Abstract Vegetation band patterns on hill slopes are studied using a mathematical model. The model applies to drylands, where the limiting resource is water, and takes into account positive feedback effects between biomass and water. Multiple band patterns coexisting in wide precipitation ranges are found. For given precipitation and slope conditions band patterns with higher wavenumbers are more biologically productive. High-wavenumber patterns, however, are less resilient to environmental changes. r 2005 Published by Elsevier B.V

Ehud Meron

Erez Gilad

Hezi Yizhaq

CiteSeerX

ARTICLE IN PRESSPhysica A 356 (2005) 139–1440378-4371/$ -
doi:10.1016/j
Correspo
University, S
E-mail adwww.elsevier.com/locate/physaBanded vegetation: biological productivity
and resilience
Hezi Yizhaqa, Erez Giladb,a, Ehud Merona,b,
aDepartment of Solar Energy and Environmental Physics, BIDR, Ben Gurion University,
Sede Boqer Campus 84990, Israel
bDepartment of Physics, Ben-Gurion University, Beer Sheva 84105, Israel
Available online 13 June 2005Abstract
Vegetation band patterns on hill slopes are studied using a mathematical model. The model
applies to drylands, where the limiting resource is water, and takes into account positive
feedback effects between biomass and water. Multiple band patterns coexisting in wide
precipitation ranges are found. For given precipitation and slope conditions band patterns
with higher wavenumbers are more biologically productive. High-wavenumber patterns,
however, are less resilient to environmental changes.
r 2005 Published by Elsevier B.V.1. Introduction
Arid and semi-arid ecosystems are characterized by patchy vegetation; the limited
availability of the water resource forces plant species to self-organize in various
vegetation patterns. A striking example is banded vegetation. Vegetation bands on
hill slopes, consisting primarily of trees and shrubs, have been observed in arid and
semi-arid regions throughout the world [1]. The bands are normally oriented
perpendicular to the slope direction and are a few tens of meters apart.see front matter r 2005 Published by Elsevier B.V.
.physa.2005.05.026
nding author. Department of Solar Energy and Environmental Physics, BIDR, Ben Gurion
ede Boqer Campus 84990, Israel. Tel.: +972 7659 6926; fax: +972 7659 6921.
dress: ehud@bgumail.bgu.ac.il (E. Meron).
ARTICLE IN PRESS
H. Yizhaq et al. / Physica A 356 (2005) 139–144140The view of vegetation patchiness as a symmetry-breaking pattern formation
phenomenon is quiet recent. It has been motivated and supported by mathematical
models that relate vegetation patterns to finite wavenumber instabilities of uniform
vegetation states [2–4]. The instabilities result from two positive feedback
mechanisms between biomass and water: (a) increased surface-water infiltration
induced by vegetation [5], and (b) soil water uptake by the plants’ roots [6].
According to the first mechanism, a growing plant induces further local infiltration
of surface water which accelerates its growth. According to the second mechanism,
as a plant grows its roots become longer. The longer the roots the more soil water
they take up and the faster the plant grows. In both mechanisms, the depletion of soil
water from the plant surrounding, and the plant competition for water resource, lead
to symmetry breaking and pattern formation.
In this paper we study the structural and dynamical aspects of banded vegetation
using a recent model that captures the two positive feedbacks discussed above.
Understanding these aspects is significant for the conservation and rehabilitation of
desertified regions [7].2. The model
The model we study has been introduced in Ref. [6]. It contains three dynamical
variables: (a) a biomass density, bðr; tÞ, representing the plant biomass above ground
level, (b) a soil-water density, wðr; tÞ, describing the amount of soil–water available to
the plant per unit area of ground surface, and (c) a surface water variable, hðr; tÞ,
describing the height of a thin water layer above ground level. The model equations
in non-dimensional form are:
qtb ¼ Gbbð1  bÞ  b þ dbr2b ,
qtw ¼ Ih  nð1  rbÞw  Gww þ dwr2w ,
qth ¼ p  Ih þ dhr2ðh
2
Þ þ 2dhrh  rzþ 2dhhr2z , (1)
where Gb represents the biomass growth rate, Gw is the rate of soil–water
consumption by the vegetation, I is the infiltration rate of surface water into the
soil, p is the precipitation rate, and the function zðrÞ describes the ground
topography.
The two positive feedback effects between biomass and water are included in the
explicit forms of the infiltration term I and the biomass growth rate Gb. The
infiltration I is assumed to depend on the biomass b according to I ¼ aðr; tÞ½bðr; tÞ þ
qf 	=½bðr; tÞ þ q	 [5,4]. The smaller the f the lower the infiltration rate at bare soil,
as compared with vegetated soil, and the stronger the water–biomass feedback [6]).
The biomass growth rate is modeled by Gbðr; tÞ ¼ n
R
O gðr; r
0; tÞwðr0; tÞdr0, where
gðr; r0; tÞ ¼ 1
2p exp 
jrr0 j2
2½ð1þZbðr;tÞÞ	2
h i
and the integration is over the entire domain O.
According to this form the biomass growth rate depends not only on the amount of
ARTICLE IN PRESS
H. Yizhaq et al. / Physica A 356 (2005) 139–144 141soil water at the plant location, but also on the amount of soil water in the
neighborhood which the plant’s roots extend to. The positive feedback effect due to
water uptake by the roots is captured by the dependence of the root length (Gaussian
width) on biomass; the larger the biomass the longer the roots and the higher the
water uptake. The water consumption rate at a point r is similarly given by
Gwðr; tÞ ¼ g
R
O gðr
0; r; tÞbðr0; tÞdr0. For more details regarding the model the reader is
referred to Ref. [6].3. Uniform and pattern solutions
Eqs. (1) have two stationary uniform states representing bare soil and uni-
form vegetation, as well as patterned states. The bifurcation diagram in Fig. 1
shows the uniform states and their stability properties. The insets show two basic
patterned states; spot patterns at low precipitation rates and band patterns at
higher precipitation rates. Stable spot patterns appear at precipitation values
where the bare soil state is still stable. Likewise, stable band patterns appear at
precipitation values where spot patterns are still stable. With appropriate para-
meter choices a tristability of bare state, spots and bands can be obtained. These0
0.2
0.4
0.6
0
bi
om
as
s 
[k
g/
m
2 ]
precipitation 
pcp1 p2
B
V
Fig. 1. A bifurcation diagram showing biomass b vs. precipitation p for a uniform slope. The branch B
represents a uniform bare soil solution and the branch V represents a uniform vegetation state. The
precipitation axis spans the range 0–1250mm/yr. The bare state becomes unstable at pc ¼ 1 (250mm/yr).
The uniform vegetation state becomes unstable at p2 ¼ 4:146 (1036.5mm/yr). The insets show typical
patterns along the precipitation gradient, spots at low p (200mm/yr) and bands at higher p (200, 250 and
500mm/yr for a single, five and nine bands, respectively). Dark shades of gray represent high biomass.
Parameter values used are: n ¼ 2, Z ¼ 5, a ¼ 160, f ¼ 0:1, q ¼ 0:05, g ¼ 5, r ¼ 1, db ¼ 0:02, dw ¼ 2,
dh ¼ 200 and slope ¼ 3 degrees.
ARTICLE IN PRESS
H. Yizhaq et al. / Physica A 356 (2005) 139–144142ranges of multistability have significant implications on desertification [3] and
habitat diversity [6].4. Multiple band states
Multistability of stable states also exists within the precipitation range of stable
bands where the bare state and spot patterns are unstable. The coexisting stable
states correspond to band patterns with different wavenumbers. At relatively low
precipitation, where the bare state is still stable, single-band patterns are possible
(second inset from the left in Fig. 1). At higher precipitations stable patterns with
higher wavenumbers appear but many of the lower wavenumber patterns remain
stable. These states are likely to be related to the discrete families of Eckhaus stable
solutions that have been found in simpler models of finite systems, such as the
Ginzburg–Landau model [8] and the complex Swift–Hohenberg model [9,10]. The
existence of a discrete family of stable banded patterns suggests two principal
responses to a gradual precipitation downshift: (i) an increase in the ratio of
the interband distance to the band width (interband–bandratio, or IBR index
[11]) without a change in the pattern’s wavenumber, and (ii) a decrease in the
pattern’s wavenumber. Fig. 2 shows how the IBR index changes with precipitation.
The numerical data fit an exponential function well, in agreement with field
observations [11].
The multistability of band states bears on the rate at which vegetation consumes
water, and as a result on the biological productivity of the ecosystem; the smaller this
rate the higher the productivity. We evaluated the water consumption rate per unitFig. 2. The effect of precipitation on the widths of vegetation bands for a pattern with a given
wavenumber. The numerically computed Interband–bandratio (IBR) (circles in part (b)) fits an
exponentially decreasing function (thick solid line) quite well. The total biomass grows linearly with
precipitation (thin solid line). The parameters are the same as in Fig. 1.
ARTICLE IN PRESS
0 2 4 6
0
0.25
0.5
0.75
1
 b
to
ta
l
0 2 4 6
0
500
1000
1500
W
to
ta
l
0 2 4 6
1500
2000
2500
k k k 
R
 
(a) (b) (c)
Fig. 3. The dependence of biological productivity on the pattern’s wavenumber k. Both total biomass,
btotal , and total amount of water consumed by the vegetation, wtotal , increase with k, but the ratio R ¼
wtotal=btotal decreases. Thus, high wavenumber patterns consume less water and are therefore more
productive. Parameters p ¼ 0:8 (200mm/year) and all other parameter values are the same as in Fig. 1.
H. Yizhaq et al. / Physica A 356 (2005) 139–144 143of biomass for all band solutions that coexist for given precipitation and slope values
by calculating the total water consumption rate wtotal ¼
R
Gwwdx, the total biomass,
btotal ¼
R
bdx and the ratio wtotal=btotal . The results are shown in Fig. 3. Both wtotal
and btotal increase with the pattern’s wavenumber (or number of bands) but wtotal
increases more slowly. As a consequence the water consumption per unit biomass
decreases as the pattern’s wavenumber increases. This result suggests that establish-
ing higher wavenumber patterns in rehabilitating desertified regions will increase the
biological productivity of these regions.5. Resilience of band states
Banded patterns with higher wavenumbers are more biologically productive but
are less resilient to environmental changes. We demonstrate this property by
studying the response of banded vegetation to a precipitation downshift. Fig. 4
shows the response of a relatively high wavenumber pattern (k0 ¼ 8:8) to a
precipitation downshift within the range of multistability of band states. The
downshift leads to an Eckhaus-type instability [12] that culminates in a significantly
lower wavenumber (k ¼ k0=2). An identical downshift of a band pattern with an
intermediate wavenumber (k0o8) leaves the wavenumber unchanged. Thus, lower
wavenumber patterns which are less productive are more resilient to precipitation
downshifts such as prolonged droughts.6. Conclusion
The results of this study are significant for rehabilitation of desertified regions. In
choosing the inter-band distances in rehabilitation projects [13] of two conflicting
factors should be carefully considered; short distances increase the biological
ARTICLE IN PRESS
Fig. 4. Low resilience of vegetation bands to precipitation downshifts. A high wavenumber pattern
(k0 ¼ 8:8) which is stable at p ¼ 2 (500mm/yr) responds to a precipitation downshift to p ¼ 1:2 (300mm/
yr) by gradual downhill elimination of every second band. Parameter values are the same as in Fig. 1.
H. Yizhaq et al. / Physica A 356 (2005) 139–144144productivity but also increase the vulnerability of the system to environmental
changes such as a precipitation downshift.Acknowledgements
We wish to thank Moshe Shachak and Efrat Shefer for helpful discussions. This
research was supported by the James S. McDonnell Foundation (grant No.
220020056) and by the Israel Science Foundation (grant No. 780/01).References
[1] C. Valentin, J.M. d’Herbès, J. Poesen Catena 37 (1999) 1.
[2] R. Lefever, O. Lejeune, Bull. Math. Bio. 59 (1997) 263;
O. Lejeune, M. Tlidi, J. Veg. Sci. 10 (1999) 201;
C.A. Klausmeier, Science 284 (1999) 1826;
T. Okayasu, Y. Aizawa, Prog. Theor. Phys. 106 (2001) 705.
[3] J. von Hardenberg, E. Meron, M. Shachak, Y. Zarmi, Phys. Rev. Lett. 87 (2001) 198101;
E. Meron, E. Gilad, J. von Hardenberg, M. Shachak, Y. Zarmi, Choas, Solitons & Fractals 19
(2004) 367.
[4] M. Rietkerk, M.C. Boerlijst, F. Van Langevelde, R. HilleRisLambers, J. Van de Koppel, L. Kumar,
H.H.T. Prins, A.M. De Roos, The Am. Naturalist 160 (2002) 524.
[5] R. HilleRisLambers, M. Rietkerk, F. van den Bosch, H.H.T. Prins, H. de Kroon, Ecology 82
(2001) 50.
[6] E. Gilad, J. von Hardenberg, A. Provenzale, M. Shachak, E. Meron, Phys. Rev. Lett. 93
(2004) 098105.
[7] M. Scheffer, S.R. Carpenter, Trends in Ecol. Evol. 18 (2003) 648.
[8] L.S. Tuckerman, D. Barkley, Physica D 46 (1990) 57.
[9] K. Tsiveriotis, R.A. Braowb, Phys. Rev. Lett. 63 (1989) 2048.
[10] L.S. Tuckerman, D. Barkley, Phys. Rev. Lett. 67 (1991) 1051.
[11] C. Valentin, J.M. d’Herbès, Catena 37 (1999) 231.
[12] M. Cross, P.C. Hohenberg, Rev. Mod. Phys. 65 (1993) 851.
[13] The inter-band distance is determined by the distance between successive contour ditches along which
the vegetation is planted. The ditches trap runoff from the bare areas uphill and thereby provide
conditions for growth.


Banded vegetation: Biological Productivity and Resilience

http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=EF8BFE475D363574C1FDA0762CE497E0?doi=10.1.1.1047.9235&rep=rep1&type=pdf

Banded vegetation: Biological Productivity and Resilience

Abstract

Similar works

Full text

Available Versions

CiteSeerX