Research ArticleMathematical Models Arising in the Fractal Forest Gap via LocalFractional Calculus

Chun-Ying Long1 Yang Zhao2 and Hossein Jafari34

1 School of Civil Engineering and Architecture Nanchang University Nanchang 330031 China2 Electronic and Information Technology Department Jiangmen Polytechnic Jiangmen 529090 China3Department of Mathematics Faculty of Mathematical Sciences University of Mazandaran Babolsar 47415-416 Iran4African Institute for Mathematical Sciences Muizenberg 7945 South Africa

Correspondence should be addressed to Yang Zhao zhaoyang19781023gmailcom

Received 8 January 2014 Accepted 10 February 2014 Published 18 March 2014

Academic Editor Carlo Cattani

Copyright copy 2014 Chun-Ying Long et al 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 forest new gap models via local fractional calculus are investigated The JABOWA and FORSKA models are extended to dealwith the growth of individual trees defined on Cantor sets The local fractional growth equations with local fractional derivativeand difference are discussed Our results are first attempted to show the key roles for the nondifferentiable growth of individualtrees

1 Introduction

Fractals had been used to describe special problems inbiology and ecology [1ndash4] because of the measure of natureobjects underlying the geometry replacing the complex real-world objects by describing the Euclidean ideas Fractaldimension was applied to describe the measure of thecomplexity in biology and ecology In forestry the fractalgeometry had been applied to estimate stand density predictforest succession and describe the form of trees [5ndash7] Thescaling of dynamics in hierarchical structure was investigatedin [8ndash12]The ecological resilience example fromboreal forestwas presented in the context [8] The quantitative theoryof forest structure was discussed [9] The allometric scalinglaws in biology were proposed in the works [10 11] Basedupon the cross-scale analysis the geometry and dynamicsof ecosystems were considered and the structure ecosystemsacross scales in time and space were discussed in [12 13]Fractal forestry was modeled by using the scaling of thetesting parameters for ecological complexity

Forest gap model (JABOWA) developed by Botkin etal [14ndash16] was the first simulation model for gap-phasereplacement It was applied to describe a forest as a mosaicof closed canopies and simulate forest dynamics based upon

the establishment growth and death of individual trees [17ndash20] The JABOWA model in the form of the FORET model(called JABOWA-FORET) was further developed in [21ndash24]The JABOWA model of the simulation of stand structure ina forest gap model was improved in [24] and the FORSKA[25] was proposed by Botkin et al In [14ndash16] the ecologicalfunctions are continuous In [10ndash13] the ecological functionswere expressed across scales in time and space However asit is shown in Figure 1 the ecological functions distinguishinghierarchical size scales in nature such as the measures of treesize and measure of soil fertility are defined on Cantor setsThe above approaches do not deal with them

Local fractional calculus theory [26ndash38] was applied tohandle the nondifferentiable functions defined on Cantorsets The heat-conduction transport Maxwell diffusionwave Fokker-Planck and themechanical structure equationswere usefully shown (see for more details [28ndash36] and thecited references therein) In order to simulate forest dynamicson the basis of the establishment growth and death ofindividual trees defined on Cantor sets the aim of this paperis to present the forest new gapmodels for simulating the gap-phase replacement by employing the local fractional calculus

The paper has been organized as follows In Section 2we review the JABOWA and FORSKA models for the forest

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 782393 6 pageshttpdxdoiorg1011552014782393

2 Abstract and Applied Analysis

0 02 04 06 08 1

1

09

08

07

06

05

04

03

02

01

0

x

y(x)

Figure 1 The ecological function 119891(119909) defined on Cantor set

succession In Section 3 we propose JABOWA and FORSKAmodels for the fractal forest succession Finally Section 4 isconclusions

2 Growth Models for Forest Gap

In this section we will revise the JABOWA and FORSKAmodels

21 The JABOWA Model The growth equation with differ-ence form is given by [15 16 24 39]

Δ119863

Δ119905= 119866 sdot 119863 sdot 120601 (119863) sdot

1

119887 (119863)sdot 119891 (119890) (1)

where the function 119863 is diameter at breast height of thetrees the parameter 119867 is tree height 119866 is a growth ratethe function 119887(119863) is a quantity encapsulating this allometricrelationship 119891(119890) is a quantity influencing the abiotic andbiotic environment on tree growth 120601(119863) = (1minus119863sdot119867(119863max sdot119867max)) and 119863max and 119867max are the maximum measures ofthe tree dimensions

The parameter 119891(119890) is simulated as follows [24]

119891 (119890) = 1198921(AL) sdot 119892

2(SBAR) sdot 119892

3(119863119863) (2)

where 1198921(AL) is a quantity of available light 119892

2(SBAR) is a

quantity of stand basal area and 1198922(SBAR) is a quantity of

the annual degree-day sum It was referred to as Liebigrsquos lawof the minimum [24]

The allometric relationship with a parabolic form iswritten as follows [24 40]

119887 (119863) = 1198871+ 1198872119863 + 11988731198632 (3)

where 1198871 1198872 and 119887

3are parameters

Leaf area index reads as follows [24]

LAI = 1205831198632 (4)

where 120583 = 119888119896 with a species-specific parameter 119888 and thescale leaf weight per tree to the projected leaf area 119896

In order to implement a newheight-diameter relationship[40] the differential formof growth equation in the JABOWAmodel was suggested as follows [41]

119889119863

119889119905= 119866 sdot 119863 sdot 120593 (119863) (5)

where the function has the following form

120593 (119863) =1 minus 119867119867max

2119867max minus 119887 (minus119904119863119887 + 2) 119890minus119863119904119887

(6)

with the parameter 119887 = 119867max minus137 and the initial slope valueof the height diameter relationship 119904

22The FORSKAModel TheFORSKAmodel was developedfor unmanaged natural forests and tree height had 119867-119863relationship with the FORSKAmodel given by [24 25 39 41]

119867 = 13 + (119867max minus 13) sdot (1 minus 119890minus119904119863(119867maxminus13)) (7)

where the parameter 119904 is the initial slope value of the heightdiameter relationship at 119863 = 0 119867max is the maximummeasure of the tree dimension119863 is diameter at breast heightof the trees and119867 is tree height

As it is known the trees in the real forest do not followthe 119867-119863 relationship The growth equation with differentialform can be written as follows [24 25]

119889119867

119889119863= 119904 sdot 119891 (119867) (8)

where

119891 (119867) =119867 minus 13

119867max minus 13 (9)

3 The JABOWA and FORSKA Models forthe Fractal Forest Succession

In this section based upon the local fractional calculustheory we show the JABOWA and FORSKA models forthe fractal forest succession At first we start with the localfractional derivative

31 Local Fractional Derivative We now give the local frac-tional calculus and the recent results

If1003816100381610038161003816119891 (119909) minus 119891 (1199090)

1003816100381610038161003816 lt 120576120572 (10)

with |119909 minus 1199090| lt 120575 for 120576 120575 gt 0 and 120576 120575 isin 119877 then we denote

[26ndash28]

119891 (119909) isin 119862120572(119886 119887) (11)

If 119891(119909) isin 119862120572(119886 119887) then we have [26]

dim119867(119865 cap (119886 119887)) = dim

119867(119862120572(119886 119887)) = 120572 (12)

Abstract and Applied Analysis 3

where 119862120572(119886 119887) = 119891 119891(119909) is local fractional continuous

119909 isin 119865 cap (119886 119887)Let119891(119909) isin 119862

120572(119886 119887)The local fractional derivative of119891(119909)

of order 120572 at 119909 = 1199090is defined as [26ndash34]

119891(120572) (1199090) =

119889120572119891 (119909)

119889119909120572

10038161003816100381610038161003816100381610038161003816119909=1199090= lim119909rarr1199090

Δ120572 (119891 (119909) minus 119891 (1199090))

(119909 minus 1199090)120572

(13)

where Δ120572(119891(119909) minus 119891(1199090)) cong Γ(1 + 120572)Δ(119891(119909) minus 119891(119909

0))

For 0 lt 120572 le 1 the increment of 119891(119909) can be written asfollows [26 27]

Δ120572119891 (119909) = 119891(120572)(119909) (Δ119909)

120572 + 120582(Δ119909)120572 (14)

where Δ119909 is an increment of 119909 and 120582 rarr 0 as Δ119909 rarr 0For 0 lt 120572 le 1 the 120572-local fractional differential of 119891(119909)

reads as [26 27]

119889120572119891 = 119891(120572) (119909) (119889119909)120572 (15)

From (14) we have approximate formula in the form

Δ120572119891 (119909) cong Γ (1 + 120572) Δ (119891 (119909) minus 119891 (1199090)) (16)

Let 119891(119909) isin 119862120572(119886 119887) The local fractional integral of 119891(119909) of

order 120572 is given by [26ndash31]

119886119868(120572)119887

119891 (119909) =1

Γ (1 + 120572)int119887

119886

119891 (119905) (119889119905)120572

=1

Γ (1 + 120572)limΔ119905rarr0

119895=119873minus1

sum119895=0

119891 (119905119895) (Δ119905119895)120572

(17)

where Δ119905119895= 119905119895+1

minus 119905119895 Δ119905 = maxΔ119905

1 Δ1199052 Δ119905119895 and

[119905119895 119905119895+1] 119895 = 0 119873 minus 1 119905

0= 119886 119905

119873= 119887 is a partition of

the interval [119886 119887]The 120572-dimensional Hausdorff measure 119867

120572is calculated

by [26]

119867120572(119865 cap (0 119909)) =

0119868(120572)119909

1 =119909120572

Γ (1 + 120572) (18)

32 The Local Fractional JABOWA Models (LFJABOWA)Here we structure the LFJABOWA models via local frac-tional derivative and difference

From (5) when

119866 sdot 120593 (119863) = 1205820119863120573minus1 (19)

the Enquist growthmodel in JABOWAmodel reads as [11 42]

119889119863

119889119905= 1205820119863120573 (20)

where 119863 is the diameter at breast height 1205820is the scaling

coefficient and 120573 is the fractal dimensionMaking use of the fractional complex transform [29] and

(20) the growth equation in the JABOWA model with localfractional derivative (LFJABOWA) is suggested by

119889120572119863

119889119905120572= 1205820119863120573 (21)

0 02 04 06 08 1

45

4

35

3

25

2

15

1

x

y(x)

Figure 2 The graph of (22) with parameters 120573 = 1 1205820= 1 and

1199050= 0

0 02 04 06 08 1

40

35

30

25

20

15

10

5

0

x

y(x)

Figure 3 The graph of (22) with parameters 120573 = 1 1205820= 2 and

1199050= 0

where 119863 is the diameter at breast height 1205820is the scaling

coefficient and 120572 and 120573 are the fractal dimensions In orderto illustrate the difference from the works presented in [42]we consider the following case when 120573 = 1 (21) can beintegrated to give

119863 (119905) = 119864120572(1205820119905120572) minus 119864

120572(12058201199051205720) (22)

For the parameters 120573 = 1 119905 = 0 the solutions of (21) withdifferent values 120582

0= 1 120582

0= 2 120582

0= 3 and 120582

0= 4 are

respectively shown in Figures 2 3 4 and 5Using the fractional complex transform (5) becomes into

119889120572119863

119889119905120572= 119866 sdot 119863 sdot 120593 (119863) (23)

Comparing (22) and (23) we have

119889120572119863

119889119905120572= 120595 (119905 119863) (24)

4 Abstract and Applied Analysis

0 02 04 06 08 1

700

600

500

400

300

200

100

0

x

y(x)

Figure 4 The graph of (22) with parameters 120573 = 1 1205820= 3 and

1199050= 0

where

120595 (119905 119863) = 119866 sdot 119863 sdot 120593 (119863) (25)

or

120595 (119905 119863) = 1205820119863120573 (26)

From (25) and (26) we could get

120593 (119863) = 119863120573minus1 (27)

1205820= 119866 (28)

Hence the local fractional JABOWA model (LFJABOWA)reads as follows

119889120572119863

119889119905120572= 120595 (119905 119863) (29)

where 120595(119905 119863) is a nondifferentiable function and 119863 is thediameter at breast height

In view of (16) from (29) we give the difference formof local fractional JABOWA model (LFJABOWA) in thefollowing form

Δ120572119863

(Δ119905)120572= 120595 (119905 119863) (30)

where 119863 is the diameter at breast height and Δ120572119863(119905) cong Γ(1 +120572)Δ(119863(119905) minus 119863(119905

0))

When the fractal dimension 120572 is equal to 1 we get thegeneralized form of (1) namely

Δ119863

Δ119905= 120595 (119905 119863) (31)

where

120595 (119905 119863) = 119866 sdot 119863 sdot 120601 (119863) sdot1

119887 (119863)sdot 119891 (119890) (32)

0 02 04 06 08 1

25

2

15

1

05

0

x

y(x)

times104

Figure 5 The graph of (22) with parameters 120573 = 1 1205820= 4 and

1199050= 0

33 The Local Fractional FORSKA Models (LFFORSKA)Here we present the LFFORSKA models via local fractionalderivative and difference

The tree height 119867 is the nondifferentiable functionnamely

1003816100381610038161003816119867 (119863) minus 119867 (1198630)1003816100381610038161003816 lt 120589120572 (33)

where for 120589 120599 gt 0 120589 120599 isin 119877 and |119863 minus 1198630| lt 120599

Following the fractional complex transform [29] from (8)we have the local fractional growth equation in the followingform

119889120572119867

119889119863120572= 119904 sdot 119891 (119867) (34)

where119891(119867) is a local fractional continuous function119867 is thetree height and 119904 is a parameter

Therefore the generalized form of (34) is suggested asfollows

119889120572119867

119889119863120572= 120594 (119863119867) (35)

where 119863 is the diameter at breast height and 119867 is the treeheight In view of (16) (35) can be rewritten as follows

Δ120572119867

(Δ119863)120572= 120594 (119863119867) (36)

where1003816100381610038161003816119863 minus 119863

0

1003816100381610038161003816 lt 120599

1003816100381610038161003816119867 (119863) minus 119867 (1198630)1003816100381610038161003816 lt 120589120572

(37)

The expression (36) is the difference form of local frac-tional FORSKA model (LFFORSKA)

Abstract and Applied Analysis 5

4 Conclusions

In this work we investigated the local fractional models forthe fractal forest succession Based on the local fractionaloperators we suggested the differential and difference formsof the local fractional JABOWA and FORSKA models Thenondifferentiable growths of individual trees were discussedIt is a good start for solving the rhetorical models for thefractal forest succession in the mathematical analysis

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the Science and TechnologyCommission Planning Project of Jiangxi Province (no20122BBG70078)

References

[1] A Fielding ldquoApplications of fractal geometry to biologyrdquoComputer Applications in the Biosciences vol 8 no 4 pp 359ndash366 1992

[2] B L Li ldquoFractal geometry applications in description andanalysis of patch patterns and patch dynamicsrdquo EcologicalModelling vol 132 no 1 pp 33ndash50 2000

[3] G Sugihara and RMMay ldquoApplications of fractals in ecologyrdquoTrends in Ecology amp Evolution vol 5 no 3 pp 79ndash86 1990

[4] J H Brown V K Gupta B L Li B T Milne C Restrepo andG BWest ldquoThe fractal nature of nature power laws ecologicalcomplexity and biodiversityrdquo Philosophical Transactions of theRoyal Society of London B vol 357 no 1421 pp 619ndash626 2002

[5] N D Lorimer R G Haight and R A LearyThe Fractal ForestFractal Geometry and Applications in Forest Science NorthCentral Forest Experiment Station Berkeley Calif USA 1994

[6] B Zeide ldquoFractal geometry in forestry applicationsrdquo ForestEcology and Management vol 46 no 3 pp 179ndash188 1991

[7] B Zeide ldquoAnalysis of the 32 power law of self-thinningrdquo ForestScience vol 33 no 2 pp 517ndash537 1987

[8] G D Peterson ldquoScaling ecological dynamics self-organizationhierarchical structure and ecological resiliencerdquo ClimaticChange vol 44 no 3 pp 291ndash309 2000

[9] G BWest B J Enquist and JH Brown ldquoA general quantitativetheory of forest structure and dynamicsrdquo Proceedings of theNational Academy of Sciences vol 106 no 17 pp 7040ndash70452009

[10] G B West J H Brown and B J Enquist ldquoA general model forthe origin of allometric scaling laws in biologyrdquo Science vol 276no 5309 pp 122ndash126 1997

[11] B J Enquist G B West E L Charnov and J H BrownldquoAllometric scaling of production and life-history variation invascular plantsrdquo Nature vol 401 no 6756 pp 907ndash911 1999

[12] C S Holling ldquoCross-scale morphology geometry and dynam-ics of ecosystemsrdquo Ecological Monographs vol 62 no 4 pp447ndash502 1992

[13] G Peterson C R Allen and C S Holling ldquoEcologicalresilience biodiversity and scalerdquo Ecosystems vol 1 no 1 pp6ndash18 1998

[14] D B Botkin J F Janak and J R Wallis ldquoRationale limitationsand assumptions of a northeastern forest growth simulatorrdquoIBM Journal of Research and Development vol 16 no 2 pp 101ndash116 1972

[15] D B Botkin J F Janak and J R Wallis ldquoSome ecologicalconsequences of a computer model of forest growthrdquo TheJournal of Ecology vol 60 no 3 pp 849ndash872 1972

[16] D B Botkin Forest Dynamics An Ecological Model OxfordUniversity Press New York NY USA 1993

[17] G L Perry and N J Enright ldquoSpatial modelling of vegetationchange in dynamic landscapes a review of methods andapplicationsrdquo Progress in Physical Geography vol 30 no 1 pp47ndash72 2006

[18] R Chen and R R Twilley ldquoA gap dynamic model of mangroveforest development along gradients of soil salinity and nutrientresourcesrdquo Journal of Ecology vol 86 no 1 pp 37ndash51 1998

[19] RW Hall ldquoJABOWA revealed-finallyrdquo Ecology vol 75 no 3 p859 1994

[20] M I Ashraf C P A Bourque D A MacLean T Erdle andF R Meng ldquoUsing JABOWA-3 for forest growth and yieldpredictions under diverse forest conditions of Nova ScotiaCanadardquoTheForestry Chronicle vol 88 no 6 pp 708ndash721 2012

[21] H H Shugart and I R Noble ldquoA computer model of successionand fire response of the high-altitude Eucalyptus forest of theBrindabella Range Australian Capital Territoryrdquo AustralianJournal of Ecology vol 6 no 2 pp 149ndash164 1981

[22] H H Shugart andD CWest ldquoDevelopment of an Appalachiandeciduous forest succession model and its application toassessment of the impact of the chestnut blightrdquo Journal ofEnvironmental Management vol 5 pp 161ndash179 1977

[23] H H Shugart A Theory of Forest Dynamics The EcologicalImplications of Forest Succession Models Springer New YorkNY USA 1984

[24] H Bugmann ldquoA review of forest gap modelsrdquo Climatic Changevol 51 no 3-4 pp 259ndash305 2001

[25] M Lindner R Sievanen and H Pretzsch ldquoImproving thesimulation of stand structure in a forest gap modelrdquo ForestEcology and Management vol 95 no 2 pp 183ndash195 1997

[26] X-J Yang Advanced Local Fractional Calculus and Its Applica-tions World Science New York NY USA 2012

[27] X J Yang Local Fractional Functional Analysis and Its Applica-tions Asian Academic Hong Kong China 2011

[28] X J Yang and D Baleanu ldquoFractal heat conduction problemsolved by local fractional variation iteration methodrdquo ThermalScience vol 17 no 2 pp 625ndash628 2013

[29] X J Yang D Baleanu and J H He ldquoTransport equationsin fractal porous media within fractional complex transformmethodrdquo Proceedings of the Romanian Academy A vol 14 no4 pp 287ndash292 2013

[30] C F Liu S S Kong and S J Yuan ldquoReconstructive schemesfor variational iterationmethod within Yang-Laplace transformwith application to fractal heat conduction problemrdquo ThermalScience vol 17 no 3 pp 715ndash721 2013

[31] A M Yang Y Z Zhang and Y Long ldquoThe Yang-Fouriertransforms to heat-conduction in a semi-infinite fractal barrdquoThermal Science vol 17 no 3 pp 707ndash713 2013

[32] Y Zhao D Baleanu C Cattani D F Cheng and X-JYang ldquoMaxwellrsquos equations on Cantor Sets a local fractional

6 Abstract and Applied Analysis

approachrdquo Advances in High Energy Physics vol 2013 ArticleID 686371 6 pages 2013

[33] X J Yang D Baleanu andW P Zhong ldquoApproximate solutionsfor diffusion equations onCantor space-timerdquoProceedings of theRomanian Academy A vol 14 no 2 pp 127ndash133 2013

[34] X-J Yang H M Srivastava J H He and D Baleanu ldquoCantor-type cylindrical-coordinate method for differential equationswith local fractional derivativesrdquo Physics Letters A vol 377 no28ndash30 pp 1696ndash1700 2013

[35] K M Kolwankar and A D Gangal ldquoLocal fractional Fokker-Planck equationrdquo Physical Review Letters vol 80 no 2 pp 214ndash217 1998

[36] A Carpinteri B Chiaia and P Cornetti ldquoStatic-kinematicduality and the principle of virtual work in the mechanics offractal mediardquo Computer Methods in Applied Mechanics andEngineering vol 191 no 1-2 pp 3ndash19 2001

[37] F Ben Adda and J Cresson ldquoAbout non-differentiable func-tionsrdquo Journal of Mathematical Analysis and Applications vol263 no 2 pp 721ndash737 2001

[38] A Babakhani and V Daftardar-Gejji ldquoOn calculus of localfractional derivativesrdquo Journal of Mathematical Analysis andApplications vol 270 no 1 pp 66ndash79 2002

[39] H K Bugmann X Yan M T Sykes et al ldquoA comparison offorest gap models model structure and behaviourrdquo ClimaticChange vol 34 no 2 pp 289ndash313 1996

[40] A D Moore ldquoOn the maximum growth equation used in forestgap simulation modelsrdquo Ecological Modelling vol 45 no 1 pp63ndash67 1989

[41] A C Risch C Heiri and H Bugmann ldquoSimulating structuralforest patterns with a forest gap model a model evaluationrdquoEcological Modelling vol 181 no 2-3 pp 161ndash172 2005

[42] D A Coomes and R B Allen ldquoEffects of size competition andaltitude on tree growthrdquo Journal of Ecology vol 95 no 5 pp1084ndash1097 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

0 02 04 06 08 1

1

09

08

07

06

05

04

03

02

01

0

x

y(x)

Figure 1 The ecological function 119891(119909) defined on Cantor set

succession In Section 3 we propose JABOWA and FORSKAmodels for the fractal forest succession Finally Section 4 isconclusions

2 Growth Models for Forest Gap

In this section we will revise the JABOWA and FORSKAmodels

21 The JABOWA Model The growth equation with differ-ence form is given by [15 16 24 39]

Δ119863

Δ119905= 119866 sdot 119863 sdot 120601 (119863) sdot

1

119887 (119863)sdot 119891 (119890) (1)

where the function 119863 is diameter at breast height of thetrees the parameter 119867 is tree height 119866 is a growth ratethe function 119887(119863) is a quantity encapsulating this allometricrelationship 119891(119890) is a quantity influencing the abiotic andbiotic environment on tree growth 120601(119863) = (1minus119863sdot119867(119863max sdot119867max)) and 119863max and 119867max are the maximum measures ofthe tree dimensions

The parameter 119891(119890) is simulated as follows [24]

119891 (119890) = 1198921(AL) sdot 119892

2(SBAR) sdot 119892

3(119863119863) (2)

where 1198921(AL) is a quantity of available light 119892

2(SBAR) is a

quantity of stand basal area and 1198922(SBAR) is a quantity of

the annual degree-day sum It was referred to as Liebigrsquos lawof the minimum [24]

The allometric relationship with a parabolic form iswritten as follows [24 40]

119887 (119863) = 1198871+ 1198872119863 + 11988731198632 (3)

where 1198871 1198872 and 119887

3are parameters

Leaf area index reads as follows [24]

LAI = 1205831198632 (4)

where 120583 = 119888119896 with a species-specific parameter 119888 and thescale leaf weight per tree to the projected leaf area 119896

In order to implement a newheight-diameter relationship[40] the differential formof growth equation in the JABOWAmodel was suggested as follows [41]

119889119863

119889119905= 119866 sdot 119863 sdot 120593 (119863) (5)

where the function has the following form

120593 (119863) =1 minus 119867119867max

2119867max minus 119887 (minus119904119863119887 + 2) 119890minus119863119904119887

(6)

with the parameter 119887 = 119867max minus137 and the initial slope valueof the height diameter relationship 119904

22The FORSKAModel TheFORSKAmodel was developedfor unmanaged natural forests and tree height had 119867-119863relationship with the FORSKAmodel given by [24 25 39 41]

119867 = 13 + (119867max minus 13) sdot (1 minus 119890minus119904119863(119867maxminus13)) (7)

where the parameter 119904 is the initial slope value of the heightdiameter relationship at 119863 = 0 119867max is the maximummeasure of the tree dimension119863 is diameter at breast heightof the trees and119867 is tree height

As it is known the trees in the real forest do not followthe 119867-119863 relationship The growth equation with differentialform can be written as follows [24 25]

119889119867

119889119863= 119904 sdot 119891 (119867) (8)

where

119891 (119867) =119867 minus 13

119867max minus 13 (9)

3 The JABOWA and FORSKA Models forthe Fractal Forest Succession

In this section based upon the local fractional calculustheory we show the JABOWA and FORSKA models forthe fractal forest succession At first we start with the localfractional derivative

31 Local Fractional Derivative We now give the local frac-tional calculus and the recent results

If1003816100381610038161003816119891 (119909) minus 119891 (1199090)

1003816100381610038161003816 lt 120576120572 (10)

with |119909 minus 1199090| lt 120575 for 120576 120575 gt 0 and 120576 120575 isin 119877 then we denote

[26ndash28]

119891 (119909) isin 119862120572(119886 119887) (11)

If 119891(119909) isin 119862120572(119886 119887) then we have [26]

dim119867(119865 cap (119886 119887)) = dim

119867(119862120572(119886 119887)) = 120572 (12)

Abstract and Applied Analysis 3

where 119862120572(119886 119887) = 119891 119891(119909) is local fractional continuous

119909 isin 119865 cap (119886 119887)Let119891(119909) isin 119862

120572(119886 119887)The local fractional derivative of119891(119909)

of order 120572 at 119909 = 1199090is defined as [26ndash34]

119891(120572) (1199090) =

119889120572119891 (119909)

119889119909120572

10038161003816100381610038161003816100381610038161003816119909=1199090= lim119909rarr1199090

Δ120572 (119891 (119909) minus 119891 (1199090))

(119909 minus 1199090)120572

(13)

where Δ120572(119891(119909) minus 119891(1199090)) cong Γ(1 + 120572)Δ(119891(119909) minus 119891(119909

0))

For 0 lt 120572 le 1 the increment of 119891(119909) can be written asfollows [26 27]

Δ120572119891 (119909) = 119891(120572)(119909) (Δ119909)

120572 + 120582(Δ119909)120572 (14)

where Δ119909 is an increment of 119909 and 120582 rarr 0 as Δ119909 rarr 0For 0 lt 120572 le 1 the 120572-local fractional differential of 119891(119909)

reads as [26 27]

119889120572119891 = 119891(120572) (119909) (119889119909)120572 (15)

From (14) we have approximate formula in the form

Δ120572119891 (119909) cong Γ (1 + 120572) Δ (119891 (119909) minus 119891 (1199090)) (16)

Let 119891(119909) isin 119862120572(119886 119887) The local fractional integral of 119891(119909) of

order 120572 is given by [26ndash31]

119886119868(120572)119887

119891 (119909) =1

Γ (1 + 120572)int119887

119886

119891 (119905) (119889119905)120572

=1

Γ (1 + 120572)limΔ119905rarr0

119895=119873minus1

sum119895=0

119891 (119905119895) (Δ119905119895)120572

(17)

where Δ119905119895= 119905119895+1

minus 119905119895 Δ119905 = maxΔ119905

1 Δ1199052 Δ119905119895 and

[119905119895 119905119895+1] 119895 = 0 119873 minus 1 119905

0= 119886 119905

119873= 119887 is a partition of

the interval [119886 119887]The 120572-dimensional Hausdorff measure 119867

120572is calculated

by [26]

119867120572(119865 cap (0 119909)) =

0119868(120572)119909

1 =119909120572

Γ (1 + 120572) (18)

32 The Local Fractional JABOWA Models (LFJABOWA)Here we structure the LFJABOWA models via local frac-tional derivative and difference

From (5) when

119866 sdot 120593 (119863) = 1205820119863120573minus1 (19)

the Enquist growthmodel in JABOWAmodel reads as [11 42]

119889119863

119889119905= 1205820119863120573 (20)

where 119863 is the diameter at breast height 1205820is the scaling

coefficient and 120573 is the fractal dimensionMaking use of the fractional complex transform [29] and

(20) the growth equation in the JABOWA model with localfractional derivative (LFJABOWA) is suggested by

119889120572119863

119889119905120572= 1205820119863120573 (21)

0 02 04 06 08 1

45

4

35

3

25

2

15

1

x

y(x)

Figure 2 The graph of (22) with parameters 120573 = 1 1205820= 1 and

1199050= 0

0 02 04 06 08 1

40

35

30

25

20

15

10

5

0

x

y(x)

Figure 3 The graph of (22) with parameters 120573 = 1 1205820= 2 and

1199050= 0

where 119863 is the diameter at breast height 1205820is the scaling

coefficient and 120572 and 120573 are the fractal dimensions In orderto illustrate the difference from the works presented in [42]we consider the following case when 120573 = 1 (21) can beintegrated to give

119863 (119905) = 119864120572(1205820119905120572) minus 119864

120572(12058201199051205720) (22)

For the parameters 120573 = 1 119905 = 0 the solutions of (21) withdifferent values 120582

0= 1 120582

0= 2 120582

0= 3 and 120582

0= 4 are

respectively shown in Figures 2 3 4 and 5Using the fractional complex transform (5) becomes into

119889120572119863

119889119905120572= 119866 sdot 119863 sdot 120593 (119863) (23)

Comparing (22) and (23) we have

119889120572119863

119889119905120572= 120595 (119905 119863) (24)

4 Abstract and Applied Analysis

0 02 04 06 08 1

700

600

500

400

300

200

100

0

x

y(x)

Figure 4 The graph of (22) with parameters 120573 = 1 1205820= 3 and

1199050= 0

where

120595 (119905 119863) = 119866 sdot 119863 sdot 120593 (119863) (25)

or

120595 (119905 119863) = 1205820119863120573 (26)

From (25) and (26) we could get

120593 (119863) = 119863120573minus1 (27)

1205820= 119866 (28)

Hence the local fractional JABOWA model (LFJABOWA)reads as follows

119889120572119863

119889119905120572= 120595 (119905 119863) (29)

where 120595(119905 119863) is a nondifferentiable function and 119863 is thediameter at breast height

In view of (16) from (29) we give the difference formof local fractional JABOWA model (LFJABOWA) in thefollowing form

Δ120572119863

(Δ119905)120572= 120595 (119905 119863) (30)

where 119863 is the diameter at breast height and Δ120572119863(119905) cong Γ(1 +120572)Δ(119863(119905) minus 119863(119905

0))

When the fractal dimension 120572 is equal to 1 we get thegeneralized form of (1) namely

Δ119863

Δ119905= 120595 (119905 119863) (31)

where

120595 (119905 119863) = 119866 sdot 119863 sdot 120601 (119863) sdot1

119887 (119863)sdot 119891 (119890) (32)

0 02 04 06 08 1

25

2

15

1

05

0

x

y(x)

times104

Figure 5 The graph of (22) with parameters 120573 = 1 1205820= 4 and

1199050= 0

33 The Local Fractional FORSKA Models (LFFORSKA)Here we present the LFFORSKA models via local fractionalderivative and difference

The tree height 119867 is the nondifferentiable functionnamely

1003816100381610038161003816119867 (119863) minus 119867 (1198630)1003816100381610038161003816 lt 120589120572 (33)

where for 120589 120599 gt 0 120589 120599 isin 119877 and |119863 minus 1198630| lt 120599

Following the fractional complex transform [29] from (8)we have the local fractional growth equation in the followingform

119889120572119867

119889119863120572= 119904 sdot 119891 (119867) (34)

where119891(119867) is a local fractional continuous function119867 is thetree height and 119904 is a parameter

Therefore the generalized form of (34) is suggested asfollows

119889120572119867

119889119863120572= 120594 (119863119867) (35)

where 119863 is the diameter at breast height and 119867 is the treeheight In view of (16) (35) can be rewritten as follows

Δ120572119867

(Δ119863)120572= 120594 (119863119867) (36)

where1003816100381610038161003816119863 minus 119863

0

1003816100381610038161003816 lt 120599

1003816100381610038161003816119867 (119863) minus 119867 (1198630)1003816100381610038161003816 lt 120589120572

(37)

The expression (36) is the difference form of local frac-tional FORSKA model (LFFORSKA)

Abstract and Applied Analysis 5

4 Conclusions

In this work we investigated the local fractional models forthe fractal forest succession Based on the local fractionaloperators we suggested the differential and difference formsof the local fractional JABOWA and FORSKA models Thenondifferentiable growths of individual trees were discussedIt is a good start for solving the rhetorical models for thefractal forest succession in the mathematical analysis

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the Science and TechnologyCommission Planning Project of Jiangxi Province (no20122BBG70078)

References

[1] A Fielding ldquoApplications of fractal geometry to biologyrdquoComputer Applications in the Biosciences vol 8 no 4 pp 359ndash366 1992

[2] B L Li ldquoFractal geometry applications in description andanalysis of patch patterns and patch dynamicsrdquo EcologicalModelling vol 132 no 1 pp 33ndash50 2000

[3] G Sugihara and RMMay ldquoApplications of fractals in ecologyrdquoTrends in Ecology amp Evolution vol 5 no 3 pp 79ndash86 1990

[4] J H Brown V K Gupta B L Li B T Milne C Restrepo andG BWest ldquoThe fractal nature of nature power laws ecologicalcomplexity and biodiversityrdquo Philosophical Transactions of theRoyal Society of London B vol 357 no 1421 pp 619ndash626 2002

[5] N D Lorimer R G Haight and R A LearyThe Fractal ForestFractal Geometry and Applications in Forest Science NorthCentral Forest Experiment Station Berkeley Calif USA 1994

[6] B Zeide ldquoFractal geometry in forestry applicationsrdquo ForestEcology and Management vol 46 no 3 pp 179ndash188 1991

[7] B Zeide ldquoAnalysis of the 32 power law of self-thinningrdquo ForestScience vol 33 no 2 pp 517ndash537 1987

[8] G D Peterson ldquoScaling ecological dynamics self-organizationhierarchical structure and ecological resiliencerdquo ClimaticChange vol 44 no 3 pp 291ndash309 2000

[9] G BWest B J Enquist and JH Brown ldquoA general quantitativetheory of forest structure and dynamicsrdquo Proceedings of theNational Academy of Sciences vol 106 no 17 pp 7040ndash70452009

[10] G B West J H Brown and B J Enquist ldquoA general model forthe origin of allometric scaling laws in biologyrdquo Science vol 276no 5309 pp 122ndash126 1997

[11] B J Enquist G B West E L Charnov and J H BrownldquoAllometric scaling of production and life-history variation invascular plantsrdquo Nature vol 401 no 6756 pp 907ndash911 1999

[12] C S Holling ldquoCross-scale morphology geometry and dynam-ics of ecosystemsrdquo Ecological Monographs vol 62 no 4 pp447ndash502 1992

[13] G Peterson C R Allen and C S Holling ldquoEcologicalresilience biodiversity and scalerdquo Ecosystems vol 1 no 1 pp6ndash18 1998

[14] D B Botkin J F Janak and J R Wallis ldquoRationale limitationsand assumptions of a northeastern forest growth simulatorrdquoIBM Journal of Research and Development vol 16 no 2 pp 101ndash116 1972

[15] D B Botkin J F Janak and J R Wallis ldquoSome ecologicalconsequences of a computer model of forest growthrdquo TheJournal of Ecology vol 60 no 3 pp 849ndash872 1972

[16] D B Botkin Forest Dynamics An Ecological Model OxfordUniversity Press New York NY USA 1993

[17] G L Perry and N J Enright ldquoSpatial modelling of vegetationchange in dynamic landscapes a review of methods andapplicationsrdquo Progress in Physical Geography vol 30 no 1 pp47ndash72 2006

[18] R Chen and R R Twilley ldquoA gap dynamic model of mangroveforest development along gradients of soil salinity and nutrientresourcesrdquo Journal of Ecology vol 86 no 1 pp 37ndash51 1998

[19] RW Hall ldquoJABOWA revealed-finallyrdquo Ecology vol 75 no 3 p859 1994

[20] M I Ashraf C P A Bourque D A MacLean T Erdle andF R Meng ldquoUsing JABOWA-3 for forest growth and yieldpredictions under diverse forest conditions of Nova ScotiaCanadardquoTheForestry Chronicle vol 88 no 6 pp 708ndash721 2012

[21] H H Shugart and I R Noble ldquoA computer model of successionand fire response of the high-altitude Eucalyptus forest of theBrindabella Range Australian Capital Territoryrdquo AustralianJournal of Ecology vol 6 no 2 pp 149ndash164 1981

[22] H H Shugart andD CWest ldquoDevelopment of an Appalachiandeciduous forest succession model and its application toassessment of the impact of the chestnut blightrdquo Journal ofEnvironmental Management vol 5 pp 161ndash179 1977

[23] H H Shugart A Theory of Forest Dynamics The EcologicalImplications of Forest Succession Models Springer New YorkNY USA 1984

[24] H Bugmann ldquoA review of forest gap modelsrdquo Climatic Changevol 51 no 3-4 pp 259ndash305 2001

[25] M Lindner R Sievanen and H Pretzsch ldquoImproving thesimulation of stand structure in a forest gap modelrdquo ForestEcology and Management vol 95 no 2 pp 183ndash195 1997

[26] X-J Yang Advanced Local Fractional Calculus and Its Applica-tions World Science New York NY USA 2012

[27] X J Yang Local Fractional Functional Analysis and Its Applica-tions Asian Academic Hong Kong China 2011

[28] X J Yang and D Baleanu ldquoFractal heat conduction problemsolved by local fractional variation iteration methodrdquo ThermalScience vol 17 no 2 pp 625ndash628 2013

[29] X J Yang D Baleanu and J H He ldquoTransport equationsin fractal porous media within fractional complex transformmethodrdquo Proceedings of the Romanian Academy A vol 14 no4 pp 287ndash292 2013

[30] C F Liu S S Kong and S J Yuan ldquoReconstructive schemesfor variational iterationmethod within Yang-Laplace transformwith application to fractal heat conduction problemrdquo ThermalScience vol 17 no 3 pp 715ndash721 2013

[31] A M Yang Y Z Zhang and Y Long ldquoThe Yang-Fouriertransforms to heat-conduction in a semi-infinite fractal barrdquoThermal Science vol 17 no 3 pp 707ndash713 2013

[32] Y Zhao D Baleanu C Cattani D F Cheng and X-JYang ldquoMaxwellrsquos equations on Cantor Sets a local fractional

6 Abstract and Applied Analysis

approachrdquo Advances in High Energy Physics vol 2013 ArticleID 686371 6 pages 2013

[33] X J Yang D Baleanu andW P Zhong ldquoApproximate solutionsfor diffusion equations onCantor space-timerdquoProceedings of theRomanian Academy A vol 14 no 2 pp 127ndash133 2013

[34] X-J Yang H M Srivastava J H He and D Baleanu ldquoCantor-type cylindrical-coordinate method for differential equationswith local fractional derivativesrdquo Physics Letters A vol 377 no28ndash30 pp 1696ndash1700 2013

[35] K M Kolwankar and A D Gangal ldquoLocal fractional Fokker-Planck equationrdquo Physical Review Letters vol 80 no 2 pp 214ndash217 1998

[36] A Carpinteri B Chiaia and P Cornetti ldquoStatic-kinematicduality and the principle of virtual work in the mechanics offractal mediardquo Computer Methods in Applied Mechanics andEngineering vol 191 no 1-2 pp 3ndash19 2001

[37] F Ben Adda and J Cresson ldquoAbout non-differentiable func-tionsrdquo Journal of Mathematical Analysis and Applications vol263 no 2 pp 721ndash737 2001

[38] A Babakhani and V Daftardar-Gejji ldquoOn calculus of localfractional derivativesrdquo Journal of Mathematical Analysis andApplications vol 270 no 1 pp 66ndash79 2002

[39] H K Bugmann X Yan M T Sykes et al ldquoA comparison offorest gap models model structure and behaviourrdquo ClimaticChange vol 34 no 2 pp 289ndash313 1996

[40] A D Moore ldquoOn the maximum growth equation used in forestgap simulation modelsrdquo Ecological Modelling vol 45 no 1 pp63ndash67 1989

[41] A C Risch C Heiri and H Bugmann ldquoSimulating structuralforest patterns with a forest gap model a model evaluationrdquoEcological Modelling vol 181 no 2-3 pp 161ndash172 2005

[42] D A Coomes and R B Allen ldquoEffects of size competition andaltitude on tree growthrdquo Journal of Ecology vol 95 no 5 pp1084ndash1097 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

where 119862120572(119886 119887) = 119891 119891(119909) is local fractional continuous

119909 isin 119865 cap (119886 119887)Let119891(119909) isin 119862

120572(119886 119887)The local fractional derivative of119891(119909)

of order 120572 at 119909 = 1199090is defined as [26ndash34]

119891(120572) (1199090) =

119889120572119891 (119909)

119889119909120572

10038161003816100381610038161003816100381610038161003816119909=1199090= lim119909rarr1199090

Δ120572 (119891 (119909) minus 119891 (1199090))

(119909 minus 1199090)120572

(13)

where Δ120572(119891(119909) minus 119891(1199090)) cong Γ(1 + 120572)Δ(119891(119909) minus 119891(119909

0))

For 0 lt 120572 le 1 the increment of 119891(119909) can be written asfollows [26 27]

Δ120572119891 (119909) = 119891(120572)(119909) (Δ119909)

120572 + 120582(Δ119909)120572 (14)

where Δ119909 is an increment of 119909 and 120582 rarr 0 as Δ119909 rarr 0For 0 lt 120572 le 1 the 120572-local fractional differential of 119891(119909)

reads as [26 27]

119889120572119891 = 119891(120572) (119909) (119889119909)120572 (15)

From (14) we have approximate formula in the form

Δ120572119891 (119909) cong Γ (1 + 120572) Δ (119891 (119909) minus 119891 (1199090)) (16)

Let 119891(119909) isin 119862120572(119886 119887) The local fractional integral of 119891(119909) of

order 120572 is given by [26ndash31]

119886119868(120572)119887

119891 (119909) =1

Γ (1 + 120572)int119887

119886

119891 (119905) (119889119905)120572

=1

Γ (1 + 120572)limΔ119905rarr0

119895=119873minus1

sum119895=0

119891 (119905119895) (Δ119905119895)120572

(17)

where Δ119905119895= 119905119895+1

minus 119905119895 Δ119905 = maxΔ119905

1 Δ1199052 Δ119905119895 and

[119905119895 119905119895+1] 119895 = 0 119873 minus 1 119905

0= 119886 119905

119873= 119887 is a partition of

the interval [119886 119887]The 120572-dimensional Hausdorff measure 119867

120572is calculated

by [26]

119867120572(119865 cap (0 119909)) =

0119868(120572)119909

1 =119909120572

Γ (1 + 120572) (18)

32 The Local Fractional JABOWA Models (LFJABOWA)Here we structure the LFJABOWA models via local frac-tional derivative and difference

From (5) when

119866 sdot 120593 (119863) = 1205820119863120573minus1 (19)

the Enquist growthmodel in JABOWAmodel reads as [11 42]

119889119863

119889119905= 1205820119863120573 (20)

where 119863 is the diameter at breast height 1205820is the scaling

coefficient and 120573 is the fractal dimensionMaking use of the fractional complex transform [29] and

(20) the growth equation in the JABOWA model with localfractional derivative (LFJABOWA) is suggested by

119889120572119863

119889119905120572= 1205820119863120573 (21)

0 02 04 06 08 1

45

4

35

3

25

2

15

1

x

y(x)

Figure 2 The graph of (22) with parameters 120573 = 1 1205820= 1 and

1199050= 0

0 02 04 06 08 1

40

35

30

25

20

15

10

5

0

x

y(x)

Figure 3 The graph of (22) with parameters 120573 = 1 1205820= 2 and

1199050= 0

where 119863 is the diameter at breast height 1205820is the scaling

coefficient and 120572 and 120573 are the fractal dimensions In orderto illustrate the difference from the works presented in [42]we consider the following case when 120573 = 1 (21) can beintegrated to give

119863 (119905) = 119864120572(1205820119905120572) minus 119864

120572(12058201199051205720) (22)

For the parameters 120573 = 1 119905 = 0 the solutions of (21) withdifferent values 120582

0= 1 120582

0= 2 120582

0= 3 and 120582

0= 4 are

respectively shown in Figures 2 3 4 and 5Using the fractional complex transform (5) becomes into

119889120572119863

119889119905120572= 119866 sdot 119863 sdot 120593 (119863) (23)

Comparing (22) and (23) we have

119889120572119863

119889119905120572= 120595 (119905 119863) (24)

4 Abstract and Applied Analysis

0 02 04 06 08 1

700

600

500

400

300

200

100

0

x

y(x)

Figure 4 The graph of (22) with parameters 120573 = 1 1205820= 3 and

1199050= 0

where

120595 (119905 119863) = 119866 sdot 119863 sdot 120593 (119863) (25)

or

120595 (119905 119863) = 1205820119863120573 (26)

From (25) and (26) we could get

120593 (119863) = 119863120573minus1 (27)

1205820= 119866 (28)

Hence the local fractional JABOWA model (LFJABOWA)reads as follows

119889120572119863

119889119905120572= 120595 (119905 119863) (29)

where 120595(119905 119863) is a nondifferentiable function and 119863 is thediameter at breast height

In view of (16) from (29) we give the difference formof local fractional JABOWA model (LFJABOWA) in thefollowing form

Δ120572119863

(Δ119905)120572= 120595 (119905 119863) (30)

where 119863 is the diameter at breast height and Δ120572119863(119905) cong Γ(1 +120572)Δ(119863(119905) minus 119863(119905

0))

When the fractal dimension 120572 is equal to 1 we get thegeneralized form of (1) namely

Δ119863

Δ119905= 120595 (119905 119863) (31)

where

120595 (119905 119863) = 119866 sdot 119863 sdot 120601 (119863) sdot1

119887 (119863)sdot 119891 (119890) (32)

0 02 04 06 08 1

25

2

15

1

05

0

x

y(x)

times104

Figure 5 The graph of (22) with parameters 120573 = 1 1205820= 4 and

1199050= 0

33 The Local Fractional FORSKA Models (LFFORSKA)Here we present the LFFORSKA models via local fractionalderivative and difference

The tree height 119867 is the nondifferentiable functionnamely

1003816100381610038161003816119867 (119863) minus 119867 (1198630)1003816100381610038161003816 lt 120589120572 (33)

where for 120589 120599 gt 0 120589 120599 isin 119877 and |119863 minus 1198630| lt 120599

Following the fractional complex transform [29] from (8)we have the local fractional growth equation in the followingform

119889120572119867

119889119863120572= 119904 sdot 119891 (119867) (34)

where119891(119867) is a local fractional continuous function119867 is thetree height and 119904 is a parameter

Therefore the generalized form of (34) is suggested asfollows

119889120572119867

119889119863120572= 120594 (119863119867) (35)

where 119863 is the diameter at breast height and 119867 is the treeheight In view of (16) (35) can be rewritten as follows

Δ120572119867

(Δ119863)120572= 120594 (119863119867) (36)

where1003816100381610038161003816119863 minus 119863

0

1003816100381610038161003816 lt 120599

1003816100381610038161003816119867 (119863) minus 119867 (1198630)1003816100381610038161003816 lt 120589120572

(37)

The expression (36) is the difference form of local frac-tional FORSKA model (LFFORSKA)

Abstract and Applied Analysis 5

4 Conclusions

In this work we investigated the local fractional models forthe fractal forest succession Based on the local fractionaloperators we suggested the differential and difference formsof the local fractional JABOWA and FORSKA models Thenondifferentiable growths of individual trees were discussedIt is a good start for solving the rhetorical models for thefractal forest succession in the mathematical analysis

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the Science and TechnologyCommission Planning Project of Jiangxi Province (no20122BBG70078)

References

[1] A Fielding ldquoApplications of fractal geometry to biologyrdquoComputer Applications in the Biosciences vol 8 no 4 pp 359ndash366 1992

[2] B L Li ldquoFractal geometry applications in description andanalysis of patch patterns and patch dynamicsrdquo EcologicalModelling vol 132 no 1 pp 33ndash50 2000

[3] G Sugihara and RMMay ldquoApplications of fractals in ecologyrdquoTrends in Ecology amp Evolution vol 5 no 3 pp 79ndash86 1990

[4] J H Brown V K Gupta B L Li B T Milne C Restrepo andG BWest ldquoThe fractal nature of nature power laws ecologicalcomplexity and biodiversityrdquo Philosophical Transactions of theRoyal Society of London B vol 357 no 1421 pp 619ndash626 2002

[5] N D Lorimer R G Haight and R A LearyThe Fractal ForestFractal Geometry and Applications in Forest Science NorthCentral Forest Experiment Station Berkeley Calif USA 1994

[6] B Zeide ldquoFractal geometry in forestry applicationsrdquo ForestEcology and Management vol 46 no 3 pp 179ndash188 1991

[7] B Zeide ldquoAnalysis of the 32 power law of self-thinningrdquo ForestScience vol 33 no 2 pp 517ndash537 1987

[8] G D Peterson ldquoScaling ecological dynamics self-organizationhierarchical structure and ecological resiliencerdquo ClimaticChange vol 44 no 3 pp 291ndash309 2000

[9] G BWest B J Enquist and JH Brown ldquoA general quantitativetheory of forest structure and dynamicsrdquo Proceedings of theNational Academy of Sciences vol 106 no 17 pp 7040ndash70452009

[10] G B West J H Brown and B J Enquist ldquoA general model forthe origin of allometric scaling laws in biologyrdquo Science vol 276no 5309 pp 122ndash126 1997

[11] B J Enquist G B West E L Charnov and J H BrownldquoAllometric scaling of production and life-history variation invascular plantsrdquo Nature vol 401 no 6756 pp 907ndash911 1999

[12] C S Holling ldquoCross-scale morphology geometry and dynam-ics of ecosystemsrdquo Ecological Monographs vol 62 no 4 pp447ndash502 1992

[13] G Peterson C R Allen and C S Holling ldquoEcologicalresilience biodiversity and scalerdquo Ecosystems vol 1 no 1 pp6ndash18 1998

[14] D B Botkin J F Janak and J R Wallis ldquoRationale limitationsand assumptions of a northeastern forest growth simulatorrdquoIBM Journal of Research and Development vol 16 no 2 pp 101ndash116 1972

[15] D B Botkin J F Janak and J R Wallis ldquoSome ecologicalconsequences of a computer model of forest growthrdquo TheJournal of Ecology vol 60 no 3 pp 849ndash872 1972

[16] D B Botkin Forest Dynamics An Ecological Model OxfordUniversity Press New York NY USA 1993

[17] G L Perry and N J Enright ldquoSpatial modelling of vegetationchange in dynamic landscapes a review of methods andapplicationsrdquo Progress in Physical Geography vol 30 no 1 pp47ndash72 2006

[18] R Chen and R R Twilley ldquoA gap dynamic model of mangroveforest development along gradients of soil salinity and nutrientresourcesrdquo Journal of Ecology vol 86 no 1 pp 37ndash51 1998

[19] RW Hall ldquoJABOWA revealed-finallyrdquo Ecology vol 75 no 3 p859 1994

[20] M I Ashraf C P A Bourque D A MacLean T Erdle andF R Meng ldquoUsing JABOWA-3 for forest growth and yieldpredictions under diverse forest conditions of Nova ScotiaCanadardquoTheForestry Chronicle vol 88 no 6 pp 708ndash721 2012

[21] H H Shugart and I R Noble ldquoA computer model of successionand fire response of the high-altitude Eucalyptus forest of theBrindabella Range Australian Capital Territoryrdquo AustralianJournal of Ecology vol 6 no 2 pp 149ndash164 1981

[22] H H Shugart andD CWest ldquoDevelopment of an Appalachiandeciduous forest succession model and its application toassessment of the impact of the chestnut blightrdquo Journal ofEnvironmental Management vol 5 pp 161ndash179 1977

[23] H H Shugart A Theory of Forest Dynamics The EcologicalImplications of Forest Succession Models Springer New YorkNY USA 1984

[24] H Bugmann ldquoA review of forest gap modelsrdquo Climatic Changevol 51 no 3-4 pp 259ndash305 2001

[25] M Lindner R Sievanen and H Pretzsch ldquoImproving thesimulation of stand structure in a forest gap modelrdquo ForestEcology and Management vol 95 no 2 pp 183ndash195 1997

[26] X-J Yang Advanced Local Fractional Calculus and Its Applica-tions World Science New York NY USA 2012

[27] X J Yang Local Fractional Functional Analysis and Its Applica-tions Asian Academic Hong Kong China 2011

[28] X J Yang and D Baleanu ldquoFractal heat conduction problemsolved by local fractional variation iteration methodrdquo ThermalScience vol 17 no 2 pp 625ndash628 2013

[29] X J Yang D Baleanu and J H He ldquoTransport equationsin fractal porous media within fractional complex transformmethodrdquo Proceedings of the Romanian Academy A vol 14 no4 pp 287ndash292 2013

[30] C F Liu S S Kong and S J Yuan ldquoReconstructive schemesfor variational iterationmethod within Yang-Laplace transformwith application to fractal heat conduction problemrdquo ThermalScience vol 17 no 3 pp 715ndash721 2013

[31] A M Yang Y Z Zhang and Y Long ldquoThe Yang-Fouriertransforms to heat-conduction in a semi-infinite fractal barrdquoThermal Science vol 17 no 3 pp 707ndash713 2013

[32] Y Zhao D Baleanu C Cattani D F Cheng and X-JYang ldquoMaxwellrsquos equations on Cantor Sets a local fractional

6 Abstract and Applied Analysis

approachrdquo Advances in High Energy Physics vol 2013 ArticleID 686371 6 pages 2013

[33] X J Yang D Baleanu andW P Zhong ldquoApproximate solutionsfor diffusion equations onCantor space-timerdquoProceedings of theRomanian Academy A vol 14 no 2 pp 127ndash133 2013

[34] X-J Yang H M Srivastava J H He and D Baleanu ldquoCantor-type cylindrical-coordinate method for differential equationswith local fractional derivativesrdquo Physics Letters A vol 377 no28ndash30 pp 1696ndash1700 2013

[35] K M Kolwankar and A D Gangal ldquoLocal fractional Fokker-Planck equationrdquo Physical Review Letters vol 80 no 2 pp 214ndash217 1998

[36] A Carpinteri B Chiaia and P Cornetti ldquoStatic-kinematicduality and the principle of virtual work in the mechanics offractal mediardquo Computer Methods in Applied Mechanics andEngineering vol 191 no 1-2 pp 3ndash19 2001

[37] F Ben Adda and J Cresson ldquoAbout non-differentiable func-tionsrdquo Journal of Mathematical Analysis and Applications vol263 no 2 pp 721ndash737 2001

[38] A Babakhani and V Daftardar-Gejji ldquoOn calculus of localfractional derivativesrdquo Journal of Mathematical Analysis andApplications vol 270 no 1 pp 66ndash79 2002

[39] H K Bugmann X Yan M T Sykes et al ldquoA comparison offorest gap models model structure and behaviourrdquo ClimaticChange vol 34 no 2 pp 289ndash313 1996

[40] A D Moore ldquoOn the maximum growth equation used in forestgap simulation modelsrdquo Ecological Modelling vol 45 no 1 pp63ndash67 1989

[41] A C Risch C Heiri and H Bugmann ldquoSimulating structuralforest patterns with a forest gap model a model evaluationrdquoEcological Modelling vol 181 no 2-3 pp 161ndash172 2005

[42] D A Coomes and R B Allen ldquoEffects of size competition andaltitude on tree growthrdquo Journal of Ecology vol 95 no 5 pp1084ndash1097 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

0 02 04 06 08 1

700

600

500

400

300

200

100

0

x

y(x)

Figure 4 The graph of (22) with parameters 120573 = 1 1205820= 3 and

1199050= 0

where

120595 (119905 119863) = 119866 sdot 119863 sdot 120593 (119863) (25)

or

120595 (119905 119863) = 1205820119863120573 (26)

From (25) and (26) we could get

120593 (119863) = 119863120573minus1 (27)

1205820= 119866 (28)

Hence the local fractional JABOWA model (LFJABOWA)reads as follows

119889120572119863

119889119905120572= 120595 (119905 119863) (29)

where 120595(119905 119863) is a nondifferentiable function and 119863 is thediameter at breast height

In view of (16) from (29) we give the difference formof local fractional JABOWA model (LFJABOWA) in thefollowing form

Δ120572119863

(Δ119905)120572= 120595 (119905 119863) (30)

where 119863 is the diameter at breast height and Δ120572119863(119905) cong Γ(1 +120572)Δ(119863(119905) minus 119863(119905

0))

When the fractal dimension 120572 is equal to 1 we get thegeneralized form of (1) namely

Δ119863

Δ119905= 120595 (119905 119863) (31)

where

120595 (119905 119863) = 119866 sdot 119863 sdot 120601 (119863) sdot1

119887 (119863)sdot 119891 (119890) (32)

0 02 04 06 08 1

25

2

15

1

05

0

x

y(x)

times104

Figure 5 The graph of (22) with parameters 120573 = 1 1205820= 4 and

1199050= 0

33 The Local Fractional FORSKA Models (LFFORSKA)Here we present the LFFORSKA models via local fractionalderivative and difference

The tree height 119867 is the nondifferentiable functionnamely

1003816100381610038161003816119867 (119863) minus 119867 (1198630)1003816100381610038161003816 lt 120589120572 (33)

where for 120589 120599 gt 0 120589 120599 isin 119877 and |119863 minus 1198630| lt 120599

Following the fractional complex transform [29] from (8)we have the local fractional growth equation in the followingform

119889120572119867

119889119863120572= 119904 sdot 119891 (119867) (34)

where119891(119867) is a local fractional continuous function119867 is thetree height and 119904 is a parameter

Therefore the generalized form of (34) is suggested asfollows

119889120572119867

119889119863120572= 120594 (119863119867) (35)

where 119863 is the diameter at breast height and 119867 is the treeheight In view of (16) (35) can be rewritten as follows

Δ120572119867

(Δ119863)120572= 120594 (119863119867) (36)

where1003816100381610038161003816119863 minus 119863

0

1003816100381610038161003816 lt 120599

1003816100381610038161003816119867 (119863) minus 119867 (1198630)1003816100381610038161003816 lt 120589120572

(37)

The expression (36) is the difference form of local frac-tional FORSKA model (LFFORSKA)

Abstract and Applied Analysis 5

4 Conclusions

In this work we investigated the local fractional models forthe fractal forest succession Based on the local fractionaloperators we suggested the differential and difference formsof the local fractional JABOWA and FORSKA models Thenondifferentiable growths of individual trees were discussedIt is a good start for solving the rhetorical models for thefractal forest succession in the mathematical analysis

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the Science and TechnologyCommission Planning Project of Jiangxi Province (no20122BBG70078)

References

[1] A Fielding ldquoApplications of fractal geometry to biologyrdquoComputer Applications in the Biosciences vol 8 no 4 pp 359ndash366 1992

[2] B L Li ldquoFractal geometry applications in description andanalysis of patch patterns and patch dynamicsrdquo EcologicalModelling vol 132 no 1 pp 33ndash50 2000

[3] G Sugihara and RMMay ldquoApplications of fractals in ecologyrdquoTrends in Ecology amp Evolution vol 5 no 3 pp 79ndash86 1990

[4] J H Brown V K Gupta B L Li B T Milne C Restrepo andG BWest ldquoThe fractal nature of nature power laws ecologicalcomplexity and biodiversityrdquo Philosophical Transactions of theRoyal Society of London B vol 357 no 1421 pp 619ndash626 2002

[5] N D Lorimer R G Haight and R A LearyThe Fractal ForestFractal Geometry and Applications in Forest Science NorthCentral Forest Experiment Station Berkeley Calif USA 1994

[6] B Zeide ldquoFractal geometry in forestry applicationsrdquo ForestEcology and Management vol 46 no 3 pp 179ndash188 1991

[7] B Zeide ldquoAnalysis of the 32 power law of self-thinningrdquo ForestScience vol 33 no 2 pp 517ndash537 1987

[8] G D Peterson ldquoScaling ecological dynamics self-organizationhierarchical structure and ecological resiliencerdquo ClimaticChange vol 44 no 3 pp 291ndash309 2000

[9] G BWest B J Enquist and JH Brown ldquoA general quantitativetheory of forest structure and dynamicsrdquo Proceedings of theNational Academy of Sciences vol 106 no 17 pp 7040ndash70452009

[10] G B West J H Brown and B J Enquist ldquoA general model forthe origin of allometric scaling laws in biologyrdquo Science vol 276no 5309 pp 122ndash126 1997

[11] B J Enquist G B West E L Charnov and J H BrownldquoAllometric scaling of production and life-history variation invascular plantsrdquo Nature vol 401 no 6756 pp 907ndash911 1999

[12] C S Holling ldquoCross-scale morphology geometry and dynam-ics of ecosystemsrdquo Ecological Monographs vol 62 no 4 pp447ndash502 1992

[13] G Peterson C R Allen and C S Holling ldquoEcologicalresilience biodiversity and scalerdquo Ecosystems vol 1 no 1 pp6ndash18 1998

[14] D B Botkin J F Janak and J R Wallis ldquoRationale limitationsand assumptions of a northeastern forest growth simulatorrdquoIBM Journal of Research and Development vol 16 no 2 pp 101ndash116 1972

[15] D B Botkin J F Janak and J R Wallis ldquoSome ecologicalconsequences of a computer model of forest growthrdquo TheJournal of Ecology vol 60 no 3 pp 849ndash872 1972

[16] D B Botkin Forest Dynamics An Ecological Model OxfordUniversity Press New York NY USA 1993

[17] G L Perry and N J Enright ldquoSpatial modelling of vegetationchange in dynamic landscapes a review of methods andapplicationsrdquo Progress in Physical Geography vol 30 no 1 pp47ndash72 2006

[18] R Chen and R R Twilley ldquoA gap dynamic model of mangroveforest development along gradients of soil salinity and nutrientresourcesrdquo Journal of Ecology vol 86 no 1 pp 37ndash51 1998

[19] RW Hall ldquoJABOWA revealed-finallyrdquo Ecology vol 75 no 3 p859 1994

[20] M I Ashraf C P A Bourque D A MacLean T Erdle andF R Meng ldquoUsing JABOWA-3 for forest growth and yieldpredictions under diverse forest conditions of Nova ScotiaCanadardquoTheForestry Chronicle vol 88 no 6 pp 708ndash721 2012

[21] H H Shugart and I R Noble ldquoA computer model of successionand fire response of the high-altitude Eucalyptus forest of theBrindabella Range Australian Capital Territoryrdquo AustralianJournal of Ecology vol 6 no 2 pp 149ndash164 1981

[22] H H Shugart andD CWest ldquoDevelopment of an Appalachiandeciduous forest succession model and its application toassessment of the impact of the chestnut blightrdquo Journal ofEnvironmental Management vol 5 pp 161ndash179 1977

[23] H H Shugart A Theory of Forest Dynamics The EcologicalImplications of Forest Succession Models Springer New YorkNY USA 1984

[24] H Bugmann ldquoA review of forest gap modelsrdquo Climatic Changevol 51 no 3-4 pp 259ndash305 2001

[25] M Lindner R Sievanen and H Pretzsch ldquoImproving thesimulation of stand structure in a forest gap modelrdquo ForestEcology and Management vol 95 no 2 pp 183ndash195 1997

[26] X-J Yang Advanced Local Fractional Calculus and Its Applica-tions World Science New York NY USA 2012

[27] X J Yang Local Fractional Functional Analysis and Its Applica-tions Asian Academic Hong Kong China 2011

[28] X J Yang and D Baleanu ldquoFractal heat conduction problemsolved by local fractional variation iteration methodrdquo ThermalScience vol 17 no 2 pp 625ndash628 2013

[29] X J Yang D Baleanu and J H He ldquoTransport equationsin fractal porous media within fractional complex transformmethodrdquo Proceedings of the Romanian Academy A vol 14 no4 pp 287ndash292 2013

[30] C F Liu S S Kong and S J Yuan ldquoReconstructive schemesfor variational iterationmethod within Yang-Laplace transformwith application to fractal heat conduction problemrdquo ThermalScience vol 17 no 3 pp 715ndash721 2013

[31] A M Yang Y Z Zhang and Y Long ldquoThe Yang-Fouriertransforms to heat-conduction in a semi-infinite fractal barrdquoThermal Science vol 17 no 3 pp 707ndash713 2013

[32] Y Zhao D Baleanu C Cattani D F Cheng and X-JYang ldquoMaxwellrsquos equations on Cantor Sets a local fractional

6 Abstract and Applied Analysis

approachrdquo Advances in High Energy Physics vol 2013 ArticleID 686371 6 pages 2013

[33] X J Yang D Baleanu andW P Zhong ldquoApproximate solutionsfor diffusion equations onCantor space-timerdquoProceedings of theRomanian Academy A vol 14 no 2 pp 127ndash133 2013

[34] X-J Yang H M Srivastava J H He and D Baleanu ldquoCantor-type cylindrical-coordinate method for differential equationswith local fractional derivativesrdquo Physics Letters A vol 377 no28ndash30 pp 1696ndash1700 2013

[35] K M Kolwankar and A D Gangal ldquoLocal fractional Fokker-Planck equationrdquo Physical Review Letters vol 80 no 2 pp 214ndash217 1998

[36] A Carpinteri B Chiaia and P Cornetti ldquoStatic-kinematicduality and the principle of virtual work in the mechanics offractal mediardquo Computer Methods in Applied Mechanics andEngineering vol 191 no 1-2 pp 3ndash19 2001

[37] F Ben Adda and J Cresson ldquoAbout non-differentiable func-tionsrdquo Journal of Mathematical Analysis and Applications vol263 no 2 pp 721ndash737 2001

[38] A Babakhani and V Daftardar-Gejji ldquoOn calculus of localfractional derivativesrdquo Journal of Mathematical Analysis andApplications vol 270 no 1 pp 66ndash79 2002

[39] H K Bugmann X Yan M T Sykes et al ldquoA comparison offorest gap models model structure and behaviourrdquo ClimaticChange vol 34 no 2 pp 289ndash313 1996

[40] A D Moore ldquoOn the maximum growth equation used in forestgap simulation modelsrdquo Ecological Modelling vol 45 no 1 pp63ndash67 1989

[41] A C Risch C Heiri and H Bugmann ldquoSimulating structuralforest patterns with a forest gap model a model evaluationrdquoEcological Modelling vol 181 no 2-3 pp 161ndash172 2005

[42] D A Coomes and R B Allen ldquoEffects of size competition andaltitude on tree growthrdquo Journal of Ecology vol 95 no 5 pp1084ndash1097 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

4 Conclusions

In this work we investigated the local fractional models forthe fractal forest succession Based on the local fractionaloperators we suggested the differential and difference formsof the local fractional JABOWA and FORSKA models Thenondifferentiable growths of individual trees were discussedIt is a good start for solving the rhetorical models for thefractal forest succession in the mathematical analysis

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the Science and TechnologyCommission Planning Project of Jiangxi Province (no20122BBG70078)

References

[1] A Fielding ldquoApplications of fractal geometry to biologyrdquoComputer Applications in the Biosciences vol 8 no 4 pp 359ndash366 1992

[2] B L Li ldquoFractal geometry applications in description andanalysis of patch patterns and patch dynamicsrdquo EcologicalModelling vol 132 no 1 pp 33ndash50 2000

[3] G Sugihara and RMMay ldquoApplications of fractals in ecologyrdquoTrends in Ecology amp Evolution vol 5 no 3 pp 79ndash86 1990

[4] J H Brown V K Gupta B L Li B T Milne C Restrepo andG BWest ldquoThe fractal nature of nature power laws ecologicalcomplexity and biodiversityrdquo Philosophical Transactions of theRoyal Society of London B vol 357 no 1421 pp 619ndash626 2002

[5] N D Lorimer R G Haight and R A LearyThe Fractal ForestFractal Geometry and Applications in Forest Science NorthCentral Forest Experiment Station Berkeley Calif USA 1994

[6] B Zeide ldquoFractal geometry in forestry applicationsrdquo ForestEcology and Management vol 46 no 3 pp 179ndash188 1991

[7] B Zeide ldquoAnalysis of the 32 power law of self-thinningrdquo ForestScience vol 33 no 2 pp 517ndash537 1987

[8] G D Peterson ldquoScaling ecological dynamics self-organizationhierarchical structure and ecological resiliencerdquo ClimaticChange vol 44 no 3 pp 291ndash309 2000

[9] G BWest B J Enquist and JH Brown ldquoA general quantitativetheory of forest structure and dynamicsrdquo Proceedings of theNational Academy of Sciences vol 106 no 17 pp 7040ndash70452009

[10] G B West J H Brown and B J Enquist ldquoA general model forthe origin of allometric scaling laws in biologyrdquo Science vol 276no 5309 pp 122ndash126 1997

[11] B J Enquist G B West E L Charnov and J H BrownldquoAllometric scaling of production and life-history variation invascular plantsrdquo Nature vol 401 no 6756 pp 907ndash911 1999

[12] C S Holling ldquoCross-scale morphology geometry and dynam-ics of ecosystemsrdquo Ecological Monographs vol 62 no 4 pp447ndash502 1992

[13] G Peterson C R Allen and C S Holling ldquoEcologicalresilience biodiversity and scalerdquo Ecosystems vol 1 no 1 pp6ndash18 1998

[14] D B Botkin J F Janak and J R Wallis ldquoRationale limitationsand assumptions of a northeastern forest growth simulatorrdquoIBM Journal of Research and Development vol 16 no 2 pp 101ndash116 1972

[15] D B Botkin J F Janak and J R Wallis ldquoSome ecologicalconsequences of a computer model of forest growthrdquo TheJournal of Ecology vol 60 no 3 pp 849ndash872 1972

[16] D B Botkin Forest Dynamics An Ecological Model OxfordUniversity Press New York NY USA 1993

[17] G L Perry and N J Enright ldquoSpatial modelling of vegetationchange in dynamic landscapes a review of methods andapplicationsrdquo Progress in Physical Geography vol 30 no 1 pp47ndash72 2006

[18] R Chen and R R Twilley ldquoA gap dynamic model of mangroveforest development along gradients of soil salinity and nutrientresourcesrdquo Journal of Ecology vol 86 no 1 pp 37ndash51 1998

[19] RW Hall ldquoJABOWA revealed-finallyrdquo Ecology vol 75 no 3 p859 1994

[20] M I Ashraf C P A Bourque D A MacLean T Erdle andF R Meng ldquoUsing JABOWA-3 for forest growth and yieldpredictions under diverse forest conditions of Nova ScotiaCanadardquoTheForestry Chronicle vol 88 no 6 pp 708ndash721 2012

[21] H H Shugart and I R Noble ldquoA computer model of successionand fire response of the high-altitude Eucalyptus forest of theBrindabella Range Australian Capital Territoryrdquo AustralianJournal of Ecology vol 6 no 2 pp 149ndash164 1981

[22] H H Shugart andD CWest ldquoDevelopment of an Appalachiandeciduous forest succession model and its application toassessment of the impact of the chestnut blightrdquo Journal ofEnvironmental Management vol 5 pp 161ndash179 1977

[23] H H Shugart A Theory of Forest Dynamics The EcologicalImplications of Forest Succession Models Springer New YorkNY USA 1984

[24] H Bugmann ldquoA review of forest gap modelsrdquo Climatic Changevol 51 no 3-4 pp 259ndash305 2001

[25] M Lindner R Sievanen and H Pretzsch ldquoImproving thesimulation of stand structure in a forest gap modelrdquo ForestEcology and Management vol 95 no 2 pp 183ndash195 1997

[26] X-J Yang Advanced Local Fractional Calculus and Its Applica-tions World Science New York NY USA 2012

[27] X J Yang Local Fractional Functional Analysis and Its Applica-tions Asian Academic Hong Kong China 2011

[28] X J Yang and D Baleanu ldquoFractal heat conduction problemsolved by local fractional variation iteration methodrdquo ThermalScience vol 17 no 2 pp 625ndash628 2013

[29] X J Yang D Baleanu and J H He ldquoTransport equationsin fractal porous media within fractional complex transformmethodrdquo Proceedings of the Romanian Academy A vol 14 no4 pp 287ndash292 2013

[30] C F Liu S S Kong and S J Yuan ldquoReconstructive schemesfor variational iterationmethod within Yang-Laplace transformwith application to fractal heat conduction problemrdquo ThermalScience vol 17 no 3 pp 715ndash721 2013

[31] A M Yang Y Z Zhang and Y Long ldquoThe Yang-Fouriertransforms to heat-conduction in a semi-infinite fractal barrdquoThermal Science vol 17 no 3 pp 707ndash713 2013

[32] Y Zhao D Baleanu C Cattani D F Cheng and X-JYang ldquoMaxwellrsquos equations on Cantor Sets a local fractional

6 Abstract and Applied Analysis

approachrdquo Advances in High Energy Physics vol 2013 ArticleID 686371 6 pages 2013

[33] X J Yang D Baleanu andW P Zhong ldquoApproximate solutionsfor diffusion equations onCantor space-timerdquoProceedings of theRomanian Academy A vol 14 no 2 pp 127ndash133 2013

[34] X-J Yang H M Srivastava J H He and D Baleanu ldquoCantor-type cylindrical-coordinate method for differential equationswith local fractional derivativesrdquo Physics Letters A vol 377 no28ndash30 pp 1696ndash1700 2013

[35] K M Kolwankar and A D Gangal ldquoLocal fractional Fokker-Planck equationrdquo Physical Review Letters vol 80 no 2 pp 214ndash217 1998

[36] A Carpinteri B Chiaia and P Cornetti ldquoStatic-kinematicduality and the principle of virtual work in the mechanics offractal mediardquo Computer Methods in Applied Mechanics andEngineering vol 191 no 1-2 pp 3ndash19 2001

[37] F Ben Adda and J Cresson ldquoAbout non-differentiable func-tionsrdquo Journal of Mathematical Analysis and Applications vol263 no 2 pp 721ndash737 2001

[38] A Babakhani and V Daftardar-Gejji ldquoOn calculus of localfractional derivativesrdquo Journal of Mathematical Analysis andApplications vol 270 no 1 pp 66ndash79 2002

[39] H K Bugmann X Yan M T Sykes et al ldquoA comparison offorest gap models model structure and behaviourrdquo ClimaticChange vol 34 no 2 pp 289ndash313 1996

[40] A D Moore ldquoOn the maximum growth equation used in forestgap simulation modelsrdquo Ecological Modelling vol 45 no 1 pp63ndash67 1989

[41] A C Risch C Heiri and H Bugmann ldquoSimulating structuralforest patterns with a forest gap model a model evaluationrdquoEcological Modelling vol 181 no 2-3 pp 161ndash172 2005

[42] D A Coomes and R B Allen ldquoEffects of size competition andaltitude on tree growthrdquo Journal of Ecology vol 95 no 5 pp1084ndash1097 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

approachrdquo Advances in High Energy Physics vol 2013 ArticleID 686371 6 pages 2013

[33] X J Yang D Baleanu andW P Zhong ldquoApproximate solutionsfor diffusion equations onCantor space-timerdquoProceedings of theRomanian Academy A vol 14 no 2 pp 127ndash133 2013

[34] X-J Yang H M Srivastava J H He and D Baleanu ldquoCantor-type cylindrical-coordinate method for differential equationswith local fractional derivativesrdquo Physics Letters A vol 377 no28ndash30 pp 1696ndash1700 2013

[35] K M Kolwankar and A D Gangal ldquoLocal fractional Fokker-Planck equationrdquo Physical Review Letters vol 80 no 2 pp 214ndash217 1998

[36] A Carpinteri B Chiaia and P Cornetti ldquoStatic-kinematicduality and the principle of virtual work in the mechanics offractal mediardquo Computer Methods in Applied Mechanics andEngineering vol 191 no 1-2 pp 3ndash19 2001

[37] F Ben Adda and J Cresson ldquoAbout non-differentiable func-tionsrdquo Journal of Mathematical Analysis and Applications vol263 no 2 pp 721ndash737 2001

[38] A Babakhani and V Daftardar-Gejji ldquoOn calculus of localfractional derivativesrdquo Journal of Mathematical Analysis andApplications vol 270 no 1 pp 66ndash79 2002

[39] H K Bugmann X Yan M T Sykes et al ldquoA comparison offorest gap models model structure and behaviourrdquo ClimaticChange vol 34 no 2 pp 289ndash313 1996

[40] A D Moore ldquoOn the maximum growth equation used in forestgap simulation modelsrdquo Ecological Modelling vol 45 no 1 pp63ndash67 1989

[41] A C Risch C Heiri and H Bugmann ldquoSimulating structuralforest patterns with a forest gap model a model evaluationrdquoEcological Modelling vol 181 no 2-3 pp 161ndash172 2005

[42] D A Coomes and R B Allen ldquoEffects of size competition andaltitude on tree growthrdquo Journal of Ecology vol 95 no 5 pp1084ndash1097 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of