Journal of Theoretical Biology - unina.it plant soil... · Negative plant soil feedback explaining ring formation in clonal plants Fabrizio Cartenıa, Addolorata Marascob, Giuliano

Journal of Theoretical Biology 313 (2012) 153–161

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Journal of Theoretical Biology

0022-51

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n Corr

E-m

marasco

m.rietke

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Negative plant soil feedback explaining ring formation in clonal plants

Fabrizio Carten�ı a, Addolorata Marasco b, Giuliano Bonanomi cI, Stefano Mazzoleni c, Max Rietkerk d,Francesco Giannino a,n

a Dipartimento di Ingegneria Agraria e Agronomia del Territorio, University of Naples Federico II, via Universit�a 100, 80055, Portici (Na), Italyb Dipartimento di Matematica e Applicazioni ‘‘R. Caccioppoli’’, University of Naples Federico II, via Cintia, 80126, Naples, Italyc Dipartimento di Arboricoltura, Botanica e Patologia Vegetale, University of Naples Federico II, via Universit�a 100, 80055, Portici (Na), Italyd Dept Environmental Sciences, Copernicus Institute, Utrecht University, P.O. Box 80115, 3508 TC Utrecht, The Netherlands

H I G H L I G H T S

c We develop a PDE model of a clonal plant response to negative plant soil feedback.c Plant sensitivity to autotoxic compounds and their decay rate induce ring formation.c We show that ring patterns could emerge also without water limiting conditions.

a r t i c l e i n f o

Article history:

Received 29 March 2011

Received in revised form

27 July 2012

Accepted 7 August 2012Available online 15 August 2012

Keywords:

Phytotoxicity

Vegetation pattern

Litter decomposition

Mathematical model

93/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.jtbi.2012.08.008

esponding author. Tel. þ39 081 2539424; fax

ail addresses: [email protected] (F. Car

@unina.it (A. Marasco), [email protected] (

[email protected] (M. Rietkerk), giannino@unina.

a b s t r a c t

Ring shaped patches of clonal plants have been reported in different environments, but the mechanisms

underlying such pattern formation are still poorly explained. Water depletion in the inner tussocks

zone has been proposed as a possible cause, although ring patterns have been also observed in

ecosystems without limiting water conditions. In this work, a spatially explicit model is presented in

order to investigate the role of negative plant–soil feedback as an additional explanation for ring

formation. The model describes the dynamics of the plant biomass in the presence of toxicity produced

by the decomposition of accumulated litter in the soil. Our model qualitatively reproduces the

emergence of ring patterns of a single clonal plant species during colonisation of a bare substrate.

The model admits two homogeneous stationary solutions representing bare soil and uniform

vegetation cover which depend only on the ratio between the biomass death and growth rates.

Moreover, differently from other plant spatial patterns models, but in agreement with real field

observations of vegetation dynamics, we demonstrated that the pattern dynamics always lead to

spatially homogeneous vegetation covers without creation of stable Turing patterns. Analytical results

show that ring formation is a function of two main components, the plant specific susceptibility to toxic

compounds released in the soil by the accumulated litter and the decay rate of these same compounds,

depending on environmental conditions. These components act at the same time and their respective

intensities can give rise to the different ring structures observed in nature, ranging from slight

reductions of biomass in patch centres, to the appearance of marked rings with bare inner zones, as

well as the occurrence of ephemeral waves of plant cover. Our results highlight the potential role of

plant–soil negative feedback depending on decomposition processes for the development of transient

vegetation patterns.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The occurrence of regularly shaped circles or rings of clonalplants has been documented since long time (Watt, 1947; Curtisand Cottam, 1950; Cosby, 1960). Clonal plant establishment starts

ll rights reserved.

: þ39 081 7760104.

ten�ı),

S. Mazzoleni),

it (F. Giannino).

with spots of highly aggregated ramets. However, as new rametsdevelop centrifugally, their density in the patch interior decreases,with senescence of older shoots and appearance of a ring belt(see Fig. 1).

Such pattern has been also reported as fairy rings (Hitchcock,1935), rings (Watt, 1947), central die-back (Adachi et al., 1996),monk’s tonsure-like gaps (Lewis et al., 2001), and often observed inresources deprived environments including water and nutrientlimited ecosystems such as deserts (Danin, 1996; Sheffer et al.,2007), peatland (Lanta et al., 2008), and primary succession over

Fig. 1. Examples of ring forming plants in different ecosystems: (a–b) Sesleria appennina in alpine conditions; (c–d) Brachypodium rupestre in mountain grassland; (e–f)

Ampelodesmos mauritanicus in semiarid Mediterranean grassland; (g–h) Scirpus holoscoenus in alluvial grassland. The last picture (h) refers to the same individual reported

in (g) after its excavation to show the tussock ring shape. All photographs by G. Bonanomi.

F. Carten�ı et al. / Journal of Theoretical Biology 313 (2012) 153–161154

bare substrate (Adachi et al., 1996). The mechanisms underlying theformation of plant rings are still poorly known. Water depletion inthe inner zone of the ring has been proposed as a possibleexplanation of central die-back and, consequently, of ring patternsformation (Sheffer et al., 2011; Ravi et al., 2008). However, clonalring perennial plants have been also observed in ecosystemswithout water limiting conditions. Examples include the grassesBrachypodium rupestre (Bonanomi and Allegrezza, 2004) (see Fig. 1),Bromus inermis in mountain grasslands (Otfinowski, 2008), thesedge Scirpus holoshoenus in alluvial grassland (Bonanomi et al.,2005) (Fig. 1), and several species of Spartina in salt marshes(Caldwell, 1957; Castellanos et al., 1994). It is quite obvious that

in these cases the water limitation hypothesis cannot be consideredas an exhaustive explanation.

Alternatively, central die-back could be induced by the build-upof negative plant–soil feedback in the inner clonal zone (Curtis andCottam, 1950; Bonanomi et al., 2005). Negative plant–soil feedbackis defined as the rise of negative conditions for plant vegetativeand reproductive performances induced in the soil by the plantsthemselves (Mazzoleni et al., 2007). Recognized mechanisms pro-ducing negative plant–soil feedback are: the soil nutrient depletion(Ehrenfeld et al., 2005), the build-up of soil-borne pathogenpopulations (Packer and Clay, 2000), the changing composition ofsoil microbial communities (Klironomos, 2002), and the release of

F. Carten�ı et al. / Journal of Theoretical Biology 313 (2012) 153–161 155

autotoxic compounds during organic matter decomposition (Singhet al., 1999). Curtis and Cottam (Curtis and Cottam, 1950) providedthe first evidences supporting the negative feedback hypothesisbased on litter autotoxicity, by studying the prairie sunflowerHeliantus rigidus. This perennial forb in the field showed cloneswith central die-back. The removal of dead roots and rhizomes,which in laboratory had autotoxic effects, improved H. rigidus

growth, while no positive effect was found after application ofmineral nutrients. Moreover, the authors replaced, in open field, thesoil present inside the ring with soil collected in the externalgrassland not previously affected by the same species. This greatlyenhanced the biomass recovery of H. rigidus in the die-back zone.Following experimental studies provided evidence that herbaceousplant with phalanx growth strategy accumulate large amount ofleaves, rhizome and root litter in the die-back zone of the clones(e.g. Watt (1947; Curtis and Cottam, 1950; Danin, 1996; Falinska,1995; Lanta et al., 2004; Bonanomi et al., 2005)), that negativelyaffect conspecific regeneration. As a consequence, negative feed-back escape strategies depend on life form and propagationpatterns. For instance, trees and shrubs can avoid the ‘‘home’’ soil(sensu Bever, 1994) via seed dispersion, thus producing a Janzen-Connell distribution of seedling emergence (Packer and Clay, 2000;Bonanomi et al., 2008), while perennial clonal plants with ‘‘pha-lanx’’ growth strategy (Lovett-Doust, 1981) can move away byvegetative growth thus forming rings (Bonanomi et al., 2005; Olffet al., 2000).

Recently, a substantial modelling effort has been done toinvestigate the mechanisms underlying the formation of severaltypes of vegetation patterns in water-limited ecosystems (Rietkerket al., 2002; Gilad et al., 2007; Meron et al., 2007; Barbier et al.,2008). Despite of a significant body of empirical studies on plantforming rings, only a few models have been developed to explainthe appearance of such patterns in clonal plants. Available modelsaddressed the possible role of water limitation on the formationof plant rings in semiarid environments (Sheffer et al., 2007; vonHardenberg et al., 2010). Here, we investigated the possibility thatnegative plant–soil feedback may be considered an additionalexplanation for ring formation in clonal plants with ‘‘phalanx’’ pro-pagation strategy. The mathematical models proposed in (Beveret al., 1997; Bonanomi et al., 2005) demonstrated, by means of non-spatial simulations, how intra-specific negative feedback can allowspecies coexistence, by creating unsuitable conditions for conspe-cifics, and suitable conditions for other species. However, to ourknowledge, no studies explored the potential effects of negativeplant–soil feedback resulting from toxic compounds during litterdecomposition for the formation of vegetation patterns.

Direction of tussock spread

Living tillers

Livingfine roots

Livingfine root

Fig. 2. Schematic representation of a clonal plant structure during vegetative propagati

rhizomes (A) the plant begins to propagate, colonizing the surrounding soil in every

accumulate in the central tussock area, where the plant is older. The accumulation of

reduces the plant growth performance (B).

In this paper, first we introduce a spatially explicit model forbiomass dynamics of one clonal plant, derived by the modeldescribed in Mazzoleni et al. (2010), to investigate the role ofintra-specific plant–soil negative feedback conceived as the pro-duct of litter toxicity (Bonanomi et al., 2006) and autotoxicity(Mazzoleni et al., 2007; Singh et al., 1999; Blok and Bollen, 1993).Then, we develop a qualitative and quantitative analysis of themodel, with special attention to the main mechanisms underlyingpattern formation. This section contains the mathematical studyof the model, including stability analysis of the homogeneoussteady-state values, linear stability analysis to spatially hetero-geneous perturbations and an estimate of the ‘‘invasion velocity’’of the plant biomass. Finally, we discuss the implications ofthe presented mechanism in the framework of water-limitationinduced pattern formation.

2. The mathematical model

Based on the work of Mazzoleni et al. Mazzoleni et al. (2010),we introduce a model for biomass dynamics of one clonal plantwith phalanx growth strategy and for the rising tussock spatialpatterns induced by negative plant–soil feedback. Beginning froman initial small tussock of tillers, roots and rhizomes surroundedby bare soil (Fig. 2, panel A), the plant starts to propagatecolonizing the soil around it in any direction. As the propagationproceeds, dead tillers, roots and rhizomes start to accumulate inthe central area of the tussock, where the plant is older. Duringdecomposition of these residues, litter degradation and microbialactivity produce phytotoxic materials (Bonanomi et al., 2006)with a direct harmful effect on plants. Also, plant resistance topathogens attack can be reduced by the phytotoxic conditions(Patrick and Toussoun, 1965; Bonanomi et al., 2007). Moreover,decaying organic matter provides the growth substrate for sapro-phytic pathogens, thus enhancing their pathogenicity (Bonanomiet al., 2011; Blomqvist et al., 2000; Hoitink and Boehm, 1999;Bonanomi et al., 2010). Recent work on phytotoxicity dynamics isclearly showing such trends of toxicity relation with litterdecomposition processes (Bonanomi et al., 2011). As previouslystated, the raise of negative conditions is concentrated in theolder (central) part of the tussock thus reducing the growthperformance of the plant (Fig. 2, panel B). In real conditions,toxicity persistence is linked to litter-decaying rate and, conse-quently, to related environmental conditions such as temperatureand water availability. For simplicity, in this model, toxicity is

Direction of tussock spread

Living tillersDead tillers, rhizomes and roots

s

on with phalanx strategy. Starting from an initial small tussock of tillers, roots and

direction. As the propagation proceeds, dead tillers, roots and rhizomes start to

dead biomass and its consequent decomposition, releasing autotoxic compounds,

Table 1List of model parameters and their units.

Parameter Description UnitAssigned

value

g Growth rate of B month�1 0.5

BmaxPlant biomass carrying

capacitykg cm�2 1

d Death rate of B month�1 0.05

s Plant sensitivity to T cm2 kg�1 month�1 Between 0.15

and 1

DPlant biomass propagation

coefficientcm2 month�1 0.05

k Decay rate of T month�1 Between 0.05

and 0.2c Proportion of toxic products

by litter decomposition

– 0.5

Table 2Definitions of the nondimensional variables and

parameters appearing in Eq. (2) in terms of their

dimensional counterparts.

Quantity Scaling

~B B/Bmax

~T Tk/(cdBmax)~x x(g/D)1/2

~y y(g/D)1/2

~t gt

a d/gb csBmax/kg k/g

F. Carten�ı et al. / Journal of Theoretical Biology 313 (2012) 153–161156

reduced in time by a constant decay/removal process of toxiccompounds (Bonanomi et al., 2006; Bonanomi et al., 2011).

2.1. Model description

The model consists of a system of two nonlinear partial differ-ential equations describing the dynamics of two state variables:plant biomass B (kg cm�2) and toxic compounds T (kg cm�2). Plantbiomass changes as a function of plant growth, mortality andvegetative propagation. In the mathematical modelling, plant rootand shoot were not considered separately because functional rootsystem and living shoot are in close proximity in clonal plants withphalanx growth strategy (Bonanomi et al., 2005). Plants growlogistically with growth rate parameter g (month�1) and carryingcapacity Bmax (kg cm�2). Plant mortality is due to a constant loss rated (month�1) and an extra loss induced by the negative plant–soilfeedback function of T concentration by means of a parameter s

(cm2 kg�1 month�1). Plant vegetative propagation is approximatedby a diffusion term of coefficient D (cm2 month�1). Toxic compoundsare produced by a fraction of the dead biomass c (dimensionless) andis reduced by litter removal or decay process simply summarised bya parameter k (month�1) which is the removal/decay rate of T. Forsimplicity no diffusion of T is considered by the model.

The model parameters descriptions are summarised in Table 1.In all simulations the parameter values are chosen according toeither Mazzoleni et al. Mazzoleni et al. (2010) or set to order-of-magnitude realistic values.

Owing to the above description, the model equations are

@B

@t¼ gB 1�

B

Bmax

� ��B dþsTð ÞþDDB,

@T

@t¼ cB dþsTð Þ�kT:

8>>><>>>:

ð1Þ

In this paper, we studied the model only for positive values ofthe parameters g, d, s, D, c, k, since for s¼0 the nonlinear partialdifferential Eq. (1) are decoupled, and for k¼0 the model refers toa system in which there is no toxicity decomposition.

In order to minimise the number of parameters involved in themodel it is extremely useful to write the system (1) in nondimen-sional form. Introducing the following dimensionless variables

~B ¼B

Bmax, ~T ¼

k

cdBmaxT, ~x ¼ x

g

D

� �1=2

, ~y ¼ yg

D

� �1=2

, ~t ¼ gt

then system (1) becomes

@B

@t¼ B 1�Bð Þ�aB 1þbTð ÞþDB,

@T

@t¼ gB 1þbTð Þ�gT ,

8>><>>: ð2Þ

where a¼d/g, b¼csBmax/k, g¼k/g and, for convenience, weomitted the superscript.

Rescaled variables and parameters are summarised in Table 2.We remark that a is plant basal mortality rate relative to growthrate; b is a composite dimensionless parameter that measuresthe impact of toxic compounds on inflation of plant mortality,combining the plant sensitivity s (per unit of concentration oftoxic compounds), with typical biomass concentration of toxiccompounds cBmax, and typical duration of such toxic influence(k�1). Finally, g measures the characteristic rate of toxicitydynamics relative to plant growth rate.

3. Results

3.1. Stability analysis of the spatially homogeneous equilibria

The first step in studying the patterns of system (2) is todetermine the equilibria of the spatially homogeneous system

@B

@t¼ B 1�Bð Þ�aB 1þbTð Þ

@T

@t¼ gB 1þbTð Þ�gT,

8>><>>: ð3Þ

i.e., the solution of the algebraic equations

B 1�Bð Þ�aB 1þbTð Þ ¼ 0,

g½B 1þbTð Þ�T� ¼ 0:

(ð4Þ

Biologically feasible equilibrium points are the non-negativesolutions of (4) in the interior of the first quadrant. System (3) hasat most two equilibria depending on the magnitude of parameter a:

�

equilibrium

ðBn,TnÞ ¼ð1þbÞ�G

2b,ðb�1Þ�2abþG

2ab2

!,

where

G¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðb�1Þ2þ4ab

q:

The Jacobian matrix J(B,T) for system (3) is given by

JðB,TÞ ¼1�a�2B�abT �abB

gþbgT �gþbgB

!,

and at the steady-state (0, 0) admits the eigenvalues l1¼1�a,l2¼�g. Moreover, at the steady-state (Bn, Tn) the Jacobian matrixJn has two eigenvalues with negative real parts (see Appendix A).

Fig(do

F. Carten�ı et al. / Journal of Theoretical Biology 313 (2012) 153–161 157

Then, the linear analysis of stability allows us to recognise that forany positive values of parameters b, g we have:

if 0oao1, and consequently a¼1 is a bifurcation value;

value of a: 0oao1.

This analysis shows that the model has two homogeneousstationary solutions representing bare soil and uniform vegeta-tion cover. Fig. 3 shows the non-trivial equilibrium values (Bn, Tn)versus b, for some fixed values of plant basal mortality raterelative to growth rate (a). Recalling that a¼d/g, the existenceand the stability character of the equilibria are ecologicallyconsistent. In fact, if the plants death rate d is higher that thegrowth rate g, the only possible solution is the complete loss ofvegetation cover. On the other hand, if the growth rate is higherthan the death rate, the biomass stabilizes in a long time on theuniform value Bn.

3.2. Linear stability analysis to spatially heterogeneous

perturbations

To study the effect of diffusion on the model system, weperform the linear stability analysis of the stationary homoge-neous solution (Bn, Tn) of the spatial model (2) to nonuniforminfinitesimal perturbations. We consider the perturbed solutions

B r,tð Þ ¼ BnþaB tð Þeir�hþcc, T r,tð Þ ¼ TnþaT tð Þeir�hþcc, ð5Þ

where r¼(x, y), h¼(h1, h2) is the wave vector of the perturbation,a(t)¼(aB(t), aT(t)) is the vector of the perturbation amplitudes and‘‘cc’’ stands for the complex conjugate.

Substituting the Eq. (5) into (2) and keeping terms to first-order only, we obtain the following system of linear ODEs for theperturbation amplitudes a(t)

daB

dt¼ 1�2Bn�h�a�abTnð ÞaB�abBnaT ,

daT

dt¼ ðgþbgTnÞaBþð�gþbgBnÞaT ,

8>><>>: ð6Þ

where h¼9h9 is the perturbation’s wave number.Assuming exponential growth for the perturbation amplitudes,

i.e.,

aB tð Þ ¼ B 0ð Þelt , aT tð Þ ¼ T 0ð Þelt , ð7Þ

we obtain the eigenvalue problem

J hð Þa¼ la, ð8Þ

where J(h) is the coefficient matrix of (6). The solutions l¼l(h) of(8) gives the dispersion relations, i.e., provide information aboutthe stability of the stationary homogeneous solution (Bn, Tn). In

0.7

0.6

0.5

0.4

0.3

0.2

0.1

01 2 3 4 5 6 7 8 9 10

B*

β

T

. 3. Spatially homogeneous equilibria of B (left panel) and T (right panel) versus b, f

tted line). At low levels of impact of toxic compounds (b) both the equilibria of pla

fact, the growth rate of a perturbation characterised by a wavenumber h is given by the largest real part of l¼l(h). In our case,when 0oao1 and the coefficients b and g are positive, all wavenumbers have negative growth rates and any perturbationdecays, i.e., the uniform solution is asymptotically stable (seeAppendix B). In ecological terms, these results show that for0oao1, and b, g positive parameters, the pattern formationalways leads to spatially uniform vegetation covers (Bn), andstable Turing patterns cannot exist.

3.3. Travelling fronts

Finally, we examine the system’s dynamics when the initialconditions of system (1) are finite, i.e.,

B x,y,0ð Þ ¼j x,yð Þ, T x,y,0ð Þ ¼c x,yð Þ, x,yð ÞAO

where j and c are suitable functions with a finite support in theplanar domain O. In this case, our model, as for some reaction-diffusion models, produce travelling waves of the plant biomass B

that travel at velocity c(t) approaching the asymptotic speed

c1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4Dðg�dÞ

p: ð9Þ

In effect, the simple equation for asymptotic invasion velocityfor the Fisher model is not restricted to logistic populationgrowth, but more generally arises as

c1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4DF 0ð0Þ

q: ð10Þ

where F(B) is a general class of population growth functions(Holmes et al., 1994). In our case, the asymptotic speed c1 of thetravelling wavefronts, can be obtained easily from heuristicarguments (Volpert and Petrovskii, 2009). In fact, at the positionof the biomass front, the toxic compounds T is absent and hencethis is a problem effectively described by a single KPP-Fisherequation (Fisher, 1937; Kolmogorov et al., 1937); its speedtherefore being given by (10), where

FðBÞ ¼ ðg�dÞB�g

BmaxB2:

Owing to relation (9), we can conclude that the asymptotic‘‘invasion velocity’’ of the biomass B is determined only by therates of net population growth (g�d) and diffusion coefficient D.Since the ‘‘coexistence’’ steady state (Bn, Tn) is locally stableagainst a nonuniform infinitesimal perturbation, the front is anarrow region that moves with constant shape and speed (Fig. 4).As can be better seen in Fig. 5, a circular invasion front spreadsout from the point at which the plant biomass is initiallylocalised, then the area inside the front is populated by plantbiomass according to the intensity of the negative feedback(synthetically represented by the dimensionless parameter

2.5

2

1.5

1

0.5

01 2 3 4 5 6 7 8 9 10

*

β

α = 0.1 α = 0.4 α = 0.7

or three different plant species: a¼0.1 (solid line), a¼0.4 (dashed line) and a¼0.7

nt biomass and toxicity are high, i.e., the equilibria decreasing as b increases.

F. Carten�ı et al. / Journal of Theoretical Biology 313 (2012) 153–161158

b¼(csBmax)/k) produced by T. Consequently, an increase in theimpact of toxic compounds on plant mortality (i.e., an increase inthe plant sensitivity to autotoxicity s or a decrease in toxiccompounds decay/removal rate k), produces a decrease in theequilibrium values as well as a decrease in height and width ofthe external biomass peaks (Fig. 6). Interestingly, for sufficientlyhigh values of b, secondary concentric rings are formed within thepatch (Figs. 5 and 6). As previously described in Section 3.2, it hasto be remembered that, for long times, inside all the ring patternsthere will always be a spatially uniform region with plant biomassand toxic compounds at their coexistence values (Bn, Tn) (Fig. 3).

4. Discussion

Previous modelling studies related the formation of rings by clonalplants to a resource shortage, i.e., water in arid ecosystems (Sheffer

β =

2.5

t = 10 t =

β =

5β

= 10 y

x

y

y

Fig. 5. Comparison of model simulations (rows) at different values of b, obtained by nu

map of biomass distribution with darker shade representing higher biomass density. T

t2.5

t20 t35 t501

0.8

0.6

0.4

0.2

0

B

x

B*

•

Fig. 4. Biomass travelling waves. Solid lines represent snapshots from subsequent

times from the initial spot centre of B (denoted by �). Lines are obtained as cross-

sections of two dimensional simulations of the model equations (Eq. (2)). The

wavefronts proceed at constant speed (Eq. 9) from the initial spot while the

biomass in the centre of the ring slowly approaches the equilibrium value Bn

(dashed line). Simulation parameters: a¼0.1, b¼1.5, g¼0.1.

et al., 2007; Rietkerk et al., 2002; Gilad et al., 2007; von Hardenberget al., 2010) which cannot explain the occurrence of ring formingplants when water is not a limiting factor. In particular, such modelsare based on scale-dependent positive and negative feedbacksbetween biomass and water (Rietkerk and Van de Koppel, 2008).Vegetation reduces the presence of soil-crust that inhibits waterinfiltration and produces shading that reduces soil water evaporation.This set of processes give rise to a positive feedback of vegetation onitself due to the increased water uptake that is only limited by theoverall water availability of the system (i.e. precipitation). In thiscontext, the biomass depression in the tussock centre, up to theformation of a clear ring, is the result of competition for water by thesurrounding plants, constituting the negative feedback. However,experimental evidences for the water depletion hypothesis are notcompelling. For instance, Sheffer et al. Sheffer et al. (2007), studyingthe grass Poa bulbosa, found in a greenhouse experiment thatseedlings allocate more biomass to the external tillers, compared tointernal ones, as water availability decreases. Indeed, evidence thatwater depletion occurs inside the rings in open field was notprovided. Later, Ravi et al. (2008) proposed that the rings of thebunchgrass Bouteloua gracilis in semiarid grassland, emerge due to theco-occurring effects of aeolian deposition, changes in soil propertyinside the clones and water limitation. Specifically, the authorsreported that water soil infiltration capacity and water content wereslightly, but significantly, reduced inside small and medium sizetussocks in the field. However, the difference in soil water contentwas very small compared to the outwards vegetated belt. Furtherstudies should clarify if a water deficit of such entity can kill adrought adapted bunchgrass. On the other hand, some studiesreported an higher water holding capacity in the die-back zone ofthe ring shaped clones because of the higher organic matter contentand changes of soil texture with an higher content of the clay fraction

25 t = 50

xx

merical integration of the model equations (Eq. (2)). Each panel shows a grey-scale

ime proceeds from left to right. Other simulation parameters: a¼0.1, g¼0.1.

1

0.8

0.6

0.4

0.2

0

2.5

2.0

1.5

1.0

0.5

0

B T

β = 2.5 β = 5 β = 10

xxx

Fig. 6. Model simulations showing responses of biomass and toxicity distribution along a central transect view across the clonal patch, according to different values of b. The

profiles are cross-sections of two dimensional simulations of the model equations (Eq. (2)). Each panel shows the distributions of B (solid black line) and T (solid grey line)

compared to their equilibrium values Bn (dashed black line) and Tn (dashed grey line). All simulations run for 50 time steps. Other simulation parameters: a¼0.1, g¼0.1.

t = 5Simulationparameters

β = 5

β = 10

t = 35t = 25 t = 45t = 15 t = 55

β = 10 y

x x x x x x

y

y

Fig. 7. Comparison of three model simulations (rows) obtained by numerical integration of the model equations (Eq. (2)), initialised with 20 spot-like patches randomly

distributed over either the entire domain (first two rows) or the first 10 columns of the lattice (last row), at two different levels of b. Each panel shows a grey-scale map of

biomass distribution, with darker shade representing higher biomass density. Time proceeds from left to right. Other simulation parameters: a¼0.1, g¼0.1.

F. Carten�ı et al. / Journal of Theoretical Biology 313 (2012) 153–161 159

(Lanta et al., 2004; Pemadasa, 1981; Pignatti, 1997; Bonanomi, 2002).In conclusion, water depletion may be a factor in the formation ofrings in arid systems, but further evidences are required to clarify ifthis process may explain the formation of rings in ecosystems wherewater is not a limiting factor. In contrast with the previouslydescribed models, our formulation is based on a negative feedbackonly. The decomposition of biomass produces an increase in soilnegative conditions that means a negative effect of vegetation onitself. Our model shows that this type of plant-soil negative feedbackcan be an additional mechanism responsible for the formation ofrings and the appearance of other vegetation patterns in clonal plants.Moreover, compared to previous works on ring formation, the spatialpatterns expected by our model are, in a limited interval of time,inhomogeneous states, always leading to long term homogeneousequilibria and uniform vegetation, in contrast with models proposedby Rietkerk et al. (2002) and Gilad et al. (2007) that provide, for someset of parameters, stationary Turing patterns.

This model output is actually reflecting real world dynamics,where clear rings are observed only in scattered plants with otherpatterns becoming evident at higher soil cover levels. In fact, clonalrings have been often reported during colonisation of bare substratesin primary successions. Examples include peatland (Lanta et al.,2008), volcanic slope (Adachi et al., 1996) and salt marsh mud

(Caldwell, 1957; Castellanos et al., 1994). In these conditions,new recruitments occur in an empty space without competition(Caldwell, 1957), thus forming clones of regular shape. Similarly, themodel provided regular shaped rings after colonisation of an emptysimulation grid followed, when the clones develop and come intocontact with other patches, by their progressive disappearance withsubsequent coalescence as observed in nature (Watt, 1947; Heslop-Harrison, 1958). Interestingly, the disappearance of regular patternsafter rings coalescence has been observed also for fungal ‘‘fairy rings’’when underground mycelia of neighbouring rings come into contact(Dowson et al., 1989). However, the mechanisms underlying thespatial rearrangement of vegetative structures during clones coales-cence are unknown. Keeping in mind such considerations, it seemsclear that the occurrence of either rings or other patterns, such aswave-like structures, during the colonisation of bare substrates,is dependent on the initial arrangement of plant recruitment.Simulations shown in Fig.7 (first two rows) indicate that randomlydistributed plant patches do produce regularly shaped rings or partof rings if new patches are initially enough spaced (i.e. the distancebetween their establishment locations is larger than the averageadult patch diameter), progressively disappearing with rings contactand coalescence. On the other hand, if initial recruitment occursin clusters or along a line (last row of Fig. 7), e.g. in a grassland

F. Carten�ı et al. / Journal of Theoretical Biology 313 (2012) 153–161160

colonisation from a woodland edge, different patterns may emergeas wave-like bands. A real pattern similar to that obtained atintermediate stages of our simulation, showing semi-rings waves,has been reported in herbaceous grasslands and related to the build-up of species-specific negative feedback (Olff et al., 2000; Blomqvistet al., 2000). In such communities, dominant species move awayfrom ‘‘home’’ soil by vegetative propagation, to escape accumulatedsoil-borne pathogens. Likely, in these cases regular rings do notemerge because all soil is covered and each plant may expand onlyinto spaces released by other species when they are affected by theirown negative feedback. Indeed, further modelling attempts mightinvestigate the effects of species-specific negative feedback on thespatial arrangement of multi-species plant assemblages.

Model analysis showed that the negative feedback intensity(parameter b ) strongly affects the biomass production in thecentral tussock zone, ranging, with increasing b, from a slightreduction of central shoots, to a complete die-back, to bare soil.This variability in degeneration levels of the internal ring zone hasbeen reported in natural ecosystems. For instance, Bonanomi andAllegrezza (Bonanomi and Allegrezza, 2004) reported, in the case ofthe ring forming grass B. rupestre, both small reductions andcomplete absence of biomass in the central parts of different clonesin four different study sites. Recently, Otfinowski (Otfinowski,2008) also found only a small reduction of living biomass in thecentre of B. inermis clones. In contrast, many studies reportedcompletely empty central areas of most investigated rings (Watt,1947; Curtis and Cottam, 1950; Lewis et al., 2001; Danin, 1996;Bonanomi et al., 2005; Caldwell, 1957; Castellanos et al., 1994).These observations are consistent with the large variability ofexperimentally observed negative plant–soil feedback intensity,ranging from small reductions of plant growth to lack of regenera-tion of conspecific individuals (Mazzoleni et al., 2007; Packer andClay, 2000; Klironomos, 2002; Kardol et al., 2007; Kulmatisky et al.,2008). We also suggest that an elevated negative feedback inten-sity, resulting from high plant sensitivity coupled with low levelsof removal/decay rates of toxic compounds, may well explainthe development of concentric rings within the same clone. Thisrarely observed pattern (Caldwell, 1957) (G. Bonanomi, pers. obs.),appeared in our model simulations if the plant suffered strongnegative feedback with, at the same time, the toxicity slowlydisappearing from the affected soil. Under these conditions, severalconcentric waves of vegetative propagation can develop in theabsence of interspecific competition because the soil can be re-colonized by the same species as soon as the detrimental effect ofnegative feedback is decreased. Experimental studies are needed totest the consistency of this hypothesis.

In conclusion, our simple single-species model demonstratesthat negative plant–soil feedback due to toxicity by the decom-position processes of accumulated litter may well explain theformation of differently shaped rings and of vegetation wavesduring substrate colonisation. Previous models (Sheffer et al.,2007; von Hardenberg et al., 2010) were also able to reproduceclonal rings, but only in arid, water-limited ecosystems. Far fromstating that plant–soil negative feedback is the only processinvolved in the formation of vegetation patterns, an interestingdevelopment on this topic could be the formulation of a modelthat considers the integration of both mechanisms (plant–waterand plant–soil toxicity feedbacks), in order to evaluate therelative importance of such processes in different environmentalconditions. Moreover, a multi-species model, with species-specific negative plant–soil feedback, can be developed for betterunderstanding plant species distribution in space and time.

Future manipulative field experiments are certainly stronglyrequired to test specific hypotheses and to clarify the mechanismsunderlying ring formation. In particular, the water limitation hypoth-esis could be verified by irrigating the central part of the rings to

check if this would allow a centripetal recolonisation by the clonalplant. On the other hand, the investigation of the toxicity hypothesisshould imply a clarification of the mechanisms producing plant–soilnegative feedback which has been attributed to nutrient depletion(Ehrenfeld et al., 2005), soil-borne pathogens (Packer and Clay, 2000),soil microbial communities (Klironomos, 2002) and autotoxicityproduced by litter decomposition (Singh et al., 1999). Fertilisationexperiments could be used to test the role of nutrient levels in ringformation, whereas, comparative chemical and microbial character-isation of the inner and outer zones of the clonal rings in the fieldshould be performed to assess the other mechanisms. In particular,following previous research work (Bonanomi et al., 2011), furtherdetailed studies by NMR-CPMAS methods could address this criticalissue, with experiments being specifically designed to verify theautotoxicity hypothesis.

Acknowledgements

The research of MR is supported by the ERA-Net on Complexitythrough the project RESINEE (‘‘Resilience and interaction of net-works in ecology and economics’’) and the project CASCADE(Seventh Framework Programme).

The authors sincerely thank the anonymous referees whose in-depth comments and detailed suggestions greatly improved thequality of the paper.

Appendix A

The non-negative uniform steady-state (Bn, Tn) is asymptoti-cally stable if and only if 0oao1 and the parameters b and g arepositive. In fact, the characteristic equation is given by

l2�ðtr JnÞlþdet Jn ¼ 0,

where

det Jn ¼�ðb�1Þ2�4abþð1þbÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðb�1Þ2þ4ab

q2b

g,

tr Jn ¼�1þb�ðb�1Þbg

2b�ðbg�1Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðb�1Þ2þ4ab

q2b

:

After some standard calculations, we derive that det Jn40 andtr Jno0 if and only if 0oao1 and the parameters b and g arepositive. Then, the Jacobian matrix Jn has two eigenvalues withnegative real parts.

Appendix B

The linear stability of the uniform state (Bn, Tn) is deduced fromthe dispersion relations

J hð Þa¼ la, ðB:1Þ

where the Jacobian matrix is

JðhÞ ¼1�a�2Bn�abTn �abBn

gþbgTn �gþbgBn

!:

Indeed, the Eq. (B.1) can be written as

l2�ðtr JðhÞÞlþdet JðhÞ ¼ 0,

where

det JðhÞ ¼ det Jnþ1

2h 1�bþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðb�1Þ2þ4ab

q� �g,

tr JðhÞ ¼ tr Jn�h:

F. Carten�ı et al. / Journal of Theoretical Biology 313 (2012) 153–161 161

Then, (Bn, Tn) is asymptotically stable if and only if the matrixJ(h) has two eigenvalues with negative real parts. Owing to h40,the stability condition takes the following form:

det JðhÞn40, tr JðhÞno0,

which holds if 0oao1 and the parameters b and g are positive.

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