# Mathematical Models Arising in the Fractal Forest Gap via Local Fractional Calculus

Authors: Chun-Ying Long; Yang Zhao; Hossein Jafari
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

The forest new gap models via local fractional calculus are investigated. The JABOWA and FORSKA models are extended to deal with the growth of individual trees defined on Cantor sets. The local fractional growth equations with local fractional derivative and difference are discussed. Our results are first attempted to show the key roles for the nondifferentiable growth of individual trees.


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## Body

## 1. Introduction

Fractals had been used to describe special problems in biology and ecology [1–4] because of the measure of nature objects underlying the geometry, replacing the complex real-world objects by describing the Euclidean ideas. Fractal dimension was applied to describe the measure of the complexity in biology and ecology. In forestry, the fractal geometry had been applied to estimate stand density, predict forest succession, and describe the form of trees [5–7]. The scaling of dynamics in hierarchical structure was investigated in [8–12]. The ecological resilience example from boreal forest was presented in the context [8]. The quantitative theory of forest structure was discussed [9]. The allometric scaling laws in biology were proposed in the works [10, 11]. Based upon the cross-scale analysis, the geometry and dynamics of ecosystems were considered and the structure ecosystems across scales in time and space were discussed in [12, 13]. Fractal forestry was modeled by using the scaling of the testing parameters for ecological complexity.Forest gap model (JABOWA) developed by Botkin et al. [14–16] was the first simulation model for gap-phase replacement. It was applied to describe a forest as a mosaic of closed canopies and simulate forest dynamics based upon the establishment, growth, and death of individual trees [17–20]. The JABOWA model in the form of the FORET model (called JABOWA-FORET) was further developed in [21–24]. The JABOWA model of the simulation of stand structure in a forest gap model was improved in [24] and the FORSKA [25] was proposed by Botkin et al. In [14–16], the ecological functions are continuous. In [10–13], the ecological functions were expressed across scales in time and space. However, as it is shown in Figure 1 the ecological functions distinguishing hierarchical size scales in nature, such as the measures of tree size and measure of soil fertility, are defined on Cantor sets. The above approaches do not deal with them.Figure 1
The ecological functionf(x) defined on Cantor set.Local fractional calculus theory [26–38] was applied to handle the nondifferentiable functions defined on Cantor sets. The heat-conduction, transport, Maxwell, diffusion, wave, Fokker-Planck, and the mechanical structure equations were usefully shown (see for more details [28–36] and the cited references therein). In order to simulate forest dynamics on the basis of the establishment, growth, and death of individual trees defined on Cantor sets, the aim of this paper is to present the forest new gap models for simulating the gap-phase replacement by employing the local fractional calculus.The paper has been organized as follows. In Section2, we review the JABOWA and FORSKA models for the forest succession. In Section 3, we propose JABOWA and FORSKA models for the fractal forest succession. Finally, Section 4 is conclusions.


## 2. Growth Models for Forest Gap

In this section we will revise the JABOWA and FORSKA models.


### 2.1. The JABOWA Model

The growth equation with difference form is given by [15, 16, 24, 39]
(1)ΔDΔt=G·D·ϕ(D)·1b(D)·f(e),
where the function D is diameter at breast height of the trees, the parameter H is tree height, G is a growth rate, the function b(D) is a quantity encapsulating this allometric relationship, f(e) is a quantity influencing the abiotic and biotic environment on tree growth, ϕ(D)=(1-D·H/(Dmax⁡·Hmax⁡)), and Dmax⁡ and Hmax⁡ are the maximum measures of the tree dimensions.The parameterf(e) is simulated as follows [24]:
(2)f(e)=g1(AL⁡)·g2(SBAR)·g3(DD),
where g1(AL⁡) is a quantity of available light, g2(SBAR) is a quantity of stand basal area, and g2(SBAR) is a quantity of the annual degree-day sum. It was referred to as Liebig’s law of the minimum [24].The allometric relationship with a parabolic form is written as follows [24, 40]:
(3)b(D)=b1+b2D+b3D2,
where b1, b2, and b3 are parameters.Leaf area index reads as follows [24]:
(4)LAI=μD2,
where μ=c/k with a species-specific parameter c and the scale leaf weight per tree to the projected leaf area k.In order to implement a new height-diameter relationship [40], the differential form of growth equation in the JABOWA model was suggested as follows [41]:
(5)dDdt=G·D·φ(D),
where the function has the following form:
(6)φ(D)=1-H/Hmax⁡2Hmax⁡-b(-sD/b+2)e-Ds/b
with the parameter b=Hmax⁡-137 and the initial slope value of the height diameter relationship s.


### 2.2. The FORSKA Model

The FORSKA model was developed for unmanaged natural forests and tree height hadH-D relationship with the FORSKA model given by [24, 25, 39, 41]
(7)H=1.3+(Hmax⁡-1.3)·(1-e-sD/(Hmax⁡-1.3)),
where the parameter s is the initial slope value of the height diameter relationship at D=0, Hmax⁡ is the maximum measure of the tree dimension, D is diameter at breast height of the trees, and H is tree height.As it is known, the trees in the real forest do not follow theH-D relationship. The growth equation with differential form can be written as follows [24, 25]:
(8)dHdD=s·f(H),
where
(9)f(H)=H-1.3Hmax⁡-1.3.


## 2.1. The JABOWA Model

The growth equation with difference form is given by [15, 16, 24, 39]
(1)ΔDΔt=G·D·ϕ(D)·1b(D)·f(e),
where the function D is diameter at breast height of the trees, the parameter H is tree height, G is a growth rate, the function b(D) is a quantity encapsulating this allometric relationship, f(e) is a quantity influencing the abiotic and biotic environment on tree growth, ϕ(D)=(1-D·H/(Dmax⁡·Hmax⁡)), and Dmax⁡ and Hmax⁡ are the maximum measures of the tree dimensions.The parameterf(e) is simulated as follows [24]:
(2)f(e)=g1(AL⁡)·g2(SBAR)·g3(DD),
where g1(AL⁡) is a quantity of available light, g2(SBAR) is a quantity of stand basal area, and g2(SBAR) is a quantity of the annual degree-day sum. It was referred to as Liebig’s law of the minimum [24].The allometric relationship with a parabolic form is written as follows [24, 40]:
(3)b(D)=b1+b2D+b3D2,
where b1, b2, and b3 are parameters.Leaf area index reads as follows [24]:
(4)LAI=μD2,
where μ=c/k with a species-specific parameter c and the scale leaf weight per tree to the projected leaf area k.In order to implement a new height-diameter relationship [40], the differential form of growth equation in the JABOWA model was suggested as follows [41]:
(5)dDdt=G·D·φ(D),
where the function has the following form:
(6)φ(D)=1-H/Hmax⁡2Hmax⁡-b(-sD/b+2)e-Ds/b
with the parameter b=Hmax⁡-137 and the initial slope value of the height diameter relationship s.


## 2.2. The FORSKA Model

The FORSKA model was developed for unmanaged natural forests and tree height hadH-D relationship with the FORSKA model given by [24, 25, 39, 41]
(7)H=1.3+(Hmax⁡-1.3)·(1-e-sD/(Hmax⁡-1.3)),
where the parameter s is the initial slope value of the height diameter relationship at D=0, Hmax⁡ is the maximum measure of the tree dimension, D is diameter at breast height of the trees, and H is tree height.As it is known, the trees in the real forest do not follow theH-D relationship. The growth equation with differential form can be written as follows [24, 25]:
(8)dHdD=s·f(H),
where
(9)f(H)=H-1.3Hmax⁡-1.3.


## 3. The JABOWA and FORSKA Models for the Fractal Forest Succession

In this section, based upon the local fractional calculus theory, we show the JABOWA and FORSKA models for the fractal forest succession. At first, we start with the local fractional derivative.


### 3.1. Local Fractional Derivative

We now give the local fractional calculus and the recent results.If(10)|f(x)-f(x0)|<εα
with |x-x0|<δ, for ε,δ>0 and ε,δ∈R, then we denote [26–28]
(11)f(x)∈Cα(a,b).
If f(x)∈Cα(a,b), then we have [26]
(12)dimH(F∩(a,b))=dimH(Cα(a,b))=α,
where Cα(a,b)={f:f(x) is local fractional continuous, x∈F∩(a,b)}.Letf(x)∈Cα(a,b). The local fractional derivative of f(x) of order α at x=x0 is defined as [26–34]
(13)f(α)(x0)=dαf(x)dxα|x=x0=lim⁡x→x0Δα(f(x)-f(x0))(x-x0)α,
where Δα(f(x)-f(x0))≅Γ(1+α)Δ(f(x)-f(x0)).For0<α≤1, the increment of f(x) can be written as follows [26, 27]:
(14)Δαf(x)=f(α)(x)(Δx)α+λ(Δx)α,
where Δx is an increment of x and λ→0 as Δx→0.For0<α≤1, the α-local fractional differential of f(x) reads as [26, 27]
(15)dαf=f(α)(x)(dx)α.
From (14), we have approximate formula in the form
(16)Δαf(x)≅Γ(1+α)Δ(f(x)-f(x0)).
Let f(x)∈Cα(a,b). The local fractional integral of f(x) of order α is given by [26–31]
(17)Ib(α)af(x)=1Γ(1+α)∫abf(t)(dt)α=1Γ(1+α)lim⁡Δt→0∑j=0j=N-1f(tj)(Δtj)α,
where Δtj=tj+1-tj,  Δt=max⁡{Δt1,Δt2,Δtj,…}, and [tj,tj+1],  j=0,…,N-1,  t0=a,  tN=b, is a partition of the interval [a,b].Theα-dimensional Hausdorff measure Hα is calculated by [26]
(18)Hα(F∩(0,x))=Ix(α)01=xαΓ(1+α).


### 3.2. The Local Fractional JABOWA Models (LFJABOWA)

Here, we structure the LFJABOWA models via local fractional derivative and difference.From (5), when
(19)G·φ(D)=λ0Dβ-1,
the Enquist growth model in JABOWA model reads as [11, 42]
(20)dDdt=λ0Dβ,
where D is the diameter at breast height, λ0 is the scaling coefficient, and β is the fractal dimension.Making use of the fractional complex transform [29] and (20), the growth equation in the JABOWA model with local fractional derivative (LFJABOWA) is suggested by
(21)dαDdtα=λ0Dβ,
where D is the diameter at breast height, λ0 is the scaling coefficient, and α and β are the fractal dimensions. In order to illustrate the difference from the works presented in [42], we consider the following case: when β=1, (21) can be integrated to give
(22)D(t)=Eα(λ0tα)-Eα(λ0t0α).
For the parameters β=1,  t=0, the solutions of (21) with different values λ0=1, λ0=2,  λ0=3, and λ0=4 are, respectively, shown in Figures 2, 3, 4, and 5.Figure 2
The graph of (22) with parameters β=1, λ0=1, and t0=0.Figure 3
The graph of (22) with parameters β=1, λ0=2, and t0=0.Figure 4
The graph of (22) with parameters β=1, λ0=3, and t0=0.Figure 5
The graph of (22) with parameters β=1, λ0=4, and t0=0.Using the fractional complex transform, (5) becomes into
(23)dαDdtα=G·D·φ(D).
Comparing (22) and (23), we have
(24)dαDdtα=ψ(t,D),
where
(25)ψ(t,D)=G·D·φ(D)
or
(26)ψ(t,D)=λ0Dβ.
From (25) and (26), we could get
(27)φ(D)=Dβ-1,(28)λ0=G.
Hence, the local fractional JABOWA model (LFJABOWA) reads as follows:
(29)dαDdtα=ψ(t,D),
where ψ(t,D) is a nondifferentiable function and D is the diameter at breast height.In view of (16), from (29) we give the difference form of local fractional JABOWA model (LFJABOWA) in the following form:
(30)ΔαD(Δt)α=ψ(t,D),
where D is the diameter at breast height and ΔαD(t)≅Γ(1+α)Δ(D(t)-D(t0)).When the fractal dimensionα is equal to 1, we get the generalized form of (1); namely,
(31)ΔDΔt=ψ(t,D),
where
(32)ψ(t,D)=G·D·ϕ(D)·1b(D)·f(e).


### 3.3. The Local Fractional FORSKA Models (LFFORSKA)

Here, we present the LFFORSKA models via local fractional derivative and difference.The tree heightH is the nondifferentiable function; namely,
(33)|H(D)-H(D0)|<ςα,
where for ς,ϑ>0ς,ϑ∈R, and |D-D0|<ϑ.Following the fractional complex transform [29], from (8) we have the local fractional growth equation in the following form:
(34)dαHdDα=s·f(H),
where f(H) is a local fractional continuous function, H is the tree height, and s is a parameter.Therefore, the generalized form of (34) is suggested as follows:
(35)dαHdDα=χ(D,H),
where D is the diameter at breast height and H is the tree height. In view of (16), (35) can be rewritten as follows:
(36)ΔαH(ΔD)α=χ(D,H),
where
(37)|D-D0|<ϑ,|H(D)-H(D0)|<ςα.The expression (36) is the difference form of local fractional FORSKA model (LFFORSKA).


## 3.1. Local Fractional Derivative

We now give the local fractional calculus and the recent results.If(10)|f(x)-f(x0)|<εα
with |x-x0|<δ, for ε,δ>0 and ε,δ∈R, then we denote [26–28]
(11)f(x)∈Cα(a,b).
If f(x)∈Cα(a,b), then we have [26]
(12)dimH(F∩(a,b))=dimH(Cα(a,b))=α,
where Cα(a,b)={f:f(x) is local fractional continuous, x∈F∩(a,b)}.Letf(x)∈Cα(a,b). The local fractional derivative of f(x) of order α at x=x0 is defined as [26–34]
(13)f(α)(x0)=dαf(x)dxα|x=x0=lim⁡x→x0Δα(f(x)-f(x0))(x-x0)α,
where Δα(f(x)-f(x0))≅Γ(1+α)Δ(f(x)-f(x0)).For0<α≤1, the increment of f(x) can be written as follows [26, 27]:
(14)Δαf(x)=f(α)(x)(Δx)α+λ(Δx)α,
where Δx is an increment of x and λ→0 as Δx→0.For0<α≤1, the α-local fractional differential of f(x) reads as [26, 27]
(15)dαf=f(α)(x)(dx)α.
From (14), we have approximate formula in the form
(16)Δαf(x)≅Γ(1+α)Δ(f(x)-f(x0)).
Let f(x)∈Cα(a,b). The local fractional integral of f(x) of order α is given by [26–31]
(17)Ib(α)af(x)=1Γ(1+α)∫abf(t)(dt)α=1Γ(1+α)lim⁡Δt→0∑j=0j=N-1f(tj)(Δtj)α,
where Δtj=tj+1-tj,  Δt=max⁡{Δt1,Δt2,Δtj,…}, and [tj,tj+1],  j=0,…,N-1,  t0=a,  tN=b, is a partition of the interval [a,b].Theα-dimensional Hausdorff measure Hα is calculated by [26]
(18)Hα(F∩(0,x))=Ix(α)01=xαΓ(1+α).


## 3.2. The Local Fractional JABOWA Models (LFJABOWA)

Here, we structure the LFJABOWA models via local fractional derivative and difference.From (5), when
(19)G·φ(D)=λ0Dβ-1,
the Enquist growth model in JABOWA model reads as [11, 42]
(20)dDdt=λ0Dβ,
where D is the diameter at breast height, λ0 is the scaling coefficient, and β is the fractal dimension.Making use of the fractional complex transform [29] and (20), the growth equation in the JABOWA model with local fractional derivative (LFJABOWA) is suggested by
(21)dαDdtα=λ0Dβ,
where D is the diameter at breast height, λ0 is the scaling coefficient, and α and β are the fractal dimensions. In order to illustrate the difference from the works presented in [42], we consider the following case: when β=1, (21) can be integrated to give
(22)D(t)=Eα(λ0tα)-Eα(λ0t0α).
For the parameters β=1,  t=0, the solutions of (21) with different values λ0=1, λ0=2,  λ0=3, and λ0=4 are, respectively, shown in Figures 2, 3, 4, and 5.Figure 2
The graph of (22) with parameters β=1, λ0=1, and t0=0.Figure 3
The graph of (22) with parameters β=1, λ0=2, and t0=0.Figure 4
The graph of (22) with parameters β=1, λ0=3, and t0=0.Figure 5
The graph of (22) with parameters β=1, λ0=4, and t0=0.Using the fractional complex transform, (5) becomes into
(23)dαDdtα=G·D·φ(D).
Comparing (22) and (23), we have
(24)dαDdtα=ψ(t,D),
where
(25)ψ(t,D)=G·D·φ(D)
or
(26)ψ(t,D)=λ0Dβ.
From (25) and (26), we could get
(27)φ(D)=Dβ-1,(28)λ0=G.
Hence, the local fractional JABOWA model (LFJABOWA) reads as follows:
(29)dαDdtα=ψ(t,D),
where ψ(t,D) is a nondifferentiable function and D is the diameter at breast height.In view of (16), from (29) we give the difference form of local fractional JABOWA model (LFJABOWA) in the following form:
(30)ΔαD(Δt)α=ψ(t,D),
where D is the diameter at breast height and ΔαD(t)≅Γ(1+α)Δ(D(t)-D(t0)).When the fractal dimensionα is equal to 1, we get the generalized form of (1); namely,
(31)ΔDΔt=ψ(t,D),
where
(32)ψ(t,D)=G·D·ϕ(D)·1b(D)·f(e).


## 3.3. The Local Fractional FORSKA Models (LFFORSKA)

Here, we present the LFFORSKA models via local fractional derivative and difference.The tree heightH is the nondifferentiable function; namely,
(33)|H(D)-H(D0)|<ςα,
where for ς,ϑ>0ς,ϑ∈R, and |D-D0|<ϑ.Following the fractional complex transform [29], from (8) we have the local fractional growth equation in the following form:
(34)dαHdDα=s·f(H),
where f(H) is a local fractional continuous function, H is the tree height, and s is a parameter.Therefore, the generalized form of (34) is suggested as follows:
(35)dαHdDα=χ(D,H),
where D is the diameter at breast height and H is the tree height. In view of (16), (35) can be rewritten as follows:
(36)ΔαH(ΔD)α=χ(D,H),
where
(37)|D-D0|<ϑ,|H(D)-H(D0)|<ςα.The expression (36) is the difference form of local fractional FORSKA model (LFFORSKA).


## 4. Conclusions

In this work we investigated the local fractional models for the fractal forest succession. Based on the local fractional operators, we suggested the differential and difference forms of the local fractional JABOWA and FORSKA models. The nondifferentiable growths of individual trees were discussed. It is a good start for solving the rhetorical models for the fractal forest succession in the mathematical analysis.



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