A bio-economic model for optimal harvest of timber and moose

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Forest Ecology and Management 206 (2005) 207–219

A bio-economic model for optimal harvest of timber and moose

Hilde Karine Wama,*, Ole Hofstada, Eric Nævdalb,c, Prem Sankhayana

aDepartment of Ecology and Natural Resource Management, Agricultural University of Norway, P.O. Box 5003, 1432 As, NorwaybWoodrow Wilson School, Princeton University, Princeton, NJ 08544-1013, USA

cDepartment of Economics and Resource Management, Agricultural University of Norway, P.O. Box 5003, Norway

Received 6 May 2004; received in revised form 18 October 2004; accepted 20 October 2004

Abstract

We present a dynamic optimization model for harvesting timber and moose when the moose inflicts costs by browsing young

trees. To illustrate its behaviour, the model was run numerically for Norwegian market conditions. Optimal management in our

model kept moose at an approximately 70% lower density than if managed without concern for timber production. Further, the

harvest of moose was slightly reduced around the time of timbering to take advantage of increased forage production in coming

years. Thereby a fluctuating pattern in moose harvest was generated over time. The amplitude of these fluctuations depended on

the productivity of the site, while the period of the fluctuations depended on the fixed costs of harvesting timber. On a particular

land, the optimal harvesting strategy for timber was determined by (i) the relationship between timber prices and fixed costs of

harvesting, (ii) the relationship between prices of timber and moose, and (iii) the length of the planning period in relation to

initial standing biomass of trees. Within a wide range of the present market regime, combined production of both timber and

moose was the most profitable strategy. Still, for all our planning scenarios more than 90% of the income came from timber. To

give this and similar models accurate applied value, more field data are needed to adequately quantify the interactions between

moose and forest vegetation.

# 2004 Elsevier B.V. All rights reserved.

Keywords: Alces alces; Bio-economic model; Forest; Harvest; Moose; Picea abies; Pine

1. Introduction

Traditionally, Norwegian forest owners with moose

on their land manage timber production and moose

harvesting separately. The lack of co-ordinated

planning typically results in a non-optimal situation:

too many moose locally retard forest regeneration

* Corresponding author. Tel.: +47 64948909; fax: +47 64948890.

E-mail address: [email protected] (H.K. Wam).

0378-1127/$ – see front matter # 2004 Elsevier B.V. All rights reserved

doi:10.1016/j.foreco.2004.10.062

(Lykke, 1964; Solbraa, 1998; Sæther et al., 1992),

while moose weights drop as forage production

changes with forest succession (Hjeljord and Histøl,

1995).

Although not equally valuable, both timber and

moose are of substantial importance for rural economy

in Norway. At present, the annual timber yield is

approximately 10 mm3, while almost 40.000 moose

are shot each year (SSB, 2002). The timber has a

market value eight to nine times the meat value of

.

H.K. Wam et al. / Forest Ecology and Management 206 (2005) 207–219208

moose (Gasdal and Rysstad, 1999). However, in

coming years we expect a higher relative value of

moose as the hunt is increasingly commercialised, and

more of its recreational value is realised (Mattson,

1990). Accordingly, the need for forest owners to

co-ordinate their harvests of timber and moose will

intensify.

We have built a bio-economic optimization model

for the combined harvest of timber and moose, where

we incorporate both the negative impact of moose on

forest trees and the positive impact of forest trees on

moose. Our objective function seeks to maximise the

profit over time, irrespectively of whether the main

income comes from harvest of timber or from moose.

Our model framework consists of a dynamic set of

logistic growth functions that can be used for both

mono-cultural and multi-species forests. We assume

that all seasonal migration of moose occurs within

the forest boundaries, and that the decision makers

are the same for the whole area. Consequently, we do

not discuss distribution issues among forest owners

(for such a discussion, see Olaussen and Skonhoft,

2002).

The greatest challenge when modelling this system

is to quantify the interactions between the herbivore

and its foraging plants. These interactions are not

simple functions of biomass densities, as quantitative

aspects of moose browsing naturally depend on the

quality of what is available (see e.g. Niemela and

Danell, 1988; Heikkila, 1990; Shipley et al., 1998;

Ball et al., 2000). Furthermore, the moose itself

Fig. 1. Model outline for the bio-economic modelli

contributes to form the quality of what is available (see

Pastor et al., 1997 and references therein). However,

quantitative data on moose–forest interactions are

accumulating (see e.g. Danell et al., 1991a, 1991b;

Bergstrom and Danell, 1987; Bergstrom and Hjeljord,

1987; Danell and Bergstrom, 1989; Edenius, 1992;

Edenius et al., 1993; Andren and Angelstam, 1993;

Heikkila and Loyttyniemi, 1992; Heikkila and

Mikkonen, 1992; Hornberg, 2001; Bergqvist et al.,

2003; Persson, 2003), making a general model

approach feasible.

2. The model

Our model is developed for forest planning when

potential timber trees are affected by moose browsing

in two ways: (i) the moose consumes part of the

biomass of young trees, and (ii) browsing damages

some trees to the extent that all their monetary value is

lost. Growth projections and interaction components

are modelled using a set of difference equations, with

the ultimate objective to maximise forest income

(Fig. 1). The model has a spatial structure where forest

growth in each individual stand is modelled separately.

Moose is supposed to distribute themselves optimally

according to density so that the relative foraging

opportunities are equalized across stands.

Let xt,s be the stock of stem volume (m3 ha�1) of

undamaged trees and yt,s the stem volume of damaged

ng of combined harvest of timber and moose.

H.K. Wam et al. / Forest Ecology and Management 206 (2005) 207–219 209

trees at time t in stand s. Let S be the total number of

stands. The stock is assumed to develop according to:

xtþ1;s ¼ ð1 � ut;sÞ½xt;s þ f ðxt;s; yt;sÞ�þ ut�c;snse

crs ½1 � bðmt;sÞ�;ytþ1;s ¼ ½yt;s þ gðxt;s; yt;sÞþ ut�c;snse

crs bðmt;sÞdðmt;sÞ�� ½1 � bðmt;sÞvðyt;sÞ� (1)

In (1) f(xt,s, yt,s) and g(xt,s, yt,s) are respectively the

growth of undamaged and damaged stem volume

during time period t, and they follow logistic growth

functions given by

f ðxt;s; yt;sÞ ¼ rsxt;s 1 � xt;s þ yt;s

Kfs

� �;

gðxt;s; yt;sÞ ¼ qsyt;s 1 � xt;s þ yt;s

Kfs

� �(2)

where rs is the intrinsic growth rate of undamaged

forest, qs is the intrinsic growth rate of damaged forest

and Kfs is the maximum volume of forest attainable in

the land in stand s.

In (1) ut,s is the fraction of the stock in stand s that is

cut at time t. Note that our model does not specify

whether a reduction in biomass density comes from

clear-felling one large or several smaller parts of the

stand. After cutting, new forest ns is recruited through

planting within the same year. The new forest does not

become part of the main stock until it reaches a certain

age of c years, i.e. the age when it first become

susceptible to moose browsing (assuming an average

snow depth of 50 cm throughout the winter). The

growth of new forest is not density-dependant within

the period of c years, and has the same growth rate rs as

the main stock.

In (1) b(mt,s) is a fraction of the young or damaged

forest that is either eaten or (further) damaged by

moose through browsing. The fraction depends on the

density of moose (mt,s) relative to its carrying capacity

(Kt,s) in the stand at time t, and is parameterised as

bðmt;sÞ ¼ asðmt;s=Kt;sÞbs , where as and bs are stand-

specific constants that quantify the amount of biomass

browsed by each moose during the time period t. The

fraction of the browsed biomass that is damaged, i.e. it

forever loses its monetary value because of stem

breakage, bark stripping or prolonged browsing of

apical leaders, is called d(mt,s). Like b(mt,s), d(mt,s)

depends on the density of moose in the stand at time t,

and it is parameterised as dðmtÞ ¼ 1 � e�gsmt;s , where

g is a stand-specific constant that quantify the amount

of biomass damaged during the time period t.

Due to repeated browsing and retarded height

growth, damaged trees may remain vulnerable

for more years than undamaged trees (Lavsund,

1987). In (1) the vulnerability of damaged forest to

further moose browsing follows the function

vðyt;sÞ ¼ 1 � e�ðls�yt;sÞ, so that the forest is no longer

vulnerable if it reaches a certain density set by the

constant ls (assuming that an increment in biomass

density to some extent corresponds to height growth).

The stock of moose also follows a logistic growth

function:

mtþ1 ¼ mt þ mtw

� 1� mPs¼Ss¼1 Kms þ esyt;s þ esns

Pj¼t�c2

j¼t�c1uj;s

� �As

h i24

35

� ht; (3)

where mt is the number of moose ha�1 at time t (counted

at the end of April each year), w the intrinsic growth rate

of moose in the area, ht the number of moose hunted

ha�1 at time t, Kms þ esyt;s þ esns

Puj

� is the carry-

ing capacity (Kt,s) for moose in stand s at time t, and As

is the proportion of stand s within the entire forest

area. We have not included natural mortality. In a real

planning scenario it can easily be incorporated, though,

either indirectly through the growth rate, or directly

as a mortality function.

In (3) Kms denotes a basic carrying capacity for

moose in stand s expressed as the density of moose

sustained over the winter without any extra forage

available from clear-cutting. This applies to the

situation when all the forest stock is close to its

carrying capacity (i.e. mature). The winter browse for

moose then must come from a patch mosaic of other

tree species, e.g. rowan (Sorbus aucuparia) or willow

(Salix ssp.).

In (3) e is a constant that transforms forest biomass

into nutritional value for moose in terms of increased

carrying capacity in winter. We assume that forest

regeneration contributes to moose carrying capacity

from (t � c1) to (t � c2) years of age, corresponding to

the heights within reach for moose with a snow depth

of 50 cm. Most of this contribution is indirectly

through increased growth of deciduous saplings after

clear-cutting.


Moose do not respect stand boundaries. In an ideal

world they will distribute themselves according to

optimal foraging theory, i.e. to go where food intake

will be higher in relation to costs (McArthur and

Pinka, 1966; Charnov, 1976). Assuming that they do

so, and that migration is effortless, the moose in our

model will disperse according to the following

condition:

mt;1

Km1 þ e1yt;1 þ e1n1

Pj¼t�c1

j¼t�c2uj;s

¼ mt;s

Kms þ esyt;s þ esns

Pj¼t�c1

j¼t�c2uj;s

8 s

¼ 2; 3; . . . ; S (4)

Eq. (4) postulates that the ratio of moose density to the

available forage is the same for all stands. We need one

more condition to determine the system:

Xs¼S

s¼1

mt;s ¼ mt 8 t: (5)

Timber is harvested and sold as different log quality

assortments. We have assigned one price for all the

timber cut in each stand, assuming an average dis-

tribution of quality assortments. A more specific price

differentiation is not applicable when harvest is simply

measured with a biomass index. The profit from

harvesting and replanting is given by

pf ¼XT

t¼1

XS

s¼1

dtðpfsut;sxt;sAs � cfsÞ; (6)

where pf, pfs and cfs, respectively are the profit,

average net price/m3 of timber cut and fixed cost of

cutting in stand s. cfs naturally is zero as long as no

timber is cut. dt is the discount factor.

Moose is harvested with a net price pm that may

include not only the meat value of the animal, but also

profits from trading the hunt itself and non-marketable

goods like recreational value for the hunters:

pm ¼XT

t¼1

dtpmht: (7)

To investigate how the model behaves for different

forest scenarios, it was run as a non-linear optimiza-

tion problem in GAMS (distribution 20.7—Windows

NT) using CONOPT21 as the solver (Brooke et al.,

1998). We first apply the model to a forest where

Scots pine (Pinus silvestris) is the only tree species

with economic value, and the pine is distributed in

stands of varying productivity. Thereafter, we apply

the model to a forest where pine, Norway spruce

(Picea abies), and birch (Betula pubescens) are

considered.

2.1. Optimal harvest in a pine forest with

varying site productivity

As a numerical example we have chosen a

hypothetical pine forest with three stands of varying

productivity: high, intermediate and low in terms of

timber production in Norway. This is a typical

situation as most forest owners in Norway have an

altitudinal gradient on their land (Eid et al., 1998).

Mathematically, the actual size of our example forest

does not matter (since all calculations are relative),

but practically it can be thought to represent one

large Norwegian moose hunting unit (i.e. 30–

50 km2, which normally includes several overlap-

ping moose home ranges, Cederlund, 1995). Note

that although the three stands are projected

separately in the model, in the real world this

may represent a diversity of different spatial

arrangements. A stand in our model is simply

outlined as a proportion of the land in question.

Therefore, it may just as well be viewed as several

smaller stands of equal productivity scattered within

the forest property. Traditionally this has been

termed a stratum in the forest management literature

(Clutter et al., 1983).

We start from a base scenario of general high

productivity (Table 1), and then re-scale parameters

from here to investigate what effect a particular

parameter has on our objective function. All model

parameters are adjusted to fit productivity and market

conditions in Norway. The planning period is set to

100 years.

The pine growth parameters Kf, rs and ns are chosen

in accordance with those given in Brantseg (1967).

There is no adequate data available on growth rates of

browsed-damaged pine, but we assume they are lower

than for undamaged trees. The actual level of qs is of

little importance here, though, as the density of

damaged pine stays negligible throughout the plan-

ning period for all realistic values (i.e. it never grows

out of reach for moose).


Table 1

Base scenario for an optimization model of harvest in a pine forest where moose inflicts costs by browsing young trees

Stand Pine growth

parameters

Moose growth

parameters

Moose browsing

impact parameters

Miscellaneous

parameters

Kf rs qs ns Kms w e as bs c1, c2 gs ls d pfs As cfs pm

1 pine 400 0.07 0.06 15 0.003 0.35 0.005 5.0 0.94 5, 15 23 40 0.03 1.0 0.5 10 50

2 pine 200 0.06 0.05 10 0.004 0.35 0.005 5.0 0.94 5, 15 23 40 0.03 0.9 0.3 10 50

3 pine 100 0.05 0.04 5 0.005 0.35 0.005 5.0 0.94 5, 15 23 40 0.03 0.6 0.2 10 50

The net value of 1 m3 timber (pf), the fixed costs of harvesting it (cf) and the price of moose (pm) is set relative to the net value of pine in the most

productive stand, which has a value of 1. Kf, Km: carrying capacity of pine and moose per ha; r, q, w: intrinsic growth rates of undamaged pine,

damaged pine and moose; a, b: constants that quantify moose browsing in undamaged pine; c: age at which saplings first become susceptible to

moose browsing; g: constant that quantify moose damage; l: constant that determine vulnerability of damaged saplings to further browsing.

The growth rate for moose ws is set to represent an

intermediately productive moose population in Nor-

way (Solberg et al., 2003). Since all moose popula-

tions in Norway are hunted, it is difficult to estimate

even rough figures for the carrying capacity Km. Our

effort is therefore a completely subjective one,

comparing harvest statistics, population levels and

forest productivity in Norway with a few studies

internationally that have shed some light on carrying

capacity for moose (Crete, 1989; Schwartz and

Franzmann, 1989; McLaren and Peterson, 1994).

Moose browsing impact parameters have been

quantified using the work of Edenius (1992),

Harkonen (1998) and Persson (2003), which are all

studies geographically close to Norway. The level of

browsing impact will in any way be site-specific, and

thus need to be adjusted if the model is implemented in

a real planning scenario. Factors that may influence

moose browsing pattern and -pressure on pine are, e.g.

site productivity (Danell et al., 1991a, 1991b), feeding

loops (i.e. present herbivory is influenced by the

Table 2

Base scenario for an optimization model of harvest in a forest with seve

browsing young trees

Stand Tree growth

parameters

Moose growth

parameters

Mo

im

Kf rs qs ns Kms w e as

1 pine 300 0.06 0.05 12 0.004 0.35 0.005 5.0

2 spruce 280 0.05 0.04 12 0.001 0.35 0.005 0.5

3 birch 180 0.07 0.06 18 0.006 0.35 0.005 3.0

The net value of 1 m3 of timber (pf), the fixed costs of harvesting it (cf) and

most productive stand, which has a value of 1. Kf, Km: carrying capacity of

trees, damaged trees and moose; a, b: constants that quantify moose br

susceptible to moose browsing; g: constant that quantify moose damage; l:

browsing.

previous) (Lyttyniemi, 1985; Bergqvist et al., 2003)

and silvacultural treatments like the use of artificial

fertilizers (Ball et al., 2000).

2.2. Optimal harvest in a multi-species forest

If we instead want to run the model for a forest with

several tree species having economic value, we let the

previous three stands denote one species each. Again

we have chosen a base scenario from which to re-scale

parameters to illustrate how the model runs (Table 2).

Tree growth parameters for spruce and birch are set

in accordance with Braastad (1966, 1975) to represent

stands of intermediate to high productivity. We have

used the work of Bergstrom and Danell (1987),

Randveer and Heikkila (1996), Danell et al. (1985,

1997), Bergstrom and Bergqvist (1999) and Jalkanen

(2001) to adjust moose browsing impact parameters

for birch and spruce compared to those given in

Table 1.

ral species having economic value, where moose inflicts costs by

ose browsing

pact parameters

Miscellaneous

parameters

bs c1, c2 gs ls d pfs As cfs pm

0.94 5, 15 23 40 0.03 1.0 0.3 10 50

0.94 5, 15 23 45 0.03 0.8 0.5 10 50

0.94 5, 15 11 35 0.03 0.9 0.2 10 50

the price of moose (pm) is set relative to the net value of pine in the

trees and moose per ha; r, q, w: intrinsic growth rates of undamaged

owsing in undamaged trees; c: age at which saplings first become

constant that determine vulnerability of damaged saplings to further


3. Results

3.1. Optimal harvest in a pine forest with varying

site productivity

For all the described scenarios, the optimal solution

in our model was a combined production of timber and

moose. The most influencing factor was the fixed costs

of cutting timber. When fixed costs were set to be

negligible the optimal harvest strategy was to cut

small amounts of the standing pine stock at a few years

intervals (Fig. 2A). With increased fixed costs the

optimal strategy approached clear-felling (Fig. 2D and

E). Note how the harvest strategy differed between

the stands. The most productive stand was cut most

frequently.

The second most important parameter to determine

the optimal harvest strategy was the impact of moose

browsing (Fig. 2A–C). With an increasingly negative

impact, the moose was kept at lower densities.

Accordingly, the amount of tree biomass removed

was kept at more or less the same level, determined by

the relative value of moose to timber. Of course, forest

productivity also influenced the optimal harvest

strategy. With less productive forest sites, less timber

was cut and the moose population was kept slightly

higher as it was relatively more valued. Within our

range of scenarios neither the price nor the produc-

tivity of moose had any noticeable influence on the

harvest strategy.

In our model the harvest of moose followed the

same cyclical pattern as the timber harvest. With

frequent cutting of pine, the moose population was

kept two to five times higher than their basic carrying

capacity (i.e. the density of moose supported when all

pine is mature) through most of the rotation period

(Fig. 2A and B). When the extra forage availability

diminished 15 years after a cutting, the population was

abruptly harvested down to the level of the basic

carrying capacity. With large timber harvests at long

intervals, however, the upper limit of moose was set by

their basic carrying capacity for most of the rotation

period (Fig. 2D and E).

Timber was the main contributor to forest income

for all our planning scenarios. Its relative contribution

varied from 94.5 to 98.5% of the net present value

(Table 3). Decreased forest productivity, increased

moose productivity, higher fixed costs of cutting and a

higher price of moose all reduced the relative

contribution of pine to forest income. Conversely,

with increasingly negative impact of moose, a greater

proportion of the income came from pine.

The factors which influenced the actual level of

total profit over time the most (apart from the price of

pine of course), was the fixed costs of cutting timber. A

five-fold increase in the fixed cost of cutting led to a

50% reduction in the net present value. Secondly,

forest productivity and the price of moose had

noticeable influence on the level of income, while a

change in the remaining parameters only had marginal

impacts.

3.2. Optimal harvest in a multi-species forest

Also in this forest we found that a combined

production of timber and moose was optimal for all the

described scenarios. The actual harvest pattern was

slightly different, though. In the multi-species forest,

more cuttings occurred simultaneously, creating an

even greater temporary effect on forage availability

for moose (Fig. 3A). Still, the relative directional

impacts of changing one or another parameter

(Fig. 3B–E) were the same as for the forest with

only pine having economic value.

As for the pine forest, timber production in the

multi-species forest was the main contributor to

forest income. Here, it accounted for 91.4–97.1% of

the net present value (Table 4), slightly lower than for

the pine forest. Again, decreased forest productivity,

increased moose productivity, higher fixed costs of

cutting and a higher price of moose all reduced the

relative contribution of timber to forest income.

Conversely, relatively more of the income came from

timber when the negative impacts of moose browsing

increased.

As before the factor which influenced the

actual level of total profit over time the most (apart

from the obvious timber prices), was the fixed costs of

cutting timber. In this multi-species forest a five-fold

increase in the fixed cost of cutting led to a 70%

reduction in the net present value. Secondly, forest

productivity and the price of moose determined the

actual level of income (in that order). Even in this case

was the influence of moose productivity and browsing

impact relatively low compared to the other three

parameters.


Fig. 2. A–E. Projected development of timber and moose in a pine forest of varying productivity. Optimization ran in GAMS (distribution

20.7—Windows NT) with CONOPT21 as solver. A: base scenario as described in Table 1; B and C: increased moose browsing impact (scenarios

I and II described in Table 3); D and E: increased fixed costs of cutting timber (scenarios III and IV in Table 3).


Table 3

Total profit per ha (ptot) from combined harvest of pine and moose in a forest of varying productivity

Base scenario Re-scaled scenarios

Forest growth parameters Re-scaling of forest growth parameters (other parameters kept as in base scenario)

Stand Kf ra ptot %pine Kf ra ptot %pine Kf ra ptot %pine Kf ra ptot %pine

1 400 0.007 300 0.006 80.6 97.0 300 0.006 200 0.006

2 200 0.006 99.5 97.9 200 0.006 150 0.005 77.8 96.8 150 0.005 70.4 96.4

3 100 0.005 100 0.005 80 0.004 50 0.004

Moose growth parameters Re-scaling of moose growth parameters (other parameters kept as in base scenario)

Stand Km w ptot %pine Km w ptot %pine Km w ptot %pine Km w ptot %pine

1 0.003 0.35 0.005 0.35 99.5 97.7 0.005 0.40 0.006 0.45

2 0.004 0.35 99.5 97.9 0.006 0.35 0.006 0.40 99.6 97.1 0.007 0.45 99.8 96.5

3 0.005 0.35 0.007 0.35 0.007 0.40 0.008 0.45

Moose browsing impact

parameters

Re-scaling of moose browsing impact parameters (other parameters kept as in base scenario)

Stand a g, l ptot %pine a g, l ptot %pine a g, l ptot %pine a g, l ptot %pine

1 5.0 23, 40 12.5 23, 40 98.9

(scenario I)

98.0 25 23, 40 25 45,55 98.7 98.5

(scenario II)2 5.0 23, 40 99.5 97.9 12.5 23, 40 25 23, 40 96.9 98.5 25 45,55

3 5.0 23, 40 12.5 23, 40 25 23, 40 25 45,55

Miscellaneous

parameters

Re-scaling of miscellaneous parameters (other parameters kept as in base scenario)

Stand cf pm ptot %pine cf pm ptot %pine cf pm ptot %pine cf pm ptot %pine

All 10 50 99.5 97.9 10 70 100.5 (scenario III) 96.6 (scenario III) 10 90 101.5 95.6 10 110 102.5 (scenario IV) 94.5 (scenario IV)


Optimization ran in GAMS (distribution 20.7—Windows NT) using CONOPT21 as the solver. Pine contribution to profit is given as percentage

(%pine). Kf, Km: carrying capacity; r, w: intrinsic growth rates; a, g: constants that quantify moose browsing; cf: fixed costs of harvesting timber;

pm: price of moose (cf and pm relative to price of pine, which has a value of 1). Shaded scenarios are illustrated in Fig. 2 along with the base

scenario.a The growth rate of damaged trees is reduced accordingly.

4. Discussion

Moose contribution to total income was very low

for all described scenarios ran with our model. At first

thought some may find this surprising, considering the

current optimistic view on potential income from

moose harvesting in Norway (see e.g. Aanesland and

Holm, 2003). However, if we set our objective

function to maximize income from moose and timber

separately instead of maximizing total income, we see

that it is in fact reasonable.

When maximizing moose income in our multi-

species base scenario (Table 2), the moose winter

population is kept at an average of 0.04 S.E. 0.001

animals per ha. Of this 0.01 S.E. 0.008 is hunted

each year. With a planning period of 100 years, the

total discounted moose value will be 9.1 (indexed

relative to the net value of 1 m3 with pine timber) per

ha for the whole period. In comparison, around 0.005

moose are harvested per ha in the most moose

productive counties in Norway today (SSB, 2002).

When maximizing income from timber, however,

the total discounted timber value will be 75.5 ha�1 over

the 100 years. The price of moose (per animal) then

must be 400 times the net value of 1 m3 pine to match

the income from timber alone. With the present timber

prices in Norway, this would be close tos12.000. Such

a high price for moose does not seem realistic. Hence,

for moose to be the main contributor to forest income,

all the factors promoting moose production rather than

timber production must be combined, i.e. low timber

productivity, high moose productivity, low fixed costs

and a high price for moose. In Norway such a scenario

may occur, e.g. on some costal islands along Trøndelag


Fig. 3. A–E. Projected development of timber and moose in a forest with several tree species having economical value. Optimization

ran in GAMS (distribution 20.7—Windows NT) with CONOPT21 as solver. A: base scenario as described in Table 2; B and C: increased

moose browsing impact (scenarios I and II described in Table 4); D and E: increased fixed costs of cutting timber (scenarios III and IV in

Table 4).


Table 4

Total profit per ha (ptot) from combined harvest of timber and moose in a multi-species forest

Base scenario Re-scaled scenarios

Forest growth parameters Re-scaling of forest growth parameters (other parameters kept as in base scenario)

Stand Kf ra ptot %pine Kf ra ptot %timb Kf ra ptot %timb Kf ra ptot %timb

1 300 0.006 77.5 96.0 200 0.005 70.3 95.4 200 0.005 62.5 94.9 200 0.005

2 280 0.005 280 0.005 200 0.004 180 0.004 54.5 94.5

3 180 0.007 180 0.007 150 0.006 100 0.005

Moose growth parameters Re-scaling of moose growth parameters (other parameters kept as in base scenario)

Stand Km w ptot %timb Km w ptot %timb Km w ptot %timb Km w ptot %timb

1 0.004 0.35 77.5 96.0 0.006 0.35 78.4 95.8 0.006 0.40 79.0 95.0 0.007 0.45 79.5 94.2

2 0.001 0.35 0.003 0.35 0.003 0.40 0.004 0.45

3 0.006 0.35 0.008 0.35 0.008 0.40 0.009 0.45

Moose browsing impact

parameters

Re-scaling of moose browsing impact parameters (other parameters kept as in base scenario)

Stand a g, l ptot %timb a g, l ptot %timb a g, l ptot %timb a g, l ptot %timb

1 5.0 23, 40 77.5 96.0 12.5 23, 40 77.3

(scenario I)

97.0

(scenario I)

25.0 23 77.0 97.1 25.0 45,55 76.8 97.0

2 0.5 23, 45 2.5 23, 45 2.5 23 2.5 45,60

3 3.0 11, 35 10.0 11, 35 20.0 11 20.0 20,50 (scenario II) (scenario II)

Miscellaneous parameters Re-scaling of miscellaneous parameters (other parameters kept as in base scenario)

Stand cf pm ptot %timb cf pm ptot %timb cf pm ptot %timb cf pm ptot %timb



Optimization ran in GAMS (distribution 20.7-Windows NT) using CONOPT21. Timber contribution to profit is given as percentage (%timb). Kf,

Km: carrying capacity; r, w: intrinsic growth rates; a, g: constants that quantify moose browsing; cf: fixed costs of harvesting timber and pm: price

of moose (cf and pm relative to price of pine, which has a value of 1). Shaded scenarios are illustrated in Fig. 3 along with the base scenario.a The growth rate of damaged trees is reduced accordingly.

and Nordland, where a highly productive moose

population thrive on very low-productive forest land

(Sæther et al., 2004). It is important to note that not only

the total income, but also the optimal management path

will depend on the initial conditions of the forest

(Crepin, 2003). For example with no mature trees at the

beginning of the planning period, and the planning

period being shorter, the moose can have a relatively

higher influence on the optimal management of a

specific property.

Although the harvest principles were the same in

the multi-species and the single-species forest, to

some degree the actual harvest pattern differed

between the two. The basic underlying factor for this

is productivity per se. All else being equal, the more

productive a stand, the more frequently it is cut

(Faustmann, 1849). Overall, the multi-species forest in

our model was not as productive as the single-species

forest, and hence it was cut less frequently. Accord-

ingly, there were fewer peaks in the moose population

within this forest, as the number and magnitude of

forest cuttings determined moose carrying capacity.

In contrast to the more regular cuttings of the pine

forest, there was a wide range in the time span between

cuttings of the multi-species forest. This made the

amplitude of moose peaks to be highly variable,

depending also on the size of cuttings. If cuttings had

been more separated in time, the moose density would

instead be kept intermediately high for more of the

planning period. Apparently, the relative value of the

moose in our described scenarios was too low for this

to be an optimal strategy.

All our model specifications and parameter

assumptions are not equally realistic. While a

biological model has to be realistically founded, some

level of detail must be omitted to give the model

general value. The major simplification in our model is

the framework of difference equations, which limits the


possibilities for having several unique classes of the

same species. Ideally, a moose–forest model should at

least distinguish between sex- and age-groups of

animals, and age- or size-groups of trees. Particularly

for moose, essential dynamics are lost when resources

are grouped into one single measure of biomass. For

timber production, using only a biomass index reduces

the opportunity to differentiate between qualitative

aspects related to age (e.g. growth rates, marked price)

and spatial scale. Biomass density may be an adequate

indicator of timber trees if stock development is

uniform throughout the stand. In Scandinavia, however,

this is often not the case, and there is an increasingly

interest in focusing on the individual tree both in

modelling (see e.g. Holthe and Solberg, 1989; Eid and

Tuhus, 2001; Andreassen and Tomter, 2003) and in

actual harvesting (Lexerød, 2003).

By increasing the number of stands in our model,

though, we can in fact make it work for more spatially

heterogeneous forests. Mathematically our model

treats a stand simply as a proportion of the land in

question. By assigning the stand label to specific age

groups for each species we can make the model work

even for mixed-species forests (i.e. forests with no

distinctive stand formation). Practically of course,

such a forest would demand selective harvest in its

uttermost form, which currently is not a prevalent

practice in Norway. Such a model alteration would

also imply some adjustments in the moose–forest

interactions. Browsing pattern and pressure on one

species may to various extent be affected by

neighbouring species in such mixed-species forests

(see e.g. Heikkila, 1990; Edenius, 1991; Heikkila and

Harkonen, 1993, 1996; Ball and Dahlgren, 2002).

In the same way as we could easily incorporate

more stands into our model, we could add different

categories of timber quality. At present all browsed

plants either escape relatively unharmed (i.e. with less

biomass) or completely damaged. Alternatively we

could assign some low marked value to parts of the

latter category (e.g. as pulpwood or firewood). This

would eventually and to some extent mitigate for the

negative impact of moose browsing, and thereby more

moose would be acceptable in the forest.

Neither does our model take stochastic events into

account. Carrying capacity for moose over time is a

stochastic and seasonal phenomenon that cannot

adequately be represented by a single number.

However, we believe that for our particular model

stochastic elements would add unnecessary complexity.

Our model is developed to aid in long-term planning.

While we believe that stochasticity may influence the

optimal management of boreal forests for shorter

periods of time, we also believe that the effects are

balanced within our planning period (as opposed to the

severity of stochastic events in more extreme areas such

as the Sahel, see e.g. Lybbert et al., 2004). The actual

outtake of a forest system like ours may in any way be

adjusted if and when stochastic events occur. If more

complexity should be added, it should preferentially be

interactions with additional species, mainly those of

great functional importance in the forest ecosystem.

Our assumptions on moose browsing are loosely

built on empirical research. At present the data we

have available are sufficient to give only a rough idea

of the mutual impacts of moose browsing. While this

may not be a major concern for the purpose of this

article, it will be if the model is implemented in a real

planning scenario. Moose browsing will in any way be

site-specific, but it should be possible to identify some

general quantitative relationships given specific

common factors.

On the basis of this work, two lines of future research

can be recommended: bio-economic modelling which

incorporates a more complex class partition, and further

studies of the feedback mechanisms between moose

and its foraging plants. With some more effort on these

aspects, we should soon be able to optimize the

combined management of timber and moose produc-

tion quite adequately on specific properties.

Acknowledgements

The authors are grateful to the Norwegian Research

Council and the Agricultural University of Norway for

financing this research project. We also appreciate the

valuable comments given by the reviewers of this

manuscript.

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A bio-economic model for optimal harvest of timber and moose