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GROWTH MODELS OF HARD MAPLE FOREST IN
SOUTHERN ONTARIO
HAWI[N SHI
A Thesis submitted in conformity with the requirements for the Degrœ of Master of sCic11ce Graduate Department ofForestry
University of Toronto
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FACULTV OF FORESTRY University of Toronto
DEPARTMENTAL ORAL EXAMINATION FOR THE DEGREE OF MASTER OF SCIENCE IN FORESTRY
Examination of Mi. Haijin SHI
Examination Chair's Signature:
We approve this thesir and affirm that it meets the depanmental oral examination requirements set down for the degrer of Master of Science in Forestry.
Date: 5 s&PJ 00
ABSTRACT
Gnmta Models of Hard Mapk Forest in Southeni Ontario
Hdjh Shi
Degret of Muter of Science, 2000
Graduate Department of Fonsty, University of Toronto
Two rnatrix powth models ofmixeci unmn-aged hard miiple forest are developed using
transition probabilities as deterministic and r d o m variables reopeaively. These two
growth models arc estimateci usine growth data (Browth period 5 years) fkom 30
permanent sample plots in southmi OntMo.
The maük growth mode1 with detedstic transition probabilities is used to obtain the
steady state ofthe atand. These two maVa growth models arc used to sirnulate the stand
dyuamics (diameter class distniutions) for 100 yean. In detemiinistic mode, stand
reaches steady state after 110 years. Stand dynadcs is almost similar in deterministic and
random modes.
The mode1 with determuiistic ttansition pmbaôiies is solved for constant yield and
nonconstant yield by linear prolpurmmg and nonconstant yidd by MAXMIN ripproach.
The cornparison ofresults sbows tbat the West regimc obtrllied by MAXMIN approacb
is better than others.
ACKNOWLEDGEMENTS
1 would üke to thuilr my supervisor Dr. Shshi Kant, for his guidance dwing my studies.
His advice h a ben vay stimulating, ud ddd a greut deai to the quaiity of my work.
He has Jso b a n very supportive to on- ideas which wae then not so clear but tumed
out to b hitfuL Additionally, 1 appdate the opportunitics he bas &en me to pursw
studics in the ana of forest economics and management.
1 dso tlmk the other m e m b ofmy cornmitte. Dr. J.C Nautiyd and Dr. David Martel
for th& helpfil suggestions and for reviewing this manuscript. 1 am gratefùl for the
assistance of the people of the Facuity of Forestry who have been openly helpAiI, uich in
their own way.
Finsacial support was greatly provided by the University of Toronto Open Feliowship
and stipmds Born Naturai Science and Engineering Rcscarch Councii, Canada. Theif
financiai apport is greatiy ickiowledged.
Edy, 1 would Lüce to thank my hmily and my fnends for th& love and support dwing
my study. This thesis is especiaiiy dedicatd to my parents, who got little education
tbemselves, but h v e aiwys ban supportive for good ducation for th& children, even
in those vmy difIiCUIt diys when they couid b d y f d their fhmily. It is theu love and
~ b i t t h a t d e m a c o m c t h U b t .
TABLE OF CONTENTS
Absttact ............ ... ...... ......... . ....... ................... ....... . .... ... ... ......... .... ~owledgements . . . . . . . . . . . . . . . . . . . .. . . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. T a b l - ~ f C ~ n t ~ ~ . . . ~ . . . . . c r . c ~ . r ~ ~ ~ ~ ~ . . . . . ~ . * ~ . + . ~ r * ~ ~ - * . ~ r . * ~ . t . r . r r r r . . . r r . . . ~ r . + . . . r . . . . .
ListofTables ... ..... .... ........ .,.....,..............................*.... .... . ......... ListofFipres ............................................ ....... ......... .................. chipa 1: Ifltroduction .... 0 . . .. . . .. +.. ... ... . . . . . . ..... . .* ..* . . . .. . ... .....*. .*. 0 . .
Chapter 2: Literature Review . . . . . . . . . . . . . . . . . . . . .... . . . . . ... . . . . . . . . . . . . . . . . . . . ...... 2.1 Growth and Yield Model . . . . . . . . . . . . . . . .. . . t., . . . . . . . . . . . . . . .. . . . . . . . . ...
2.1.1 Single Tree Modeh and Stand Models ... . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Ma& Growth Models .. . . . .. . . . . . ... . . . . . . . . . . . . . . . . . . . . . ..... . . . ...
2.2 Linau Progrpmmllig and Forest Management . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Random Variables in Lincar PrognmMng Problems ... . . . . . . . . . . . . . 2.4 Mximhtion ofMinimm (WUMIN) Approach and Forest
Management . . . . . . . . . ... . . . . . . . . . . . . . . . .. . . . ..... ........ ....... ... .. .............. . . . . . .. Chapter 3: Tbcory and Mahodology .......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
3.1 Thwry .......o........... t t t t....c r.. 9.. +*. ...o.* + * * .+. ..*..*.-.+.*.-........
3.1.1 Detsnniiristic Transition Probability Matrix Growth Mode1
(DTPGM) .................... .. .................................. .......... 3 . 1.2 Random Transition Probaôii hhtrix Growth Model
(RDPGM) .............. . .. ... ... .. . . . . ... ... . .. ... ... . .. . . . . .. .......... .. 3.1.3
O * *on of the Minimun 0 Apptoach ..... . . .
3.2 MettiOdoIow ...... ............ ... ... ... .......................... ............. .....
3.2.1 nie Suad Dynumcs ud the Steady State of a Stand in the
DetenninjsticEnvirommt ................................................... 40
3.2.2 The Stand Dynamics rad the S t d y Strte of a Stand in the Random
Envkoment .................... ,,t ........ ,,., ...................................... 42
3.2.3 Comparative Study of the Hard Maple Stand in the DetettnjlUstic and
the Randorn Environment ..................................................... 43
3.2.4 The Comparative Study of the Hard Maple Stand for Conventional
Equity (Constant Yield Condrahts) and Rawlss Equity (MAXMIN)
Criterion .................... ,,,,. ................ .., . 44
3.2.4.1 Maxhhtion of NPV with Constant Harvests ................
3.2.4.2 M;aximization of the Net Present Value (NPV) with Non-
............................. ...........*.....,. .. mnstant Hamest ... ......
3.2.4.3 Mu<imUatioa of the Minimum NPV for Each Cutting
Cycle .............................................................................
Chapta 4: Data ...............................................................................
4.1 Stand Growth Data: SowcesS Interpretation, and Preliminary
AnaIysis ......................................................................
4.2 Price Data ....................................................................
Chaptcr 5: Matrix Gmwth Modd Estimation .............................................
5 . 1 Components of the M.tmt Growth Mode1 ..................... .... ..
5.1.1 hgrowth Equations ..................................................
5.1.2 EqUIIfions for Transition Proôabilities ...........................
5.1.3 Mortrlity Equatiom ................... ... ..........................
5.2GrowthMiitrixG .................... ...................... .................
5.3 Variances ofTransition PmbabilititS ......................................
Chapter 6: Studics and ResuIts ............................................................
6.1 Anaiysis of the Hud @le Stand with Daaministic Transition
Probability Gmwth Modd (DTPGM) . ...........-.......................
6.1.1 The Steady State of the Stand .....................................
6.1.2 S t d y State and the Stand DyiiMics ............................ 6.1.3 The State of the Stand at Dierent Periods ......................
6.2 Analysis of Stand Orowth with Random Transition Probaôiiity
......................... Growth Modei (RTPGM) .. ... .. .............
6.2.1 The StandDynamics ................................................
6.2.2 State ofthe Stand in the Random Transition Probability
Growth Mode1 (RTPGM) .................... ... .. .... ......
6.3 Cornpuison ofthe States of the Hard Maple Stand, Obtained
......................... by Using DTPGM and RTPGM, at 110 y w s
6.4 The Comparative Study of Economic Outcornes under Constant
Harvtst, Non-constant Harvest, and Martimization of Minimum
..................................................................... HaWest
Chapta 7: Conclusions .....................................................................
..................................................................................... Refiim
LIST OF TABLES
Table 4.1 : The Y-. A m and Age of Establishment and the Remmement Yeam
of the Plots ......... ,.. ............... ..., ...................................... Table 4.2. The Bugc Description of Data Set .................. ... ..... .. ........... Tuble 4.3. Per Trce Stumpage for Diffkmt Diameta Classes (Unit: CADWtree) ....
Table 5.1 : Ingrowth Equations for Hud Maple. White Ash, Bladc Che ny, and 0th
s pcoies ............................................................................... ... .. Table 5.2: T d t i o n Probabüity Equations for ELvd Maple. White Ash, Black Cherry.
and ûther Species .......................... .. ....................................
Table 5.3: The Mean of Transition Probabilities of Hard Maple, White Ash, Black
Cheny, and Mer Species ......................... ... ..........................
T ' l e 5.4: Mortnüty Equations for Hard MPple. White Ash, Black Cherry. and
0th- SpeGie3 ............................................................................
Table 5.5: The Means of Mortuiity for Hard Maple. White Ash, Black Cherry and
0th- Speçies ............................................................................
Table 5.6: Growth Mtrix G. for B = 21.6 m2/ha, and for a Time Intemal of 5
Y .........*.......... ........... .......*.....*.....................*..................
Tabk 5.7: Table 5.7 Standard Deviation of T d t i o n Probabiüties of Hard Maple.
White Ash, Blrk Cherry. and ûther Species ..................... ... ........... ....... Table 6.1 : The Initial Strte (Dimeta distribution) of the Hard Maple Stand
............ Table 63: S t d y Statc mameter Distri'bution) of the Hud Maple Stand
Table 6.3: The Aggregate Dimeter Dismiiiution (of AU Species Together)
Table 6.4: The R a n p ofNumk of T m in Difkent Diameter Classes (a 5%
Sienificance Level) with RTWM at 5. 50. and 100 y- ................
Table 6.5: The Statc o f Hard Miple Stand (Diuneter Distribution) at 5.50. and 100
Yeus (For the Case ofNPV Muamurti N D
'on and Using RTPGM) .........
Tabk 6.6: The Harvcsts Lswls at 5.50, rad 100 Y u n (For the Case of NPV
......................................... Wximhtion and Using RTPGM)
Table 6.7: NPVs and the Cumnt Net Rehim with Constant Hawe* Non-constant
Harvcst and MAXMIN Approach (Unit: CADS) ............................
Table 6.8. Harvest Levels in the Case ofNon-constant Harvests ......................
Table 6.9: *est Levels in the Case of Maximization of Minimum Harvests
(MMMiN) Approach ......................... .. .............................
Table 6.1 O: Harvests Levels in the Case of Constant Harvests .........................
vii
Figue 4.1 : The location of petmanent sample plots @la@ are the location of
pennancnt sample plots) ...........................................................
Figure 6.1. The Diameta Distriiution of Hud Maple at S t d y State .................
Fi- 6.2. The Diameter Distribution of White Ash at Stecdy State .................
Figure 6.3 : The Diuneta Distribution of Blrdc Chary at Steady State .............
Figure 6.4. Tbe Diameter Diatribution of ûther SpeQes at S t d y State ............
.................... Figure 6.5. The Diameter Distribution at 5 Yean .. .... .. .........
Figure 6.6. The Diameter Distnibution at 50 Yeats ........................................
Figure 6.7. The Diameter Distribution at 100 Y m .......................................
Figure 6.8. The Diameter Distribution ofHud Maple ....................................
' F ~ W 6.9. The ~ i a m e t a Distribution of white Ash ......................................
Figure 6.10. nie Diameter Distniution of Black Cherry ................... .. ......... Figure 6.1 1: The Diameter Distri'bution of Otha Species ...............................
... Figure 6.12: The Cornparison bctween DTPGM and RTPGM with AU Species
....... Figure 6.13 : Cornpuison of DUmeter Distniution of AN Spies at 5 Yean
..... Figure 6.14: Cornpuison of DWneter Distribution of AU Species at 50 Years
... Figure 6.14: Cornparison of Diameter Distniîbution of AU Species at 100 Years
CHAîTER 1 INTRODUCTION
As w d known, Canada is a forest nation. The whole forest land is 453.3 miliion ha. The
totai vobne U B,92 1 million m' (Hegyi, 199 1). More thui haif of the iand uci in the
ten provinces of Canada is covercd with forests. For txample, the forest land in Quebec,
Ontario and British Columbii is 83 -9.60.6 and 58 million ha, respectively, wwch is
54.44%, 63.92% and 54.26% of total provincial land rerpectively (Natural Resource
Canodr, 1999).
The forests of Canada are divided into eight regions. Each region is a major geographic
zone cbaractcriwd by broaâ dormity in physical features and in the composition of the
dominrnt tree species (Rowe, 1972). About 140 native t ne species are found in Canadian
forests (Hosie, 1%9). W trar fd into one of two groups - the softwbod trees, which
hold th& nde-üke leaves for two scssons or longer. and the hardwood trees, whose
broad leaves change color in autwnn and are shed fkom the tree, uwdiy before winter.
Ody 31 of the native species in Canada ire sofhivood, yet they dominate Canada's
forests, accountin~ for about 80?? of the totd volume of merchantable timber (Northcott,
1981).
With cusfiü management, we can can that forests wül thrive and wntînuc to provide
the muiy bendits to winch we have becornt accustomad. Foresters can caicuiate an
'aHowaôIe cut! of &ces per yeu for any @en fonded am that will ensure a sutUncd
yidd in psrpeduity. B d on vaious a~sumptr~onr, incIudin8 a bigh Ievel of forest
2
management and a stable fores& land base, the maximum aîlowable cut of both hardwood
and softwood spccies in Canada has bœn estimated at 276 million rn3 (Northcott, 1981).
In the culy 1980% the v o h ofwood buvested per ysu was roughly 57% of th9s
amount. Tnditionai attitudes Bvored men-aged management of s o A w d forests that
produccd large and pndiaable yields, howmr, the harvest of economicaliy accessible
softwaod m i e s is vesy n u r the upper W. Shoortages offonsts, which could be
harvested. are emsrghg in evety region (Northcott, 1981). The increasing focus on
biologial divenity and ecologicai processes hu stimulateci interest in managing for
hardwood and mixed forests (MacDonald, 1996).
In Canada, the hardwood forest is 15% that is srnail proportion compared to sofbvood
forests (NRC, 1999). Hardwood forests are, normally, mixed hardwood forests. At the
provinciai Ievei, hardwd fonst constitutes 23% in Ontario. In Saskatchewan, Alberta,
Nova Scotia, and Prince Edward Island hudwoad forest are 36,33,33, and 300/. of the
total fonstland respedively. In New Bninswiclc, Manitoba, and Quebec, proportions of
hardwood fonst are 24.21, and 19?6 respcctively (NRC, 1999).
Cher the nuct two d d c s , the management focus on puMc forest lands is atpeaed to
SM away fiom thber and towards greater emphasis on naturat system p r e ~ t l ~ t i o n and
non-commodiry output. These prospective trends in pubiic management policies mry act
to ampw demuds on mixeci hardwood fbrests* This sbiA dexnands that the f i s of
fomt ~aylamait pmctkes iIro k dùeciad towds hardwood for- Furthamore,
First, commercial intuest in untven-aged mixed hardwood forests is increasing, because
thcy represmt an eaaiily ~ i l e source of highqurüty fiber. MiKd hardwood forest
sites am the most fertile d productive. Mixd hardwood forest management reduces
market volatiiity for industryD h s e hardwoods and softwoods generate dinerent
produds. The structures of many mVred hsrdwood stands produce large trees that
constitutt an attractive source of sawlog and vcneer materiai (Oppet 1981). Some mixed
hudwood stands have higher yields thPa rnonodtures of a component species. For
example, mature mixeci hardwood stands in north-central Ontario produce about 268
m3& cornpurad with 188 r n 3 h for average black spnice stand (Opper 1981). The
Mxed-species e R i is most pronound for vertidy stratifiecl mixtures (Burkhart and
Thua 1992).
Second, uneven-aged mixed hardwood forests are eculogically resüient. The loss of a
singie component does not threaten the integrity of a species-rich ecosystem. Cornpanion
species differ in theu limiting environmentai factors, growth habits, and physiologicai
proœssa, maximiEng biologid activity per unit area (Chon et 1 1988; Schder and
Smith 1988). Mixeci hardwood stands are dso more resistant than monocultures to
damage by whcl, sun, insccts, and h g i (Bedell1962; Navratil et ai. 1991). Physid
sepurtion of susceptible @es inhi'bits the s p d of many biotic pests (Burkhart and
Tham 1992). ûrowing a mixture of species over thne on cach site h a cornmon
agcicultural p d œ to miiauin Bte productivity. Soii nutrient s t ias can be similady
enband in forcsüy ôy promothg speciCes mïxtum. For exampIeD hardwood uee litter
impmves mil propeftics in European &cd bardwood forests (Nyyssonen 1991;
Panerson 1993). The soii-improving benefits of mixed hardwood stands have ban
tccogiiizcd in western Canada (Namtil a al. 1991).
F M y , muiy benefits can be reaüzed firom mixd hardwood management because of its
emphasis an species mixtures and partiai d n g . Aestheticaiiy vlried mixed hardwood
forest landscapes provide opportunities for raxeation and towism, and the succession of
vegefitioa protects watershed stability, ensuns an even flow of crop trees to wood-
processing industries, and many species of birds and rnaMnals are favored by the variety
of successional stages typical of Mxed hardwood forests (Boyle 1992). The diversity of
flon and fwna dso supports rnany abonginai values, such as oppomuiities for hunting,
fishing, trapping, and securing traditionai mcdicid plants. The promotion of species
divasity ensures adaptabiiity to the changing needs of society (Schütz 1990).
The traditionai mcthod to manage miwd uneven-aged hardwood forests is the selection
system This system is excciitnt where environmental conditions, protection
considcrations, or aestbetic considerations wuire that forest cover r d n continuously
on the landscape (Kinmiinr 1992). W~ers (1977) offaed the foiiowing definition of
selection systun:
suaiincd yidd - i d d y over a whole forest or wocicing chle, but
&ocn p d c a i coupos of cutting suies; regontfation mainiy Ruunl
rad a a p ideally aii-aged.
Hcng selection system accommodates aistainad yield by maintainhg uneven-ageâ
stand8 in such a condition as to fostu the production of d o r m periodic harvest
volumes. In the ii@ of tecent deveiopments mentioned More, thme is an urgent n a d
for predictive twls to d s t hardwd forest managers in using selection systern
(Burkhart and Tham 1992). The variety of possible species mixtures coupled with the
range of management options neassitics a modeiiig approach. Models of growth and
yield for mixeci hardwood stands are essentiai for making sound management decisions.
Howmr, the pra*ices, studies, and information bases in Canada do not adquately
support mixd hardwood forest management.
VMous approaches, such as caiculus approach, optimal control technique and matrix
growth mode1 approach bove been used to mode1 different aspects of uneven-aged
hardwood forest Howevet, the most cornmon approach has beea the matrix growth
modd that wu Brst p r o p o d and u d by Leslie (1945) to study animai populations.
U k (1966) inaoducd it into forestry rad Buongiorno and Michie (1980) uscd it to find
out nistaiaed-yield management r q h c s thaî wouid maximhe the net prrsent value of
pecidc hamsts for a sinde sp ies hardwd fom. 'RB mat& growth modd is based
on pmbabilities ofmnrition of- between diameter ciasses and ingrowth of new =S.
Buongiorno et ai. (1994), ushg the &mework of Buongiorno and Michie (1980),
pmented a mcxiel to compte cconomic cutting policies in regdateci uneven-aged
fomts. Bwngiorno et al. (1995) devtloped a mmLr mdd for mixd forest, and midied
stand dynamics, the stcdy state, the maximum divenity and mnvimum cconomic
efIicieacy using Iinear propmnhg. In the Carindian contcxt, Kaat (1999) uscd the
mitra growth mode1 to examine the impact of thce eumomic factors--the rate of tirne
p r t f i c t , incorne and pro- taxes, and the subsidy for nhabilitation ofdegmded
forests-on sustainable management of unewn-agd pnvate woodlot.
However, the- growth matm rnodels and th& solutions have two weaknesses: (1)
transition pmbabüities arc issumai to be deterministic; and (2) the objective function 0 .
(eg., profit) is mwunued subject to ''constant yield wllStraintsn. Constant yield
constraints are implicitly rosigned infiaite 'talut" and are thus met (if possible)
regardlas of the opportunii cost. Also such constnints do not d o w harvest levels to go
dom even a smaii amount unless predetennincd (and therefore cqually arbitrary)
deviations fiom tirne pend to tirne period are prespecified. The ma of these constraints
cen be enonnws in temis of foregone objective hction attainment.
The logk behind constant yield constraints aeems to foliow one of two rationales:
s t a b i i and the c o d o n . The rationale of consewation implics an equitable
distniution oftimber hwests acroro generations (or timber paiod). However, other
view for inter-gcndon equity is Rnivls' @ty criterion (1971), which means that
goods shouid k disbi'buted to most M t the wont-off individual. Applying UW equity
aitarion to the timkr hmt-scheduling pmblem would suggest that, over a time
horizon that is assumeci to be adquate, the minimum hanmt levd of the given time
In this thesis, an attanpt has kai d e to ddnss these two weakncsses of the matrk
growth m d d and th& use in uneven-aged hudwood forcas. Hcnce, this reseorch has
two mrin objectives: to incorponte d o m nature of transition probabüities in rnatrix
growth modei, a d to sdve the matrix growth mode1 for non-constant yield constmints
thut satisf;l Rawl's equity criterion.
in this thesis, rnatrOt growth models have been fomuiated and solved for mixed Hard
Maple (Acer s~;%churn) forests of muthem Ontario. Data fkom 30 permanent sampie
plots of hard maple foms, tnaintained by the Ontario MUüstly of Natural Resources,
have km used. To adcûcss the two objectives, total nsearch is divided in four
componaits. Fust, a matrix growth mode1 with deterministic probabüities is fomulated
and solveâ for steady state and stand dynamics. Sccond. randomness of transition
probabiities h incorpotated in the maÉrix growth modei, and stand dynamics is examineci
in this environment of randomneu. Third, outcornes ofdeterministic transition
probability d e l and random tnngtion probability modei are compareci. Fourth,
detssministic transition probability mode1 is wlvd for constant yield constraints and
yield constdnts thrt ds fy Raws cquity criterion, and d t s are comparai.
In tbU thesis, tbe Iiteranin about the p w t b mode! and management ofuncven-aged
mimi fôrtst ir reviewai in Chapter 2. In Chipter 3, a theoretical fhmtwork of the
mrtmc growth modd of uneven-aged mixed hardwood fomts and a matrix growth mode1
with Mdom tnnsition ptobabiiity are presented and the methodologies for dysis are
inttodud. In Cbapter 4, the data is introdud and interprcted. In Chaptar 5, asthutcd
parameters of the gmwth d e l are pmmted. In Chapter 6, nsults ofaii four
componenta of raauch are dioarssed. Finally, r-ch is concludeâ in Chapter 7.
CHAPTER 2 LITERATURE REYLEW
Approack to p w t h and yield mdels, use of Lwu Progmming to forest
management problmu, random variables in LP problems and approaches to th&
~olutioiy and the maxhhtion of minimum d u e s (WUMIN) are four brod
theoretid aspects relevant to this research, Hence, litcnture d e w is divided in four
sectjolls*
2.1 Gmwth and Yitld M d &
Empincal models and proctss models are two broad groups of growth and yield models.
Empind rnodels are used when little is hown about the process that is being modeled.
Pmcess models ( a h d d theoretical models) an based on a complete knowledge of
physicai, chernicol. or biological mechanisms invalvecl in the process that is behg
modeled (Box and Hunter, 1962; Blake et al., 1990). Empincal models use corretations
betwcen variables to make a prdiction (for example, the correlation between tree size
and age can bc used to constnact a growth and yield model) whenas process rnodeis use
uusband-efkt nlationships to desaibe a system (for aample. a lack ofmoishm
causes stress in tms) (Blake et ai., 1990). Most mdels are a compromise betwan these
two approaches and consequeatly have some anpinul and some proccss aspects in them
(Blake et al., 1990).
Pmea modds have muiy advantagcs compareci to anpiricil modds such as -ter
~~phmtory power, wida applicsbility, gmtm s e f i w b adnpohting byond the
spatial ud temporal mage of data usai in dweloping the modcIs (Ludsberg, 1987).
The main disadvanta8e with p r m mdels, porticululy biologicai modcls, is that the
pmcesses iwolved am oAen not W y understocxi a d bencc mrne ernpiricism must be
intmâuced into the madel. Aiso, the cornplex proceeses require complut rnodels and
amounts ofdata thrt are difEcult to obtrin. In the forest modeling, thae hs b a n a
trend fkom developing anpirical models to proass modds. Cunmly, the empirical
modcls, based on obsemd dationships, are the most açcumte (Sharpe, 1990). However,
because ofth& great potential for accuacy, u more work is dont in developing and
r e g proeess models they wül eventudy have better predictive abilities than
empincal rnodels.
2.1.1 Single T m Modeh and Stand Modeb
At present, many kinds of growth and yield models have b a n developed ta describe
d u s attn'butes of forest dywnics (e.g., basai uea or volume) and the way they change
o v u time, howevcr most ofthem are empuical mode1s. Munro's (1974) ciassified these
empincal growth and yield models u single tree (now d e d individual trec) models and
whole stand models.
Sin& trec models are bascd on individuai tne characteristics (e.g. tree diameter*
dutance to nearcst neighbor* cornpetitive position in the stand). These models are fitrther
classihd as baag distance-indqmdent or distandependent, baseâ on wàaher the
dinrnce bnweai trees is roquirrd or not (Mme, 1974).
Mowr (1972) dmloped distance-independent single trrt modd based on a set of
difhntiai equations repft~enting d v o r growth, ingtowth and moWty ntes as
fundons of stand basal a m and n e o f m . The ainimation ofindividual
componcnt gcowth nter yidded the net p w t h nte for the entire stand. Moser applied
numericd methods to solve thc systcm of difftttntial equations and update initial model
stand basai a m and number of trsas pa unit ner. Distance independent single-tree
mdels devdoped by Botkin et ai (1972)' and Shugart and West (1977) have proved to be
very powerfbl means of ceptesenthg cornpetition baween trees, mortatity, variations in
species composition, and environmentai infiuences on forest growth.
Opie's (1972) mode1 for mn-aged Eucui)p&s reegnrmr is another acample of distance-
independent single-trec model, which comprised two parts. The &a Meen yean were
modeîied using a whole stand approach, afta which individual tree diameters were
estimated and subsequently modeîied using a single-tree approach The mua1 cycle of
dimeter incnment (dlowing for heteroscedasticity and serial correlation), tree death,
and optional thinning w u implernented through men key fiinctiow. These included a
trœ hifit-age fhction, a basai ami inmement fundon, an increment docation d e , a
bei&t-diameter hction, and a stackîng guide* The model was subsequently enhanced
(Campbell a ai, 1979) anâ continues to fom the buis o f forest management models in
Viaoria and c 1 h t t t in A u e (Rayner and Tumer, 1990).
EL and Monsentde (1977) dnnloped one ofthe fint distura-dcpendent sin#e-trec
moddr for mixed fores&. Thoy used height ntha than diameter, as the k y variable.
Potentirl beight inaenient ofany tree was assumeci eqwl to that ofa dominuit trœ, and
potartiil diuneter hcrement w u the comesponding dimeter incnmait of an o p
p w n tme of the ruw height. nicK potentiai inc~cmmts wen reduced for individual
mes iccording to th& mwn ratio and cornpetition atpaienced. MortaÜty wu mdelled
using a threshold incrunent dependent on trœ size. Regaieration was modelied fiom
seeû in a sub-modei and recniited to the main mode1 when it reached brsost height.
Although single-tra models have the potential to k more accurate, they are at a
disdvantage ôccause additionai information about every t n e in a m p l e must be
dcctd. In practice, this wül ükely be a prohiiitively procedure. Whole-stand models,
in s td , arc by name much more aggregiated, representing forest stands with very few
panmeters. Newrtheless, the amount of intemution they provide is usually sufficient to
answer key questions of importance ta forest managers.
Ek (1974) and Adams and Ek (1974) proposai whole-stand models. These types of
models provide information on distribution of tms, basal ara and volume by diamcter
c h . The modeis represent difbent approaches in projeaing stand development. EK
(1974) and Adams and Ek (1974) projected in~fowth, mort- and upgrowth of trees by
5.0- diameter CI- usine noniincar regrasion equations. hicpendent vanables
such as totai number of tr- stand ôasal am, average diameter, numbcr of mes in a
chmeter dass, c b midpoint diameter .ad site index contmlled the growth componarts.
Ek (1974) Urcluded the ratio of average ôasai area par tree for a specific diameter cl- to
the average buai uei pa tiw for the entire stand as a measwe ofdominance. This
domiruncc meiaire Eictorcd into the mortaiity and upgrowth projections. Componcnt
p w t h equations d c s c n i tnt mortality, ingrowth of stems to the s d e s t diietcr
Jru and growth of- h m one diarnetct cl- to the nact over a 5-ycar pcriod. T b a
modd stand table could be updatcd over a 5-yerr growth period. Adamil and EK (1974)
applied non-lincar programming to study the detennination of op- stand stnicture for
a given forest and the optimai cutting scheduk to amive at the tupet structure. One ofthe
basic fmtwes of this paper is that there is a strong intadependence between optimal-
stand structure and stocltiag. and has ban later highlighted by Adams (1976). However,
al those mdels l d to exponential growth of the number oftrees in eoch sUe class. This
outcorne may k acceptable for short-tenn projections, but it does not permit global
opthkation ofbest ing strategits.
M o r d and C d (1977) developed mortality, s u ~ v o r tree diameter growth and
ingrowth subroutines to project ftture measurements for individual permanent sampie
plots. The estimated probability of a tree dying reflected tree diarneter and past diameter
gtowth, T m d.au was detecmined by comparing the estimated probability with a
random numbcr between O and 1. Trees were cotisidered "dead" if the number was less
than or equal to the estimateû probability and "alive" othefwise. Vigor class, tree DBK
defoliation class and tirne interval determined fiture survivor tra DBH. Number of
ingrowth stems was projeaed uPng the variables average basai uu per stem on the plot
ad tallies ofsapiings below the minimum measUrable diameter. hgrowth stem diameters
wae gentratecl h m an atponmtiil probabiiity distniution. ingrowth specier
composition reflected e i k specia composition of sapüngs beiow the minimum
meamrable diameter or over story species composition.
West (1981) dcvised a fnunewotk for 8 @d, proeess-basad model for mes in pure
stands. His rnodel is an actemion of the general stand-1cvd mechanistic model of
MoMumie and Wolf(1983). West modelleâ stem biomass, ond leafand mot biomoss by
annd classes, assuming thrt they wodd lin for t h m and two yein respcctively. Grou
photosynthesis of an individual tree is predicted &om the site's potential grou
photosynthctic production (per unit ara), mdtiplied by the leafarea ofthe tree and an
ernpirid modifier to account for shade within and between trees. Respintion (a constant
times leafbiomrss) is subtncted fiom this to give net photosynthesis, available for
maintenance and growth of other tree parts. The model assumes that trees die when net
photosynthesis fds to zero. West (1993) developed the model fkther ta examine mon
rcalistic ways to model photosynthate partitionhg in response to ftnctional relatiomhips
W a n tree parts. This mode! provideci rasonable preâictions for an even-aged
Euca(Lptus regnanr plantation in Victoria (Austdia).
The models discussed above demonstrate the wide variety of approaches tha! have ben
taken in modeling even-aged and uneven-agd stands. Each hu advantages and
disadvantages in terms ofinput data requhments, d d of stand output data and modd
operathg wsts. Individusi tnadisbnce dependent models cm provide vcry detailed
information on the & à of inteme cornpetition on trœ wwth but qpue large
imamsc ofcornpitu stomge spicc ad much cornputer t h . Whole stand models
economize on compta stongc and time nquircmcnts yet proâuce only a m t e stand
meamres ( M m 1974). The lit- contains a d e t y of opinions on the
appropriateness of crch modeiing phiiosophy.
Hurn rad Ban (1979) advocated the use ofwhole stud models in studying uneven-aged
forest management. They -cd that this type ofmodel, "is casier to devdop, chcapet tu
i n i W and cun, and takm Icss cornputer corc than do the single trœ varîeties." A b ,
individuil tree ~gowth data rnay not be necded to answer moa o f the uneven-aged
management questions. The smaller con requirements permit interfishg these types of
modes with computcrlled optimization techniques.
Clutter (1980) adopted a pragmatic view towardo evaluating mode1 suitability based on
the model's capabüity to help solve a specific problem and its operathg costs. He stated,
If a given mode1 leaâs to the correct uuwa for a particular problem, it
is a ''goocln mode1 in the context of that problem although it may not be
a "good" mode1 in the wntext of some other problem. If several models
wüi produce the correct answcr to a givcn problem, the one involving
Clutter pointed out that treslevel and struid-Iwel models might be most appropriate for
diff*cr~af types of pmblaar. Stand-level moddr may oE&r the most eflicient approach to
mrLe the mury growth pmjcctions a d e d to iden- an optimal stand West schedule
fOt an eatirr focest property. The study of stand 8rowth rtsponse to différent types of
cutting trtatments amy requirr t&e detail OW by an inâividuaî tree model.
matrix p w t h modd appmach to whok staad modd hc b w popular in the ment
decade. Hcnce, now 1 tum to tbir a p p r o d
2.1.2 Mi* Growth Modela
The matrices were first used by Leslie (1945,1948) to study the animal populations. He
puped an animai population by age classes, and then he calculated the elements ofa
matrix that wouid lgik the n u m k s of the animals in the various age classes at successive
times. The busic element in the matrix is the probability h t a femie animal in the ith
age class WU be alive in the (i+l)th age class. Using this ma* he studied the
movement of the animai populations.
Basing on the matrices introdud by Leslie (1945, 19481, Usber (1966) developed a
matrix to study the forest growth. In his study, the matrices represented diameter classes,
instead of age classes. He d e W the elements of the rnatrix as the probabiiity of a
remaining in the same diameter clam and the probabüity ofa trce moving &om one
diameter class to the nact diameter clw duing a growth p e r i d With an Usher mat&,
only som of the sumivllig mes grow into the next class, whiist those with little or no
growth remain in the same c l w . In order to duce the large n imba of panmeters
r q u M to be estimateci in his he mide an assumption - "Choosing the t h e
interval and dass width so that no tree cm gmw mon than one class during the period
(for convenic~~cq d e d the U s k assumption).~ He aiso usecl two other standard
rnentioning thaa. The Uukov usumption quites that the probab'üïty ofany event
depends only on the initiai state, and not on any pmvious state. The statioauy assumption
~~ tbrt the probrbities do not change over the- He inu~mted this moLr model
by r d è m c e to a Scots phe forest-
Usher matrices, perhaps btcause of the &ciencies inhuent in the Usher assumption,
hive ban widdy used. U k (1976) used these matrices to estimate optimum yield and
rotation laigth for P. syhstns plantations in Bntah. R o m (1978) continued thU
analysis to prove th.1 the optimai aiSullied yield huvesting regime is a cutting limit
mghe which removes aU the s t e m in only one class, removes a proportion of the stems
in rmnl smaUer classes, and leaves ail the remaining smdest classes untoucbed. This is
consistent with some odection harvesting guidelines, but at odds with harvesting
pncticts in many counüies. However, a basic problem Uiherent in these models in ternis
ofrepresenting the behavior ofa stand of trees is that they Iead to exponential growth of
the number of trees in each size closs. This problem is solved by Buongiorno and Michie
(1980) by modifying Usher's model to d e uigrowth only partiaiiy dependent upon
harvest, and to aiiow in~rowth to raspond to changes in stand density and diameter
disaiiutiori. As a d t ofthis mechanism, the stand can gmw at an increasing, a
decreasing or a constant rate dependhg upon conditions. Buongiorno and Michie (1980)
constructecl a mtrix growth mode1 fot a single species hardwood fomt, ushg the
stationary ad Ushu assmptions, but not the Markov wsumption, because t h y
incorporated the ingrowth data i to the state of the stand into th& mafrices.
la Buongiorno and Michie's (1980) study. aithou& the movements are indicated by
prabrb'ities are maund to k detamùiistic for evay 5-year paid. They u d i
sepafate ma& to represent the barvca, ro that thy could effktively examine
alternatives. B d on thir modei, they detcfmined the optimal harvest, residud stock,
diameter distribution, and cutting cycle using heu prognmmllig. A M a r model for
Indonesian fonsts (Mendoza and Setgarso, 1986) indiutcd that sdection logging (Le.
harvesting a proportion of trees in each merchantable size class) would sustain higher
yidds than simple cutting iimit basai on diameta.
Solomon et ai. (1986) developed a two-stage matrix model. They assumed the
proportions oftrees rrmoining in a diuneter clas, the proportions of trees growing out of
a diameta class, and the rnortaiity rate are dso related to tree size and stand density. In
the fint stage* a set of hcar regrestions was used to estimate all those proportions âom
independent stand variables+ In the second stage, the results obtained from the linear
regasions were uscd to project the diameta d is tr i ion of the stand. Tests for this two-
stage maaix modd agillist Iarge indepmdent data sets indicated t h t the mode1
pcrformcd wen in pdicting growth, volume yield and specics-diameter distribution
changer over both the short and long nms.
Buongiorno et ai. (1995) dcveloped r maük growth model for mixai hrrdwood forest,
and used thh gmwth modd with nonlinear prognmmiag to study the stand dmcs, the
mocimum divctsity and auxhum economic &cicncy+ The resuits
aiggested that a light management could improve trœ divcrsity, relative to Mtunl stand
p w t h . MiuQmuni economic dficiency wodd require a mbstantiai rduction in basai
MI and average riae of trœs, and it would lead to low tme diversity. n#re wodd k
more large she trees when the stand maches r steady state.
The terms used by mihors are d i f f i ~ ~ ~ t for descn'bing the tree muvernent trom one
Ainmcter c h to another diameta clus. For example, Usher (l%6, 1976), and
Buongiorno and Michie (1980) used the terni of transition probabilities, howenr.
Solomon et al. (1986) and Buongiomo et al. (1995) used the h d o n of aee movement.
So thîs toiminology can be debated, but we continue to use the tenn used in the onginai
models of Usher (1966). and Buongiomo rad Michie (1980).
These mMix growih models have ban uKd to mdy the outcornes of various
management options and the impact of Merent kesting regimes on stand structures.
In these studies, the main mathematicai tool has been iinear programmhg. Hence, next,
iitmture on applications of tincw progromming in forest management is rcvitwed in
briff*
2.2 Linear h.grimming and Forcit Management
NIutjd and Peirse (1967) introduced multi-paiod dynamic hear programming to
forest management problems. T h y used dynamic linear programdng for the conversion
to mstahed yidd forestry. Since uim. liaur prognmming has been used for almost
every aspect of focest nmagemeiit. For example, NWteU (1982) studied the impact of
fh on forest management, rnd Bcumelle et d. (1990) reviewed thber v e r n e n t
modeîs tht incorporateci ri& 60m timba p w t h , timber priccs, or losses from fin.
iiwcts, or 0th cuuer. Hace, it wül not be possiilt ta review the whole literature on
linear programming and forest management in this section. ûnly some important aspects,
ad literature foaiscd on unevm-uged hardwood forets is rcviewed.
Buongiorno (1%9) cxBmined the appücabüity of linau prognmming to the probkm of
accelerated ait management to achieve a regulated forest in a shorter penod of time than
with conventional management regimes. His objective -ion was the maximhtion of
voIume output of the forest. He discriminateci threc sets of constraints, (1) the maximum
volume ait in p * o d j and cornpartment i is the volume present in that clus at that time,
(2) harvest fbw constrPints accordhg to the management regime select64 and (3) long
mn stabüity constnints.
Buongiorno and Michie (1980) used a lineu programmhg method to determine
sustained-yield management regimcs timt would maximize the net present value of
psriodic harvests. This mahod aiioweû for the joint detemination of optimum hawest,
teSidu81 stock, diameter diotnktion, and cutting cycle.
Lu and Buongiom (1993) presented a lin- programhg mode1 subject to a matrix
p w t h modci, wbich recognized d i f f i c e s in tnt species, qualjty and sizc. This mamt
growth d e i was d b n t e d with drtr fiom permanent sample plots in hardwod forests
in Wmnsin, cbs@kg trœs in N v e sizes, uch dmded iato bigh value, low-due
and non-corntnercial trsss. The b a r progmnmhg madd wu used to compare rite
cutting guides in termr of the mil reat and eeotogicai divmity they would obtain a steady
state. The r ~ ~ ~ l t s showad hi@-gradhg a stand wouid give high short-tenn mtums, but it
led to a very poor stand structure, with low divemity and negative rrtumr. in the long
term. The forest value U not abie to r e v d this long-term deficiency, because through
discounting, it is affccfed very litde by dirt.nt wents. For thrt nsson, steady *te
considerations should continue to play a major role in cornpuing forest management
ah etnatives.
Buongiorno a al. (1994) siudied tree sire diversity and economic retum in uneven-aged
forest stands. In their study, Shannon-Wiener index was used to measwe the tree size
divetsity of forest stands, and linear prognunming and nonlinear prograrnming
incoiponted with a matrix growth mode1 wen appiied ta northem hardwood forests.
Thai. midy sug,gestd that a naîunl, undisturbed stand would mach the highest possible
sustainable diversity oftree size. Any intervention wodd decrease that diversity. In
particulrv, economic huMstmg poticies wodd nduce tree size diversity by 10-20%.
However, econodcs and divttSity did not conQict.
In thcw lin- progmmnhg formulations technid or yield coefacicnts (coefficients of
MTiabIcs in constraints) and nght hand side variables ut assumcd to k dacnninistic. in
mmgt p w t h models, tndtion pmbabiities am yield d c i e n t s , and 1 want to
hcoiporate the nadomess of these probaiditics in growth models. Henct, next a
litcnaut on Mdom vdabIt8 in LP pmb1ems U disad.
In the LP pmblmq mdomness of the coefficient mry ocair in the objective hction as
wdl as in the coastnists. Howmr, d o m objcaive fùnction d c i e n b do aot cause
any d pmblem in Lmur progam. The expected values of the nndom coefficients cui
be substituted into the objective function and the result is the maxhhd apected value
of that objective function (Hoc 1993, p67). In the comtnhts, ranâom variabla may be
on dgbt hand aide or as technical d c i e n t s on l & h d ride. The Chames and Cooper
method (1%3) is the classic approach to deai with random variables on nght band side of
constrahts in LP problems. In this approach, they derive liaear, dctcnninistic LP
formulations that are quivalent to probabüistic problems. This approach is quite
powanil, but applies only to certain cases, in particular, when one wishes to constrain the
m d e i such that each nght-hand side (input or output) is obtained with pre-specined
probability* Mülw and Wagner (1%5) proposeci an alternative approach that would apply
when it is desired not to lirnit each constraint to m a t its ri@-hand side with a certain
probability but to constrain the joint probability of meeting ali random right-hand sides to
be at least some pre-specified constant. These approaches are known as Chance
Constrained Programming, the Chames and Coopen apptorch as Udividual chance-
constrained prognniming, and Miller and Wagner's approach as joint probabiity chance-
coDstMwd propnmhg* Hof(1993, p.77) proposai the total probability chance-
constmined progmmhg as another approach to mdom right-band side problans.
Hunter a ai. 1976. and Hofancl Pichais (1991) used chance constnint progmnming to
nuuni cemurce docation pcoblsma Howcver, Hof(1993, p.77) pointai out that in -
d 1 e msource management problamr, fdbility (meethg output targets d o t
exhiuisting input availaôility) may o h k a very high pnonty. in which case it may be
d& to murimia the chance of meeting the d o m right-hand sider. Hence, Churo
martimiPng approaches should be u d instcad of Chance-coiwtrained approaches. Hof
(1993) analyzed a forestry aumple with c h a n c e m g (using tmchhation of the
minimum - MAXMIN) and chuice-conshwed approadies. This example demonstrated
thrt Mient approaches could yield substantialiy diffèrent tesuits.
In chance constraint prognmmiag and chance-maxhkins approaches. technical
coefficients are assurneci to be detedstic. Van de Panne and Popp (1963) extended the
Charnes and Cooper (1963) approach to the we of random technical coefficients.
Let us tlke a general fodation ofthe LP problem. This problem an be fonnulated as
Van de Puine and Popp (1963) assumd that the aq (COCflticimts) are stochastidy
indepadent. Miller and Wagner (1965) rrlrxed this rwumption by applying the
covariance mrtrix. Assuming that the q have normal distn'butions with means au and
variances/~vachces ok, a chance constraint which nquirrs a y, probabiiity of meeting
the fi* b d side b,, Cui be wntten as:
a 1
whme 6, is the standard deviate (romctimes d e d the "tabulor value")
co~ctsponding to the cquired probabiity, such thot: y, = Sr@) z 1 - F(4) , and F(G,)
rt 1 -- bas the standud dennition: ~ ( 6 , ~ ) = Tla ah .
2%
Cm de Panne and Popp (1963) used this random yield (technid) coefncients approach
to mdy the minimum-cost cattle fad mder probabilistic protein coastraints. Hof et al
(1988) stuclied optimization with mdom yieid coefficients in renewable resource ünear
programmîng. Theu study has demonstrateci somc implications of this practice when
yidd cafnciaas are stochastic. It also shom thrt yieid constnints have potential to
substcuitidy cxacerbate the complications intrduced by stochastic yields, and a tenable
pmblem formulation and solution with these constnllits is much more clusive than
without them.
Hofand Pichcas (1992) M e r studied the chance constrahts and chance mruamization
with mdom yidd dcients in mewaôle CCSOUTC~S. Th& cx11zbp1t was based on Hof
and P i c h (MN), and addressad a foteary land ailocation pmblan whcre the amounts
ofoutputs that ~ ~ t d âom a varîety of matmgment regha were randorn. The output
approoc&r wuid yidd substantidy diffèrent m d t s , h niggestd how the resuits might
As dimusui, linau programming Y the most conmion mcthod of optimieng thber
h e s t scheduies. It is puticuluty powanil when tirnba harvests are to be schcduld
subject to constraints as a kd planning horizon, multiple initial age and productivity,
non-declining yielâ, tetminal uea coatroi, output targetq and management requiremcnts
(such as maintainhg certain acreage of old growth). It also can be used to solve chance
consvrin prognmming &et some tniipfomiition is conducteci. However, one potential
weakuess of the linear programmllig approaches Y thit discrete t h e periods mus be
deheâ, and noaiinear thber yield fiinctions are piecewise approximated using these
discrett the peciods. This introduces two possible sources of enor: (1) yields are
misestimatecl by the p i d s c approximations, and (2) o p W timing of the scheduled
h e s t r is not adequately caphued by the di- time periods. Also, viewing in a way
that dll.e*ly incorporates the nonlinear timber growth provides a dierent insight into the
timba kat scheduiing probIems. Hence, some authon such as Roise (1986), and
Wtintraub and Abramovich (1995) have uscd nonlincar pcogramming techniques to
solve the fores management problems.
2.4 Muimization dMinimum @fMWN) Appmach and Forest M.nrgemuit
The traditionai approach to the o p W huvest is to mp0mue the objective bction
(eg., profit) subject to a set of consüahs that prevent deciining (or uneven) harvest fiom
one time pdod to the nexî. Hof et ai (1986) m k e d this rppmicb and afguad that
nondcclining yidd constmints un uuse substaatiil ~ u c t i o n s in prcscnt net worth when
scheduliiig timbcr harvC8tS of forest tbrt is not in a "mgulated" sute. N o n d c c ~ g yield
constraints dm do not aiiow hawest leveis to go down s ~ n a smrll mount uniess
p d e t d e d deviations fmm time pdod to timc @od in prcspecified.
In order ta whre tliis problcm, Hof et al (1986) suggested MAXMIN approach. The
bMXMN approach will diow hmst levels to go up or down frorn period to period, but
WU tend towuds even flow. It al00 tends towrrds profit martimizIItion, becsuse it
maximias the minimum time penod's ait (iicludin8 t h e periods carly in the planning
horizon). It wouid mate an equitaôle distribution of timba harvest across generations, so
the MAXMIN approach essentiaily foliows the ntionak of sustainability.
Hof a ai. (1986) tested the MAXMIN rpproach in a case study involving conversion of
an unreguiated mn-aged forest hto a regulated state. Their example provided how the
MAXMIN approach can be formulated in a typical t i m k harvest-scheduling model. The
r d t s demonstrateâ that the pdionmance ofthe MAXMIN approach is highly sensitive
to "initial stand structuren, and may or may not be sensitive to the Werminai steady state
structure." B d on th& study, tky pointed out that usefid solutions could be obtaiued
âom the MAXMiN apprthch, whm the tnditional approach ofmaxhhhg profit subject
to a nondeclining yidd constmht rtsuIts in a barvesting pattern of increasing Bow.
Howmr. dicm U one wcakness of the MAXMIN apptoach. The MAXMIN approach
mi* cut naina9y imnraun mes ifthe raitiil age structure is such that no mtwe
t m s are avaiiabic for hamst h the suly t h e puiods. For this nuon, it mut be
eniphasized tbat the suggested use of the MAXMIN a p p m h is in comôiition with
otha appr~~:hes to brmrt schduliag.
Up to now, the MAXMIN approach is only used by Hof et ai. (1986) for the &est
scheâuling of nmi-aged fonst. There is no literahin which incorporates this approach
into unmn-aged fonst management.
CHAPTER 3 THEORY AND METHODOLOGY
As mentioncd in Chapter 2, there have ban many lcinds of modelling approaches to
develop powth models of uneven-agd hardwood for-. Compuitiveiy, wholastand
modeis seem to have mon advantages thui single-trnc models, because additional
infiormation about evay tne in a sample must be d e a d for single-tne modeh, in
practice, this will Wely k too cornplat to do. Hence, one wbole-stand model - matrix
growth mode1 - has b a n used for different shidies in this research. in the fint part of this
chapter, theoretical foundations of the ma& growth model an provided. In the
mahodology pari, mathematical formulations used to d y z e dinerat scenarios are
prrsented.
3.1 Tbeory
In this chapter, first, one wholemstand model - matk growth model and its applications to
forest management are d i s a i d in detaii. The matrix growth model was developed for
single species by Buongiorno and Michic (1980). and then enhancd for muitiple species
by Buongirono et ai (1995). In these mai& growth models, tree movements an indicated
by probabilities of transitions between diameter classes and ingrowth ofaew trees.
However, thew probabilities are usumed to be dctcrmhistic. Hencc, second, a matrix
growth modd with random tratisition probabilities L htroduced. Fmally, the theory of
MAXMIN appraadi is d i s a i d .
3.1.1 DetcrainMie Tniritkn RobabUty Matris Grawth Modd (DTPGM)
F i a shgie speciu mitiix modd is diicursed. La us assurne that an uneven-qed
fonst stand comprises a hite numbar (n) of diameter classes. The number of living tnes
within a diameter clus, 1 to n, at a speded point in tirne (0 is denoted by yJt, y% . . . and
y,,,, respeaiveiy. Hence, the total stand at time t is reprtsented by a column vect~ry&~],
where/-1.2 ... n. During a specific growth period of T yem, trees in a given diameter
c h j may remain in the same clus, move to a hi* diameter clws, or may be
harvested. Suppose the n u m k oftrees harvested firom a diameta class j during the
period T is denoted by hfi Therefon the enth W e s t is represented by a column vector
h, =[h,& w h e r e ~ l . 2 .. . n.
The gcowth of i stand is specified by the probab'iity of a tree mnaining in the same
diameter clms and the probabiity of a tree moving âom one diameta class to the next
dimeter class during a growth period of T years and ingrowth in the lowest diameter
class. In this spsafication, it is assurneci that the selected time intenial of T yean is sa
smaii and the ran~e of dimeter classes is so large that a tne can not move more than one
diameta drrs d h g this paiod. Let q denote the probability of a tne in diameter class j
at tirne t, which is not barvesteci durin8 the inttrval T, to continue to be in the sune
diameta dam at time t+R and b denote the pmbabii of a tree in diameter d u s j-1 at
time t, wbich is not harvosted durin8 the intavil T, to move to the diametet cl= j at time
t+T. Let II denote the expectad ingrowth, i-e., the number of tries entering the d c s t
diametu c lm durhg the htcnial T. The n e of trees in diffemt diameter classes at
timar+Tisthenacprcsscdu:
y1t+r=&a1@1t - ~ I I )
The system of equations (3.1)' wnsisting of n equations - one comsponding to each
diameter cluo - npresents the movement of trr# in n diameter classes. The speQncation
of ingrowth fùnction L is necessary to complete the specification of growth system.
FoUowing Eks' (1974) method, Buongiorno and Michie (1980) aiggested that ingrowth
was invenely related to the basal area of the stand; and for a given basal ana, ingrowth
was M y related to the number of trees. FoUowing this argument, the ingrowth
nuiction is specined as:
whae I , 2 0, BI is the basai a m of a tree of average diameter in class j, while d and e are
coefficients which one is m p c t to be ncgative, and positive, respectively, becwse the
ingrowth shouid incrcast with die number oftrtts, and decrtast with the b d area. CO is
acpcct to k positive, because there must be sorne tries which enter into the smaiicst
diameter class.
nie systun of equations (XI) can be d e n in mmix fonn:
yt+r= Gb- U+C (3 .4)
wbae G ic the growth matrix and c is a colwlill vector of constant d c i e n t s
Equation (3.4) gives the number ofttees in n diameter classes der a growth period of T
y a n giwn the initial number of tries (y,) at time t, growth matrix G, constant vcctor e,
and the hawest vector Ar. The mortaiity is not considered in this single species model. It
is assumai to be zero.
Next, let us extend this model to the multiple species. Assume there are m species in an
un--aged forest stand and other assumptions are the Mme as before. Therefore, at
time t, the totai stand of living trees P repfcscntecl by a oalumn vcctor where
hl,?. .. rn.j=l,l...n. yy is the nurnôer of tree of species i at dimeter class j at tirne t. The
entire h c s t is reptc~eated by a column vcctor rkr[h@], where iel.2.. . m, pl ,2 .. .n. hM
is the h e s t of species f in diameta class j a time t.
Let cÿr denote the probabiity of a tm ofa species i in diametet ciam j at tirne t, which is
not hnnstcd dwing the iatav;il T, to continue to be in the same diameter c h at thne
t+T; d bu denote the probribility ofa tree ofapecie i in diamcter class j-1 at time t.
which is not barvesteci d u h g the intervrl T. to m v e to the diameter class j at the t+ T.
S i to the sîngie species models, the pmbabilitics of trier rraiiining in a dimeter
class, the probab'ities of- growing out of a diameter class, and the mortaüty rate an
rdated to tnt s k and stand density (Solomon a ai, 1986). Generdy, the upgrowth
tnnrition probabiity is directiy proportional to the tne siÿe (e.g., diameter) and inversely
proportional to stand density (e.g.. stand basal am). Following this relationship,
transition probabiüity ninaions wen developed by Solomon et al (1986). and Mengel and
Roise (1990). SimiSulys the up-growth transition probabiiity (bu) is posited to be a
hction ofthe stand basai a m , and of tree s ize (Buongiono et al, 1995). Hence. bu is
expresd as:
foi j=1
where Bj is the b d arer of the average tnt in size-class j , Dj is its diameter, and pf ,
q, and 4 are parametm. As mentioned by Buongiorno et ai. (1995), q, is espected to be
negative, reflecting a s10wer growth me at higher stand density. A similar relationship
govems the probabiiïty m, thrt a trœ of species i and j wiii die (typically ofter
whac u,,v, and w, arc panmeten, u, is expected to be positive. reficcting that a
morblity is pate r thm zero. v, is e~pected to be ncgativq refiecthg a less mortiüty at
hiehg stand denrity. w, ir orpected to be positive, since larger and taller trces are more
prone to windfkü. Thai the probability a# can be obtained as:
ay= l -by -m, , for j<n
au = 1- mv for j = n
Let 1 denote the ~cpected ingrowth of spccies i, i-e., the acpected number of tres
entering the srnallest dimeter c1ass during the intervil T. Ingrowth ftnction is baseû on
the hypothmis that ingrowth for a particular species is r direct hear fiindon of the
number of tras of that species, and an inverse Iinear hction of the basai area of trees in
erch species. It is an extension of the ingrowth mode1 for single species forests by
Buoneiomo and Michie (1980) to muhiple species stands (Buongiorno et al, 1995). The
ingrowth equation is:
whae b is the ingrowth. is expected to k ncgative, so that ingrowth is lower at higher
stPad density, regardles, of spccîer, but the q t t u d e of the efféct may vuy by spies .
el is expected to be positive, ingrowth of @es i increasing when the stand has more
tries ofthat species, otba things king equal. Tbe constant cf is atpected to be
nomipeitive, meaaing tbit roms inpowth may ocw. independent of stand stw. due to
w c d ~ f i . O m ~ g s t a a d s ~
Now, for multiple @es tbrests, the mmOr p w t h d e l , similu to 3.4 for single
specisr &as, can kexpresseâ as:
y ~ = ~ ( B 3 b r & d + e (3.10)
w b G(&)=A(&)+R, in which the mitmtA(&) is the upgrowth ma- which depends
on the stand ôasai area & afttr the cut, & is a vector which inchides the basai a m of
cach species iit eadi JiEe c h . R is the ingrowth mtry and c is a constant vector
derived b m the hgrowth equations. Thew matrices and vcctors are dcfined as follows:
Each subrnatnx Rÿ. which represents the effects oftrees of spccies i on the ingrowth of
specits &, h buiit with the panmetas ofequation (3.7), as foliows:
The c vector is:
each subvector ct contains the constant of the ingrowth equation (3.9). i.e.,
Equation (3.10) is the rnatrix growth model for multiple species uiieven-aged forests, md
it wiiî m e as the basic mode1 for our rcsearch. The stand dynamics and the neady state
of the hard mapk stands wül k analyzed, in Chapta 6. using this model. Howam. as
mentioned earlier, in this modd, üansition probabilities are usumed to bc deterministi~~
th is not the case in d t y . H~mce. in the nad section, this model is modifiecl to
incofporate the d o m nature oftransition probabilities.
3.13 Rudom Trraritioa Pmbibüity Matri. Gmwth Modd (RDPGM)
In the mrtmt growth mdds, u d by Buongiorno and others, transition probabüities are
dcthtcd âom p w t h data ofa nmibat of~mple plots. Howcvu~ aü these studies have
uscd the mern tmdioa probibilities. The mean tnd ion pmbabilitia am, nonnaüy,
caiculated, by avengin8 the üansition probrbilitiisr o v a the selected sample plota
Haicq the dances in tnnsition probrb'ities laou tbe iclected sample plots are not
acamted h tbew modeIr In thk section, a mmOt p w t h modd is dewloped thit
incorpontes meam and variances of tnnsition pmbabüitics.
In the case of grow&h data king used âom a large numba of sample plots, we cm
assume the transition piobabiility ofevery diameter clus has a normai distribution. Now
let us stül talc6 bu as the upgrowth transition probability of i spaies and j diamaer class,
and let z, denote the standard deviation of bb Comspondingly, assume ou is the
probabiüty of nmaining in the rame diameter class, and let s, denote the stand deviation
of q.
Now, the tm movement during a t h intemai Twül depmd upon the mean and
variances of transition probabilitia. Fint, let us set this movement for one diameter class
(B and @es (0. Aftu the intaval T, the n u m k of t r w of species (0 in the diameter
(3.14)
wbere ( is the tabulu vrlw contsponding to the required co&dmcc intemi, aad F(()
has the standard dation:
In equation (3.14). the fint component is corrcsponding to the mean of transition
probrbiiities iad the second cornponent is for the variance of transition probabiiitics.
Heacq this fomulrtion captufes the d o m e s a oftnnrition probrb'ities. NOW. the
moditwd fom of equation 3.14 for ail species and diameter chses can be written in
maük form:
y,, = G ( B ~ MY, - Cc,) W ( y t - MY^ - 4 ) ' ~ ; j3 +
6' is the standard normal deviate (sometimes d l e d the "tabular value") correspond in^ to
thc nquired confidence interval. Other parameters are the same as in the matrix growth
modd (3.10). Hence, equation 3.16 rtpctstnts a matrix growth model, for multiple
species forests, that incorporates the mdom nature of transition probabiiitia. This model
wiii be used to d y z e stand dynunics and steady state of the hard rnaple fotests in
soutbeni Ontario.
3.1.3 Miriabation of the Minimum (MMMïN) Approrieb
An appfoacb that tends to reach even harvesting pattern naturally can k derived from
the Wtay set" and '"nusy god programming" literature (ignizo 1982). A fiury set is a
set whose n#mbenbip is d&ed coatinuoudy, rathm than distinctIy. Borrowhg Born
Bcllmlui and Zadeh (1970) and Zmunennuin (1976), ifX=(x} is a cdleEtion of objectq
tbcnafiizzysetAinXOasetoforddpairs:
A = {(x, ~ , ( a l x E X) (3.11)
wbere MA is d d 8 maabadiip function applying to cach x. It indicatcs the de8ra or
grade of membahip of each x in A, as a hction of x. The membership tùnction ofany
two Aiay sets (say, A and B) is defined as:
MM = MNM'sMn) (3.18)
Ifmembership in a fuzzy set is of interest to a decision maka, thm the membership
fùnction c m k interpreted as an objective bct ion. Zimmermann (1976) suggested that
ifa decision maker w u interestcd in m o n than one tiiny set (e.g., A and B), then a
"rnembership fùnction ofthe deasion," Ma can be defineâ as:
MD = MAd =Min(MA,M,) (3.19)
Put simpiy, the MD indicates the minimum grade of membership across al1 of the f b q
sets of W s t . A solution which m a x h h MO is often niggested as a means of
optimiPng a fiiay decision. This 1 4 s to the concept ofa MAXMIN operator, because
the MD beiig marrimized io itselîa minimum of (MA, a). This MAXMIN operator is
dcrived rigorousiy in Bell- and Zadel(1970).
Ifa set of variables (xi) enter h o separate membenhip functions Mt (xi) and if these
membedip functiom are linau ad similarly dehed, then the MAXMïN opcntor can
bc applied dirsctly to the xt. For example, assume that the minimum xi is to be t C mmarmd. Then, d&hg an arbitmdy large target k, the problem can be solveâ with
the fdowing aaerr program
when the 4 serves as dcviations betwœn eachq and k nie 1 sawr as the minimum
attainsble to be n-ummd C 0 . This program un d y k formadateci so that the R is scaled
to be behNccn O Md 1 and thus uin be intcrpreted as a normalized "membership function
of the decision" Naturslly, oher coIISttaints would be included, or the solution wodd be
tmnd (ali xt =k).
This concept of MAXMIN will be used in this thesis to address the Rawl's equity
critaion in hmsting decisiom of unewn-aged hard maple forests in southem Ontario.
3.2 Methodology
Thcsc theoretical concepts ofrnatrix growth mode1 with deterministic and m d o m
transition probabilities, and MAXMIN approsdi are used to ddress the two objectives of
this thesis. DiBiorent mathematid formulations used to addnss these objectives are
d i m d ncxt.
F h , the mathematical fonnuiations usai to midy the stand dynamics and steady state
arc prcscnted in the detcrmitiistic environment @ a d on DTPGM). Second,
m a î b d c a l formulations useâ in the mdom eavironment @rwd on RTPGM) are
(DTPGM) ad d o m (RTPGM) models are enumemted. Fourth, mathematical
formuiations d to corn- the cconomic outcorno rnd the stand fa- are
d i d for coiuant harvest, non-comtant harvest, and hvvest Ievels to satisty the
Rawl's equity aitaion (biised oa MAXMIN).
3.2.1 Tbc Stand Dynuics and the Study State of r Stand in the Deterministic
Eaviroamen t
Gendy , the static propaties of a forest are ury to see, but it is very difncult to see the
dynamk properties of a forest. It is even harder to h o w ifwhat we think we understand
acMunts for what we cm observe, without a tool that demonstrates the impüutions of
that knowledge. However, matrix growth models are one of important tools that could be
used to understand some of these compkxities. Matrix growth mode1 (3.10) cm be used
rccwsivdy to forecast the state ofthe stand over any number of pe~iods (w).
Suppose, the initial state of a stand is given by y* and a sequence of harvest is h ~ , ha ..., k.
The statt of the stand at ciiffient t h e periods ( 1.2 .. . w) can be expresseci rs:
YI = G(B3(idd+c (3.21)
y2 - G(B1)@1-h)+c
An important cc0Iogiai concept in forest management is the s t d y state of a stand Le. a
sute tht w d d mwitrin itseif. wah staad gmwthjust nplachg the mortality, without
bamst. In the stuây state, number of trœs of diiirsnt @sr in different diameter
clama shouM k the sune .t différent tirne Uitemls. Hence, y*~ = Y, = Substitution
of thtw stitionuy vaiues ofy in equation (3.10) wiU M to:
y*=G(wi8-h L)+c. (3 .23)
In the case of hear ecpttions, system of equations (3 .Z3) caa be solveû easily to finâ the
steady state. How- inclusion of S, in the model(3. 10). ad haice in quation (3.23).
introduces the non-ünearity in the model. Even though the model becornes linear for the
h e d basai area, but this iinear mode1 may not have a solution for no harvest cases
(h*=û). An aitcrnativt is to use a ümu programming model in which the conttol, k*, is
kept as d l as possible:
A solution of eqpation (3 24) would correspond to a naturai steady state, including
mortality done, ifthe totai barvest of iive t m s , 2 . were n w zero. Hence, the systcm of
equitiom (3.23) and the lineu prosMmyng problem formutated as equation (3.24) will
be used to rtudy the stand dyarmics and the steady statc of hard mple forest stand in the
detcLmjIUlSfic envita-
3.2.2 The Stand Dynamica and the Sttrdy Statt of a Stand in tbe Random
Environment
As d i m s d in the matrix growth madd with -dom probabilities, transition
probrbilities are assumeci to have n o 4 disüi'bution. Aeturl transition probabi ia miy
k bdow the mcaa d u e or above the m e ~ d u e . Henœ, stand dynamics, obullied by
simulation USUlg mrtrix modei with random proôaôilitics (RTPGM)* wül give a range of
nu- of tras of species in M i diameta classa insted of a single vrhie. Two
equations (3.25 and 3.26), given Mow, are used to find the lower and upper iixnits of the
range of numba oftrees of different spocies in Merent diameter classes at dEerent tirne
perioâs. These limits are caicuiated for 95% confidence Ievel(6, = 1.645).
This outcorne, range ofnumber of trees instead ofa single nu*, maka the concept of
study state non-fêasiile in this type of formulation of nndorn mvironment. However,
the arte of a stand, aibject to any management objectives such u profit maximhation, at
any ginn time can k determincd using reainive method. Hence, 6nl. the tirne requind
to mach the s t d y state in the deterministic environment wu dctermined (Modd 3.24).
Laîa on, îhe state ofthe stand was determincd at îhat time in the =dom envifocunent
using rsarnivc mcthod (quaîion (3.25) md (3.26)).
In order to find out the maximum NPV and the co~fcspondent strtc of the stand in the
d m environment, the foliowing non-ihmr fodation wu u d , and non-limu
prograamhg techniques were UA to rolve tlw problea
Yt - 4 2 0
h, 2 0
where Ph=Ipri] is the pr ia vector. pvis the nlue (as stumpage) of a live uee of species i
and sizc j. It would be fiuthm ucplained in Chapter 4. The hard maple stand reached the
stcady state IAa 1 IO y m in the detcrministic environment (Chapter 6.1.2). Hcnce, this
fodation was solved for 22 tirne intemais (each of five years period) to h d out the
state ofthe stand at 110 yeon in the mdom environment.
3.2.3 Comparative Study of the Hird Mipk Stand in the Dtterministie and
-dom Environment
The harâ map1e stand, in the dctenninistic C Z W i t C ) m -fies the s t d y sute
(OutCome: of the soIution ofthe LP modd 3.24) rtta 110 y- The state ofthe stand,
obtained from equation (3.25) and (3.26) in the random environment, ifta 110 y m was
cornparmi with the stcndy state in tbe daerministic envitonment.
3.2.4 The Comparative Study of tbe HIrd Maple Stand for Conventionai Equity
(Constaat Yidd Conrtrriats) and Rawrr Equity (MAXMIN) Criterioa
To compare the stand structure and economic rrtuins under constant yield constraints,
nonanstant yidd CO- and yield constraints that sat i4 the Rawl's equity
criterion, the detenninistic probab'i matrix growth mode1 (DTPGM) are used. Three
hear pmgrammhg problems an formulated. in ail three formulations, the objective
fbnction is b a d on the net present value ofthe thber harvest. Three LP formulations
are nuct.
3.2.4.1 Mdmization of NPV witti Constant Hantests
wherer is the intemt rate. z U the huvest puiod. ûthet parauneters are srme as before.
This b a r pmpmmh8 madei is a case of M d y state in which the growth and harvests
are the same (CO-) ove d timt intmds.
for 22 CU* cycles of five yeus eich to cover tbe total paiod of 110 y-.
324.3 Muimktion of the Minimum NPV for Eaeh Cuttina Cyck
B a d on the MAXMIN rpproacô, m&nhtion over ail nitting cycles of the minimum
NPV for each aming cycle, the foiiowin~ probkm was forrnulritd.
MMamizc1
CHAPTER 4 DATA
In Ontuio, tolerant hudwood fonitr arc miinly confinai to the Great Lakcs-St.
Lawrence region (southan Ontdo), and owas about 3.6 million hectares with 563
minion aibic meters of gross machantable volume (OMNR, 1996). These forests are
divided in k m working groups - Hud Maple Working Gioup, Yellow Birch Working
Group, and ûther Hardwoods Workhg Group. The Hard Wple Working ûroup is the
kgest that comprises 73% of the am and 75% of the l p o ~ volme (0- 19%). The
bits of the ara arc show in Figure 3.1. High valued hard mple forests of southern
ûntario an sc1ccted for this study. Sources, interprctation, and preliminary anaiysis of
two types of &ta - stand growth data and price data - are disawsed next.
4.1 Stand Growth Data: Sources, Interpretation, and Preliminary Analysis
The data for my study came fiom permanent sample plots (PSP) established by the
Ontario Msüy of N a d Resources (OMNR) in southem Ontario. The OMNR
established 180 PSPs, in hiud maple forests of 0.04 ha, 0.081 ha or 0.0101 ha between
1968 and 1987. Trees in these PSP an, normally, measured at 5 yean internai.
At each measurement, the DBH (brust height is 1.30 m) of each tree is recordeci to the
nauert 0.01 crn. The height (to the nearest 0.1 m) and DBH of the 5 tallest trier for each
species are also ncordeâ. The DBH ofdead tree is also mrded . Ethe DBH of a tree
wu Smallcr at the second mennutcment thn the first mersurement, it is marked by a code
@). M g the anaiysis of data, the DBH with code (D) ue moved fiom the data set.
In the hard mapb ( A a r ~ l i Û p u n r ) PSPs, otha common spscics arc bl.clt cherry
(Pnmus sebhrm), white ash (Fdnus amen'cc~lyl), Ironwood (0- vitgz'niam), and
white b o i (Betda pqgqyera).
In this rc~carch, we seiected only îhose 30 PSPs in which two main usaaite species of
hud mapk am black cherry and white & In thas PSPs a b , only those muauanents
that waa talcen at five ycln intcrvai wae iacluded In total, 70 muauements nom 30
PSPs am useû in this study. The am, age of establishment, and yeus of meaarrernents of
thesc PSP are given in Table 4.1. The plot location and number an also labeled in Figure
4.1. This growth data âom 70 mcwrement points were used to calailate transition
probabiities, ingrowth, mortaiity, and initial state of the stand.
In each PSP, trees are grouped into five diameter classes, every class of 6cm each (8-14,
14-20.20-26.26-32, above 32cm). The data set of each measutement point gave the
numbcr of tries for thtee cetegories: (1) tnes that remaineci in the m e diameter class,
(2) trœs tbat movd fiom one diametu class to the n a d higher diameter class, and (3)
trœs that died during the intaval of 5 yean. In the data set, we do not find a singie case
in wbidi a trce moved more thn one dianmer clyr during the period of 5 yern.
A cornputer program, written in the Visurl Basic, is u d to anaiyze data which
aimmuutd the data h m each plot and cmted a sinde data me with one rewrd per
plot. The sutmmuy drti consisted of plot aumba, totaî number o f m . total b d ara
(m2), proportion of hud maple trcer, proportion ofwhite ash mes, proportion ofblrck
cherry aa~r, ud pmportion o fo tk species tms. The d t s oôtsined hm above
calcuiations are $ivcn in Table 42.
Tabk 4.1 nia yeu. uci and age of establwhrnent and the rc-masurement ytur of the plots
B d 6 3
B a d 6 4 CIearl04 CloatlO5 Collia26 Derby13 1 Decby132 -133
Derby134 Derby135 Derby136
Derby137 Derby138
Daymdl22 Daymd 123 Drorno 100
-32 Grah128 Hiudal07
HardalO8
Murpw5 Nagel48 Piers01 P i d 2 P0w;crlS 1 Sydeasl Sydm82 Sydm83 Syden84
50
Table 4 2 The basic deraiption of data Set
Beard64 B d 6 4 Bead64 Clearlû4 Clearl û4 Clearl OS ClervlOS Couin26 Coilin26 Collia26 Daymd 122 Daymdl22 DaymdlU Daymdl23 Daymd 123 Daymd 123 Derby131 Derby13 1 Derby132 Derby132 Derby132 Derby133 Derby133 Derby133 Derby134 M y 1 3 4 Derby135 M y 1 3 5 Derby135 Duby136 -136 -137 Dabyl37
Plot Number NUM. Pmaple Posb Pchary Pother ~A(rn~/plot)
1
Derby138 . . 0,6508 0.0000 0.0000 0,3492 -
-138 Dtomoloo hm0100
-32 Dnuy32 -128 -128 HardalO7 Hardal07 HIvdalOS Ehdal08
MurphyaJ MW w s Nage148 Nagel48 Nage148 Piers6 1 Piers6 1 P i e d 2 Piers62 Powerl 5 1 Powerl 51
Powerl 5 1 Syden81 Syda18 1 Syd-82 Syden82 Syden83 Sydtn83 Sydanss Syden84 SydaisS Syden8S Vivia38 V i 8 V i 8
4.2 Rice Data
Prices of trier of Mêrent spccics and diuneta cLsser are crlcuiated using the stumpagc
modds for aown focest luds in southem Oncirio developed by Nautiyal et al. (1995)
rad Jmwy to Much 1999 muket prices of these rpecies L routhern Ontuio (OMNR,
1999) Pncer per trœ of dtffotent species and diameter classes are given in Taôle 3.3. An
hnpficit assumption, in ushg these prices is that the pice perme is independent of the
total volume or total mimba of trcm sold âom a lot which is a reasonable assumption for
a stand-levd dysis .
Table 4.3 Pa tne stumpage for diierent diameter classes (unit: CADS/trce)
Diameter Clam (cm)
Species 8-14 14-20 20-26 26-32 > 32
Hard Maplc 70 167 306 487 709
White Ash 31 73 133 212 309
Black Cheny 43 102 186 297 432
Othcr Speclcs 11 26 47 75 1 09
CEAPTER S MATRiX GROWTH MODEL ESTIMATION
Data h m 70 mcasurement points of 30 permanent sample plots (PSP) of mixd hard
maple forest waa and@ to estimate the Mierit componcnts of the mtrix growth
modd. As mmtioned in Chapter 4, trees wen grouped into fin diameter classes, each
ciau of 6cm (8-14, 14-20,ZO-26,2632. above 32). For each species (hyd maple, black
ch-, white ash, and othm (JI o t k specics grouped as o h ) ) , transition
probaôüities for each diameta class, ingrowth into lowest diameter class, ~d mortality
were cilcrilateci fkom &ta set of each measunment point. These cplnilateâ vaiues mre
u d to estimate the puuneters ofthe equations for ingrowth, transition probabilhies, and
mortaüiy.
In this chapter, fint, three wmponents of ma& growth modd wiîh detemilliistic
transition probabüities- ingrowth equations, transition probability equations, and
mortaiity equations, for ail four specîes, arc discussed. Second, the ~rowth matmt (O)
based on the outcornes of the estimatecl equations is presenteâ. Findy, the variance of
the transition probrbilities is given.
5.1 Compownb of the M a t h Growth Modd
Equitioas comrponding ta aü thne componaits ofthe ma& growth mode1 - ingmwth,
rnnsition pmôaôiies, and mortrlity - were estimated ushg the linau regression
method; SHAWM (7.0) software wu wcd foc these estimations.
Ingmwth equations (equation 3.9) were estimated for four species, and nwihs w given
in Table 5.1,
In the bard maple ingrowth quation, the cocEcient of nwnba of trœs of bard maple is
positive and diffttttlt ftom zero at the 5% sigdicancc I d The eocfficients of basai
ami ofmapk, cheny, and 0 t h wes are negative and diffèrent tiom zero at the 5%
significance levd. However, the coafncicat of basai m a of ush is positive but it is not
diffèrent fiom zero a the 5% signifiama Id. The basal a m of hPrd mple has the
iargest effect on its ingrowth whüe the basal area ofwhite ash does not have my effect
on hud mapk's hgrowth. R'of hard maple ingrowth quation is 0.3094.
In the white ash hgrowth quation, the coefficient of nurnber of trœs of white ash is
positive and diiffient fiom zero at the 5% significance leval. However, only the
d c i e n t of basai area of white ash is difftrent fiom zero at the 5% signiûcance levd
and it is negative whik other coefficients of basai area are not different âom zero at the
5% sigdicance Id. The d c i e n t of basai a m of black cherry does not have th
expected sign, but it is not statisîicaily signifiant. P o f th^ equation is ais0 very low
In the black cherry ingrowth quation, the coefficient of number of tries of black cherry
is poritiw lad diffbent h m mo at the 5% significance I d The COCfIicients of basai
a m of b&ack cherry, h a d mple, rad white as& are ofexpectd sip, but only the
d c i c a t of basai a m of Ma& cherry is diffcrcnf fiom zen, at tbe 5% s i ~ c a n c e
levd. The C0tflcici-t ofBA ofothec swes ù positive anâ d W i t fimm zero at the 5%
sieeificatlct lani. R' is 0.2649.
Table 5.1 Ingrowth equations for bud maplc, white ash, b k k cheny, and 0th- @es
Hard hhp1e (tred5 y)
Hard Indepetdent variabla Mople Basal uer (m2h)
Statistics (-1 Made Asb Cherry Others Constant
White Independent variable Ash Basai cvei (m2/ha)
Statistics (tr*) Maple Ash ch- ûthers Constant
Black Cherry ( W 5 yr)
Black Independent variable Ch- B d ana (m2/ha)
statistics Or*) ~ p l e Ash ch- Others Constant
Mer independent variabte S@es Basal a m (m2/ha)
statistics ( W h ) wpie ~ s h cherry mers constant CocfIicic11t 4-0143 -1.0434 0,579 1 4.9902 . 4.2707 27.872
In the ingrowth q d o n ofother species, the d c i e n t s ofBA ofhud mrple, black
ch-, and other spec*es, arc of atpeaed sign and M i t fiom zero at the 5%
sigdicance level. The COttIjcient ofBA of white ash is not ofeqected sign but it is also
not c l i f f i h m zero at the 5% signinuna levd. The sign of the numbcr oftrees of
othcr spscjts is dw not of enpected sign and it is different nom zero at the 5%
significanœlsnl. R'is 0.3055.
In sumnmy, R' values ofthese equation are not very high (between 0.0796 and 0.3094),
however these values are in the similu range as they were obtained in other studies. It is
most likely due to deficiencies of the model. But, in the absena of any other viable
dtematives, we are forced to continue with these outwmes.
5.1.2 Equitionr for Transition Probablitia
Transition piobability @robab*ity that a trœ grew from one diameter clas to the next
diameter class in 5 years) equations [equation (3.611 were estirnateci for ail four species.
The d t s of estimation are &en in Table 5.2.
In the use of hard mapk equation, the coefficient of BA is of expeaed sign and diffbrent
fiom zero at the 5% signiflcance l a d , but the coefficient of diameter is not Maent
fiom zero at the 5% signifmnce kvd. Similady, in the case of white ash, coefficient of
BA is diff~rtllt âom zero at the 5% sigdic~ce b e l but not of acpected Ygn, and the
ca86cicnt of dimeter is not différent h m zero at the 5% si@canct level. In the case
ofbiack ch- and otha @es quations, none ofthe d c i e n t s ut di&rcnt h m
zao at 5% SiHcancc levd. In rll the f o u r ~ 0 ~ VaIuer 0fR' are dso very low
Table 5.2 T d t i o n probeb'ity equations for hard miple, white ash, black cherry. and other
Hard Maple hdependeat variable
Statistics Basal wr (m2/ha) Dimeta (cm) Constant C d c i e n t 4,0086 4.0008 0.3476
SE 0.0043 0.0025 O. 1291
T -1,998' 6 . 3 149 2.6930e
R' 0.0571
W t e Ash Independent variabh
Statistica Basal a m (m2/ha) Diameter (cm) Constant
Coefficient 0.0 170 0.1789 4.4042
SE 0.005 1 0,0029 0.1541
T 3.3 160* 0,5987 -2.6240a
3 O. 1438
Black Cheny Independent variable
Statistics Basai a m (m2h) D W e r (cm) Constant
Coefljcient 0.0096 0.0066 -0.2999
SE 0,0075 0,0044 0.2265
T 1.2770 1.492 -1.3240
3 0.0530
othaspecies
Mependait variable Statistics Basai a m (m2h) Diameter (an) Constant
Coenici~ 0,0043 4.0001 4.0747
SE 0.0045 0.0026 O. 1347
T 0,9672 4.0456 -0.5544
R' 0.0 13%
(bdow 0.15). Hencc, these equations are unable to explain the variations in transition
probrbilities ofmes.
In h view of these d t s of estirnation of tnnrition probabiiity quitions, the mean
values of transition proôabüitics âom 70 rnemrernents points ire used in the growth
ma& The mean vdum oftransition probab'ilitics are given Li Table 5.3. The table
contrint two probabitities - probability of remaining in the same diameter class (au), and
probability of movhg fkom one diameter class to the next higher diameter clms (bu). For
exampk the tne movement of hsrd maplc baween diameter class 1 I cm and 1 7cm. 0.85
muns that 85% trees ternain in diameter class 1 lcm, and 12% trees grow into diameter
cl- l7cm within tirne intervai of5 years.
5.1.3 MorWity Equatioos
The mortrüty equations (quition 3.7) for four species were estimiited and the nsults are
givai in Table 5.4.
The results ofmortility equitio~w arc dso sixnilu to the rrsults of tnnsition probabiüty
equrtions. Fht, in the mortality aputions of h r d maple and 0th- species neither the
d c i « ~ f d of BA nor the d c i e n t s of diameter are different fiom zero at the 5%
d~~ I d . In the case of white u h .ad bhck c m , only the d c i e n î s of
d is exüexntly low, the highest is 0.05 15 for white ash. These mtimated equations are
unable to explaiil the vlvirbility in moirtality of M h n t qmitb. Hencq in this case aisa,
we d the mean of the mortaiity caiculrtd nom 70 meaauement points of30 PSPs.
Thcie values are given in Taùle 5.5.
Table 5.3 The mean of transition probab'ities of hud maple, white ash, black cherry, and
other species
HiUd Maple D 11 17 23 29 35+ 11 0.85 O O O O 17 0.12 0.82 O O O 23 O O. 14 0.79 O O 29 O O 0.12 0.8 1 O
35+ O O O O 0.93 White Ash
D 11 17 23 29 35+ 11 0.85 O O O O 17 0.05 0.91 O O O 23 O 0.07 0.87 O O 29 O O 0.12 0.9 1 O 35+ O O O 0.08 0.90
Table 5.4 Morulity equations tOr hrd maplq white ash, black ch-, and othr @es
Hard Maple rndcpadentvaciable
Statistics Basai a m (m'ha) Diametu (cm) Constant
Cdcient 0.5233 4,0011 0.02479
SE 0,0004 0,0007 0.0 188
t 1 .270 -1,3960 1.3210
3 0.0148
White Ash
Inde~endent variable
Statistics Basai m a (m2/ha) Diameter (cm) Constant Coefficient 0,0024 -0.00 18 -0.0095
SE 0.0007 0.001 1 0.0305
T 3.527* -1 .6090f -0.3 108
R' 0.05 15
Black Cheny Indcpendent wiabk
Statistia Basai ara (m2/ha) Diameter (cm) Constant Coefficient 4.00002 4,0002 0.0046
SE 0,00006 0,00009 0 .O026
t 4.2769 -1.7430* 1,768*
otha specics
Independent mCab1e
Statistics Basai ara ( m h ) Diameta (an) Constant
Coefficient 4.0018 4.0019 O. 1966
SE 0.0015 0,0024 0.0665
t -1 -2440 6,8082 2.958*
Table 5.5 The meuu ofrnortrlity for hrd maple, white asb, blick cherry, and otha
species
Ingrowth matrix (R) is caldateci using the resuhs of four ingrowth equations given in
Table 5.1. Up growth mat& (A) is d d a t c d using the mean values of transition
probabilities given in Table 5.3, and the mean value of moruüty given in Table 5.5.
These two matrices - ingrowth and upgrowth - are combineâ togaher to get the growth
mitrix (G), and it is given in Table 5.6. The basai area is set at 21.6m2/ha. In this ma*
the nrst row shows, for example, that an additional hard maple tree of at Ieast 35 cm
would lead to 0.5 1 fewm hard maples per ha in the siw-class 1 1 cm, 5 yean later. It dso
shows that the probabüity that a hard maple of size-class 11 cm moves to size-class 17
cm in 5 yeus was O. 12, whiie the probabüity that a hard mple of size-class 17 cm stays
5.3 Variinces o f Transition Probrbüititr
The mitr0r p w t h modd with tandom transition probaôilities, as discussed in 3.1.2,
indudes the wirnce of transition probaôiies. Hence, the variances oftnnsition
pmbabiies am caiculrted. The standard d d o n oftransition probabilities of four
spdm and five dirunder classes are @m in Table 5.7. In thh table, for aumplq the
standud dcviation of the transition probability of- which move Born diameter clrw
1 1 cm to cüamcter clam 17- ir 0.005, as given in the ~ ~ o d row. Similady, the
rtrabrd devi.tion of transition probabiiity of tnsr remabhg in diameter c l w 17 cm is
0.0052.
Table 5.6 Growtb mitrirr G, for B = 2 1.6 m2/ha, and for a t h e intavll of 5 y-
Hard 1 7 , i z , l u o 0 0 o o o O O o o o o o o o o o o M q k 2 3 0 . i r . n o O O o o O O o o o o O o o o o o
2 9 0 0 . 1 2 . 8 1 0 0 0 O O O o o o o o o o o o o
2 9 0 0 0 0 0 0 0 . 1 2 . 9 i o 0 0 0 0 O 0 0 o o o
C b a y 2 3 0 0 0 0 0 0 0 0 0 0 0 . 0 2 . 9 4 0 0 0 0 0 0 0
2 9 0 0 0 0 0 O O O O O 0 0 . 0 6 . 9 s o o o o o o
0 0 0 O C
O O O O C
O O O O C
e o o o c
3 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
O O O O C
o o e o c
O O O O C
O O O O C
0 0 0 0 I
O O O O C
O O O O C
e o o o c
O O O O C
O O O O C
e o o o c
O O O O C
O O O O C
O O O O C
o o e o c
CHAPTER 6 RESULTS AND DISCUSSION
AB mentioned in Cbrpter 1 ad Chpter 3, al m h is âivided in four p a s . Hence, in
this chaptsr, resuits cocmponding to each part are p m t d sqmmteiy. First, the rrsults
probability growth modd (DTPGM) are pmented. Second, the nsults of the d y s i s of
stand with random transition probability growth mode1 (RTPGM) are dimssed. Third, a
comparative pichire of the nsults with DTPGM and RTPGM is given. Forth,
comparative piautes of economic outcornes achieved under three options - constant yield
hruvest, non-constmt yield hantest, and maximhtion of minimum harvest - are
discussed.
6.1 Analysis of the Hard Maple Stand with Detcrministic Transition Probabilities Growth
Modd @T'MM)
6.1.1 Tbt S t d y State ofthe Stand
Accordin8 to the ecdogy theory, ifa stand is not distwbed, it wodd grow xutudy and
wcntually mch a s t d y -te, in wbich growth just replaces the mortality. Mode1 (3.24)
is useci to study this icind of deady state. in this modd, the objective b to mkimh the
- harvest (condition inttodud to find optllnil solution). The mode1 (3.24) is oolved
for^' = 2L6 m 2 h The initial state of the hird maple stand k giwn in Tabh 6.1. The
diameter distribution of the hud maple sbid in the s t d y state is given in Table 6.2.
t
In the rieady state; the aggegate di- distri.bUtion (di spectFes togcther) and the
diamcter M o n ofthe domhant species @ud mapie) are stül like a mem "J"
m. Tht diunttm dWbutions of white idi and othu specim are aiso close to a
state dud stmctwe haa muiy distinct h t w u as compued to the pnseat s û u c t u n ofthe
stand. ki the sterdy statt, the totai numbct of treci Y 737 tht is 65 % of 1 134 - the
numbar of trar in initial state. in this procaq trees of hrd maple, white ash, and other
speci*cs duced by 27.2%. S9.8%, and 72.6%, but munber of tnes of black cherry
i n d by 30.1Y0.
Table 6.1 The initial state (diameter distriiution) of the hard maple stand
Diimaer CIUS (cm)
SpaCia 11 17 23 29 35+
Hard Maple 341 180 79 27 21
White A& 56 46 28 8 4
BI& Cherry 54 25 20 9 2
Otha Specks 93 71 38 18 14
Total 544 322 165 62 41
Table 6.2 S t d y State @iameter Distribution) of the Hard Maple Stand
Diameter Class (cm)
SpOats 11 17 23 29 35+
tlrrd Uiple 174 116 77 49 56
White Asti 22 12 7 9 7
Black Ch- 65 78 1 O O
ot&iS@at 48 11 3 2 O
Total 309 217 88 60 63
In the stedy statq one of the isnm b bat how long it would take the stand to reach thu
state. TO h d out that the, the rtud dynamics modd (3.10) is uscd d v e I y . In l a s
t h intemi- For a ~ m p l e . for the sp8cics i. y,(,,, = 1, t a,& + 4) implies that a lower
baiadrry for y,,,,, is $& - ha). The d t s of i t d o n are given in Table 6.3. In this
table, oaly the mimbas of totai trecs (in @a), in difkmt diuneter c~~ at
G e n d y , u the stand moves towards steady me, the number of big trœs should
incrtast and number of s d e r trees should dccresse. The n u m k of trees in different
diameter cluscs at subsequcnt time paiods confimu this g e n d trend. The number of
tmes in the lowest thra diameter classes (8-14,1420, and 20-26cm) deaeued
wntinuoudy in every successive t h puiod, except one exception in the 20-26cm
diameter class at the 95' ycir. The number oftrccs in the two highest diameter classes in
the steady is definitely higher thn the initial state of the stand, but inaease in these
diameter classes has not ban contirmous. Mer 100 y u n , there is no signifiant
différence ofthe number of trœs between each 5-yau period. T b number oftrees in
din i t diameter classes &et 100 years bccame almost statiomy, and very close to the
numbei of becs in M i t diamter classes at the s t d y state. Hence, we assumeâ that
the stand nrches the s t d y state at 110 yws. Howmr, to confimi that this isnimption
of 110 y t ~ is masonable, diameter-class distn'buton ofci8Rcrent spccics at 110 yean is
compared with the stady-state diuaetu~lrss distniutioa of diffctcllt specier. The
steady state diameter class distnion and the diameter clus distriiution at the age 110
years, obtained by recunive solution ofthe modd 3.10, for hard maple, white ash, black
chay. and other species arc givca in Figures 6.1.6.56.3, and 6.4 mpe*ivcly. These
figures indiate that the aumkr oftnes in diffbrent diameter classes at the steady state is
mtthesameuthenumberoftrrcrin~crcntdiuneterclassesat 110 yain. But, ford
the fw species, the traad of vuiuion in the numba oftna acrou five âîamettr ciasses
is aûnost the rrme h r s t t d y state and the stand-me at 110 yeus. Hcnce. 110 yeus is a
rcasonable approximation of the time to reach stcidy statt for hrd miph stand.
Thse nnihr of the stand dynamïcs and the steady state must be used cautioudy, since
they in based on a mode1 c i l iratd with growth data of 5-year p e n d only. Howcver,
tby do not contradict previous trœ 8fowth and ecologiul knowledge, and it semu then
remonable to use the modal to study forart management guideha.
Tabk 6.3 The Aggregaîe diameter distriiution (of JI species togaha) a 5-years inteml
Diameter Clam (cm) Year 11 17 23 29 35+
5 526-09 3 10.48 162,47
-t Real Growth + Steady State
Figure 6.1
- - - -
Tho DiameCr Distribution of White Aah at Steady State
-t Real Gmwth + Steady State
The Olime1.r Distribution of Othoi Spoks at Stoady State
6.1.3 Tbt State o f the Stand at Differeat Petioda
In order to get further insights of the stand dyiumics, the diameta distributions of each
species at 5.50 and 100 years in givm in Figure 6.4,6.5 and 6.6 mpectively. After 5-
year growth, thae is no important difference abut distributions from the initial diameta
disrrion (iabie 6.1). With time pauing by, an inmase in the number of big mes
would tend to close the stand and hamper regeneration, provoking a dedine in the
number of trew of d e s t sb. In 50 years pmiod, this decline is more apparent in white
ash than any other species. The number oftrees of black cherry and other species also
becorne less than that at initial state. However. the number of smaiier trees of hard maple
incrcasts a littls, it might result from the character of hard maple which could live under
less light. This aging of the stand continues leading to a forest with more large hard
map1es in 100 yean.
r- - -
The Diameter Distribution at 6 Yeam
+Diameter Dbtrikition of Hard Maple +Ohmetet Distribution of Wh& Abh +Dirimeter Dbtributkn of Balck Cherry +Dkmeter Obtributkn of m e r Spedes
Figure 6.6
The Diameter Di8trlbuUon at 100 Y0w8
6.2 And@ of Stand Growth ~4th hadom Triarition RobabUity Gmwth Modd
6.2.1 The Stind Dynrmicr
In the RTPGM disaissed in Cbipter 3, it is assumd tbit tnnsition probabiüties an
n o d y distributed. Hence, the value oftnnsition probabüity may lie below or above
the mtui d u e . Hence, in îhe Mdom environment. it wüi not be possible to fiad the
wict number oftnes in any diuneter class for rny sp8acs. Howawr, with the help of
Equations 3.25 and 3.26, the lower and uppa iimits ofnumber oftrees in dEerent
diameter classes can k obtained for the given significance level. Henct, these two
apations 3.25 and 3.26 are used recursively to iind the lowa and upper limits of the
number of trœs in di ient ciiarneta classes at 95% signiticance level. The results are
shown in Table 6.4.
As ucpcctd, the nuniber of bigger trees increases with the stand growth, whiie the
numkr of d a trœs dccrews with the stand growth, thc tnnd that was present in the
deterdistic tl~vironment. The two intcrcsfing ftritures of diameter distribution in
Mdom environmcnt me: (i) the mge of numk of- in any diameter clasr increases
with tirne; Ci) in some cases, such as the lowcst diameter class of hard msple at agc 50,
the n&ve deviation &es the u p p iimit rad the positive dmWation gives the lower
lima ofthe nnge of mmiba ofüœs. We think thrt is due to the combined e&ct of t m s
Table 6.4 The nage of numbet of trias in di&mit chuneter classes (at 5% signBcanct I d ) with RTPGM at 5,50, and 100 yeprs
6.2.2 Stite of the Stand in the Riradom Transition Pmbrbiiities Gmwth Modd
4 m M )
In the case of RTPGM, a concept hilu to the s t d y state in the DTPGM is not
f-ik However, the RTPGM can be used to 6nd out the state of the stand at any giwn
point for a given objective. Hence, the mdel(3.27) is used to find the sute of the stand
at difnasnt aga given the objective of mrucimization of net prestnt vaiue (NPV). This
modal was solved recursively for evay five y- intervrl and total t h e horizon of 110
years. The basai uu (B.) is fixecl at 21.6 rn/ha. Non-ünrr programming solver of
GAMS was useâ to solve this formulation. The results of this solution - the diameter
distribution at 5, 50, and 100 y u n - an givm in Table 6.5, and the harvest levels at
Mèrent times are given in Table 6.6.
The main fatwe of these r d t s is that the major change toker place Erom 5 to 50 years.
The number of d tnes of hard maple hy hcreased during this perioâ, howmr the
numba of big tnts of hard maplc b decfcased. It results fiom the harvest which mostly
focuses on hird maple uable 6.6). Ahhough then are some harvests of white ash and
b k k cherry, cornpuecl with hard maplq they are s d . The number of s d trees of
hard mipk at 5 years is 360, der 50 ycur, it is 469. This is simiiar to the inaease with
DTPGM (6.1.2). M g thb perbd, most smrll mes of wbite a& are dead, howcva
there M dl some big treer which wuid kœp in the uppa aown. The diameter
~ o n s of blrdc c m and 0th swes bave not changeci too much It sœms tht
theymthestcdycompoacntrinthesimp1estand.
Mer 50 y- aN specim rlmost kœp in stcady m e . ' h m is no apparent incniw or
deaeisc with the numkr of trœ of each s@m. Only the numbct of m d u m trecs of
hudmiplebalittkincreasc.
Table 6.5 The state of hard maple stand (diameta distribution) at 5.50, and 100 yeaus
(For the case of NPV mxhht ion uid using RTPGM)
Year Hard Mapie White Ash Black Ch- Other Species
11 360.029 47.012 53.699 89.985
17 191.840 45.204 29.199 63.589
23 45.977 28,059 19,518 3 5.870
29 3,094 11.000 9.970 19.48 1
35+ O 4,340 0.548 12.670
11 469.440 0,905 53.468 77.693
17 292.856 6.4 16 13.614 30.924
50 23 44,072 22.600 13.884 18.753
29 0,258 27.838 15,527 20,899
35+ O 6.901 0.926 6.603
11 464.547 0,057 56.979 68.05 1
17 329,133 0.004 49,548 21.639
100 23 49.001 6.471 14.167 10.321
29 O 23.414 17,514 14.254
3S+ O 9.447 3.213 4.089
Table 6.6 The hrrvests levds at 5.50, and 100 yan
(For the case of NPV maxiniization and usiag RTPGM)
D 5 Y m 25Yean 5OYcm 75Yesn 1ûûyear~
11 Hirrd 17 6 Maple 23 31 44 3 1 41 49
29 3 2 4 35+ 11
White 17 6 Ash 23
29 35+ 7 1 11
Black 17
char^ 23 29
35+ 1 1 2
11 ûther 17
SpcCies 23
29 35+
6.3 Coiapuiroa o f the Statu of the Eard Mipk Stand, obtiined by Using DTPGM
and R'ïPGM, rt 110 ymn
Ilr diissed in th sedion 6.2, the concept of the s t d y statt of the stand d a s not seem
f i i e in the fonnulrton ofRTPGM, Hace, to compare the outcornes ofDTPGM and
RTPGM, we seleaed the point of 110 yeus, the point at which the stand wüi k close to
the steady state in the case of det enmnUtic ptobab'ies (DTPGM). Thrre diameter
distributions - distn'butioe with detmmhhtic pmbabiüty, dinriitioa with positive
d-on (RTPGMPD)B ad rnddis'bution with negative deviatioa(RTPGMND) - for hard
the dirmatar distribution of white r s h
Ln~~o~wouldarpsctthtthtaimkroftnaisachdiamcterdrssobtiincd
with the hcip of positive deviation coadnint (RTPGMPD) sbould be more tbin thit with
det~c(msui)nkKcoartrPmDTPGM Simüuty, numbaoftrees obuhû a
erch dizuntter cLsr obtained by the n@ve d d o n coartnint d e i (RTPGMND)
should & kss than that of DTPGM However, genedly, tbis is not the case for most of
the diameter chues for ail four @esfS Ifw take a look the randam transition
probability rnodel (RTPGM) (given by apation 3.10), we cm undaotlad the proccss.
The deviation in the this mdd is the mot of the produa oftransition probabiiity's
d a o n tirne the number oftrees, ratha than the deviation of the number of m e s .
Thdore, ifwe MC a positive deviation oftransîtion probabüity, thae wouid be more
trea which move h m one diuneter clus 0) to the ncxt Iarger diameter clus ÿ +1). The
number of trrcs remain in the d i i w t a clru 0' would becorne less, c o m p d with the
n m k of trce forecasted with DTPGM In our case, whm MPDTP nins 22 timcs (till
110 years), there must be less tncs in the low diameter classes and more m e s in hi@
diameter ciasses, though thrr is an annuai ingrowth. AdualIy, it alsa refiects the
dynarnics of the stand. W&n th- are more big trees, generaily they are in the main
canopy ofthe stand. Those trccs tend to hcmm yearly in hcight, bote size, kngth of
each giowing spacq and numkr ofluves, m it lads to the natwal thuining. Some d
trees wdd not sucvive, rad thcn cüe. It d t s in lem rmill ~TCCS in the stand. I fwe take
the negative deviation of transition probib ies in the mdom model, tbe opposite
phenornena wouid happea. Thaî msrnr them are more small t m s at low diameta ciasses
and less big trœs at hi@ chneter dures thui tht forecasted with DTPGM. However, in
téw dilimcttr-classes, the numborofmcs with RTPGMPD k more than tbrt ofDTPGM
and the nurnber of trees with RTPGMND is less tbat that of DTPGM, This is in lower
diameter cluser dut can k attri'buted to ingrowth eqyation because they do not have
random demcnts.
The diametu distniution of ail species togetba is giwn in Figure 6.12. The genenl
trends of tm dishiution with respect to DTPGM, RTPGMPD, and RTPGMND are the
srw as for individuai specier dimsrod above. The main fèatwe is that the number of
trœs in thna upper diameter cluses for RTPGM (for both RTPGMPD and RTPGMND)
is not much di&rent fiorn the number for DTPGM. However, this was not the case for
individuai specics, specindy white ash Hence, the use of deterministic transition
probabiiity mode1 may be able to predict with reasonable accwacy the totai number of
tries (aii species togaha) in the more vduaôle diameter classes. But it wül be unable to
predict the stand structure and number of tms in lower diameter classes acairately given
the mdom natwe of transition probabilities.
6 4 The Comparative Study oCEconomk Outeomg under Canatant Hamrt, Non-
coratut LIIrvat, and Muimbtion of Mmimui (MAXMIN) H.mrt
Tbe thme modds discussed in tbe section 3.2.4 ure soLvd for prices givcn in TabIe 4.3,
rate of discount of 3%, and total time @oâ of 110 y-. The totai NPV for 110 years,
ad m e n t net nhrrnr at di8krent points of time are given in Trble 6.7. In the case of
constant barvat, current retums at 5 y- intend are $3327.84, and the NPV for 1 10
y m paiod U S20.93 1.40. T b NVP for non-constant harvut and for MAXMIN
approachcs are S89.897.98 and $8 1,972.22 respcctively. Heace, NPV for nonconstant
harvests is the maximum and for constant hamst, it is minimum. The dinemice between
NPV for non-constant harvest and MAXMIN approach is only S7,925. However, Li the
world of resource (forest) conservation, hwesting scheduies should also be wmpared.
The West levels for non-constant harvest, maximization ofMiillnum brrvest, and
constant huvest, are given in Table 6.8,6.9, and 6.10 respdvely.
In the case of non-constant harvest, munly one species - hard mapk - is hantesteci
mguiariy. Wcnct, in the lone-nan, nonanstant harvest will c b g e the structure of stand
itseE In the case of constant harvests, four trres Born highest dietcf clus of bud
maple rad 7 trar &oml7cm diameter clatm of white ash are harvested at every five y w r
btsml. In the case ofMAXWN approach, hammhg is dimibuteci over aü species and
muiy diameter classes. In addition, différent number of tmcs h m difftrent diameter
ciasses and Spcaes ire harvested at ~~ thne paiodr. Hace, the bvvesting pattem
obtained h m MAXMIN approach would be more desirable Born the point of
mdntrining the pccsuü rtncbrr o f b stad or the ~~nservatiou ofspecies dBnnity in
NPVI ud the aimat net retums with constant harvest, non-cunstant hantest ad MAXMIN approich (unit: CADS)
OtrjcctiveFUnction NPVhhmhhtion MAXMIN NPV Mruàniization
Constant Yidd No No Yes
NPV 89897.98 8 1972.22 2093 1.40
r i p e n d
5 45 720-98 36250.13 3327.84
10 8485.13 7688.204 3327.84
15 8987,09 8406.29 3327.84
20 9342.56 8975.64 3327.84
25 9832.04 9491 -74 3327.84
30 10340.40 8720.86 332734
35 10303 .O3 10198,Sl 3327.84
40 10748.68 75 17.9 3327.84
45 10962.19 10804.48 3327.84
50 10178,90 1 1059.48 3327.84
55 6289.3 8 1 1272-69 3327.84
60 10898.70 1 1450.91 3327.84
65 11 169,31 1 1829.9 1 3327.84
70 1 1 169.76 1 1702.67 3327.84
75 1 1450.89 1 1787.89 3327,84
80 9753.51 10971 -46 332734
85 11 130.28 9 174.74 3327.84
90 1 103 -48 8622.0 1 3327.84
95 842 1 -79 937SO 3327.84
100 10689.22 9909.54 3327.84
105 10530.19 11488.01 3327.84
110 10384.66 133 17.78 332734
Black
Cherry
White
Ash
Table 6.9 Harvest levds in the case of maxhkation of minimum hatvests
app-h
D (cm) 5 Y ~ M 25 Yuus 5OYcin 75 Years 100 Yean
11
White
Ash
Black
Cherry
the stand. In addition, as mentioned d e r , banciai mtum h m MAXMIN approach
are bigher thui that of coartint huvest, ad not much d a ttom nonanstant harvest.
aitaion instead of the conventionai constant hanest cpuity aiterion.
Table 6.10 Harvest levels in the case of coIWtlllIt hmests
Diameter Clam (cm)
Specim 11 17 23 29 35+
Hard Maple O O O O 4
White Ash O 7 O O O
Black Cherry O O O O O
otha Speaes O O O O O
T& diameter distibutior~~ o f l species at 5,5& and 100 yurs for non-constant yield
horvests and MAXMIN approach are shown in Figure 6.13.6.14 and 6.15. The number
of tries with MAXMIN approach is less thpa that with non-constant harvests at the
d e r diameter classes, but it is more for the large diameter classes. It indicates that
MAXMeJ approach is aiso good to get hi* number ofbig tnes towards tnatunty of
fomts. Hence, the MAXMIN approrch seems to have the potmtid to gentfate uscAi1
solutions to rustainable forest management of uneven-agd forcsts.
Figure 6-14
CHAPTER 7 SUMMARY AND CONCLUSIONS
nie matrix growth modd approach has ken usecl to analyze the p w t h of mixeci hard
maple stands in from wutheni Ontario. The ertisting ma&k growth modd approach,
b w d on dctermimstio traiwition probabilities, is modifiecl to incorponte m d o m nature
of transition probabüitics. The steady state, stand dynimics, and the state of the stand in
steady state are d y z e d using the matrix p w t h mode1 with detesministic transition
probaôiities. The stand dynarnics and the state ofthe stand at Mient h e s are ais0
anaiyzed with the matrix growth mode1 with nndom transition probabilities, and the
outcomes are cornperd with the outcomes ofDTPGM. DTPGM model is used to
determint the ecanomic outcomes and hawesting schedules for constant yield
constraints, non-constant yield constraints, and mwimization of minimum hawests, and
the results of these thnt approaches are comparecl.
The growth model estimation is bued on the growth data (growth paiod-5 years) nom
30 petmanent sample plots of hird maple forcsts in southern Ontario. Plot size varies
fiam 0.04 ha to 0.101 hi. The main species are hud maplc, white ash and black cherry in
these PSPs. The d t s am gaKnl in nature, but ue subject to postulates made such as
Usher ~ssumption, and the n o d distn'bution of the transition proWity. The d o
should k used ody to understand the expected trends and not to take the s p d c
numerical numbers into account for focest munag~113t11t decisions.
(1) In the detenninistic environment, the mrtrBr gmwth d e i can k useû to fonust the
staaâ growth and the s t d y state. But, it is cW6CUIt to detamiae the time at which the
stud maches the stoldy state- The gnphiaî cornpuison mi@ k a better metbod to Bnd
out this time but oniy approxhatdy.
(2) in reality, forest p w t h environment hm mdom elemaitr. in the matrix growth
moâd, Mdom ci- of growth can bc incorporateci by trsithg transition probabilities
as rarrdom variables.
(3) In the case of bard mple stands of southan Ontario, the total nurnber of trees in five
chneter clweq der 110 years ofgrowth, obtaineû Born daenninistic and random
transition probab'ities mode1 are not mch diffèrent. But, the diameter distriiution of
each specics is differcnt fmm #ch 0th~. Hena, ignorance of mdom nrtwc rnay not
cause problans at the aggrcgate level but the projection of stand structure d have
problems.
(4) In the case of uneven-aged forests, harvestin8 decisions based on the maximiaiion of
niinimum @UMMN) barvests - the concept equivdent to the Rawl's equity aiterion - mry provide Wer harvesting dechions &om consuvation as welî as financiai
ptfspcctive-
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