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A tree-growth model to optimize silviculture
Patrice Loisel ∗†
, Jean François Dhôte ‡
2011
Abstract : In this paper, we present the description of a simplified model of the dynamic of
a mono-specific even-aged forest. The model studied is a tree-growth model based on a system
of two ordinary differential equations concerning the tree basal area and the number of trees. The
analytical study of this model permits us to predict the behavior of the system solutions. We are
trying to highlight the influence of economic parameters and growth parameters on the system
solutions, in the framework of the optimization of silviculture.
Keywords : growth model; optimization ; control
1 Introduction
The forest management, because of its impact on our environment, is a topic that now involves
researchers of many disciplines : forestry, economy, and ecology. These various communities have
models adapted to the questions they wish to tackle. The economists usually study the best age at
which to cut down a tree or stand of trees, the simultaneous management of several forest stands.
As to foresters, they are moreover interested in silviculture at the stand level. We will focus on the
models developed by foresters.
The models of growth for silviculture, expanded rapidly these last few years, and represent a
significant part of the developed models. Here, we are focusing on a particular type of forest (thus
a particular type of model) : a mono-specific even-aged forest where all trees belong to the same
species and are the same age.
∗INRA, UMR 729 MISTEA, 2 place Viala, F-34060 Montpellier, France
†SupAgro, UMR 729 MISTEA, 2 place Viala, F-34060 Montpellier, France
‡Office National des Forêts, R&D Departement Boulevard de Constance, F-77300 Fontainebleau, France
1
Models built by forest modellers are based on statistical adjustments of dendrometric data [1] :
these models accurately describe the evolution of a forest, but the analytical study of those models
is made difficult due to their complexity. The analytical study of models allows us to predict the
influence of different parameters : for instance, if by modifying parameters to take into account the
climate change and analyzing so the potential consequences is provide. To allow analytical studies
while having realistic model, we decide in this paper to consider a simplified model.
The models, on which we are focusing here, are tree centered distance independent models
where the trees are not spatialized. In this type of model each tree is characterized by its basal area
at the height of 1.3meters : sand eventually by its height h. The model here described is based
on the concepts developed in the growth model “Fagacées” [2] [3] for Oak or Beech forest. In
this model, the link between the stand level and the individual tree level is explicit. This modeling
allows us to describe the evolution of a forest of high density. “Fagacées” was broadcast through
the project Capsis [4].
In order to allow analytical studies, we are starting with the simplified hypothesis in which we
consider that all trees have the same basal area. We then consider a forest of ntrees with basal area
swhich needs management such as thinning ethroughout time.
The tree growth (due to the observed densities) is not independent of its neighbor’s growth.
There is a competition for the available resources : photosynthesis and access to the light on one
hand, and mineral nutrients on the other hand. Thus, cutting a tree implies the increase of its
neighbors’ growth, and cutting no tree limits individual growth. That shows how a forest is not just
a juxtaposition of trees.
This phenomenon is considered through an assessment equation that allows us to distribute
the energy resources of the forest stand level between the various trees. This equation coupled to
an equation describing the evolution of the number of trees leads us to a dynamic system of the
forest. The studied model takes into account the characteristics that the foresters consider as the
most important and as required : the basal area at 1.3meters, the height, and the number of trees.
In Section 2 we will present the designing of the model. In Section 3 we will present general
results on the behavior of the system solutions, then we will look for strategies which permit to
leave the viability domain in minimal or maximal time. Finally in Section 4 we will highlight the
influence of economic parameters and growth parameters for silviculture within the resolution of
an optimization problem.
2
2 Designing the model
The trees density and the RDI of forest stand.
Let’s consider a forest with a given area. It is intuitively clear that the tree number which this
area can bear is limited. The environmental conditions (type of soil, local climatic conditions) are
also factors to be taken into account for the maximum tree capacity. Foresters have established a
law called “self-thinning”, described hereafter, to evaluate this maximum capacity. Let’s note s∗
the average tree basal area (at the height of 1.3meters) of the forest. Reineke [5] observed
monospecific forests with various densities and various species. Out of these observations he
claimed the maximum tree number nmax(s∗)that a stand can bear is given by the following self
thinning relation :
log nmax(s∗) = C0−q
2log s∗,
where C0>0and 1< q < 2are characteristic constant values of the forest species and of its
environment, and in particular the ground fertility.
As for a given s∗, beyond the number nmax(s∗)the trees die, the forest stand (in terms of tree
number) has to remain under this limit. To simplify we will here make the assumption that all the
trees have the same basal area s. For a forest of where the effective tree number is n, taking into
account the relation of self-thinning, the density ris the ratio of the tree number and the maximum
tree number of basal area s that the forest can support, ris then defined by :
r(n, s) = n
nmax(s)
is written this way :
r(n, s) = nsq
2
eC0=Ansq
2
where A:= e−C0. This ratio ris called RDI (Relative Density Index or Reineke Density Index).
By definition this ratio is always less than 1.
Competition between trees : from forest stand level to individual tree level
We will now describe the temporal evolution of state variables s, n and now r, of the considered
forest.
3
The growth of a tree depends on its neighbors. There is competition for the resources and
the death of a tree, natural or due to cuttings, implies an increased growth for its neighbors . We
make the assumption, that in the course of time, silviculture makes it possible to maintain the trees
uniformly distributed on the area. The model is characterized by the existence of two levels in the
modeling.
At the forest stand level the available energy for the considered forest, is considered globally,
due to photosynthesis or due to nutrients in the soil. This supplied energy makes it possible to
ensure at the same time the maintenance and the growth of the trees. The share reserved for main-
tenance increases with the tree height, therefore with time, which limits all the more so the available
part for growth. The energy left for growth is therefore a decreasing time function and allows the
increase of basal area of the forest stand. The increase of basal area of the forest stand at its peak of
density r(n(t), s(t)) = 1 is given by the function V(.). We assume that V(t)verify the following
properties :
(H1):V(.)is a positive, decreasing, convex function of t.
For a lower density (r(n(t), s(t)) <1), the effective increase of the basal area for the forest is
reduced by a factor dependent on this same density : g(r(n(t), s(t))) at any time. Thus the energy
actually used at time tis given by :
g(r(n(t), s(t)))V(t).
The function g(.)is supposed to satisfy the following properties :
(H2):g(.)is an increasing, concave function of rsuch that g(r)> r for r∈(0,1),g(0) =
0, g(1) = 1.
The concavity of gis related to crown development in relation to basal area.
On the individual tree level, tree growth is characterized by the evolution of tree basal area
and therefore by the evolution of the function s(t): the instantaneous increase is thus ds(t)
dt . As
mentioned in the hypothesis all trees have the same basal area, the total sum increase of basal areas
of all trees is n(t)ds(t)
dt . This total increase is obtained from the available energy resources. We
thus obtain the equation which describes the link between the forest stand level and the individual
tree level :
g(r(n(t), s(t)))V(t) = n(t)ds(t)
dt for n(t)>0,for all t
4
For any t, this enables us to establish the first dynamic equation of our model :
ds(t)
dt =g(r(n(t), s(t)))
n(t)V(t)
In addition, the evolution of the tree number depends on several factors. To permit analytical study
of the model, we’ve decided to simplify and we suppose the only cause of tree mortality is due to
fallings that foresters could operate. We noted e(t)the instantaneous rate of trees cutting at time
t. Thus the evolution of the tree number is given by :
dn(t)
dt =−e(t).
We wish to preserve a minimum tree number in the forest stand, which implies n(t)≥n >
0, for any t. Technologically and to ensure a provisioning not too irregular, the thinning rate is
limited : 0≤e(t)≤e, for any t.
The forest is therefore described using the two state variables s, n and its evolution follows the
following dynamic :
(S0)
ds(t)
dt =g(r(n(t), s(t)))
n(t)V(t)
dn(t)
dt =−e(t)
with the constraints 0≤e(t)≤e,n(t)≥n, r(n(t), s(t)) = An(t)s(t)q
2≤1for any t.
Foresters built this type of model from observed forest data. The available data only allows
us to validate the model on a limited period of time. The system (S0)has therefore a time limit
domain : t∈[0, T∗].
It is a dynamic system in the state variables nand s, controlled by the control variable e. For a
cutting policy, i.e. the data of a particular function e(.), and for each initial condition (s(0), n(0)),
this system has a single solution : we will suppose the functions g(.), V (.)are regular enough for
it to happen. We will specify these trajectories in the following paragraph.
To finish with the model description, the tree height his supposed to depend only on the tree
basal area sand on the dominant height h0(average height of the 100 largest trees), h0is a concave
function of time tand shouldn’t depend on silviculture (cuttings in the course of time) and thus
depends only on time t. The height hhas therefore no influence on s(t)and its evolution, his
consequently an output of the model.
5
If, as supposed earlier, the basal area sat time tis the same for all trees, the height his also the
same. We therefore deduce h(t) = h0(t), for any t.
3 Studiing the solutions
3.1 Model properties
The solutions of the dynamic system (S0)must satisfy in particular the constraint r(n(t), s(t)) ≤
1for any t. If there is no cutting, i.e. if e(t) = 0 for any t, we deduce that n(t) = n(0),s(.)
and r(.)are increasing with respect to time t. Let’s suppose there is one time τ < T∗such as
r(n(τ), s(τ)) = 1, we deduced t > τ if we apply a control identically null then r(n(t), s(t)) >1
and the constraint is no longer satisfied. In order to let us know which control we should apply we
are led to study the evolution of the density function r(.):
dr(n(t), s(t))
dt =r0
s(n(t), s(t))g(r(n(t), s(t)))
n(t)V(t)−r0
n(n(t), s(t))e(t)
=r(n(t), s(t))
n(t)[q
2
g(r(n(t), s(t)))
s(t)V(t)−e(t)] (1)
Out of this last equation we can deduce that in order to respect the constraint r(n(t), s(t)) ≤1
for t>τ, we should apply a non-identically null control on the system. Thus the cutting e(t) =
q
2
V(t)
s(t)for t > τ respects the constraints by binding, i.e. r(n(t), s(t)) = 1. If we define the function
er(., .)by : er(s, t) := q
2
V(t)
s, for any s > 0,t > 0, the solutions, independently of the cutting
function e(.)applied to the system, are only valid if the constraint : er(s(t), t)≤eis satisfied. We
are therefore led to formulate the following assumption (H3):
(H3):er(sm(t), t) = q
2
V(t)
sm(t)< e for all t∈(0, T∗)
where sm(t)is the minimal value s(t)can reach at the time t.
Remark 3.1 sm(t)is not specified at the moment but will be specified later on, however we can
take an approximate lower bound for now : sm(t)> s(0).
We noted previously that the system (S0)is considered only for t∈[0, T∗]. It is advisable to
specify now, the behavior of the solutions in this interval.
6
Definition 3.1 The function V(.;.)is defined by V(t;T) = ZT
t
V(u)du and represents the energy
that has been available for growth in the period [t, T ].
The following Lemma shows us that the system validity field depends on this energy value :
Lemma 3.1 Assuming (H2),(H3), then :
(i) if V(0; T∗)is large enough then there exists a time τ < T∗such that r(n(τ), s(τ)) = 1 and
n(τ) = n. The dynamical system is only valid on the interval [0, τ ]. This time τdepends on the
evolution of the cutting e(.).
(ii) conversely if V(0; T∗)is small enough then the dynamical system is valid throughout the
entire interval [0, T∗].
Proof : (i) From g(r)≥rwe deduce : ds(t)
dt ≥r(n(t), s(t))
n(t)V(t) = As(t)q
2V(t)hence :
s(T∗)1−q
2≥s(0)1−q
2+A(1 −q
2)V(0; T∗)
From r(n(T∗), s(T∗)) ≤1and n(T∗)≥nwe deduce : s(T∗)≤1
(An)2
q
. If V(0; T∗)is large
enough, we obtain a contradiction.
(ii) Let’s set τthe first period where r(n(t), s(t)) reaches 1, then for 0< t ≤τwe deduce :
ds(t)
dt ≤V(t)
n(t)≤V(t)
nand therefore s(τ)≤s(0) + V(0; τ)
n. If V(0; T∗)is small enough, we
deduce r(n(τ), s(τ)) <1in contradiction with the assumption.
Specific trajectories easily expressed in terms of control, will play an important role, we are
introducing them here : let’s consider the system of equations (S0), for trajectory E0from a fixed
initial condition (s(0), n(0)) we apply the maximum cutting e(t) = euntil we reach the value n
for the tree number, t0,n is the time needed to go from the tree number n(0) to n. By definition, we
therefore have t0,n =n(0) −n
e.
For trajectory E0, starting from the same initial condition (with r(n(0), s(0)) <1) we apply
the minimum cutting e(t)=0until reaching the value 1for the RDI r, then we apply the control
er(s(t), t)until n=n.t0is the time needed to go for the RDI from r(n(0), s(0)) to 1and T0the
final time. By definition, we therefore have t0and T0respectively solutions of :
7
q
2n(0)2
q−1A
2
qV(0; t0) = Z1
r(n(0),s(0))
u
2
q−1
g(u)du (at constant n)
n1−2
q=n(0)1−2
q+A
2
q(1 −q
2)V(t0;T0)(at constant r)
Notations. To summarize we note the following definitions of the specific trajectories :
E0:e(t) = (eif t < t0,n
0if t > t0,n
E0:e(t) = (0if r(n(t), s(t)) <1,i.e. t < t0
er(s(t), t)if t0< t < T 0
For t0,n < T < t0we can also define an intermediate trajectory ET
ET:e(t) = (0if t<T−t0,n
eif t>T−t0,n
if t0,n < T < t0,n +t0
ET:e(t) =
0if t<t0
er(s(t), t)if t0< t < t∗
eif t∗< t < T
if t0,n +t0< T < T 0
where t∗is defined by : n(t∗)1−2
q=n(0)1−q
2+A(1 −2
q)V(t0;t∗)and (T−t∗)e=n(t∗)−n.
We note that ET0=E0and by extension if T < t0,n then ET=E0.
Figure 1 : Phase plane in the coordinates sand n.
The functions obtained by just following the trajectories E0, ETand E0will be noted by the
indices 0,Tand 0.
From the increasing of the basal area sand the non increasing of the number nwe deduce
that the system solutions have no choice but to move to the right bottom in the phase plane. The
Lemma 3.1 (i) has shown that, if V(0; T∗)is large enough, the solutions are not valid throughout the
entire interval [0, T∗]. That implies that as from a time τ, the solution doesn’t belong to the validity
8
domain defined by the constraints r(n(t), s(t)) ≤1and n(t)≥n. The only point which makes it
possible to leave this validity domain is the point such as r(n(τ), s(τ)) = 1 and n(τ) = n, this
point is represented by a square on Figure 1. As the solution remains valid basal area s(t)verifies
for all t < T :s(t)≤s=1
(An)2
q
(is deduced from r(n(t), s(t)) ≤1).
We define T(resp. T) as the minimum (resp. maximum) time necessary to reach the point
defined by r(n(T), s(T)) = 1 and n(T) = n. Then :
- if T≤Tthe solution remains valid whatever the trajectory (i.e. whatever the evolution of the
cutting e(.)).
- if T < T ≤Tthe system has a solution on [0, T ]for certain controls e(.).
- if T > T the system has no solution on [0, T ]whatever the controls e(.).
We noted that, from Lemma 3.1, if V(0; T∗)is small enough then Tand especially Tcan no
exist.
A particular case
We could consider the particular function g(r) = gθ(r) = r1−θ,0< θ < 1. In that case
ds(t)
dt =A1−θs(t)q
2(1−θ)
n(t)θV(t). The basal area sis explicitly deduced from the tree number n:
s(t)1−q
2(1−θ)=s(0)1−q
2(1−θ)+A1−θ(1 −q
2(1 −θ)) Zt
0
V(u)
n(u)θdu.
In this class of functions gθ(.)we consider the extreme case (θ= 0) for which some of the
properties of the hypothesis (H2)are not satisfied : g0(r) = r. In this last case G(r)≡0and the
evolution of the basal area sis independent from the evolution of the tree number n:
s(t)1−q
2=s(0)1−q
2+A(1 −q
2)V(0; t).
In this particular case, provided that V(0; T∗)is large enough, Tand Tare equal, don’t depend
on the cutting e(.)and are the unique solution of the following equation in T:
s(0)1−q
2+A(1 −q
2)V(0; T) = s1−q
2.
9
3.2 Minimum and maximum time necessary to reach the point (r, n) =
(1, n)
In order to succeed in the conclusion of the study of the minimum and maximum time needed
to reach this point, we will need the following properties and definitions related to the function
g(.). The increase in basal area of each tree is given by g(r(n(t), s(t)))
n(t)V(t). We will to know
thereafter the evolution of g(r(n(t), s(t)))
n(t), for the same aim, we will need to define the functions
G(.),γ(.):
Definition 3.2 The function G(.)is defined by : G(r) = d
dr [r
g(r)] = g(r)−rg0(r)
g2(r). The function
γ(.)is defined by : γ(r) = rg0(r)
g(r).
In the “Fagacées” model, g(r) = (1 + p)r
r+pwith p > 0,Gis constant G(r)≡G=1
1 + p.
From G(.)and γ(.)definitions, we can establish the following properties for the model :
Lemma 3.2 Assuming the hypothesis (H2), then :
(i) The function r
g(r)is an increasing function of rand G(.)satisfies 0<G(r)g(r)≤1for any
r > 0.
(ii) The function g(r(n, s))
nis a decreasing function of n.
(iii) The function g(r(n, s)) is an increasing function of s.
(iv) The function γ(.)satisfies γ(r)≤1for any r > 0.
(v) if n(t)and s(t)are solutions of systems (S0)the function g(r(n(t), s(t)))
n(t)is an increasing
function of t.
Proof : (i) From the concavity of g(.),d
dr [g(r)−rg0(r)] = −rg00(r)>0for any r > 0and
from g(0) = 0, we deduce that g(r)−rg0(r)>0and G(r)>0. From g0(r)>0,G(r)≤1
g(r).
10
(ii) ∂
∂n [g(r(n, s))
n] = r(n, s)g0(r(n, s)) −g(r(n, s))
n2=−G(r(n, s)) g2(r(n, s))
n2<0.
(iii) ∂g(r(n, s))
∂s =g0(r(n, s))r0
s(n, s)>0.
(iv) From g(r)−rg0(r)>0we deduce the result.
(v) d
dt[g(r(n(t), s(t)))
n(t)] = (g0(r)r0
s)(n(t), s(t)))
n(t)
ds(t)
dt +G(r(n(t), s(t)))g2(r(n(t), s(t)))
n2(t)e(t)
and from (i) we deduce the result.
We are thus focusing on the trajectories and also on the strategies which allow us to reach
respectively in a minimum and maximum time the point (1, n)in the (r, n)coordinates.
The minimum time T(resp. the maximum time T) is reached by solving the problem of optimal
control : min
e(.)T(resp. max
e(.)T) with the set of admissible values for the control variable [0, e].n(.)
and s(.)are the state variables governed by the system (S0)of initial condition (n(0), s(0)) and
satisfying constraints, for all t∈[0, T ),r(n(t), s(t)) ≤1,n(t)≥nand the right end time
constraints r(n(T), s(T)) = 1, n(T) = n.
Proposition 3.1 Assume (H2),(H3). If n(.)and s(.)are the solutions of the system (S0)for a
control e(.)then, ∀t∈[0, T ]:
(i) n0(t)≤n(t)≤n0(t)
(ii) s(t)≥s0(t)
(iii) if g(r) = gθ(r) = r1−θ,s(t)≤s0(t)
If we assume that the final tree-number n(T)is equal to nthen :
(iv) n(t)≤nT(t)
(v) if g(r) = gθ(r) = r1−θ,s(t)≤sT(t)
Proof (i) and (iv) Follows from the definition.
(ii) From Lemma 3.2 (ii) we deduce : ds(t)
dt =g(r(n(t), s(t))
n(t)V(t)≥g(r(n0(t), s(t))
n0(t)V(t).
For t≤t0we deduce : ds
g(r(n(0), sq
2)) ≥V(t)
n(0) dt then by integration of the inequality :
11
Zs(t)
s(0)
dx
g(r(n(0), xq
2)) ≥V(0; t)
n(0) =Zs0(t)
s(0)
dx
g(r(n(0), xq
2)) and we deduce s(t)≥s0(t).
For t > t0,ds(t)
dt ≥V(t)
n0(t)=ds0(t)
dt and from s(t0)≥s0(t0)we deduce by integration
s(t)≥s0(t).
(iii) From the previously stated expression of the basal area sand n(t)≥n0(t)we deduce :
s(t)1−q
2(1−θ)−s(0)1−q
2(1−θ)
A1−θ(1 −q
2(1 −θ)) =Zt
0
V(u)
n(u)θdu ≤Zt
0
V(u)
n0(u)θdu =s0(t)1−q
2(1−θ)−s(0)1−q
2(1−θ)
A1−θ(1 −q
2(1 −θ))
and hence the result.
(v) From the previously stated expression of the basal area sand n(t)≤nT(t)we deduce :
s(t)1−q
2(1−θ)−s(0)1−q
2(1−θ)
A1−θ(1 −q
2(1 −θ)) =Zt
0
V(u)
n(u)θdu ≥Zt
0
V(u)
nT(u)θdu =sT(t)1−q
2(1−θ)−s(0)1−q
2(1−θ)
A1−θ(1 −q
2(1 −θ))
If we remark that to reach the point (r, n) = (1, n)in minimal time (resp. in maximal time) is
equivalent to reach s=sin minimal time (resp. in maximal time), we deduce the the trajectory
that allows to reach the point (r, n) = (1, n)in minimal or maximal time :
Corollary 3.1 Assume (H2),(H3). Let Tthe minimal time (resp. Tthe maximal time) necessary
to reach the point (r, n) = (1, n)using the control e(.). Then :
(i) if Tis finite and g(r) = r1−θ, the trajectory that allows to reach the point (r, n) = (1, n)in
minimal time Tis the trajectory E0.
(ii) if Tis finite, the trajectory that allows to reach the point (r, n) = (1, n)in maximal time T
is the trajectory E0. Maximal time Tis then the solution of :
n1−2
q−n(0)1−2
q=A
2
q(1 −q
2)V(t0;T).
12
4 Optimization of silviculture
In order to optimize the silviculture, we are interested in problems which consist in seeking the
minimal and maximum values of a variable function depending on the state variables nand s.
4.1 Preliminary results
We consider the hypothesis (H4):
(H4): there exists a constant γ > 0such as γ≤γ(r)for any r∈(0,1).
We obtain the following Lemma (with proof in Annex A) :
Lemma 4.1 Assume (H2),(H3). If n(.)and s(.)are the solutions of the system (S0)for a control
e(.)then, ∀t∈[0, T ]: (with the convention that the inequalities including s0(t)are valid only if
g(r) = gθ(r) = r1−θ)
(i) the function g(r(n, s))
nsatisfies :
g(r(n0(t), s0(t)))
n0(t)≤g(r(n(t), s(t)))
n(t)≤g(r(n0(t), s0(t)))
n0(t)
.
Morever, assume (H4):
(iia) if 0< b < 1−q
2γ
1−γthen n0(t)s0(t)b≤n(t)s(t)b≤n0(t)s0(t)b
in particular, for b=q
2the RDI r(n, s)satisfies :
r(n0(t), s0(t)) ≤r(n(t), s(t)) ≤r(n0(t), s0(t))
(iib) if b>b∗=1 + q
2(1
g(r(n,s(0))) −γ)
1−γthen n0(t)s0(t)b≤n(t)s(t)b≤n0(t)s0(t)b
(iii) the relative increase ξof the basal area ssatisfies ξm(t) = s00(t)
s0(t)q
2s1−q
2
≤ξ(t). Moreover,
if g(r) = gθ(r) = r1−θ,ξm(t) = s00(t)
s0(t)q
2s0(t)1−q
2
≤ξ(t).
13
Remark : If we assume that the final tree-number n(T)is equal to nthen the result obtained in
the Lemma 4.1 remains valid if we replace respectively n0, s0by nT, sT.
4.2 The optimization problem
We are focusing here, on the setting in the wood market of a forest whose evolution is set by
the model studied in the previous paragraphs. We are introducing the price (minus the cost of
thinning) which depends only on the basal area sand the height h: we noted P0(s, h, t). Owing to
the fact that the height hdoes not depend on the basal area sand is equal to a fixed function h0
of time t, the price can be written in a new function Pof sand t:P(s(t), t) := P0(s(t), h(t), t).
In other words we will set the price function in the following form : P0(s, h, t) = p(s)he−δt
where p(.)is a price function for the basal area sand δis the actualisation parameter. We deduce
P(s, t) = p(s)h0(t)e−δt and if we define the function δh(.)by : δh(t) = δ−h0
0(t)
h0(t)for any t > 0
then P0
t(s, t) = −δh(t)P(s, t). Mostly to simplify we’ll assume p(s) = ksα, α > 0.
We are assuming that at each time ta quantity e(t)is taken and that at the end of the period of
exploitation Tthe remaining trees would have been cut. The instantaneous value of the trees that
would have been cut is P(s(t), t)e(t)and the final value is P(s(T), T )n(T).
The criterion which we suggest to maximize consists of an integral term corresponding to the
cuttings that would have occurred during the interval [0, T ]and the final term corresponding to the
final cuttings at time T.
The optimization problem, relating to the cuttings e(.), on the interval [0, T ], is therefore writ-
ten :
(P) : max
e(.)ZT
0
P(s(t), t)e(t)dt +P(s(T), T )n(T)
with 0≤e(t)≤eand nand ssolutions of (S0)with initial conditions (n(0), s(0)) and
fulfilling the constraints : n(t)≥net r(n(t), s(t)) ≤1.
Intuitively, from the fact that the function g(r(n, s))
nis a decreasing function of n(Lemma 3.2
(ii)), we are tempted to suggest the following assertion :
In order for the trees to get the best benefits from the nutrients, one should, from the beginning
cut a significative number of trees, so that in the end of the exploitation timescale, one should get
14
a limited tree number of good quality.
We will try to validate or invalidate according to the cases this assertion and we will also try to
answer the complementary yet important questions for management :
1) Does optimal silviculture depend on the term T?
2) Which role the various parameters of the model play : economic parameters p(.), δ and
growth parameters g(.),q?
The optimization problem (P)can be rewritten just by replacing e(t)by −dn(t)
dt :
max
n(.)∈C −ZT
0
P(s(t), t)dn(t)
dt dt +P(s(T), T )n(T)
where Cis the whole set of curves :
C={n(.)∈C1([0, T ])| − e≤dn(t)
dt ≤0 & An(t)s(t)q
2≤1}
By an integration by part we deduce :
max
n(.)∈C ZT
0
dP (s(t), t)
dt n(t)dt +P(s(0),0)n(0)
under the same constraints as in the initial problem.
We are here defining the function ξ(.), the relative increase of the basal area s, by ξ(t) = s0(t)
s(t).
By applying the results of Lemma 4.1 (iii) we deduce the following Proposition (with proof in
Annex B) :
Proposition 4.1 Assume (H1),(H2),(H3),(H4),T≤T, then
(i) if g(r) = r1−θ,α > α∗= 1 + (b∗−q
2)(1 −θ)and δh(t)≤α(1 −θ)ξm(t),then the optimal
trajectory is E0.
(ii) if α < 1−q
2γ
1−γand δh(t)≤αγξm(t), then the optimal trajectory is E0.
From the remark following the Lemma 4.1 we deduce :
15
Corollary 4.1 If we assume that the final tree-number n(T)is equal to nthen :
if α < 1−q
2γ
1−γand δh(t)≤αγξm(t), then the optimal trajectory is ET.
The condition on αin Proposition 4.1 (i) implies pmust be sufficiently convex. Thus, under
the conditions mentioned in the Proposition 4.1 (i) (if pis sufficiently convex and the parameter
of actualization not too high), one may find it beneficial to cut the maximum tree number at the
beginning to ensure a high rate for the remaining tree basal area at the end of Tas foretold in the
stated assertion. Similar results were obtained with the full model “Fagacées” [6]. Moreover, in the
studied cases in Proposition 4.1, silviculture, i.e. cuttings policy e(.), does not depend on final time
T. The conditions depend on economic parameters : a sufficient convexity of the price function
relative to the basal area sand a small enough parameter of actualization. The stated assertion
however is no longer satisfied under the conditions of the Proposition 4.1 (ii).
5 Conclusion
In that article, starting with a tree-growth model governed by the tree basal area and the number
of trees, we study the viability properties of the system solutions. We highlighted the importance
of the economic parameters and growth parameters on silviculture. Hence for a price (minus the
thinning costs) is sufficently convex and a parameter of actualization not too high, it is optimal to
cut the trees at the beginning of the period of exploitation.
6 Proof of Lemma 4.1
(i) the result is a consequence of Lemma 3.2 (ii) (iii) and Proposition 3.1 (ii) (iii).
(iia) d(n(t)sb(t))a
dt =abn(t)a−1s(t)ab−1g(r(n(t), s(t)))V(t)−ae(t)n(t)a−1s(t)ab.
If we denote y(n, s)the expression of d(nsb)a
dt , then, if we assume 0<a<1:
16
y0
n=an(t)a−2s(t)ab−1(b[rg0(r)+(a−1)g(r)](n(t), s(t))V(t) + (1 −a)e(t)s(t))
≥abn(t)a−2s(t)ab−1(γ+a−1)g(r(n(t), s(t)))V(t)
y0
s=abn(t)a−1s(t)ab−2([q
2rg0(r)−(1 −ab)g(r)](n(t), s(t))V(t)−ae(t)s(t))
≤abn(t)a−1s(t)ab−2(q
2γ+ab −1)g(r(n(t), s(t))))V(t)
hence, if we choose asuch that 1−γ< a < min(1−q
2γ
b,1), we deduce q
2γ+ab −1<0<
γ+a−1then y0
n>0and y0
s<0and :
d(n0(t)s0(t)b)a
dt =y(n0(t), s0(t)) ≤d(n(t)s(t)b)a
dt ≤y(n0(t), s0(t)) = d(n0(t)(s0(t))b)a
dt
by integration, we obtain the result.
(iib) From the expression of y0
nand y0
sand using e(t)≤er(s(t), t), we deduce that, if a < 1−γ:
y0
n≤an(t)a−2s(t)ab−1(b(γ+a−1)g(r(n(t), s(t))) + q
2(1 −a))V(t)
y0
s≥abn(t)a−1s(t)ab−2((q
2γ+ab −1)g(r(n(t), s(t))) −q
2a)V(t)
and, if b>b1(a) = q
2
1−a
1−a−γ
1
g(r(n, s(0)) then y0
n<0.
if b>b2(a) = q
2
1
g(r(n, s(0)) +1−q
2γ
athen y0
s>0.
To obtain the minimal limit value for b, as b1is increasing in a, and b2is decreasing in a, we
choose the value asuch that b1(a) = b2(a), this value is a∗=(1 −γ)(1 −q
2γ)
1 + q
2(γ
g(r(n,s(0)) −γ)hence the
result if b>b∗=bi(a∗).
(iii) From Lemma 3.2 (i), g(r)
ris a decreasing function of rthen :
s0(t)
s(t)=Ag(r(n(t), s(t)))
r(n(t), s(t))
V(t)
s(t)1−q
2
≥Ag(r(n0(t), s0(t)))
r(n0(t), s0(t))
V(t)
sM(t)1−q
2
. From sM(t)≤swe de-
duce the result, if g(r) = gθ(r) = r1−θ,sM(t) = s0(t).
17
7 Proof of Proposition 4.1
(i) Let’s consider the auxiliary problem which consists in maximizing, at each time t, the inte-
grand :
dP (s(t), t)
dt n(t) =P0
s(s(t), t)g(r(n(t), s(t)))V(t) + P0
t(s(t), t)n(t)
=kh0(t)e−δt (αs(t)α−1g(r(n(t), s(t)))V(t)−δh(t)n(t)s(t)α)
We denote y=nsband z(y, s, t)by : z(y, s, t) = αsα−1g(Ays q
2−b)V(t)−δh(t)ysα−bthen :
z0
y=αs(t)α−1[rg0(r)](n(t), s(t))
y(t)V(t)−δh(t)s(t)α−b
≥s(t)α−b−1(α(1 −θ)g(r(n(t), s(t)))
n(t)V(t)−δh(t)s(t)) = s(t)α−b(α(1 −θ)ξ(t)−δh(t))
From δh(t)< α(1 −θ)ξm(t)we deduce z0
y>0. Moreover if b≥α:
z0
s=s(t)α−2(α([(q
2−b)rg0(r)+(α−1)g(r)](n(t), s(t)))V(t)−(α−b)δh(t)y(t)s(t))
≥αs(t)α−2((q
2−b)(1 −θ) + α−1)g(r(n(t), s(t)))V(t)
Due to α > α∗we can choose bsuch that max(b∗, α)< b < α−1
1−θ+q
2. Then z0
s>0, from
Lemma 4.1 (iib) we deduce the result.
(ii) We then denote y=nsαand z(y, s, t)by : z(y, s, t) = αsα−1g(Ays q
2−α)V(t)−δh(t)y.
z0
y=αs(t)α−1[rg0(r)](n(t), s(t))
y(t)V(t)−δh(t)
≥αγ g(r(n(t), s(t)))
n(t)s(t)V(t)−δh(t) = αγξ(t)−δh(t)
From δh(t)< αγξm(t),z0
y>0. Moreover :
18
z0
s=αs(t)α−2[(q
2−α)rg0(r)+(α−1)g(r)](n(t), s(t))V(t)
=αs(t)α−2[((q
2−α)γ(r) + α−1)g(r)](n(t), s(t))V(t)
From α < 1−q
2γ
1−γ, we deduce : z0
s< αs(t)α−21−q
2
1−γ(γ−γ(r))g(r(n(t), s(t)))V(t)≤0then
z0
s<0. Moreover, from z0
y>0, Lemma 4.1 (iia) with b=αand z0
s<0we deduce the result.
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de la production en volume chez le Chêne sessile. Ann. For. Sci. 121-142.
[3] Le Moguedec G. and J.F. Dhôte (2011) Fagacées a tree-centered growth and yield model for
Sessile Oak (Quercus petraea L.) and common Beech (Fagus sylvatica L.). Accepted with
revisions in Annals of Forest Science.
[4] De Coligny, F., (2005) Capsis : Computer-Aides Projection for Strategies In Silviculture, a
software platform for forestry modellers. Workshop on Information Science for Agriculture
and Environment (ISAE), 3-4 June 2005, GuiZhou Normal University, Guiang, P.R. China.
[5] Reineke, L.H. (1933) Perfecting a stand-density index for even-aged forest. Journal of Agri-
cultural Research 46, 627-638.
[6] Le Moguedec G. and P. Loisel (2011) Formalisation of silvicultural schedule for optimisa-
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Fagacées. Submitted for publication.
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