Research Article Rich Spatiotemporal Dynamics of a ...downloads.hindawi.com/journals/ddns/2014/218053.pdf · Research Article Rich Spatiotemporal Dynamics of a Vegetation Model with

Research ArticleRich Spatiotemporal Dynamics of a Vegetation Model withNoise and Periodic Forcing

Xia-Xia Zhao12 and Jian-Zhong Wang12

1 National Key Laboratory for Electronic Measurement Technology North University of China Taiyuan Shanxi 030051 China2 Key Laboratory of Instrumentation Science and Dynamic Measurement Ministry of Education North University of ChinaTaiyuan Shanxi 030051 China

Correspondence should be addressed to Xia-Xia Zhao zxxzbdxsinacn

Received 1 September 2013 Revised 16 December 2013 Accepted 4 January 2014 Published 3 March 2014

Academic Editor Ryusuke Kon

Copyright copy 2014 X-X Zhao and J-Z Wang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The growth of vegetation is undeniably subject to random fluctuations arising from environmental variability and internal effectsdue to periodic forcing To address these issues we investigated a spatial version of a vegetation model including seasonal rainfallnoise and diffusion By numerical simulations we found that noise can induce the pattern transition from stationary pattern toother patterns More specifically when noise intensity is small patch invasion is induced As noise intensity further increaseschaotic patterns emerge For the system with noise and seasonal rainfall it exhibits frequency-locking phenomena Patternstransition may be a warning signal for the onset of desertification and thus the obtained results may provide some measures toprotect vegetation such as reducing random factors or changing irrigation on vegetation

1 Introduction

Understanding the effect of external variability on vegetationsystems is a problem of great interest An obvious and mostimportant source of external variability is seasonality Theconsequences of the cyclic variation of seasonality have beenwell investigated [1ndash4]The importance of understanding theeffect of the seasonal rainfall on the biological productivity ofa region within a specificmodel is equal to understanding theeffect of additional fluctuations such as interannual variations[5] On the other hand the environments inmodels and labo-ratories aremuch less complex than ecological environmentsand thus vegetation systems can be modeled as open systemsin which the interaction with the environment is noisy [67] Stochasticity can give rise to counterintuitive behaviorssuch as stochastic resonance noise-enhanced stability noise-delayed extinction and noise-induced pattern formation [8ndash10]

In the past years the influences of noise and periodic forc-ing have been well studied in ecosystems [11ndash13] They foundfrom ecological models that noise can induce the emergenceof chaotic patterns Moreover noise and external periodic

forces may cause instability and enhance the oscillation ofthe species density In addition interactions of noise andexternal periodic forces can give rise to resonant patterns andfrequency-locking phenomena

Regular patterns of vegetation have been observed inmany arid and semiarid regions of the world The formationof regular vegetation bands on hillsides of semiarid catch-ments is often attributed to a low scale process of waterredistribution by runoff [14ndash20] It is well recognized thatlocal feedbacks along with dispersion can induce regularvegetation patterns to develop as a result of Turing-likeinstability This phenomenon motivated many recent studiesconcerning the stability of the soilmoisture andplant biomassbalance equations [20] Mathematical analysis showed thatwhen the vegetation system is unstable an even slightly het-erogeneous initial distribution of the vegetation can induceregular spatial patterns Now it is natural to ask whetherthe noise and seasonal rainfall can induce spatial patterns invegetation systems

Themain purpose of this paper is to investigate the effectsof noise and seasonal rainfall on the vegetation patterns In

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014 Article ID 218053 7 pageshttpdxdoiorg1011552014218053

2 Discrete Dynamics in Nature and Society

particular we want to check whether pattern transition orfrequency locking emerges The rest of our paper is arrangedas follows In Section 2 we give the vegetation model withnoise seasonal rainfall and spatial diffusion In Section 3it is found that there is pattern transition from stationarypattern to patch invasion or chaotic pattern Furthermorethe interactions of noise and periodic forces can give riseto frequency-locking phenomena Finally we present someconclusion and discussion

2 Main Model

VonHardenberg et al proposed a partial differential equationmodel with equations for the biomass density 119899(119903 119905) and theground water density 119908(119903 119905) in both space and time [21]

120597119899

120597119905

=

120574119908

1 + 120590119908

119899 minus 1198992minus 120583119899 + nabla

2119899 (1a)

120597119908

120597119905

= 119901 minus (1 minus 120588119899)119908 minus 1199082119899 + 119863nabla

2(119908 minus 120573119899)

minus ]120597 (119908 minus 120572119899)

120597119909

(1b)

where nabla2 = 12059721205971199092+ 12059721205971199102 is the Laplacian operator in

two-dimensional space with 119903 = (119909 119910) (120574119908(1 + 120590119908))119899 isused to describe plant growth at a rate that grows linearlywith 119908 for dry soil minus120583119899 presents mortality and herbivoryminus1198992 represents saturation due to limited nutrients nabla2119899 is the

spread of plants 119901 corresponds to precipitation minus(1minus120588119899)119908 isevaporationminus1199082119899 is local uptake of water by plants119863nabla2(119908minus120573119899) is the feedback term ](120597(119908minus120572119899)120597119909) is used tomodel thedrop of runoff in vegetated areas due to increased infiltration

When combined with noise and seasonal rainfall theoriginal spatially extended model is written as the followingsystem

120597119899

120597119905

=

120574119908

1 + 120590119908

119899 minus 1198992minus 120583119899 + 120578 (119903 119905) + nabla

2119899 (2a)

120597119908

120597119905

= 119901 [1198600+ 1198601sin (120596119905)] minus (1 minus 120588119899)119908 minus 1199082119899 + 120578 (119903 119905)

+ 119863nabla2(119908 minus 120573119899) minus ]

120597 (119908 minus 120572119899)

120597119909

(2b)The seasonal rainfall is assumed to be sinusoidal with ampli-tude 119860

1and frequency 120596 which is considered as an additive

version for the reason that toxins produced by differentpopulations have a significant role in shaping the dynamicalbehavior of ecosystems [22] 120596 is related to a seasonal rainfallvariation There are 12 months in one year and rainfall maybe different in different months so the value of 120596 is 1205876 Inthat case the rainfall reaches its maximum value in Marchand minimum value in September This phenomenon can befound in Southern China

In (2a) and (2b) the stochastic factors are taken intoaccount as the term 120578(119903 119905) which is obtained in the contin-uum limit from themaster equation arising frommicroscopic

interaction in the space [23 24] where the typical white noisewill emerge Recently colored noise and white noise haveboth been used in describing ecological evolution [25 26]White noise is the limiting case of colored noise so weconsider the more general case-colored noise in the presentpaperThe noise term 120578(119903 119905) is introduced additively in spaceand time which is the Ornstein-Uhlenbech process [13 27]The colored noise 120578(119903 119905) which is temporally correlated andwhite in space satisfies

⟨120578 (1199031 1199051) 120578 (1199032 1199052)⟩ =

120576

120591

exp(minus10038161003816100381610038161199051minus 1199052

1003816100381610038161003816

120591

) 120575 (1199031minus 1199032) (3)

where 120591 controls the temporal correlation and 120576measures thenoise intensity Here ⟨⟩means inner product and 120575 presentsimpulse function

Before proceeding to the spatially explicit case the firststep is to have a look at the properties of the local dynamicsThe local system of systems (1a) and (1b) is as follows

119889119899

119889119905

=

120574119908

1 + 120590119908

119899 minus 1198992minus 120583119899 = 119867

1(119899 119908)

119889119908

119889119905

= 119901 minus (1 minus 120588119899)119908 minus 1199082119899 = 119867

2(119899 119908)

(4)

We assume that the local system has a stable equilibrium119864lowast= (119899lowast 119908lowast)which can be obtained by solving119867

1(119899 119908) = 0

and1198672(119899 119908) = 0 The Jacobian matrix corresponding to this

equilibrium point is

119869 = (

1198861111988612

1198862111988622

) (5)

We make the following substitute 119899 = 119899lowast+ 119899( 119903 119905) and

119908 = 119908lowast+ 119908( 119903 119905) into systems (1a) and (1b) and assume that

|119899| ≪ 119899lowast |119908| ≪ 119908

lowast Then in the linear approximation weobtain that

120597119899

120597119905

= 11988611119899 + 11988612119908 + nabla

2119899 (6a)

120597119908

120597119905

= 11988621119899 + 11988622119908 + 119863nabla

2119908

minus 119863120573nabla2119899 minus ]nabla119908 + 120572]nabla119899

(6b)

The initial conditions are assumed as 119899|119905=0

= 119891( 119903) and119908|119905=0

= 119892( 119903) where the functions 119891( 119903) and 119892( 119903) decayrapidly for 119903 rarr plusmninfin Following the standard approach letus now perform a Laplace transformation of the linearizedequations over the two independent variables 119903 and 119905 For 119903

we use the so-called two-sided version of the transformationThe relations for the forward and backward transforms are[28 29]

119899120582119902= int

infin

0

119890minus120582119905119889119905int

+infin

minusinfin

119899 ( 119903 119905) 119890minus119902 119903119889 119903 (7)

119899 ( 119903 119905) = minus

1

41205872int

120573+119894infin

120573minus119894infin

119890120582119905119889120582int

119894infin

minus119894infin

119899120582119902119890119902 119903119889119902 (8)

Discrete Dynamics in Nature and Society 3

where 120582 and 119902 are complex variables After this transforma-tion we have that

(120582 minus 11988611minus 1199022) 119899120582119902minus 11988612119908120582119902= 119865 (119902) (9)

(120582 minus 11988622minus 1198631199022+ ]119902)119908

120582119902

+ (minus11988621+ 119863120573119902

2minus 120572]119902) 119899

120582119902= 119866 (119902)

(10)

where 119865(119902) and 119866(119902) are the transforms of 119891( 119903) and 119892( 119903)The transition between these two cases happens at the criticaladvection coefficient that can be determined from the set ofequations [28 29]

119871 (120582 119902) = 0 (11)

By solving the linear equations (9) and (10) we find 119899120582119902

and then use the backward transformation (8) to obtain thefollowing formal solution

119899 ( 119903 119905) = minus

1

41205872int

120573+119894infin


119890120582119905119889120582

times int

119894infin

minus119894infin

(120582 minus 11988622minus 1198631199022+ ]119902) 119865 (119902) + 119886

12119866 (119902)

119871 (120582 119902)

119890119902 119903119889119902

(12)

Then we obtain the linear stability of this state which isdescribed by the dispersion relation

119871 (120582 119902) = (120582 minus 11988611minus 1199022) (120582 minus 119886

22minus 1198631199022+ ]119902)

+ 11988612(minus11988621+ 119863120573119902

2minus 120572]119902)

(13)

For the sake of convenience we firstly pay attention to the onedimension case By setting 119902 = 119894120581 we have that

120582 =

11988611+ 11988622minus 1205812minus 1198631205812

2

+

]1205811198942

plusmn

radic1198621+ 1198622119894

2

(14)

where 1198621= (119863 minus 1)

21205814+ (211988611119863+ 2119886

22minus ]2 minus 2119886

22119863minus 2119886

11+

411988612119863120573)1205812+ (11988611minus11988622)2+ 41198861211988621and 119862

2= minus2119886

11]120581+ 2]1205813 minus

2119863]1205813 + 211988622]120581 + 4119886

12120572]120581

Straightforward manipulation of (14) yields

Re (120582) =11988611+ 11988622minus 1205812minus 1198631205812

2

plusmn

1

2

radic1

2

(radic1198622

1+ 1198622

2+ 1198621)

Im (120582) =

]1205812

plusmn sign (1198622)

1

2

radic1

2

(radic1198622

1+ 1198622

2minus 1198621)

(15)

where Re(120582) and Im(120582) are the real and imaginary parts of 120582respectively

The condition for a spatial mode 119902 (in one- or two-dimensional space) is to be unstable and thus systems (1a) and(1b) grow into a pattern that is Re(120582) gt 0

3 Main Results

In this section extensive testing was performed throughnumerical integration to describe systems (2a) and (2b)In simulation zero-flux boundary conditions are used andthe time step is Δ119905 = 10

minus5 time unit The space step isΔ119909 = Δ119910 = 01 length unit and the grid sizes in theevolutional simulations are 119872 times 119873 (119872 = 119873 = 500) TheFourier transform method is used for the deterministic partin (1a) and (1b) On the discrete square lattices the stochasticpartial differential equations (2a) and (2b) are integratednumerically by applying the Euler method Several differentdiscrete methods were checked and the results indicate thatthe Fourier transform accurately approximates solutions of(2a) and (2b) On the other hand the Fourier method offersa speed advantage over other numerical methods

31 Pattern Formation of Systems (1a) and (1b) without Noiseand Periodic Force In order to well show the effects of noiseand seasonal rainfall we pay attention to the spatial patternof systems (1a) and (1b) without noise and periodic forceParametersrsquo values are used as 120574 = 16 120590 = 16 120583 = 05120588 = 15 119863 = 10 120572 = 3 and 120573 = 3 which can ensure thatRe(120582) gt 0 for the simulation

In Figure 1 we show the spatial pattern of the vegetationin two-dimensional space The initial density distributioncorresponds to random perturbations around the trivialstationary state One can see that for the cases 119901 = 045 and] = 0 systems (1a) and (1b) can show spotted pattern (seeFigure 1(a)) And for the cases119901 = 013 and ] = 50 the stripe-like and spotted patterns coexist in the space (see Figure 1(b))The pattern structures are consistent with the previous work[30]

For the systems (1a) and (1b) one can see that therandom initial distribution leads to the formation of regularpatterns which means that the distribution of vegetation isself-organized

32 Emergence of Pattern Transition of Systems (1a) and(1b) with Noise In recent years noise-sustained and noise-induced spatial pattern formations have been discussed inecological systems [31 32] Now we firstly investigate thespatial pattern of the systems (1a) and (1b) with ] = 0 whenthe additive noise is turned on It can be seen from Figure 1(a)that typical stationary Turing pattern is shown when thereis only diffusion However combined with additive noiseTuring pattern disappears and crescent moon-like patternemerges (cf Figure 2(a)) When the noise intensity andtemporal correlation further increase patch invasion willemerge which can be seen from Figure 2(b)

For the case ] = 0 we found that there is chaotic patternfor some appropriate values of noise intensity and tempo-ral correlation shown in Figure 3 Chaos may lead to theextinction of or disorder in the vegetation density [19] whichimplies that noise may induce the onset of desertification

In order to see the effects of noise intensity and tem-poral correlation on pattern dynamics we give regions ofpattern structures with respect to the two parameters in


(a)

018

02

022

024

026

028

03

032

(b)

Figure 1 Snapshots of contour pictures of the vegetation at 119905 = 1000 with 120574 = 16 120590 = 16 120583 = 05 120588 = 15 119863 = 10 120572 = 3 and 120573 = 3(a) 119901 = 045 and ] = 0 (b) 119901 = 013 and ] = 50

(a)

01

02

03

04

05

(b)

Figure 2 Snapshots of contour pictures of the vegetation at 119905 = 1000 with noise Other parametersrsquo values are the same as in Figure 1(a)(a) 120576 = 0001 and 120591 = 5 (b) 120576 = 01 and 120591 = 11

Figures 4 and 5 One can see that for ] = 0 small valuesof noise intensity and temporal correlation cannot inducepatch invasion And for ] = 0 one can choose the appropriatevalues of noise intensity and temporal correlation to ensureemergence of chaotic patterns

33 Frequency Locking of Systems (2a) and (2b)withNoise andPeriodic Force It iswell known that an external periodic forceapplied to a nonlinear pendulum can cause the pendulum tobecome entrained at a frequency which is rationally relatedto the applied frequency a phenomenon known as frequencylocking It is useful to reveal the complexity of the ecosystem

It can be found that systems (2a) and (2b) produce oscilla-tions about period 119879out with respect to the external period119879in = 2120587120596 this phenomenon is called frequency lockingor resonant response that is when the system produces onespike within each of the 119865 (119865 = 1 2 3 ) periods of theexternal force that is 119865 1 resonant response In the presentpaper the output period 119879out is defined as follows 119879

119894is the

time interval between the 119894th spike and the 119894 + 1th spike 119902spikes are taken into account and their average value is 119879outwhere 119879out = sum

119902

119894=1119879119894(119902 minus 1) [33]

It is checked by numerical simulations that when systems(1a) and (1b) are only combined with periodic forcing there


02

025

03

035

04

Figure 3 Snapshots of contour pictures of the vegetation at 119905 = 1000with 120576 = 007 and 120591 = 6 Other parametersrsquo values are the same as inFigure 1(b)

0 5 10 150

02

04

06

08

1

I

II

120576

120591

Figure 4 Regions of pattern structures of systems (1a) and (1b) withrespect to noise intensity and temporal correlation for ] = 0 I nopatch invasion II patch invasion

are no frequency locking phenomena As a result we do notshow the dynamical behavior of systems (1a) and (1b) onlywith periodic forcing When there are seasonal rainfall andnoise term we show that there are 1 1 and 2 1 frequencylocking in the systems (2a) and (2b) which we plot in Figures6 and 7 Note that the initial conditions are chosen as in thefollowing form

119899 (119903 0) = 119899lowast+ 1206011119909 + 1206012119910 + 120593 (16a)

119908 (119903 0) = 119908lowast (16b)

where 1206011 1206012 and 120593 are positive constants

0 5 10 150

02

04

06

08

1

I

II

120591

120576

Figure 5 Regions of pattern structures of systems (1a) and (1b)with respect to noise intensity and temporal correlation for ] = 0 Ichaotic pattern II no chaotic pattern

Temporal correlation 120591 of the colored noise plays impor-tant role in the spatial pattern formation and transition ofthe different patterns In order to well understand the phasetransition by the influence of temporal correlation 120591 we givephase diagram of the 120591 minus 120576 parameter space in Figure 8by performing a series of simulations that is fixing 120591 andscanning the noise intensity 120576 when the frequency lockingevidently changes in the long term It can be seen from thisfigure that there are 1 1 and 2 1 frequency locking in differentregions

4 Discussion and Conclusion

In this paper we investigated a vegetation model combinedwith seasonal rainfall noise and spatial diffusion By per-forming a series of numerical simulations we found thatthere was emergence of pattern transition from stationarypattern to patch invasion What is more chaotic pattern willappear if noise intensity is large And for the system withboth noise and periodic forcing it exhibits frequency-lockingphenomena The results showed that noise and seasonalrainfall play an important role in vegetation patterns

Climate fluctuation is also considered as a source of veg-etation spatial pattern which means that all the parametersin systems (2a) and (2b) can show temporal and spatialvariations And it is believed that climate effect can enhancethe likelihood of catastrophic shifts to the desert state orcontrol the transitions between preferential states in bistabledynamics [34ndash36] Furthermore in [37] it was shown thatstatic disorder in terms of environmental variability hadinfluence on the pattern dynamics in a spatially ecologicalsystem Then one may want to check whether this phe-nomenon can occur in vegetation systems These issues needto be well addressed in the further investigation

The mechanisms inducing the change of structure ordynamics of vegetation populations are among the most


0 20 40 60 80 100

Time

01

02

03

04

01

02

03

⟨n⟩

(a)

0 20 40 60 80 100

Time

minus04

minus02

0

02

04

A1sin

(120596t)

(b)

Figure 6 The 1 1 frequency-locking oscillation with the values of the parameters 1198601= 035 120596 = 1205876 120576 = 011 and 120591 = 3 (a) The mean

concentration of vegetation (b) periodic forces

0 20 40 60 80 100

Time

01

02

03

04

01

02

03

⟨n⟩

(a)

0 20 40 60 80 100

Time

minus04

minus02

0

02

04

A1sin

(120596t)

(b)



0 5 10 1501

02

03

04

05

06

07

08

(A)

(B)

120591

120576

Figure 8 Phase diagram in 120591 minus 120576 parameter space with 1198601= 035

and 120596 = 1205876 Regions (A) and (B) are corresponding to the 1 1 and2 1 frequency locking

challenging research areas in ecology [38ndash40] However avariety of rich behaviors observed in vegetation populationsare far from being well understood [19] In this sense richdynamical behaviors emerging from our work contribute

to a better understanding of wetland ecosystems From anecological standpoint the results reported in this paperindicate that noise and external forcing can cause patternstransition in vegetation dynamics which implies that theymay induce the onset of desertification As a result we needto take measures to decrease stochastic factors and changeirrigation to protect vegetation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

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[2] B T Grenfell B M Bolker and A Kleczkowski ldquoSeasonalityand extinction in chaotic metapopulationsrdquo Proceedings of theRoyal Society B vol 259 no 1354 pp 97ndash103 1995

[3] A A King and W M Schaffer ldquoThe geometry of a populationcycle a mechanistic model of snowshoe hare demographyrdquoEcology vol 82 no 3 pp 814ndash830 2001


[4] S Altizer A Dobson P Hosseini P Hudson M Pascual and PRohani ldquoSeasonality and the dynamics of infectious diseasesrdquoEcology Letters vol 9 no 4 pp 467ndash484 2006

[5] V Guttal and C Jayaprakash ldquoSelf-organization and produc-tivity in semi-arid ecosystems implications of seasonality inrainfallrdquo Journal of Theoretical Biology vol 248 no 3 pp 490ndash500 2007

[6] A Provata IM Sokolov and B Spagnolo ldquoEditorial ecologicalcomplex systemsrdquoEuropean Physical Journal B vol 65 no 3 pp307ndash314 2008

[7] O Richter ldquoSpatio-temporal patterns of gene flow and dispersalunder temperature increaserdquoMathematical Biosciences vol 218no 1 pp 15ndash23 2009

[8] G Q Sun Z Jin L Li and Q X Liu ldquoThe role of noise ina predator-prey model with Allee effectrdquo Journal of BiologicalPhysics vol 35 pp 185ndash196 2009

[9] G-Q Sun L Li Z Jin and B-L Li ldquoEffect of noise on thepattern formation in an epidemic modelrdquo Numerical Methodsfor Partial Differential Equations vol 26 no 5 pp 1168ndash11792010

[10] J G Vilar and R V Sole ldquoEffects of noise in symmetric two-species competitionrdquo Physical Review Letters vol 80 pp 4099ndash4102 1998

[11] Q X Liu and Z Jin ldquoResonance and frequency-lockingphenomena in spatially extended phytoplanktonndashzooplanktonsystem with additive noise and periodic forcesrdquo Journal ofStatistical Mechanics vol 5 Article ID P05011 2008

[12] F Rao W Wang and Z Li ldquoSpatiotemporal complexity ofa predator-prey system with the effect of noise and externalforcingrdquoChaos Solitons amp Fractals vol 41 no 4 pp 1634ndash16442009

[13] G-Q Sun Z Jin Q-X Liu and B-L Li ldquoRich dynamics ina predator-prey model with both noise and periodic forcerdquoBioSystems vol 100 no 1 pp 14ndash22 2010

[14] O Lejeune M Tlidi and P Couteron ldquoLocalized vegetationpatches a self-organized response to resource scarcityrdquo PhysicalReview E vol 66 no 1 Article ID 010901 2002

[15] S B Boaler and C A H Hodge ldquoVegetation stripes inSomalilandrdquo Journal of Ecology vol 50 pp 465ndash474 1962

[16] F T Maestre and J Cortina ldquoSpatial patterns of surface soilproperties and vegetation in aMediterranean semi-arid stepperdquoPlant and Soil vol 241 no 2 pp 279ndash291 2002

[17] C A Klausmeier ldquoRegular and irregular patterns in semiaridvegetationrdquo Science vol 284 no 5421 pp 1826ndash1828 1999

[18] M Rietkerk S C Dekker P C De Ruiter and J Van DeKoppel ldquoSelf-organized patchiness and catastrophic shifts inecosystemsrdquo Science vol 305 no 5692 pp 1926ndash1929 2004

[19] M Rietkerk and J van de Koppel ldquoRegular pattern formationin real ecosystemsrdquo Trends in Ecology and Evolution vol 23 no3 pp 169ndash175 2008

[20] C Valentin J M DrsquoHerbes and J Poesen ldquoSoil and watercomponents of banded vegetation patternsrdquo Catena vol 37 no1-2 pp 1ndash24 1999

[21] J Von Hardenberg E Meron M Shachak and Y ZarmildquoDiversity of vegetation patterns and desertificationrdquo PhysicalReview Letters vol 87 no 19 Article ID 198101 2001

[22] K L Kirk and J J Gilbert ldquoVariation in herbivore response tochemical defenses zooplankton foraging on toxic cyanobacte-riardquo Ecology vol 73 no 6 pp 2208ndash2217 1992

[23] T Reichenbach M Mobilia and E Frey ldquoNoise and correla-tions in a spatial population model with cyclic competitionrdquoPhysical Review Letters vol 99 no 23 Article ID 238105 2007

[24] T Reichenbach M Mobilia and E Frey ldquoMobility promotesand jeopardizes biodiversity in rock-paper-scissors gamesrdquoNature vol 448 no 7157 pp 1046ndash1049 2007

[25] RMankin A Ainsaar A Haljas and E Reiter ldquoTrichotomous-noise-induced catastrophic shifts in symbiotic ecosystemsrdquoPhysical Review E vol 65 no 5 Article ID 051108 2002

[26] R Mankin A Sauga A Ainsaar A Haljas and K PaunelldquoColored-noise-induced discontinuous transitions in symbioticecosystemsrdquo Physical Review E vol 69 no 6 Article ID 0611062004

[27] D J Higham ldquoAn algorithmic introduction to numericalsimulation of stochastic differential equationsrdquo SIAM Reviewvol 43 no 3 pp 525ndash546 2001

[28] G Q Sun Z Jin Q X Liu and L Li ldquoDynamical complexityof a spatial predatorndashprey model with migrationrdquo EcologicalModelling vol 219 pp 248ndash255 2008

[29] G Q Sun L Li Z Jin and B L Li ldquoPattern formation in aspatial plant-wrackmodel with tide effect on thewrackrdquo Journalof Biological Physics vol 36 pp 161ndash174 2010

[30] G-Q Sun J Li B Yu and Z Jin ldquoNoise induced patterntransition in a vegetation modelrdquo Applied Mathematics andComputation vol 221 pp 463ndash468 2013

[31] C Zimmer ldquoLife after chaosrdquo Science vol 284 pp 83ndash86 1999[32] B Blasius A Huppert and L Stone ldquoComplex dynamics and

phase synchronization in spatially extended ecological systemsrdquoNature vol 399 no 6734 pp 354ndash359 1999

[33] F N Si Q X Liu J Z Zhang and L Q Zhou ldquoPropagationof travelling waves in sub-excitable systems driven by noise andperiodic forcingrdquo The European Physical Journal B vol 60 pp507ndash513 2007

[34] P DrsquoOdorico F Laio and L Ridolfi ldquoNoise-induced stability indryland plant ecosystemsrdquo Proceedings of the National Academyof Sciences of the United States of America vol 102 pp 10819ndash10822 2005

[35] P DrsquoOdorico A Porporato and L Ridolfi ldquoTransition betweenstable states in the dynamics of soil developmentrdquo GeophysicalResearch Letters vol 28 pp 595ndash598 2001

[36] V Isham D R Cox I Rodrıguez-Iturbe A Porporato andS Manfreda ldquoRepresentation of space-time variability of soilmoisturerdquo Proceedings of the Royal Society A vol 461 no 2064pp 4035ndash4055 2005

[37] M Fras and M Gosak ldquoSpatiotemporal patterns provoked byenvironmental variability in a predatorndashprey modelrdquo Biosys-tems vol 114 pp 172ndash177 2013

[38] L Ridolfi P DrsquoOdorico and F Laio ldquoVegetation dynamicsinduced by phreatophyte-aquifer interactionsrdquo Journal of The-oretical Biology vol 248 no 2 pp 301ndash310 2007

[39] D Tilman Dynamics and Structure of Plant CommunitiesPrinceton University Press Princeton NJ USA 1988

[40] J P Grover Resource Competition Chapman and Hall NewYork NY USA 1997

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


particular we want to check whether pattern transition orfrequency locking emerges The rest of our paper is arrangedas follows In Section 2 we give the vegetation model withnoise seasonal rainfall and spatial diffusion In Section 3it is found that there is pattern transition from stationarypattern to patch invasion or chaotic pattern Furthermorethe interactions of noise and periodic forces can give riseto frequency-locking phenomena Finally we present someconclusion and discussion

2 Main Model

VonHardenberg et al proposed a partial differential equationmodel with equations for the biomass density 119899(119903 119905) and theground water density 119908(119903 119905) in both space and time [21]

120597119899

120597119905

=

120574119908

1 + 120590119908

119899 minus 1198992minus 120583119899 + nabla

2119899 (1a)

120597119908

120597119905

= 119901 minus (1 minus 120588119899)119908 minus 1199082119899 + 119863nabla

2(119908 minus 120573119899)

minus ]120597 (119908 minus 120572119899)

120597119909

(1b)

where nabla2 = 12059721205971199092+ 12059721205971199102 is the Laplacian operator in

two-dimensional space with 119903 = (119909 119910) (120574119908(1 + 120590119908))119899 isused to describe plant growth at a rate that grows linearlywith 119908 for dry soil minus120583119899 presents mortality and herbivoryminus1198992 represents saturation due to limited nutrients nabla2119899 is the

spread of plants 119901 corresponds to precipitation minus(1minus120588119899)119908 isevaporationminus1199082119899 is local uptake of water by plants119863nabla2(119908minus120573119899) is the feedback term ](120597(119908minus120572119899)120597119909) is used tomodel thedrop of runoff in vegetated areas due to increased infiltration

When combined with noise and seasonal rainfall theoriginal spatially extended model is written as the followingsystem

120597119899

120597119905

=

120574119908

1 + 120590119908

119899 minus 1198992minus 120583119899 + 120578 (119903 119905) + nabla

2119899 (2a)

120597119908

120597119905

= 119901 [1198600+ 1198601sin (120596119905)] minus (1 minus 120588119899)119908 minus 1199082119899 + 120578 (119903 119905)

+ 119863nabla2(119908 minus 120573119899) minus ]

120597 (119908 minus 120572119899)

120597119909

(2b)The seasonal rainfall is assumed to be sinusoidal with ampli-tude 119860

1and frequency 120596 which is considered as an additive

version for the reason that toxins produced by differentpopulations have a significant role in shaping the dynamicalbehavior of ecosystems [22] 120596 is related to a seasonal rainfallvariation There are 12 months in one year and rainfall maybe different in different months so the value of 120596 is 1205876 Inthat case the rainfall reaches its maximum value in Marchand minimum value in September This phenomenon can befound in Southern China

In (2a) and (2b) the stochastic factors are taken intoaccount as the term 120578(119903 119905) which is obtained in the contin-uum limit from themaster equation arising frommicroscopic

interaction in the space [23 24] where the typical white noisewill emerge Recently colored noise and white noise haveboth been used in describing ecological evolution [25 26]White noise is the limiting case of colored noise so weconsider the more general case-colored noise in the presentpaperThe noise term 120578(119903 119905) is introduced additively in spaceand time which is the Ornstein-Uhlenbech process [13 27]The colored noise 120578(119903 119905) which is temporally correlated andwhite in space satisfies

⟨120578 (1199031 1199051) 120578 (1199032 1199052)⟩ =

120576

120591

exp(minus10038161003816100381610038161199051minus 1199052

1003816100381610038161003816

120591

) 120575 (1199031minus 1199032) (3)

where 120591 controls the temporal correlation and 120576measures thenoise intensity Here ⟨⟩means inner product and 120575 presentsimpulse function

Before proceeding to the spatially explicit case the firststep is to have a look at the properties of the local dynamicsThe local system of systems (1a) and (1b) is as follows

119889119899

119889119905

=

120574119908

1 + 120590119908

119899 minus 1198992minus 120583119899 = 119867

1(119899 119908)

119889119908

119889119905

= 119901 minus (1 minus 120588119899)119908 minus 1199082119899 = 119867

2(119899 119908)

(4)

We assume that the local system has a stable equilibrium119864lowast= (119899lowast 119908lowast)which can be obtained by solving119867

1(119899 119908) = 0

and1198672(119899 119908) = 0 The Jacobian matrix corresponding to this

equilibrium point is

119869 = (

1198861111988612

1198862111988622

) (5)

We make the following substitute 119899 = 119899lowast+ 119899( 119903 119905) and

119908 = 119908lowast+ 119908( 119903 119905) into systems (1a) and (1b) and assume that

|119899| ≪ 119899lowast |119908| ≪ 119908

lowast Then in the linear approximation weobtain that

120597119899

120597119905

= 11988611119899 + 11988612119908 + nabla

2119899 (6a)

120597119908

120597119905

= 11988621119899 + 11988622119908 + 119863nabla

2119908

minus 119863120573nabla2119899 minus ]nabla119908 + 120572]nabla119899

(6b)

The initial conditions are assumed as 119899|119905=0

= 119891( 119903) and119908|119905=0

= 119892( 119903) where the functions 119891( 119903) and 119892( 119903) decayrapidly for 119903 rarr plusmninfin Following the standard approach letus now perform a Laplace transformation of the linearizedequations over the two independent variables 119903 and 119905 For 119903

we use the so-called two-sided version of the transformationThe relations for the forward and backward transforms are[28 29]

119899120582119902= int

infin

0

119890minus120582119905119889119905int

+infin

minusinfin

119899 ( 119903 119905) 119890minus119902 119903119889 119903 (7)

119899 ( 119903 119905) = minus

1

41205872int

120573+119894infin


119890120582119905119889120582int

119894infin

minus119894infin

119899120582119902119890119902 119903119889119902 (8)




(120582 minus 11988622minus 1198631199022+ ]119902)119908

120582119902

+ (minus11988621+ 119863120573119902

2minus 120572]119902) 119899

120582119902= 119866 (119902)

(10)


119871 (120582 119902) = 0 (11)



119899 ( 119903 119905) = minus

1

41205872int

120573+119894infin


119890120582119905119889120582

times int

119894infin

minus119894infin

(120582 minus 11988622minus 1198631199022+ ]119902) 119865 (119902) + 119886

12119866 (119902)

119871 (120582 119902)

119890119902 119903119889119902

(12)



22minus 1198631199022+ ]119902)

+ 11988612(minus11988621+ 119863120573119902

2minus 120572]119902)

(13)


120582 =

11988611+ 11988622minus 1205812minus 1198631205812

2

+

]1205811198942

plusmn

radic1198621+ 1198622119894

2

(14)

where 1198621= (119863 minus 1)

21205814+ (211988611119863+ 2119886


22119863minus 2119886

11+

411988612119863120573)1205812+ (11988611minus11988622)2+ 41198861211988621and 119862

2= minus2119886

11]120581+ 2]1205813 minus

2119863]1205813 + 211988622]120581 + 4119886

12120572]120581


Re (120582) =11988611+ 11988622minus 1205812minus 1198631205812

2

plusmn

1

2

radic1

2

(radic1198622

1+ 1198622

2+ 1198621)

Im (120582) =

]1205812


1

2

radic1

2

(radic1198622

1+ 1198622

2minus 1198621)

(15)



3 Main Results










(a)

018

02

022

024

026

028

03

032

(b)


(a)

01

02

03

04

05

(b)





119894is the


119902

119894=1119879119894(119902 minus 1) [33]



02

025

03

035

04


0 5 10 150

02

04

06

08

1

I

II

120576

120591



119899 (119903 0) = 119899lowast+ 1206011119909 + 1206012119910 + 120593 (16a)

119908 (119903 0) = 119908lowast (16b)


0 5 10 150

02

04

06

08

1

I

II

120591

120576








0 20 40 60 80 100

Time

01

02

03

04

01

02

03

⟨n⟩

(a)

0 20 40 60 80 100

Time

minus04

minus02

0

02

04

A1sin

(120596t)

(b)



0 20 40 60 80 100

Time

01

02

03

04

01

02

03

⟨n⟩

(a)

0 20 40 60 80 100

Time

minus04

minus02

0

02

04

A1sin

(120596t)

(b)



0 5 10 1501

02

03

04

05

06

07

08

(A)

(B)

120591

120576







References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(120582 minus 11988622minus 1198631199022+ ]119902)119908

120582119902

+ (minus11988621+ 119863120573119902

2minus 120572]119902) 119899

120582119902= 119866 (119902)

(10)


119871 (120582 119902) = 0 (11)



119899 ( 119903 119905) = minus

1

41205872int

120573+119894infin


119890120582119905119889120582

times int

119894infin

minus119894infin

(120582 minus 11988622minus 1198631199022+ ]119902) 119865 (119902) + 119886

12119866 (119902)

119871 (120582 119902)

119890119902 119903119889119902

(12)



22minus 1198631199022+ ]119902)

+ 11988612(minus11988621+ 119863120573119902

2minus 120572]119902)

(13)


120582 =

11988611+ 11988622minus 1205812minus 1198631205812

2

+

]1205811198942

plusmn

radic1198621+ 1198622119894

2

(14)

where 1198621= (119863 minus 1)

21205814+ (211988611119863+ 2119886


22119863minus 2119886

11+

411988612119863120573)1205812+ (11988611minus11988622)2+ 41198861211988621and 119862

2= minus2119886

11]120581+ 2]1205813 minus

2119863]1205813 + 211988622]120581 + 4119886

12120572]120581


Re (120582) =11988611+ 11988622minus 1205812minus 1198631205812

2

plusmn

1

2

radic1

2

(radic1198622

1+ 1198622

2+ 1198621)

Im (120582) =

]1205812


1

2

radic1

2

(radic1198622

1+ 1198622

2minus 1198621)

(15)



3 Main Results










(a)

018

02

022

024

026

028

03

032

(b)


(a)

01

02

03

04

05

(b)





119894is the


119902

119894=1119879119894(119902 minus 1) [33]



02

025

03

035

04


0 5 10 150

02

04

06

08

1

I

II

120576

120591



119899 (119903 0) = 119899lowast+ 1206011119909 + 1206012119910 + 120593 (16a)

119908 (119903 0) = 119908lowast (16b)


0 5 10 150

02

04

06

08

1

I

II

120591

120576








0 20 40 60 80 100

Time

01

02

03

04

01

02

03

⟨n⟩

(a)

0 20 40 60 80 100

Time

minus04

minus02

0

02

04

A1sin

(120596t)

(b)



0 20 40 60 80 100

Time

01

02

03

04

01

02

03

⟨n⟩

(a)

0 20 40 60 80 100

Time

minus04

minus02

0

02

04

A1sin

(120596t)

(b)



0 5 10 1501

02

03

04

05

06

07

08

(A)

(B)

120591

120576







References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a)

018

02

022

024

026

028

03

032

(b)


(a)

01

02

03

04

05

(b)





119894is the


119902

119894=1119879119894(119902 minus 1) [33]



02

025

03

035

04


0 5 10 150

02

04

06

08

1

I

II

120576

120591



119899 (119903 0) = 119899lowast+ 1206011119909 + 1206012119910 + 120593 (16a)

119908 (119903 0) = 119908lowast (16b)


0 5 10 150

02

04

06

08

1

I

II

120591

120576








0 20 40 60 80 100

Time

01

02

03

04

01

02

03

⟨n⟩

(a)

0 20 40 60 80 100

Time

minus04

minus02

0

02

04

A1sin

(120596t)

(b)



0 20 40 60 80 100

Time

01

02

03

04

01

02

03

⟨n⟩

(a)

0 20 40 60 80 100

Time

minus04

minus02

0

02

04

A1sin

(120596t)

(b)



0 5 10 1501

02

03

04

05

06

07

08

(A)

(B)

120591

120576







References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










02

025

03

035

04


0 5 10 150

02

04

06

08

1

I

II

120576

120591



119899 (119903 0) = 119899lowast+ 1206011119909 + 1206012119910 + 120593 (16a)

119908 (119903 0) = 119908lowast (16b)


0 5 10 150

02

04

06

08

1

I

II

120591

120576








0 20 40 60 80 100

Time

01

02

03

04

01

02

03

⟨n⟩

(a)

0 20 40 60 80 100

Time

minus04

minus02

0

02

04

A1sin

(120596t)

(b)



0 20 40 60 80 100

Time

01

02

03

04

01

02

03

⟨n⟩

(a)

0 20 40 60 80 100

Time

minus04

minus02

0

02

04

A1sin

(120596t)

(b)



0 5 10 1501

02

03

04

05

06

07

08

(A)

(B)

120591

120576







References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 100

Time

01

02

03

04

01

02

03

⟨n⟩

(a)

0 20 40 60 80 100

Time

minus04

minus02

0

02

04

A1sin

(120596t)

(b)



0 20 40 60 80 100

Time

01

02

03

04

01

02

03

⟨n⟩

(a)

0 20 40 60 80 100

Time

minus04

minus02

0

02

04

A1sin

(120596t)

(b)



0 5 10 1501

02

03

04

05

06

07

08

(A)

(B)

120591

120576







References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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