Research article Special Issues

Linking state-and-transition simulation and timber supply models for forest biomass production scenarios

  • We linked state-and-transition simulation models (STSMs) with an economics-based timber supply model to examine landscape dynamics in North Carolina through 2050 for three scenarios of forest biomass production. Forest biomass could be an important source of renewable energy in the future, but there is currently much uncertainty about how biomass production would impact landscapes. In the southeastern US, if forests become important sources of biomass for bioenergy, we expect increased land-use change and forest management. STSMs are ideal for simulating these landscape changes, but the amounts of change will depend on drivers such as timber prices and demand for forest land, which are best captured with forest economic models. We first developed state-and-transition model pathways in the ST-Sim software platform for 49 vegetation and land-use types that incorporated each expected type of landscape change. Next, for the three biomass production scenarios, the SubRegional Timber Supply Model (SRTS) was used to determine the annual areas of thinning and harvest in five broad forest types, as well as annual areas converted among those forest types, agricultural, and urban lands. The SRTS output was used to define area targets for STSMs in ST-Sim under two scenarios of biomass production and one baseline, business-as-usual scenario. We show that ST-Sim output matched SRTS targets in most cases. Landscape dynamics results indicate that, compared with the baseline scenario, forest biomass production leads to more forest and, specifically, more intensively managed forest on the landscape by 2050. Thus, the STSMs, informed by forest economics models, provide important information about potential landscape effects of bioenergy production.

    Citation: Jennifer K. Costanza, Robert C. Abt, Alexa J. McKerrow, Jaime A. Collazo. Linking state-and-transition simulation and timber supply models for forest biomass production scenarios[J]. AIMS Environmental Science, 2015, 2(2): 180-202. doi: 10.3934/environsci.2015.2.180

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  • We linked state-and-transition simulation models (STSMs) with an economics-based timber supply model to examine landscape dynamics in North Carolina through 2050 for three scenarios of forest biomass production. Forest biomass could be an important source of renewable energy in the future, but there is currently much uncertainty about how biomass production would impact landscapes. In the southeastern US, if forests become important sources of biomass for bioenergy, we expect increased land-use change and forest management. STSMs are ideal for simulating these landscape changes, but the amounts of change will depend on drivers such as timber prices and demand for forest land, which are best captured with forest economic models. We first developed state-and-transition model pathways in the ST-Sim software platform for 49 vegetation and land-use types that incorporated each expected type of landscape change. Next, for the three biomass production scenarios, the SubRegional Timber Supply Model (SRTS) was used to determine the annual areas of thinning and harvest in five broad forest types, as well as annual areas converted among those forest types, agricultural, and urban lands. The SRTS output was used to define area targets for STSMs in ST-Sim under two scenarios of biomass production and one baseline, business-as-usual scenario. We show that ST-Sim output matched SRTS targets in most cases. Landscape dynamics results indicate that, compared with the baseline scenario, forest biomass production leads to more forest and, specifically, more intensively managed forest on the landscape by 2050. Thus, the STSMs, informed by forest economics models, provide important information about potential landscape effects of bioenergy production.


    First, consider a kind of variation-inequality problem

    $ {Lu0,(x,t)ΩT,uu0,(x,t)ΩT,Lu(uu0)=0,(x,t)ΩT,u(0,x)=u0(x),xΩ,u(t,x)=0,(x,t)Ω×(0,T),
    $
    (1)

    with the non-Newtonian polytropic operator,

    $ Lu=tuum(|um|p2um)γ|um|p.
    $
    (2)

    Here, $ \Omega \subset {{\mathrm{R}}^N}(N \ge 2) $ is a bounded domain with an appropriately smooth boundary $ \partial \Omega $, $ p \ge 2 $, $ m > 0 $ and $ u_0 $ satisfies

    $ {u_0} > 0, u_0^m \in {W^{1, p}}(\Omega ) \cap {L^\infty }(\Omega ). $

    The theory of variation-inequality problems has gained significant attention due to its applications in option pricing. These applications are discussed in references [1,2,3], where more details on the financial background can be found. In recent years, there has been a growing interest in the study of variation-inequality problems, with a particular emphasis on investigating the existence and uniqueness of solutions.

    In 2022, Li and Bi considered a two dimension variation-inequality system [4],

    $ \left\{ min{Liuifi(x,t,u1,u2),uiui,0}=0,(x,t)ΩT,u(0,x)=u0(x),xΩ,u(t,x)=0,(x,t)Ω×(0,T),
    \right. $

    involving a degenerate parabolic operator

    $ {L_i}{u_i} = {\partial _t}{u_i} - {\rm{div}}(|\nabla {u_i}{|^{{p_i} - 2}}\nabla {u_i}), i = 1, 2. $

    Using the comparison principle of $ L_i u_i $ and norm estimation techniques, the sequence of upper and lower solutions for the auxiliary problem is obtained. The existence and uniqueness of weak solutions are then analyzed. While reference [5] considers the initial boundary value problem under a single variational inequality, the author explores more complex non-divergence parabolic operators

    $ Lu = {\partial _t}u - u{\rm{div}}(a(u)|\nabla u{|^{p(x) - 2}}\nabla u) - \gamma |u{|^{p(x)}} - f(x, t). $

    Reference [5] constructs a more intricate auxiliary problem and proves that the weak solutions are both unique and existent by using progressive integration and various inequality amplification techniques. Readers can refer to references [6,7,8] for further information on these interesting results.

    In the field of differential equations, there are various literature available on initial boundary value problems that involve the non-Newtonian polytropic operator. In [9,10], the authors focused on a specific class of initial boundary value problems that feature the non-Newtonian polytropic operator

    $ \left\{ tu(|um|p2um)+h(x,t)uα=0,(x,t)ΩT,u(0,x)=u0(x),xΩ,u(t,x)=0,(x,t)Ω×(0,T).
    \right. $

    To investigate the existence of a weak solution, they made use of topologic degree theory.

    Currently, there is no literature on the study of variational inequalities under non-Newtonian polytropic operators (2). Therefore, we attempt to use the results of partial differential equations from literature [5,9,10] to investigate the existence and blow-up properties of weak solutions for variational inequalities (1). Additionally, considering the degeneracy of the operator $ Lu $ at $ u = 0 $ and $ \nabla {u^m} = 0 $, some traditional methods for existence proofs are no longer applicable. Here, we attempt to use the fixed point theorem to solve this issue, and obtain the existence and blow up of generalized solutions.

    We first consider the case of variation-inequality in corn options. During the harvest season, farmers face the problem of corn storage, while flour manufacturers are concerned about the downtime caused by a lack of raw materials.

    In exchange for the farmer storing the raw materials in the warehouse, the flour manufacturer promises the farmer the following contract:

    $ \mathrm{Farmers \ at \ any \ time \ within \ a \ year \ have \ the \ right\ to \ sell \ corn \ at \ the \ agreed \ price} \ K. $

    Assuming that the current time is 0, the corn price $ S_t $ follows the time interval $ [0, \ T] $, and is given by:

    $ {\rm{d}}{S_t} = \mu {S_t}{\rm{d}}t + \sigma {S_t}{\rm{d}}{W_t} , $

    where $ S_0 $ is known, $ \mu $ represents the annual growth rate of corn price, and $ \sigma $ represents the volatility rate. $ \{ {W_t}, t \ge 0\} $ stands for a winner process, representing market noise.

    In addition, to avoid significant economic losses for flour manufacturers due to rapid increases in raw material prices, obstacle clauses are often included in the following form: if the price of corn rises more than $ B $, the option contract becomes null and void. According to literature [1,2,3], the value $ V $ of the option contract at any time $ t \in [0, T] $ satisfies

    $ {min{L0V,Vmax{SK,0}}=0,(S,t)(0,B)×(0,T),u(0,x)=max{SK,0},S(0,B),u(t,B)=0,t(0,T),u(t,0)=0,t(0,T),
    $
    (3)

    where $ L_0V = {\partial _t}V + \frac{1}{2}{\sigma ^2}{S^2}{\partial _{SS}}V + rS{\partial _S}V - rV $, $ r $ is the risk-free interest rate of the agricultural product market; $ B $ is the upper bound of corn prices, which prevents flour manufacturers from incurring significant losses due to rising corn prices. On the one hand, if $ x = lnS $, then (3) can be written as

    $ \left\{ min{L1V,Vmax{exK,0}}=0,(x,t)(,lnB)×(0,T),u(0,x)=max{exK,0},x(,lnB),u(t,lnB)=0,t(0,T),u(t,0)=0,t(0,T),
    \right. $

    where $ L_1V = {\partial _t}V + \frac{1}{2}{\sigma ^2} {\partial _{xx}}V + r {\partial _x}V - rV $. It can be seen that problem (4) is a constant coefficient parabolic variational inequality problem, which has long been studied by scholars (see [1,2,3] for details). On the other hand, when there are transaction costs involved in agricultural product trading, the constant $ \sigma $ in the operator $ LV $ is no longer valid and is often related to $ {\partial _S}V $, as well as $ V $ itself. For instance, the well-known Leland model [5] adjusts volatility $ \sigma $ into a non-divergence structure represented by

    $ σ2=σ20(1+Lesign(Vx(|xV|p2xV))),p2,
    $
    (4)

    where $ \sigma $ denotes the original volatility and $ Le $ corresponds to the Leland number.

    Inspired by these findings, we aim to explore more intricate variation-inequality models in (1). When $ m = 1 $, the non-divergence polytropic structure $ {u^m}\nabla (|\nabla {u^m}{|^{p - 2}}\nabla {u^m}) $ in model (1) degenerates into a similar n-dimensional structure as model (4). It's worth noting that while model (4) only considers one type of risky asset and is defined in a 1-dimensional space, model (1) studies the problem in an n-dimensional space.

    Variation-inequality (1) degenerates when either $ u = 0 $ or $ \nabla {u^m} = 0 $. Classically, there would be no traditional solution. Following a similar way in [1,3], we consider generalized solutions and first give a class of maximal monotone maps $ G:[0, + \infty) \to [0, + \infty) $ satisfies

    $ G(x)=0ifx>0,G(x)>0ifx=0.
    $
    (5)

    Definition 2.1 A pair $ (u, \xi) $ is called a generalized solution for variation-inequality, if for any fixed $ T > 0 $,

    (a) $ u^m \in {L^\infty }(0, T, W^{1, p}_0 (\Omega)) $, $ {\partial _t}u \in {L^2}({\Omega _T}), $

    (b) $ \xi \in G(u - {u_0})\; {\rm{for}}\; {\rm{any}}\; (x, t) \in {\Omega _T} $,

    (c) $ u(x, t) \ge {u_0}(x), \; u(x, 0) = {u_0}(x)\; {\rm{for}}\; {\rm{any}}\; (x, t) \in {\Omega _T} $,

    (d) for every test-function $ \varphi \in {C^1}({\bar \Omega _T}) $, there holds

    $ \int {\int_{{\Omega _T}} {{\partial _t}u \cdot \varphi + {u^m}|\nabla {u^m}{|^{p - 2}}\nabla {u^m}\nabla \varphi {\rm{d}}x{\rm{d}}t} } + (1 - \gamma )\int {\int_{{\Omega _T}} {|\nabla {u^m}{|^p}\varphi {\rm{d}}x{\rm{d}}t} } = \int {\int_{\Omega_T} {\xi \cdot \varphi {\rm{d}}x{\rm{d}}t} } . $

    As far as what was mentioned above, $ {u^m}\nabla (|\nabla {u^m}{|^{p - 2}}\nabla {u^m}) $ degenerates when $ u^m = 0 $ or $ \nabla {u^m} = 0 $. We set and use a parameter $ \varepsilon \in [0, 1] $ to regularize $ {u^m}\nabla (|\nabla {u^m}{|^{p - 2}}\nabla {u^m}) $ in operator $ Lu $ and the initial boundary condition. Meanwhile, we use $ \varepsilon $ to construct a penalty function $ \beta _\varepsilon (\cdot) $ and use it to control the inequalities in (1) that the penalty map $ {\beta _\varepsilon }:{{\rm{R}}_ + } \to {{\rm{R}}_ - } $ satisfies

    $ βε(x)=0ifx>ε,βε(x)[M0,0)ifx[0,ε].
    $
    (6)

    In other words, we consider the following regular problem

    $ {Lεuε=βε(uεu0),(x,t)QT,uε(x,0)=u0ε(x),xΩ,uε(x,t)=ε,(x,t)QT,
    $
    (7)

    where

    $ {L_\varepsilon }{u_\varepsilon } = {\partial _t}{u_\varepsilon } - u_\varepsilon ^m\nabla ({(|\nabla u_\varepsilon ^m{|^2} + \varepsilon )^{p - 2}}\nabla u_\varepsilon ^m) - \gamma {(|\nabla u_\varepsilon ^m{|^2} + \varepsilon )^{p - 2}}|\nabla u_\varepsilon ^m{|^2}. $

    Similar to [4,5], problem (8) admits a solution $ {u_\varepsilon } $ satisfies $ u_\varepsilon ^m \in {L^\infty }(0, T; {W^{1, p}}(\Omega)) $, $ {\partial _t}u_\varepsilon ^m \in {L^\infty }(0, T; {L^2}(\Omega)) $, and the identity

    $ Ω(tuεφ+umε(|umε|2+ε)p22umεφ+(1γ)(|umε|2+ε)p22|umε|2φ)dx=Ωβε(uεu0)φdx,
    $
    (8)

    with $ \varphi \in {C^1}({\bar \Omega _T}) $. Meanwhile, for any $ \varepsilon \in (0, \; 1) $,

    $ u0εuε|u0|+ε, uε1uε2forε1ε2.
    $
    (9)

    Indeed, define $ {A_\theta }({u_\varepsilon }) = \theta u_\varepsilon ^m + (1 - \theta){u_\varepsilon } $,

    $ Lθ,ωεuε=tuεAθdiv((|Aθ(uε)|2+ε)p22Aθ(uε))γ(|Aθ(uε)|2+ε)p2|Aθ(uε)|2.
    $

    One can use a map based on Leray-Schauder fixed point theory

    $ M:L(0,T;W1,p0(Ω))×[0,1]L(0,T;W1,p0(Ω)),
    $
    (10)

    that is,

    $ {Lθ,ωεuε=θβε(uεu0),(x,t)ΩT,uε(x,0)=u0ε(x)=u0+ε,xΩ,uε(x,t)=ε,(x,t)ΩT,
    $
    (11)

    so that by proving the boundedness, continuity and compactness of operator $ M $, the existence result of (6) can be established. For details, see literature [11], omitted here.

    In this section, we consider the existence of a generalized solution to variation-inequality (1). Since $ u_\varepsilon ^m \in {L^\infty }(0, T; {W^{1, p}}(\Omega)) $, $ {\partial _t}u_\varepsilon ^m \in {L^\infty }(0, T; {L^2}(\Omega)) $, by combining with (9), we may infer that the sequence $ \{ u_\varepsilon, \varepsilon \ge 0\} $ contains a subsequence $ \{ u_{{\varepsilon _k}}^{}, k = 1, 2, \cdots \} $ and a function $ u $, $ {\varepsilon _k} \to 0\; {\rm{as}}\; k \to \infty $,

    $ uεkua.e.inΩTask,
    $
    (12)
    $ umεkweakuminL(0,T;W1,p0(Ω))ask,
    $
    (13)
    $ tumεkweaktuminL2(ΩT)ask.
    $
    (14)

    From (9), one can easily show that $ {u_{{\varepsilon _k}}} \le u $, $ \forall (x, t) \le {\Omega _T} $, $ k = 1, 2, 3, \cdots $. So, one can infer that for all $ (x, t) \in {\Omega _T} $,

    $ βεk(uεku0)ξask.
    $
    (15)

    Next, we pass the limit $ k \to \infty $. It follows from (13), that for any $ (x, t) \in {\Omega _T} $, $ k = 1, 2, 3, \cdots $,

    $ umεk(|umεk|2+εk)p22umεkweakχ1inL1(Ω)ask,
    $
    (16)
    $ (|umεk|2+εk)p22|umεk|2weakχ2inL1(Ω)ask.
    $
    (17)

    so that pass the limit $ k \to \infty $,

    $ Ωtuφ+χ1φdx+(1γ)Ωχ2φdx=Ωξφdx.
    $
    (18)

    Choosing $ \varphi = {u_{{\varepsilon _k}}} - u $ in (8) and turning $ \varepsilon $ into $ \varepsilon _k $, one can infer that

    $ Ωtuεkφ+umεk(|umεk|2+ε)p22umεkφ+(1γ)(|umεk|2+εk)p22|umεk|2φdx=Ωβεk(uεku0)φdx.
    $
    (19)

    Subtracting (18) from (19) and integrating it from 0 to $ T $,

    $ ΩT(tuεktu)φ+[umεk(|umεk|2+εk)p22umεkχ1]φdxdt+(1γ)ΩT[(|umεk|2+εk)p22|umεk|2χ2]φdxdt=t0Ω[βεk(uεku0)+ξ]φdxdt.
    $
    (20)

    From (32), we infer that

    $ limkt0Ω[βεk(uεku0)+ξ]φdxdt=0,
    $
    (21)
    $ ΩT[(|umεk|2+εk)p22|umεk|2χ2]φdxdt=0.
    $
    (22)

    Recall that $ {u_{{\varepsilon _k}}}(x, 0) = {u_0}(x) + {\varepsilon _k} $ for any $ x \in \Omega $. Then

    $ \int {\int_{{\Omega _T}} {({\partial _t}{u_{{\varepsilon _k}}} - {\partial _t}u) \cdot \varphi {\rm{d}}x{\rm{d}}t} } = \frac{1}{2}\int_\Omega {{{({u_{{\varepsilon _k}}} - u)}^2}{\rm{d}}x} - \frac{1}{2}\varepsilon _k^2 \ge 0. $

    Note that $ {\varepsilon _k} \to 0\; {\rm{as}}\; k \to \infty $. So we may infer that $ \int {\int_{{\Omega _T}} {({\partial _t}{u_{{\varepsilon _k}}} - {\partial _t}u) \cdot \varphi {\rm{d}}x{\rm{d}}t} } \ge 0 $ if $ k $ is large enough. Then, removing the non negative term on the left hand-side in (20) and passing the limit $ k \to \infty $, it is clear to verify that

    $ limkΩT[umεk(|umεk|2+εk)p22umεkχ1]φdxdt0,
    $
    (23)

    Note that $ u_{}^m|\nabla {u^m}{|^{p - 2}}\nabla {u^m} = |\nabla {u^{\mu m}}{|^{p - 2}}\nabla {u^{\mu m}} $, $ \mu = \frac{p}{{p - 1}} $. As mentioned in [12], it follows from (9) that

    $ [umεk(|umεk|2+εk)p22umεkum|um|p2um](uμmεkuμm)[|uμmεk|p2uμmεk|uμm|p2uμm](umεkum)C|uμmεkuμm|p0.
    $
    (24)

    Thus, by using $ {\mathop{\rm sgn}} \left({\nabla u_{{\varepsilon _k}}^{\mu m} - \nabla {u^{\mu m}}} \right) = {\mathop{\rm sgn}} (\nabla \varphi) $, leads to

    $ [(|umεk|2+εk)p22umεk|um|p2um]φ0.
    $
    (25)

    Subtracting (24) from (25), one can see that

    $ ΩT[um|um|p2umχ1]φdxdt0.
    $
    (26)

    Obviously, if we swap $ {u_{{\varepsilon _k}}} $ and $ {u } $, one can get another inequality

    $ ΩT[χ1um|um|p2um]φdxdt0.
    $
    (27)

    Combining (26) and (27), we obtain (28) below and give the following Lemma.

    Lemma 3.1 For any $ t \in (0, T] $ and $ x \in \Omega $,

    $ χ1=um|um|p2uma.e.inΩT,
    $
    (28)
    $ uμmεkuμmLp(Ω)0ask.
    $
    (29)

    Proof. One can deduce that (29) is an immediate result of combining (23), (24) and (29).

    Following a similar proof showed in (16)–(28), one can infer that

    $ χ2=|um|pa.e.inΩT.
    $
    (30)

    Further, we prove $ \xi \in G(u{\rm{ }} - {\rm{ }}{u_0}) $. When $ {u_{{\varepsilon _k}}} \ge {u_0} + \varepsilon $, we have $ {\beta _{{\varepsilon _k}}}({u_{{\varepsilon _k}}} - {u_0}) = 0 $, so

    $ ξ(x,t)=0u>u0.
    $
    (31)

    If $ {u_0} \le {u_{{\varepsilon _k}}} < {u_0} + {\varepsilon _k} $, $ {\beta _\varepsilon }({u_\varepsilon } - {u_0}) \le 0 $ and $ {\beta _\varepsilon }(\cdot) \in {C^2}({\rm{R}}) $ imply that

    $ ξ(x,t)0u=u0.
    $
    (32)

    Combining (31) and (32), it can be easily verified that $ \xi \in G(u - {u_0}) $.

    Further, passing the limit $ k \to \infty $ in the second line of (44) and the third line of (6),

    $ u(x, 0) = {\rm{ }}{u_0}(x) \ {\rm{ in }} \ \Omega , \ u(x, t) \ge {\rm{ }}{u_0}(x) \ {\rm{ in }} \ {\Omega _T}. $

    Combining the equations above, we infer that $ (u, \xi) $ satisfies the conditions of Definition 2.1, such that $ (u, \xi) $ is a generalized solution of (1).

    Theorem 3.1 Assume that $ u_0^m \in {W^{1, p}}(\Omega) \cap {L^\infty }(\Omega) $, $ \gamma \le 1 $. Then variation-inequality (1) admits at least one solution $ (u, \xi) $ within the class of Definition 2.1.

    In this section, we consider the blowup of the generalized solution when $ \gamma > 2 $ and try to prove it by contradiction. Assume $ (u, \xi) $ is a generalized solution of (1). Taking $ \varphi = {u^m} $ in Definition 1, it is easy to see that

    $ 1m+1ddtΩu(,t)m+1dx+(2γ)Ω|um|pumdx=Ωξumdx.
    $
    (33)

    It follows from (5), (9), and $ \xi \in G(u{\rm{ }} - {\rm{ }}{u_0}) $ that $ \int_\Omega {\xi \cdot {u^m}{\rm{d}}x} \ge 0 $. Let

    $ E(t) = \int_\Omega {u{{( \cdot , t)}^{\omega m + 1}}{\rm{d}}x}. $

    It follows from (c) in Definition 2.1 that $ E(t) \ge 0 $ for any $ t \in (0, T] $, so one can infer that

    $ ddtE(t)(m+1)(γ2)Ω|um|pumdx.
    $
    (34)

    Using the Poincare inequality gives

    $ Ω|um|pumdx=p(ω+p)mΩ|u(1p+1)m|pdxp(p+1)mΩ|u|(p+1)mdx.
    $
    (35)

    Here, $ \int_\Omega {|u{|^{(p + 1)m}}{\rm{d}}x} $ need to keep shrinking. By the Holder inequality

    $ E(t) \le C(|\Omega |){\left( {\int_\Omega {|u{|^{(p + 1)m}}{\rm{d}}x} } \right)^{\frac{{m + 1}}{{(p + 1)m}}}}, $

    so that

    $ Ω|u|(p+1)mdxC(|Ω|)E(t)(p+1)mm+1.
    $
    (36)

    Combining (34)–(36), one can find that

    $ ddtE(t)C(|Ω|)p(m+1)(γ2)(p+1)mE(t)(p+1)mm+1.
    $
    (37)

    Note that $ mp > 1 $. Using variable separation method, we have that

    $ ddtE(t)1mpm+1C(|Ω|)p(γ2)(1mp)(p+1)m,
    $
    (38)

    such that

    $ E(t)1(E(0)1mpm+1C(|Ω|)p(γ2)(mp1)(p+1)mt)m+1mp1.
    $
    (39)

    This means that the generalized solution blows up at $ {T^*} = \frac{{E{{(0)}^{\frac{{1 - mp}}{{m + 1}}}}(p + 1)m}}{{p(\gamma - 2)(mp - 1)C(|\Omega |)}} $.

    Theorem 4.1 Assume $ mp > 1 $. if $ \gamma > 2 $, the generalized solution $ (u, \xi) $ of variation-inequality (1) blows up in finite time.

    In this study, the existence and blowup of a generalized solution to a class of variation-inequality problems with non-divergence polytropic parabolic operators

    $ Lu = {\partial _t}u - {u^m}\nabla (|\nabla {u^m}{|^{p - 2}}\nabla {u^m}) - \gamma |\nabla {u^m}{|^p}. $

    We first consider the existence of generalized solution. Due to the use of integration by parts, $ -\gamma |\nabla {u^m}{|^p} $ becomes $ (1-\gamma)|\nabla {u^m}{|^p} $. In the process of proving $ u_\varepsilon ^m \in {L^\infty }(0, T; {W^{1, p}}(\Omega)) $ and $ {\partial _t}u_\varepsilon ^m \in {L^\infty }(0, T; {L^2}(\Omega)) $ in [4,5], $ (1 - \gamma)|\nabla {u^m}{|^p} $ is required to be greater than 0, therefore eliciting the restriction $ \gamma \le 1 $. Regarding the restriction of $ p $, the condition $ p \ge 1 $ is required in (24) and the above formula. As what mentioned, we have used the results $ u_\varepsilon ^m \in {L^\infty }(0, T; {W^{1, p}}(\Omega)) $ and $ {\partial _t}u_\varepsilon ^m \in {L^\infty }(0, T; {L^2}(\Omega)) $ in literature [4,5] where $ p \ge 2 $ is required, therefore we require the restriction that $ p \ge 2 $. The results show that variation-inequality (1) admits at least one solution $ (u, \xi) $ when $ \gamma\le 1 $.

    Second, we analyzed the blowup phenomenon of a generalized solution. In (38), $ mp $ must be big than 1, otherwise (39) is invalid. The results show that the generalized solution $ (u, \xi) $ of the variation-inequality (1) blows up in finite time when $ \gamma \ge 2 $.

    The author sincerely thanks the editors and anonymous reviewers for their insightful comments and constructive suggestions, which greatly improved the quality of the paper.

    The author declares no conflict of interest.

    [1] Sedjo RA, Sohngen B (2013) Wood as a Major Feedstock for Biofuel Production in the United States: Impacts on Forests and International Trade. J Sustain For 32: 195-211. doi: 10.1080/10549811.2011.652049
    [2] Goh CS, Junginger M, Cocchi M, et al. (2013) Wood pellet market and trade: a global perspective. Biofuels Bioprod Bioref 7: 24-42. doi: 10.1002/bbb.1366
    [3] Immerzeel DJ, Verweij PA, van der Hilst F, et al. (2014) Biodiversity impacts of bioenergy crop production: A state-of-the-art review. GCB Bioenergy 6: 183-209. doi: 10.1111/gcbb.12067
    [4] McDonald RI, Fargione J, Kiesecker J, et al. (2009) Energy sprawl or energy efficiency: Climate policy impacts on natural habitat for the United States of America. PLoS One 4: e6802. doi: 10.1371/journal.pone.0006802
    [5] Stoms DM, Davis FW, Jenner MW, et al. (2012) Modeling wildlife and other trade-offs with biofuel crop production. GCB Bioenergy 4: 330-341. doi: 10.1111/j.1757-1707.2011.01130.x
    [6] Dale VH, Kline KL, Wiens J, et al. (2010) Biofuels: Implications for Land Use and Biodiversity. The Ecological Society of America Biofuels and Sustainability Reports. Available from: http://www.esa.org/biofuelsreports/files/ESA%20Biofuels%20Report_VH%20Dale%20et%20al.pdf.
    [7] Wiens J, Fargione J, Hill J (2011) Biofuels and biodiversity. Ecol Appl 21: 1085-1095. doi: 10.1890/09-0673.1
    [8] Daystar J (2014) Environmental Impacts of Cellulosic Biofuels Made in the South East: Implications of Impact Assessment Methods and Study Assumptions. North Carolina State University: 264 pages.
    [9] Wear D, Abt R, Alavalapati J, et al. (2010) The South's Outlook for Sustainable Forest Bioenergy and Biofuels Production. The Pinchot Institute Report. Available from: http://www.pinchot.org/uploads/download?fileId=512.
    [10] Fletcher RJ, Robertson BA, Evans J, et al. (2011) Biodiversity conservation in the era of biofuels: risks and opportunities. Front Ecol Environ 9: 161-168. doi: 10.1890/090091
    [11] Riffell S, Verschuyl J, Miller D, et al. (2011) Biofuel harvests, coarse woody debris, and biodiversity – A meta-analysis. For Ecol Manage 261: 878-887. doi: 10.1016/j.foreco.2010.12.021
    [12] Dale VH, Lowrance R, Mulholland P, et al. (2010) Bioenergy Sustainability at the Regional Scale. Ecol Soc 15: 23.
    [13] Wear DN, Huggett R, Li R, et al. (2013) Forecasts of Forest Conditions in U.S. Regions under Future Scenarios: A Technical Document Supporting the Forest Service 2010 RPA Assessment. Gen Tech Rep SRS-170.
    [14] Lubowski RN, Plantinga AJ, Stavins RN (2008) What Drives Land-Use Change in the United States? A National Analysis of Landowner Decisions. Land Econ 84: 529-550.
    [15] Daniel CJ, Frid L (2012) Predicting Landscape Vegetation Dynamics Using State-and-Transition Simulation Models. Proc First Landsc State-and-Transition Simul Model Conf June 14-16 2011: 5-22.
    [16] Bestelmeyer BT, Herrick JE, Brown JR, et al. (2004) Land management in the American southwest: a state-and-transition approach to ecosystem complexity. Environ Manage 34: 38-51.
    [17] Costanza JK, Hulcr J, Koch FH, et al. (2012) Simulating the effects of the southern pine beetle on regional dynamics 60 years into the future. Ecol Modell 244: 93-103. doi: 10.1016/j.ecolmodel.2012.06.037
    [18] Wilson T, Costanza J, Smith J, et al. (2014) Second State-and-Transition Simulation Modeling Conference. Bull Ecol Soc Am 96: 174-175.
    [19] Halofsky J, Halofsky J, Burscu T, et al. (2014) Dry forest resilience varies under simulated climate-management scenarios in a central Oregon, USA landscape. Ecol Appl 24: 1908-1925. doi: 10.1890/13-1653.1
    [20] Provencher L, Forbis TA, Frid L, et al. (2007) Comparing alternative management strategies of fire, grazing, and weed control using spatial modeling. Ecol Modell 209: 249-263. doi: 10.1016/j.ecolmodel.2007.06.030
    [21] Abt R, Cubbage F, Abt K (2009) Projecting southern timber supply for multiple products by subregion. For Prod J 59: 7-16.
    [22] Abt KL, Abt RC, Galik CS, et al. (2014) Effect of Policies on Pellet Production and Forests in the U.S. South: A Technical Document Supporting the Forest Service Update of the 2010 RPA Assessment. Gen Tech Rep GTR-SRS-202.
    [23] U.S. Geological Survey National Gap Analysis Program (2013) Protected Areas Database-US (PAD-US), Version 1.3. Available from: http://gapanalysis.usgs.gov/padus/.
    [24] Terando A, Costanza JK, Belyea C, et al. (2014) The southern megalopolis: using the past to predict the future of urban sprawl in the Southeast U.S. PLoS One 9: e102261. doi: 10.1371/journal.pone.0102261
    [25] Noss RF, Platt WJ, Sorrie BA, et al. (2015) How global biodiversity hotspots may go unrecognized: lessons from the North American Coastal Plain. Richardson D, ed. Divers Distrib 21: 236-244. doi: 10.1111/ddi.12278
    [26] Southeast Gap Analysis Project (SEGAP) (2008) Southeast GAP regional land cover [digital data]. Available from: http://www.basic.ncsu.edu/segap/.
    [27] Burke S, Hall BR, Shahbazi G, et al. (2007) North Carolina's Strategic Plan for Biofuels Leadership. Available from: http://www.ces.ncsu.edu/fletcher/mcilab/publications/NC_Strategic_Plan_for_Biofuels_Leadership.pdf.
    [28] Forisk Consulting LLC (2014) Wood bioenergy US database 2013. Available by subscription.
    [29] Lal P, Alavalapati JRR, Marinescu M, et al. (2011) Developing Sustainability Indicators for Woody Biomass Harvesting in the United States. J Sustain For 30: 736-755. doi: 10.1080/10549811.2011.571581
    [30] Evans A, Perschel R, Kittler B, et al. (2010) Revised assessment of biomass harvesting and retention guidelines. For Guild, St Fe, NM, USA: 33.
    [31] Janowiak MK, Webster CR (2010) Promoting Ecological Sustainability in Woody Biomass Harvesting. J For 108: 16-23.
    [32] Apex Resource Management Solutions (2014) ST-Sim state-and-transition simulation model software. Available from: http//www.apexrms.com/stsm.
    [33] Rollins MG (2009) LANDFIRE: a nationally consistent vegetation, wildland fire, and fuel assessment. Int J Wildl Fire 18: 235-249. doi: 10.1071/WF08088
    [34] Comer P, Faber-Langendoen D, Evans R, et al. (2003) Ecological Systems of the United States: A Working Classification of U.S. Terrestrial Systems. Arlington, VA, USA. NatureServe, 82 pages.
    [35] Costanza JK, Terando AJ, McKerrow AJ, et al. (2015) Modeling climate change, urbanization, and fire effects on Pinus palustris ecosystems of the southeastern U.S. J Environ Manage 151: 186-199. doi: 10.1016/j.jenvman.2014.12.032
    [36] LANDFIRE (2014) LANDFIRE 2008 (version 1.1.0) Succession Class (S-Class) Layer. U.S. Department of Interior, Geological Survey. Available from: Http://landfire.cr.usgs.gov/viewer.
    [37] Multi-Resolution Land Characteristics Consortium (MRLC) (2011) National Land Cover Database, USFS Tree Canopy Cartographic, 2014. Available from: http://www.mrlc.gov/nlcd11_data.php.
    [38] Mackie R, Mason J, Curcio G (2007) LANDFIRE biophysical setting model for Southern Piedmont Dry Oak(-Pine) Forest. Available from: http://www.landfire.gov/national_veg_models_op2.php.
    [39] USDA Forest Service (2012) Forest Inventory and Analysis Data. Available from: http://apps.fs.fed.us/fiadb-downloads/datamart.html.
    [40] Young T, Wang Y, Guess F, et al. (2015) Understanding the Characteristics of Non-industrial Private Forest Landowners Who Harvest Trees. Small-scale For 1-13.
    [41] Hardie I, Parks P, Gottleib P, et al. (2000) Responsiveness of Rural and Urban Land Uses to Land Rent Determinants in the U.S. South. Land Econ 76: 659. doi: 10.2307/3146958
    [42] USDA Natural Resources Conservation Service (2000) 1997 National Resources Inventory Data, Revised December 2000.
    [43] Dale VH, Kline KL, Wright LL, et al. (2011) Interactions among bioenergy feedstock choices, landscape dynamics, and land use. Ecol Appl 21: 1039-1054. doi: 10.1890/09-0501.1
    [44] Evans JM, Fletcher RJ, Alavalapati JRR, et al. (2013) Forestry Bioenergy in the Southeast United States: Implications for Wildlife Habitat and Biodiversity. Availbale from: http://www.nwf.org/News-and-Magazines/Media-Center/Reports/Archive/2013/12-05-13-Forestry-Bioenergy-in-the-Southeast.aspx.
    [45] Owens AK, Moseley KR, McCay TS, et al. (2008) Amphibian and reptile community response to coarse woody debris manipulations in upland loblolly pine (Pinus taeda) forests. For Ecol Manage 256: 2078-2083. doi: 10.1016/j.foreco.2008.07.030
    [46] Otto CR V, Kroll AJ, McKenny HC (2013) Amphibian response to downed wood retention in managed forests: A prospectus for future biomass harvest in North America. For Ecol Manage 304: 275-285. doi: 10.1016/j.foreco.2013.04.023
    [47] Davis JC, Castleberry SB, Kilgo JC (2010) Influence of coarse woody debris on herpetofaunal communities in upland pine stands of the southeastern Coastal Plain. For Ecol Manage 259: 1111-1117. doi: 10.1016/j.foreco.2009.12.024
    [48] Wood P, Sheehan J, Keyser P, et al. (2013) Cerulean Warbler: Management Guidelines for Enhancing Breeding Habitat in Appalachian Hardwood Forests. American Bird Conservancy. The Plains, VA, USA. 28 Pages.
    [49] Perry RW, Thill RE (2013) Long-term responses of disturbance-associated birds after different timber harvests. For Ecol Manage 307: 274-283. doi: 10.1016/j.foreco.2013.07.026
    [50] Wilson MD, Watts BD (2000) Breeding bird communities in pine plantations on the coastal plain of North Carolina. Chat 64: 1-14.
    [51] Peet RK, Allard DJ (1993) Longleaf Pine Vegetation of the Southern Atlantic an Eastern Gulf Coast Regions: A Preliminary Classification. In: Hermann SM, ed. Proceedings of the Tall Timbers Fire Ecology Conference, No. 18, The Longleaf Pine Ecosystem: Ecology, Restoration and Management. Tallahassee, FL, USA: Tall Timbers Research Station.; 1993: 45-81.
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