Djeumen et al., 2018b, Govender et al., 2006), thereby preventing trees and shrubs to depress grass production through competition for light and nutrients. The grass-fire feedback is widely acknowledged in literature as a force able to counteract the asymmetric competition of trees onto grasses, at least for climatic conditions that enables sufficient grass production during wet months.
Dynamical processes underlying savanna vegetation have been the subject of many models. Some of them explicitly considered the influence of soil water resource on the respective productions of grass and woody vegetation components (see the review of Yatat Djeumen et al. (2018b)). Most of the models also incorporated the grass-fire positive feedback, several of them distinguishing fire-sensitive small trees and shrubs from non-sensitive large trees (Higgins et al., 2000, Beckage et al., 2009, Baudena et al., 2010, Staver et al., 2011, Yatat Djeumen et al., 2014, Yatat Djeumen et al., 2018b), while the rest stuck to the simplest formalism featuring just grass and tree state variables (Van Langevelde et al., 2003, D’Odorico et al., 2006, Higgins et al., 2010, Accatino et al., 2010, Beckage et al., 2011, Yu and D’Odorico, 2014, Tchuinté Tamen et al., 2014, see also the review of Yatat Djeumen et al. (2018b) and Fig. 1).
Models featuring the grass-fire feedback have shown that complex physiognomies displaying tree–grass coexistence (i.e. savannas) may be stable (Van Langevelde et al., 2003, D’Odorico et al., 2006, Baudena et al., 2010, Accatino et al., 2010, Yatat Djeumen et al., 2014, Tchuinté Tamen et al., 2014) along with more “trivial” equilibria such as desert, dense forest or open grassland. There are also field observations that report contrasted savanna–forest mosaics at landscape scale (see e.g. Fig. 2) that suggest bistability which is indeed among the outcomes of some models (see for instance Accatino et al. (2010), Staver et al. (2011), Tchuinté Tamen et al. (2014), Yatat Djeumen et al., 2014, Yatat Djeumen et al., 2018b and references therein). However, the ability to predict, along the whole rainfall gradient, all the physiognomies that are suggested by observations as possible stable or multi-stable outcomes was not fully mastered and established. Indeed, most models focused on specific contexts or questions and often feature parameters difficult to assess over large territories, especially in Africa (Accatino et al., 2010, Higgins et al., 2010, Baudena et al., 2010, De Michele et al., 2011, Beckage et al., 2011, Yu and D’Odorico, 2014). Nonetheless, the (Accatino et al., 2010)’s attempt was a seminal step in that direction but with some notable imperfections.
The Accatino et al. (2010) model was pioneering in the sense that it allowed these authors to provide a “broad picture”, by delimiting stability domains for a variety of possible vegetation equilibria or steady states as functions of gradients in rainfall, the most limiting resource, and fire frequency, the most widespread disturbance. This result was especially interesting and the considered model was sufficiently simple (two vegetation variables, i.e. grass and tree covers) to provide analytical projections. However, results from Accatino et al. (2010) were questionable regarding the role of fire return time. In fact, all over the rainfall gradient their model predicted that increasing fire frequency would lead to an increase in woody cover which contradicts empirical knowledge on the subject. The features of the model that led to this problem were barely debated in the ensuing publications. And more recent papers instead either devised more complex models or shift to stochastic modeling (see the review of Yatat Djeumen et al. (2018b)) that did not allow much analytical exploration of their fundamental properties.
In this paper, we aim to account for a wide range of physiognomies and dynamical outcomes of the tree–grass interactions system as observable at both regional and continental scales by relying on a simple model that explicitly address some essential processes that are: (i) limits put by rainfall on woody and grassy biomasses development, (ii) asymmetric interactions between woody and herbaceous plant life forms, (iii) positive feedback between grass biomass and fire intensity, and decreased fire impact with tree height.
Starting from Yatat Djeumen et al. (2018b), we explicitly express the growth of both woody and herbaceous vegetation as functions of the mean annual rainfall, with the aim to study model predictions in direct relation to rainfall and fire frequency gradients. Through the present contribution we aim at extending and improving a framework for modeling vegetation in the savanna biome through an ODE-based model, that is minimal (in terms of state variables and parameters), mathematically tractable and generic in the sense that its structure does not pertain to particular locations in the savanna biome.
An idiosyncrasy of our minimalistic tree–grass model is that we considered the fire-induced loss of woody biomass by mean of two independent non-linear functions, namely (see (3)) and (see (4)). Introducing these two functions, Tchuinté Tamen et al. (2017) showed that the previous model substantially improve previously published results on tree–grass dynamical systems (see also Yatat Djeumen et al. (2018b). For example, they showed that increasing fire return period systematically leads to woody biomass build-up with possible switch from grassland/savanna to forest. This result is entirely consistent with field observations (Bond et al., 2005, Yatat Djeumen et al., 2018b) and references therein). From this sound basis, we introduced improvements in the model which are exposed in the present paper. Notably, we now let influences of trees on grasses range from facilitation to competition according to climate.
The goal of the present paper is to present the improved version of the minimalistic tree–grass ODE model (Section 2) and show through a complete theoretical analysis (Section 3 and appendices) that it is able to provide, at broad scales, an array of sensible predictions about possible vegetation physiognomies that was not to date attained by tree–grass models of similar levels of complexity (in terms of the number of equations and/or the types of non-linearities). Predictions sensitivity to parameters ranges was assessed in Section 4. Relying only on qualitative results, we will construct a bifurcation diagram (Section 6) depicting the possible vegetation types along the rainfall vs. fire frequency gradients. Last but not least, in order to render our approach easy-to-use, we have developed a R-Shiny application (Section 5) to build the bifurcation diagram taking into account all the model parameters as to let them been changed easily according to the reader’s wish.
Section snippets
The minimalistic ODE model formulation
Our model features two coupled ordinary differential equations (Eq. (6) below) expressing the dynamics of tree and grass biomasses. Each equation entails a term of logistic growth (with parameters depending on MAP, Section 2.1) and terms of biomass suppression by external agents (e.g. herbivory grazers or browsers) and fire. Coupling of the equations occurs because fire intensity impacts woody biomass as a non-linear increasing function of grass biomass (see Section 2.3), while the grass
Qualitative analysis results
Our approach has kept the model amenable to a complete qualitative analysis of equilibria and stability thereof, as developed in the appendices. Equilibria embodying the long-term behavior of system (6) are summarized in Table 2, Table 3 in the case of competitive and facilitative influences of trees on grasses, respectively. Tables 2–3 result from the theoretical analysis of system (6) provided in Appendix A. For reader convenience, we recall in the following some key findings from the
Sensitivity analyses of model (6)
Interpretation of results from mathematical models of biological systems is often complicated by the presence of uncertainties in experimental data that are used to estimate parameter values (Marino et al., 2008). Moreover, some parameters are liable to vary in space, even in a given reference area. Sensitivity analysis (SA) is a method for measuring uncertainty in any type of complex model by identifying critical inputs and quantifying how input uncertainty impacts model outcomes. Different SA
Bifurcation diagrams and numerical simulations
We first provide bifurcation diagrams, based on the thresholds computation for the following set of parameters (see Table 4). We secondly present numerical simulations (also based on Table 4 values) to illustrate bifurcations in relation to mean annual rainfall (W) and fire frequency ().
Thanks to the qualitative analysis of system (6)) (see Appendix A), any version of the bifurcation diagrams (see for instance Figs. 7–8, in terms of the fire frequency and the MAP, summarize the outcomes of the
Meaningful and diversified outcomes from a simple ODE framework
The present line of modeling aimed at demonstrating that meaningful and diversified outcomes can be expected from a full qualitative analysis of a parsimonious 2-dimensional models of grassy and woody biomasses interactions in the savanna biome. On the basis of a simple ODE framework, realistic results were indeed reached regarding how vegetation physiognomies change in relation to MAP and fire frequency. The model is liable to predict “trivial” equilibria, i.e. desert, grassland and forest as
Conclusion
In this paper, we presented and analyzed an improved version of a ‘minimalistic’ tree–grass model that addresses the influence of fire and rainfall (MAP) in tree–grass ecosystems. The model is minimalistic in terms of state variables and parameters, by only explicitly addressing essential processes that are: logistic growth of woody and grassy biomasses, asymmetric direct interactions thereof (both MAP-modulated), positive grass-fire feedback and decreased fire impact on large woody biomass. To
Supplementary materials
The R package containing the source code of the ‘Tree–Grass’ application is available at https://gitlab.com/cirad-apps/tree-grass.
CRediT authorship contribution statement
I.V. Yatat Djeumen: Conceptualization, Methodology, Formal analysis, Software, Visualization, Writing - original draft, Writing - review & editing. Y. Dumont: Conceptualization, Methodology, Software, Visualization, Writing - original draft, Writing - review & editing. A. Doizy: Software, Visualization, Writing - original draft, Writing - review & editing. P. Couteron: Conceptualization, Methodology, Validation, Writing - original draft, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
VY and YD were supported by the DST/NRF SARChI Chair in Mathematical Models and Methods in Biosciences and Bioengineering at the University of Pretoria, South Africa (grant 82770). This work benefited from ongoing field investigation in Cameroon supported by Nachtigal Hydropower Company, Cameroon (Contract n C006C007-DES-2017). YD is funded by the European Union Agricultural Fund for Rural Development, by the Conseil Régional de La Réunion, the Conseil Départemental de La Réunion, and by the
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