Formation of Vegetation Patterns and Hysteresis Phenomena in Arid and Semiarid Zones

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CHIN.PHYS.LETT. Vol. 24, No. 12 (2007) 3586

Formation of Vegetation Patterns and Hysteresis Phenomena in Arid andSemiarid Zones ∗

HUA Da-Yin(���)∗∗, WANG Lie-Yan(���)Department of Physics, Ningbo University, Ningbo 315211

(Received 26 August 2007)

Considering the spatial effect of the complicated interactions between plant and resource (soil water) and thestochastic character of the complicated ecological processes in an ecosystem, we introduce a lattice gas modelto investigate the vegetation pattern formation in the arid and the semiarid ecosystems with Monte Carlo sim-ulations. With consistency of some previous mean-field models, the model reproduces a wide range of patternsobserved in the arid and the semiarid regions. The sequences of vegetation states are also investigated with

scanning three parameters, i.e., the precipitation and the plant growth rate, and then the parameter ranges areidentified where a hysteresis loop appears and two different stable states coexist. The simulation results denotethat the complex structures occur only when the precipitation parameter decreases. Moreover, based on themodel, it is found that an increase of the competition of the plant for the resource along the slope directionresults in the regular stripes parallel to the hill’s contours. The model provides an interesting mechanism tounderstand the formation of the other regular vegetation stripe in a limited resource condition.

PACS: 87. 23.Cc, 89. 75.Kd

Ecosystems are exposed to all kinds of environ-mental changes. How ecosystems evolve with such en-vironmental changes is one of the main frontiers inecology and biophysics. It has been found that theenvironmental changes can lead to a sudden catas-trophic shifts in the structure and function of ecosys-tems although the changes are slow and gradual.[1]

Such catastrophes are commonly linked to the exis-tence of two alternative stable states in ecosystems,which has been defined as bistability.[1,2] Many meanfield model studies have proposed that an occurrenceof many complex self-organized patterns of vegeta-tion is associated with the bistability.[2−6] When re-source decreases, the spatial self-organized patterns ofplant and resource may be developed. Once resourcescarcity is beyond a threshold, the system jumps toa desertification state with a little vegetation. In-creasing resource again does not recover these local-ized structures, because the initial vegetation and re-source distributions are different. This phenomenon iscalled the hysteresis, which denotes the importance ofself-organized patterns for a better understanding ofcatastrophic shifts. Therefore, understanding the un-derlying mechanism of such patterns is considered asan important step toward a comprehension of the veg-etation degeneration and the desertification process.

Vegetation patterns are regarded as a typical ex-ample of one species (plant) and one resource (water)system and field studies have discovered a wide vari-ety of stable spontaneous self-organized patches in alimited resource condition.[7−13] Thiery et al.[14] haveproposed a random model to simulate the formationof banded vegetation patterns. Recent several models

of vegetation growth based on mean field reaction-diffusion equations[2,4,15,16] can yield stripes, spots,labyrinth, and other ordered arrangements attributedto two positive-feedback mechanisms, i.e., (1) vegeta-tion increases surface-water infiltration, and (2) thelonger the roots, the more soil-water they take up andthe faster the plant grows. On the other hand, somestudies proposed that the positive feedback of theplant in short range and negative competition of plantfor resource in a long range are necessary conditionsfor the formation of the self-organized patterns.[5,17,18]

The situation becomes more complicated if the inter-actions between the plant and resource admit spatialeffects. Furthermore, the effects of the soil conditionand stochastic feature of an ecological process takesa very important role in the plant birth and death.Therefore, in order to understand the mechanism ofthe pattern formation deeply, it is needed to further in-vestigate the effect of complicated ecological processesand the local vegetation and spatial landscape.

In this Letter, we introduce a lattice gas model toinvestigate the pattern formation of the vegetation in alimited resource condition. There are two ingredientsin our model, one is the introduction of the spatial ef-fect of the complicated interactions between soil waterand biomass including the effect of the soil condition.Another ingredient is that the stochastic character of adynamical process in an ecosystem is considered verywell. The model can reproduce a wide range of pat-terns observed in the arid and the semiarid regionsand it is found that an increase of the competition ofthe plant for resource along the slope direction resultsin the regular stripes perpendicular to the hill slope.

∗Supported by the National Natural Science Foundation of China under Grant No 10575055.∗∗ whom correspondence should be addressed. Email: Huadayin@nbu.edu.cnc© 2007 Chinese Physical Society and IOP Publishing Ltd

No. 12 HUA Da-Yin et al. 3587

It provides an interesting mechanism to understandthe formation of the complex vegetation stripes. Us-ing this model, we also investigate the sequences ofvegetation states with scanning two parameters, i.e.,the precipitation parameter and the plant growth rate,and identify the parameter ranges where a hysteresisloop appears and two different stable states coexist.

Model: We employ an L×L square lattice to modela surface. As shown in Fig. 1, the lattice site is calledthe plant position v(i, j), every site can be occupiedby vegetation (v(i, j) = 1) or vacant(v(i, j) = 0). Inorder to describe the interactions between the waterand biomass, we assume that the soil water is locatedin the centre hollow site surrounded by four latticesites and the hollow site is called the water positionw(i, j) (w(i, j) = 1 or 0, it denotes that there is wateror not on the water position). Therefore, a plant po-sition is also surrounded by four water positions. Ourmodel takes into account of the complicated interac-tions between plant and soil water including the effectof the soil condition.

Fig. 1. Sketch of the square lattice surface. Solid linesindicate the links of the plant positions and dotted linesrepresent the links of the water positions.

The distribution of the soil water is changed dueto the effect of the competition between the plants forresource, rainfall and evaporation. Firstly, due to thecompetition of the vegetation for the water resourcein the surroundings, we assume that the soil watercan diffuse to one of the four nearest neighbour (NN)water positions if the sum of the plants on the fourNN plant positions around the new water position islarger than that around the initial one. Therefore, inour model, it is considered that a local aggregation ofvegetation can increase the competition strength andlead to an exhaust of the resource at a short rangeposition. On the other hand, rainfall and evaporationcan lead to a redistribution of the soil water, theseprocesses are related with the precipitation and theprotection of vegetation for the soil water. We assumethat due to the co-effect of rainfall and evaporation, awater position is occupied by water with a probabilitypw = 1.0−exp[−I(Σv(i, j)+v0)], where Σv(i, j) is thesum of plants on the four NN plant positions around

the water position and it denotes a local positive feed-back effect of the vegetation on the soil water, the di-mensionless parameter I includes the rainfall strengthand the protection of the soil water and v0 describesthe effect of the soil itself on the protection of the soilwater. In our model, v0 is regarded as a dimensionlessconstant for a homogeneous soil condition.

The distribution of the vegetation is changed dueto the plant growth and death. For simplicity, we as-sume that a new plant can grow with a probabilitypb = 1.0 − exp[−bΣv(i, j)] on a vacant plant positionwhen one of the four water positions around the va-cant plant position is occupied by water at least, whereΣv(i, j) is the sum of the plants on the four NN plantpositions around the vacant plant position and con-stant b describes the growth ability of the vegetation.On the other hand, a positive feedback between localsoil water and vegetation survival is considered, we as-sume that a plant can be alive with a unit probabilityif the four NN water positions around the plant areoccupied, otherwise a plant can survive with a prob-ability pc = 1.0 − exp[−cΣw(i, j)], where Σw(i, j) isthe sum of the soil water on the four NN water po-sitions around the plant and c is a constant. In ourmodel, we mainly hope to understand the effect ofthe change of environmental condition and vegetationgrowth ability on the evolutional behaviour, then wecan assume that the grazing effect is very light andcan be neglected. If it is needed, it can be consideredconveniently in the survival probability pc.

Before describing the simulation algorithm, we de-fine several parameters for the simulation. Here q de-scribes a relative change rate between soil water andplant because the soil water can change much rapidlythan the plant growth and death. Here q1 describesthe relative effect of the plant competition for resourceon the water redistribution.

Simulation process A random number ρ0 (0 < ρ0 <1) is generated firstly. If ρ0 ≤ q, a trial of soil waterredistribution is taken following item (a), otherwise acorresponding step of plant change is carried out fol-lowing item (b):

(a) A water position w(i, j) is selected randomlyand a random number ρ1 (0 < ρ1 < 1) is chosen. (1)If ρ1 ≤ q1, a water competition process is carried out.Then, if w(i, j) = 1, another NN water position is cho-sen randomly, if the second water position is occupiedthen trial ends, otherwise the water can diffuse to thesecond water position if the sum of plants on the fourNN plant positions around the new water position islarger than that around the initial one. (2) If ρ1 > q1,the effect of rainfall and evaporation is carried out,then w(i, j) = 1 with a probability pw or w(i, j) = 0with a probability 1.0 − pw.

(b) A plant position v(i, j) is selected randomly,(1) if v(i, j) = 0 and if there is water on one of the

3588 HUA Da-Yin et al. Vol. 24

four NN water positions around the vacant plant po-sition, a new plant grows on the plant position with aprobability pb, (2) if v(i, j) = 1 and the four NN wa-ter positions around the plant position are occupiedby water, the plant can survive with an unit prob-ability, otherwise the plant can survive only with aprobability pc.

With Monte Carlo simulation, we can discuss thevegetation patterns under different conditions. Weemploy an L × L (L = 200) square lattice under aperiodic boundary condition. Under the initial con-dition, 30% of water positions w(i, j) are chosen ran-domly and are occupied by water (w(i, j) = 1) and thecorresponding v(i, j) = 1. In our simulation process,the evolution time uses the Monte Carlo step (MCS)(an MCS means L × L Monte Carlo trials). In Fig. 2,typical simulation results for a flat plain are presentwhen the system enters into a steady state. With in-crease of parameter I, the vegetation pattern changesfrom a spot pattern into a labyrinth stripe and thenthe surface is covered almost by the vegetation.

Fig. 2. Patterns of the vegetation on a plain by MonteCarlo simulations. The black place is covered by vegeta-tion. (a) The irregular spot pattern, I = 0.47; (b) theirregular stripe-spot pattern, I = 0.50; (c) the labyrinthstripe pattern, I = 0.53; (d) the surface is almost coveredby the vegetation when I = 0.56. The other parametervalues used are q = 0.75, q1 = 0.99, v0 = 0.00015, b = 1.1,c = 0.001.

Compared to the irregular patterns of the vegeta-tion, the regular patterns, such as the regular stripeson a hillside, have attracted much more interesting.In the previous mean field models,[2,4,15,16] a diffusionterm of water only along the hill slope is introduced,it is very difficult to describe the competition of theplant for the resource along the hillside. On the otherhand, there also exists a water competition betweenplants along the contour. Due to the hillside, once aplace is bare, much more soil water can be soaked in

by the next vegetation of the downhill than the neigh-bouring vegetation along the hill’s contours, the plantscan aggregate much more rapidly and then causes anexhaust of the soil water in the next place along thehillside. The competition between vegetation alongthe hillside is increased. As a result, the slope in-creases the indirect interaction (through the competi-tion for soil water) between the vegetation along thehillside.

Fig. 3. Regular stripe on a hill slope: (a) p1 = 0.75;(b)p1 = 0.85; (c) p1 = 0.95; (d) p1 = 0.99. The otherparameters are the same as those in Fig. 2(c).

In our simulation of the competition process forsoil water on a hill slope, the second NN water po-sition is chosen with a probability p1 from the twoNN sites randomly in the x direction which denotesthe hillside direction and with a probability 1 − p1

from the other two NN sites randomly. In Fig. 3, weshow the simulation results with the increasing param-eter p1. Compared to the pattern in Fig. 2(c) wherep1 = 0.5, the vegetation pattern exhibits a gradualchange from the complex labyrinth stripe into a reg-ular stripe parallel to the y direction which denotesthe parallel direction of the hill’s contours. Comparedwith the simulation patterns in Fig. 2, these simula-tion results show that an increase of the competitionfor resource along the slope direction leads to the ori-entation of the vegetation stripes. Therefore, we canunderstand the labyrinth stripe deeply and can predictother special patterns of vegetation stripes on differ-ent local landscapes, such as a ring stripe. In the sim-ulation process of the competition of the vegetationfor resource on a flat plain, the second site is chosenrandomly from the four NN sites and then the com-petitions in two directions are equal, as a result, thedirection of the stripe can change randomly to form alabyrinth pattern.

Based on our model, we can predict other com-

No. 12 HUA Da-Yin et al. 3589

plex patterns of the vegetation under other disturbedfactors which can change the source competition ondifferent directions. For an example, under a deser-tification condition, soil is an important resource andwind can change the local soil distribution. Alongthe main wind direction, the competition for soil isincreased and a regular stripe perpendicular to themain wind direction is formed.[19]

Fig. 4. Hysteresis loop for a flat plane. (a) The biomasscoverage vs parameter I, b = 1.1; (b) the biomass coveragevs parameter b, I = 0.53. The other parameters are thesame as those in Fig. 2.

In some mean field models,[2,4,15,16] it is shown thatthe sequences of the nonequilibrium steady states ofthe vegetation with the change of precipitation pa-rameter exhibits a bistability and then it predicts acatastrophic shift according to the vegetation struc-ture. We discuss the effect of the parameter I and bby scanning two parameters with the decreasing pa-rameter firstly and then with the increasing param-eter. In Fig. 4, it is shown that the system exhibitsa bistability state and hysteresis behaviour with thescanning of I and b. Under different scanning rate,we can also obtain a hysteresis loop. Furthermore, inthe decreasing process of the parameter, we find thatthe vegetation exhibits a series of complex patterns

as shown in Fig. 2. However, the vegetation exhibitsan aggregation pattern as those as the right inserts inFig. 4 when the parameter increases.

In conclusion, we have introduced a lattice gasmodel to simulate the dynamical process consideringthe spatial feature of the complicated interactions be-tween soil water and biomass. The model can repro-duce a wide range of patterns observed in the aridand the semiarid regions. It is discovered that an in-crease of the indirect interactions between the vegeta-tion along the slope direction leads to the formationof the regular stripes parallel to the hill’s contours. Itprovides an interesting mechanism to understand theformation of the complex vegetation patterns undera limited resource condition. We also investigate thesequences of vegetation states with scanning the pre-cipitation parameter I, the plant growth rate b andthe soil condition parameter v0 to identify the param-eter ranges where a hysteresis loop appears and twodifferent stable states coexist. The scanning simula-tion results denote that the complex structures occurwhen the two parameters decrease, but not when thetwo parameters increase.

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