www.biomathforum.org/biomath/index.php/biomath

ORIGINAL ARTICLE

A tribute to the use of minimalistic
spatially-implicit models of savanna vegetation

dynamics to address broad spatial scales in
spite of scarce data

Ivric Valaire Yatat Djeumena,b, Alexis Tchuinté Tamenc,∗, Yves Dumonta,d,e, Pierre Couteronc,f
aUniversity of Pretoria, Department of Mathematics and Applied Mathematics, Pretoria, South Africa

bUniversity of Yaoundé 1, National Advanced School of Engineering, Yaoundé, Cameroon
cIRD, Umr AMAP, Lmi DYCOFAC, Yaoundé, Cameroon

alexis.tchuinte@yahoo.fr; yatat.valaire@gmail.com; ivric.yatatdjeumen@up.ac.za
dCIRAD, Umr AMAP, Pretoria, South Africa

eAMAP, Université de Montpellier, CIRAD, CNRS, INRA, IRD, Montpellier, France
yves.dumont@cirad.fr; yves.dumont@up.ac.za

f AMAP, IRD, Cirad, CNRS, INRA, Université de Montpellier, Montpellier, France
pierre.couteron@ird.fr

∗The first two authors contributed equally

Received: 5 September 2018, accepted: 16 December 2018, published: 20 December 2018

Abstract—The savanna biome encompasses a va-
riety of vegetation physiognomies that traduce com-
plex dynamical responses of plants to the rain-
fall gradients leading from tropical forests to hot
deserts. Such responses are shaped by interactions
between woody and grassy plants that can be either
direct, disturbance-mediated or both. There has
been increasing evidence that several vegetation
physiognomies, sometimes highly contrasted, may
durably coexist under similar rainfall conditions
suggesting multi-stability or at least not abrupt tran-
sitions. These fascinating questions have triggered
burgeoning modelling efforts which have, however,
not yet delivered an integrated picture liable to

furnish sensible predictions of potential vegetation
at broad scales. In this paper, we will recall the
key ecological processes and resulting vegetation
dynamics that models should take into account. We
will also present the main modelling options present
in the literature and advocate the use of minimalistic
models, capturing only the essential processes while
retaining sufficient mathematical tractability and
restricting themselves to a minimal set of parameters
assessable from the overall literature.

Keywords-Biogeography; Rainfall; Fires; Ordi-
nary differential equations; Impulsive differential
equation; Tree-Grass interactions; Multi-stability.

Copyright: © 2018 Yatat Djeumen et al. This article is distributed under the terms of the Creative Commons Attribution
License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original
author and source are credited.
Citation: Ivric Valaire Yatat Djeumen, Alexis Tchuinté Tamen, Yves Dumont, Pierre Couteron, A tribute
to the use of minimalistic spatially-implicit models of savanna vegetation dynamics to address broad spatial
scales in spite of scarce data, Biomath 7 (2018), 1812167, http://dx.doi.org/10.11145/j.biomath.2018.12.167 Page 1 of 29

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I.V. Yatat Djeumen, A. Tchuinté Tamen, Y. Dumont, P. Couteron, A tribute to the use of minimalistic ...

I. INTRODUCTION

Savannas have been identified by biogeogra-
phers as a biome corresponding to warm mean
annual temperatures (> 20°C) and a broad range
of intermediate mean annual rainfall (100–2000
mm.yr−1) (Sarmiento (1984) [76], Youta Happi
(1998) [114], Abbadie et al. (2006) [1], Lehmann
et al. (2011) [58]). Such climatic context predom-
inates along rainfall gradients leading from sub-
equatorial wet climates to hot arid climates. The
wider definition to which we refer here tends to
integrate climatic variant allowing for nearly desert
vegetation or on the contrary seasonal tropical
forests. Savannas display specific interplays of
natural constraints that prevent or at least impede
closure of woody cover and ensuing suppression
of light-demanding herbs and grasses. A central,
albeit non-exclusive cause for this is the ’ideal fire
climate’ (Trollope (2011) [98]) that characterizes
tropical regions with seasonal droughts alternating
with warm and wet rainy seasons producing high
herbaceous biomass that once dried-up becomes
highly ignitable and fuels fires (Frost et al. (1986)
[39], Thonicke et al. (2001) [93], Govender et al.
(2006) [44]). High frequency of lightning storms
which is a characteristic of Africa (Abbadie et al.
(2006) [1], Trollope (2011) [98]) also contributes
to make it the ”Fire continent” even though present
fire regimes mostly rely on human-made ignitions
(Archibald et al. (2009) [7], Govender et al. (2006)
[44], Trollope (2011) [98]).

Dynamics of vegetation within the savanna
biome has long interested ecologists as it clearly
departs from the classical post-disturbance succes-
sion pathways that are expected to rapidly bring
back closed canopy forest, as observed in most of
temperate and wet tropical climatic zones (Bond et
al. (2005) [22]). The last decades have witnessed
burgeoning efforts of modelling as to account for
the possibly long-lasting coexistence of grassy and
woody components and try to predict potential
shifts from two-phased vegetation physiognomies.
These efforts have, however, not yet delivered an
integrated picture liable to furnish at broad scales
(i.e., for fractions of continents) sensible predic-

tions of possible vegetation dynamics. Such a big
picture is nevertheless desirable for figuring out
the future of vegetation in the face of climate and
anthropic change scenarios (Mayaux et al. (2004)
[60], Bond et al. (2005) [22], Archibald et al.
(2009) [7], Accatino et al. (2010) [4], Favier et al.
(2012) [36]). It is also necessary for applications
to territories devoid of reference data and long-
term observation sites, as it is the case for most of
tropical Africa.

The objectives of the present contribution are
fourfold. It first aims at recalling and synthetizing
the main array of facts about ecological processes
and resulting vegetation dynamics that models
should aim to capture and render (see section II).
Second, in order to claim genericity, we synthetize
the main modelling options present in the liter-
ature, and put emphasis on minimalistic models,
capturing only essential processes while retaining
sufficient mathematical tractability and restricting
themselves to a minimal set of assessable pa-
rameters (see section III). Thirdly, on this basis,
we argue that such models have now become
more comprehensive, and useful for meaningful
predictions (see section IV). Finally, we discuss
how those models may now help guiding data
collection for improved calibration and testing of
dynamical hypotheses (see section V).

II. A BRIEF REVIEW ON SPACE-IMPLICIT
TREE-GRASS INTERACTIONS MODELLING

A. Tree-grass coexistence and possible alternative
stable states

Over very large tropical territories, field ob-
servers have documented long lasting coexistence
of notable levels of grass and woody biomass
(Backéus (1992) [9]). The most frequently re-
ported form of coexistence is observed locally
through vegetation physiognomies that associate
fairly continuous grassy cover and more or less
scattered populations of trees and shrubs of vary-
ing clumping levels. This is referred to as sa-
vanna physiognomy (see Figure 1). Such vege-
tation types mixing both lifeforms are manifold
and progressively merge in space or through time

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without clear-cut boundaries (Torello-Raventos et
al. (2013) [95]). Another modality of long lasting
association between herbaceous and woody life-
forms occur at landscape scale under the form of
mosaics featuring forests (usually closed canopy
ones) and open savannas or grasslands (e.g. Figure
1; Bond and Parr (2010) [21]). In those landscapes
that pertain to moist-wet climates, normally seen
as favourable to forests, the mosaics appear highly
contrasted and among the most ”emblematic vege-
tation transitions” in the world (Oliveras and Malhi
(2016) [72]): outside the closed forest, woody
vegetation is of low biomass and the dominant
physiognomies relate mainly to grassland. More-
over, boundaries between forest and grassland are
generally sharp (Hoffmann et al. 2012 [48], Cuni-
Sanchez et al. (2016) [27]).

Our interpretations of those various physiog-
nomies are limited by the length of the observation
windows we can rely on for distinguishing trends
against fluctuations. For field observations, this
window length barely extend over some decades
and this only for a very small number of sites
where invaluable data have been gathered. At the
scale of extensive territories, representativeness of
those sites remains yet an open question. Remote
sensing is progressively broadening our observa-
tional means. But the best nowadays space-borne
sensors for estimating woody cover (Buccini and
Hanan (2007) [26]) or biomass (Mermoz et al.
(2014) [64], Bouvet et al. (2018) [24]) are recent
and do not allow tracking changes far back. More-
over, the accuracy of those estimations, notably
for woody cover is limited, due to the difficulty
to separate grass vs. tree in signal responses in
mixed stands. This is particularly true regard-
ing long diachronic series that mainly feature
optical images of insufficient spatial resolution.
Apart from blatant changes, e.g., forest encroach-
ment or recession, Mitchard and Flintrop (2013)
[68], subtle evolution of the grass-tree balance
in mixed physiognomies are still beyond reach.
Remote sensing, however, recently brought two
interesting contributions to the savanna debate.
First, broad scale assessment of woody cover at

regional (Central Africa, Favier et al. (2012) [36])
to continental/global scales (Hirota et al. (2011)
[47]) clearly showed that contrasted levels of cover
can coexist under the same ranges of climatic
conditions, making the existence of multi-stable
states at least plausible. Second, in both Central
and West Africa, comparison between ancient air
photographs from the 50s and satellite images
from the 80-90s frequently evidenced a progress of
forest over savannas/grasslands in landscape fea-
turing contrasted mosaics of the type exemplified
in Figure 1 (Youta Happi (1998) [114], Mitchard
et al. (2011) [69]).

Even though there is still no conclusive evidence
that alternative stable states may exist within the
savanna biome, models should be able to account
for them as plausible outcomes of tree-grass in-
teractions. The same applies to savanna physiog-
nomies locally associating trees and grasses that
may be seen as either stable or transient two-
phase states. Since those mixed physiognomies are
observable at broad scale, there is no reason to a
priori rule out that some observed mixtures may
be stable under their local environmental context.
Indeed, hypothesis testing is a fundamental role
of models, though this use is not so widespread
in ecology. And to this aim, the wider the array
of reasonable predictions the more relevant is the
model.

B. Lines of thoughts

Most authors agree on the fact that soil water
budget, herbivory (i.e. grazing and/or browsing)
and fires are the principal factors influencing
growth of woody and herbaceous plants and their
dynamical interactions (Scholes and Archer (1997)
[79], Higgins et al. (2000) [45], Scholes (2003)
[78], Van Langevelde et al. (2003) [101], Bond
et al. (2005) [22], Bond (2008) [17], Abbadie et
al. (2006) [1], Accatino et al. (2010) [4], Staver
and Levin (2012) [84], Baudena et al. (2010) [11],
(2014) [12], Jeffery et al. (2014) [51]). Authors
however diverge on the relative importance of
those factors in shaping dynamical outcomes of
tree-grass interactions. This is not surprising con-
sidering the broad extent of the savanna biome

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I.V. Yatat Djeumen, A. Tchuinté Tamen, Y. Dumont, P. Couteron, A tribute to the use of minimalistic ...

Figure 1: Landscape-scale mosaic between dense forest and herbaceous savanna (grassland) observed in central Cameroon (Ayos). Brown-
pink tree crowns indicate marshy forests in talwegs. Note the weak congruence between topography and the occurrences of forest vs. grassland.

Airborne photo from N. Barbier, June 2017.

and the variety of both environmental conditions
and anthropogenic pressures that apply therein. A
factor appearing pervasive in a given context is not
systematically due to prevail elsewhere. One group
of authors has been insisting on direct interactions
among or between plant-types (i.e. tree-tree or
tree-grass) such as competition for light or for
soil limiting resources (often moisture via root
systems) (e.g. Scholes and Archer (1997) [79],
Scholes (2003) [78]). It is obvious that the tree–
grass interaction is highly asymmetric: trees have
a strong competitive effect on grasses, but grasses
have a weak competitive effect on mature trees,
although they may have a strong effect on saplings
that have not grown above the grass layer (Scholes
(2003) [78], Figure 2-a).

Another group of authors has been emphasizing
that woody vegetation would be likely to reach a
closed canopy situation and suppress grasses in the
absence of recurrent disturbances induced by fires
or browsers (or both sources) that delay or block
the build-up of woody biomass by destroying the
aerial part of seedlings and saplings (e.g. Bond
et al. (2005) [22], Bond (2008) [17], Staver and
Levin (2012) [84], Baudena et al. (2010) [11],
(2014) [12], Jeffery et al. (2014) [51]; Figure 2-
b & -c). Browsers impact, though undoubtedly
pervasive in certain situations (McNaughton and
Georgiadis (1986) [62], Scholes and Walker (1993)

[80], Van Langevelde et al. (2003) [101], Holdo
et al. (2009) [49]) is not systematic across the
savanna biome and the generality of the distur-
bance hypothesis relies mainly on fire. Indeed,
experimental fire suppression systematically leads
to the thickening-up of the woody vegetation and
to the development of dense woodlands or thickets.
For sufficient annual rainfall, shifts toward close
canopy forests are also observed (Bond et al.
(2005) [22], Jeffery et al. (2014) [51]).

Literature may sometimes overemphasize the
distinction between ’interaction’ (between plant
types for limited resource) and ’disturbance’ hy-
potheses (see Scholes and Archer (1997) [79])
as to make them appear as alternatives, though
they are by no means mutually exclusive. It is
widely acknowledged that to have notable impact
on vegetation, fire disturbance requests sufficient
intensity through enough dry grass biomass as
main source of fuel. Under a certain level of
grass biomass, owing to insufficient rainfall or
intense grazing, fires tend to spread difficultly
and, where occurring, have modest impacts on
woody plants. Logically, most authors tend now
to distinguish disturbance-limited (i.e., under moist
climate) vs. water limited (i.e., arid) savannas (e.g.
Bond et al. (2003) [20]). Inter-tree competition
shapes the second type (Sankaran et al. (2005)
[75]), while asymmetric and fire-meditated tree-

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Figure 2: Three facets of woody plant resprouting just after fire and rainfall onset in the humid savannas of the Sanagha basin (Cameroon,
Central Africa). Note that tufts of perennial grasses did also systematically resprout. Seedling struggling in a middle of a grass tuft a).

Seedling resprouting after topkill either at ground b) or stem c) level. Photos: Pierre Couteron (March 2018).

grass interactions is central to the first one. But
less clear-cut situations obviously occur under
intermediate rainfall (Diouf et al. (2012) [31])
or because of modulation by edaphic conditions,
grazing and anthropogenic pressures. Grazing may
lead savannas toward physiognomies and function-
ing looking less fire-prone, i.e. more ”arid-like”,
than expected from the only climate features as an
emergent consequence of dynamical amplification
of external forcing.

III. MAIN PUBLISHED MODELLING OPTIONS

The questions raised by observed or putative
dynamics within the savanna biome have trig-
gered an increasing interest in terms of modelling.
Pioneering works (Walker et al. (1981) [103],
Walker and Noy-Meir (1982) [104]) first used
systems of ordinary differential equations (ODE)
to address the particular case of arid, fire-immune
savannas in which excessive grazing fosters bush
encroachment (Skarpe (1990) [82]). This line of
modelling featured grass and woody biomasses as
state variables and aimed at explicitly depicting
their interactions in relation to soil moisture dy-
namics. As such, it became a paradigm for ’inter-
action models’ involving a limited resource, but

the central assumption of soil niche partitioning
between the two plant forms called Walter’s (1971)
hypothesis [105] has been ever since hotly debated
and is obviously not verified in all ecological
contexts where savannas, dry thickets or grasslands
are observable.

Another line of ODE-based modelling built on
the application to savannas of the initial concept of
asymmetric competition of (Tilman (1994) [94])
through a simple framework that allows consid-
ering both direct and disturbance-mediated plant
interactions. Tilman’s framework reinterpretation
(see Accatino et al. (2010) [4]), De Michele et
al. (2011) [28] used two states variables, namely
cover-fractions of grass (G) and tree (T ) assumed
exclusive and summing between zero and one. It
modelled their interacting dynamics in a system of
two ODE.



dT

dt
= cTT(1 −T) −δTT,

dG

dt
= cGG(1 −T −G) − cTTG−δGG,

(1)

where, T and G are dimensionless and denote
the fractions of sites occupied by tree and grass

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respectively. cT and cG are the colonisation rates
of tree and grass respectively. δT and δG represent
the mortality rates of tree and grass respectively.
In the sequel, we refer to system (1) as Tilman’s
model.

Logistic growth of the inferior competitor
(grasses plus herbs) is bounded and depressed
by the cover of the superior competitor (woody
plants) which logistic growth is not directly af-
fected by grasses (asymmetric competition). In
system (1) there is no fire-mediated retroaction
of G on T . This was however introduced by
subsequent authors (Van Langevelde et al. (2003)
[101], Beckage et al. (2009) [14], Accatino et al.
(2010) [4], Beckage et al. (2011) [13], De Michele
et al. (2011) [28]) via a linear function of G. Thus,
explicitly including the impact of fire on T in
Tilman’s model, the first equation of (1) becomes:

dT

dt
= cTT(1 −T) −δTT −δFfω(G)T, (2)

where, δF represents the trees vulnerability to fire,
f is the fire frequency (inversely proportional to
fire return time period) and ω(G) is a function
of grass biomass that represents the fire impact.
Through ω(G), there is thus indirect, fire-mediated
negative feed-back of grass cover onto tree cover
that counterbalance direct, tree-grass asymmetric
interactions.

A larger array of models (see Tables I and II)
took a leaf from the previous modelling frame-
work (system 1 and equation 2). Main sources
of variations between models were: (1) nature
of the equations and temporal treatment of fire
disturbance (time-continuous forcing, i.e. ODE vs.
time-discrete or impulsive occurrences); (2) nature
of the function expressing grass-fire feedback on
trees (linear vs. nonlinear); (3) integration of her-
bivory in addition to fire; (4) facultative explicit
treatment of water availability through models
with one (and sometimes more) additional state
variables expressing water resource in interaction
with vegetation variables. We will refer to such
models as ’ecohydrological’ (see Table I), among
which is system (3) proposed by Accatino et al.

(2010) [4] that features a first equation devoted to
the dynamics of a soil moisture variable (S)




dS

dt
=

p

w1
(1−S)−εS(1−T−G)−τTST

−τGSG,

dT

dt
=cTST(1−T)−δTT−δFfTω(G),

dG

dt
=cGSG(1−T−G)−cTSTG−δGG−fG,

(3)

where
p

w1
(per year) represents the rainfall rate

normalized with respect to root zone capacity, ε
(per year) is the evaporation, τT and τG (per year),
are water uptake parameters for tree and grass
respectively. cT , cG, δT , δG, δF and f are defined
as in system (1) and equation (2).

Note that setting ω(G) = 0 in the second equa-
tion of (3) makes Tilman’s model (1) analogous
to the system coupling the second and the third
equations of (3). Moreover, if the S variable is held
constant, the main difference between systems (1)
and (3) is that Accatino et al. (2010) [4] considered
ω(G) = G (i.e., impact of fire on trees as a
linear function of grass biomass) while in Tilman’s
model, this function is equal to zero (no impact of
fire).

Taking ω(G) as any increasing function of the
grass cover, provides a more general expression of
the fire impact on trees. Without loss of generality,
we referred to Holling type functions (Holling
(1959) [50], Augier et al. (2010) [8], Tewa et al.
(2013) [92], see equation (4) for generic ones).
The general form of ω(G) reads as

ω(G) =
Gθ

Gθ + αθ
, (4)

where, G in tons per hectare (t.ha−1) is grass
biomass, α is the value takes by G when fire
intensity is half its maximum, and the integer θ
determines the steepness of the sigmoid. Non-
linear response was retained by some other authors
(Scheiter and Higgins (2007) [77], Higgins et al.
(2010) [46], Staver et al. (2011) [83], Touboul et

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al. (2018) [96]). Accatino and De Michele (2016)
[3] also introduced a piecewise linear function
of grass biomass (indeed qualitatively mimicking
extreme non-linearity) in their non-equilibrium
model (NEM) as the probability of the occurrence
of fire.

Considering a nonlinear (sigmoidal) shape for
ω(G) allows for the existence of up to three tree-
grass coexistence (i.e. savanna) equilibria, while
two of them may be simultaneously stable (i.e.
bistability) and forest-savanna-grassland tristabil-
ity is reachable (Yatat et al. (2014) [112], (2018)
[109], Tchuinté Tamen et al. (2014) [91], (2017b)
[88]). Conversely, we proved that for linear ω(G)
functions, tristability is unreachable (Yatat et al.
(2018) [109], Tchuinté Tamen et al. (2018) [88]).
As pointed out in Yatat et al. (2018) [109], possible
tristability is in good agreement with results of
Favier et al. (2012) [36] obtained along a general
climatic transect over central African (latitude in
the range of 3 – 4◦ north). These results concerned
a very large range of woody cover (wc) variations
(from very low values approaching grassland phys-
iognomies to nearly 80% cover (i.e. forest) through
wc of 40%, i.e. savanna) which suggests grass-
land/savanna/forest tristability as at least plausible.
ODE models have been criticized on the basis that
they only predict abrupt transitions between two
alternative stable states (Accatino and De Michele
(2016) [3]) that are deemed unrealistic. However,
tristability of equilibria as well as bistability of
two savanna equilibria suggests that shifts from
one stable state to another may be less spectacular
than hypothesized from previous models and that
models may render more complex pathways of
vegetation changes (see Yatat et al. (2018) [109]
for further discussion).

In our earlier works (Yatat et al. (2014) [112],
(2017) [110], (2018) [109], Tchuinté Tamen et
al. (2014) [91], (2016) [89], (2017) [90]), we
chose to use above-ground biomasses instead of
covers as state variables in contrast to Accatino
et al. (2010) [4] and most of the models which
have been proposed on the subject (reinterpretation
of Tilman (1994) [94], Baudena et al. (2010)

[11], Staver et al.(2011) [83], De Michele et al.
(2011) [28], Synodinos et al. (2015) [85]) that
considered cover fractions. Modelling biomasses
help accounting from the fact that plant types
are not mutually exclusive at a given point in
space since field studies suggested that grass often
develop under tree crowns (Belsky et al. (1989)
[16], Belsky (1994) [15], Weltzin and Coughenour
(1990) [107], Abbadie et al. (2006) [1], Dohn et
al. (2012) [33], Moustakas et al. (2013) [70]).
Moreover, biomasses directly refer to the cycle
of carbon and can be assessed from radar remote
sensing in savanna ecosystems that correspond to
woody biomasses below the saturating level of the
backscatter L-band radar signal (Mermoz et al.
(2015) [65], Bouvet et al. (2018) [24]).

The minimal configuration of the published
models (Tables I & II) featured only two vege-
tation state variables (e.g., Van Langevelde et al.
(2003) [101], Tchuinté Tamen et al. (2014) [91],
(2016) [89], (2017) [90], (2017b) [88], Synodinos
et al. (2018) [86]) as in Tilman’s (1994) [94] initial
framework, but several models distinguished size
classes within the woody component of vegetation
(e.g. Favier et al. (2004) [37], Baudena et al.
(2010) [11], Staver et al. (2011) [83], Yatat et
al. (2014) [112], (2017) [110], Touboul et al.
(2018) [96]). Some models used more than two
size classes through matrix population models (e.g.
Accatino et al. (2013) [2], (2016) [5] and refer-
ences therein). Simpler models used only two size-
related variables in addition to grass and simply
account for the asymmetric nature of tree-grass
interactions as discussed previously (see also Sc-
holes (2003) [78], Yatat et al. (2014) [112], (2017)
[110]). Other models separate large trees, having
top buds above the flame zone and therefore facing
limited risks of topkill, from smaller trees and
shrubs which have high probability of having their
aerial systems destroyed (Beckage et al. (2009)
[14], Staver et al. (2011) [83], Yatat et al. (2014)
[112], (2017) [110]). This distinction stems from
field observations (Trollope (1984) [97], Trollope
and Trollope (1996) [99]) that evidence rapid
decline of percent topkill with tree height (see

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Table I: Comparison of several models of tree–grass dynamics with respect to some modelling options.
Walter’s hypothesis refers to the differences in root depth of herbaceous and woody vegetation in water
seeking while ecohydrological frameworks stand for models that consider additional state variables
expressing water resource in interaction with vegetation variables. From Tchuinté Tamen (2017) [87]
and Yatat (2018) [113]. The symbol ∗ means that we refer to system (1)

State Tree Herbivory Walter’s Ecohydro-

Authors variables state variables perturbation hypothesis logical

Cover Biomass All sizes lumped Size-structured applied frameworks

Walker et al. (1981) [103] X X X X

Tilman (1994)∗ [94] X X

Higgins et al. (2000) [45] X X X

Van Langevelde et al. (2003)[101] X X X X

D’Odorico et al. (2006) [32] X X

Beckage et al. (2009) [14] X X

Baudena et al. (2010) [11] X X

Higgins et al. (2010) [46] X X X

Accatino et al. (2010) [4] X X X

De Michele et al. (2011) [28] X X X X

Staver et al. (2011) [83] X X

Beckage et al. (2011) [13] X X

Yu and D’Odorico (2014) [115] X X X X

Touboul et al. (2018) [96] X X

Synodinos et al. (2018) [86] X X X X

Yatat et al. (2014) [112] X X X

Tchuinté Tamen et al. (2014)[91] X X X

Tchuinté Tamen et al. (2016)[89] X X X

Tchuinté Tamen et al. (2017)[90] X X X

Yatat et al. (2017) [110] X X X

Yatat et al. (2018) [109] X X X

Tchuinté Tamen et al. (2018)[88] X X X X

also Figure 3). ODE models featuring two woody
variables in addition to grass proved analytically
tractable (Beckage et al. (2009) [14], Staver et
al. (2011) [83], Yatat et al. (2014) [112], (2017)
[110]) as long as other complexities were not
introduced.

A strong objection against ODE models is that

fire is not a forcing that continuously removes a
small fraction of biomass through time as per the
previous ODE equation systems. Instead, fire ac-
tually suppresses a substantial fraction of biomass
at once through punctual outbreaks that shape
ecosystem aspect and immediate post-fire func-
tioning (Figure 4). This principle was implemented

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Table II: Summary of the characteristics of tree–grass interactions models with respect to fire modelling
options (continued Table I). From Tchuinté Tamen (2017) [87] and Yatat (2018) [113]. Some references
(unticked) do not model fire. The symbol ∗ means that we refer to system (1).

Fire perturbation Impact of Fire

Authors time- time- time- time- linear sigmoidal
continuous stochastic discrete impulsive forms

Walker et al. (1981) [103]

Tilman (1994)∗ [94]

Higgins et al. (2000) [45] X X

Van Langevelde et al. (2003) [101] X X

D’Odorico et al. (2006) [32] X

Beckage et al. (2009) [14] X X

Baudena et al. (2010) [11] X X

Higgins et al. (2010) [46] X X

Accatino et al. (2010) [4] X X

De Michele et al. (2011) [28] X X

Staver et al. (2011) [83] X X

Beckage et al. (2011) [13] X X

Yu and D’Odoricco (2014) [115] X X

Touboul et al. (2018) [96] X X X

Synodinos et al. (2018) [86] X X

Yatat et al. (2014) [112] X X

Tchuinté Tamen et al. (2014) [91] X X

Tchuinté Tamen et al. (2016) [89] X X

Tchuinté Tamen et al. (2017) [90] X X

Yatat et al. (2017) [110] X X

Yatat et al. (2018) [109] X X

Tchuinté Tamen et al. (2018) [88] X X X

via time-discrete recurrence equation models by
Scheiter and Higgins (2007) [77], Higgins (2010)
[46]. But another framework of impulsive differ-
ential equations (IDE) also proved relevant to gain
realism regarding nature and consequences of fire
while keeping a high level of analytical tractability
thanks to the ODE modelling of inter-fires vegeta-
tion dynamics (Yatat et al. (2017) [110], Tchuinté
Tamen et al. (2016) [89], (2017) [90]).

IV. REACHING SENSIBLE PREDICTIONS FROM
MINIMAL MODELS

A. A seminal ”big picture” at biogeographic scale

The model from Accatino et al. (2010) [4]
was pioneering and inspiring in that it first ven-
tures into generically predicting vegetation phys-
iognomies (in terms of percent covers of woody vs.
herbaceous plants) over the entire savanna biome.
The authors used bifurcation analysis (Accatino

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0

10

20

30

40

50

60

70

80

90

100

HEIGHT CLASSES − m

T
O

P
K

IL
L

 −
 %

0 − 0.50 0.51 − 1.00 1.01 − 1.50 1.51 − 2.00 2.01 − 2.50 2.51 − 3.00 3.01 − 3.50 3.51 − 4.00 4.01 − 4.50 4.51 − 5.00

4
6

8

13
16

29

48

70

87

92

(a)

0

10

20

30

40

50

60

70

80

90

100

HEIGHT − m

T
O

P
K

IL
L

 −
 %

0.5 1 2 3 4 5

9

27
29

43

49

69

75

89
91

9999

59

(b)

Figure 3: Illustration of the effect of height on the frequency of topkill of individual trees subjected to fires in the Kruger National Park
(panel (a)) and in the central highlands of Kenya (panel (b)). In the panel (b), continuous bars denote head fire while black-dash bars represent

back fire (reproduced from Trollope and Trollope (2010) [100], © Trollope and Trollope (2010) [100] ).

Figure 4: Aspects of two nearby savannas both located close to a forest boundary in the Sanagha Basin, Cameroon (Central Africa)
depending on recent fire occurrence (left) or not (right). Photos were taken the same day. On the left, note the general resprouting of both

the herbaceous stratum and topkilled woody plants. Photos: Pierre Couteron, March 2018.

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et al. (2010) [4], Tchuinté Tamen et al. (2018)
[88], see also Figure 7) based on two important
parameters of strong intuitive meaning, namely
mean total annual rainfall and fire frequency. They
thereby achieved delineation of domains in which
main physiognomies (i.e., grassland, savanna, for-
est) can be expected as stable. Bistability situations
(forest-grassland and forest-savanna) were also
highlighted for sufficiently high fire frequencies.
For low fire frequencies, a sensible gradient of
increasing woody cover with increasing annual
precipitation was found. But one may note that
only dense woodlands (i.e. still two-phase vegeta-
tion) were obtained even for the highest rainfall
range, while forest stricto -sensu (mono-phase,
with no tall, light-demanding savanna grasses in
the understory) is widely observed for the cor-
responding ranges of precipitations. Moreover,
transitions between vegetation types in relation
to fire frequencies proved tricky. Indeed, in the
high rainfall range, increasing fire frequency leads
from the aforementioned woodlands to forest.
In the intermediate rainfall range, increasing fire
frequency leads from savanna mono-stability to
forest-savanna bistability (see Figure 7). Analo-
gously, for fairly low rainfall, grassland stability
shifts to grassland-forest bistability. Hence, all
over the rainfall gradient it looks as if sufficient
frequency of fire were a necessary condition to
reach forest (bi)-stability. This is contradicted by
empirical knowledge according to which frequent
fires are known to jeopardize or at least delay
woody biomass build-up, but never favour it (Bond
et al. (2005) [22], Archibald et al. (2009) [7], Bond
and Parr (2010) [21]).

Where did this critical problem come from?
Most of subsequent papers barely evoked the ques-
tion. A large share of them investigated different
modelling options, often more complex and/or
less tractable; or they assumed particular biogeo-
graphic conditions. In a further contribution, Ac-
catino and De Michele (2016) [3] argued about in-
trinsic limitations of ODE-based modelling. They
also put forward that there is no evidence accord-
ing to which observed vegetation physiognomies

may be close to a stable equilibrium point. It is
in fact undisputable that climate is likely to vary
through time, and there is no guaranty that woody
vegetation can track such variation with enough
celerity. They also underline as questionable the
assumption according to which the parameter f
of fire frequency should be treated as a constant
forcing, independently of vegetation characteris-
tics. All these arguments brought them to propose
a ’non-equilibrium model’, based on stochastic
difference equations, as alternative to the time-
continuous model of Accatino et al. (2010) [4]
referred to as ’equilibrium-model’ (EM). In their
non-equilibrium (NEM) model, fire occurrence is
a stochastic event all the more likely to occur in a
given dry season that ignitable dry grass biomass
abundantly built-up in the foregone rainy seasons.

Accatino and De Michele (2016) [3] compared
the predictions of their (EM) vs. (NEM) models.
They argued that separation of fire-immune vs.
fire-prone savannas as an indirect consequence of
rainfall is an emergent property with their NEM
while it is artificially induced by the choice of the
f parameter with EM. They also pointed out that
when considering high rainfall, the NEM is able to
reproduce the ”bimodality” of woody cover extent
observed in remote sensing studies. But in fact,
their NEM was not a straightforward time-discrete
analogue of the ODE based EM of Accatino et
al. (2010) [4], since it features several novelties.
Therefore, the differences they reported between
EM and NEM are not a simple consequence of
time-continuous fire forcing vs. time-discrete fire
occurrences. In fact, several aspects altogether
contribute to the more satisfactory results obtained
with the NEM by Accatino and De Michele
(2016) [3]. We will subsequently illustrate the fact
that predictions that are qualitatively satisfactory
can be obtained by directly improving the ODE
based ’EM’ framework, notably regarding the fire-
mediated feedback of grass onto tree dynamics.

B. Fire frequency, grass biomass and fire impact

The way in which fire impact is modelled in the
previous equation systems (1 and 3) is obviously
a crucial question. As stated by Scholes (2003)

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0 2 4 6 8 10
0

2

4

6

8

10

Grass biomass (t.ha−1)

F
ir

e
 i
m

p
a

c
t 

(w
(G

))

 

 

Holling type I

(a)

0 2 4 6 8 10
0

0.2

0.4

0.6

0.8

1

Grass biomass (t.ha−1)

F
ir

e
 i
m

p
a

c
t 

(w
(G

))

 

 

Holling type II

Holling type III

(b)

Figure 5: Graphical representation of the function shapes w(G). The fire impact term in the ODEs is given by λfT fω(G).

Table III: Functions involving ω(G).

Functions Models (ODE) Coexistence

equilibria

ω(G) = 0 Tilman (1994) [94]
One savanna

monostability

ω(G) = G
Accatino et al. (2010) [4] 2 savannas

Van Langevelde et al. (2003) [101] bistability

Holling
type II

Tchuinté Tamen et al. (2014) [91]
2 savannas

bistability

Holling
type III

Staver et al. (2011) [83] 3 savannas

Tchuinté Tamen et al. (2014) [91] bistability

Yatat et al. (2014) [112] 3 savannas
bistability &
tristabilityTchuinté Tamen et al. (2018) [88]

[78], modelling savanna dynamics in fire-prone
contexts actually requires introducing an ”equation
predicting the effect of grass biomass, via fire
intensity, on tree biomass”.

In fact, non-linearity in ω(G) may be justified
on various, non-exclusive grounds, since what is
important is to properly model, as a whole, the
causal chain that leads from grass abundance and
ignition regime to woody biomass suppression.
As steps in this chain we may identify: (i) fire
frequency to be seen as an external forcing upon

the tree-grass system (think about a targeted fire
regime in a managed area such as a ranch or a
protected area); (ii) actual yearly fire probability
(or frequency) of occurrence in any arbitrary small
piece of land once (i) has been set; (iii) fire
potential impact (intensity and flame height) on
woody vegetation; (iv) fire actual impact that also
depends on features intrinsic to woody vegetation
(see below section IV-C).

Fire intensity which is strictly speaking a quan-
tity of energy released (Bond and Keeley (2005)

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Figure 6: Example of spatial heterogeneity of fire propagation in an altitude mosaic of forests and low biomass grasslands in Cameroon.
Here local community hunters tend to set fire every year at landscape scale (i.e. f = 1) as to flush small game from spots of dense grass

cover. But all the area does not burn every year because fire actually do not propagate everywhere (From P. Couteron, February 2017, Mount

Cameroon National Park).

[18]) appears empirically as a fairly linear function
of grass biomass. But impact on trees also depends
on flame height which is reported as increasing
exponentially with observed quantity of dry grass
biomass (Scheiter and Higgins (2007) [77], Staver
et al. (2011) [83], Synodinos et al. (2018) [86]),
though one may suppose some levelling off for
maximal grass biomass (and height) values. In
earlier works (Yatat et al. (2014) [112], (2017)
[110], (2018) [109], Tchuinté Tamen et al. (2014)
[91], (2016)[89], (2017) [90]), we systematically
assumed fire impact on woody vegetation as a
non-linear, increasing bounded function of grass
biomass (Yatat et al. (2014) [112], (2017) [110],
(2018) [109], Tchuinté Tamen et al. (2014) [91],
(2016) [89], (2017) [90], (2018) [88]), w(G).

In our modelling, the f parameter was kept as
constant multiplier of w(G), but we interpret it
as a man-induced ”targeted” fire frequency (as for
instance in a fire management plan), which will not
translate into actual frequency of fires of notable
impact as long as grass biomass is not of sufficient
quantity. With this interpretation, the actual fire
regime may substantially differ from the targeted
one, as frequently observed in the field. And,
whatever f values, any hypothetical piece of land

will actually be fire-prone only if other forcing fac-
tors (climate, herbivory, etc.) allow for sufficient
grass biomass. Most previous modelling papers
including those from our group did not elaborate
much regarding the successive steps involved in
the grass-fire feedback. Distinction between (i)
and (ii) may appear subtle and to our knowl-
edge was never emphasized before. It directly re-
sults from space-implicit savanna modelling. Fire
regime, which is nowadays overwhelmingly man-
induced (Govender et al (2006) [44], Archibald
(2009) [7]) is a forcing at landscape scale since
people do not go and set fire in every piece of
land. They instead count on fire propagation that
depends on abundance and spatial evenness of dry
grass. In presence of low and unevenly distributed
grassy fuel, fire will barely propagate leaving a
large share of the area unburnt. This makes the
difference between steps (i) and (ii), as frequently
observed in the field (see Diouf et al. (2012) [31];
Figure 6). The response of percent area burnt to
grass abundance is likely to be sharply nonlinear,
as suggested by the impressive results reported by
McNaughton (1992) [61] at the scale of the entire
Serengeti complex in Tanzania. In this remarkable
study, local fire frequency dwindled over a decade

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following grass biomass suppression by soaring
herbivore populations, while the ignition regime by
people dwelling around the park likely remained
more or less the same. In fact, since we are here
dealing with mean-field models the ω(G) function
is also due to embody the difficult spreading of fire
in presence of fuel of overall low quantity keeping
in mind natural spatial variability of grass biomass
(Figure 6). Non-linearity of ω(G) seems therefore
a necessary feature for adequately capturing the
fire-mediated grass-tree feedback.

C. Tree survival

To be relevant, the most parsimonious models
featuring just grass and tree state variables must
overcome the limitation pointed out in sub-section
IV-A for the precursor model of Accatino et al.
(2010) [4]. All things being equal, any increase
in fire frequency should never increase the woody
component of vegetation. Fire, if any, is expected
to be of no substantial consequence over the driest
stretch of the rainfall gradient while for the moister
part, it is widely observed that extending the
average time between successive fires (decreas-
ing frequency) favours the building up of woody
vegetation. Accounting for that proved to be a
challenge for minimal two-variable models that
do not distinguish between fire sensitive and fire
insensitive woody fractions. Non-linearity of the
ω(G) function, though important proved not suf-
ficient to overcome this problem. Tchuinté Tamen
et al. (2017) [90] further introduced a second
non-linear decreasing function, which directly ex-
presses that high woody biomasses, corresponding
to tall trees proportionately experience far less fire-
related losses than low woody biomasses relating
to seedlings, saplings and shrubs (see Figure 2).
We hence proposed the following function to de-
note the fire-induced tree/shrub mortality:

ϑ(T) = λminfT + (λ
max
fT −λ

min
fT )e

−pT , (5)

where λminfT (in yr
−1) is minimal loss of tree

biomass due to fire in systems with a very large
tree biomass, while λmaxfT (in yr

−1) is maximal loss
of tree/shrub biomass due to fire in open vegetation

(e.g. for an isolated small woody individual having
its crown within the flame zone), p (in t−1.ha) is
proportional to the inverse of biomass suffering an
intermediate level of mortality.

This general form was suggested by experimen-
tal observations showing dwindling rate of topkill
with increasing tree height, since tall trees are
likely to have their upper parts above the flame
zone, even for high grass biomass (Trollope and
Trollope (2010) [100]).

Notice that taking into account a nonlinear and
decreasing function of tree biomass is a way to by-
pass introducing size classes as to keep the model
minimal and retain mathematical tractability (see
inspiring examples in Meron et al. (2004) [67],
Lefever et al. (2009) [55]). The addition of the
ϑ(T) function was indeed decisive in ensuring that
a two-equation ODE system provides predictions
that qualitatively agree with the general ecological
knowledge about the role of fire return period.
On this basis, Tchuinté Tamen et al. (2017) [90]
designed a model that also implemented punctual
fire events through impulsive differential equa-
tions. But predictions of the ODE model itself
were already satisfactory.

D. Relating to water resource

Several savanna models have explicitly mod-
elled soil water resources, via a dedicated equation
as in system (3). But the soil moisture dynamics
is very rapid compared to change in vegetation.
Soil moisture variations linked to a given rainfall
event are damped within a few days (Barbier et
al. (2008) [10]), while vegetation growth proceeds
over months for grasses and even years for woody
plants. It therefore makes sense to consider veg-
etation dynamics in relation to a level of soil
water resource that is approximately constant for
a given level of total annual rainfall or, ideally,
water deficit (rainfall minus evapotranspiration).
Martinez-Garcia et al. (2014) [59] reached sim-
ilar conclusions for a partial differential equa-
tion model of non-local plant-plant interactions
for water. This justifies letting parameters in the
vegetation dynamics equations directly depend on

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climatic parameters. This principle sustains the
model expressed through the following system:




dG

dt
= rG(W)G

(
1 −

G

KG(W)

)
−δGG

−ηTGTG−λfGfG,

dT

dt
= rT (W)T

(
1 −

T

KT (W)

)
−δTT

−fϑ(T)ω(G)T,

G(0) =G0,T(0) = T0,

(6)

where,
• G and T are grass and tree biomasses respec-

tively;
• W is the mean annual precipitation (in mil-

limeters per year, mm.yr−1);

• rG(W) =
γGW

bG + W
and rT (W) =

γTW

bT + W
are annual productions of grass and tree
biomasses respectively, where γG and γT
(in yr−1) express maximal growths of grass
and tree biomasses respectively, half satura-
tions bG and bT (in mm.yr−1) determine how
quickly growth increase with water availabil-
ity;

• KG(W) =
cG

1 + dGe−aGW
, and KT (W) =

cT
1 + dTe−aT W

are carrying capacities of

grass and tree respectively, where cG and
cT (in t.ha−1) denote the maximum values
of the grass and tree biomasses, aG and aT
(mm−1yr) control the steepness of the curves
of KG and KT respectively, and dG and dT
control the location of their inflection points;

• δG and δT respectively express the rates of
grass and tree biomasses loss by herbivores
(grazing and/or browsing) or by human ac-
tion;

• ηTG denotes the asymmetric influence of trees
on grass for light (shading) and resources
(water, nutrients) which relate to competitive
or facilitative influences;

• λfG is the specific loss of grass biomass due
to fire;

• f = 1
τ

is the fire frequency, where τ is the
fire return period.

Submitting model (6) to bifurcation analysis
provides Figure 7-(b) that is to be compared to
Figure 7-(a) from Accatino et al. (2010) [4], which
has been reobtained using Matcont (see Govaerts
(2000) [42], Dhooge et al. (2003) [29], Govaerts
et al. (2007) [43] and references therein). Both
Figures 7-(a) and 7-(b) are sensible regarding low
fire frequencies for which increasing MAR leads
to a sequence of physiognomies of increasing
woody biomass (i.e., grassland, savanna, forest).
But in Figure 7-(b), the improvement resulting
from introducing ω(G) and ϑ(T) is apparent when
increasing fire frequency (f) at different levels
of the rainfall gradient. For high MAR values,
the expected physiognomy shifts from monostable
forest to forest-grassland bistability. Indeed, in
presence of high MAR, it is known that grasslands
are due to be encroached by forest under fire
prevention or even just because of decreasing
fire frequencies (Jeffery et al. (2014) [51]). Our
model accords with field observations in that a
high fire frequency is indeed a necessary condition
to perpetuate the grassland (or savannas of low
woody biomass) physiognomies. Moreover, large-
scale observations of bimodality between high and
very low woody cover situations (Hirota et al.
(2011) [47], Favier et al. (2012) [36]) can be
accounted for by the forest-grassland bistability,
though the converse is not necessarily true. In
fact bimodality may stem from either transient
situations or topographical heterogeneity and does
not automatically implies bistability. For low to
intermediate MAR values, say 600 − 1000 mm,
fire is known to be less pervasive, though field
observations or experiment results depict woody
vegetation thickening in case of fire frequency
decrease (Brookman-Amissah (1980) [25]). The
model is able to render such thickening as a
shift from the grassland to the savanna stability
domain. The model also predicts forest-savanna or
savanna-grassland bistability, and even tristability
thereof for restricted domains in the MAR-fire
frequency plane that were situated around 1000

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mm of MAR. This value is of course dependent
on parameter values used for computations un-
derlying figure 7-(b). Refined calibrations relating
to a specific regional context may displace the
thresholds. Notably, the parameter expressing the
influence of woody biomass on grasses proved to
be influential on the thresholds between vegetation
states (Tchuinté Tamen et al. (2018) [88]).

E. Impulsive time-periodic occurrences of fire
events

In previous models, the traditional time-
continuous fire forcing formalism is often used.
However, it is questionable to model fire as a
permanent forcing that continuously removes frac-
tions of fire sensitive biomass all over the year.
Indeed, several months and even years can pass
between two successive fires, such that fire may
be considered as an instantaneous perturbation of
the savanna ecosystem (Yatat et al. (2017) [110],
Tchuinté Tamen (2016) [89], (2017) [90]; see also
Table IV, page 18). Several recent papers have
proposed to model fires as stochastic events while
keeping the continuous-time differential equation
framework (Baudena et al. (2010) [11], Beckage
et al. (2011) [13], Klimasara and Tyran-Kamińska
(2018) [54], Synodinos et al. (2018) [86]) or using
a time-discrete model (Higgins et al. (2000) [45],
Accatino and De Michele (2013) [2], Accatino et
al. (2016) [5]). However, a drawback of most of
the aforementioned recent time-discrete stochastic
models (Higgins et al. (2000) [45], Baudena et al.
(2010) [11], Beckage et al. (2011) [13]) is that they
barely lend themselves to analytical approaches.

Based on Table IV, page 18, we further consider
in our group (Yatat et al. (2017) [110], (2018)
[109], Yatat and Dumont (2018) [111], Tchuinté
Tamen (2016) [89], (2017) [90], (2018) [88]) im-
pulsive time-periodic fire events which is an ap-
proximation that keeps the potential of analytical
investigation as large as possible while modelling
discrete fires. An impulsive differential equations
system can be used to express fire through impul-

sive periodic occurrences (e.g. system (7), below):




dG

dt
= rG(W)G

(
1−

G

KG(W)

)
−δGG

−ηTGTG,

dT

dt
= rT (W)T

(
1−

T

KT (W)

)
−δTT,

t 6= nτ,

∆G(nτ) = −λfGG(nτ),

∆T(nτ) = −ϑ(T(nτ))ω(G(nτ))T(nτ),

n = 1, 2, ...,Nf,

(7)

where,
• for π ∈{G,T}, ∆π(nτ) = π(nτ+) −π(nτ)

and π(nτ+) = lim
θ→0+

π(nτ + θ);

• τ = 1
f

is the period between two consecutive
fires;

• Nf is a countable number of fire occurrences;
• nτ, n = 1, 2, ...,Nf , are called moments of

impulsive effects of fire, and satisfy 0 ≤ τ <
2τ < ... < Nfτ.

Properties of models (6) and (7) have been
analysed in Tchuinté Tamen et al. (2018) [88].
Below we provide some numerical simulations as
to illustrate the bifurcations between the stable
domains delineated in Figure 7-b as a consequence
of increasing the fire frequency for two particular
values of MAR, i.e. W = 946 mm.y−1 and W =
1003 mm.y−1. For each of the two MAR values,
we compare the consequences of increasing f
with the ODE (system (6)) and the IDE (system
(7)) frameworks, by comparing Figure 8 against
Figure 9 and Figure 10 against Figure 11. For
the lower MAR situation, ODE and IDE frame-
works qualitatively agree and show the bifurcation
from monostable savanna to monostable grassland
through an intermediate bistable situation (Figure
8-b; Figure 9-b). For the higher MAR case, both
frameworks also show the transition from monos-
table forest to forest-grassland bistability through
tristability involving savanna. Qualitatively, the

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(a)

(b)

Figure 7: Bifurcation diagrams using Matcont. (a) Accatino et al. (2010) [4] model re-implementation. (b) Implementation of system (6).
Single red, green and black symbols (rectangles in panel (a), dots in panel (b)) stand for grassland, forest and desert respectively. Twinned red

and green symbols stand for savanna (coexistence state). Size of the symbols qualitatively denote grass and tree cover fractions in panel (a)

and biomass levels in panel (b). Parameters used to compute Figure 7-(a) are from Accatino et al (2010) [4] (see also Table V in appendix).

The parameter values used in 7-(b) are from Tchuinté Tamen et al. (2018) [88] (see also Table VI in appendix). (Color in the online version).

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Table IV: Average fire period (τ, in yr) ranges of values found in literature with respect of the mean
annual rainfall (MAR). Ranges of values of MAR are from Yatat et al. (2017) [110] and Tchuinté
Tamen et al. (2016) [89]

Ranges References

Low MAR: 5 – 50 Frost and Robertson (1985) [40]

MAR ≤ 650 mm.yr−1 4 – 8 Trollope (1984) [97]

Intermediate MAR: 3 – 5 February et al. (2013) [38]

650 mm.yr−1 ≤ MAR ≤ 1100 mm.yr−1 5 – 7 Van Wilgen et al. (2004) [102]

High MAR: 0.5 – 3 Jeffery et al. (2014) [51]

MAR ≥ 1100 mm.yr−1 0.5 – 2 Bond and Keeley (2005) [18]

Accatino et al. (2010) [4]

1 – 5 Abbadie et al. (2006) [1]

1 Menaut and Cesar (1979) [63]

Gignoux et al. (2009) [41]

predictions of the two frameworks agree about the
predicted sequence of vegetation physiognomies
when increasing the fire frequency. However, the
IDE model systematically predicted bifurcations
for lower values of f than for the ODE. This
indicates that shifting to the conceptually more
satisfactory IDE framework will introduce speci-
ficities in forthcoming stages concerning refined
calibration and comparison with real-world obser-
vations.

V. DISCUSSION AND PROSPECTS

In the present paper we emphasize that ecolo-
gists did probably not yet exploit all the potential
of simple ODE systems for modelling vegetation
dynamics in the savanna biome to which most sea-
sonal tropical ecosystems pertain. We showed that
reasonable, non-trivial predictions can be obtained
in reference to hypothetical situations directly re-
lating to rainfall and fire frequency gradients. Ap-
plication to specific contexts and locations would
request refined calibration for the parameters of
the generic minimalistic model. But it may also
invite to better address specific processes deemed
influential in a particular situation under study.
This would mean complexifying the model to
match a specific piece of reality. Though this is

actually a natural and sensible trend in science,
parsimony is an opposing principle that tells us to
keep complexification under control. A meaning-
ful, balanced modelling approach should restrict to
what we strictly need to account for a well-defined
array of empirical facts in a particular situation.

On the empirical side, the ongoing development
of remote-sensing techniques and derived products
is providing avenues to better depict the spatiotem-
poral variation of environmental factors, such as
rainfall (e.g. via CHELSA, Karger et al. (2017)
[52]), topography (via the SRTM, Farr et al. (2007)
[35], published at increased spatial resolution by
NASA in 2013) or fires (http://modis-fire.umd.edu;
e.g. Archibald et al. (2009) [7], Diouf et al. (2012)
[31]). This also applies to the monitoring of some
vegetation variables, though disentangling effects
on most remotely-sensed signals from grasses vs.
trees in mixed savanna physiognomies is still
challenging. Improving and diversifying sources
of remote sensing information will obviously help
sorting out relevant predictions from unrealistic
ones and refine the benchmarking of models.
But most of the parameters expressing vegetation
dynamics will remain out of reach of remotely-
sensed assessment and will remain dependent on
field information. An increased effort of field data

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I.V. Yatat Djeumen, A. Tchuinté Tamen, Y. Dumont, P. Couteron, A tribute to the use of minimalistic ...

0 5 10 15 20

Grass biomass (t/ha)

0

10

20

30

40

50

60

70

80

90

100

T
re

e
 b

io
m

a
s

s
 (

t/
h

a
)

(W=946, f=0.4)

(a)

0 5 10 15 20

Grass biomass (t/ha)

0

10

20

30

40

50

60

70

80

90

100

T
re

e
 b

io
m

a
s

s
 (

t/
h

a
)

(W=946, f=0.51)

(b)

0 5 10 15 20

Grass biomass (t/ha)

0

10

20

30

40

50

60

70

80

90

100

T
re

e
 b

io
m

a
s

s
 (

t/
h

a
)

(W=946, f=0.7)

(c)

Figure 8: Illustration of a bifurcation due to f with the ODE model of system (6) for a constant MAR value of W=946 mm.y−1. When
the fire frequency f increases, the system shifts from a savanna monostable (see panel (a)) to a grassland monostable (see panel (c)) passing

through a bistability between savanna and grassland (see panel (b)).

(a) (b) (c)

Figure 9: Illustration of a bifurcation due to fire frequency f with the impulsive model (7) for a constant MAR value of W=946 mm.y−1.
This figure is based on same parameters values as Figure 8 but with the impulsive IDE framework savanna-grassland bistability is already

observable for f = 0.4 (panel (a)) and give way to monostable grassland for f=0.51 (panel (b)). These values are to be compared to f = 0.25

and f = 0.7, respectively when using the ODE model (system (6)).

0 5 10 15 20
Grass biomass (t/ha)

0

20

40

60

80

100

120

140

160

180

200

T
re

e
 b

io
m

a
s
s
 (

t/
h

a
)

(W=1008, f=0.4)

(a)

0 5 10 15 20

Grass biomass (t/ha)

0

20

40

60

80

100

120

140

160

180

200

T
re

e
 b

io
m

a
s

s
 (

t/
h

a
)

(W=1008, f=0.5)

(b)

0 5 10 15 20

Grass biomass (t/ha)

0

20

40

60

80

100

120

140

160

180

200

T
re

e
 b

io
m

a
s

s
 (

t/
h

a
)

(W=1008, f=0.56)

(c)

Figure 10: Numerical simulations with the ODE model (6) illustrating a bifurcation induced by increasing the fire frequency (f) in
presence of a constant MAR (W) value of 1008 mmy−1. Vegetation shifts from monostable forest (left) to savanna-forest bistability (center)

to grassland-savanna-forest tristability (right).

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0 5 10 15 20

Grass biomass (t/ha)

0

20

40

60

80

100

120

140

160

180

200

T
re

e
 b

io
m

a
s
s
 (

t/
h

a
)

(W=1008, f=0.3)

(a) (b)

(c) (d)

Figure 11: Numerical simulations with the IDE model (7) illustrating a bifurcation induced by increasing the fire frequency (f) in presence
of a constant MAR (W) value of 1008 mm−1. This figure is analogous to Figure 10 obtained with the ODE model. Note that with the ODE

model, there is still tristability for f = 0.56 while forest is still monostable for f = 0.4.

collection is obviously desirable but insufficient
means for research in most tropical countries is
enduring reality. That strong data limitation is alas
probably here to stay finally pleads for parsimony
in modelling. It also underlines the importance for
modelling to be sufficiently convincing and acces-
sible to ecologists as to guide data acquisition and
orient scarce resources towards assessing param-
eters proven as the most influential by sensibility
analyses.

On the modelling side, within the class of mod-
els that distinguished size classes for the woody
component of vegetation, some used more than
two size classes through matrix population models
(e.g. Accatino et al. (2013) [2], (2016) [5]). How-
ever, such models remain generally simulation-
based and usually involve a fairly large number of
parameters. Thus, it is not easy/possible to assess
how model parameter variations may influence the
model outcomes (Yatat et al. (2018) [109]). ODE

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and IDE models featuring two woody variables,
in addition to grass, proved analytically tractable
(Beckage et al. (2009) [14], Staver et al. (2011)
[83], Yatat et al. (2014) [112], (2017) [110]) as
long as other complexities were not introduced.
For example, Yatat et al. (2014) [112] (resp. Yatat
et al. (2017) [110]) studied a ODE-like (resp.
IDE-like) tree-grass interactions model where in
addition to grass they considered two classes of
woody plants: fire-sensitive like seedlings and
fire insensitive. But, based on recent publications
of our group, we found that even with models
that feature only one state variable for woody
component, meaningful results are obtained and,
to some extent, are qualitatively similar to those
obtained with models that used two size-related
variables for woody component (Tchuinté Tamen
et al. (2014) [91], (2016) [89], (2017) [90], (2018)
[88], Yatat et al. (2018) [109]).

The IDE framework is an obvious improvement
that expresses a reasonable trade-off between in-
creased realism and decreased analytical tractabil-
ity. Within the framework of IDE, a further step
in that direction could be considering patterns of
fire occurrences featuring stochastic components
instead of the deterministic periodic regime we
used (Yatat et al. (2017) [110], (2018) [109],
Tchuinté Tamen et al. (2016) [89], (2017) [90],
(2018) [88]). But one may note here that peri-
odicity of fire outbreaks is not that unrealistic
in the context of subequatorial humid savannas
for which fires can only occur at the end of dry
seasons which are of short duration. Here the
annual climatic cycle strongly defines the temporal
window for fires while in less humid savannas
fire is simultaneously less frequent and liable to
occur all over extensive dry seasons (Diouf et al.
(2012) [31]). We believe that while mathematical
tractability or theoretical study of a model is not
an absolute requirement or is not always possible,
it remains nevertheless desirable at least for two
reasons. First, it can appear as a kind of guarantee
that numerical simulations displayed by the model
are not the result of some numerical artifacts. In
other words, the choice or the construction of

a suitable algorithm to solve a given (complex)
model strongly relies on its qualitative study or,
when this study is not possible, on the analysis
of some sub-models, that can be mathematically
tractable. Nowadays, there are more and more
works that point out some spurious behaviors that
may appear when using some ’classical’ schemes
for model simulations (see for example Anguelov
et al. (2012) [6]). Second, any theoretical analysis
may provide useful informations about the role of
some particular parameters in the dynamics of the
system.

We have here focused on spatially-implicit mod-
els because we believe that such models have
still important insights to provide and also be-
cause spatially-explicit models are far more de-
manding in terms of parametrization and more
difficult to study theoretically. Substantial efforts
to design and run spatial models of savannas
have however been made during the last decade
(Borgogno et al. (2009) [23]). Most of them re-
lied on individual-based models such as cellular
automata. At this step of the discussion, it seems
meaningful to point out that there are some authors
who pleaded for a mutualistic or complementary
relationship between mathematical tractability-
based and simulation-based formalisms (Omohun-
dro (1984) [73], Wolfram (1985) [108], Weimar
(1997) [106], Narbel (2006) [71], Dietrich et al.
(2014) [30], Dumont et al. (2018) [34]) and we
also agree with that. As an illustration, it may be
more difficult to achieve a very deep theoretical
analysis of a partial differential equations model
when taking into account spatial heterogeneity
while it seems more easy to handle it when using
for example an individual-based model or cellular
automaton formalism. Independently, partial dif-
ferential equations (PDE) have also widely been
used to account for the genesis of vegetation
regular spatial patterns (bare soil vs. dense shrubby
cover) in the particular context of arid savannas
(Lefever and Lejeune (1997) [56], Klausmeier
(1999) [53], Sherratt (2005) [81], Borgogno et al.
(2009) [23], Lefever et al. (2009) [55], Meron
(2011) [66], Lefever and Turner (2012) [57]). In

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this line, there was a recent attempt by Yatat et al.
(2018) [109] to account for the dynamics of forest-
savanna boundaries by introducing local diffusion
operators for herbaceous and woody biomasses in
a parsimonious space-implicit model, with time-
continuous fire events forcing, closely related to
the one of Tchuinté Tamen et al. (2017) [90].
The analysis of this model shows that there ex-
ists monostable or bistable travelling solutions,
related to the boundary movements in the forest-
grassland mosaic. And the authors showed that
depending on fire-return time as well as difference
in diffusion coefficients of woody and herbaceous
vegetation, fire events are able to greatly slow
down or even stop the progression of forest in the
humid part of the savanna biome. This kind of
results, obtained from theoretical analysis, are of
great interest for practical needs or management
policies. However, as an improvement of Yatat
et al. (2018) [109], there are also some ongoing
works that aim to deal with existence of travelling
waves for system of Impulsive Partial Differential
Equations (IPDE) that model savanna dynamics
(e.g. Yatat and Dumont (2018) [111]). This type of
modelling is of great interest considering that the
forest-savanna ecotone is the most widespread in
the tropics and that forests have been encroaching
during the last decades in many humid savannas
of West and Central Africa, and to a lesser ex-
tent in other regions of the world (Oliveras and
Malhi (2016) [72]). On the other hand, forest
encroachment, which is of great consequence for
the global carbon balance of terrestrial ecosystems
proved heterogeneous in space and time (Oliveras
and Malhi (2016) [72]). Hence, modelling should
help better understand the hierarchy of processes
and forces accounting for such heterogeneity. This
makes spatially-explicit modelling desirable. But,
interpretation and calibration of local, diffusion
operators as used in Yatat et al. (2018) [109]
cannot rely on much empirical knowledge in plant
ecology. And one may note that this also applies
to colonization rates between adjacent cells that
are central to cellular automata models. In the
case of PDEs, modelling non-local plant-plant

interaction processes (e.g. through interaction ker-
nels) is mathematically more challenging, though
attempts in Martinez-Garcia et al. (2014) [59]
and in Lefever et al. (2009) [55] or Lefever and
Turner (2012) [57] nevertheless provide sources of
inspiration.

A recent line of criticism questions the rele-
vance of reasoning in reference to equilibrium
states. There is indeed no compelling evidence that
physiognomies presently observable correspond or
even are close to predictable equilibria determined
by current environmental conditions. In fact, pa-
rameters reflecting environmental variables, no-
tably climate are due to fluctuate or even change
through time. And vegetation, especially its woody
component may be unable to keep pace with
such variations and rather track them with delay
thereby remaining distant from any equilibrium
state. Long-lasting consequences of past climate
periods probably still mark present vegetation.
For instance, in Central Africa, concomitant to,
drier period occurred some centuries ago (Europe’s
”Little Ice Age”, 500-200 years BP), which proba-
bly provoked forest cover recession and fragmen-
tation (Oslisly et al. (2013) [74]). The trend of
widespread forest boundaries displacement within
savannas, as observed during the last decades may
be a delayed recovery after this past drier period
that is progressing at slow and unequal pace owing
to the counteracting influence of fires. Increased
CO2 availability that favours C3 woody plants
against C4 savanna grasses (Bond and Midgley
(2000) [19]) may also reinforce this trend.

VI. CONCLUSION

Minimal savanna models using ODE systems
have been criticized from different standpoints.
The first one was the poor realism of the overall
picture made by the predicted stable equilibria.
The present paper shows that some unrealistic
predictions are not a direct drawback of the
ODE framework, but rather derive from inadequate
modelling of the crucial fire-mediated negative
feedback of grassy biomass onto woody vegeta-
tion. Using nonlinear functions such as ω(G) and
ϑ(T) (as in equations 5) is not only justified by the

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nature of the mechanisms at play, but also proved
sufficient to get a meaningful ”big picture” of
vegetation physiognomies predicted as stable for
varying mean annual rainfall and fire frequency (as
in Figure 7-(b)). Another argument against ODE
models is that they predict too contrasted stable
states meaning that shifts between them would ap-
pear as more catastrophic than actually observed.
But some strong contrasts such as landscape mo-
saics of forest and grassland are indeed observable
in the field (see Figure 1). Moreover equilibria that
have attraction domains ”adjacent” in Figure 7 do
not systematically show contrasted biomass values.
Transitions may actually be progressive in terms
of state variables. Indeed, at low fire frequency, the
transition from savanna to forest along the MAR
gradient corresponds to a continuous increase of
tree biomass with concomitant decrease of grass
biomass. The same applies to the transition from
savanna to grassland via increased fire frequency
(see Figures 8 and 9). Here, the shape of the non-
linear functions embodying the grass-fire feedback
matters as shown in previous works (Yatat et al.
(2014) [112], Tchuinté Tamen et al. (2014) [91]).
Strongly nonlinear shapes (e.g. Holling functions
of higher order see Table III) allow for multiple
coexistence equilibria (i.e. multiple savanna phys-
iognomies) of different woody biomass values,
which may be seen as ”stepping stones” between
grassland and forest. Thus, the ODE framework
does not automatically imply abrupt changes be-
tween equilibria of very contrasted biomass values.

On the other hand, it is undisputable that mod-
elling fire as an external forcing continuously
suppressing small amount of biomass through time
is not satisfactory. Models based on punctual fires
impacting large shares of biomasses are more
relevant. This is implemented in time-discrete
stochastic models which are however of limited
analytical tractability. Impulsive differential equa-
tion systems are a good compromise since they
permit to model time-discrete fire impact while
keeping ODE for modelling vegetation growth. As
such, they remain analytically tractable to a large
extent while being more realistic.

In this paper, we show that minimal savanna
models are able to provide a wide array of
meaningful and relevant predictions of savanna
dynamics while retaining sufficient mathematical
tractability and restricting themselves to a minimal
set of parameters assessable from the overall litera-
ture. Moreover, simplicity is overarching whatever
the level of tractability. With a simple model,
simulations can claim a thorough exploring of all
parameters space. Conversely, it is difficult to be
sure that sufficient exploration of model behaviors
has been carried out for overcomplicated models
which tend to flourish in ecology. Moreover, be-
cause there is naturally substantial uncertainty for
many parameter values, it is difficult to conclude
whether results are due to the ranges taken for
parameters or to the structure of the model itself.
Therefore, using complex models, it becomes even
more illusory to test hypotheses, while this is one
of the fundamental roles of modelling.

ACKNOWLEDGEMENTS
A. Tchuinté Tamen and I.V. Yatat Djeumen

are grateful to the French National Institute for
Research for Sustainable Development (IRD), the
French Agricultural Research Centre for Interna-
tional Development (CIRAD), the French Institute
for Research in Computer Science and Automation
(INRIA, EPI MASAIE), the Department of Coop-
eration and Cultural Action (SCAC) of the French
Embassy in Yaoundé, the International Center for
Pure and Applied Mathematics (CIMPA), the In-
ternational Laboratory for Computer Sciences and
Applied Mathematics (LIRIMA), the annual inter-
national conference on Mathematical Methods and
Models in Biosciences (BIOMATH 2016 & 2017),
for their financial supports during the preparation
of their Ph.D theses. I.V. Yatat Djeumen also
acknowledges the support of the SARChI Chair
in Mathematical Models and Methods in Bioengi-
neering and Biosciences (University of Pretoria,
South Africa).

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I.V. Yatat Djeumen, A. Tchuinté Tamen, Y. Dumont, P. Couteron, A tribute to the use of minimalistic ...

APPENDIX

Table V: Parameter values used to obtain Figure 7-(a), page 17. From Accatino et al. (2010) [4] (here see model
(3)).

w1, − ε, yr−1 cT , yr−1 cG, yr−1 γT , yr−1 γG, yr−1 δT , yr−1 δG, yr−1 δF , −

345 20 30 10 2 180 0.04 2.8 0.35

Table VI: Parameter values used to get Figure 7-(b), page 17. From Tchuinté Tamen et al. (2018) [88] (here see
model (6)).

cG, t.ha−1 cT , t.ha−1 bG, mm.yr−1 bT , mm.yr−1 aG, yr−1 aT , yr−1

20 450 501 1192 0.0029 0.0045

dG, − dT , − γG, yr−1 γT , yr−1 δG, yr−1 δT , yr−1

14.73 106.7 2.5 1 0.01 0.1

λfG, − λminfT , − λ
max
fT , − p, t

−1ha α, t.ha−1 ηT G, ha.t−1yr−1

0.3 0.05 0.7 0.01 1 0.01

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http://dx.doi.org/10.11145/j.biomath.2018.12.167

	Introduction
	A brief review on space-implicit tree-grass interactions modelling
	Tree-grass coexistence and possible alternative stable states
	Lines of thoughts

	Main published modelling options
	Reaching sensible predictions from minimal models
	A seminal "big picture" at biogeographic scale
	Fire frequency, grass biomass and fire impact
	Tree survival
	Relating to water resource
	Impulsive time-periodic occurrences of fire events

	Discussion and prospects
	Conclusion
	References