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Simulation of growth, yield and management of aspen Bella, Imre E 1970

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SIMULATION OF GROWTH, YIELD AND MANAGEMENT OF ASPEN by IMRE E. BELLA B.S.F., Un i v e r s i t y of B r i t i s h Columbia, 1958 M.F. Un i v e r s i t y of Washington, 1962 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Faculty of FORESTRY We accept t h i s t h e s i s as conforming to .J the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1970 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e tha t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d tha t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Co lumbia V ancouver 8, Canada Supervisor: Professor J.H.G. Smith i i ABSTRACT A semi-stochastic model was developed to simulate tree growth and stand y i e l d information required f o r managing aspen (Populas  tremuloides (Michx.)). The model i s based upon new approaches f o r evaluating i n t e r - t r e e competition e f f e c t s , representing a c t u a l tree s p a t i a l arrangement, defining i n t e r a c t i o n s between•increments of height and d.b.h. and competition measures, and representing random components of v a r i a t i o n i n tree growth and m o r t a l i t y . In b u i l d i n g the model, components of tree growth and m o r t a l i t y were i d e n t i f i e d and described mathematically or represented d i r e c t l y i n the computer.. Inter-tree competition, the most important component i n the model, was extensively studied. The maximum zone of influence of a tree was derived from estimates of f u l l y open grown crown width. A new hypothesis was advanced and mathematically expressed to describe competition e f f e c t s on tree growth by exponential terms of r a t i o s based on r e l a t i v e tree s i z e . The model simulates height increments as i f each tree was dominant or open-growing. The rate of height growth i s a function of s i t e q u a l i t y . Simulated " p o t e n t i a l " height increment i s reduced, according to the tree's competitive status, to obtain height increment. D.b.h, increment i s based on data from open grown aspen then reduced i n proportion to tree competition. Reductions from the maximum rate of growth are based on tree growth and mortality expected i n normal aspen stands. In the model mo r t a l i t y i s d i r e c t l y r e l a t e d to the t r e e ' s competitive status and i n v e r s e l y to i t s current increment, including random v a r i a t i o n , i n r e l a t i o n to a s p e c i f i e d threshold value.. i i i The model was c a l i b r a t e d with data from a normal stand growing on an above average aspen s i t e i n Saskatchewan. Input data were from a suitable permanent sample p l o t having f a i r l y uniform c l o n a l s t r u c t u r e . A f t e r a few c a l i b r a t i o n runs and model refinements, simulated stand growth s t a t i s t i c s showed s a t i s f a c t o r y correspondence with a c t u a l growth on the permanent sample p l o t , and with comparable y i e l d table s t a t i s t i c s . Simulations were made also f o r normal stands growing on poor s i t e s and on the best s i t e s i n the same region. The model was generally s a t i s -f a c t o r y and could replace normal y i e l d t a b l e s . A f t e r c e r t a i n extensions, the r e v i s e d model also simulated growth and development of i n i t i a l l y open stands reasonably w e l l . The model also was used to simulate aspen stand growth and p r o d u c t i v i t y i n terms of tree component dry matter weights f o r normal stands on average s i t e s with weight regressions determined i n an. associated study. For maximum production of wood f i b r e , the optimum r o t a t i o n was 33 years f o r e i t h e r volume or weight. Although tested only f o r pure aspen stands, the model can be modified f o r use with other species. With further refinement, i t may be possible also to simulate the growth of mixed stands and uneven-aged stands. i v ACKNOWLEDGEMENTS I w ish to thank my s u p e r v i s o r . D r . J . H . G . Smi th , f o r h i s con t inued i n t e r e s t , adv i ce and encouragement i n t h i s s tudy , and the members o f my Graduate Committee: D r s . C.W. Boyd, D. H a l e y , C . S . R o l l i n g , A . Kozak and D.D. Munro f o r u s e f u l d i s c u s s i o n s on v a r i o u s aspec ts o f the t h e s i s p r o j e c t and f o r t h e i r c r i t i c a l rev iew o f t h i s d i s s e r t a t i o n . Some o f the data used here was r e t r i e v e d from aspen t h i n n i n g p r o j e c t s , e s t a b l i s h e d by the Canadian F o r e s t r y Se rv i ce i n Mani toba and Saskatchewan. I am g r a t e f u l to Mr . G . A . S teneker , Research F o r e s t e r , f o r h i s c o - o p e r a t i o n i n o b t a i n i n g t h i s d a t a , and to M r . J . P . DeF rancesch i , Fo res t Research T e c h n i c i a n , f o r h i s a s s i s t a n c e i n the v a r i o u s phases o f data c o l l e c t i o n , a n a l y s i s and d r a f t i n g work. The Aspen i t e D i v i s i o n o f M a c M i l l a n B l o e d e l L t d . , Hudson Bay, Saskatchewan, co -opera ted i n o b t a i n -i n g he igh t d i s t r i b u t i o n data i n mature aspen s t a n d s . I thank M iss L . Cowdel l and M r s . K. H e j j a s f o r t h e i r ready h e l p w i t h the ex tens i ve computer a n a l y s i s r e q u i r e d f o r t h i s s tudy . I am o b l i g e d to my employer , the Canadian F o r e s t r y S e r v i c e , f o r f i n a n c i a l support du r ing my e d u c a t i o n a l l eave and to the F a c u l t y o f F o r e s t r y f o r g r a n t i n g me the Van Duesen Graduate F e l l o w s h i p . F i n a l l y , I am most g r a t e f u l to my w i f e , L e s l i e , whose chee r -f u l companionship, i n t e r e s t and s t i m u l a t i n g unders tand ing c o n t r i b u t e d g r e a t l y f o r making t h i s e n t i r e under tak ing an en joyab le and rewarding e x p e r i e n c e . V INDEX OF CONTENTS TITLE PAGE i ABSTRACT i i ACKNOWLEDGEMENT i v INDEX OF CONTENTS v INDEX OF TABLES v i i i INDEX OF FIGURES i x CHAPTER 1. INTRODUCTION 1 1.1 Trembling aspen as a stand component i n Canada 1 1.2 The economic importance of poplar species i n the present, and future expectations 1 1.3 Problems re q u i r i n g e a r l y a t t e n t i o n i n aspen management: stocking, stand density and structure, and t h e i r i n -fluence on growth and y i e l d 3 1.4 Solving aspen management problems r e l a t e d to growth and y i e l d 5 CHAPTER 2. DESCRIPTION OF THE SPECIES AND ITS HABITAT ' 8 2.1 Taxonomy, range and d i s t r i b u t i o n 8 2.2 E c o l o g i c a l and s i l v i c a l c h a r a c t e r i s t i c s 9 2.3 Damaging agents 13 2.k Anatomy, p h y s i c a l and chemical wood properties Ik CHAPTER 3- STATE OF KNOWLEDGE ABOUT ASPEN STAND DEVELOPMENT 18 3.1 Regeneration and development of juvenile stands (up to age 10 years) 18 3.11 Factors a f f e c t i n g the stocking and q u a l i t y of regeneration 18 3.12 Growth and development of j u v e n i l e stands 21 3-13 Is early (under 10 years of age) spacing c o n t r o l desirable? 23 3.2 Growth and development of young and intermediate aged trees and stands 2k v i 3 . 2 1 Growth and y i e l d studies i n untreated stands: s i t e index curves and y i e l d tables 25 3 . 2 2 Thinning studies 30 CHAPTER 4 . APPROACHES IN STAND DEVELOPMENT STUDIES: SYSTEMS MODELING AND SIMULATION 46 4 . 1 A review of approaches i n stand development and other population dynamic studies 46 4 . 2 Systems models and simulation 49 4 . 3 A review and c r i t i q u e of fo r e s t stand growth simulation models and suggestions f o r improvement 51 CHAPTER 5» DEVELOPMENT OF A STAND GROWTH SIMULATION MODEL FOR ASPEN 59 5 .1 The purpose and uses of the model 59 5 . 2 Methods of model b u i l d i n g and t e s t i n g 6 0 5 . 3 Basic components of growth and stand development with s p e c i a l emphasis on aspen 6 l 5 . 4 Development and t e s t i n g of an improved i n t e r t r e e competition model 6 3 4 . 4 1 Background 63 5 . 4 2 A new model: Competitive Influence-zone Overlap (CIO) . 65 5 . 4 3 Estimating model parameters f o r aspen 67 5 . 5 A general flow diagram f o r the stand growth model 75 5 . 6 Detailed flow diagram, programming and model refinement 79 5 . 7 Stand growth simulation experiments with the model 88 5 . 7 1 C a l i b r a t i o n with check p l o t (normal stand density) data from above average s i t e s 88 5 . 7 2 Simulating growth of normal density aspen stands growing on poor and on excellent s i t e s • 104 5 . 7 3 Simulation of stand growth on above average s i t e s for two subnormal stand de n s i t i e s 109 v i i CHAPTER 6. SIMULATING DRY MATTER PRODUCTION IN ASPEN STANDS ' 115 6.1 " Uses of biomass data 115 6.2 Development of aspen a e r i a l tree component dry weight estimating equations 116 6.3 Simulating growth and dry matter p r o d u c t i v i t y of normal density aspen stands on above average s i t e s 121 CHAPTER 7. ASPEN STAND MANAGEMENT ALTERNATIVES AND ECONOMIC IMPLICATIONS 126 CHAPTER 8. SUMMARY AND CONCLUSIONS 132 LITERATURE CITED lk2 APPENDIX I DETAILED FLOW DIAGRAM OF THE STAND GROWTH MODEL 1^ 9 A l i s t and de s c r i p t i o n of the more important variables i n the model „ 150 APPENDIX II STAND GROWTH MODEL FORTRAN PROGRAM (MAIN PROGRAM AND SUBROUTINES) 176 APPENDIX III SIMULATED STAND TREE STATISTICS 18U APPENDIX IV ALBERTA NORMAL YIELD TABLES FOR ASPEN (FROM MACLEOD 1952) 18;'8 v i i i INDEX OF TABLES 3.1 Annual d.b.h. increment (inches) over d.b.h. regressions f o r aspen stands aged 11 to k-5 years at treatment (50 observations per p l o t ) , 37 5.1 C o e f f i c i e n t s of determination f or Dine-CIO c o r r e l a t i o n s f o r aspen. F i r s t increment period; age Ik to 20; 1^ 3 study trees 73 5.2 Optimum model parameters f o r the three increment periods, and r e l a t e d c o e f f i c i e n t s of determination f or Dine-CIO multiple c o r r e l a t i o n s ; also Dinc-D simple c o r r e l a t i o n s 7^+ 5.3 Aspen stand growth simulation summaries f o r above average s i t e c l a s s i n Manitoba and Saskatchewan tyk 5*k Stand development s t a t i s t i c s from a permanent sample p l o t , and from the Alb e r t a normal y i e l d t a b l e s, above average aspen s i t e ( S i t e Index 75, MacLeod 1952) 101 5.5 Aspen stand growth simulation summaries f o r poor and excellent s i t e classes i n Manitoba and Saskatchewan 106 5.6 Aspen stand growth simulation summaries f o r the two subnormal stands on above average s i t e c l a s s i n Manitoba and Saskatchewan Ilk 6.1 Data summary showing means and measures of dispersions of tree component dry weights ( i n grams) and l i n e a r tree s i z e v a r i a b l e s f o r 152 aspen trees 117 6.2 Regression analyses f or dectecting possible trends i n aspen a e r i a l component weight d i s t r i b u t i o n , based on 152 trees (weights i n grams) 119 6.3 Regressions f o r estimating a e r i a l component weights of aspen based on 152 trees (weights i n grams) 120 6.h Aspen stand growth and dry matter p r o d u c t i v i t y simulation summaries, above average s i t e i n Manitoba and Saskatchewan ... 123 INDEX OF FIGURES i x 3.1 Aspen s i t e index curves for medium s i t e classes i n c e n t r a l and western Canada, and the Lake States 27 3.2 T o t a l basal area over age curves for aspen on medium s i t e classes i n c e n t r a l and western Canada, and the Lake States .. 29 3.3 Ten year d.b.h. increment (1950-1960) by d.b.h. classes and treatment. From Steneker and J a r v i s 1966; study 5 - stand 23-year-old at thinning 35 3.4 Annual d.b.h. increment - d.b.h. r e l a t i o n s of aspen on thinned and check p l o t s at various ages (data from Steneker and J a r v i s 1966) 38 3.5 Frequency diagrams of d.b.h. increment and that of d.b.h. of dead trees i n the period they succumb, and d.b.h. frequency diagrams of surviving t r e e s . Two sample p l o t s ; increment period from 25 to 30 years (Data from Steneker and J a r v i s 1966, study 2.) 1+3 4.1 Newnham's (1964) Competition Index: angle summation of p o t e n t i a l crown overlaps 53 5.1 Influence-zone overlaps between a competing subject tree (#3) and four competitors. These values are used f o r c a l c u l a t i n g CIO (see text) 68 5.2 Crown width-d.b.h. r e l a t i o n of open grown aspen trees i n western Canada 72 5.3 General flow diagram of the stand growth model 77 5.4 D,b.h./height - height r e l a t i o n of open grown aspen 85 5.5 Relationship between diameter increment adjustment and height increment adjustment 85 5.6 Average d.b.h. and basal area per acre trends from permanent sample p l o t s and y i e l d tables ( S i t e Index 75, MacLeod 1952), and from stand growth simulation for above average s i t e s i n Manitoba and Saskatchewan 95 5-7 Cumulative tree height frequency d i s t r i b u t i o n s (10 classes between minimum and maximum values) from a stand growth simulation f o r above average s i t e s i n Manitoba and Saskatchewan 97 5.8 Cumulative tree d.b.h. frequency d i s t r i b u t i o n s (10 classes between minimum and maximum values) from a stand growth simulation f or above average s i t e s i n Manitoba and Saskatchewan 98 X 5.9 Average d.b.h. and basal area per acre trends from y i e l d tables ( S i t e Index 55? MacLeod 1952) and from stand growth simulation f o r poor s i t e s i n Manitoba and Saskatchewan 107 5.10 Average d.b.h. and basal area per acre trends from y i e l d tables ( S i t e Index 90, MacLeod 1952) and from stand growth simulation for excellent s i t e i n Manitoba and Saskatchewan 108 6.1 Above ground dry matter p r o d u c t i v i t y by components on age 12h 6.2 D i s t r i b u t i o n of above ground dry matter components over stand age 124 7.1 Stem wood y i e l d curves for an aspen stand growing on above average s i t e c l a s s (Manitoba - Saskatchewan), and ages of maximum average annual increment 129 CHAPTER 1. INTRODUCTION 1 1.1 Trembling aspen as a stand component in Canada Trembling aspen Populus tremuloides Michx. i s one of the most widely distributed tree species in Canada. It occurs in commer-c i a l quantities in British Columbia, in the three Prairie Provinces and the Yukon and Northwest Territories, in Ontario and in Quebec. Trembling aspen with the four other native poplar species — large tooth aspen P. grandidentata Michx., balsam poplar P. balsamifera L., eastern cottonwood P. deltoides Marsh, and black cottonwood P. tricho-carpa Torr. and Gray — contain an estimated net merchantable volume (stem wood over 4-inch diameter excluding stumps and c u l l ) of 65.6 b i l l i o n f t and comprise 54$ of the tot a l volume of a l l hardwoods. Poplars represent 9$ of the entire net merchantable forest resources in Canada (Fitzpatrick and Stewart. 1968), in which trembling aspen i s generally the most important constituent contributing well over half of a l l volume. Trembling aspen i s a particularly important stand component in the extensive Boreal Mixedwood Forest of the Prairie Region and the Territories. It may form pure stands after a disturbance, but successional trends favor the development of a conifer admixture with advancing age. This study w i l l concentrate, however, only on pure, younger aspen stands, which may be of greatest potential interest to future forest management tending toward short rotations. 1.2 The economic importance of poplar species i n the present, and  future expectations Although hardwoods, and particularly poplars, have been used for lumber and pulp manufacture for many years, the actual amount of 2 of wood thus u t i l i z e d was negligible compared to the more favored conifers. The abundant supply of accessible and economically harvestable (sound wood, high stand density, sometimes associated with large tree size) high quality conifers in Canada slowed the ut i l i z a t i o n of poplars, with their soft, relatively low strength wood, short fibre, and the relatively high incidence of decay characteristic of mature stands. Technical developments achieved after World War II in the forest industry and the rapidly increasing demand for wood fibre and other forest products in the recent years which i s expected to continue (Fowler 1966) are showing effects on poplar u t i l i z a t i o n . During the 5-year period 1961 to 1965 about 76 million f t were used annually, about 70% of this for pulp and paper (Fitzpatrick and Stewart 1968). Aspen has already achieved considerable commercial importance in the Lake States (e.g., Blyth. 1968), where the proportion of aspen as pulp raw material has risen to about half of the total amount (4f-million cords) of wood u t i l i z e d in 1967. A l l United States production of hardwood pulpwood has recently been rising much more rapidly than that of softwoods, and the trend i s expected to continue (Hair 1968). As the wood-fibre needs of the Canadian pulp and paper industry i s expected to increase about fivefold by the year 2000 (Fowler 1966), Canadian poplar resources no doubt w i l l have a major role as raw material source. With the depletion of old growth natural stands, i t w i l l also become necessary to grow trees in a managed forest. This w i l l require substantial investments which would be tied up in the growing stock u n t i l harvest. Trembling aspen appears to be a particularly desirable species from this point of view, because i t 3 regenera tes r e a d i l y by roo t suckers w i thou t much c u l t u r a l a s s i s t a n c e , and grows ve ry r a p i d l y a t younger ages . These c h a r a c t e r i s t i c s makes i t i d e a l l y s u i t e d f o r management on r e l a t i v e l y shor t r o t a t i o n s ; an a t t r a c t i v e p r o p o s i t i o n i n t imes o f h i g h i n t e r e s t r a t e s and r a p i d l y advancing t echno logy . I t i s p o s s i b l e t ha t l a r g e d imension s tock f o r lumber and veneer may a l s o be grown economica l l y i n aspen p l a n t a t i o n s u s i n g f a s t growing hyb r i ds o r s u p e r i o r c lones o f n a t u r a l s tock (Smith 1968). Because o f the s p e c i e s ' r a p i d growth and moderate s i t e requi rements i t c e r t a i n l y deserves i n c r e a s e d c o n s i d e r a t i o n i n the f u t u r e . Much r e l e v a n t i n f o r m a t i o n i s a v a i l a b l e f rom Europe on i n t e n s i v e management o f aspen (Borse t i960; J a r o i960; K a l d i 1961, P o s p i s i l 1967, Solymos i960) where the wood o f Populus t remula L . i s the main source o f match s t o c k . !-3 Problems r e q u i r i n g e a r l y a t t e n t i o n i n aspen management: s t o c k i n g , s tand d e n s i t y and s t r u c t u r e , and t h e i r i n f l u e n c e on growth and  y i e l d Aspen management i s a t a v e r y e a r l y stage i n Canada. I n d i -c a t i o n s a r e , however, t ha t management e f f o r t s w i l l be i n t e n s i f i e d i n the f o reseeab le f u t u r e to meet i n c r e a s i n g raw m a t e r i a l demand o f the f o r e s t i n d u s t r y . Some i n f o rma t i on i s a l r e a d y a v a i l a b l e on growth and y i e l d o f n a t u r a l , o l d growth , un t rea ted stands i n Canada (Johnson 1957; K i r b y et al.,1957; MacLeod 1952; P l o n s k i 1956), and i n the U n i t e d S ta tes (Brown and Gevo rk ian t z 1934; Graham et_ a l . 1963). Some aspen t h i n n i n g exper iments have a l s o been conducted t o study the e f f e c t o f s tand d e n s i t y on t r e e and s tand growth ( B i c k e r s t a f f 1946; h Heinselman 1 9 5 2 ; Sorensen 1 9 6 8 ; Steneker and J a r v i s . 1 9 6 6 ; Strothmann and Heinselman 1 9 5 7 ; Zehngra f f , 1 9 ^ 9 ) . Whi le the above s t u d i e s have p rov i ded some u s e f u l i n f o rma t i on on s tand development p rocesses f o r o l d growth stands o r i g i n a t i n g a f t e r h o t , w i l d f i r e s , the r e s u l t s may not be d i r e c t l y a p p l i c a b l e to s tand c o n d i t i o n s which p r e v a i l a f t e r the o r i g i n a l s tand has been c u t . F i r e s i n the pas t g e n e r a l l y k i l l e d a l l . the s tand ing aspen t r e e s , and consumed the brush and a l a r g e p o r t i o n o f the o rgan i c matter i n the upper s o i l h o r i z o n . Abundant sucke r i ng f o l l o w i n g w i l d f i r e s commonly r e s u l t e d i n f u l l s t o c k i n g and s tand d e n s i t i e s as h i g h as 5 0 , 0 0 0 stems pe r a c r e . Cond i t i ons f o r sucke r i ng are g e n e r a l l y l e s s s u i t a b l e a f t e r l o g g i n g . A s u b s t a n t i a l amount o f o rgan ic d e b r i s i s l e f t beh ind ; some t r e e s a re l e f t s t a n d i n g , and the brush l a y e r and the s o i l o rgan i c h o r i z o n are but l i t t l e a f f e c t e d . These w i l l a f f e c t the s t o c k i n g , s tand d e n s i t y and s t r u c t u r e o f the new sucker s t a n d . There fore growth , y i e l d and s tand development i n cu tover stands may d i f f e r from those stands o r i g i n a t i n g a f t e r f i r e . F o l l o w i n g l o g g i n g sucker s tands a re g e n e r a l l y pa tchy — s tocked a reas i n t e r r u p t e d by open ings . But even i n the s tocked areas s tand d e n s i t y i s l i k e l y to be lower than on comparable bu rns . Con-s i d e r i n g tha t the a c t u a l t i m i n g and the t ime p e r i o d over which sucke r i ng . takes p l a c e may be q u i t e v a r i a b l e , the number o f s tand s t r u c t u r e com-b i n a t i o n s - - each o f which has a c h a r a c t e r i s t i c s tand development p a t t e r n - ' c o u l d be ve ry numerous i n d e e d . S t o c k i n g , s tand d e n s i t y and s t r u c t u r e g e n e r a l l y have s i g n i f i c a n t e f f e c t s on growth and y i e l d o f f o r e s t s t ands . As these s tand v a r i a b l e s a l s o are r e l a t i v e l y easy to manipulate by c u l t u r a l p r a c t i c e s , improved unders tand ing o f these r e l a t i o n s i s impera t i ve f o r more e f f e c t i v e f o r e s t management p r a c t i c e . 1 .h S o l v i n g aspen management problems r e l a t e d to growth and y i e l d The gene ra l p r a c t i c e i n pas t growth and y i e l d i n v e s t i g a t i o n s has been to s e l e c t i n na tu re the s tand c o n d i t i o n s one d e s i r e d to s tudy , t o e s t a b l i s h sample p l o t s , and to take the necessary measurements or o b s e r v a t i o n s . Growth and m o r t a l i t y s t a t i s t i c s were ob ta ined e i t h e r by re -examina t i on o f the same p l o t s a t g i ven i n t e r v a l s , o r by e s t i m a t i n g pas t o r f u tu re t r e e s i z e s based on increment core d a t a . (The l a t t e r , however, p rov i des no m o r t a l i t y e s t i m a t e s . ) The r e s u l t s were then p i e c e d toge ther f o r d i f f e r e n t age groups acco rd ing to d e f i n e d c l a s s e s ( s i t e , d e n s i t y , e t c . ) , and the a n a l y s i s g e n e r a l l y conducted i n terms o f average s tand or t r e e s t a t i s t i c s . There are s e v e r a l drawbacks to t h i s method, some o f which are insurmountable c o n s i d e r i n g the p resen t p rob lem. The method i s expens i ve , i t i s s low i n p roduc ing r e s u l t s ( g e n e r a l l y a t l e a s t one 5-year increment p e r i o d i s r e q u i r e d ) , and because o f the e lapsed t ime between measurements sample p l o t s may be des t royed be fore y i e l d i n g the i n f o r m a t i o n r e q u i r e d . E x t r a p o l a t i o n s from the r e s u l t s are frowned upon, so a l a r g e number o f s tand c o n d i t i o n combinat ions o f p o t e n t i a l i n t e r e s t to the f o r e s t manager would need to be sampled and s t u d i e d . Th is i s not on l y a v e r y b i g t a s k g e n e r a l l y , but i t i s c l e a r l y u n f e a s i b l e because a t the p resen t t ime on l y ve ry young cu tover s tands are a v a i l a b l e f o r samp l i ng . Another drawback i s tha t even a f t e r ex tens i ve sampl ing the r e s u l t s tend to be main ly d e s c r i p t i v e ( i n a form o f average s tand s t a t i s t i c s ) and u s u a l l y 6 do not reveal causal factors i n f l u e n c i n g tree growth. Stand growth and development i s determined by the growth and development of i n d i v i d u a l t r e e s . I d e a l l y then, stand growth and development should be studied on an i n d i v i d u a l tree b a s i s . This would require that various influences (environmental g e n e t i c a l , etc.) and int e r a c t i o n s (competition e f f e c t s , etc.) a f f e c t i n g tree growth proces-ses could be q u a n t i t a t i v e l y evaluated and described. A synthesis of such quantitative information i s c a l l e d a mathematical model. A f t e r t r a n s l a t i n g the model i n t o a computer language, i t can be fed i n t o a computer along with relevant tree and p l o t information, and by a c t i v a t i n g the model, tree growth can be simulated. Stand growth i s obtained from summing tree growth. n Newham and Smith ( 1 9 6 4 ) were the f i r s t to use t h i s approach to study and describe stand growth and development i n terms of i n d i v i d -u a l tree performance. They developed and tested t h e i r model f o r Douglas f i r Pseudotsuga menziesii (Mirb.) Franco and extended i t to lodgepole pine Pinus contorta Dougl. Lee ( 1 9 6 7 ) made further improvements on the lodgepole pine model. M i t c h e l l ( 1 9 6 7 ) used a s i m i l a r approach with several new concepts f o r h i s white spruce Picea glauca (Moench) Voss model. While these researchers demonstrated the tremendous p o t e n t i a l of t h i s approach to study complex e c o l o g i c a l systems l i k e f o r e s t stands, much more improvement and refinement are needed to u t i l i z e t h i s poten-t i a l f u l l y . In t h i s d i s s e r t a t i o n a stand model i s presented f o r trembling aspen and used to test a range of management a l t e r n a t i v e s f o r given objectives by simulating stand development of various stand condition 7 combinations and i n t e r p r e t i n g the outcome i n economic terms. This model resembles the Newnham and Smith (1964) model i n p r i n c i p l e , although no p a r t o f t h e i r model was used d i r e c t l y . The present model was developed a f t e r a thorough f a m i l i a r -i z a t i o n w i t h the s p e c i e s . I t i s based on the height growth tre n d o f the dominant p o r t i o n o f aspen stands, which i s considered a f a i r l y s t a b l e expression of growth p o t e n t i a l o f the species on a given s i t e . I n t e r -t r e e competition - - a n important concept i n stand model work -- has been examined i n d e t a i l , and a new hypothesis was advanced and mathe-m a t i c a l l y d e s c r i b e d . The model provides f o r s u b s t a n t i a l f l e x i b i l i t y i n r e p resenting t r e e s p a t i a l p a t t e r n s as w e l l as c l o n a l s t r u c t u r e , c h a r a c t e r i s t i c o f aspen sucker stands. R e s i d u a l v a r i a t i o n i n t r e e growth was represented by random elements generated on the computer. M o r t a l i t y was generated by a v a r i a b l e t h r e s h o l d based on a minimum l e v e l o f height increment and competition s t a t u s o f the i n d i v i d u a l t r e e . In a d d i t i o n to stem volume y i e l d s , dry matter production o f above ground components were simulated f o r young aspen stands. I t i s b e l i e v e d that the present model i s s u f f i c i e n t l y g e n e r a l f o r use to describe stand development o f other t r e e s p e c i e s , or even mixtures a f t e r appropriate model parameters have been de f i n e d f o r the s p e c i e s . CHAPER 2 . DESCRIPTION OF THE SPECIES AND ITS HABITAT 8 2 . 1 Taxonomy, range and d i s t r i b u t i o n The genera Populus L. belongs to the Salicaceae family and S a l i c a l e s order. The genus Populus i s divided i n t o f i v e groups: Leuce, A i g e i r o s , Tacamahaca, Leucoides and Turanga; the f i r s t three are native to Canada. There are two native aspen species: P. tremuloides Michx. (trembling) and P. grandidentata Michx. (large tooth) which belong to the Leuce group (Anon. 1 9 6 1 ) . In addition to the t y p i c a l form of P. tremuloides Michx. found i n eastern and north-central Canada, several of i t s v a r i e t i e s and forms have been di s t i n g u i d e d (summarized by Maini 1 9 6 8 ) : var. aurea (Tid.) Dan. (golden aspen) and Magnifica V i c t . , form pendula Jaeger and Beissner, reniformis (Tid.) and Vancouveriana ( T r e l . ) Sarg. Along with the t y p i c a l form, the most widely d i s t r i b u t e d i s the golden aspen occur-r i n g i n the West, ( i n the following sections of t h i s d i s s e r t a t i o n , no d i s t i n c t i o n w i l l be made between v a r i e t i e s and formsof trembling aspen, and the term aspen alone w i l l mean trembling aspen.) Trembling aspen i s the most widely d i s t r i b u t e d tree species i n North America, covering 110° of longitude and 4 7 ° of l a t i t u d e (Fowells 1 9 6 5 ) from the A t l a n t i c coast to a l l but the most western edge of B r i t i s h Columbia. Its northern l i m i t i s close to the f o r e s t tundra; along the northern shores of Hudson Bay i n c e n t r a l and eastern Canada, and i n the West i t extends i n a north-westerly d i r e c t i o n from the south-west corner of Hudson Bay to northwestern Alaska. The southern boundary of the species extends w e l l i n t o the United States i n the East, and i n the mountainous regions of the West at higher elevations as f a r as 9 nor the rn M e x i c o . In the P r a i r i e P r o v i n c e s (Man i toba , Saskatchewan and A l b e r t a ) , s tands and groves o f t r emb l i ng aspen c o n s t i t u t e the Aspen Grove type - a t r a n s i t i o n between the B o r e a l Fo res t i n the no r th and g r a s s l a n d i n the s o u t h . M a i n i ( 1 9 6 8 ) s y n t h e s i z e d d i s t r i b u t i o n maps f o r t remb l ing aspen and o ther ind igenous pop la r s p e c i e s from in fo rma t i on a v a i l a b l e up to 1 9 6 7 . 2 .2 E c o l o g i c a l and s i l v i c a l c h a r a c t e r i s t i c s Aspen i s an i n t o l e r a n t p i onee r spec ies t ha t becomes e s t a b -l i s h e d a f t e r such d i s tu rbances are f i r e , l o g g i n g , e t c . I t occu rs over a wide range o f h a b i t a t c o n d i t i o n s , but f avo rs f r e s h , w e l l d r a i n e d , up land s i t e s w i t h loamy s o i l r i c h i n l i m e , and ach ieves best development where some seepage i s p resent (Jameson 1 9 6 3 , S t o e c k e l e r i 9 6 0 ; Strothmann I960). M a i n i ( 1 9 6 8 ) s t u d i e d the growth o f aspen i n Saskatchewan a long a 750 m i l e sou th -no r th t r a n s e c t . He sampled 96 s t ands , s t a r t i n g i n the g r a s s l a n d i n the south ( 4 9 ° N l a t . ) and ex tend ing ac ross the v a r i o u s v e g e t a t i o n and c l i m a t i c zones to the no r the rn l i m i t o f the B o r e a l f o r e s t near the no r the rn boundary o f Saskatchewan ( 6 0 ° N l a t . ) . M a i n i found optimum he igh t growth f o r aspen i n the B o r e a l F o r e s t between 55° and 56° N l a t . , where mature, dominant aspen has average he igh t between 70 and 80 f e e t and aspen t r ees over 9 0 f e e t are not uncommon. He suggested tha t s i m i l a r e x c e l l e n t growth may be expected f o r aspen a t these l a t i t u d e s i n eas te rn A l b e r t a , Mani toba and nor thwestern O n t a r i o . Aspen t r e e s are r e l a t i v e l y s h o r t e r southward i n the f o r e s t - g r a s s l a n d t r a n s i t i o n as w e l l as toward the no r the rn d i s t r i b u t i o n l i m i t s o f the s p e c i e s . Growth o f aspen i s l i m i t e d by mois ture towards the south and 10 by temperature and other edaphic factors towards the north. The area i n which aspen seems to a t t a i n optimum growth i n the P r a i r i e Provinces has a continental climate, characterized by long, co l d winters and short, r e l a t i v e l y warm summers. The mean annual temperature i s about 30°F (McKay 1965). J u l y i s generally the warmest and January the coldest, with mean d a i l y temperatures between 60 to 65°F and -5 to -10°F, r e s p e c t i v e l y . However, summer temperatures may exceed 95°F and winter temperatures lower than -55°F have been recorded i n the area. The growing season (degree-days above k2°F) i s about 150 days. The mean annual t o t a l p r e c i p i t a t i o n i s around 16 inches i n eastern Alberta and Saskatchewan, and increases eastward i n Manitoba and northwestern Ontario to 18-20 inches. Most of the p r e c i p i t a t i o n f a l l s during the growing season (Anon. 1957). Aspen i s a major stand component i n the Boreal Mixedwood Forest (Rowe 1959) o f the P r a i r i e Provinces. I t may form pure stands a f t e r a disturbance on upland s i t e s , or i t may grow i n a s s o c i a t i o n with balsam poplar, white spruce, black spruce Picea mariana ( M i l l . ) BSP, balsam f i r Abies balsamea (L.) M i l l . , jack pine Pinus banksiana Lamb., and lodgepole pine Pinus contorta Dougl. i n the Subalpine Forest of A l b e r t a . Barring disturbance, succession w i l l proceed toward replacing the i n t o l e r a n t s h o r t l i v e d hardwoods and pines with tole r a n t spruce and f i r . Many shrubs, herb, moss and l i c h e n species are found i n these f o r e s t s . Their composition i s mainly a function of stand and canopy density and structure, i n add i t i o n to s i t e i n f l u e n c e s . Aspen i s normally dioecious. Flowering s t a r t s at about 15 years of age. The flowers appear before l e a f i n g out i n the spring. P o l l i n a t i o n i s accomplished by wind. The f r u i t , catkins of capsules, 11 ripens about 5 weeks a f t e r flowering. Each capsule contains about 10 small brown seeds, each of which i s surrounded by a t u f t of large white s i l k y h a i r ; Thus equipped, the l i g h t seed of aspen may be dispersed long distances from the mother tree with the help of wind or water i The number of seeds produced i n a good seed year may be as many as several m i l l i o n on a mature female aspen. Good seed crops can be expected i n every second or t h i r d year with a f a i r l y high proportion of abortive seeds (Maini 1 9 6 8 ; Fowells 1 9 6 5 ) . Within a day or two a f t e r d i s p e r s a l , r i p e aspen seeds w i l l germinate on a s u i t a b l e , moist seedbed. I f lack of moisture prevents germination, the seed w i l l generally lose i t s v i a b i l i t y under f i e l d conditions a f t e r 2 to k weeks (Moss 1 9 3 8 ) . ( V i a b i l i t y may be pro-longed, however, by s t o r i n g the seed under appropriate conditions i n a l a b o r a t o r y ) . Even i f germination takes place i n the f i e l d , seedling establishment i s r e s t r i c t e d to moist, f r e s h l y exposed mineral s o i l (Maini 1 9 6 8 ) . During the f i r s t year, the young seedlings are suscep-t i b l e to damage from heat, drought and f u n g i . Aspen reproduces vigorously by means of root suckers (ad-v e n t i t i o u s shoots on r o o t s ) , and to a l e s s e r extent and only on younger trees, by root c o l l a r and stump sprouts (Maini 1 9 6 8 ) . R o o t a b i l i t y of aspen stem cuttings i s generally poor, although a recent study i n C a l i f o r n i a reported up to 100$ rooting of stem cuttings, depending upon the time of year and g e n e t i c a l c h a r a c t e r i s t i c of the source material (Barry and Sachs 1 9 6 8 ) . Good rooting, of presumably wind broken aspen shoots which landed i n a moist d i t c h , was also observed i n Vancouver, B.C. (J.H.G. Smith, personal communication). 12 Suckers u s u a l l y o r i g i n a t e on roots l e s s than 1 inch i n d i -ameter and which are within 3 to k inches of the s o i l surface. Although dormant sucker buds formed i n the previous years may be present and l o c a l l y numerous on a root, laboratory t e s t s showed that 95% of 174 successful suckers o r i g i n a t e d from buds formed i n the same season (Sandburg and Schneider 1 9 5 3 ; Maini 1 9 5 8 ) . Sucker formation v a r i e s considerably by clones and i s a f f e c t e d by temperature. Stand treatment which r e s u l t s i n higher s o i l temper-atures, e.g., cutt i n g of trees, removal of vegetation, f i r e , s c a r i f i c a -t i o n , etc., w i l l generally promote aspen suckering (Maini and Horton 1 9 6 6 ) , although there i s a d i f f e r e n t i a l response to s p e c i f i c temper-ature regimes by clones (Maini 1 9 6 7 ) . This contradicts the "a p i c a l dominance" sucker formation theory (Farmer 1962 ) that adventitious buds on aspen roots develop into suckers when the i n h i b i t i n g influence of shoot apex ( i . e . , the auxin produced therein) i s removed. This a p i c a l dominance theory i s only based on greenhouse experiments, and would need to be substantiated i n nature by further experiments . Repeated vegetative reproduction may r e s u l t i n formation of male or female clones having from a few to several hundred ramets ( i . e . , trees of same genotype). Members of one clone can be distinguished from those of a neighboring clone by a v a r i e t y of morphological and physio-l o g i c a l c h a r a c t e r i s t i c s . Apparently, young ramets up to 25 years of age, depend h e a v i l y on the parent root system while developing t h e i r own independent root systems (Barnes 1 9 6 6 ) . Ramet-parent root systems may become separated i f the l a t t e r i s a f f e c t e d by decay (Sandberg and Schneider 1 9 5 3 ) ; but parent root conections apparently are maintained among healthy trees (DeByle 1 9 6 4 ) . Even roots under dead stumps can 13 survive, f o r a considerable length of time i f connected to l i v i n g trees, because of the t r a n s l o c a t i o n of materials between interconnected root systems (DeByle 1 9 6 4 ) . Interconnected stems of a clone may function as a single unit f o r n u t r i t i o n and water blanace, and thus can generally outcompete independent i n d i v i d u a l s located within such a group. With-i n an interconnected group, trees of any crown c l a s s seem to translocate materials, both i n proximal d i r e c t i o n (to root section towards parent tree) and i n d i s t a l (away from parent tree) (DeByle 1 9 6 4 ) . Wo quantitative data are a v a i l a b l e so f a r , however, on the s i g n i f i c a n c e of t h i s t r a n s l o c a t i o n on three growth i n r e l a t i o n to environmental and stand density-competition i n f l u e n c e s . 2.3 Damaging agents A large number of insect and disease species attack aspen i n Canada. The Forest Insect and Disease Survey reported at l e a s t 300 species of insects and 150 fungi on l i v i n g aspen. Many of these, however, cause l i t t l e , i f any, damage (Davidson and Prentice 1 9 6 8 ) . The most important diseases of aspen are caused by fungi, among which the decay of heartwood ranks f i r s t . Over l / 4 of the merchantable volume of aspen may be a f f e c t e d by decay (Davidson and Prentice 1 9 6 8 ) . Heart rot can take place i n any tree, subject to i n f e c t i o n and having exposed heartwood as a r e s u l t of i n j u r y or natural pruning. The volume of decay increases with stand age, but within a stand, the amount and incidence of decay seem to be associated with c l o n a l c h a r a c t e r i s t i c s (Wall 1 9 6 9 ) . Studies to date present contradicotory r e s u l t s on the influence of s i t e condition on decay (Basham 1 9 5 8 ; Thomas et a l . i 9 6 0 ) . Most of the l o s s i s caused by: Ik Radulum casearium (Morg.) Lloyd, Fomes i g n i a r i u s (L. ex F r . ) Kickx and Peniophora polygonia (Pers. e x F r . ) Bourd. & Galz. (Basham 1 9 5 8 , Thomas et a l i 9 6 0 ) . As most heartwood decay enters v i a branchwood i n f e c t i o n s (Thomas et a l . i 9 6 0 ) , a r t i f l c l a l . p r u n i n g may be an e f f e c t i v e c o n t r o l measure on selected t r e e s . Regeneration favoring disease r e s i s t a n t clones (Wall 1 9 6 9 ) may also be f e a s i b l e with more intensive aspen management. Shorter rotations could also l i m i t the amount of decay by cutting trees before losses become serious. Cankers are fungi that i n f e c t areas of the bark u n t i l they g i r d l e and k i l l the stem. Hypoxylon canker (Hypoxylon sp.) i s the most important and causes extensive damage i n aspen across Canada (Davidson and Prentice 1 9 6 8 ) . Young stands (15 to kO years) are most susceptible. Higher s u s c e p t i b i l i t y i s associated with lower stand density and stocking, but r e l a t i v e tree v i g o r (dominance or suppression) seems to have no e f f e c t (Anderson and Anderson 1 9 6 8 ) . Growing aspen on good s i t e s i n f a i r l y dense, f u l l y stocked stands may be a desirable c o n t r o l measure. Periodic and generally l o c a l i z e d damage by other agencies may include h a i l ( R i l e y 1 9 5 3 ) , f r o s t (Cayford et a l . 1 9 5 9 ) , glaze . (Cayford and Haig 1 9 6 1 ) and browsing by mammals. However, these have only minor s i g n i f i c a n c e i n aspen management. 2 .k Anatomy, p h y s i c a l and chemical wood properties Aspen i s a low density, diffuse-porous hardwood with uniform texture and no g r a i n pattern. Its wood i s l i g h t i n color; the sap i s white and the heartwood i s off-white to creamy colored with a g r a y i s h tinge ( I r v i n and Doyle 1 9 6 1 ) . The"wet wood"has a disagreeable odor. 15 Tens ion wood, the r e a c t i o n xy lem o f hardwoods, i s found e x t e n s i v e l y i n aspen . I t s presence i s a s s o c i a t e d w i t h l e a n i n g s tems. The g e l a t i n o u s f i b r e s which c h a r a c t e r i z e t e n s i o n wood are u s u a l l y concen t ra ted on the upper s ide o f l e a n i n g stems, p a r t i c u l a r l y at lower l e v e l s i n the b o l e , but may occur a t g r e a t e r h e i g h t s , e s p e c i a l l y i n the crown ( T e r r e l l 1952). Tens ion wood may be reduced by e l i m i n a t i n g l e a n i n g stems from the s t a n d , by g e n e t i c a l s e l e c t i o n o f d e s i r a b l e c l ones (Kennedy 1968), o r by us i ng sho r t e r r o t a t i o n s as young t r e e s seem to produce l e s s t e n s i o n wood (Sachsse 1965). Prun ing shou ld be a v o i d e d , as i t may s u b s t a n t i a l l y i n c r e a s e t e n s i o n wood fo rmat ion (Sachsse 1965). F i b r e l e n g t h , d i r e c t l y r e l a t e d to pu lp s t r e n g t h ^ v a r i e s i n d i f f e r e n t Populus h y b r i d s (Johnson 19^2) acco rd ing to the t ime o f fo rmat ion i n any one y e a r ; w i t h la tewood f i b r e s be ing 10-20 pe r cent l onge r than those o f e a r l y wood. F i b r e l eng th was found to i n c r e a s e r a p i d l y w i t h i n c r e a s i n g age (number o f r i n g s ) from the p i t h i n aspen (E inspahr et a l . 1968) and i n o ther Populus s p e c i e s . In b l a c k c o t t o n -wood, f i b r e l eng ths n e a r l y doubled from 1 to 15 years o f age (Kennedy 1957). F a s t growth i s g e n e r a l l y a s s o c i a t e d w i t h l onger f i b r e s i n pop la r s (Kennedy 1968) i n c l u d i n g aspen (E inspahr et a l . I 9 6 8 ) , a l though some s t u d i e s found no such c o r r e l a t i o n s i n aspen (Brown and V a l e n t i n e I 9 6 3 ) . A s t rong g e n e t i c i n f l u e n c e on f i b r e l eng th has been observed i n aspen, w i t h t r i p l o i d s and t r i p l o i d h y b r i d s hav ing an average f i b r e l eng th 26% g r e a t e r than d i p l o i d s (Bu i j t enen et a l . 1958, E inspahr et a l . I968). S u b s t a n t i a l v a r i a t i o n e x i s t s i n f i b r e l e n g t h between c lones o f the same spec ies w i t h s i m i l a r growth r a t e s (Brown and V a l e n t i n e 1963). In most p rev i ous s t u d i e s , f i b r e l e n g t h o f aspen was 16 between 0 . 4 ram and 1 .4 mm; mature trees generally have f i b r e lengths close to the upper l i m i t . S p e c i f i c g r a v i t y of aspen wood i s highly v a r i a b l e and may be subject to environmental and/or genetic c o n t r o l (Kennedy 1 9 6 8 ) . Past studies report s p e c i f i c g r a v i t y (green volume) f o r aspen wood between 0 . 3 1 - 0 . 4 7 . Result's so f a r are inconclusive on growth rate - s p e c i f i c g r a v i t y r e l a t i o n s f o r aspen (Brown and Valentine 1 9 6 3 ; Einspahr et a l . 1 9 6 8 ) . Numerous studies reviewed by Kennedy (I968) on other poplar species suggest that f a s t e r growth rate may lead to lower s p e c i f i c g r a v i t y . But the generally small differences may not have much p r a c t i c a l importance when compared to the amount of v a r i a t i o n present. Much v a r i a t i o n e x i s t s i n s p e c i f i c g r a v i t y between trees of the same clone (Brown and Valentine 1 9 6 3 ; Buijtenen et a l . 1959j Einspahr e t _ a l . 1 9 6 8 ) , suggesting strong environmental e f f e c t s . Other studies (Wilde and Paul 1 9 5 1 ; Benson 1 9 5 6 ) show r e l a t i v e l y minor environmental ( s i t e ) influence on s p e c i f i c g r a v i t y , although i n these studies the confounding e f f e c t of c l o n a l v a r i a t i o n has not been considered. Thick gelatinous layers which are present i n tension wood f i b r e s can cause an increase i n s p e c i f i c g r a v i t y of 0 . 0 2 to 0 . 0 6 u n i t s (Kennedy 1 9 6 8 ) . The carbohydrate content of poplars, including aspen, i s remarkably high and the l i g n i n content correspondingly low, a t t r i b u t e s that r e s u l t i n easy pulping of these species. Current experiements showed that these a t t r i b u t e s also r e s u l t i n good d i g e s t i b i l i t y of aspen wood by ruminants a f t e r inexpensive steam treatment (Bender et a l . 1 9 7 0 ) . The above r e s u l t s were obtained from experiments with mature aspen wood 17 and bark, and even better d i g e s t i b i l i t y may be expected i f young, more succulent and less woody components l i k e branches, twigs and f o l i a g e were used. Regarding mechanical properties, aspen wood i s generally weak because of i t s low s p e c i f i c g r a v i t y . Yet tension wood with higher s p e c i f i c g r a v i t y tends to have even les s desirable strength properties (Kennedy 1968). Aspen has poor n a i l holding capacity, but i t s low tendency to s p l i t makes i t desi r a b l e f o r boxes and crates (Ir\pin and Doyle 1961; Kennedy 1968). Low s p e c i f i c g r a v i t y and porous structure r e s u l t i n les s than i d e a l machining properties worsened by the presence of tension wood (Cantin 1965), while the low density i s an advantage i n p a r t i c l e board manufacture (Baldwin and Yan 1968). L i v i n g aspen trees have high moisture content with summer low and winter high values (Gibbs 1935; Bendsten and Rees 1962). Those studies obtained summer low values between 65 and 80%, and winter highs between 113 and 130%. Moisture content i s f a i r l y stable within seasons (Bendsten and Rees 1962; Be l l a 1968). Seasonal change i s much greater i n the sapwood than i n the heartwood (Bendsten and Rees 1962). Aspen and poplars i n general, have high r a t i o s of tange n t i a l to r a d i a l shrinkage (average 2.2) thus they are more subjected to seasoning defects (Kennedy 1968). In addition, tension wood with high l o n g i t u d i n a l shrinkage ( T e r r e l l 1952) r e s u l t s i n even more seasoning problems. 18 CHAPTER 3 . STATE OF KNOWLEDGE ABOUT ASPEN STAND DEVELOPMENT 3 . 1 Regeneration and development of .juvenile stands (up to age 10 years) 3 . 1 1 Factors a f f e c t i n g the stocking and q u a l i t y of regeneration Because most aspen stands at present regenerate by suckers, the number and the q u a l i t y of suckers are l i k e l y to be a f f e c t e d by the c h a r a c t e r i s t i c s (e.g., stand density and structure, age, c l o n a l v a r i a t i o n , etc.) and the condition (e.g., health, vigor, etc.) of the parent stand; by environmental influences, e s p e c i a l l y those r e l a t e d to weather; and by the nature and timing of the disturbance that destroyed the parent stand. Graham et a l . ( 1 9 6 3 ) found that the number of suckers per acre one year a f t e r cutting was more or l e s s d i r e c t l y r e l a t e d to basal 2 area density of the parent stand. Up to 50 f t basal area the number of suckers averaged 5 , 2 0 0 per acre, and from parent stands having 100 2 and more f t of basal area, the number of suckers averaged 9,900 per acre. They also showed that with increasing number of trees i n the parent stand, the number of suckers per tree r a p i d l y declined. No d i r e c t information i s a v a i l a b l e on the possible e f f e c t of tree s p a t i a l arrangement i n the parent stand on suckering, but i t i s expected to be of l e s s e r importance than stand density. Aspen stands generally have extensive s u p e r f i c i a l root systems, which can i n i t i a t e suckers and which, because of t h e i r c l o n a l habit, v i z . , interconnected ramets, can survive f o r some years even a f t e r some of the belonging stems have been cut or died (DeByle 1 9 6 4 ) . This means that l i v e roots may f u l l y i n t e r s e c t the area, regardless of the a c t u a l arrangement of trees i n the parent stand. 19 Clonal c h a r a c t e r i s t i c s i n aspen stands appear to have an important e f f e c t on suckering. Garrett and Zahner ( 1 9 6 4 ) found that inherent v a r i a t i o n i n suckering between clones i n a stand can be over 6OO/0 ( 4 , 0 0 0 vs. 2 8 , 0 0 0 suckers per acre) i n the f i r s t year a f t e r c u t t i n g . Although i n t e r t r e e competition quickly reduces the v a r i a t i o n , they found, four years a f t e r c u t t i n g , a difference i n the number of s u r v i v a l s as high as h00%. As suckering i s under strong environmental c o n t r o l (Maini and Horton 1 9 6 6 ) , these r e l a t i v e differences between various clones i n any one year are very much influenced by a c t u a l weather condition i n the years immediately a f t e r disturbance. The age and vigor of the parent stand also have influences on aspen suckering. In the Lake States, best suckering occurred i n 3 5 - to 5 0 - year-old cutover stands (Graham et a l . 1 9 6 3 , Stoeckeler and Mason 1 9 5 6 ) , but suckers are produced i n cutover stands much older than t h i s . Generally, healthy, vigorous parent trees produce numerous suckers, whereas few suckers are produced from roots of decadent or dying trees (Graham et a l . 1 9 6 3 ) . The nature of the disturbance has important e f f e c t s on suckering. Most of the present day mature aspen stands i n the P r a i r i e Provinces o r i g i n a t e a f t e r hot, w i l d f i r e s which destroyed the trees i n the former aspen stand as w e l l as consuming brush and weeds, and generally creating very favorable conditions f o r suckering. At the present, f i r e no longer plays such an important r o l e i n the renewal of aspen f o r e s t s , because current f i r e c o n t r o l measures prevent extensive w i l d f i r e s . Attempts to regenerate aspen by prescribed burning have been generally unsuccessful, because of the f u e l conditions, v i z . , 20 high moisture content and generally slow i g n i t i o n of hardwoods, that p r e v a i l on aspen types i n the Boreal Forest (Tucker and J a r v i s 1 9 6 7 ) . I f an aspen stand i s harvested, the degree of cuttin g has a d e f i n i t e e f f e c t on suckering. The more trees l e f t standing i n an aspen cutover, the poorer w i l l be the suckering and the s u r v i v a l of the suckers (Zehngraff 1 9 4 7 : Graham et a l . 1 9 6 3 ; Garrett and Zahner 1 9 6 4 ) . Stoeckeler and Macon ( 1 9 5 6 ) found only one h a l f as many sprouts ( 1 , 4 8 0 per acre) where the r e s i d u a l stand basal area was approximately 10 f t , compared to stands completely c l e a r cut ( 2 , 7 2 0 per a c r e ) . To ensure vigorous suckering, a l l trees should be cut, including species other than aspen. F a i l u r e to do t h i s w i l l not only hinder the establishment of a new aspen stand, but also i s l i k e l y to enhance the invasion of more tolerant tree species. The presence of hazel and other shrubs may also i n h i b i t aspen suckering, so t h e i r removal may be necessary. The time of logging, or the time of disturbance i n general, also has an influence on aspen suckering. Logging during the dormant season can r e s u l t i n as much as 25$ more sprouts at the end of the f i r s t growing season, than i f logging was done during the growing season (Zehngraff 1 9 4 6 ; Stoeckeler and Macon 1 9 5 6 ) . But at the end of the second growing season, t h i s difference tends to disappear as the roots continue to send up more suckers (Graham et a l . 1 9 6 3 ) . Regardless of the degree and time of c u t t i n g , the number of sprouts a f t e r c u t t i n g w i l l g enerally not be as high as that a f t e r a hot f i r e . A f t e r c u t t i n g , 1 0 , 0 0 0 sprouts per acre may be a good upper average, but even 6 , 0 0 0 w e l l d i s t r i b u t e d sprouts can provide adequate stocking per acre (Graham e_t a l . 1 9 6 3 ) . An aspen regeneration survey 21 on current large scale aspen cutovers at Hudson Bay, Saskatchewan (personal communication with S. Horvath and T. Stringer, MacMillan Bloedel Ltd.) found that 93% of the area had more than 6,000 suckers per acre, ranging up to as high as 17,000. Although r e s u l t s are la c k i n g at present on the growth and y i e l d of aspen cutovers i n Canada, lower i n i t i a l sucker density than that of previous stands of f i r e o r i g i n may not n e c e s s a r i l y r e s u l t i n decreased y i e l d . On the contrary, i t may be possible that lower i n i t i a l density w i l l cause r a p i d i n d i -v i d u a l tree growth and r e s u l t i n increased y i e l d i n cutover stands, provided that the area i s adequately stocked. 3-12 Growth and development of juvenile stands Aspen suckers drawing upon the extensive root system of the parent stand grow very r a p i d l y r i g h t from the s t a r t . Vigorous suckers can grow as much as f i v e feet t a l l by the end of the f i r s t growing season. Average height increment of about two feet per annum was found f o r the f i r s t f i v e years i n the Lake States (Strothmann and Heinselman 1957). Similar growth may be expected on average, or better than average, aspen stands i n the Boreal Mixedwood Forest of the P r a i r i e Provinces. Regardless of the coppice nature of the aspen f o r e s t , the development of young aspen stands i s very much l i k e that of other i n t o l e r a n t tree species. Graham et a l . (1963) showed that the l i f e of an aspen stand, from the time of sucker i n i t i a t i o n u n t i l the trees reach commercial maturity, i s characterized by successive periods of slow growth with heavy mortality, followed by intervening periods of rapid growth. On the basis of breast height basal area increment, data 22 obtained from a si n g l e sound dominant or co-dominant t r e e from each of 90 p l o t s (trees granging i n age from 22 to 50 years), they concluded tha t c y c l i c growth v a r i a t i o n was r e l a t e d to age.only and not influenced by weather or other p e r i o d i c f a c t o r s . While t h i s c y c l i c growth pattern may indeed e x i s t i n aspen stands, i t i s probably much l e s s dependent on age than Graham and his co-workers reported. Smith (personal communications) suggests that severe c l i m a t i c stress (e.g. drought) at a period of extreme crowding and competition could cause such c y c l i c development pattern i n most fo r e s t stands. Following sucker i n i t i a t i o n , r a p i d l y growing aspen shoots w i l l f u l l y occupy the s i t e i n a r e l a t i v e l y short time, p a r t i c u l a r l y i f the i n i t i a l density i n terms of number of trees was above 1 0 , 0 0 0 trees per acre. As a r e s u l t of t h i s r a p i d growth, such stands quickly become over crowded, with intense competition between trees f o r moisture or other l i m i t i n g f a c t o r s . Moisture i s generally the most c r i t i c a l need, e s p e c i a l l y on l i g h t textured s o i l s . I f a dry year occurs at the period of high crowding and intense competition, then t h i s added c l i m a t i c stress could be s u f f i c i e n t to cause heavy tree m o r t a l i t y . A f t e r n atural thinning, the r e s i d u a l trees enter a period of r e l a t i v e l y r a p i d growth u n t i l the stand again becomes crowded. Depending on i n i t i a l stand density, the f i r s t heavy mortality i n aspen stands can take place between 3 to 8 years of age (Graham elb a l . 1 9 6 3 ) . Although diameter increment declines with increasing crowding of trees, height growth apparently i s much less a f f e c t e d over a r e l a t i v e l y wide range of stand density l e v e l s . The 23 f i r s t f i v e year r e s u l t s i n an aspen sucker density study that was i n i t i a t e d at age one year (Strothmann and Heinselraan 1957), showed no density e f f e c t on height growth between treated p l o t s of up to 1,500 trees per acre and check plo t s averaging 10,000 trees. Although 10,000 trees per acre may represent an average sucker density i n cutover stands, d e n s i t i e s much higher than t h i s may occur frequently i n aspen stands at t h i s age, and then stand density may have sub-s t a n t i a l influence on height growth of i n d i v i d u a l t r e e s . 3.13 Is e a r l y (under 10 years of age) spacing c o n t r o l desirable? At present, the only published information a v a i l a b l e on ear l y stand growth and development i n r e l a t i o n to stand density was presented by Strothmann and Heinselman (1957). They showed that on good s i t e s at l e a s t , the f i r s t f i v e year's height growth was not af f e c t e d by density. A f t e r the remeasurement of the same experiment at 15 years of age, Sorensen (1968) reported that the la r g e s t 100, 200, and U00 trees per acre had about the same average d.b.h., while the average d.b.h. of a l l trees was in v e r s e l y r e l a t e d to stand density. This suggests that on good s i t e s , i n t e r t r e e competition e f f e c t s on diameter growth are n e g l i g i b l e on as many as hOQ of the largest trees per acre up to age 15 for a f a i r l y wide range of stand d e n s i t i e s . Sorensen (1968) also reported that t o t a l height was not influenced by treatment. As may be expected, the check p l o t s had the highest f i n a l volume and also the greatest amount of mo r t a l i t y during the f i r s t 15 years. Although i t would be unwise to generalize from the r e s u l t s of one experiment, i t should be noted that they are reasonable and 2k seem to correspond with thinning r e s u l t s reviewed l a t e r . Because of the extreme intolerance of aspen, heavy na t u r a l thinning occurs i n dense young stands. Stagnation i s uncommon. In the following section i t i s shown that thinning aspen generally does not increase f i n a l volume y i e l d ; on the contrary, f i n a l volume y i e l d i s generally lower i n thinned stands. This point, and i t s economic implicationSjare discussed i n more d e t a i l i n that section, most of which applies to spacing c o n t r o l i n j u v e n i l e stands. While there may not be any s i g n i f i c a n t advantage i n t h i n -ning aspen stands under 10 years of age, there would be at l e a s t two drawbacks: ( l ) the much greater number of trees that would need to be cut i n very young stands compared to older stands, would l i k e l y r e s u l t i n high cost of thinning operation, and (2) the longer the time required f o r the investment (cost of thinning) to mature, the higher would be the cost ( i n t e r e s t ) of investment and lower the return --assuming comparable r o t a t i o n ages a f t e r both thinnings. 3.2 Growth and development of young and intermediate aged trees and  stands Two main sources of information are a v a i l a b l e on development of intermediate aged trees and stands: ( l ) growth and y i e l d studies compiled as y i e l d tables, and (2) thinning studies. Y i e l d tables contain average stand growth and y i e l d s t a t i s t i c s f o r a given area or region. They are u s e f u l , among other things, f o r evaluating differences i n average growth and y i e l d of a given species between d i f f e r e n t regions. Y i e l d tables generally show stand averages from 10 years of age to maturity (up to 100 years, or even above f o r some l o n g - l i v e d species), or to harvest. 25 Thinning studies are conducted to determine r e l a t i o n s h i p s between tree and stand growth, and stand density. While most t h i n -ning study r e s u l t s are presented i n the form of sample p l o t or stand averages, they generally provide some tree growth information, as w e l l as stand development data f o r i n d i v i d u a l p l o t s . In addition to providing d e s c r i p t i v e information, some of these studies attempt to reveal the processes of change involved i n these r e l a t i o n s h i p s . As these processes and r e l a t i o n s h i p s are e s s e n t i a l l y the same f o r a large portion of the geographical range of a species, the r e s u l t s of t h i n -ning studies may have much wider a p p l i c a b i l i t y than y i e l d t a b l e s . 3.21 Growth and y i e l d studies i n untreated stands: s i t e index curves and y i e l d t a b l e s . As the prime goal of fo r e s t management at present i s wood production, i t i s u s e f u l to describe stand development i n terms of the amount of wood produced per un i t area. Normal y i e l d tables are the most sui t a b l e f o r t h i s purpose, because they provide a common base of comparison. These tables are based on samples taken i n f u l l y stocked stands of normal density. This i s advantageous when comparing p r o d u c t i v i t y at a ce r t a i n age, as the influence of stand density -- which has a strong e f f e c t on t o t a l y i e l d may be con-sidered constant. Empirical y i e l d tables which represent a c t u a l stand conditions i n an area, provide volume production trends which are l e s s u s e f u l f o r comparison purposes. Apparent differences i n wood production may r e f l e c t stand density v a r i a t i o n s rather than differences i n grow-ing conditions or species v a r i a t i o n . At the same time, both types of y i e l d tables are prone to have biased wood volume estimates, because 26 of p o s s i b l e differences i n sampling procedures, i n c u l l deductions and i n a n a l y t i c a l procedures. Therefore, y i e l d tables may not provide very r e l i a b l e comparisons. Fortunately, y i e l d tables generally contain a stand s t a t i s t i c , the average height of the dominant portion of the stand at a given age, which i s an i n d i r e c t expression of the p r o d u c t i v i t y p o t e n t i a l of the s i t e f o r a given species . The most often used comparative value i s " s i t e index", which often i s the average dominant height of the stand at reference age. Using average dominant height, or s i t e index, f o r comparison has several advantages ( l ) i t i s f a i r l y independent of stocking and stand density e f f e c t s , ( 2 ) i t i s r e l a t i v e l y easy to de-termine, and ( 3 ) i t i s l e s s a f f e c t e d by sampling, decay status, and a n a l y t i c a l procedures than are volume s t a t i s t i c s . Three y i e l d tables have been compiled f o r pure aspen stands of the Boreal Mixedwood Forests of the P r a i r i e Provinces: normal y i e l d tables f o r Alberta (MacLeod 1 9 5 2 ) , and empirical y i e l d tables for Saskatchewan (Kirby et_ a l . 1957 ) and f o r Manitoba (Johnson 1 9 5 7 ) . Two other normal y i e l d tables, representing important aspen growing areas i n the v i c i n i t y of the Great Lakes, are relevant to the present study. These are f o r north-western Ontario (Plonski 1956 ) and the Lake States (Brown and Gevorkiantz 1 9 3 * 0 -Figure 3 -1 shows medium s i t e index curves from these f i v e y i e l d t a b l e s . (Only medium curves are presented, as these are l e a s t a f f e c t e d by the number of s i t e index classes -- or subdivisions --a c t u a l l y used i n constructing a given table.) From t h i s f i g u r e , aspen growth a p p e a r s ^ 0 be poorest i n Manitoba, better i n Saskatchewan and Alberta, and best around the Great Lakes i n the Lake States and north-FIGURE 3.1 Aspen s i t e index curves f o r medium s i t e classes i n central and western Canada, and the Lake States. western Ontario. On the whole, aspen growth seems remarkably s i m i l a r i n Alberta and Saskatchewan, also around the Great Lakes — Ontario to the north and the Lakes States to the south. Average dominant height values at maturity i n Ontario are greater than i n the Lake States, which may be the r e s u l t of better health and continued t h r i f t y growth of aspen stands i n Ontario as compared to that of the Lake States. Any f i n e r comparisons than these may not be meaningful or j u s t i f i e d , because of the a r b i t r a r y nature of the s i t e index curves. Only Plonski (1956) used stem analysis to e s t a b l i s h the shape of height growth curves. T o t a l per acre basal area values presented i n y i e l d tables are also u s e f u l for growth and p r o d u c t i v i t y comparison. They are easy to c a l c u l a t e accurately, are c l o s e l y r e l a t e d to wood production, yet they remain free from bias that can creep i n t o volume s t a t i s t i c s from differences i n estimating procedures and p o s s i b l e decay adjustments-The main drawback of using basal area f o r comparisons i s i t s d i r e c t dependence on stocking and stand density. Figure 3-2 shows t o t a l basal area per acre growth curves f o r medium s i t e classes from the f i v e y i e l d t a b l e s . Ranking of basal area growth curves i s s i m i l a r to that of the s i t e index curves ( F i g . 3-2), except f o r A l b e r t a , which has the highest values. The reason f o r the high Alberta values probably l i e s i n the f a c t that MacLeod (1952) only sampled "well stocked" stands, having obviously higher d e n s i t i e s than those on which the Ontario, and Lake States y i e l d tables were based; As expected, the two empirical y i e l d tables (Manitoba and Saskatchewan) C M . a> o o a> a. a a to o 1 5 0 1 4 0 1 3 0 1 2 0 1 1 0 1 0 0 9 0 8 0 7 0 6 0 5 0 Alberta (MacLeod 1952) — — «— Ontario (Plonski 1956) - — • — = • • — » • Lake States (Brown and Gevorkiantz 1934) Saskatchewan (Kirby et JIL 1957 ) Manitoba (Johnson 1957) JL _L 2 0 3 0 4 0 5 0 6 0 Age 7 0 8 0 FIGURE 3.2 Total basal area over age curves for aspen on medium s i t e classes i n central and western Canada, and the Lake States. 30 had the lowest basal area per acre. 3 . 2 2 Thinning studies C y c l i c patterns of growth and m o r t a l i t y were mentioned under juvenile stand development. According to Graham e_t a l . ( 1 9 6 3 ) , the f i r s t p e r i o d of heavy mortality may occur around 6 years of age --e a r l i e r i n denser stands and l a t e r i n open ones. A f t e r several years of r a p i d growth which i s followed by increased competition and de-c l i n i n g growth rate of i n d i v i d u a l t r e e , the second p e r i o d of heavy mor t a l i t y may take place around age 1 5 . A f t e r another of these cycles, they found the t h i r d and g e n e r a l l y f i n a l period of heavy mortality to occur i n the e a r l y twenties of an aspen stand's l i f e . Graham et a l . ( 1 9 6 3 ) pointed out that during the heavy m o r t a l i t y periods many of the slow growing l e s s vigorous trees may become i n f e c t e d with various decay organisms which hasten the dying of trees and r e s u l t i n an unhealthy appearance of the stand.. However, the r e s i d u a l trees generally recover and resume r a p i d growth. While t h i s c y c l i c development pattern appears reasonable, i t could not be confirmed from the r e s u l t s presented i n any of the reviewed aspen thinning studies. The main purpose of thinning i n North America generally i s to r e d i s t r i b u t e growth onto, selected r e s i d u a l t r e e s . To t h i s end, i t i s necessary to l e a r n about the r e l a t i o n s h i p e x i s t i n g between growth and development, and stand density. While the best way to study stand . development would be to observe and record changes i n the selected stands from year one t o maturity or harvest, t h i s method i s generally considered too expensive and.at the same time very slow i n y i e l d i n g r e s u l t s . Rather than t h i s , studies are conducted i n stands of d i f f e r e n t 31 ages representing stages of development and pieced together according to some selected c r i t e r i a , e.g., age, s i t e , e t c . This introduces a source of error i n these studies, as regardless of how c a r e f u l t h i s p i e c i n g together may be, there i s no guarantee that trends thus obtained indeed represent a c t u a l development of any r e a l stand. Stand density may be considered as a continuous v a r i a b l e ( e s p e c i a l l y when expressed i n terms of basal area or volume), and as such i t can take on an i n f i n i t e l y large number of values under d i f f e r -ent stand conditions. As stand density influences are gene r a l l y studied together with age and s i t e e f f e c t s , the number of d i f f e r e n t stand condition combinations of p o t e n t i a l i n t e r e s t to fo r e s t management can be very l a r g e . Thinning and growth studies i n general, therefore only involve a few d i s c r e t e classes of density conditions and interpolate and extrapolate f o r conditions not covered. The saving i n experimental cost and time i n many cases outweights the gain i n information which i s achieved by increasing the number of stand condition combinations studied, unless there i s a corresponding improvement i n the method of analysis used f o r evaluation. For the purpose of t h i s d i s s e r t a t i o n , two categories of t h i n -nings may be distinguished: ( l ) only one thinning i s c a r r i e d out during the l i f e of the stand, (2) more than one thinning i s c a r r i e d out, with thinnings taking place at regular i n t e r v a l s . Two types of thinnings are generally d i s t i n g u i s h e d regarding i t s timing ( t h i s r e l a t e s to stand age and tree s i z e ) : ( l ) precommerical thinning and (2) commercial thinning - only the l a t t e r produce commerical material at the time of treatment. 32 For the present a n a l y s i s , thinning r e s u l t s r e l a t i n g to tree growth are most important. Thus i n reviewing thinning studies, the following aspects of growth received p a r t i c u l a r attention: (1) response to release i n diameter growth of various tree sizes (corresponding to tree or crown classes) i n a stand; ( 2 ) a s s o c i a t i o n of release growth response with stand age, i . e . , over what age aspen trees growing i n dense stands lose t h e i r a b i l i t y to respond to release; ( 3 ) a s s o c i a t i o n of release growth response with stand density p r i o r to thinning, i . e . , whether and how g r e a t l y tree competitive status, absolute and r e l a t i v e , a f f e c t s the t r e e s ' a b i l i t y to respond to release; (4) the e f f e c t of s i t e q u a l i t y on response; (5) release response i n height growth; (6) undesirable thinning e f f e c t s ; e.g., shock, sun s c a l d , disease. A l l experimental thinning r e s u l t s on aspen a v a i l a b l e up to 1966 from several studies i n Manitoba and Saskatchewan have been pooled and analyzed by Steneker and J a r v i s (1966). The age of these stands v a r i e d between 11 and 45 years at f i r s t t hinning. Some stands were thinned only once, others several times at prescribed i n t e r v a l s . The i n t e n s i t y of thinning v a r i e d from l i g h t to heavy. Thinning removed mainly suppressed and intermediate trees, also trees having undesirable c h a r a c t e r i s t i c s and tree species other than aspen. Generally, stands studied i n Manitoba and Saskatchewan were growing on better than medium s i t e s f o r t h i s area, judging from the average dominant height of aspen trees on the p l o t s compared to Manitoba and Saskatchewan s i t e index curves. 33 In Canada, B i c k e r s t a f f ( 1 9 ^ 6 ) was the f i r s t to p u b l i s h r e s u l t s on the e f f e c t of thinning upon the growth and y i e l d of aspen stands. His studies were conducted at Petawawa, Ontario. Stand ages v a r i e d from 17 to kO years at the time of thinning. Only one thinning operation was conducted i n each stand, to remove between 70 and 80% of the o r i g i n a l stand. Site conditions sampled ranged from poorer than medium to good. Zehngraff ( 1 9 ^ 9 ) who reported on a s e r i e s of thinning exper-iments conducted i n Minnesota suggested that h i s r e s u l t s were generally applicable f o r aspen a l l across the Lake States. His experimental stands at the time of f i r s t thinning ranged i n ages from 1 3 to 31 years. A l l the stands studied were growing on good s i t e s , with s i t e index 70 or over (at 50 years). The k i n d and i n t e n s i t y of thinning v a r i e d according to the f i n a l product desired: from 5 by 5 f t to 10 by 11 f t spacing i n a 2 0-year-old; and a combination of thinning from above and below with 100 to 300 overstory crop trees i n a 3 1-year-old stand. The r e s u l t s of another experiment i n Minnesota (Heinselman 1 9 5 2 ) were reported a f t e r 13 years of growth following crown thinning (from above) i n 3 0-year-old aspen stands of medium s i t e q u a l i t y . A l l of the above mentioned studies reported --or implied i f the analysis was done on a stand basis some p o s i t i v e response to release i n diameter increment. Although a l l tree s i z e classes responded, the greatest per cent increment response was among the smaller trees ( B i c k e r s t a f f 1 9 ^ 6 ; Steneker and J a r v i s 1 9 6 6 ) , which obviously were the ones most a f f e c t e d by crowding or competition and had very low growth rates before thinning. In absolute terms, however, the l a r g e r trees maintained s u b s t a n t i a l l y higher growth rates than the smaller ones 34 although per cent response may not have been very high. Thus i t i s not s u r p r i s i n g that crown thinnings, which remove the larger trees from the stand, generally r e s u l t e d i n decreased basal area and volume increment i n aspen (Zehngraff 1 9 4 9 ; Heinselman 1 9 5 2 ) . Figure 3 . 3 shows release response by d.b.h. classes to d i f f e r e n t i n t e n s i t i e s of low thinning i n a 2 3-year-old stand growing on good s i t e i n Manitoba. Steneker and J a r v i s ( I 9 6 6 ) also found, from stem a n a l y s i s , that i n a 1 9-year-old stand growing on medium q u a l i t y s i t e s i n Manitoba, r a d i a l increment of dominant and co-dominant trees increased immediately ( i . e . the f i r s t growing season) following thinning; the maximum r a d i a l increment was reached i n the t h i r d year a f t e r thinning, and the value of t h i s increment was proportional to the degree of release. Unfortunately, none of the studies reviewed analyzed the r e l a t i o n s h i p between age and response to release i n trees of d i f f e r e n t sizes i n thinned stands. I t i s reasonable to expect that trees of i n t o l e r a n t species growing i n a dense stand lose t h e i r a b i l i t y to respond to release a f t e r they pass a c e r t a i n stage of development. This may be p a r t i c u l a r l y pronounced f o r trees i n the intermediate and suppressed crown c l a s s e s . Steneker and J a r v i s ( 1 9 6 6 ) presented i n -d i r e c t evidence, i n the form of p e r i o d i c annual basal area increment of the 200 l a r g e s t trees per acre p l o t t e d over r e s i d u a l basal area, which suggested that post thinning increment of the l a r g e s t r e s i d u a l trees, at l e a s t , i s independent of age; while the increment of the same trees showed a decline with increasing r e s i d u a l basal area. To further illuminate the e f f e c t of age on growth response, appropriate data were taken from the Manitoba-Saskatchewan aspen thinning studies (described by Steneker and J a r v i s 1966 ) and analyzed 35 u E a> o 2.0 J C g 1 0 I C |2 Treatment In 1950 - 3 . 0 — Control • •Thinned to 8'x8' spacing • — " " 10,'xlO' " " " 12x12 " Residual basal area/acre after thinning, 1950 (sq.ft.) Number of Trees 2 3 4 D .b .h . c l a s s e s ( inches) in 1 9 5 0 FIGURE 3.3 Ten year d.b.h. increment (1950-1960) by d.b.h. classes and treatment. From Steneker and J a r v i s 1966; study 5 (" stand 23-year-old at thinning. 36 us ing r e g r e s s i o n methods. The necessary t ree growth data were ob ta ined from a th inned and a check p l o t f o r f ou r age g roups . The th inned p l o t s were s e l e c t e d to represen t approx imate ly the same degree o f r e l e a s e , so as to l e s s e n r e l e a s e response v a r i a t i o n from the s o u r c e . I t would have been more d e s i r a b l e a l s o to compare the growth of the same t r ee before and a f t e r t h i n n i n g r a t h e r than u s i n g check p l o t t r ees f o r comparison as i t i s p o s s i b l e tha t some inheren t d i f f e r e n c e s i n t r e e growth e x i s t e d between t r ees on th inned and on check p l o t s d e s p i t e the h i g h degree o f u n i f o r m i t y i n s tand and s i t e c o n d i t i o n s a t p l o t e s t a b -l i shment . Diameter increment data from fou r s t u d i e s , hav ing s tand ages o f 1 1 , 14, 19 and 45 years a t the t ime o f f i r s t t h i n n i n g , were ana l yzed by r e g r e s s i o n t e c h n i q u e s . Annual p e r i o d i c d iameter increment was c a l c u l a t e d f o r each sample t r ee e i t h e r from 5 - , 9"> ° r 10-year d iameter increment v a l u e s , depending on the t ime of f i r s t remeasurement. Con-densed r e s u l t s o f the r e g r e s s i o n ana lyses are p resen ted i n Table 3 - l 5 i n c l u d i n g on l y the s i g n i f i c a n t (a t l e a s t a t the 0.05 p r o b a b i l i t y l e v e l ) independent te rms. The r e l a t i o n s h i p s were p l o t t e d (F igu re 3 .4) f rom increments es t imated by s o l v i n g these r e g r e s s i o n s f o r s p e c i f i c d . b . h . va lues (between minimum and maximum d . b . h . ) . Trends shown i n F i g u r e 3.4 are i n agreement w i t h those i n F i g u r e 3.3 and seem to s u b s t a n t i a t e the same c o n c l u s i o n s . F i gu re 3.4 shows p o s i t i v e response to r e l e a s e i n d iameter increment f o r a l l s i z e c l a s s e s , w i t h g r e a t e s t per cent increment response among the s m a l l e r t r e e s . As expec ted , d iameter increment g e n e r a l l y d e c l i n e s w i t h age . The f o l l o w i n g p r e l i m i n a r y conc lus ions may be drawn rega rd ing a p o s s i b l e r e l a t i o n s h i p between age and response to r e l e a s e i n t r e e s o f d i f f e r e n t TABLE Annual d.b.h. increment (inches) over d.b.h. regressions f o r aspen stands aged 11 to 45 years at treatment (50 observations per p l o t ) . Age Group Growth Period • (yrs) Treatment a Indep. variable D Regression coe s and t h e i r s i g n i f D 2 f f i c i e n t s and F-ra b F 2 2 icance E t i o s 3 F 3 R or r S.E. of Est. ( i n / y r ) 11 5 Control .004 .0248 ^ 9 .712 .044 Thinned - . 1 6 4 • 3225 - . 0 4 7 2 4 . 7 - - . 850 .050 l 4 6 Control - . 0 8 3 - - .1087 33 - . 0 2 8 5 21 .818 .030 Thinned .111 - - - - .0062 30 .619 .034 19 10 Control . 828 - . 9 7 2 9 11 • 3733 13 - . 0 4 3 3 14 .849 .027 Thinned .084 - - . 0080 22 - - .564 .037 45 9 Control . 020 _ _ _ .0001 12 .446 .019 Thinned - . 0 3 4 .0073 7 . 2 - . 0 0 0 7 4 . 1 .652 .024 00 - 0 Annual db.h. increment (inches) r - l O CO e ft> i—1 < ft) r-1 O c co ft) 0Q rt> CO ft) 0 ft o ft) f-( ro b fl) B m 0 cn i rt (D &• 0 • (6 a* ft> i-i H-1 ft) C-H rt ft) H-n o <! O H" CO CO o H-* Hi ON ft) ON co • CD 0 o 0 b- b 3 o rr a> co is «a O H-0 0 CD a-ft) 0 O 0* fl> o M o rr CO CO b O oo b ro o ~T~ O J o -r-O T O H O H O - t p o rr o rr O rro 5*2. 5*3. 5 3-rj ^  n ^  s -, a> o a> o <r> o a.— Q.- o.— rr rs i a> o cn to (D (O •53 CD o 0 1 0 1 " "9 . 0 ) A N co =1 « CO * l Q 2. -x cr < « </>" -» CO CD cn 9£ 39 s i z e s : ( l ) there may be good response to release even i n 4 5-year-old aspen stands, ( 2 ) the greatest absolute and r e l a t i v e increase i n d i -ameter growth rate i s among the intermediate s i z e trees, i . e . , co-dominants and some intermediates (these are the trees which are inc r e a s i n g l y a f f e c t e d by competition, but are not yet "over the hump"), and ( 3 ) the smaller trees i n older stands may show a l i m i t e d response i n d.b.h. increment a f t e r thinning, but the main e f f e c t of thinning i s probably a prolonged dying p e r i o d f o r these t r e e s . None of the reviewed thinning studies presented information on the e f f e c t of a c t u a l stand density, or degree of crowding, p r i o r to thinning, on the trees' a b i l i t y to respond to release. Again, the a b i l i t y to respond may vary by tree status (described by tree s i z e ) within the same stand, the suppressed trees having the l e a s t capacity to respond and the l a r g e r dominant trees the greatest. This e f f e c t i s l i k e l y to have a strong i n t e r r e l a t i o n s h i p with age, with older sup-pressed trees having much l e s s a b i l i t y to respond than s i m i l a r trees at a younger age. The thinning studies reviewed contained no information on the e f f e c t of s i t e q u a l i t y differences on release response i n i n d i v i d u a l t r e e s . This i s p a r t l y because past thinning studies were intended mainly to obtain stand information with no emphasis on i n d i v i d u a l tree growth, and also because thinning studies i n a region were generally concentrated on one or two of the most extensive s i t e types which looked promising f o r thinning from a growth and y i e l d viewpoint. Stand data from these studies suggest s u b s t a n t i a l release response i n trees growing on good s i t e s (e.g., B i c k e r s t a f f 1 9 4 6 ; Steneker and J a r v i s 1 9 6 6 ; and p a r t i c u l a r l y Zehngraff 1 9 4 9 ) , and a much 4o smaller response i n trees on poorer s i t e s , Because growth rate of trees i n f u l l y stocked stands of s i m i l a r density i s a function of s i t e q u a l i t y at a given age, i t seems that p o t e n t i a l release response of trees i s more or l e s s d i r e c t l y r e l a t e d to t h e i r current growth r a t e . Much l e s s r e l i a b l e information i s a v a i l a b l e on the e f f e c t of release on height increment. This i s p a r t l y because ( l ) height, par-t i c u l a r l y height increment, i s much more d i f f i c u l t to determine accurately than diameter increment, and (2) height increment generally i s much l e s s a f f e c t e d by stand density v a r i a t i o n than diameter i n c r e -ment. Thus l e s s e f f o r t has been expended f o r determining height increment. Two of the aspen thinning reports reviewed ( B i c k e r s t a f f 1946; Steneker and J a r v i s 1966), which d i d contain height growth i n f o r -mation, stated that height increment of the dominant stand, at l e a s t , d i d not appear to be s i g n i f i c a n t l y a f f e c t e d by thinning. Unfortunately, no data are presented on height growth of the r e s t of the stand. Although thinning stimulated diameter growth of i n d i v i d u a l aspen trees, stand increment and t o t a l volume y i e l d generally were not increased by thinning. On the contrary, heavier low thinning and any kind of crown thinning r e s u l t e d i n lower t o t a l volume y i e l d ( B i c k e r s t a f f 1946; Heinselman 1952; Steneker and J a r v i s 1966). Similar t o t a l volume y i e l d s were achieved, however, a f t e r a decade or more of growth following low thinning i n young stands to spacings up to 10 by 10 f t on good or the best aspen s i t e s i n Manitoba (Steneker and J a r v i s 1966), Ontario ( B i c k e r s t a f f 1946) and Minnesota (Zehngraff 1949). For these same conditions, merchantable volumes ca l c u l a t e d f o r s p e c i f i c diameter l i m i t s of u t i l i z a t i o n were considerably higher, because of the greater s i z e of i n d i v i d u a l t r e e s . hi B r i e f l y , a s u b s t a n t i a l amount of tree growth response information i n r e l a t i o n to release by thinning i s a v a i l a b l e at present, but most of the information i s fragmented and much of i t i s buried i n f i l e s . Some of the d i f f i c u l t i e s i n these studies may be a t t r i b u t e d to the tremendous v a r i e t y of stand conditions and r e l a t e d development trends that a r i s e from differences i n age, stand density and s i t e , not mentioning stand and clone s t r u c t u r e . Past thinning studies, t r y i n g to cut down the problem i n t o manageable s i z e , concentrated on only a few classes of conditions, and presented thinning r e s u l t s mainly i n terms of d e s c r i p t i v e stand averages, rather than t r y i n g to define and explain basic tree growth r e l a t i o n s h i p s . As f o r e s t growth a c t u a l l y occurs i n the t r e e , i t seems obvious that causal growth processes should be studied on a t r e e , rather than on a stand b a s i s . Studying f o r e s t growth thus, not only seems more f e a s i b l e and rewarding, but i t should also be the basis of understanding and explaining stand growth and development. However, meaningful synthesis of tree growth information into stand growth and development processes i s a very complex problem, and only recently -- with the advent of high speed, large storage capac-i t y computers -- could t h i s problem be tackled. The main purpose of t h i s work i s to attempt such synthesis through the development of a mathematical growth model f o r aspen. Undesirable thinning e f f e c t s do not seem to be major problems i n aspen thinning. Sun scald i s probably the most serious i n j u r y associated with opening up dense aspen stands, and as much as 15% of a l l r e s i d u a l trees can be a f f e c t e d a f t e r a heavy thinning ( B i c k e r s t a f f 1946). Even i f such i n j u r y causes no d i r e c t tree mortality, i t creates an entry f o r i n f e c t i o n by various micro-organisms. Thus sun s c a l d and various 1+2 mechanical i n j u r y from thinning operations, e.g., axe and f e l l i n g wounds, logging wounds, etc., may w e l l be the cause of apparently greater incidence of Hypoxylon canker i n thinned aspen stands ( B i c k e r s t a f f I9I+6; Anderson and Anderson 1968) -- better tree v i g o r and f a s t e r growth, notwithstanding. Anderson and Anderson (1968) suggested however, that the higher incidence of canker i n thinned stands i s probably due to environmental f a c t o r s . Along with tree growth, mortality i s the other most important component of stand growth and development. Although most thinning and growth studies conclude that competition mortality i s concentrated among the smaller, slow growing trees, studies reviewed here d i d not present any d e t a i l e d quantitative information. Two kinds of quantitative information as to mortality i s needed most f o r the present study: ( l ) frequency d i s t r i b u t i o n of increment of dying trees i n the period they a c t u a l l y succumb, and (2) frequency d i s t r i b u t i o n of dead trees by d.b.h. i n r e l a t i o n to that of surviving trees i n the begin-ning of the period. Such frequency d i s t r i b u t i o n s are presented i n Figure 3«55 which were based on appropriate remeasurement data from two, 1/5 acre, undisturbed, aspen permanent sample p l o t s , f o r an increment p e r i o d of 25 to 30 years. Figure 3-5 shows that i n the studied stand, between ages 25 to 30 years, about 75$ of the trees that died i n that period had p r a c t i c a l l y n i l d.b.h. increment i n the same period; and that the maximum increment before death was 0.3 inches. Considering d.b.h., on the other hand, about 75$ of a l l dead trees had smaller d.b.h, than about 20$ of the smallest surviving trees i n the stand, while about 90$ of a l l dead trees had d.b.h. under the average d.b.h. of s u r v i v a l s . 43 lOO—i u § 5 0 -a-Mortality Surviving V///JWA lOO-i 50-r n r 0 .1 .2 .3 D.b.h. incr. (inches) Sample plot 6 mil t i i c q 4.0 5.0 6.0 D.b.h. (inches) lOO-i >» o c _ _ . © 5 0 -cr l O O - i 5 0 -i • i - t r l T 0 .1 .2 .3 D.b.h. incr. (inches) 1.0 J m • i i « i Z0 3.0 4.0 D.b.h. (inches) Sample plot II 5.0 FIGURE 3.5 Frequency diagrams of d.b.h. increment and that of d.b.h. of dead trees i n the period they succumb, and d.b.h. frequency diagrams of surviving trees. Two sample p l o t s ; increment period from 25 to 30 years. (Data from Steneker and J a r v i s 1966, study 2.) 6.0 A common view was expressed i n these thinning reports that only young, vigorously growing aspen trees under 30 years of age --even safer under 20 years -- w i l l respond s i g n i f i c a n t l y to release. So the f i r s t thinning should be done at such ages, provided thinning i s also economically f e a s i b l e . It i s now generally accepted f o r most forest tree species that i f thinning i s to be c a r r i e d out to improve growth, i t should be conducted while the stand i s young and f a s t growing, so i n d i v i d u a l crowns can enlarge - - a p r e r e q u i s i t e to main-t a i n i n g increased growth. This i s p a r t i c u l a r l y important considering i n t o l e r a n t species l i k e aspen, i n d i v i d u a l s of which quickly lose t h e i r a b i l i t y to respond to release. Even at a r e l a t i v e l y young age, however, only the f a s t e s t growing aspen trees -- the dominants and the co-dominants -- w i l l have s u f f i c i e n t l y high growth rate a f t e r thinning to compensate, at l e a s t p a r t i a l l y , f o r the l o s s i n growth of removed i n d i v i d u a l s . Crown thinning, which removes the l a r g e r trees from the stand, w i l l r e s u l t i n decreased growth and lower y i e l d ; therefore, such p r a c t i c e i s d e f i n i t e l y undesirable i n aspen stands. It would not be r e a l i s t i c to discuss frequency of thinnings (the length of time i n t e r v a l between thinning operations), as i t i s extremely u n l i k e l y that aspen stands i n the P r a i r i e Provinces w i l l be thinned more than once on a p r a c t i c a l basis . Whether, and what kind of thinning w i l l be used i n aspen growing depends u l t i m a t e l y on the aim of management. This question of thinning i s c l o s e l y r e l a t e d to: ( l ) the s i z e of stems needed to be produced, (2) the length of r o t a t i o n , or the time a v a i l a b l e f o r growing a f o r e s t crop, and (3) the cost of thinning, or the investment needed for conducting the thinning operation. These are just some of the more important points to be considered, and the f i n a l decision on thinning i n a s p e c i f i c area has to be based on a d e t a i l e d cost -benefit a n a l y s i s . In some s i t u a t i o n s , however, the decision i s r e l a -t i v e l y simple. Let us consider, f o r instance, thinning aspen i n the Boreal Mixedwood Forests of the P r a i r i e Provinces. In t h i s area tree growth i s r e l a t i v e l y slow, at the same time, very large areas are a v a i l a b l e f o r wood production. Under these circumstances i t i s very l i k e l y that thinning w i l l not be done because only extensive forest management w i l l be p r a c t i c e d f o r the purpose of producing wood f i b r e f o r pulp manufacture, ( i t has been pointed out i n Chapter I that t h i s segment of the forest economy has been, and i s expected to be, expand-ing most r a p i d l y i n the foreseeable future.) I t i s also reasonable to expect that wood harvesting and u t i l i z a t i o n w i l l be geared to smaller tree sizes than at the present time, thus the cost advantage of large tree size should have l e s s economic importance. With the exception of stands growing on the best s i t e s , thinning aspen i n the Boreal Mixed-wood Forest generally r e s u l t i n decreased f i n a l volume y i e l d , consid-ering close u t i l i z a t i o n f o r pulp manufacture. Thus i t seems safe to conclude that n a t u r a l aspen stands i n t h i s area w i l l not be thinned. Nevertheless, the r e s u l t s of past thinning studies are extremely valuable i n studies of tree growth i n r e l a t i o n to stand density, and i n various stand development studies. 46 CHAPER 4. APPROACHES IN STAND DEVELOPMENT STUDIES: SYSTEMS MODELING AND SIMULATION 4.1 A review of approaches i n stand development and other population  dynamic studies Perhaps on no other aspect of f o r e s t r y has been gathered more data than on growth, y i e l d and stand development. Enormous amounts of such data are a v a i l a b l e i n Europe from sample p l o t measure-ments and remeasurements, and a great deal of s i m i l a r data are being gathered i n North America. Regardless of how (e.g., i n d i v i d u a l tree measurement records, only stand tables, etc.) the growth data have been obtained i n these studies the analyses were generally conducted i n terms of tree and stand averages. While these average trends were us e f u l to forest managers to provide answers to s p e c i f i c questions, they mainly served as stopgap measures and d i d not n e c e s s a r i l y improve gen-e r a l understanding of actu a l tree growth and stand development processes. In f a c t i n some cases, having such averages might have hindered progress i n growth studies, as such averages tended to obscure i n t r i c a t e cause-e f f e c t r e l a t i o n s h i p s between tree growth and i t s c o n t r o l l i n g f a c t o r s . Furthermore, a general comparison and evaluation of the r e s u l t s of these studies i s made d i f f i c u l t by t h e i r fragmented nature, and differences i n t h e i r methods of data coLlexrb.ionand a n a l y s i s . Not unjustly, Czarnowski (1961) r e f e r r e d to past reports on growth and y i e l d information as a "cemetery of numbers". It i s i n t e r e s t i n g to note that i n most of these studies, tree growth analysis was generally an appendage, rather than an i n t e g r a l part of the i n v e s t i g a t i o n of stand growth and development. On the other hand, numerous studies were conducted only to study tree 47 growth i n r e l a t i o n to competition or density e f f e c t s from surrounding trees, with the e f f e c t s expressed e i t h e r i n terms of the subject t r e e s ' r e l a t i v e crown s i z e , or some measure of stand density of the surrounding t r e e s . Comprehensive recent reviews of crown size and other stand density measures were given by Curtin ( 1 9 6 8 ) and Osborn ( 1 9 6 8 ) . The ins i g h t s about tree growth provided by these studies, however, could not be f u l l y u t i l i z e d because u n t i l very recently no methods were av a i l a b l e to synthesize information of t h i s nature. Smith ( 1 9 6 4 ) summed up the s i t u a t i o n t h i s way: "Growth i n diameter, height, and form of many i n d i v i d u a l trees has been studied without techniques f o r t r a n s l a t i n g tree data into models of stand development". The recent advances i n t r a n s l a t i n g tree data into d e s c r i p t i o n of stand develop-ment processes involve computer oriented systems modelling and simulation (Newnham 1 9 6 4 ; Newnham and Smith 1 9 6 4 ; Lee 1 9 6 7 ; M i t c h e l l 1 9 6 7 ) . Concurrently with the progress i n tree studies, important advances have been made i n stand growth analysis (Buckman 1 9 6 2 ; Turnbull and Pienaar 1 9 6 6 ; Moser and H a l l 1 9 6 9 ) , again p a r t l y due to the powerful a n a l y t i c a l and computational techniques made pos s i b l e by the advent of large, high-speed e l e c t r o n i c computers. For example, the stand growth functions of Turnbull and Pienaar ( 1 9 6 6 ) f i t t e d to actu a l data provide a b i o l o g i c a l l y meaningful d e s c r i p t i o n of stand development at a c e r t a i n l e v e l of abstraction, considering the stand as a massive organism rather than a complex b i o l o g i c a l system made up of i n d i v i d u a l organisms, i . e . , t r e e s . The ch i e f drawback of t h i s approach i s inherent i n i t s philosophy i n considering f o r e s t stands as massive organisms. As actu a l growth processes occur at the i n d i v i d u a l (tree) k& l e v e l , i t seems that t h i s i s where cause-effect r e l a t i o n s h i p s between growth and i t s c o n t r o l l i n g factors may best be studied. For t h i s reason, functions based on stand growth information can only be con-sidered d e s c r i p t i v e rather than explanatory mathematical structures. Somewhat analogous s i t u a t i o n s p r e v a i l e d i n other f i e l d s of population dynamics, or population ecology. H o l l i n g (1968), Morris (1968) and Watt (1968) gave good summaries of the various approaches used mainly i n entomology to describe changes i n insect populations . H o l l i n g (1968) discussed the dilemma such population studies were faced with because of the inadequacies of the concepts and the techniques a v a i l a b l e f or a n a l y s i s . As he pointed out, population ecologists e i t h e r r e s t r i c t e d themselves i n t h e i r studies to s p e c i f i c c h a r a c t e r i s t i c s of the population, or worked with simple populations i n the laboratory, i n order to r e t a i n p r e c i s i o n while s a c r i f i c i n g g e n e r a l i t y . A l t e r n a -t i v e l y , they sought broad generalizations on complex populations and s a c r i f i c e d p r e c i s i o n . H o l l i n g c a l l e d the adherents of the l a t t e r philosphy " s t r a t e g i s t " , who were generally more successful i n t h e i r aims than the " t a c t i c i a n s " , because the t a c t i c i a n s were faced with the h i g h l y complex t a c t i c a l s i t u a t i o n s of population ecology which just d i d not lend themselves to methods of analysis and synthesis that were a v a i l a b l e and generally used i n these studies. The root of the problem i n analyzing population systems i s t h e i r tremendous complexity. The most important features of such systems - - l i k e i n t e r a c t i o n s between various components, s t r u c t u r a l and s p a t i a l c h a r a c t e r i s t i c s , thresholds and timelags --cannot be handled by s t a t i s t i c a l techniques and are also beyond the realm of formal mathematical a n a l y s i s . As of today, only computer oriented h9 quantitative systems modeling can provide the complexity required f o r r e a l i s t i c representation and analysis of population systems. h.2 Systems models and simulation Models are as o l d as human h i s t o r y . Abstract v i s u a l models were the basis of geometry. P h y s i c a l models of various kinds have been used i n architecture and engineering. Mathematical models representing objects or processes are probably the most abstract, and i n some ways, the most f l e x i b l e models. Working with them may not involve anything more than a p e n c i l and a piece of paper. In t h i s study, mathematical models have d i r e c t importance, and they are under-stood to mean a set of mathematical functions representing the compo-nents, subcomponents and elements of the studied system and t h e i r i n t e r a c t i o n s . The main common c h a r a c t e r i s t i c of a l l models i s that they imitate or represent things and processes of the r e a l world. They are abstractions, representing or describing only the e s s e n t i a l features of the phenomenon. In f a c t , "too" r e a l i s t i c models, p a r t i c u l a r l y mathematical ones, would be excessively complicated and completely i n t r a c t a b l e . Simulation, or more p r e c i s e l y computer simulation, i s more d i f f i c u l t to define and i t has several important r o l e s . As a model, simulation also means representation of things and processes of the r e a l world. In simulation, however, the verb form (to simulate ', or emulate a c t i v i t y over time), should be emphasized. In that sense, simulation provides another dimension to mathematical models, a t t r i b u t i n g to them dynamic p r o p e r t i e s . 50 Mathematical systems models of d i f f e r e n t complexity have been developed i n a v a r i e t y of d i s c i p l i n e s , from the behavioral sciences to population ecology and f o r e s t r y . Lee ( 1 9 6 7 ) presented a review of the d i f f e r e n t kinds of systems models a v a i l a b l e up to 1 9 6 7 . The most s t r i k i n g common c h a r a c t e r i s t i c s of these models are t h e i r complexity. Also, considerable s i m i l a r i t y seem to e x i s t i n the concepts of model bu i l d i n g , regardless of the kind and nature of the modeled system. Model building' generally begins by i d e n t i f y i n g the e s s e n t i a l components of the system and t h e i r underlying s t r u c t u r e . This can be best represented on a flow chart. This basic model i s then enlarged, or r e f i n e d , by incorporating subcomponents. Concurrently, each of these components are tr a n s l a t e d into a language (e.g., FORTRAN) acceptable by computer. These segments of programs or "subroutines" then are pieced together (interfaced) to obtain the complete system model. This modular model b u i l d i n g approach not only c l o s e l y corre-sponds to the organization of the modeled system, but i t also enor-mously f a c i l i t a t e s the computer programming task. I t should be noted at t h i s point that the ro l e of computer i n t h i s type of systems model i s not only that of a very f a s t c a l c u l a t i n g machine, but perhaps j u s t as important i s i t s r o l e i n representing an o v e r a l l system structure i n the form of the stored program. H o l l i n g ( 1 9 6 6 ; 1968 ) and Watt ( 1 9 6 8 ) presented the concepts and strategy of bu i l d i n g models of complex b i o l o g i c a l systems and also discussed the various features of the FORTRAN language which make i t p a r t i c u l a r l y s u i t a b l e f o r modeling such systems. 51 4 .3 A review and c r i t i q u e of forest stand growth simulation models and  suggestions for improvement The f i r s t f o r e s t stand growth simulation models were developed by Newnham ( 1 9 6 4 ) , and Newnham and Smith ( 1 9 6 4 ) . These models described the growth of the forest stand on an i n d i v i d u a l tree b a s i s . They were developed to represent the growth of Douglas f i r and lodgepole pine plantations, with spacings 3 - 3 by 3 - 3 , 6 . 6 by 6 . 6 , 9 S by 9 - 9 , a n d 1 3 . 2 by 1 3 . 2 f t . Tree locations i n the hypothetical stand were represented i n the computer by a 15 by 15 matrix ( i . e . , 225 t r e e s ) , thus the size of the hypothetical p l o t f or the four spacings v a r i e d from a minimum 4 9 - 5 by 4 9 . 5 f t . to 198.O by 198.O f t . These models were based on two main assumptions: ( l ) a tree w i l l maintain a rate of diameter growth s i m i l a r to an open growing tree as long as i t i s free from competition from surrounding trees, and ( 2 ) a f t e r competition sets i n , diameter increment i s reduced by an amount proportional to the amount of competition, u n t i l the diameter i n c r e -ment f a l l s below a c e r t a i n minimum value and then the tree i s assumed to have died. Growth was simulated with the model f o r f i v e year i n t e r v a l s from age 10 to 100 years (or any r o t a t i o n age des i r e d ) . Lee ( 1 9 6 7 ) improved the model f o r lodgepole pine by enlarging the i n i t i a l tree matrix to 30 by 30 (900 trees) and by streamlining the computer program. The model, i n the form of a computer program, contains a set of equations f o r p r e d i c t i n g p e r i o d i c diameter increment, f o r estimating crown width, d.b.h., height and volume of every t r e e . For each simulation run an appropriate set of regression c o e f f i c i e n t s i s read into the computer. The four basic equations are: 52 (1) Crown width/d.b.h. equation f o r open growing trees; (2) D.b.h. increment/age regression, with appropriate adjustment f o r the amount of competition; (3) Height/d.b.h. regression, and (4) Volume estimating regression using d.b.h. and height. Other inputs f o r a simulation run included: d.b.h. of trees i n the i n i t i a l matrix, matrix s p e c i f i c a t i o n s , s t a r t i n g and f i n a l (harvest) age of trees, minimum diameter increment values of surviving trees (DINC) and two "simulation v a r i a b l e s " to reduce tabulated crown width to competitive crown width (REDFAC and REDINC). It i s now f a i r l y w e l l accepted that -•- i n a pure, even-aged stand growing on f a i r l y uniform s i t e and made up of g e n e t i c a l l y s i m i l a r i n d i v i d u a l s -- the greatest amount of v a r i a t i o n i n tree d i -ameter growth i s r e l a t e d to i n t e r t r e e competition e f f e c t s , and these e f f e c t s are also the most important causes of d i r e c t or i n d i r e c t tree m o r t a l i t y . Thus the evaluation of i n t e r t r e e competition i s of c r u c i a l s i g n i f i c a n c e i n stand growth simulation models. Various crown measures and r a t i o s (e.g., Crown Width/D.b.h.; Maximum C.W./Actual C.W.; T o t a l Height/C. Length) cannot be used e f f e c t i v e l y as these themselves are the r e s u l t , and not the cause, of past competitive i n f l u e n c e s . Newnham used a competition index c a l c u l a t e d from the angle summation of poten-t i a l (open-growing) crown overlaps ( F i g . 4.1). He assumed that each tree i n h i s hypothetical stand has a competition c i r c l e p r oportional to i t s d.b.h., and i s equal to the crown area of that tree as i f i t were open grown. Then competition index of a tree was a measure of the proportion of the circumference of i t s competition c i r c l e occupied by the c i r c l e s of surrounding competitors. While, the index appears b i o l o g i c a l l y reasonable 53 Competition circles CI = 2 T T FIGURE 4.1 Newnham's (1964) Competition Index: po t e n t i a l crown overlaps. angle summation of and also i s easy to compute, i t s effectiveness i n accounting f o r competition r e l a t e d tree growth v a r i a t i o n i n Douglas f i r and lodgepole pine cannot be evaluated d i r e c t l y as no actu a l tests were conducted f o r these species. Newnham (1964) and Lee (1967) showed that a f t e r a few c a l i b r a t i o n runs, combined with appropriate manipulation of simu-l a t i o n variables REDFAC and REDINC, the output obtained by the simulation model was i n reasonably close agreement with published y i e l d t a b l e s . While recognizing the merits of Newnham's model i n pioneering a new approach to describe stand development i n terms of tree growth, i t must also be recognized that t h i s was but the i n i t i a l 5h step i n a new l i n e of studies which have almost unlimited scope for expansion. Further development i s required i n concepts and approaches of stand growth model building and improvements and refinements also are needed i n s p e c i f i c features of the model. Holling's ( 1 9 6 6 ; 1968 ) "experimental component a n a l y s i s " approach may be r e a d i l y adaptable to stand growth modeling. More w i l l be s a i d about t h i s i n the next chapter. Improvements are needed i n evaluating competition e f f e c t s between trees, i n representing s p a t i a l tree d i s t r i b u t i o n s , i n generating or simulating growth of i n d i v i d u a l trees by basing the model on height growth rather than on diameter growth, and introducing stochastic v a r i a b l e s where appropriate. It was mentioned e a r l i e r that the evaluation of i n t e r t r e e competition e f f e c t s i s the most c r u c i a l part of stand growth modeling. Therefore, the whole concept of evaluating i n t e r t r e e competition needs to be c r i t i c a l l y reviewed and tested using actual tree growth data. As a r e s u l t of such work a better understanding of competition should emerge that would lead to the development of a more r e a l i s t i c competition index or model. Permanent sample p l o t records, f o r the required species and conditions ( s i t e s , ages, etc.) which contain tree growth data along with tree maps, provide information i d e a l l y suited f o r analyzing competition e f f e c t s . Newnham ( 1 9 6 4 ) used only a regular g r i d (a square tree matrix) to describe tree l o c a t i o n s , which s u i t a b l y represents the s p a t i a l pattern of p l a n t a t i o n s . However, natural stands have i r r e g u l a r s p a t i a l pattern and representing them with a regular one i n the model, could r e s u l t i n a p o s i t i v e growth b i a s . It i s now generally accepted that 55 r e g u l a r l y spaced p l a n t a t i o n show better development than i r r e g u l a r n a t u r a l stands. Therefore, models developed for n a t u r a l stands would need to have a greater f l e x i b i l i t y i n representing tree s p a t i a l patterns, e.g., random and clumped patterns, as w e l l as a v a r i e t y of t r a n s i t i o n a l ones (random tending to uniform, e t c . ) . This can be most e a s i l y accomplished by using orthogonal tree c o o r d i n a t e s . With such a system, i t would be p o s s i b l e to "read i n " f o r simulations a c t u a l tree sizes and locati o n s obtained from sample p l o t s , whose growth records were also a v a i l a b l e from remeasurements. The comparison of a c t u a l growth trends from p l o t s with simulated trends would be a good method of model evaluation. Using tree height growth as a basis of stand growth simu-l a t i o n may be more suitable f o r some species than using diameter growth, because the former i s generally much more stable and p r e d i c t a b l e . This i s p a r t i c u l a r l y so when considering stands of i n t o l e r a n t species l i k e aspen, where over-stocking and stagnation occurs r a r e l y , and height growth shows a remarkable s t a b i l i t y over a wide range of d e n s i t i e s (e.g., see Steneker and J a r v i s 1966). M i t c h e l l (1967), whose work i s reviewed l a t e r , based h i s model on dominant height growth. V a r i a b i l i t y i s a c h a r a c t e r i s t i c feature of i n d i v i d u a l organisms i n b i o l o g i c a l populations. Some of t h i s v a r i a t i o n i n growth and size may be r e a d i l y accounted f o r , but there always remains a c e r t a i n amount of un-accountable v a r i a t i o n . Part of t h i s unaccounted v a r i a t i o n i s due to gen-e t i c a l differences between i n d i v i d u a l s , while the remaining p o r t i o n may stem from inadequacies of current knowledge about growth of organisms and simply random e f f e c t s . In some e c o l o g i c a l systems, unaccounted v a r i a t i o n 56 i s considerable and important, so i t should also be included i n the model as an appropriate random component. That way, i t i s possible to evaluate the e f f e c t and r e l a t i v e importance of such components by simulation. Although M i t c h e l l ( 1 9 6 7 ) included a random v a r i a b l e i n his white spruce stand growth model, he d i d not analyze the e f f e c t of t h i s component. M i t c h e l l ( 1 9 6 7 ) developed a model which simulates the growth of white spruce stands i n terms of crown expansion of i n d i v i d u a l t r e e s . The rate of crown (or branch) expansion within the l i m i t a t i o n s of the size and l o c a t i o n of competing trees, i s dependent on concurrent height growth and may be predicted from a height-age r e l a t i o n s h i p of dominant trees on the p a r t i c u l a r s i t e . The p l o t f o r simulation i s represented i n the memory of computer by a matrix containing an entry for each square p l o t of growing space. Each entry contains a code denoting the number of the tree i f the p a r t i c u l a r area i s occupied. The number of entries occupied by each tree increases as the crowns of i n d i v i d u a l trees expand into vacant growing space. A random v a r i a b l e i s generated to represent unexplained sources of v a r i a t i o n , e.g., heredity. The height, competitive status, suppression and mortality of each tree depend on the average height of dominants and the tree's r e l a t i v e crown width ( i . e . , a c t u a l C.W./maximum C.W. the tree could have attained without competition). Diameter and volume of the bole are estimated from crown area and height. M i t c h e l l compared his model against permanent sample p l o t growth data from white spruce plantations between ages 10 to 45 years and obtained good correspondence between a c t u a l and simulated stand s t a t i s t i c s . 57 This model's main weakness seems to be i n describing tree competitive status. Average dominant height and r e l a t i v e crown width used to describe competitive status more or l e s s amounts to the same thing as using r e l a t i v e tree size d i r e c t l y f o r t h i s purpose. Although M i t c h e l l , l i k e many others, recognized that r e l a t i v e crown siz e i s a good integrated expression of the amount of competition a tree has been subjected to i n the past, he ignored the f a c t that the same crown mea-sure may be useless to describe current competitive status and increment of i n d i v i d u a l trees (e.g., Smith et a l . 196l, Smith 1966a). This means that i n evaluating competitive status M i t c h e l l ' s model does not take into account the i n t e r a c t i o n e f f e c t s which are the r e s u l t of the r e l a t i v e size of a given tree compared to that of i t s competing neigh-bors, as w e l l as the neighbors' distance and l o c a t i o n . In other words, two intermediate trees of the same size (both crown and height) at a given age may have quite d i f f e r e n t competitive status depending on the s i z e and l o c a t i o n of t h e i r surrounding competitors. M i t c h e l l ' s model simulates the growth of white spruce stands i n terms of h o r i z o n t a l crown expansion ( i . e . , branch elongation) of i n d i v i d u a l t r e e s . -The necessary r e l a t i o n s h i p s of branch length to height above the p a r t i c u l a r branch were obtained from d i s s e c t i o n studies of i n d i v i d u a l t r e e s . While such methods are s u i t a b l e f o r conifers with symmetrical crowns, they are obviously l e s s suitable f o r broadleaved trees with asymmetrical crowns. In summary, both of the approaches reviewed constitute a major pioneering step, i n d i f f e r e n t ways, to stand growth simulation models. For developing an aspen stand growth model with s u f f i c i e n t g e n e r a l i t y that the basic model would be e a s i l y adaptable to most native f o r e s t tree species, Newnham's approach seems to have more of the desired f l e x i b i l i t y . So i n developing the present model, some of the concepts o r i g i n a l l y advanced by Newnham ( 1 9 6 4 ) and Newnham and Smith ( 1 9 6 4 ) were u t i l i z e d and expanded along the l i n e s suggested i n the e a r l i e r part of t h i s s e c t i o n . 59 CHAPTER 5. DEVELOPMENT OF A STAND GROWTH SIMULATION MODEL FOR ASPEN 5.1 The purpose and uses of the model The purpose of t h i s work to develop an aspen stand model f o r simulating growth and y i e l d of na t u r a l aspen stands on an i n d i v i d u a l tree basis has been stated p r e v i o u s l y . The basic aim, v i z . , construct-ing a stand growth model, i s e s s e n t i a l l y the same as that for the two e a r l i e r models (Newnham 1964: M i t c h e l l 1967) developed f o r some con i -ferous species; but i n constructing the present model some a d d i t i o n a l objectives were considered s i m i l a r l y important. F i r s t , that the model should have ge n e r a l i t y , so that i t may be e a s i l y adapted to (or at l e a s t the same approach can be used) other f o r e s t tree species growing ei t h e r i n n a t u r a l stands or plantations; and second, that the model should be r e a l i s t i c and allow the representation of s p e c i f i c c h a r a c t e r i s t i c s of aspen stands., Generality i s r e l a t e d to model features such as i n t e r -tree competition and tree s p a t i a l arrangement i n the stand. S p e c i f i c c h a r a c t e r i s t i c s , f o r instance, include c l o n a l habit of aspen, which may have important e f f e c t s on growth and stand development. The model w i l l be used to determine how growth and y i e l d of aspen are a f f e c t e d by stocking and stand density, stand structure and conal v a r i a t i o n and structure. Information already a v a i l a b l e on the e f f e c t s of stand structure on growth and y i e l d w i l l be used i n model construction and t e s t i n g . Simulation w i l l be used to determine the ef f e c t of stand structure, c l o n a l v a r i a t i o n and structure on growth and y i e l d . Stand development and r e l a t e d growth and y i e l d w i l l be evaluated f o r a range of possible stand conditions, which can be h e l p f u l i n s e l -l e c t i n g optimum management strategies f o r given management obje c t i v e s . 6o While the p r e d i c t i v e use of the model i s rather obvious, perhaps j u s t as important i s i t s explanatory r o l e . In model b u i l d i n g , f o r instance, the in v e s t i g a t o r has to i d e n t i f y and q u a n t i t a t i v e l y describe a l l the e s s e n t i a l components of the system, as w e l l as to define i t s o v e r a l l s t r u c t u r e . This alone requires a much improved i n s i g h t i n t o the system. Then by performing simulation experiments on the model and with appropriate changes i n model parameters or i n stand conditions, the e f f e c t and r e l a t i v e importance of each component can be determined. This " s e n s i t i v i t y a n a l y s i s " would indicate the components i n which further refinement would most l i k e l y r e s u l t i n improved model performance. 5.2 Methods of model b u i l d i n g and t e s t i n g In developing the present model, the approach used i s s i m i l a r to Holling's (1966) "experimental component a n a l y s i s " , and the f i n a l model has c e r t a i n common features with that of Newnham (1964). Using experimental component analysis the f i r s t task i n the study i s the i d e n t i f i c a t i o n of u n i v e r s a l p r o p e r t i e s , or "basic compo-nents" of the process i n question by observation and experimentation, or by ana l y s i s of already a v a i l a b l e data. These basic components always operate where the p a r t i c u l a r process occurs, e.g., i n tree growth s i t e e f f e c t s , density, competition e f f e c t s , e t c . In a s i m i l a r way, ad d i t i o n a l or subsidiary components are i d e n t i f i e d , e.g*, c l o n a l e f f e c t s operate i n aspen and not i n , say, Douglas f i r stands. Using experi-mental and s t a t i s t i c a l techniques, the operations of these components are analysed and q u a n t i t a t i v e l y described, which w i l l form the basic segments of the systems model. Besides having s t a t i s t i c a l l y s a t i s -f a c tory f i t to the data, such quantitative descriptions are expected to 6 l provide a r e a l i n s i g h t into the phenomenon. Simultaneously with t h i s a n a l y s i s , the o v e r a l l structure of the model i s i d e n t i f i e d and defined, and represented f i r s t by a flow chart and then with an appropriate computer program. Each component i s generally described by a computer subroutine. This approach not only f a c i l i t a t e s the necessary computer programming, but also allows the researcher to conduct computer experimentation (and s e n s i t i v i t y a n a l y s i s ) on each component subroutine or submodel to improve his under-standing of the operation of the p a r t i c u l a r component. The goodness, or accuracy of the e n t i r e model may be evaluated by comparing appropriate simulated r e s u l t s with a c t u a l data. I t i s also p o s s i b l e , and generally desirable at t h i s stage, to conduct further computer experiments and s e n s i t i v i t y analysis on the model and make necessary refinements. This procedure i s analogous to c a l i b r a t i n g a man-made co n t r o l system, e.g., an a i r - c o n d i t i o n i n g system of a new, large b u i l d i n g , 5.3 Basic components of growth and stand development with s p e c i a l emphasis  on aspen Although the present model i s to describe growth and stand development i n terms of i n d i v i d u a l trees, the growth and development processes involved have t h e i r f u l l meaning only when considered, within the stand context. For example, complex i n t e r a c t i o n s between i n d i v i d u a l trees can be expressed only within a stand, and such i n t e r a c t i o n s are fundamental i n stand growth and development. For the purpose of t h i s model, the following basic components of growth and stand development were i d e n t i f i e d : - environment: s o i l and climate 62 - species c h a r a c t e r i s t i c s : tolerance, growth habits ( i n d i v i d u a l s , clones, etc.) - i n t e r t r e e competition: stocking, stand density and structure - age The influence o f environment on the growth of the trees w i l l be described by the height growth of the dominant trees i n the stand. This i s an expression of a l l environmental influences and can be rea-dily determined (e.g., from y i e l d t a b l e s , stem a n a l y s i s , etc.) f o r most tree species. Periodic f l u c t u a t i o n s i n dominant height growth gen e r a l l y r e f l e c t c l i m a t i c patterns. S o i l influences, on the other hand, are more stable and determine the average growth of a species at a given age. In closed stands, the degree of tolerance of the species has important e f f e c t s on tree growth, on mo r t a l i t y , and on response to release by i n d i v i d u a l t r e e s . Aspen i s a very i n t o l e r a n t species. Its r e l a t e d c h a r a c t e r i s t i c s are described i n Chaper 3- In the model, however, the tolerance e f f e c t s must be q u a n t i t a t i v e l y expressed. As tolerance " " e f f e c t s are c l o s e l y r e l a t e d to i n t e r t r e e competition (and lose t h e i r importance when the trees are open grown), these e f f e c t s should be included i n the i n t e r t r e e competition component. The c l o n a l habit of aspen has an important e f f e c t on growth and development of aspen stands. In the model, t h i s e f f e c t i s consid -ered only i n s o f a r as i t i s manifested i n d i f f e r e n t i a l growth rate between various clones. Assigning clone i d e n t i f i c a t i o n to each tree i n the model, tree s p a t i a l pattern on a p l o t w i l l simultaneously describe c l o n a l structure. 63 Intertree competition i s the most important component of growth and stand development, thus of any stand growth simulation model. It i s a f f e c t e d by stand v a r i a b l e s such as stocking, stand density and structure; by environmental influences; as well as tree species c h a r a c t e r i s t i c s l i k e tolerance. Yet, the r e l a t i v e competitive status, and to a large extent, current growth of a tree i n a pure, even-aged stand on uniform s i t e , i s determined by i t s own s i z e , and the size and distance of i t s competing neighbours.. This i s obviously a very complex component. In the following section j some approaches and concepts used f o r describing i n t e r t r e e competition e f f e c t s are evaluated and expanded into a new, general competition model, or submodel, to be incorporated i n the stand growth model. Tree or stand age i s not considered as a causal component but rather a dimension i n which growth and development occurs. Because of the dynamic nature of the processes involved, age has to be included i n stand growth models, e.g., rate of height growth of dominants i s strongly r e l a t e d to age. 5.h Development and t e s t i n g of an improved i n t e r t r e e competition moder 5.kl Background Two main approaches have been used to describe competition influences on growth of i n d i v i d u a l t r e e s . One of these used various d e f i n i t i o n s of stand density around the subject t r e e . The other ap-proach was based on the concept of influence zone (Aaltonen 1926); an area over which a tree presently obtains or competes f o r resources of the s i t e . A manuscript has been prepared on t h i s aspect of the work ( B e l l a 1970) and submitted f o r j o u r n a l p u b l i c a t i o n . 64 Using various d e f i n i t i o n s of stand density as a measure of competition f o r i n d i v i d u a l trees i s based on the simple premise that i n closed stands, other things being equal, the higher the density the greater the competition and the slower the growth of i n d i v i d u a l s . Basal area (Steneker and J a r v i s 1 9 6 3 ) , angle count (Lemmon and Schumacher 1 9 6 2 ) , point density (Spurr 1 9 6 2 ) , and "area p o t e n t i a l l y a v a i l a b l e " (Brown 1965 ) were used i n t h i s context; along with simple counts l i k e number of sides f r e e . Opie ( 1 9 6 8 ) merrtioned some of the d i f f i c u l t i e s inherent i n using these expressions as a competition index. The main source of the problem here, however, i s t r y i n g to use stand variables to describe i n d i v i d u a l tree growth. Influence zone models evaluate competition from the amount of influence zone overlap between competing subject tree and surround-ing competitors. Several investigators studying tree competition and growth recognized or used a version of t h i s concept. Staebler's ( 1 9 5 1 ) measure of competition i s the sum of l i n e a r overlaps within competition c i r c l e s , although he r e a l i z e d that area overlap would be a more d i r e c t measure. Newnham's ( 1 9 6 6 ) angular competition measure has already been described i n Chapter 4. In the model of Opie ( 1 9 6 8 ) and Gerrard ( 1 9 6 9 ) , competition e f f e c t i s evaluated d i r e c t l y from the area of influence • zone overlap r e l a t i v e to the t o t a l influence zone of the subject t r e e . A close r examination of the previous models reveals t h e i r basic s i m i l a r i t y i n that they a l l assume a l i n e a r l y a d d i t i v e type of competition e f f e c t between competing tree and competitors. Such models would be r e a l i s t i c i f s i z e differences between competing tree and 'competitor, had no e f f e c t on competitive i n t e r a c t i o n s between these i n d i v i d u a l s - - a n assumption t h i s w r i t e r r e j e c t e d . 65 5.42 A new model: Competitive Influence-zone Overlap (CIO) When two trees of d i f f e r e n t sizes compete i n a f o r e s t stand they do not equally a f f e c t one another. For maintaining a higher rate of growth with a generally lower rate of metabolic e f f i c i e n c y (e.g., B a s k e r v i l l e 1965)5 the l a r g e r tree must expl o i t the s i t e considerably beyond i t s proportional share; the smaller tree with low growth rate and higher e f f i c i e n c y can e x i s t on l e s s resources than would be pro-p o r t i o n a l to i t s s i z e . Thus i n describing and evaluating i n t e r t r e e competition e f f e c t t h i s d i f f e r e n t i a l nature of the phenomenon should also be considered. The present competition model i s an extension of the influence zone concept. In developing t h i s model a somewhat empirical approach was used for three reasons: (1) the present lack of basic t h e o r e t i c a l knowledge on i n t e r t r e e and generally on i n t e r p l a n t competition, (2) the nature of the a v a i l a b l e t e s t data and (3) the intended use of the model. This present l a c k of knowledge i s quite unlike that found i n other branches of biology, e.g., i n zoology, where numerous studies have been done on competition, and large amount of information i s a v a i l a b l e on t h i s kind of i n t e r a c t i o n between animals (e.g., M i l l e r , 1967). Although some of the r e s u l t s from these studies could be of value i n an analysis of plant competition e f f e c t s , e.g., r e s u l t s on the nature and form of " e x p l o i t a t i o n " of a resource, l e s s quantitative work has been done on these aspects of animal competition than on the more q u a l i t a t i v e aspects, e.g., t e r r i t o r i a l i t y , i nterference, aggression, etc., which are s p e c i f i c to animal communities. 66 In bu i l d i n g the model i t was f i r s t assumed that the i n f l u -ence zone of a tree i s p r o p o r t i o n a l to i t s size -- a c h a r a c t e r i s t i c conveniently and w e l l described by d.b.h.; Relative tree si z e e f f e c t mentioned above was considered i n the model by a r a t i o of the d.b.h. of the competitor and that of the competing subject t r e e . Such a r a t i o i n the model could modify the competition e f f e c t estimated from i n f l u -ence zone overlap and give greater competitive "weight" to large trees than to small ones, while the e f f e c t between trees of s i m i l a r sizes would remain e s s e n t i a l l y unweighted. However, just exactly how much more, or l e s s , extra weight should be assigned to a competitor of a given r e l a t i v e s i z e i s l i k e l y to be the function of such factors as species c h a r a c t e r i s t i c s , age, s i t e , e t c . . To allow for t h i s d i f f e r e n t i a l weighting -- which would also r e s u l t i n a more general competition model -- an exponential term was added to the above d.b.h. r a t i o . I f a value of one i s used f o r exponent, the weighting r a t i o w i l l remain unchanged; a value greater than one would increase the weighting e f f e c t and a value l e s s than one would decrease i t . Defining the phenomenon as a hypothesis: the t o t a l  competition e f f e c t on each tree i s a function of r e l a t i v e i n fluence-zone overlap between i t and i t s competitors, whereas the e f f e c t of  i n d i v i d u a l competitors depends, i n an exponential form, on t h e i r s i z e  r e l a t i v e to that of the subject t r e e . Symbolically, as a mathematical model: Z O . . D. E X C I O i = * C C ^ ) * ( D 1 ) ] .....(5.1) j = i i i CTC\ = Competitive influence-zone overlap f o r competing tree i . 67 n Number of competitors whose zone i n t e r s e c t s that of the competing t r e e . ZO. . Area of zone overlap between competing tree i and competitor j . ZA. Influence-zone area of competing tree i . 1 D. 3 D.b.h. of competitor j . D. D.b.h. of competing tree i . l EX Exponent, generally greater than one and charac-t e r i s t i c of the species tolerance. Figure 5-1 shows schematically the p o t e n t i a l zone of influence of a subject tree (#3)? those of four competitors and the zone overlaps. I t i s i m p l i c i t i n the model that competition e f f e c t s between trees are evaluated as though the i n t e n s i t y of competition was uniform within a given tree's influence zone, rather than having a de-c l i n e i n competitive i n t e n s i t y towards the edge of the zone. Although t h i s may be considered a weakness of the model from a t h e o r e t i c a l standpoint, i t was not considered an important weakness when the intended purpose of t h i s model i s taken into consideration, i . e . , stand growth modeling, where the r e l a t i v e competitive status (rank) of a tree rather than an absolute amount of competitive e f f e c t s i s required. However,in further refinements of the competition model, t h i s p a r t i c u l a r aspect of the phenomenon could warrant further study. 5.^3 Estimating model parameters f o r aspen Ind i v i d u a l tree diameter growth data f o r determining model parameters were obtained from permanent sample pl o t remeasurement records. Two kinds of data were e s s e n t i a l : ( l ) d.b.h. measurement of Figure 5.1 Influence-zone overlaps between a competing subject tree (#3) and four competitors. These values are used f o r c a l c u l a t i n g CIO (see t e x t ) . 6 9 tagged trees, and (2) tree maps to describe tree locations at e s t a b l i s h -ment. The p l o t selected for t h i s purpose was a thinning study c o n t r o l (for d e t a i l e d d e s c r i p t i o n see Steneker and J a r v i s 1966, study 2). The trees on t h i s p l o t showed a r e l a t i v e l y homogeneous g e n e t i c a l character considering growth r a t e . (A cursory examination i n the summer of 1968 revealed no s t r i k i n g differences i n tree sizes between clones present on the p l o t . ) Comparing height data a v a i l a b l e from t h i s p l o t with s i t e index curves from Manitoba and Saskatchewan, i t seemed that the sample stand was growing on somewhat better than medium s i t e . In t e s t i n g the competition model, maximum zone of influence of the species was r e l a t e d to i t s open-grown crown siz e for a given stem diameter, adjusted by a f a c t o r c h a r a c t e r i s t i c of that species, and p o s s i b l y a f f e c t e d by s i t e and age. A f a c t o r greater than one means that competition would commence before open growing tree crowns come i n contact with each other. In symbolic form: CR = (a + bxD)/2 .(5.2) E = CR x CZ (5-3) where: CR = Open grown crown radius D = D.b.h. a, b^ = Regression c o e f f i c i e n t s R = Influence or competition radius CZ = Adjusting f a c t o r A computer program was written that c a l c u l a t e d CIO values f o r each tree from tree co-ordinates, i n i t i a l d.b.h. and selected values of CZ and EX, on the basis of equations (5.1), (5-2) and (5.3). (Border e f f e c t was eliminated by excluding as competing trees, i n d i v i d u a l s nearer to the border than about three times the open grown crown r a d i i 70 of the larger trees on the p l o t . ) Intermediate output included d.b.h. and CIO at the s t a r t of the increment period along with diameter (Dine) and basal area (BAinc) increment figures f o r trees that survived through the period. The output was analyzed by multiple regression and c o r r e l a t i o n techniques. To f a c i l i t a t e analysis and i n t e r p r e t a t i o n , a simple growth-competition regression was f i t t e d . D.b.h. was not e x p l i c i t l y included i n these regressions, because the CIO independent terms accounted f o r v a r i a t i o n associated with tree size (see also equation (5.2)) to such extent that a d d i t i o n a l d.b.h. terms were generally n o n - s i g n i f i c a n t . Site v a r i a t i o n was minimized by analyzing study trees from a r e l a t i v e l y small sample p l o t located i n a uniform stand. The general form of the regression i s shown below: Dine = a + b ^ I O + b g CIO 2 + b ^ I O 3 (5-4) Various combinations of FC and EX were t r i e d u n t i l the corresponding CIO values combined i n equation (5-^) accounted f o r the 2 2 l a r g e s t proportion of the v a r i a t i o n i n Dine (highest R or r v a l u e s ) . I n i t i a l l y , a wide range of CZ, and EX values with large class i n t e r v a l s (course g r i d ) were t r i e d . I t was then possible to "narrow i n " with further i t e r a t i o n s on the best parameter values allowed by the s e n s i t i v i t y of t h i s method. This approach also provided for a b u i l t -i n s e n s i t i v i t y analysis of model parameter values, as w e l l as f o r a ready comparison with Opie's zone count, without the exponential term. The usefulness of d.b.h. alone for p r e d i c t i n g Dine was also b r i e f l y explored. The open grown crown radius - d.b.h. r e l a t i o n (equation (5.2)) was based on data c o l l e c t e d i n western Canada, from Manitoba to B r i t i s h Columbia. These data are p l o t t e d , and relevant regression s t a t i s t i c s are presented i n Figure 5.2. The data indicate, remarkable uniformity i n crown size - d.b.h. r e l a t i o n across western Canada, for f u l l y open grown aspen. These data and regression d i f f e r from those presented by Smith (1966b) for aspen i n i n t e r i o r B r i t i s h Columbia. While the intercepts of the two regressions are s i m i l a r (3.63 and 4.0, present and Smith's, r e s p e c t i v e l y ) , the smaller "b" c o e f f i c i e n t s (1 .30 vs. 1.6l) i n Smith's regression, which indi c a t e r e l a t i v e l y smaller crowns f o r bigger diameter trees, suggests that h i s bigger sample trees, although open growing at the time of measurement, may have been exposed to competition at younger ages . The present r e s u l t s thus confirm the importance of stand density influences on crown width -d.b.h. r e l a t i o n of aspen. 2 2 Table 5.1 shows c o e f f i c i e n t s of determination (R and r ) f o r Dine-CIO c o r r e l a t i o n s f o r a "coarse g r i d " of cz. and EX parameters f o r the f i r s t measurement period. These c o r r e l a t i o n combinations include independent terms s i g n i f i c a n t at l e a s t at the 0.05 l e v e l p r o b a b i l i t y . The subscripts of c o e f f i c i e n t s denote the independent terms a c t u a l l y used, e.g., i n 0.721 0 (-1) denotes negative l i n e a r term and (2) p o s i t i v e quadratic term. The highest R values are under l i n e d , i n d i c a t i n g optimum region of CZ and EX parameters. To evaluate the effectiveness of CIO i t i s u s e f u l to compare i t to Opie's (1968) model, who demonstrated h i s zone count to be at l e a s t as e f f i c i e n t and precise as any previous models. For the aspen data, Opie's model i n combination with equation (5.2^ ) accounted f o r generally l e s s than 10$ of the v a r i a t i o n i n diameter increment (see Table 5.1) as compared to over 50$ accounted by the CIO model (Tables 72 5 10 15 D.b.h. (Inches) FIGURE 5.2 Crown width-d.b.h. r e l a t i o n of open grown aspen trees i n western Canada. TABLE 5.1 C o e f f i c i e n t s of determination f o r Dinc-CIO c o r r e l a t i o n s — for aspen. F i r s t increment period; age Ik to 19; 14-3 study trees. CZ EX 0.0- 1.0 1.5 2.0 2.5 3.0 1.0 .028_3 .202_x • 2 8 2-2,3 •^-1,2 • U° 7-l,2,3 •^-1,2,-3 1.5 •°8l-l,2,-3 .332_ 2 j 3 • t o 7 - l,2 •k66-12 -3 • 5 l 8-l,2,-3 • 5 t o - l,2,-3 2.0 .052_3 •*71-2,3 •501-1,2 • 5 3 2 - l 2 -L>':-> -3 • 5 5 9-l,2,-3 • 5 6 l - l,2,-3 2.5 .052^ • 5 3 8-2,3 • * 3 - l,3 •550 -3 •566-1.2,-3 • 5 6 5 - l,2,-3 3-0 .056^ • 5 5 6-2,3 • 5 W-1,3 •550.x -3 • 5 6 6 - l,2,-3 — A l l c o r r e l a t i o n s are s i g n i f i c a n t at the 0.01 p r o b a b i l i t y l e v e l ; subscripts and t h e i r signs denote the independent terms a c t u a l l y used. parameter values greater than 3.0 were also t r i e d . = 1.0 74 5.1 and 5-2). Optimum model parameters f o r aspen between ages l4 to 30 (one 6-year and two 5-year measurement periods) are presented i n Table 5.2. These optimum parameters are used i n the stand growth simulation model. The table also shows r e l a t e d c o e f f i c i e n t s of determination f o r Dine -CIO multiple c o r r e l a t i o n s ; and for Dinc-D simple c o r r e l a t i o n s . Optimum CZ parameters showed remarkable s t a b i l i t y over d i f f e r e n t measurement periods, while the EX values appeared to decrease with age. B e l l a (1970) compared optimum EX parameters of four species: aspen, jack pine, red pine and Douglas f i r , and found that the value of EX at comparable ages declined with increasing tolerance for these species, suggesting that t h i s parameter i s an expression of the species tolerance. This parameter q u a n t i t a t i v e l y expresses what i s more or l e s s common knowledge, that the e f f e c t of r e l a t i v e tree size on tree growth depends on the species tolerance; the more i n t o l e r a n t the species the greater the influence of r e l a t i v e tree s i z e difference on growth. TABLE 5-2. Optimum model parameters f o r aspen for the three increment periods, and r e l a t e d c o e f f i c i e n t s of determination f o r Dine-CIO multiple c o r r e l a t i o n s - ; also for Dinc-D simple c o r r e l a t i o n s . Age at beginning CIO C o e f f i c i e n t s (and at end) of Number parameters of determination increment period of trees :cz., EX Dine-CIO Dinc-D 14 (to 20) Ik 3 2.7 2.7 • 5 6 8 - l , 2 , - 3 .385 20 (to 25) 120 3.0 2.4 • 5 0 5-l,2,-3 • 545 25 (to 30) 98 3-0 2.0 • 5 0 7 - l , 2 , - 3 •439 - A l l c o r r e l a t i o n s are s i g n i f i c a n t at the 0.01 p r o b a b i l i t y l e v e l . 75 Furthermore, B e l l a (1970) showed that growth v a r i a t i o n i n suppressed and intermediate aspen trees i s much more dependent on competition than that i n l a r g e r , more dominant trees, and that smaller trees generally represent a much wider range of competition l e v e l s than larger t r e e s . Competition e f f e c t s on a small tree apparently depend very much on the s p a t i a l arrangement of i t s mainly bigger neighbors, which may be quite v a r i a b l e . Although competition slows the growth of the l a r g e r trees, they have a r e l a t i v e l y stable competitive status (and index), as they are l e s s a f f e c t e d by t h e i r mainly smaller neighbors. Decrease i n EX at higher ages could mean increasing tolerance as the trees get older, or that the e f f e c t of r e l a t i v e s i z e difference between competing tree and competitor diminishes with increasing s i z e . The value of FC parameter indicates that competition i n t e r -a ction f o r aspen extends w e l l beyond the open-growing crown width of competing trees and competitors. The b u i l t - i n s e n s i t i v i t y analysis of model parameters shows just how much l e s s of the v a r i a t i o n i n growth i s accounted f o r i f no competition i s assumed u n t i l open growing crowns touch, e.g., the approximate figures f o r aspen at Ik years of age are hVJo vs. 57% (Table 5.1). Preliminary tests ( B e l l a 1970), conducted on growth data from poorer s i t e s i n d i c a t e d that the value of CZ i s r e -l a t e d to environmental ( s i t e ) influences, and increases with decreasing s i t e q u a l i t y . 5.5 A general flow diagram f o r the stand growth model In Section 5-3 the major components of growth and stand development were i d e n t i f i e d and t h e i r respective roles were b r i e f l y described. The previous section shows the development and t e s t i n g of an i n t e r t r e e competition submodel. The submodel describes the complex 76 competitive i n t e r a c t i o n s which occur between i n d i v i d u a l trees growing i n closed stand, insofar as t h i s i n t e r a c t i o n a f f e c t s tree growth. The next step i n developing a stand model i s the construction of a very-general flow diagram that includes the e s s e n t i a l features of the modeled system, thus i t provides a framework or o v e r a l l structure, as w e l l as representing system dynamics. Figure 5.3 shows such flow diagram for the present model. (Notations and symbols i n the flow diagram are s i m i l a r to those used i n FORTRAN programming and are h e l p f u l i n v i s u a l -i z i n g system dynamics.) The flow diagram i s more or l e s s s e l f explanatory, so only b r i e f a d d i t i o n a l d e s c r i p t i o n should s u f f i c e . At the s t a r t , the READ statement encompasses the necessary d e s c r i p t i v e s t a t i s t i c s , regression c o e f f i c i e n t s , and model parameters whichhave to be s p e c i f i e d for'each simulation run. Tree s p a t i a l arrangement i s defined by the nature of the data. E i t h e r permanent sample p l o t data are used and tree l o c a t i o n s (tree map or rectangular co-ordinates) and sizes are a v a i l a b l e and read i n for simulation, or a r t i f i c i a l tree data are generated representing random s p a t i a l pattern. Another a l t e r n a t i v e i s to manually construct a r t i f i c i a l tree s p a t i a l patterns and use the tree co-ordinates and d.b.h. values for the simulation runs. Using t h i s method, the e f f e c t of tree s p a t i a l arrangement could be tested over a great d i v e r s i t y of patterns from very regular (plantation) to clumped conditions. Yet another advantage of using a r t i f i c i a l p l o t data i s the ease of defining c l o n a l v a r i a t i o n ; an important advantage because such data are generally l a c k i n g . Trees i n the same clone can be assigned the same clone i d e n t i f i c a t i o n , or simply clone number. As the trees would already 77 ( START') READ I DESCRIPTIVE STATISTICS (AGE, PLOT SIZE, ETC.), REGRESSION COEF., AND MODEL PARAMETERS (CR, EX, ETC) AND il PERMANENT SAMPLE PLOT DATA READt TREE X-Y CO-ORD- AND SIZES ARTIFICIAL PLOT DATA GENERATE! TREE X-Y CO-ORD.H AND D YES READS CLONE 10 OF TREES =9 o INCREASE AGE BY ONE PERIOD GENERATE! POTENTIAL H AND D INC PER TREE CALCULATE I COMPETITION INDEX FOR EACH TREE OBTAIN!ACTUAL H AND D INC-(ADJUST POTENTIAL INC- BY COMPETITION) YES LIVING TREES CALCULATE > NEW TREE DIMENSIONS PLOT BA AND VOLUME FIGURES NO. DEAD TREES CALCULATE! PLOT BA AND VOLUME FIGURES OBTAIN '• DESCRIPTIVE TREE STATISTICS AVERAGES, MIN, MAX, SD , AND FREO- DISTR. WRITE! AGE, AVERAGES, MIN ,MAX, AND SD VALUES, FR. DISTR. FOR H INC, D INC , H , D , FOR LIVING AND DEAD TREES NO CALCULATE I TOP HEIGHT PER ACRE VALUES FOR NT. B A , VOLUME FOR LIVING TREES AND MORTALITY STORE» PER AC-VALUES ON DISK (YIELD TABLE LIKE OUTPUT) YES c WRITE i DISK VALUES ( W ) FIGURE 5.3 General flow diagram of the stand growth model, 78 have t h e i r l o c a t i o n on the p l o t , c l o n a l structure would thus automat-i c a l l y be defined. In t h i s model,clonal e f f e c t s are considered only insofar as they are manifested i n tree growth rate d i f f e r e n c e s . This f i r s t section of the flow diagram and the corresponding model may be considered as the input segment for transmitting the necessary data f o r stand growth simulation. The next, c e n t r a l segment i s the f u n c t i o n a l part of the model, where the process simulation i s produced. This includes: generating p o t e n t i a l height and d.b.h. increment f o r each t r e e ; evaluating each tree's competitive status; adjusting p o t e n t i a l tree increment values by t h e i r respective competitive status; generating mortality as a function of current increment and tree competitive status (a threshold'); and f i n a l l y obtaining new tree dimen-sions of l i v i n g and dead trees at the end of the increment period. The f i n a l , output segment of the model i s designed to c a l -culate appropriate summary s t a t i s t i c s and produce written output. The stand s t a t i s t i c s c a l c u l a t e d are temporarily stored during each i n c r e -ment period ( i t e r a t i o n ) and are written out a f t e r the simulation run i s completed i n a form s i m i l a r to y i e l d t a b l e s . This general flow diagram represents only the skeleton of the stand growth model. It has to be tremendously expanded step by step, and each significant feature of the process i d e n t i f i e d and q u a n t i t a t i v e l y described. I n i t i a l l y , some of these mathematical descriptions may be quite crude, because of the i n s u f f i c i e n t knowledge on c e r t a i n aspects of the process. Computer experimentation and i t e r -a t i v e , numerical methods can be h e l p f u l at t h i s phase of the work. 79 5• & Detailed flow diagram, programming and model refinement The f i n a l d e t a i l e d flow diagram and the corresponding computer program are presented i n Appendix I and I I . Comments included i n both the flow diagram and the ..program should enable the reader to follow how the model functions. The input and output segments of the model are operational i n nature, and as such has no e f f e c t on I t s functioning. These segments are s e l f explanatory and w i l l only be covered very b r i e f l y . The c e n t r a l f u n c t i o n a l core of the model w i l l be explained i n d e t a i l . To f a c i l i t a t e f a m i l i a r i z a t i o n with the model, i t seemed desirable to present and discuss f i r s t the c e n t r a l model segment, as both input and output are dependent on t h i s . Also, t h i s c e n t r a l seg-ment i s the most massive part of the model. I t forms the major part of the main program as well as two complete subprograms; one f o r generating height growth, the other f o r evaluating competitive status of each t r e e . The height growth (HGRO) subprogram generates p o t e n t i a l height increment of each tree as though i t was a dominant. Average dominant height i s estimated f o r each peri o d using von Bertalanffy's growth curve ( f i t t i n g program from Fabens 1965), based on appropriate dominant height-age information. Height increment values f o r a given period are obtained by subtracting estimated average dominant height at the beginning of the period, from that at the end of the period. A random normal deviate i s generated f o r each tree and added to the average increment, to imitate the random (or at least, unaccounted) v a r i a t i o n present i n height increment. The amount of v a r i a t i o n , or spread, i n generated p o t e n t i a l height increment values i s defined by the appropriate standard deviation value expressed i n terms of average height increment 8o i n a given period. Increasing the number of standard deviation here w i l l r e s u l t i n smaller spread of increment values. The c l o n a l c h a r a c t e r i s t i c s of aspen was taken into account by introducing a c e r t a i n amount of v a r i a t i o n i n growth associated with t h i s source. The subprogram contains p r o v i s i o n to generate, i n the f i r s t i t e r a t i o n , inherent differences i n growth rate between clones; and c a l c u l a t e f o r each clone a clone performance r a t i o (CPR) and use these i n the subsequent i t e r a t i o n s to adjust p o t e n t i a l height growth of each i n d i v i d u a l tree according to i t s c l o n a l i d e n t i t y . This feature would make possible the quantitative study of clone dynamics; i n p a r t i c u l a r , to determine the magnitude of difference i n growth rate necessary f o r dominance and a r e a l expansion of some clones at the expense of others, during the l i f e of a stand. In the simulation runs so f a r only l i m i t e d use was made of the height growth subprogram, and no t e s t has been conducted to evaluate the e f f e c t of c l o n a l growth rate d i f f e r e n c e s , i . e . , only one clone per p l o t was assumed. Much future work i s planned on these aspects of aspen stand development with the help of the model. Model c a l i b r a t i o n and t e s t i n g was f a c i l i t a t e d using a s i m p l i f i e d model which also reduced the amount of computer time used f o r the simulation runs. In t h i s s i m p l i f i e d model the height growth subprogram was removed, and i t s chief function performed by the main program. So i n place of s t a t e -ments 0066 and OO67 i n the main program (App. I l ) , 11 statements were transplanted from the height growth subprogram in t o the main program (shown following main program i n App. I I ) . This s i m p l i f i e d model produces i d e n t i c a l p o t e n t i a l height increment for each tree (no random v a r i a b l e ) , and contains no p r o v i s i o n for growth v a r i a t i o n 81 associated with c l o n a l d i f f e r e n c e s . The function of the competition subprogram (COMPE) i s to evaluate competitive status of each tree, which values are l a t e r used to reduce p o t e n t i a l tree increment to actual increment. This i n t e r -tree competition submodel has already been discussed i n some d e t a i l i n the previous section. The subprogram used i n the stand model i s e s s e n t i a l l y the same as the program used f o r t e s t i n g the CIO model, except f o r the method by which border e f f e c t i s eliminated. This i s done i n the subprogram by p r o j e c t i n g , or mirroring trees as competitors outside the p l o t adjacent to i t s boundary from the opposite side of the p l o t ( e a s i l y v i s u a l i z e d by imagining an endless, two dimensional s t r i p , l i k e a conveyor b e l t ) . This mirroring i s done i n two d i r e c t i o n s p a r a l l e l to two-two sides of the square p l o t . The four s t i l l empty outside corners are " f i l l e d i n " s i m i l a r l y . In the following section of the model (main program statements 0077 to 0128; App. II) act u a l height and diameter increment are obtained by reducing p o t e n t i a l increment values according to the trees' compet-i t i v e status. This reduction i s based on four basic assumptions which were derived from current knowledge (Chapter 3) of tree growth and mortality within aspen stands of normal density. These assumptions are: (1) the height increment, and to a l e s s e r extent diameter increment, of the largest trees (most dominants) i n a stand i s not, or only very l i t t l e a f f e c t e d by competition or stand density; (2) very l i t t l e height and diameter increment, sometimes almost none, together with greatest proportion of mo r t a l i t y occur among suppressed trees; (3) "intermediate" trees -- i n between the above two extreme conditions --82 grow i n proportion to t h e i r r e l a t i v e competitive p o s i t i o n i n the stand; and (4) as a r e s u l t of competition, the per cent diameter increment of these "intermediate" trees may be reduced to almost h a l f of t h e i r per cent height increment. Increment adjustment of trees growing under the two extreme conditions i s r e l a t i v e l y simple; none or l i t t l e reduction i n growth of the la r g e s t trees, and almost complete reduction (no growth) f o r the smallest suppressed t r e e s . The d i f f i c u l t problem i s the develop-ment of appropriate height and diameter increment reductions f o r the intermediate trees, which make up by f a r the greatest portion of a stand. The use of a competition index f o r t h i s type of adjustment i s quite complex, but i t depends mainly on the range of possible competition index values and t h e i r frequency d i s t r i b u t i o n . I d e a l l y , the index values should have a symmetrical (normal) type of d i s t r i b u t i o n , with most values f a l l i n g close to the mean, and fewer and fewer towards the two extremes. Then the required adjustment could be made propor-t i o n a l to competition index values, with l i t t l e or no reduction at the very low competition l e v e l s and up to 100 per cent reduction at high l e v e l s . Most trees, having competition indices close to the mean, would have height increment reductions around 50 per cent. Competition index values c a l c u l a t e d using the CIO model ( i n wines oj the t e s t runs to obtain optimum^parameters CZ and EX; see Section 5.4) showed a very pronounced p o s i t i v e skewness. Competition values f o r the trees on the tes t p l o t at age 14 ranged from about 5 to 315, with mean value of 35. Competition index values having such an asymmetric type of frequency d i s t r i b u t i o n cannot be r e a d i l y used f o r the intended purpose of adjustment. It seemed, however, that logarithmic transformation of these values may produce the desired symmetry of the frequency d i s t r i -bution. Simulation runs l a t e r proved that logarithmic transformation was indeed appropriate. (Some of these frequency d i s t r i b u t i o n s both f o r o r i g i n a l and transformed competition index values are shown i n App. I I I . ) Height increment adjustment of each tree was based on i t s transformed competition index value, using a m u l t i p l i e r (BADJ; main program statement 0092, App. II ) - - a n inverse of the range (RNG) of . transformed competition i n d i c e s . When using the m u l t i p l i e r f o r adjust-ing, minimum competition index value (CIRN) was always subtracted, thus the tree that had the lowest competition index would have zero height increment adjustment. On the other hand, the tree having the maximum competition index would have adjusting value 1.0, meaning complete increment reduction or no growth. Instead of a simple inverse, a c o e f f i c i e n t named BNO was put i n the nominator of the r a t i o f o r c a l c u l a t i n g BADJ. Thus i t was possible to manipulate the amount of adjustment by a l t e r i n g BNO -- the kind of f l e x i b i l i t y which i s important i n simulation models. For example, lower value f o r BNO r e s u l t s i n smaller m u l t i p l i e r (BADJ) and smaller increment reduction. While the r e l a t i v e e f f e c t of change i n BNO i s the same across the e n t i r e range of competition statuses, the absolute e f f e c t i s much greater on trees having higher competition i n d i c e s . P o t e n t i a l height increment reduction i s r e l a t i v e l y s t r a i g h t forward and i s d i r e c t l y p r o p o r t i o n a l to the given tree's competition index. The p r i n c i p l e s involved are already explained i n the previous 84 paragraph. Diameter increment reduction i s somewhat more complex, as the per cent reduction f o r i t may exceed height increment reduction nearly twice. Before getting into that problem, however, i t should be explained how p o t e n t i a l diameter increment values are obtained. Diameter increment values are estimated using a D.b.h./ Height over Height r e l a t i o n s h i p , based on open growing tree measurements (Figure 5-4), from p o t e n t i a l height increment values already a v a i l a b l e . F i r s t , open growing D / H r a t i o was estimated from a c t u a l tree height, which r a t i o m u l t i p l i e d by p o t e n t i a l height increment provides respec-t i v e p o t e n t i a l diameter increment. These p o t e n t i a l increment values were used d i r e c t l y f o r open growing trees, i . e . , when competition indices approach zero. The a d d i t i o n a l (to height) per cent reduction i n diameter increment r e s u l t i n g from competition-effects was c a l c u l a t e d using a mathematical expression (main program 0124 to 0126 App. II) that provides maximum adjustment f o r trees of intermediate competitive status, and minimum or no adjustment f o r trees at the two eoctremes of competition. This expression, developed through numerical experimen-t a t i o n , i s an algebraic representation of a f a i r l y w e l l established r e l a t i o n s h i p between height and diameter increment and i n t e r t r e e compe-. t i t i o n , which however, has not been previously expressed i n quantitative terms. Height increment adjustment of a tree, ADJH(l), can range from 0.0 to 1.0, i . e . , no reduction i n increment to t o t a l reduction. Diameter increment adjustment i s the function of ADJH(l), the two extremes (0.0 and 1.0) being i d e n t i c a l . Between the two extremes, however, reduction i n diameter increment always exceeds that of height increment. The r e l a t i o n s h i p between height increment adjustment and that of diameter increment i s i l l u s t r a t e d i n Figure 5.5. . 2 0 o o - . 1 5 CT X . 1 0 1 D/H = .084 + . 0 0 2 I U H n = 20 r = .78 1 0 2 0 4 0 6 0 Height (ft.) FIGURE 5.4 D.b.h./height - height r e l a t i o n of open grown aspen. 1.0 r c E "co o | o . 5 (U u c 0> E o O u 0 . 5 1.0 Height increment adjustment Dominant Suppressed FIGURE 5.5 Relationship between diameter increment adjustment and height increment adjustment. 86 Three c o e f f i c i e n t s were included i n the above expression to be used f o r minor adjustment of the r e l a t i o n between height and diameter increment and competition, v i z . , FCP, AY, and ADJP. FCP was designed f o r a l t e r i n g the r e l a t i o n s h i p between height and diameter increment; FC?>1.0 would r e s u l t i n greater o v e r a l l diameter increment reduction, and FCP< 1.0 i n lower reduction. AY and ADJP would allow a d d i t i o n a l minor reduction i n diameter increment of the l a r g e r trees, should assumption ( l ) prove u n r e a l i s t i c i n that diameter increment d i d decrease for the larges t trees because of competition. Calculating diameter increment reduction values f o r Figure 5.5, FCP was equal to one, and AY and ADJP to zero, so a c t u a l l y these terms had no e f f e c t . I t should be r e a l i z e d that t h i s adjusting, or c o n t r o l , mechanism i s the most c r u c i a l part of the modeled system, and i t i s c e r t a i n l y the one most d i f f i c u l t to quantify. By making these adjust-ments f l e x i b l e , they w i l l allow further model refinement through simulation. At t h i s point the reader may s t a r t to wonder about the v a l i d i t y and accuracy of the quantitative descriptions of these i n t e r -r e l a t i o n s h i p s . Some of these may appear crude, or even a r b i t r a r y . But l i t t l e r e f l e c t i o n should reveal that these i n t e r r e l a t i o n s h i p s are i n accord with current knowledge on tree growth and stand development. It should also be r e a l i z e d that the f i n a l mathematical d e s c r i p t i o n i s generally reached a f t e r t e s t i n g many other a l t e r n a t i v e s by computer simulation experiments. A f t e r the height and diameter increment of each tree has been adjusted, these "actual" increment values f o r the current p e r i o d are added to the respective tree heights ( H ( l ) ) and diameters ( D ( l ) ) 87 to obtain the new tree s i z e s . The model generates competition dependent mo r t a l i t y based on the assumption that the chance of a tree dying i s d i r e c t l y r e l a t e d to i t s competitive status, and i n v e r s e l y r e l a t e d to i t s current growth, including a c e r t a i n amount of random v a r i a t i o n . In other words, trees most l i k e l y to die are the ones with high competition index and slow growth; but there i s a c e r t a i n amount of chance as to which among these trees w i l l a c t u a l l y succumb. The dispersion of t h i s random normal component i s a function of the r a t i o of current average p o t e n t i a l height increment (AHIN)anda s p e c i f i e d standard deviation value (SDD). I f the "death r a t i o " (DER) thus obtained exceeds that of a c e r t a i n s p e c i f i e d value (DERT(K)) f o r that period, the tree i s assumed to have died (main program; statements 131-135, App. I I ) . The remaining f i n a l segment of the model performs outputing functions. Two main kinds of output are calculated: (1) d e t a i l e d tree s t a t i s t i c s of the hypothetical, stand on a p l o t basis, which include averages and measures of dispersion ( v i z . , standard deviation, coef-f i c i e n t of v a r i a t i o n , minimum and maximum values and a frequency d i s -t r i b u t i o n ) for t r e e increment and tree s i z e (height and diameter), (2) y i e l d table l i k e summaries on a per acre basis that includes top height, average diameter, basal area and stem volume per acre values. Tree summaries are written out f o r each increment period. For the y i e l d table summaries, top height was c a l c u l a t e d from the height of the t a l l e s t k-0 trees per acre, and t o t a l stem and merchantable volumes were estimated from tree height and d.b.h. using Honer's (1966) volume regressions for aspen. 88 In addition, an output of tree competition index s t a t i s t i c s was obtained for each increment period. These included averages and measures of dispersion both f o r the o r i g i n a l and f o r transformed competition index values. Now, only the input segment needs further d e s c r i p t i o n . As mentioned i n Section 5.5, three kinds of input are required f o r each simulation run. These are ( l ) d e s c r i p t i v e s t a t i s t i c s , (2) regression c o e f f i c i e n t s , and (3) model parameters. Input of d e s c r i p t i v e s t a t i s t i c s includes: p l o t s i z e ( l i n e a r dimension of a square p l o t ) , i n i t i a l number of trees, s t a r t i n g and harvest age, length of growth period, the nature of tree data and r e l a t e d s p a t i a l arrangement, and the number of clones present. The following regression c o e f f i c i e n t s are read i n : open growing crown width - d.b.h. regression, open growing d.b.h./height -height regression, height growth regression of dominant trees, and height - d.b.h. regression when i n i t i a l tree data ( l o c a t i o n and d.b.h.) comes from sample p l o t measurement. Input of the following model parameters i s required: influence zone c o e f f i c i e n t s and exponents f o r the competition submodel and death r a t i o thresholds f o r each increment period. In addition, i n i t i a l values of random deviates and t h e i r respective standard deviations and c o e f f i c i e n t s f o r minor adjustment of height and diameter increment of s p e c i f i c tree size classes are read. 5.7 Stand growth simulation experiments with the model 5.71 C a l i b r a t i o n with check p l o t (normal stand density) data from above average s i t e At t h i s point i t should be quite c l e a r that the system being modeled i s of considerable complexity. Thus even i f a l l the major system 89 components and t h e i r underlying structure have been c o r r e c t l y i d e n t i f i e d and r e a l i s t i c a l l y described, i t would s t i l l be very l i k e l y that some parts of the model had some weaknesses. Some of these weaknesses may a r i s e from inadequate knowledge about c e r t a i n aspects of the system, or from inaccurate quantitative descriptions, or both. Likewise, s t r u c t u r a l flaws as w e l l as programming errors ( c a l l e d "bugs" by programmers) may occur i n model b u i l d i n g and r e s u l t i n uns a t i s f a c t o r y model performance. Therefore very few, i f any, simulation models are l i k e l y to produce s a t i s f a c t o r y r e s u l t s f o r the f i r s t time they run. So a f t e r the model has been programmed and assembled, a f i n a l "debugging" and refinement, which may also be c a l l e d c a l i b r a t i o n , i s required and can only be done through a c t u a l simulation runs. Closely r e l a t e d to t h i s c a l i b r a t i o n procedure are the exper-imental simulation runs conducted to "narrow i n " on some model para-mater values ( v i z . , death r a t i o threshold, DERT), as well as on some simulation c o e f f i c i e n t s (e.g., standard deviation values f o r random va r i a b l e s ; c o e f f i c i e n t s f o r minor adjustment of diameter increment -height increment r e l a t i o n s of large trees) whose accurate values would be very d i f f i c u l t to determine through conventional experiments... or d i r e c t observations and measurements. This procedure i n f a c t i s a " s e n s i t i v i t y a n a l y s i s " , as i n the process of obtaining desired optimum values the researcher w i l l f i n d out how ce r t a i n changes i n these parameters a f f e c t model performance; i n other words, how se n s i t i v e the model i s to c e r t a i n changes i n s p e c i f i c parameters. The i d e a l data, f o r stand growth model c a l i b r a t i o n and.for the kind of s e n s i t i v i t y analysis described above, are long term permanent sample p l o t records of the species and stand conditions studied. The 90 records required at each measurement include: i n d i v i d u a l tree d.b.h.; appropriate number of height measurements; and a tree map at p l o t establishment. Such permanent sample p l o t data can be used to determine, by i t e r a t i v e techniques described i n Section 5-4, the two most important model parameters (CZ and EX) necessary In evaluating i n t e r t r e e competition e f f e c t s during stand growth simulation. Because one of these parameters (EX) i s age dependent, long term records are necessary to e s t a b l i s h a d e f i n i t e parameter-age trend. The problem with using short term records (one or two growth periods) i s that v a r i a t i o n s .from other sources may obscure the trend one i s t r y i n g to define. The other, and s i m i l a r l y important use of long term permanent sample p l o t information that includes a c t u a l stand growth and develop-ment and tree si z e s t a t i s t i c s , i s i n providing d i r e c t comparison with s i m i l a r simulated s t a t i s t i c s . In t h i s context, actual growth i n f o r -mation i s used as a standard, and i n c a l i b r a t i n g the model appropriate parameters and c o e f f i c i e n t s are r e f i n e d u n t i l a c t u a l and simulated stand s t a t i s t i c s are i n reasonable agreement. The permanent sample p l o t records a v a i l a b l e f o r the present study (see also Section 5.4) covered three growth periods from age l4 to 30 years; one 6-year and two 5-year i n t e r v a l s . Thus model parameters CZ and EX, and the necessary tree size and stand s t a t i s t i c s required f o r comparison i n model c a l i b r a t i o n , could only be obtained f o r these three periods. For the time period permanent sample p l o t data were a v a i l a b l e , model c a l i b r a t i o n turned out to be r e l a t i v e l y simple. It was also f a i r l y s t r a i g h t forward to determine and r e f i n e model parameters and c o e f f i c i e n t s whose values were obtained through experimental simulation runs and r e l a t e d s e n s i t i v i t y a n a l y s i s . Beyond 30 years of age, stand growth simulation was 9 1 extended period by period through extrapolating model parameters and r e f i n i n g them by simulation experiments. Act u a l tree and stand s t a t i s t i c s required f o r comparisons were taken from Alberta and Saskatchewan y i e l d t a b l e s . Numerous simulation runs were conducted with the model. Many of these were required f o r debugging the computer program, and others f o r r e f i n i n g model parameter values and analysing t h e i r e f f e c t . Some of the combinations t r i e d w i l l be b r i e f l y discussed, but only the f i n a l simulation run i s described i n d e t a i l . As mentioned previously, three kinds of input values are used i n t h i s simulation model; d e s c r i p t i v e s t a t i s t i c s , regression co-e f f i c i e n t s and model parameters. Because some of the parameters may change with increasing age (e.g., EX), the model was constructed to allow input of appropriate parameter values f o r every increment period. Understandably, d e s c r i p t i v e s t a t i s t i c s and regression c o e f f i c i e n t s remain unchanged through a simulation run, or even several runs on any one set of conditions (e.g., s i t e and d e n s i t y ) . Input values s p e c i f i e d f o r the f i n a l c a l i b r a t i o n run are given below. Descriptive s t a t i s t i c s : - simulation p l o t s i z e , PS = 6 6 . 0 f t . square ( 0 . 1 acre) - i n i t i a l number of trees on the p l o t , NT = 409 - stand age at s t a r t , ISA = Ik years - harvest age (terminate simulation), LHA = ^ 9 years - length of growth period LGP = 5 years - X-Y coordinate option, IXY = 1 (tree coordinates from permanent sample p l o t tree map) - number of clones, NC = 1 (assumed) 92 - number of classes WCC i n competition index frequency d i s t r i b u t i o n ; either 30 or 40 was used - acre m u l t i p l i e r , AC = 10 ( f o r a .10 acre p l o t ) - number of trees f or c a l c u l a t i n g top height, WTTH = 4 (on t h i s . 10 acre p l o t ) Regression c o e f f i c i e n t s : - open growing crown width-d.b.h. regression, A l = 3 . 6 3 ; B l = 1 . 6 l 4 ; as i n F i g . 5 - 3 - open growing d.b.h./height-height, regression A2 = . 0 8 5 ; B2 = . 0 0 2 1 1 ; as i n F i g . 5 . 4 - height-age growth function A = 9 0 . 0 7 ; B4 = . 9 6 l l 4 ; B5 = . 02017 - height-d.b.h. regression of trees at p l o t establishment A3 = 1 1 . 1 6 ; B3 = 6 . 1 4 6 . Model c o e f f i c i e n t s and parameters: - c o e f f i c i e n t of height increment adjustment BWO = 0 . 8 - m u l t i p l i e r f o r minor adjustment of diameter increment, FCP = 1 . 0 - c o e f f i c i e n t s to adjust (decrease) the l a r g e s t trees diameter increment, AY = 0 . 0 ; ADJP = 0 . 0 - standard deviation of the death r a t i o threshold random v a r i a b l e , SDD = 0 . 5 - influence, or competition zone c o e f f i c i e n t s CZ(K), for the competition submodel (from f i r s t to l a s t period) 0 . 0 , 2 . 7 3 - 0 3 . 0 3 . 0 3 . 0 3 - 0 3 - 0 - competition submodel exponents EX(K) 0 . 0 2 . 7 2 . 4 2 . 1 1 ,9 1 .82 1 . 7 5 1 .71 - death r a t i o thresholds, DERT(K) 9 , 0 9 . 0 9 . 0 9 - 0 9 . 0 9 . 0 9 . 0 9 . 0 93 From the above input values, the height growth function needs some c l a r i f i c a t i o n . The sample p l o t data provided only four average dominant height values between ages Ik and 30 years, which were i n -s u f f i c i e n t f o r curve f i t t i n g . I t was found a f t e r comparing these four dominant height values to y i e l d table s i t e index curves that they f e l l almost exactly on the Lake States medium s i t e index curve (see also Figure 3»1). It was decided then to extend the dominant height growth curve on the p l o t by reading height values at higher ages from the Lake States medium s i t e index curve. The height growth function used i n t h i s simulation run was based on these combined height-age information. The same dominant height-age curve would roughly coincide with that of the better than average s i t e c l a s s f o r A l b e r t a , and good-medium (between medium and best, perhaps closer to best) s i t e class f o r Manitoba and Saskatchewan. It has been mentioned before that model parameters CZ and EX were a v a i l a b l e only f o r three increment periods -- excluding i n i t i a l zero values f o r s t a r t i n g age. Extrapolating CZ values presented no d i f f i c u l t y as they showed no trend with age. I t required numerous simulation runs, however, to f i n d appropriate EX values for each period. These l a t t e r values would, of course need to be v e r i f i e d i n the future with a c t u a l data. Values f o r the death r a t i o threshold, DERT, were also r e f i n e d by simulation and t h e i r r e l a t i v e s t a b i l i t y with age was one of the surprises of t h i s study. The main r e s u l t of t h i s simulation run are given i n Table 5.3 i n a y i e l d t a b l e - l i k e format and Figure 5.6 shows g r a p h i c a l l y stand development trends i n average d.b.h. and basal area per acre. Detailed tree s t a t i s t i c s of the h y p o t h e t i c a l stand ( i . e . , p l o t ) for each TABLE 5.3 Aspen stand growth simulation summaries f o r above average s i t e class i n Manitoba and Saskatchewan, (per acre). AGE TOP NUMBER CF TREES AVE • CBH BASAL AREA V. YIELD V. MOAT GROSS V. H L.TOT. L.4+ MORT ALL 4 + TOT. 4 + TOT. 4+ CURRENT YIELD YEARS FT INCHES FT 2 3 FT 14 28.4 4090 0 0 1.6 60. 0. 577. 0. 0. 577. 19 34.3 3420 20 670 2.2 3.8 88. 2. 1037. 13. 43. 1080. 24 39.8 2930 150 490 2.7 4.1 115. 14. 1593. 147. 80. 1715. 29 44.7 2490 520 440 3.2 4.2 135. 49. 2111. 557. 132. 236 5. 34 49.2 2010 850 480 3.6 4.3 144. 87. 2502. 1106. 255. 3011. 39 53.2 1690 1080 320 4. 1 4.5 154. 121. 2914. 1734. 245. 3668. 44 56.8 1340 1060 350 4.6 4.8 151. 136. 3099. 2186. 381. 4234. 49 60^ .0 1040 950 300 5.0 5.2 144. 139. 3159. 2480. 434. 4729. 4H © o 53 © o_ © > < I--150 100 o v., © Q, © 5 0 S o CD 0 H r 10 2 0 3 0 Average db.h. SP or Y T statistics Simulated statistics SP or Y T statistics Basal area per acre A Simulated statistics SP Sample plot Y T Yield table 40, Age 5 0 Figure 5-6 Average d.b.h. and basal area per acre trends from permanent sample plots and y i e l d tables ( S i t e Index 75, MacLeod 1952), and from stand growth simulation f o r above average s i t e s i n Manitoba and Saskatchewan. 96 simulated increment period are presented i n App. I l l , along with tree competition index s t a t i s t i c s both f o r the o r i g i n a l and f o r the transformed competition index values. Cumulative tree d.b.h. and t o t a l height frequency d i s t r i b u t i o n s of the hypothetical stand are shown i n Figures 5.7 and 5,8. In model c a l i b r a t i o n and refinement a l l three kinds of output are used simultaneously, but f o r the purposes of p r a c t i c a l f o r e s t management the y i e l d table format alone w i l l generally provide most growth and y i e l d information needed. The d e t a i l e d tree s t a t i s t i c s (App. I l l ) i ndicate that current height increment values of surviving ( l i v i n g ) trees generated by the model follow a more or l e s s symmetrical frequency d i s t r i b u t i o n ; whereas comparable diameter increment d i s t r i b u t i o n s have a p o s i t i v e skewness, i . e . , a r e l a t i v e l y high frequency of low values. This would suggest that the model f u l f i l l s assumption (k) (see p. 82). Although increment values of current mortality trees do not show very d i s t i n c t frequency d i s t r i b u t i o n s , appropriate averages and minimum-maximum values i n d i c a t e that death occurs among trees having r e l a t i v e l y low increment (d.b.h. and height) that generally f a l l s below the cal c u l a t e d average current increment of surviving t r e e s . The same applies considering t o t a l height and d.b.h. of mortality trees, rather than t h e i r increments. The l i v i n g trees'height and d.b.h. frequency d i s t r i b u t i o n at d i f f e r e n t periods are s i m i l a r to comparable d i s t r i b u t i o n s of act u a l stands (Figures 5-7 and 5.8). The range of tree sizes also appears reasonable, as w e l l as the c o e f f i c i e n t of v a r i a t i o n values which gradually decrease with higher-age. Another u s e f u l s t a t i s t i c i n model c a l i b r a t i o n i s the height/ d.b.h. ( H / D ) r a t i o . I t can be ca l c u l a t e d from comparable average, minimum, or maximum values given i n App. I I I . For example, H / D r a t i o s Height (feet) Figure 5.7 Cumulative tree height frequency d i s t r i b u t i o n s (10 classes between minimum and maximum values) from a stand growth simulation f o r above average s i t e s i n Manitoba and Saskatchewan. D.b.h. ( i n c h e s ) Cumulative tree d.&Jfc. frequency d i s t r i b u t i o n s (10 classes between minimum and ma^  values) from a stand, growth simulation for above average s i t e s i n Manitoba and Saskatchewan. 99 c a l c u l a t e d from average height and d.b.h. values of l i v i n g trees decrease from 13.2 at age lk, to 9-8 at age k<9. Decrease i n t h i s r a t i o was also gradual and may be considered a good approximation of the trend occuring i n a c t u a l stands. Similar r a t i o s of mortality trees are generally higher f o r comparable periods and they also decrease with age, e.g., from 17.5 at age 19, to 11.5 at age k9. The above analysis of tree s t a t i s t i c s revealed no obvious s t r u c t u r a l or computational flaws i n the model. This i s an i m p l i c i t proof, and based on the s i l e n t assumption that only a sound model can produce c o n s i s t e n t l y reasonable simulated r e s u l t s ( t h i s i s almost l i k e saying "the end j u s t i f i e s the means"). Making such an assumption may be j u s t i f i e d by the nature of the model, i t s development and i t s complexity. Simulation r e s u l t s summarized on a stand, or per acre basis allow further a n a l y s i s of the model. The wide use and f a m i l i a r i t y with stand s t a t i s t i c s makes them p a r t i c u l a r l y suitable f o r comparisons. In t h i s study, three stand s t a t i s t i c s - were used f o r t h i s purpose: t o t a l number of l i v i n g trees; average d.b.h.. of a l l trees; and t o t a l basal area. Actual stand data obtained from the remeasurement of the afore-mentioned permanent sample p l o t afforded an i d e a l comparison up to age 30. Over that age, development trends i n the Alberta normal y i e l d tables (MacLeod 1952) seemed most appropriate and were used. On the basis of the f i r s t 30 years' growth, the sample (and the simulated) stand would f a l l i n the intermediate s i t e c l a s s , i . e . , between medium and best s i t e , of t h i s normal y i e l d table. Thus, the necessary stand s t a t i s t i c s over 30 years were obtained from t h i s y i e l d table by i n t e r p o l a t i o n . Those values, as well as stand s t a t i s t i c s from the sample p l o t , are given i n 100 Table 5-4 and i l l u s t r a t e d i n Figure 5-6. A comparison of simulated r e s u l t s and a c t u a l stand development trends i n Figure 5-6 revealed remarkable s i m i l a r i t y . This suggests that the model approximates q u a l i t a t i v e stand development processes to a s a t i s f a c t o r y degree. P a r t i c u l a r l y r e l i a b l e comparison can be made up to age 30 using permanent sample p l o t data. I t i s open to question, though, how close simulated r e s u l t s should agree with a c t u a l stand development trends. Attempts to achieve too close agreement may be f r u s t r a t e d , or even rendered meaningless by border e f f e c t s , unaccounted v a r i a t i o n i n growth and mortality, e t c . . The confounding e f f e c t of such factors could be a l l e v i a t e d somewhat by using l a r g e r p l o t s with more trees, but such p r a c t i c e would also increase simulation cost i n geometric progression. A simulation run l i k e the one described here takes up nearly 4 minutes on an IBM System 360/67 computer. It was mentioned before that numerous simulation runs were conducted to debug the program, c a l i b r a t e the model, and r e f i n e some of the model parameters and c o e f f i c i e n t s . The r e f i n i n g procedure was use-f u l f o r f i n d i n g out about the e f f e c t of s p e c i f i c changes i n a p a r t i c u l a r parameter on model performance, i . e . , on simulated stand growth and development. The e f f e c t of death r a t i o threshold, DEBT, i s the easiest to understand. As can be i n f e r r e d from the program l i s t i n g (App. I I ) , i t s values determine, u l t i m a t e l y the amount of mortality i n a given period. The higher t h i s threshold, the fewer trees w i l l d i e . Considering a DERT value of 9.0, as i n the present simulation run, a h a l f unit change i n t h i s value may not have more than 1 to 2 per cent change i n the amount of mortality at any one period. The e f f e c t i s non-linear, however, and 101 • increases with greater change. TABLE 5-4 Stand development s t a t i s t i c s from a permanent sample p l o t , and from the Alberta normal y i e l d tables, above average aspen s i t e ( S i t e Index 75, MacLeod 1952).. Source Stand Age Number of trees Ave. d.b.h.. (inches) Basal area ( f t . /acre) 14 4090 1.6 60 Sample 20 3330 2.4 101 Plot 25 2590 2.9 121 30 2190 i4o 35 - _ -Y i e l d 40 2060 3.6 142 Table 45 - - -50 1450 4.3 149 While the amount of mortality i s the function of DERT, the d i s t r i b u t i o n of tree mortality i n r e l a t i o n to tree competition status and current increment i s determined by the r a t i o of AHIN and SDD (a s p e c i f i e d standard, deviation; 'see also p. 87). For the present simulation run SDD=0.5 was used which r e s u l t e d i n a rfuadom--component with considerable d i s p e r s i o n . Thus i t was possible to generate mor-t a l i t y among trees with higher than the minimum increment and greater than minimum s i z e (see App. I l l ) ; a much more r e a l i s t i c approximation of actual stand growth and development (see also Figure 3-5) than having a l l trees eliminated with increment or size below a c e r t a i n minimum value. Raising SDD to 1, 2 or 3 r e s u l t s i n correspondingly l e s s e r dispersion of the deviates, and an increased concentration of mor t a l i t y among trees with low increment and small s i z e . In f a c t , with SDD values 3-0 or over, the spread i n increment or s i z e of dead trees becomes so narrow that generating mortality w i l l amount to complete 102 elimination of the slowest growing, smallest t r e e s . That such elimination i s u n r e a l i s t i c was proven by simulation runs with SDD=3«0. A f t e r only one or two increment periods, the height and d.b.h. frequency d i s t r i b u t i o n of the simulated stand resembled a truncated normal d i s t r i b u t i o n with no values at the lower end; a condition not found i n nature. These computer generated random normal deviates are not t r u l y random, as they are c a l c u l a t e d by deterministic rule and a s e r i e s of deviates are reproduceable from the same i n i t i a l number (Naylor et_ a l . 1968, give d e t a i l e d discussion on random number and stochastic v a r i a b l e generation). While t h i s procedure may be c r i t i c i z e d on t h e o r e t i c a l grounds, using such "pseudo" random numbers has important advantages in simulation. For example, i t becomes possible to study the e f f e c t of c e r t a i n parameters without the d i s t u r b i n g (confounding) e f f e c t of random v a r i a t i o n ; on the other hand, the e f f e c t of random v a r i a t i o n can be studied simply by doing simulation runs on the model with d i f f e r e n t sets (from d i f f e r e n t i n i t i a l numbers) of random deviates. This l a t t e r i s analogous to sampling from a population, with the random v a r i a t i o n corresponding to sampling e r r o r . Three simulation runs with d i f f e r e n t i n i t i a l random numbers resul t e d i n very s i m i l a r tree growth and stand development. V a r i a t i o n i n t o t a l number of trees and basal area between simulated trends was under 2 per cent at harvest age of 49 years, although differences were as high as 5 per cent at younger ages. The e f f e c t s of the other two model parameters, CZ and EX, on tree growth and stand development are much more complex and so i s the analysis of t h e i r r e l a t i v e e f f e c t s . The analysis of competition sub-model i n Section 5.4 revealed that an increase i n either (or both) of these parameters increases tree competition index values. But while a 103 higher CZ r e s u l t s i n a ce r t a i n per cent increase i n each tree's competition index, a higher EX value increases the range i n index values and the e f f e c t i s p a r t i c u l a r l y strong at the two extremes of competition. As the analysis of competition submodel showed CZ parameter values quite stable with age, i . e . , close to 3-0> the same value (3.0) was used a l l through every simulation and no a d d i t i o n a l s e n s i t i v i t y analysis was conducted on t h i s parameter. In contrast to CZ, the EX parameter i s age dependent, v i z . , i t s values decrease with age. The rate of decrease i n EX values i s also age dependent and i s much higher i n young stands than i n older ones. Simulation runs conducted to r e f i n e extrapolated EX values f o r increment periods over age 30 showed that increasing the value of t h i s parameter re s u l t s i n increased mortality among the slow growing t r e e s . While t h i s increase i n mortality was comparatively small following the change (which never exceeded 0.1) i n EX, i t was s u f f i c i e n t , at l e a s t i n these simulation runs, to slow the accumulation of growing stock and prevent excessive mortality at a l a t t e r p e riod. The e f f e c t of height increment adjustment, BNO, was studied by t r y i n g a range of BNO value from O.65 to 1.0. An optimum value, found near 0.8 was used i n the f i n a l simulation run. Values under 0.8 discounted competition influences too much and re s u l t e d i n excessive growth rates. Values close to 1.0 produced the opposite e f f e c t -- too slow growth f o r the suppressed and intermediate t r e e s . S e n s i t i v i t y analysis was also conducted to determine the r e l a t i v e e f f e c t of adjusting (lowering) the la r g e s t trees diameter increment using a range of AY and ADJP values. For both of these co-e f f i c i e n t s , values up to 0.06 were t r i e d , which meant as much as 6% reduction i n diameter increment even for the largest trees of the simu-i c 4 l a t e d stand. The r e s u l t of t h i s analysis turned out to be quite a s u r p r i s e . The t o t a l f i n a l y i e l d of hypothetical stands having t h i s reduced diameter increment was a c t u a l l y higher (as much as 10% i n t o t a l basal area and volume) than s i m i l a r stands without diameter increment reduction. This apparent contradiction seems to a r i s e from the greater uniformity i n tree diameters; i n p a r t i c u l a r , from the l a c k of a few r e l a t i v e l y too large ( i n d.b.h.) t r e e s . It should be r e a l i z e d that even a few large trees may have a s i g n i f i c a n t e f f e c t on the competition' index, and growth, of the majority of smaller trees on the p l o t . Because so many trees may be a f f e c t e d , even a comparatively minor tree increment diffe r e n c e associated with competition may add up to a s i g n i -f i c a n t difference i n y i e l d f o r the whole stand. These r e s u l t s also have important implications regarding p l a n t a t i o n development. I t i s now generally accepted that plantations out produce natural stands of the same species on comparable s i t e s . Although appropriate stand development studies so f a r d i d not prove conclusively the cause of t h i s d i f f e r e n c e , improved crown and root devel-opment associated with regular spacing i s believed to be of primary importance. Stand growth simulation r e s u l t s described i n the previous paragraph suggest, however, that t h i s difference i n production between plantations and n a t u r a l stands may, to a c e r t a i n extent, be due to an i n t e r a c t i o n between tree size uniformity ( i . e . , smaller dispersion i n d.b.h. frequency d i s t r i b u t i o n ) , lower competition e f f e c t s and consequently f a s t e r growth of medium s i z e t r e e s . 5.72 Simulating growth of normal density aspen stands growing on poor and on excellent s i t e s For simulating growth and stand development on a poor s i t e , the necessary tree input data (d.b.h., tree l o c a t i o n , d.b.h.-height 105 r e l a t i o n s h i p ) were obtained from a .10 acre sample p l o t established i n an appropriate, uniform, even-aged, 15-year-old, pure aspen stand of f u l l stocking and "normal" density. Again, only one clone was assumed to be present on the p l o t . The necessary height growth function was based on the Lake States curves and i t follows t h e i r shape ( t h i s f a l l s almost exactly i n between Alberta s i t e index class 50 and 60) . The l e v e l of the p a r t i c u l a r poor s i t e curve was selected on the basis of average dominant height on the sample p l o t at age 15. The same model c o e f f i c i e n t s were used as i n the f i n a l c a l i -bration run previously described. Model parameter EX values were also the same, as stand age at the s t a r t of simulation were nearly i d e n t i c a l (15 compared to Ik years). Appropriate CZ and DERT parameters were selected through experimental simulations. Optimum CZ values used were: 0 .0 , 3 .5 , 3 .5 , 3-5, 3-5, 3-5, 3-5, 3-5, and those of DERT were: 19 . 0 , 19 .0 , 19 .0 , 19 .0 , 19 .0 , 19 .0 , 19 .0 , 19 .0 . Table 5-5 presents simulated stand s t a t i s t i c s i n a y i e l d table l i k e format, and Figure 5.9 i l l u s t r a t e s average d.b.h. and basal area per acre trends. For simulating growth and stand development on an excellent  s i t e the same tree data input was used as f o r the above average . s i t e , a f t e r lowering i n i t i a l stand age from ik to 10 years. In other words, i t was assumed that tree sizes and stand density on the best s i t e s at age 10 would be s i m i l a r to tree sizes and stand density at age Ik i n stands growing on good medium s i t e s . One clone per p l o t was s p e c i f i e d . The shape of the height growth function followed the Lake States s i t e index curves, and i t s value at age 10 was equal to the average dominant height on the p l o t at that age. (The height growth curve roughly corresponds to Alberta s i t e index c l a s s 90•) TABLE 5.4 Aspen stand growth simulation summaries for poor and excellent s i t e classes i n Manitoba and Saskatchewan.;, (per acre). POOR SITE AGE TOP NUMBER OF TREES A V E . D B H BASAL AREA V . Y I E L O V.MORT GROSS V. H L . T O T . L.4+ MORT ALL T O T . 4+ T O T . •• 4 + CURRENT Y I E L D YEARS FT INCHES 2 FT FT 3 15 21 .1 5860: 0 0 1.3 5 5 . 0 . 3 8 3 . 0 . 0 . 3 8 3 . 20 25.6 4360 0 1500 1.8 7 4 . 0* 6 3 9 . 0 . 4 2 . 6 8 1 . 25 2 9 . 7 3670 0 690 2 .1 9 1 . 0 . 9 3 0 . 0 . 4 6 . 1 0 1 8 . 30 3 3 . 4 3 120 9o: 550 2 . 5 3.9 1 0 7 . 7 . 1 2 4 2 . 5 9 . 7 3 . 1 4 0 4 . 35 36.6 2800 260 320 2.8 4 . 0 12 0 . 2 3 . 1 5 3 8 . 2 0 1 . 6 3 . 1 7 6 2 . 40 39.6 2330 480 470 3 .1 4 . 1 1 2 5 . 4 4 . 1748 . 4 2 7 . 1 5 2 . 2 1 2 5 . 45 4 2 . 2 1970 720 360 3 . 5 4 . 2 1 2 8 . 6 8 . 1 9 2 4 . 7 2 3 . 1 6 2 . 2 4 6 3 . 50 4 4 . 5 1610 810 . 3 6 0 3 . 8 . . 4 . 3 126 . 8 3 . 2 0 0 9 . , 9 7 0 . 2 0 9 . 2 7 5 6 . EXCELLENT SITE A G E TCP N U M e E R OF TREES AVE.DBH EASAL AREA V.YIELD V.MORT GROSS V . H L.TOT. L.4+ MOPT ALL 4+ TCT. 4+ TCT. 4+ CURRENT YIELD . 10 28 .4 4090 0 C 1.6 6C . C. 577. C. C. 577. 15 36. 5 2 9 5 0 9C 114C 2 .4 3.8 95 . 7. 1208. 59. ICS. 1317. 20 43.8 225C 44C 7CC 3.1 4. 1 121. 4C. 18 69. 442 . 234 . 2212. 25 50.2 1800 910 450 3.8 4.3 139. 94. 2484. 1219. 29C. 3117. 3C = 6. C 147C 1CSC 330 4.3 4.7 151 . 130. 3033. 20 19 . 333 . 40CG. 35 61.2 12CC 1060 27C 5.0 5. 2 163. 1*5. 3619. 2799. 354. 4939 . 40 65.8 1010 970 190' 5.5 5.6 1,6 7 . 164. 3992. 33 14. 387. 57CC. 45 *9.9 8 6C S6C 15C 6. C 6.C ' 171 . 171 . 4384 . 3795. 364. 6455. 50 73.6 710 7 10 15C 6.6 6 . 6 167. 167. 4565. 405 1. 47£. 7114. 150 4 -CM 100 8 Q . O - 5 0 o o CO 0 S -^10 Average d.b.h. YT statistics Simulated statistics A YT statistics Basal area per acre • Simulated statistics Y T Yield table 2 0 3 0 4 0 , Age 5 0 Figure 5.9 Average d.b.h. and basal area per acre trends from y i e l d tables ( S i t e Index 55, MacLeod 1952) and from stand growth simulation for poor s i t e s i n Manitoba and Saskatchewan. 7 -6 - - 2 0 0 tp 5 o c -150 4 -0) o Q) 3 — 2 -V -O O CX. O IOO S a m 5 0 10 Average d.b.h. Y T statistics Simulated statistics A Y T statistics Basal area per acre • Simulated statistics Y T Yield table 2 0 3 0 4 0 5 0 Figure 5^ -10 Average d.b.h. and basal area per acre t-rehds from y i e l d tables ( S i t e Index 90, MacLeod 19-52) and from stand growth simulation f o r excellent s i t e s i n Manitoba and Saskatchewan. 109 Model c o e f f i c i e n t s were also the same as f o r the above average s i t e c l a s s . Parameter EX was extended from age 14 to 10, otherwise the same EX values were used at s i m i l a r ages. CZ and DERT parameter values were selected again through experimental simulation runs. The f i n a l op-timum CZ values were: 0.0, 2.5, 2.5, 2.5, 2.5, 2.5, 2.5, 2.5, 2.5; and those of DERT 4.0, 4.0, 4.0, 4.0, 4.0, 4.0, 4.0, 4.0. The lower portion of Table 5-5 contains simulated stand s t a t i s t i c s f o r t h i s s i t e , and Figure 5.10 shows average d.b.h. and basal area per acre trends. Simulation r e s u l t s f o r both s i t e classes showed good agreement with comparable stand development trends i n y i e l d tables (Figure 5-9 and 5.10). For comparison the Alberta normal y i e l d tables are the most appropriate (given i n App. IV). The r e s u l t s of tree s t a t i s t i c s were i n close correspondence with those of the above average s i t e s previously described. The analysis revealed no conspicuous i r r e g u l a r -i t i e s e i t h e r i n increment, or i n tree si z e d i s t r i b u t i o n s and t h i s applied both to l i v i n g and m o r t a l i t y t r e e s . Calculated H/D r a t i o s at d i f f e r e n t increment periods showed, the same type of gradual decrease with age as described previously f o r the good medium s i t e . While these stand growth simulation r e s u l t s seem reasonable on the basis of the above analysis and comparisons with appropriate y i e l d table trends, they should s t i l l be further confirmed using long term data from per-manent sample p l o t measurements, which would also a f f o r d d i r e c t estimation of i n t e r t r e e competition parameters CZ and EX. 5.73 Simulation of stand growth on above average s i t e s f o r two subnormal stand de n s i t i e s In t h i s section, growth and stand development i s simulated fo r two stands having subnormal den s i t i e s at the s t a r t of the simulation. 110 The main purpose of these simulations was to f i n d out whether the model can be used i n i t s present form, a f t e r making appropriate c o e f f i c i e n t changes, to simulate growth and stand development i n stands of subnormal density; or i f some major s t r u c t u r a l changes would be required i n the model. Assuming the model performed s a t i s f a c t o r i l y , then the simulations should provide some preliminary quantitative information on stand development a f t e r c l e a r c u t t i n g and logging when i n i t i a l stand density i s generally lower than that a f t e r w i l d f i r e s . These simulated r e s u l t s would be also u s e f u l to study the e f f e c t of a precommercial thinning on subsequent growth, y i e l d and stand development. The tree input data used i n these simulations are the same as those used f o r model c a l i b r a t i o n (Section 5.71), except that some of the smallest trees were eliminated from the p l o t s . For the denser stand option, only trees with d.b.h. under 1.2 inches were eliminated; i n the more open stand option, a l l trees under l.k inches. While t h i s meant a s u b s t a n t i a l stand density reduction i n terms of number of trees (e.g. over 30 per cent of the trees i n the second option) compared to the o r i g i n a l p l o t , the reduction was l i t t l e i n terms of basal area, because the small trees removed constitute but a minor portion of the stand's basal area. More precise comparison of density l e v e l s may be done d i r e c t l y on the basis of stand summaries (at s t a r t i n g age Ik) from Tables 5.3 and 5.6. To study the growth and development of stands of subnormal density, i d e a l l y one would need some a c t u a l tree data from such stands. Obviously, no n a t u r a l , undisturbed stand would have such truncated tree s i z e frequency d i s t r i b u t i o n as obtained by eliminating a c e r t a i n number of small trees from a dense, Ik-year-old stand. However, the a r t i f i c i a l I l l low density stands w i l l s u f f i c e f o r preliminary a n a l y s i s , u n t i l appro-p r i a t e data are c o l l e c t e d i n na t u r a l stands. In developing the model, four basic assumptions were made r e -garding tree height and diameter growth i n r e l a t i o n to i n t e r t r e e com-p e t i t i o n and mo r t a l i t y . Those assumptions, and t h e i r quantitative representation i n the model, applied to normal density stands. Two of these assumptions which i n essence are: ( l ) the growth of the biggest trees i n the stand i s v i r t u a l l y unaffected by competition, and (3) the growth rate of "intermediate" trees i s proportional to t h e i r r e l a t i v e competitive p o s i t i o n i n the stand, would hold f o r stands with subnormal density. The other two assumptions (2 and k) had to be somewhat extended. Assumption (2) i s now stated thus: while height and d.b.h. increment are lowest among the smallest, generally suppressed trees, even these trees can maintain a c e r t a i n minimum growth rate because of the le s s intense competition i n subnormal stands. Assumption (k) defines the r e l a t i o n s h i p between height and diameter growth rate, i n p a r t i c u l a r , how much more diameter increment i s reduced than height increment because of competition. This assumption i s e s s e n t i a l l y unchanged, but because the general l e v e l of competition i s lower i n subnormal stands, there w i l l be l e s s r e l a t i v e difference i n these two growth rates ( i . e . , increment reductions converge). The model was o r i g i n a l l y designed to allow f l e x i b i l i t y regarding minimum growth rate i n r e l a t i o n to competition (using the BNO c o e f f i c i e n t , explained on page 83), and to a l t e r the r a t i o between height and diameter increment reduction also i n r e l a t i o n to competition (with FCP c o e f f i c i e n t , on page 85). One of each of these c o e f f i c i e n t s were s p e c i f i e d f o r a simulation run and used f o r each i t e r a t i o n ( i n -112 crement p e r i o d ) . While i t may have been reasonable to use the same c o e f f i c i e n t values from s t a r t i n g to harvest age f o r stands which attained a c e r t a i n degree of s t a b i l i t y , the s i t u a t i o n i s much more complex and dynamic i n subnormal stands which are approching normality with i n -creasing age. Obviously, the value of these c o e f f i c i e n t s at a c e r t a i n age then should depend on the r e l a t i v e status of the stand, and the c o e f f i c i e n t values should converge to those found f o r appropriate normal stands. Accordingly, the computer program was a l t e r e d to incorporate FCP and BNO values f o r each increment p e r i o d and use them during simulation. Most input values used f o r simulation were i d e n t i c a l to those already described f o r the f i n a l c a l i b r a t i o n run (see page 91), except f o r i n i t i a l number of trees and BNO and FCP c o e f f i c i e n t s . I n i t i a l number of trees was 326 f o r the denser stand option ( l i g h t thinning) and 271 f o r the more open stand option (medium t h i n n i n g ) . Appropriate values f o r BNO and FCP were estimated by experi-mental simulations. As no s i m i l a r a c t u a l stand development data were av a i l a b l e from permanent p l o t records to use f o r comparison, thinning study r e s u l t s (Steneker and J a r v i s 1 9 6 6 ) provided some crude guide l i n e s as to how stands thinned to s i m i l a r l e v e l of stand density would grow and develop. Stand development trends a v a i l a b l e f o r a comparable normal stand (both simulated and actual) were u s e f u l to i n d i c a t e a possible upper l i m i t of production. A f t e r a few experimental runs the following coef-f i c i e n t s were selected, which gave s a t i s f a c t o r y simulated s t a t i s t i c s : BNO f o r denser stand 0 . 0 , 0 . 7 0 , 0 . 7 3 , 0 . 7 5 , 0 . 7 7 , - 0 . 8 0 , 0 . 8 0 , 0 . 8 0 l e s s dense stand 0 . 0 , 0 . 6 0 , 0 . 6 6 , 0 , 7 0 , 0 . 7 3 , 0 . 7 5 , 0 . 7 7 , 0 , 7 7 FCP f o r denser stand 0 . 0 , O.kO. 0 . 6 0 , 0 . 7 7 , 0 . 8 9 , 0 . 9 6 , 1 . 0 , 1 . 0 l e s s dense stand 0 . 0 , 0 . 0 , 0.50, O . 6 5 , 0 , 7 5 , 0 . 8 5 , O . 9 5 , 1 . 0 113 Stand s t a t i s t i c s from the two f i n a l simulation runs f o r each stand density l e v e l s are presented i n Table 5.6. Development trends depicted by these s t a t i s t i c s appear reasonable i n the l i g h t of current knowledge of growth and y i e l d i n comparable stands. Tree si z e and i n -crement s t a t i s t i c s (not presented) were found a l s o , both f o r l i v i n g and fo r dead trees, to correspond with s i m i l a r s t a t i s t i c s i n a c t u a l stands. H/D r a t i o s c a l c u l a t e d f o r consecutive increment periods•showed a gradual decrease with increasing age, a usual trend i n most stands. I t was mentioned i n the f i r s t part of t h i s section that simulating growth and development of stands having subnormal density was mainly f o r study purposes and, i n a d d i t i o n , to provide some pre-liminary information to forest management. The model di d simulate growth and development of subnormal stand reasonably w e l l , and i t was possible to study and evaluate i n experimental simulation runs some i n t e r -r e l a t i o n s h i p s between height and diameter increment and i n t e r t r e e competition using what i s generally c a l l e d the "black box" approach. Perhaps the most important .result of these simulations was f i n d i n g out what part of the model should have f i r s t p r i o r i t y f o r refinements and extension. This turned out to be the c o n t r o l mechanism that determines growth and to a large extent mortality of i n d i v i d u a l t r e e s . This c o n t r o l r e l a t e s to i n t e r t r e e competition and further improvement i n the c o n t r o l mechanism w i l l have to be based on competition sub •••model refinement. At the same time, ways have to be found also to make the c o n t r o l more s e l f - s u f f i c i e n t and automatic, so that growth simulation fo r d i f f e r e n t stand conditions would require fewer manual controls and adjustment. TABLE 5.6 Aspen stand growth simulation summaries.for two subnormal stands on above average s i t e class i n Manitoba and Saskatchewan.(per acre). DENSER STAND (LIGHT THINNING) A G E TOP NUMBER OF T R E E S A V E . D 6 H B A S A L A R E A V . Y I E L D V . M O R T G R O S S V . H L . T O T . 1 . 4 + MORT A L L T O T . 4+ T O T . 4 + C U R R E N T Y I E L D YEARS ' FT INCHES 2 FT 3 FT 14 2 8 . 4 3 2 6 0 0 0 1 . 8 5 6 . 0 . 5 4 6 . 0 . 0 . 5 4 6 . 19 3 4 . 3 3 0 8 0 20 180 2 . 3 3 . 8 8 9 . 2 . 1 0 4 4 . 1 3 . 2 2 . 1 0 6 6 . 24 3 9 . 8 2 7 9 0 1 9 0 2 9 0 2 . 8 4 . 0 1 1 8 . 1 7 . 1 6 1 1 . 1 7 1 . 6 3 . 1 6 9 6 . 2 9 4 4 . 7 2 4 1 0 5 9 0 3 8 0 3 . 3 4 . 2 1 3 9 . 5 5 . 2 1 4 5 . 6 1 7 . 1 3 8 . 2 3 6 8 . 34 4 9 . 2 2 0 1 0 9 7 0 4 0 0 3 . 7 4 . 3 1 5 3 . 9 9 . 2 6 2 4 . 1 2 4 4 . 2 3 6 . 3 0 8 3 . 3 9 5 3 . 2 1 6 9 0 1180 320 4 . 2 4 . 6 1 6 2 . 1 3 4 . 3 0 2 8 . 1 8 9 6 . 3 0 0 . 3 7 8 7 . 44 5 6 . 8 1 2 7 0 1 0 4 0 4 2 0 4 . 7 5 . 0 1 5 4 . 1 4 0 . 3 1 3 1 . 2 2 9 8 . 5 8 7 . 4 4 7 7 . 49 60 .1 1 0 6 0 9 5 0 2 1 0 5 . 2 5 . 3 1 5 5 . 1 4 8 . 3 3 7 7 . 2 6 8 5 . 2 9 1 . 5 0 1 4 . LESS DENSE STAND (MEDIUM THINNING) A G E TOP NUMBER OF T R E E S A V E . O B H B A S A L A R E A V . Y I E L D V . M O R T G R O S S V . H L . T O T . L . 4 + MORT A L L 4+ T O T . 4+ T O T . 4+ C U R R E N T Y I E L D 14 2 8 . 4 2 7 1 0 0 0 1 . 9 5 1 . 0 . 5 0 7 . 0 . 0 . 5 0 7 . 19 3 4 . 3 2 6 7 0 2 0 4 0 2 . 5 3 . 8 8 8 . 2 . 1 0 4 9 . 1 3 . 7 . 1 0 5 6 . 24 3 9 . 7 2 5 2 0 2 1 0 150 2 . 9 4 . 0 1 1 8 . 1 9 . 1 6 1 4 . 1 8 4 . 4 6 . 1 6 6 7 . 2 9 4 4 . 6 2 3 6 0 5 9 0 1 6 0 3 . 3 4 . 2 1 4 4 . 5 6 . 2 1 9 5 . 6 3 0 . 7 4 . 2 3 2 2 . 34 4 9 . 0 1 9 9 0 1 0 3 0 3 7 0 3 . 8 4 . 3 1 5 4 . 1 0 4 . 2 6 1 1 . 1 2 8 5 . 2 6 5 . 3 0 0 3 . 39 5 3 . 0 1 5 5 0 1 1 7 0 4 4 0 4 . 3 4 . 6 1 5 4 . 1 3 3 . 2 8 5 1 . 1 8 6 0 . 3 9 1 . 3 6 3 4 . 44 5 6 . 6 1 2 9 0 1 1 3 0 2 6 0 4 . 7 4 . 9 1 5 6 . 1 4 7 . 3 1 3 0 . 2 3 1 8 . 3 0 1 . 4 2 1 4 . 49 5 9 . 8 9 8 0 9 4 0 3 1 0 5 . 2 5 . 3 1 4 4 . 1 4 2 . 3 1 1 4 . 2 5 1 0 . 5 4 5 . 4 7 4 4 . 115 CHAPTER 6. SIMULATING DRY MATTER PRODUCTION IN ASPEN STANDS 6.1 Uses of biomass data Past f o r e s t p r o d u c t i v i t y , or y i e l d studies, were c h i e f l y concerned with estimating stem wood volume production to c e r t a i n mer-c h a n t a b i l i t y standards, i . e . , above a c e r t a i n minimum stem diameter. Increase i n raw material demands of the wood using industry and r e l a t e d changes and i n t e n s i f i c a t i o n of f o r e s t management pr a c t i c e s can make some of the o l d standards outdated or inadequate f o r t h e i r intended use. It has already been mentioned i n the introductory chapters that the greatest expansion i n raw material demand i s expected i n the pulp manu-factu r i n g industry, where merchantability l i m i t s of u t i l i z a t i o n are changing p a r t i c u l a r l y f a s t . In f a c t , i t i s predicted that i n the f o r e -seeable future tree components other than stem wood -- v i z . , branches, wood, bark, or even leaves -- may be u t i l i z e d economically* Obviously, when the amount of wood f i b r e produced becomes a major consideration, stem wood volume, or wood volume i n general w i l l be l e s s u s e f u l to express forest production than dry weight of tree components. At the same time, i n t e n s i f i c a t i o n of forest management would l i k e l y be- g r e a t l y helped ' by the increased use of dry weight as a unit of production and y i e l d . For instance, i n the evaluation of are energy t r a n s f e r rates and how these^affected by various c u l t u r a l p r a c t i c e s can be done best on a dry weight b a s i s . Also, i n forest f i r e c o n t r o l and i n prescribed burning, dry matter weights provide d i r e c t estimates of the amount of f u e l . Although there are many advantages i n using dry weight y i e l d estimates, i t w i l l be some time before they become widely accepted and used e i t h e r i n the industry, or i n f o r e s t management and research. Only 116 a l i m i t e d amount of weight data have been accumulated so far on any of the commercial tree species i n t h i s country. At the same time, f a m i l i a r i t y with the use, and the large amount of information r e a d i l y a v a i l a b l e on stem volume estimation and y i e l d make volume measures important f o r some time. The stand growth model developed and tested i n the previous chapter provided stem volume y i e l d and mortality estimates. These volumes were estimated from d.b.h. and tree height using Honer's (1966) stem volume regressions. In the same way, the model may be used f o r p r e d i c t i n g dry matter production by tree components (e.g., stem, branches, leaves, roots) a f t e r appropriate dry weight estimating equations are i n s t a l l e d i n the model, which estimate weight i n terms of d.b.h. and tree height. As no such dry weight estimating equations were a v a i l a b l e f o r aspen these had to be developed. This work i s out-l i n e d i n the following section. Then these equations were incorporated i n the stand growth model and used to estimate dry matter production i n the subsequent simulation runs. 6.2 Development of aspen a e r i a l tree component dry weight estimating  equations This part of the work was reported i n d e t a i l by B e l l a (1968); here only a b r i e f summary i s given. The i n i t i a l sample i n that report covered only younger stands up to 25 years of age, with maximum sample tree d.b.h. of h .2 inches. Since that report was written, further sampling has been undertaken i n older aspen stands, up to age 60. The l a r g e s t sample tree had 8.2 inches d.b.h. Regression equations presented i n the l a t t e r part of t h i s section are based on the combined data. Sample trees were taken from pure, even-aged, f u l l y stocked 117 aspen stands growing on average q u a l i t y s i t e s i n western Manitoba and eastern Saskatchewan. Stands or trees showing excessive browsing or other damage were not included. F i e l d work was c a r r i e d out i n l a t e J u l y and August i n 1967 and 1968. It was assumed that, by mid-July, l e a f expansion had ceased, and that most of the current year's height and diameter growth had also been completed. Procedures regarding f e l l i n g , d i s s e c t i n g , drying and weighing ; tree components were described by B e l l a (1968). For small trees up to about two inches d.b.h., a l l above gound tree components were d r i e d and weights obtained; f o r l a r g e r trees, component dry weights were estimated from subsamples taken from each tr e e . A t o t a l of 152 trees were included i n the a n a l y s i s . Table 6.1 i s a data summary including means, standard deviation and range. TABLE 6.1 Data summary showing means and measures of dispersion of tree component dry weights ( i n grams) and l i n e a r tree size v a r i a b l e s f or 152 aspen t r e e s . Standard Variable Mean Deviation Minimum Maximum Leaf 305 428 8 2,958 Branch 879 2,126 36 18,485 Stem wood 5,686 13,797 . 0 94,719 Stem bark 1,465 3,320 0 24,768 Stem t o t a l 7,150 17,090 119,487 TOTAL 8,334 19,515 71 140,930 D.b.h. ( i n ) I . 8 9 1.45 0.2 8.2 Height ( f t ) 21.00 12.58 6.2 60.0 Simple l i n e a r c o r r e l a t i o n c o e f f i c i e n t s c a l c u l a t e d between the variables studied were a l l p o s i t i v e and h i g h l y s i g n i f i c a n t . The best 2 2 independent v a r i a b l e s , D and D H, generally accounted for over 90$ of 118 the v a r i a t i o n i n aspen component weight. The best independent v a r i a b l e 2 describing l e a f weight, and to a c e r t a i n extent branch weight, was D ; whereas stem components and t o t a l tree weight were most strongly cor-2 r e l a t e d with D H. As a n t i c i p a t e d , strong p o s i t i v e c o r r e l a t i o n s were found between various tree component weights . To a s c e r t a i n whether the r e l a t i v e d i s t r i b u t i o n of d i f f e r e n t components changes with tree weight or tree dimension, a series of multiple regression analyses were conducted with t o t a l weight as i n -dependent v a r i a b l e and weight of components as dependent v a r i a b l e (Table 6.2). Significance of the quadratic t o t a l weight term (T ) would indicate a r e l a t i v e increase or decrease -- depending on the sign --of the p a r t i c u l a r tree component i n r e l a t i o n to t o t a l weight. The basic premise of t h i s a nalysis i s that, i f the proportion of one component . (e.g., branches) increases or decreases, there must be a complementary decrease or increase i n the proportion of the remaining components, as they are a l l expressed i n terms of the sum of components. Condensed r e s u l t s of these multiple regression analyses are presented i n Table 6.2. This includes regression c o e f f i c i e n t s (b.), 2 2 t h e i r s i g n i f i c a n c e ( F x j ) , c o e f f i c i e n t s of determination (R or r ) and standard error of the estimate. Only the s i g n i f i c a n t independent var i a b l e terms were included i n the t a b l e . A l l regressions were highly s i g n i f i c a n t and a combination of t o t a l weight terms generally accounted for more than 90% of the v a r i a t i o n i n component weights. The l i n e a r t o t a l weight term (T) was most important. The s i g n i f i c a n t p o s i t i v e 2 quadratic term (T ) i n the branch regression indicates that the r e l a t i v e amount of branches increases with greater tree s i z e ; whereas the s i g -n i f i c a n t negative quadratic term f o r stem wood indicates a decrease i n 119 the proportion of that component. The quadratic e f f e c t i s much more important f o r the branch component, f o r which, i n f a c t , i t accounts for more v a r i a t i o n than the l i n e a r term. However, the inherently large v a r i a t i o n i n branch weight (see standard error of estimate) makes generalizations somewhat r i s k y . A series of multiple regression analyses were conducted f o r developing component weight estimating equations f or aspen i n terms of stem variables D and H and t h e i r combinations. The .two most important 2 2 independent terms were D and D H, and the i n c l u s i o n of further terms, although s t a t i s t i c a l l y s i g n i f i c a n t f o r one or two components r e s u l t e d only i n minor decrease of the standard e r r o r . TABLE 6.2 Regression analyses f o r detecting possible trends i n aspen a e r i a l component weight d i s t r i b u t i o n , based on 152 trees (weights i n grams). Dependent Var i a b l e -Component Weight Indep. v a r i a b l e s & t h e i r s i g n i f i c a n c e X2 = Xl X,=Total Weight Regn. C o e f f i c i e n t s and F - r a t i o s b„ a b l Fx * 134.89 .0204 985 218.29 .0512 101 450.28 - --407.93 .7588 16098 -200.02 .7062 66642 57.65 .1688 9519 -346.59 .9267 19752 -142.37 .8750 91214 Fx, R or SE of Es t . (gr.) Leaf Branch I ! Stem Wood M t l Stem bark Stem To t a l .00000052 .00000096 .00000051 -.00000051 127 193^ 89 70 .868 .957 .928 •998 .998 .984 • 999 •998 156.1 442.9 572.1 520.7 656.O 4i4.9 574.0 694.8 * Sig n i f i c a n c e l e v e l s : 150) 3.915 H F ( l j l 5 Q ) = 6.81 120 TABLE 6.3 Regressions f o r estimating a e r i a l components weights of aspen based on 152 trees (weights i n grams). Indep. variables & t h e i r s i g n i f i c a n c e Dependent X.. = T o t a l Weight X R 2 SE Variable - c • i\ •of Component Regn. C o e f f i c i e n t s and F - r a t i o s or 2 E s t . Weight a b l Fx * b 2 F x 2 r (gr.) Leaf 19.179 76.737 110 -.65010 25 .910 129.5 11 76.929 i+o.691 1274 - - 139.4 Branch -46.062 73.127 3.56 2.3156 11 .897 685.7 11 77.230 - - .3.6069 1277 • 895 691.6 Stem Wood 60.035 81.976 2.77 23.21+9 700 .996 871.6 t i 11 198.250 - - 2l+ .696 3721+5 .996 876.7 Stem bark -67.701 132.1+26 27 3.5530 62 .982 448.6 11 t i 155.57 - - 5.8913 6889 •979 486.3 Stem t o t a l -7.61+0 214.385 13 26.802 616 .996 1071.0 I ! I I 353.82 - - 30.587 355^9 .996 1111.5 TOTAL -34.547 36U.265 21 28.1+67 1+08 •995 1398.2 I T 579.61 - - 34 .899 25765 • 994 1489-6 Significance l e v e l s : % = 3-91; 1$ = 6.81 The regression combinations selected f o r estimation are presented i n Table 6.3. Regression s t a t i s t i c s included are: regression c o e f f i c i e n t s (b ) and t h e i r s i g n i f i c a n c e (Fx.), and standard error of the estimate. A l l regressions presented were highly s i g n i f i c a n t , and generally accounted f o r over 9°$ of the v a r i a t i o n i n any tree component weight. Of the two 2 independent terms included i n these regressions, D H was more important 2 i n a l l but the l e a f component regression, for which D was more s u i t a b l e . In some component regressions only one independent term was s i g n i f i c a n t ( v i z . , f o r branch and also f o r stem wood), so the other term need not be included. However, f o r the purposes of t h i s study i t was necessary to obtain additive component weight estimates, i . e . , component 0 121 estimates which would add up to estimated t o t a l weight. Such estimates may be obtained from regressions having exactly the same form and number of independent v a r i a b l e s . Only then would the sum of c o e f f i c i e n t s of component regressions be equal to the corresponding c o e f f i c i e n t s of the t o t a l weight regression, and the sum of estimated component weights equal the estimated t o t a l weight. So, component weight estimating regressions used i n the model included both independent terms regardless of t h e i r s i g n i f i c a n c e . (For more d e t a i l s on the analysis and f o r discussion, see B e l l a 1968). 6.3 Simulating growth and dry matter p r o d u c t i v i t y of normal density  aspen stands on above average s i t e s . The model, described i n d e t a i l and tested f o r normal stands i n the previous chapter, was used a f t e r some minor a l t e r a t i o n s . Re-gression equations f o r estimating stem wood volumes (main program statements 0lk2, and 0155-6, 0160-1; App. I l ) were deleted from the model and dry matter component weight estimating equations presented i n the previous section were i n s t a l l e d . Appropriate changes were made also i n the output. The y i e l d table l i k e stand summary output was changed to include tree component dry weight production estimates instead of stem volumes (main program 0217 to 0230). The f i r s t part of the yield, table l i k e output ( i . e . , age, top height, number of trees per acre, average diameter, and basal area per acre values), however, remained unchanged. Input values for the present simulation were i d e n t i c a l , with the exception of the i n i t i a l value of death threshold random var i a b l e i n (DRND), to those s p e c i f i e d i n the previous chapter f o r simulating growth and stem volume i n a normal stand (p • 88) on a s i m i l a r s i t e c l a s s . 122 Because of the d i f f e r e n t DRND value, another set of random deviates was generated and used i n the simulation. As these random deviates determine, to a c e r t a i n extent, which t r e e s a c t u a l l y die i n a given period, a d i f -ferent set of random deviates implies differences i n subsequent stand growth and development. It was mentioned e a r l i e r (p. 102) that such differences a r i s i n g due to random deviates may be comparable to sampling error, and f o r the t e s t runs described (p. 102), the difference i n y i e l d between sets of random deviate ser i e s would be under 5$. Simulated stand summaries are presented i n Table 6.4. Growth and stand development trends i n t h i s simulation are nearly i d e n t i c a l (differences under 5$) to those given i n Table 5.6, where p r o d u c t i v i t y was expressed i n terms of stem volume. Sl i g h t differences (e.g., i n number of trees and basal area per acre) have r i s e n from using a d i f -ferent set of random deviates i n generating tree mortality; which, i n c i d e n t a l l y , provide an example of the magnitude- of difference i n stand development and y i e l d that may be expected from t h i s source. Figures 6.1 and 6.2 present g r a p h i c a l l y some dry matter p r o d u c t i v i t y trends i n r e l a t i o n to age. The t o t a l above ground biomass of the simulated stand at age 49 i s about 55 short tons; of which about 71$ i s stem wood, 17% i s stem bark, 10% i s branch and only 1% i s l e a f . The amount of stem wood, both i n absolute and i n r e l a t i v e terms, increases with age, whereas the amount of leaves d e c l i n e s . The absolute amount of stem bark and branches increases with age, but the r e l a t i v e amounts are f a i r l y stable -- between 17.2 and 18.6$ for bark, and 9.9 to 11.0$ f o r branches. The rate of t o t a l dry matter, and stem wood and bark production i s highest between 20 and 25 years of age, and the rates s t a r t to decline between 25 and 30 years. TABLE G.h Aspen stand growth and dry matter productivity simulation summaries, above average site Manitoba and Saskatchewan. (Per acre values.) D R 1 M A T T S R V. S I G K T (IS LB S.) TO? i\i L":43E P. OF T R3SS AV3.D3H; BA . H L.T0T. '1.4+ !<!0RT; ALL 4- TOT . 4+ L 17 1 !'• G -I 0 R T A_L I T Y G R OSS 2 ST 3-1 'BRANCH LEAF | TOTAL ST 3M BRANCH TOTAL 5T EPANCH , TOTAL EARS .FT IHCHES FT ; HOOD ' BARK 'WOOD BARK WOOD 14 28.4 409C 0 0 1.6-0. 0" 60 15243 4560 2633 4090 26526 0 0 0 0 1524 3 4560 26 33 22436 19 34.3 3360 20 730 2.2 3. 8 87 1 26031 7655 4508 3360 41554 1315 338 188 1841 27346 7993 4696 40 0 35 24 3 9. 8 2870 150 490 2.7 4. 1 1 13 13 38554 10923 6500 2870 58847 2020 572 332 2924 41889 11833 7020 60 7 42 29 44^ .8 2430 52C 440 3.2 4. 2 134 49 50774 13868 8311 2430 75383 3729 1054 623 5406 57838 15832 9454 83 124 34 49.2 2010 860^ ' 420 3.7 4. 3 146 88 60292 15953 9610 2010 87865 5692 1564 933 8189 73048 19481 1 1686 104215 3.9 53.2 1640 1070 3.7.0 4.1 4. 6 150 120 .67209 17294 1046-2 1640 96605 7524 2018 1211 107J5J .87489 22840 13749 124078 44 56.8 1340 1080 300 4.6 4. 8 151 137 72214 18147 11015 1340 102716 7709 2024 1219 10952 100203 257 17 15521 141441 49 60. 1 1140 1020 200 5.0 5. 1 154 147 77851 19175 11671 1140 1C9837 6069 94,8 8587 111909 23315 17125 157349 ro 12h 1 0 0 v> JQ O o O f 5 0 5 a E >» a Per Acre • • Stem wood • • • • — Stem bark 14 19 2 4 2 9 3 4 3 9 4 4 Age 4 9 Figure 6.1 Above ground dry matter pr o d u c t i v i t y by components on age. 1 0 0 o E >> T3 a "5 5 0 CO U 14 Figure 6. 2 19 Stem wood 3 9 2 4 2 9 3 4 D i s t r i b u t i o n of above ground dry matter components over stand age. 4 4 Age 4 9 125 These r e s u l t s apply only f o r s i m i l a r stands growing on above average s i t e s i n Manitoba and Saskatchewan, and are subject to further refinement of the model and the dry matter component estimating equations. 126 CHAPER 7. ASPEN STAND MANAGEMENT ALTERNATIVES AND ECONOMIC IMPLICATIONS At t h i s point i t would be timely to consider the problem of • optimizing timber p r o d u c t i v i t y , i . e . , determining r o t a t i o n length that r e s u l t s i n maximum returns from the f o r e s t . Only very general guide-l i n e s w i l l be presented here on t h i s problem, which should serve as a s t a r t i n g point, or basis, f o r d e t a i l e d analysis of s p e c i f i c management a l t e r n a t i v e s . Under c e r t a i n circumstances the problem i s s i m p l i f i e d and becomes that of maximizing p r o d u c t i v i t y i n terms of stem volume or dry weight. Because estimates f o r these may be obtained through simulation, a b r i e f analysis of such estimates are presented. Because of the longevity of forest enterprise, optimizing timber production ref e r s to a multi-period s i t u a t i o n . Therefore, the present value of a l l future p r o f i t s i s maximized, rather than p r o f i t s i n any one pe r i o d . Assuming t h i s objective, f i n a n c i a l maturity occurs when a stand's a n t i c i p a t e d future value growth w i l l not increase the firm's present net worth (Bentley and Teeguarden 1965). In other words, a longer r o t a t i o n can be j u s t i f i e d only i f the marginal increase i n the value of the timber (marginal value growth) i s greater than, or at le a s t equal to, the marginal costs of holding the timber. The current value of the timber i s expressed as stumpage; t h i s i s the value of the trees on the stump, net of harvest cost. Stumpage value can be determined through d e t a i l e d analysis i f tree sizes and costs are known, and summarized i n the form of revenue functions or f i n a n c i a l y i e l d tables . Rel i a b l e records of s i m i l a r stands harvested i n the past could also be used.. P a r t i c u l a r l y i n the past, stumpage value tended to increase with greater age and l a r g e r tree size because of lower harvest cost and p o t e n t i a l l y better u t i l i z a t i o n of the la r g e r logs f o r lumber. 127 Generally, the Faustmann formula should be used to determine optimum r o t a t i o n because i t takes into account both the growing stock --the cost ( i n t e r e s t ) of holding the c a p i t a l i n the form of timber inventory -- and the land rent. Land i s considered as a f i x e d factor of production, and consequently a l l economic surplus accrues to i t . Thus optimum r o t a t i o n i s c a l c u l a t e d i n d i r e c t l y , and i t coincides with maximum land value. In algebraic form the formula i s as follows: L = - (7.1) (1+1)* - 1 L = land value R(t) = stumpage i n year " t " C = regeneration cost i = i n t e r e s t rate Assuming that values of R ( t ) , C and i are a v a i l a b l e , i t e -r a t i v e procedure may be used to obtain optimum ro t a t i o n age ( t ) when land value (L) i s maximum. The i n t e r e s t rate may be looked upon as an a l t e r n a t i v e highest rate of return on the invested c a p i t a l ( a f t e r adjusting f o r r i s k , uncertainty, etc.) expressed as a decimal f r a c t i o n . Then the problem of optimum r o t a t i o n becomes, e s s e n t i a l l y , that of determining the value growth function R(t) and regeneration cost C. Considering aspen management, some important s i m p l i f i c a t i o n s are possible i n c a l c u l a t i n g optimum r o t a t i o n . F i r s t , regeneration costs may be dropped because aspen stands regenerate r e a d i l y by root suckers as soon as the parent stand i s harvested. And secondly, i f f i b r e production f o r pulp manufacture i s the ch i e f management objective and close u t i l i z a t i o n i s assumed with mechanized multistem harvesting techniques, conventional y i e l d curves showing t o t a l stem volume or weight over time may be used i n l i e u of value growth functions because (a) 128 a l l the stem wood i s u t i l i z e d regardless of stem diameter, and (b) harvesting cost (and stumpage) i s more or les s independent of i n d i v i d u a l tree s i z e . Because of the above s i m p l i f i c a t i o n , p r o d u c t i v i t y alone would tend to determine r o t a t i o n age, which would coincide with maximum average annual increment''". This means that simulated stand development trends and y i e l d values may be used d i r e c t l y for optimizing f o r e s t p r o d u c t i v i t y . The u n i t measure a c t u a l l y used for optimization generally depends on the kind of timber u t i l i z a t i o n ; weight measure i s undoubtedly more meaningful than stem wood volume when the main purpose of timber growing i s f i b r e production for pulp manufacture. Yet, volume measures may s t i l l be preferred o c c a s i o n a l l y because of t h e i r wide spread use i n the past. Rotation age i s determined i n Figure 7.1, incorporating the above s i m p l i f i c a t i o n s , for a normal density aspen stand growing on above average s i t e c l a s s for Manitoba and Saskatchewan. The smoothed y i e l d curves show t o t a l stem wood p r o d u c t i v i t y i n terms of cubic foot volume (from Table 5.6) and weight (from Table 6.4) from s i m i l a r stands. The amount and nature of di f f e r e n c e between the two stand growth simulations i s explained i n section 6.3. The tangents (T^ and T ) drawn from the i n t e r s e c t i o n of the two axes to the y i e l d curves i n d i c a t e at the tangent points (A^ and A^) maximum average annual increment and optimum r o t a t i o n length. On the basis of stem wood weight production, optimum r o t a t i o n i s 33 years of age, whereas on-the basis of volume y i e l d optimum r o t a t i o n f a l l s at 34 years. These two values may be considered i d e n t i c a l f o r p r a c t i c a l purposes. The contact between y i e l d curves and the tangents at tangent points i s not sharply defined, which in d i c a t e s a c e r t a i n f l e x i b i l i t y as to when a c t u a l l y harvest should take place; a year or two ei t h e r way ^ With i n t e r e s t rates greater than three per cent, the f i n a n c i a l r o t a t i o n would be shorter. 701 - 6 0 + O O O 50 H 5 £ 4 0 + 2 " O O O 5 30H E 2 20+1 CO 10-3 _ H -o o o CD E o > •o o o •s E <u CO + Per acre Dry weight •Volume A 2 10 2 0 30 4 0 FIGURE 7.1 Stem wood y i e l d curves f o r an aspen stand groining on good medium s i t e Saskatchewan), and ages of maximum average annual increment. 50 class (Manitoba -Age 130 would have hardly any e f f e c t on production maximization. It should be kept i n mind, however, that i n these simulations no allowance was made for decay l o s s e s . As pointed out i n Chapter 2, decay i n aspen stands begins to occur at around 30 years of age, or even e a r l i e r , and the amount of los s from t h i s cause increases r a p i d l y as the stand matures. Appropriate adjustment of stem wood y i e l d f o r decay would mean lower net increase and a f l a t t e n i n g out of these y i e l d curves at ages over 35-40 years. I f decay losses occured before the age of maximum average annual increment, they would tend to reduce r o t a t i o n length. In the above analyses only stem wood p r o d u c t i v i t y was considered, f o r i t not only makes up by far the greatest portion of the above ground biomass (approximately 70$), but i t a l s o , at l e a s t f o r the time being, i s the sole economic resource. No doubt bark, branches or even leaves w i l l eventually be u t i l i z e d , and then they w i l l also have to be considered i n optimizing p r o d u c t i v i t y . From Figures 6.1 and 6.2 i t appears, however, that t h e i r e f f e c t on optimum r o t a t i o n would be r e l a t i v e l y minor: ( l ) because they make up only a small portion of the t o t a l biomass, and (2) the two most important components a f t e r stem wood, i . e . , stem bark and branches, contribute an approximately constant r e l a t i v e amount to t o t a l dry matter throughout the age studied. Although the proportion, as well as the absolute amount of leaves show a strong decline with age, which would tend to shorten rotations i f leaves were also u t i l i z e d , t h e i r contribution to t o t a l biomass i s too small, e s p e c i a l l y during the c r i t i c a l age period, to be of consequence i n optimizing production. The above example i s a hypothetical one which would have implications only i f most of the above ground wood f i b r e a v a i l a b l e i n an aspen stand was u t i l i z e d and a multiple stem harvesting technique 131 was used. In t h i s case, i n d i v i d u a l tree size would have but minor e f f e c t on harvest cost. Such close u t i l i z a t i o n would probably involve pulp manufacture and p o s s i b l y the use of crown components for feed processing f o r ruminants (Bender et a l . 1970). Of p a r t i c u l a r importance considering such short rotations i s the very rapid early growth of aspen sucker stands, which r e s u l t s i n complete occupancy of the s i t e and r e l a t i v e l y high t o t a l f i b r e y i e l d at young ages. At present, most of the aspen harvested i n the P r a i r i e Provinces i s used f o r p a r t i c l e - b o a r d manufacture. One company engaged i n such an enterprise i n e a s t - c e n t r a l Saskatchewan conducts most of i t s logging i n stands of about 70 years o l d having average d.b.h. close to s i x inches with a range of d.b.h. from h to 10 inches. Only s i x foot s t i c k s with a minimum of k inch outside diameter are u t i l i z e d . Y i e l d information f o r a d e t a i l e d economic analysis based on present management objectives w i l l be provided by the present model with simulations extended to 70 or 80 years of age, when a d d i t i o n a l data i s a v a i l a b l e on the growth-competition r e l a t i o n s h i p s of aspen. It w i l l also be necessary to consider possible decay losses i n these stand growth simulations. The increase i n volume and decrease i n logging cost of conventional logging can be expected to increase r o t a t i o n length well beyond those based on maximization of dry matter y i e l d s . 132 CHAPTER 8 . SUMMARY AND CONCLUSIONS The model as presented simulates growth and development of aspen stands i n terms of i n d i v i d u a l tree growth and mortality. The major objective i n bu i l d i n g t h i s model was to provide a f l e x i b l e t o o l to predict growth and y i e l d f o r a range of stand conditions of po-t e n t i a l importance i n aspen management. This stand model was designed to have two main a t t r i b u t e s : realism and g e n e r a l i t y . In t h i s context, realism means that the major components of tree growth and development had to be quantitatively-described or represented i n the model. Generality r e l a t e s to o v e r a l l model structure. The present model i s intended to be s u f f i c i e n t l y general f o r use i n the future to simulate stand growth and development of most forest tree species growing under a wide range of conditions, a f t e r the r e q u i s i t e tree growth r e l a t i o n s h i p s and parameters have been determined. The f i r s t step i n t h i s type of model bui l d i n g i s a thorough f a m i l i a r i z a t i o n with the modeled system. In bu i l d i n g t h i s stand growth model, only those components, f a c t o r s , or variables which had s i g n i f i -cant e f f e c t on tree growth and mortality were considered. The following components were then i d e n t i f i e d as "basic" (the term used in the context advanced by H o l l i n g 1966) i n growth and stand devel-opment : - environment: s o i l and climate - species c h a r a c t e r i s t i c s : tolerance, growth, habits ( i n d i v i d u a l s , clones, etc.) - i n t e r t r e e competition: stocking, stand density and structure - age 133 Concurrently with the foregoing, a preliminary flow diagram (Figure 5.3) was constructed depicting general system structure. This flow diagram was expended as the model b u i l d i n g progressed and was also t r a n s l a t e d into a computer language (FORTRAN). The basic component i n the model had to be q u a n t i t a t i v e l y expressed, or represented d i r e c t l y i n the computer, e.g., i n d i v i d u a l tree s p a t i a l arrangement by orthogonal co-ordinates. The e f f e c t s of environmental, or s i t e , influence on tree growth were described by the height growth of the dominant trees i n the stand. This expression --which i s unaffected by competition, r e l a t i v e l y stable and easy to determine -- i s p a r t i c u l a r l y u s e f u l to characterize the growth of in t o l e r a n t species l i k e aspen. Such species are u n l i k e l y to stagnate even i n dense stands, thus the measure may be used s a f e l y f o r a range of stand density conditions. Because of these c h a r a c t e r i s t i c s , stand growth simulation was based on the height growth of dominant t r e e s . In closed stands, the degree of tolerance of the species has important e f f e c t s on tree growth, on mortality and on the i n d i v i d u a l trees response to release. In the model, these tolerance e f f e c t s also had to be q u a n t i t a t i v e l y expressed. As they are c l o s e l y r e l a t e d to competition, these e f f e c t s were included i n the i n t e r t r e e competition component. Clonal habit of aspen was considered i n the model only i n -sofar as i t i s manifested i n d i f f e r e n t i a l growth rate between various clones. Thus assigning clone i d e n t i f i c a t i o n would be equivalent to specifying r e l a t i v e growth rate of each clone. Intertree competition i s probably the most important f a c t o r determining the r e l a t i v e growth rate of trees i n a pure even-age stand I3h growing on a uniform s i t e . This competition e f f e c t i s the function of the size of a given tree and r e l a t e d to the s i z e and distance of neighbours, and to species c h a r a c t e r i s t i c s l i k e tolerance and to environmental influences. The competition sub-model developed and used i n t h i s study i s an extension of the influence - zone concept advanced by e a r l i e r workers. The parameters of the competition model were obtained by i t e r a t i v e techniques i n combination with multiple regression and c o r r e l a t i o n analysis using i n d i v i d u a l tree growth records from a per-manent sample plot.. The influence-zone of a tree of a c e r t a i n s i z e (d.b.h.) was assumed to be r e l a t e d to i t s open grown crown width. The necessary regression f o r aspen was based on data from western Canada (Figure 5.2). Competition index values c a l c u l a t e d using the competition model having optimum parameters and combined i n a simple tree increment -- competition regression accounted f o r generally over 50$ of the v a r i a t i o n i n tree d.b.h. increment. Of the two model parameters one ( v i z . , EX) i s an expression of the species tolerance, and the other (CZ) of s i t e i n fluences. A f t e r the analysis and quantitative d e s c r i p t i o n of these components, they had to be i n t e r f a c e d and synthesized as w e l l as translated into a suitable computer program. This i n t e r f a c i n g was probably the most d i f f i c u l t part of model b u i l d i n g , as the analysis of components di d not n e c e s s a r i l y provide the kind of mathematical de s c r i p t i o n that plugs d i r e c t l y into the model. This was found e s p e c i a l l y so for the most important and most complex component, v i z . , i n t e r t r e e competition. This stand model i s based on the height growth of, dominants, 135 which approches that of open growing t r e e s . The model i s programmed to estimate height increment of every tree as though i t was open grown. This increment i s then reduced i n proportion to the i n d i v i d u a l ' s r e l -a t i v e competitive status. P o t e n t i a l diameter increment i s estimated from p o t e n t i a l height increment using a d.b.h./height over height regression based on open grown aspen tree data from western Canada (Figure 5.4). P o t e n t i a l diameter increment reduction for most trees exceeds that of height increment reduction and i s correspondingly more complex, except for the large s t and f o r the smallest trees, f o r which the two increment reductions are s i m i l a r . These reductions were based on assumptions derived from current knowledge of tree growth and mortality i n normal aspen stands. They are (1) the height increment, and to a l e s s e r extent diameter increment, of the l a r g e s t trees (most dominants) i n a stand i s not, or only very l i t t l e a f f e c t e d by competition or stand density; (2) very l i t t l e height and diameter increment, sometimes almost none, together with greatest proportion of mortality occur among suppressed trees; (3) "intermediate" tree -- i n between the above two extreme conditions --grow i n proportion to t h e i r r e l a t i v e competitive p o s i t i o n i n the stand; and (k) as a r e s u l t of competition, the per cent diameter increment of these "intermediate" trees may be reduced to almost h a l f of t h e i r per cent height increment. The model generates suppression, or competition dependent mortality f o r each increment period. It i s assumed that such m o r t a l i t y 136 i s d i r e c t l y r e l a t e d to the tree's competitive status, and i n v e r s e l y r e l a t e d to current tree increment -- including a c e r t a i n amount of random v a r i a t i o n . Trees most l i k e l y to die are the ones with high competition index and slow growth. I f the "death r a t i o " , c a l c u l a t e d by d i v i d i n g tree competition index with current height increment, exceeds that of a c e r t a i n s p e c i f i e d value c a l l e d "death r a t i o threshold" f o r that period, the tree i s assumed to have died. The model was c a l i b r a t e d f o r a normal density stand growing on above average aspen s i t e i n Saskatchewan. Tree s p a t i a l arrangement and i n t i a l tree sizes were obtained from appropriate permanent sample p l o t data. I n i t i a l number of trees was 409 on a 66 by 66 foot p l o t . Starting age, i . e . , age at p l o t establishment, was l 4 , and harvest age was s p e c i f i e d at 4-9- Five-year increment periods were used and one clone was assumed to be present on the p l o t . Competition submodel parameters used were those derived from increment-competition analysis of i n d i v i d u a l tree growth data from the permanent sample p l o t f o r the increment periods covered. For periods not covered by a c t u a l data, the parameters were extrapolated through t r i a l simulation runs. Dominant height growth curves were derived from a c t u a l p l o t data and from appropriate s i t e index curves. Death r a t i o threshold values were also r e f i n e d through simulations. A f t e r a few c a l i b r a t i o n runs and model refinements, simulated stand growth s t a t i s t i c s showed good correspondence with actual growth on the permanent sample p l o t ; and with comparable y i e l d table s t a t i s t i c s (Tables 5.3 and 5.4, and Figure 5.6). These c a l i b r a t i o n runs also provided a s e n s i t i v i t y analysis of the various model parameters, showing how c e r t a i n changes i n these 137 parameters a f f e c t the performance of the modeled system. An unexpected discovery was made i n the process of c a l i b r a t i o n that has implications both on pl a n t a t i o n and on natural stand development. Simulation r e s u l t s suggested that greater y i e l d on plantations i s the r e s u l t of l e s s e r competition e f f e c t s associated with tree si z e uniformity. The growth of normal density aspen stands on poor and on excellent s i t e s was also simulated between ages 10 and 50 years. The necessary tree data f o r poor s i t e were obtained from a .10 acre (66 by 66 f t ) sample p l o t . For the excellent s i t e , the same tree input data were used as for the medium s i t e c l a s s , but using a lower i n i t i a l stand age (10 years instead of ih). I t was assumed that tree sizes and stand density on the best s i t e at age 10 would be s i m i l a r to tree sizes and stand density at age Ik i n stands growing on medium s i t e s . Height growth functions f o r these simulations were based on appropriate y i e l d table s i t e index curves. A f t e r adjusting model parameters CZ and DERT through experimental simulations, stand s t a t i s t i c s thus predicted showed good correspondence with comparable y i e l d table values (Table 5-5 and Figure 5-9). At the same time, tree growth and tree size s t a t i s t i c s and d i s t r i b u t i o n s were also reasonable i n comparison to s i m i l a r values i n a c t u a l aspen stands. On the basis of these growth and stand development simulations of normal stands, the model may be considered s a t i s f a c t o r y f o r general use i n natural stands. This means that normal y i e l d tables could r e a d i l y be generated with the model f or any given s i t e c l a s s defined by height over age r e l a t i o n s h i p . Although so f a r the model was tested only f o r aspen stands, i t could undoubtedly be used f o r other species a f t e r the 138 r e q u i s i t e model parameters have been determined. Simulation of growth and development of subnormal stands were t r i e d a f t e r c e r t a i n r e v i s i o n s of the i n i t i a l model. E s s e n t i a l l y , t h i s amounted to extending two of the basic assumptions (2 and k) and incorporating appropriate changes into the model (computer program). Revised assumption (2) states that while height and d.b.h. increment are lowest among the smallest, generally suppressed trees, even these trees can maintain a c e r t a i n minimum growth rate because of the l e s s intense competition i n subnormal stands. And former assumption (U), which defines the r e l a t i o n s h i p between height and diameter growth rate i n response to competition, loses importance because with lower compe-t i t i o n l e v e l the extra reduction i n d.b.h. increment diminishes. Tree data f o r these simulations were obtained by eliminating a c e r t a i n number of the smallest trees from the p l o t representing a normal stand on medium s i t e . Two density l e v e l s were thus obtained. While these a r t i f i c a l l y created low density stands approximated thinned stand conditions, they were much l e s s suitable to approximate n a t u r a l , undisturbed, subnormal stands that may become established a f t e r logging. The revised model simulated growth and development of sub-normal stands reasonably well (Table 5-6) , and made i t possible to study and evaluate by experimental simulation runs some i n t e r - r e l a t i o n -ships between height and diameter increment and i n t e r t r e e competition. Besides providing preliminary information on the dynamics of s i m i l a r stands, perhaps the most important r e s u l t of these simulations was i n f i n d i n g out what part of the model should have f i r s t p r i o r i t y f o r refinement and extension. This seems to be the c o n t r o l mechanism that determines the growth, and to a large extent mortality, of i n d i v i d u a l t r e e s . The improvements should include further refinement of the 139 competition submodel, as w e l l as making the c o n t r o l more s e l f - r e g u l a t i n g and general f o r use to simulate growth under a v a r i e t y of stand conditions. The model developed was also used to simulate growth and p r o d u c t i v i t y i n terms of tree components dry matter weight, a f t e r the stem volume estimating equations were replaced by appropriate dry matter weight regressions. These regressions estimated weight from d.b.h. and tree height, and were developed from tree weight data c o l -l e c t e d on medium q u a l i t y s i t e s i n western Manitoba and eastern Saskatchewan. Dry matter production simulation was conducted only f o r a normal stand growing on a medium s i t e . Input values f o r t h i s simu-l a t i o n were i d e n t i c a l , except f o r the i n i t i a l values (DRND) of death threshold random v a r i a b l e , to those used f o r simulating growth and stem volume production i n a normal stand on the same s i t e . This difference i n DRND r e s u l t s i n another set of random deviates to be used f o r generating tree mortality, which i n turn has implications f o r subsequent stand growth and development. Differences i n stand growth and development a r i s i n g from t h i s cause, which may be comparable to sampling error i n conventional studies, were under 5%-Simulated r e s u l t s f o r the stand (Table 6.4, Figures 6.1 and 6.2) at 49 years of age showed t o t a l above ground tree biomass of approx-imately 55 short tons; which was made up of about 71% stem wood, 17% stem bark, 10% branch and only 1% l e a f . The amount of stem wood, both i n absolute and i n r e l a t i v e terms, increased with age; whereas the amount of leaves declined. The absolute amount of stem bark and branches increased with age, but the r e l a t i v e amount remained constant. The rate of t o t a l dry matter, stem wood and bark production was highest between 20 and 25 years of age, and the rates started to decline between 25 and iko 30 years. The problem of optimizing timber production f o r aspen stands was b r i e f l y considered. Under'a c e r t a i n type of aspen management (e.g. f i b r e production f o r pulp manufacture combined with mechanical multistetn harvesting techniques) t h i s problem becomes simply maximizing p r o d u c t i v i t y i n terms of stem volume or dry weight. There i s no regeneration cost because aspen stands r e - e s t a b l i s h themselves n a t u r a l l y and quickly by root suckers. Optimum r o t a t i o n length was determined, assuming the above type of management f o r a normal density aspen stand growing on medium s i t e , from simulated y i e l d s as expressed i n stem volume and i n stem wood dry weight. On the basis of weight p r o d u c t i v i t y optimum r o t a t i o n i s 33 years, whereas on the basis of stem volume y i e l d i t i s 34 (Figure 7.1). The analysis showed a c e r t a i n f l e x i b i l i t y , a year or two e i t h e r way, as to when the harvest should take place. No allowance was made f o r pos s i b l e decay l o s s e s , however, and i f such losses occurred before the i n d i c a t e d r o t a t i o n age, they would tend to reduce optimum r o t a t i o n . B r i e f l y , the present model was found su i t a b l e f o r simulating growth and development of pure, even-aged aspen stands. It may r e a d i l y be used as presented to p r e d i c t growth and y i e l d i n normal stands i f the r e q u i s i t e height and age r e l a t i o n s h i p s are a v a i l a b l e ; thus i t provides a convenient a l t e r n a t i v e to normal y i e l d t a b l e s . One of the main advantages of using such models instead of y i e l d tables i s greater f l e x i b i l i t y i n providing y i e l d estimates i n any desired-unit (volume or weight) and merchantability standard, assuming estimating equations are a v a i l a b l e on a tree b a s i s . This f l e x i b i l i t y i s p a r t i c -u l a r l y important at times l i k e the present, from the standpoint of i n t e n s i f i c a t i o n of forest management pr a c t i c e s and research, and because Ikl of the possible r a p i d changes i n wood f i b r e demands, i n u t i l i z a t i o n p r a c t i c e s and r e l a t e d merchantability standards. The model was also used to simulate growth and development of subnormal stands with a reasonable degree of success, but i t was found that further refinements w i l l be necessary before i t can be used for t h i s purpose on a p r a c t i c a l basis . Although at the present stage of development the model i s r e l a t i v e l y crude and can only be considered a preliminary one, i t i s hoped that i t represents e i t h e r d i r e c t l y (e.g., tree s p a t i a l arrange-ment) or i n d i r e c t l y i n mathematical form (e.g., s i t e influence,com-p e t i t i o n e f f e c t s ) , the major component and i n t e r a c t i o n s i n a f o r e s t system as they a f f e c t tree growth and m o r t a l i t y . The model thus o f f e r s almost unlimited scope f o r extension and improvement. A f t e r further study and refinement of the most important i n t e r a c t i o n , i . e . , i n t e r t r e e competition, i t may be possible to simulate the growth of mixed stands as w e l l as uneven aged stands. This i s possible because i n the model the stand i s represented i n terms of i n d i v i d u a l trees, with each having i t s s p e c i f i c c h a r a c t e r i s t i c s defined, or evaluated i n r e l a t i o n to i t s neighbours. 142 LITERATURE CITED Aaltonen, V.T. 1926. On the space arrangement of trees and root competition. J . Forest. 24:627-644. Anderson, G.W. and R.L. Anderson. 1968. Relationship between density of quaking aspen and incidence of Hypoxylon canker. Forest S c i . 14:107-112. Anon. 1957. A t l a s of Canada. Can. Dep. Mines Tech. Surv., Geogr. Br. Publ., Ottawa. Anon. I 9 6 I . Native trees of Canada. 6th Ed. Can. Dep. Forest. B u l l . 61. 291 pp. Baldwin, S.H. and M.M. Yan. 1968. U t i l i z a t i o n of poplar i n fibreboard and p a r t i c l e b o a r d . In Maini, J.S. and J.H. Cayford Ed. "Growth and u t i l i z a t i o n of poplars i n Canada", pp. 191-200. Can. Dep. Forest. Rural Develop., Forest Br., Publ. 1205. Barnes, B.V. 1966. 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Zehngraff, P.J. 1947. Response of young aspen suckers to overhead shade. U.S.D.A., Forest Serv., Lake States Forest Exp. Sta., Tech. Note 278, 1 p. Zehngraff, P.J. 1949. Aspen a forest crop. J . Forest. 47: 555-565. APPENDIX I DETAILED FLOW DIAGRAM OF THE STAND GROWTH MODEL 150 A l i s t and des c r i p t i o n of the more important variables i n the model A Intercept f o r H over Age regression. AC Acre m u l t i p l i e r .-ADJP A value of ADJH(l) above which large trees have no extra diameter growth rate decrease AY An intercept value to adjust (decrease) the largest tree diameter increment. A l Intercept f o r open growing CW over H regression. A2 Intercept f o r open growing D/H over H regression. A3 Intercept f o r H over D regression. B l C o e f f i c i e n t f o r open growing CW over H regression. B2 C o e f f i c i e n t f o r open growing D/H over H regression. BADJ C o e f f i c i e n t f o r c a l c u l a t i n g increment adjusting values from the tree's competition index. B3; C3 Coefficients f o r H over D regression. "Eh; B5 C o e f f i c i e n t s f o r H over Age regression. CZ(K) Influence zone c o e f f i c i e n t at age K. DERT(K) Death r a t i o threshold at age K. DRND I n i t i a l value f o r death threshold random v a r i a b l e . FCP Factor f o r minor adjustment of diameter increment reduction. HRND I n i t i a l random number f o r height increment random v a r i a b l e . IHA Age at harvest. ISA Age at the st a r t of simulation. IXY X-Y co-ordinate option ( l PSP data, 2 random, 3 other). KW Number of growth period a f t e r which output i s wr i t t e n . LGP Length of growth period. NC Number of clones on the p l o t . NCC Number of classes i n the CI frequency d i s t r i b u t i o n . NGP Number of growth periods simulated. NT Number of trees at the s t a r t of simulation. NTTH Number of trees f o r c a i u l a t i n g top height. PS Plot s i z e . SDC Standard deviation value f o r clone height increment random component. SDD Standard deviation value f o r the random component i n the death threshold. SDH Standard deviation value f o r tree height increment random component. SEX(K) Influence zone exponent at age K. 151 ( S T A R T } R E A D ; P S , N T , I S A , I H A , L G P , IXY, N C , IGAM,NCC (descriptive plot and stand statistics, and options) NGP = (IHA-ISA)/LGP + I BAF= 0-005454 R E A D : HRND,SDH,SDC, IHO BNO, FCR AY, ADJP, SDD, DRND Al , 6 1 ^ 2 , 6 2 ^ , 8 4 , 6 5 ^ 3 , 8 3 (CZ(K), K = I,NGP) (SEX(K),K = I,NGP) (GLMIK),K = I,NGP) (RM(K),K = I,NGP) (DERT(K), K = l, NGP) AC, IMTTH, KW (Regression coefficients,model parameters and simulation options) WRITE » Output table headings K= I KR8 = 0 iVIM = 0 VPM = 0.0 VGS =0.0 SVPM = 0.0 DRD = RANDN(DRND) A 6 ® © 152 I 1 R E A D : ( X ( I ) , Y ( D , D U ) , I = I . N T ) (PSP data) 4 1= I,NT H(I)= A 3 + B 3KD ( I ) I C ( I ) = 1 L R E A D : D N , S X , S Y (For generating artificial random tree co-ordinates) Other tree spatial structure option KRN = (Kt KW-I) / KW KT = ISA+LGP* (K-1) 3 8 I = I , N T I i 153 CI (I) =0.0 HIM(I) = 0.0 DIM(I)*0.0 HIL(I) = 0.0 DIL(I) = 0.0 ADJH(I) = 0.0 PHIN(I) = 0.0 3 9 J= I.NCC ADJH(J)= 0.0 WRITE: K,KT and table headings CALL HGRO (K, A,B 4, B5T ISA, IXY, HRND F SDH.SDC.IHO.AHIN.KT) YES 154 36 ' EX = SEX(K) —»-(I = ?JNT) D(I) = (A2 + B2* H(I))*H(I) CALL COMPE (AI,BI,CIZ,EX,CLS,CIMN, NFC,GLD,R, AN,TDIS) 130) STORE (3)K,CIMN,GLD,R,AN,CLS(NFCU),I = I.NCC) (CI statistics) (REDUCE TREE INCREMENT BY COMPETITION AND GENERATE MORTALITY) CRI =0.0 SCR =0.0 CIRN = 0.0 CIRX=0.0 YES 155 l X C R = ALOGIO(CKI)) Y E S C I R N = XCR NO i Y E S C I R X = X C R NO SCR = SCR + XCR C I R ( I ) = XCR C R I = C R I + 1.0 ----- - f 8 AC1R = S C R / C R I RNG = C I R X - C I R N B A D J = B N O / R N G C L A S = R N G / F L O A T ( N C C ) J J = 0 156 NFC(I) = 0 P = C I R N + F L O A T ( J - I ) * C L A S Q = P + C L A S Y E S Y E S V NO NFC(J) = NFC(J) + I J J = J J + I NFC(NCC) = NFC(NCC) + I 157 f S T 0 R E ( 3 ) « CIRN, CIRX, ACIR, BADJ, CRI, CLAS, (NFC(I), I = I, NCC) r YES NO ADJH(I) = (CIR(I) - CIRN) * BADJ YES NO ADJH(I) = |.0 -4) 30 i L = 0 L 4 = i 0 M=0 BA = 0 0 BA4 = 0 0 V= 0 - 0 V4 = 0 - 0 158 Y E S NO 18 • DHR = A 2 + B 2 * H(I) A D J = (1.0 - AOJH(I)) * ADJH(I) * F C P 16 ADJI = ADJI + A D J H ( I ) DIN(I) = DHR * P H I N * ( 1 . 0 - A D J I ) PHIN(I)= P H I N ( I ) * ( 1 . 0 - A D J H ( J ) ) H(I) = H(I) + PHIN(I) D(I) = D(I) + D I N (I) DRD = RANDN(O.O) Y E S 159 DEN = DRD * A H I N / S D D + PHIN(I) + . 0 0 0 0 0 9 DER = C K D / D E N (Mortality t rees) M = M + l MM = M M + I HIM(M) = PHIN(I) DIM(M) = DIN(1) HM(M)= H(I) D M ( M ) = D ( I ) V P M = V P M + D ( I ) * * 2 / ( - . 3 l 2 + + ( 4 3 6 . 6 8 3 / H ( I ) ) ) H(I) = - 1-0 D ( I ) = - I . O 161 (Living trees) L = L + 1 HIL(L) = PHIN(I) DIL(L) = DIN(I) H L ( L ) = H(I) D L ( L ) = D(I) BAI = D ( I ) * * 2 B A = B A + B A I V T = B A I / ( - . 3 l 2 + ( 4 3 6 - 6 8 3 / H ( l ) ) ) V = V + V T L 4 = L 4 + I B A 4 = B A 4 + BAI 2 =((3.0/D(I)) * * 2 ) * ( l . 0 + (0 .5/H(I)) ) V 4 = V 4 + V T * ( 0 - 9 6 0 4 + Z * ( - 0 - I 6 6 - . 7 8 6 8 * 2 ) ) C A L L STAT ( H I L , L , A H I L , S D H I L , C V H I L H I L N , H I L X , N F H I L , C L H I L ) (Descriptive tree s t a t i s t i c s ) C A L L S T A T ( D I L , L , A D I L , S D D I L , C V D I L , D I L N , D I L X , N F D I L , C L D I L ) C A L L S T A T ( H L , L , A H L , S D H L , C V H L , H L N , H L X , N F H L , C L H L ) C A L L S T A T ( D L , L , A D L , S D D L , C V D L , D L N , D L X , N F D L , C L D L ) W R I T E ! (Descriptive tree statistics for living) 6 W R I T E : No mortality j WRITE : ( M o r t a l i t y only one t ree) C A L L S T A T ( H I M , M , A H I M , S D H I M , C V H I M , H I M N . H I M X . N F H I M . C L H I M ) C A L L S T A T (DIM, M , A D I M , S DDI M , C V D I M , D I M N , D I M X , N F D I M , C L D I M ) C A L L STAT ( H M , M , A H M , S D H M , C V H M , H M N , H M X , N F H M , C L H M ) C A L L S T A T ( D M , M , A D M . S D D M , C V D M , D M N , D M X , N F D M , C L D M ) , " — W R I T E '. (Descript ive tree stat. for morta l i ty ) Y E S NO IS HL(I) > HL(J) Y E D M Y = HL(I) HL(I) •-•  H L ( J ) HL( J) = DMY Y E S NO 1 J = J +1 S T H = 0 . 0 16k S T H = S T H + H U I ) ATH= S T H / F L O A T ( N T T H ) . 1 ~ (Obtain yield table like output) F L = L AD = S Q R K B A / F L ) B A = B A * B A F * A C V = V * A C V P M = V P M * A C S V P M = S V P M + V P M V G S = S V P M + V J A C = AC + 0-2 L = L * J A C M = M M * J A C (-) OR (0) 3 4 F L 4 = L 4 A D 4 = S Q R K B A 4 / F L 4 ) B A 4 = B A 4 - * B A F * A C L 4 = L 4 * J A C V 4 = V 4 * AC 35  f STORE(I) : K T , A T H , L , L 4 t M , A D , A D 4 , B A , B A 4 , V , V 4 T V P M t V G s " 165 MM = C O V P M = 0 - 0 Y E S WRITE Yield table-l ike table headings I^WRITE (I) 1 K T , A T H , L , L 4 , M , A D , A D 4 , B A , B A 4 t V , V 4 , V P M , V G S j'wRITE : CI statistics table headings |WRITE ( 3 ) : K,CIMN,GLD,R,AN,CLS(NFC(I),I = I,NCC) ( END ) 166 167 Y E S N O , > C R S = ( C R ( I ) + C R ( J ) ) * * 2 • 5 0 L = 1.4 1 • T D I S ( L ) = 0 . 0 T D I S ( I ) = ( X ( I ) - X ( J ) ) * * 2 + ( Y ( I ) - Y ( J ) ) * * 2 • T D L S ( 2 ) = ( X ( I ) - - X ( J ) + P S ) * * 2 + + ( Y ( I ) - Y ( J ) ) * * 2 T D I S ( 4 ) = ( X ( I ) - X ( J ) + P S ) * x 2 + + ( P S - A B S ( Y ( I ) - Y(J)}) * * 2 T D I S ( 2 ) = ( X ( D - X ( J ) - P S ) * * 2 + + ( Y ( D -- Y ( J ) ) * * 2 T D I S ( 4 ) - ( X ( D -• X ( J ) - P S ) * * 2 + + ( P S - A B S ( Y ( I ) - Y ( J ) ) ) * K 2 I 168 T D I S ( 3 ) = ( ( X ( I ) - X ( J ) ) * * 2 ) + + ( ( Y ( I ) - Y ( J ) + P S ) * * 2 ( - )OR(O) RI = C R ( I ) R2= C R ( J ) TDIS(3) = ( ( X ( I ) - X ( J ) ) * * 2 ) + + ( ( Y ( I ) - Y ( J ) - P S ) * * 2 ) - H I 3 H TD = S Q R T ( T D I S ( D ) S U = RI + T D - R 2 Y E S (300) CR(J) '2„ • R l = CR(J) R 2 = C R ( I ) U Y E S 169 18 17 S = 0 . 5 * ( R I + R 2 + T D ) S S = S * ( S - R I ) * ( S - R 2 ) * ( S - T D ) ( - ) 4 5 , • ( W R I T E : S S , R I , T D , R 2 , S (0)0R(+) 46_ A 9 = S Q R T ( S S ) V= 2 . 0 * A 9 / T D VRI = V / R I V R 2 = V / R 2 Q= A R S I N ( V R I ) Z = A R S I N ( V R I ) R C S = R 2 * * 2 - V * * 2 T D 2 = T D * * 2 IS (0)0R(+) 14 • T D 2 A 2 = ( R I * * 2 ) * ( P I * Q ) —*>~f AT= P I * ( R I * * 2 ) R C S 15 • A 2 = ( R I * * 2 ) * Q \—< A 3 = ( R 2 * * 2 ) - Z AT = A 2 + A 3 - 2-0 * A9 R J I = ( D ( J ) / D ( I ) * * E X CI(I )=CI(I ) + ( A T / ( P I * ( C R ( I ) . x * 2 ) ) * R J I C l ( J ) = CI(J) + (AT/(PI^(CR(J)HH2))jt(l.0/RJD) JL (Obtain C I M E A N , M I N , M A X , V A R ) C I M N = 9 9 9 9 9 9 . C I M X = 0 - 0 S C I = 0 - 0 S S C I = 0 - 0 A N = 0 . 0 A N = A N + 1.0 171 J S C I = S C I + C K I ) S S C I = S S C I + C I ( I ) * * 2 RNG = C I M X -- C I M N C L A S = RNG / F L O A T ( N C C ) J J = 0 N F C ( J ) = 0 P = C I M N + F L O A T ( J - l ) * C L A S Q = P + C L A S I N O , N F C ( J ) = N F C ( J ) + 1 J J = J J + 1 172 A C I = S C I / A N C I V A R = ( S S C I - S C I * * 2 / A N ) / ( A N - I . O ) GLD = ( A C I - C I M N ) / C I V A R R= ( A C I - C I M N ) / G L D R E T U R N X M N = 9 9 9 . X M X = 0 . 0 SDX = 0 . 0 C O V X = 0 . 0 A V X = 0 . 0 CS = 0 . 0 IFC(J) = 0 sx = o.o S X 2 : - 0 - 0 <-' \ • 1 X V I = XV(I) i l h N O N O JL S X = S X + X V I S X 2 = S X 2 + X V I * * 2 Y E S X M N = X V I Y E S 1 X M X = X V I Z Z T J i V N = N V A V X = S X / V N S D X = S Q R T ( ( S X 2 - ( S X « K 2 ) / V N ) / ( V N - I . O ) ) C O V X = I O O . O * S D X / A V X R N G = X M X - X M N C S = R N G / 1 0 . K K = 0 2 j = i,IO 175 P = X M N + F L O A T ( J - l ) * C S Q = P + C S IFC(J) = I F C ( J ) + I K K = K K + I APPENDIX II STAND GROWTH MODEL FORTRAN PROGRAM (MAIN PROGRAM AND SUBROUTINES) 177 C * * S T A N D GROWTH MODEL C * * OY 1 E B E L L A 0 0 0 1 1)1 MENS ICN X I 1000 1 . Y ( 1 0 0 0 ) .01 1 0 0 0 1 , ICI 1000 I . C I I 1 0 0 0 ) , C M N M ! 5 0 ) 1 , C 1 XI 50) . P H I N ! 1000 1 , A U J H ( 1000 1 , B B U O O O I , OE RT ( 50 1 , C 1 AX ( 50 > 2 , C R ( 1 0 0 0 ) , TO I St 4) , N F C ( 100 ) ,H ( 1000 ) ,01 N( 1000 ) , H I L I 1 0 0 0 ) ,TJTL I ' l0001 3 , H I M ( ] 0 0 0 1 , D I M ! 1 0 0 0 1 , HL ( 1 0 0 0 ) , O H 1000 ) , HM t 1 0 0 0 ) , OM ( 1000 1 , B A 0 J 0 ( 50 ) 4 , N F H I L ( I 0 > , N F D I L ( 1 0 I . N F H I M ( 1 0 t , N F D l M ( l 0 l , N F H L ( 1 0 ) , N F 0 L ! 1 0 ) 5 , NFHMI 10 ) , N F O M ( 10 ) . C P R I 201 , S E X I 50 ) , C i. I 50) , C 1 R ( 1000 ) , I RNK 1 1 0 0 0 ) 6 , C C I N ( 5 0 I 0 0 0 2 COMMON P S , N T , N C , I \ C C , N G P , X , Y , H , 0 , I C , PH 1 N , CPR , C I 0 0 0 3 R F A C ( 5 , 9 1 ) PS , N T , I S A , I H A . L G P , 1 X Y . N C , NCC 0 0 C 4 WR I TE (6 ,81 ) P S , N T , I S A , I H A . L G P , I X Y . t v C , NCC 0 0 0 5 81 F O R M A T ( F 5 . 1 , 1 5 , 6 12) c * * I N P U T : PS= P L O T S I Z E , NT=NUMBER OF T R E E S , 1 S A - S T A R T INO AGE c * * I H A = H A R V F S T A G E , L G P = L E N G T H UF GROWTH P E R I O O , c * * [ X Y = X - Y C C - G R O . O P T I O N ( 1 - P S P O A T A , 2 - R A N D O M , 3 - C L U M P E D ) c * * NC=NUMRER OF C L O N E S , " c « * NCC=NUMHER OF C L A S S E S IN CI F R E O . O I S T R . ( U S E 4 0 OR L E S S ) . ' 0 0 0 6 NGP=( I H A - I S A I / L G P * \ c * * NOP NUMBER CF GROWTH P E R I O D 0 0 0 7 B A F = 0 . 0 0 5 4 5 4 OCCfi RE AO ( 5 , 8 5 ) H R N D . S D H . S O C , I HO O 0 C 9 W R I T E 1 6 , H 5 > H R N O . S D H . S D C l h P 0 0 1 0 85 F O R M A T ( F 1 0 . 7 . 2 F 5 . 1 , 1 2 ) C * * 1 N P U T : H R N D = 1 N I T . R A N D O M NUMBER FOR H E I G H T , C * * SOU ANO S O C = T R E E AND C L O N E H E I G H T I N C . S T A N O A R D O E V . U N I T S . c * * I H 0 = D I S P E R S I 0 N OF T R E E H . I N C . ( 1 - D 1 S P . P R O P . H I N C , 2 - P R C P . T O MINI 0 0 1 1 R E A D ( 5 , 8 7 ) B N O , F C P , A Y , A 0 J P , S D D » D R N 0 0 0 1 2 W R I T E ( 6 , 8 7 ) B N O , F C P , A Y , A D J P , S O O , D R N O r.** I N P U T : B N U = A V A L U E G E N E R A L L Y BfTWEF.M 0 . 7 AND 1 . 0 U S E O FOR C A L C U L A T I N G c « * C O E F F I C I E N T OF H E I G H T A D J U S T M E N T , c * * F C P = F ACT CR FOR MINOR A D J U S T M E N T OF D I A M E T E R I N C R E M E N T R E O U C T I O N c « * AY= AN I N T E R C E P T V A L U E T O A D J U S T 1 0 E C R E A S E ) T H E L A R G E S T T R E E S c * * D I A M E T E R I N C R E M E N T , c » * A D J P = V A L U E OF A D J H l I ) ABOVE WHICH L A R G E T R E E S HAVE NO E X T R A c » * D I A M E T E R GROWTH R A T E O E C R E A S E , c » * SDO= S T A N D A R D D E V I A T I O N FOR THE RANOCM COMPONENT IN THE O E A T H c * * T R E S F O L D , . c * * DRND= I N I T I A L V A L U E FOR D E A T H T R E S H G L O RANOCM C O M P O N E N T . 001 3 R E A D I 5 . 8 6 ) A l . B l , A 2 , B 2 , A , B 4 , B 5 , A 3 , B 3 , C 3 0 0 1 4 W R I T E ( 6 , 8 6 1 A l , B 1 , A 2 , B 2 , A , B 4 , B 5 , A 3 , H 3 , C 3 0 0 1 5 86 F O R M A T ! 1 1 F 7 . 3 ) c * * I N P U T : A l AND B I R O P E N GROWING C W - H R E G N . C O E F S . c « * A2 AND B2 = D / H - H R E G N . C C E F S . FOR E S T I M A T I N G D FROM H . c * * A , B 4 ANO B5= H - A G E GROWTH F U N C T I O N C O E F S . c * * A 3 , B 3 , C 3 =H-0 R E G N . C O E F S • IXY O P T I O N 1 . 0 0 1 6 R E A D ( 5 , 8 7) 1 C Z ( K ) , K = l . N G P ) 0 0 1 7 REAR ( 5 , 8 7 1 ( S E X ( K ) , K * U , N G P I 0 0 1 8 W R I T E I 6 . R 7 ) ( C Z ( K ) , K = l , N G P ) 0 0 1 1 W R I T F . ( 6 , 8 7 ) ( S E X ( K ) , K = 1 , N G P ) 0 0 2 0 R E A D ( 5 , 8 7 ) ( D E R T ( K ) , K = 1 , N G P ) 0 0 2 1 W R I T E ( 6 , 8 7 ) ( O E R T ( K ) , K = l , N G P | c * * I N P U T : C Z ( K I AND S E X ( K1= I N F L U E N C E ZONE C C C F . ANO E X P O N E N T c * « C I X ( K ) = MAXIMUM C I V A L U E S AT AGE K . c * « DER T I K ) = 0 E A TH R A T I O T H R E S H O L D 0 0 2 2 87 F O R M A T ! 1 3 F 7 . 3 I 0 0 2 3 R E A D ( 5 , 8 8 ) A C N T T h ' f K W 0 0 2 4 88 F O R M A T ( F 6 . 2 . 2 I 3 1 C * * IMPUT: .AC= A C R E M U L T I P L I E R , NTTH= NUMBER OF T R E E S FOR C A L C . TOP H E I G H T C * * KW= NUMBER OF I N C . P E R I O D S A F T E R WHICH O U T P U T W R I T T E N . 0 0 2 5 W R I T E I 6 . 9 1 0 ) 0 0 2 6 9 1 0 F O R M A T ! 1 H 1 6 X , ' S T A N D GROWTH S I M U L A T I O N M O D E L ' ) 0 0 2 7 W R I T E ( 6 , 9 U ) P S . N T . N C 0 0 2 8 911 FORMA T ( / 6 X , ' P L O T 01 M R N S I C N ' F 5 . 1 , 3 X , 2 0 H I N I T I A L N O . OF T R E E S 1 5 , 1 3 X , 20HNUUBEP. OF C L O N E S . . . . 13 ) C C 2 S WRITE I 6 , 9 1 2 1 I S A , I H A . L G P 0 0 3 0 9 1 2 FORMAT 1 5 X , 1 7 H AGE S T A R T I N G . . . . I 3 , 3 X , 2 1 H A G E AT H A R V E S T 1 4 , 1.3X . 2 0 H L E N G T H OF I N C R . P E R . 13 ) 0 0 3 1 W R I T E ( 6 , 9 1 3 ) I X Y , N G P 0 0 3 2 9 1 3 F O R M A T ( 5 X , 1 6 H X - Y C O - O R D . O P T ION 1 2 , 3 X , 2 2 H N U M B E R OF I N C R . P E R . . . 1 3 ) 0 0 3 3 K= 1 0 0 3 4 KRB = 0 0 0 3 5 • MM = 0 0 0 3 6 V P M = 0 . 0 0 0 3 7 V G S = O . C 0 0 3 8 S V P M = 0 . 0 0 0 3 9 DRD=RANDN(DRNDI 0 0 1 0 I F I I X Y - 2 ) 1 , 2 , 3 0 0 4 1 1 R E A D ( 4 , 8 2 ) < X ( [ ) , Y ( I ) , D 1 I ) , I = 1 , N T I 0 0 4 2 82 F O R M A T ! 3 0 1 2 F 3 . 1 , F 2 . 1 ) ) C * * I N P U T : I N U I V . T R E E X - Y C O - O K O S . ANO C B H . IXY 1 P S P O A T A . C * » T H I S IS IXY O P T I O N 1 , P E R M A N E N T S A M P L E P L O T O A T A ARE U S E O 0 0 4 3 00 4 1 = 1 , N T 0 0 4 4 HI I 1= A 3 * B 3 * 0 < I) 178 C * * ESTIMATE HEIGHT FROM DIAMETER BY H -C REGRESSION 8ASE0 ON OATA C * * OBTAINED AT ESTABL1SHMFNT,A3 AND B3 ARE REGN. COEFFICIENTS. 0045 4 IC C I » = I C * * ONLY ONE CLONE. 0046 GO TO 5 0C47 2 CONTINUE c** IXY OPTION 2,RANDOM X-Y CO-OROS GENERATEO WITH ON MIN.TREE DISTANCE 0048 GO TO 5 . 0049 3 CONTINUE C * * IXY OPTION 3.FOR CLUMPED TREE DISTRIBUTION 0050 5 CONTINUE 0051 51 KRN=(K+KW-1)/KW 0052 KT=ISAt LGP*(K-1 1 0053 CD 38 1=1,NT 0C54 cm 1=0.0 0055 K1MU 1=0.0 0C56 DIM! I 1=0.0 0C57 HILII 1=0.0 0058 DILI I 1 = 0.0 0059 ADJHI I 1=0.0 0060 38 PHINII I=0.0 006 1 WB |TE(6 ,920I K,KT 0062 9Z0 F O R M A T ! / / / 6 X 1 T R E E SUMMARY STATISTICS AND DISTRIBUTIONS FOR INCR. P 1 E R I O D . . • 1 2 , ' , ' 3X ' A G E . . . ' 13, • *********•****<I 0063 WR|TF(6,923I 0064 923 FORMAT!/6X • •NAME ,9X,3HAVE4X,2HSD5X,3HC0V4X>3HMIN4X,3HMAX4X, 1 'CLW 1 2 3 4 5 6 7 8 9 10 • ) 0065 TK = KT C * * TK IS AGE 0066 CALL HGR0IK,A ,B4 ,B5 , ISA, IXY ,HRND,SDH,SOC, IH0 ,AH1N,KT) 0067 I F I K . G T . l ) GO TO 36 0068 IF ( IXY .NE .2 1 GO TO 30 0069 DO 37 1=1,NT 0070 37 D(II= <AZ+B2*H(I))*H(I) 0071 GO TO 30 0072 36 EX=SEX(K) 0073 CU=CZ(K I 0074 CALL C 0 M P E ( A I , B 1 , C I Z , E X , C L S , TDIS,CIMN,C1 M X , A C I , C V R . N F C , AN1 0075 WRITE(3 ,997I K ,C IMN,CI MX,AC 1 , C VR , A N , C L S , (NFC 11 1 ,I= 1,NCC1 0076 997 FORMAT( 5X I 3 , 4F 1 1 . 3 , F 7 . 1, F 11 . 4/401 3 ) C * * NCC=NUMBER OF CLASSES FOR CI FREO. D ISTR. , CIMN=CI MIN, c** NFCII)=NUMBFR OF TREES IN FREO. CLASS I, CIMX=CI MAX, c** AN=MIMBER OF COMPETING TREES. C * * C * * CALCULATE ADJUSTING VALUES TO REDUCE OPEN GROWING HEIGHT AND CIAMETER C * * INCREMENT IN PROPORTION TO TREE COMPETITIVE STATUS I C I ( I I ) . 0077 CR1=0.0 0078 SCR=0.0 0079 CIRN=9999. 00B0 CIRX=0.0 008 1 on R 1=1,NT 008<i- IF (HI I 1 . L E . 0.0 1 GO TO 8 00B3 XCR = AL0G10(CI I I 1 1 c** LOGARITHMIC TRANSFORMATION IS USED FOR NORMALIZING THE FREQUENCY c** D IS TR I BUT ION OF C H I 1-S. 0084 IF (XCR.LT .C IRN) CIRN=XCR 0085 IF (XCK.GT.C IRX) CIRX=XCR 0C86 SCR=SCR+XCR 0087 CIRII)=XCR 0088 CRI=CRI+1.0 0C89 8 CCNTINUE 0090 ACIR=SCR/CRI 009 1 RNG=CIRX-CIRN 0092 13 BADJ= BNO/RNG C * * BADJ IS A COEFFICIENT FOR CALCULATING INCREMENT AOJUSTINf. VALUES C * * (ADJHI K - S ) FROM TREE COMPETITIVE STATUS I C K I I I . 0093 14 CLAS=RNG/FLnAT(NCC) 0094 J J = 0 0C95 DO 42 J=1,NCC 0096 NFC(J ! =0 0097 P=C IRNtFLOAT! J - 1 I * C L A S 0098 Q=P+CLAS 0099 00 4 3 1=1,NT 0100 I F ( H I I 1 . L T . I .0) GO TO 43 0101 I F ( C I R ( I ) . L T . P . O R . C I R I I ) . GE.Q) GO TO 43 0102 N F C IJ ) = N F C (Jl + 1 0103 JJ=JJ+ 1 01C4 43 CONTINUE 0105 42 CONTINUE 01C6 IF I ( F L O A T ( J J I + 0 . 2 ) . L T . C R I 1NFCINCC>=NFC(NCC1+1 0107 WRITE!?.9961 r. IRN.C IRX.ACI R.BAOJ.CRI ,CLAS , INFCI I 1 , 1 = 1, NCC) 0108 996 F U R M A T!BX , 4 F 1 1.7,F7 . I , F 1 1.7/ 4013) 0109 DO 9 1=1,NT 0110 I.FICIl I I . L E . 0.01 GO* TO 9 O i l ! A O J H ! 1 ) = ( r . I R I I I - C I R N I * H A r > J c** A n j H ( l l = A D J U S T M E N T F O R H E I G H T I N C R E M E N T . 0 1 1 2 i r (An.iHi 11 . c r . i . o ) A n j H i n = i . o 0 1 1 3 9 C C N T I N U F c** A I U U S I D I A M E T E R D I N A N D H E I G H T P H I N I N C R E M E N T F D R C O M P E I [ H O N E F F E C T . C * * D I A M E T E R I N C R K M F N 1 D I N I S O B T A I N E D F R O M A D J U S T E D P H I N A N D D/H R A T I O C * * G E N E R A T E C O M P E T I T I O N M O R T A L I T Y l!Y T H R E S H O L D D E A T H R A T I O D E R T . c <•* FOR L I V I N G G E N E R A ! F D I N A N ! ) C E T E R M l N E N E W H A N D D 0 1 1 4 3 0 L = C 0 1 1 5 L 4 = 0 0116 M = 0 0 1 1 7 U A = 0 . 0 O l i d 8 A 4 = 0 . 0 0 1 1 9 V = C 0 0 1 2 0 V 4 = 0 . 0 0 1 2 1 D O 1 5 1 = 1 ,NT 0 1 2 2 I F I I H 1 I . L T . O . O ) G O T O 1 5 0 1 2 3 1 8 O H R = A 2 + B 2 * H ( 1 ) 0 1 2 4 A D J 1 = ( ( 1 . O - A D J H ! I M * A O J H ( I 1 I * F C P 0 1 2 5 IF 1 A D J H I 1 » . L T . A O J P ) A D j t = A D j 1 • ( A Y - ( A Y / A D J P 1 * A D J H ( I)1 0 1 2 6 A C J 1= A O J I > A D J F ( I I -0 1 2 7 D I M 1 l = O H R * P H I N ( I ) * I 1 . 0 - A r . J l l C * * F O R O P E N G R O W I N G T R E E C M I I = 0 . 0 , T H E A D J U S T I N G T E R M B E C O M E S I . 0 1 2 8 P H I N I 1 1-PH[N( 1 1 * l 1 . O - A O J H I I 1 1 0 1 2 9 H( 1 )=HI I 1 + P H 1 N l I ) 0 1 3 0 D! I ) = D ( I 1 • [> I N ! 1 1 0 1 3 1 1 6 D R D = R A N D N ( O . O I 0 1 3 2 I F I O R D . L E . 0 . 0 1 C O T O 1 6 0 1 3 3 C * * D E N = n R D * A H I N / S n D + P H I N I I 1 • 0 . 0 0 0 0 0 9 D R C = R A N D C M V A R I A B L E F U R G E N E R A T I N G M O R T A L I T Y . 0 1 3 4 O E R = C I ( I I / D E N 0 1 3 5 c** I F t D E f ' . L T . n E R T I K ) ) G O T 0 1 6 1 M O R T A L I T Y T R E E S 0 1 3 6 M=M + 1 0 1 3 7 *P = MMH' 0 1 3 8 H I M ( M I = P H [ M I 1 0 1 3 9 D I M ( M 1 = D I N ( I 1 0 1 4 C HM(M1 = H( I 1 0 1 4 1 O M ( M I = 0 ( 1 1 0 1 4 2 V P M=VPM+ O i l 1 * * 2 / 1 - 0 . 3 1 2 * 1 4 3 6 . 6 8 3 / H ( I I I I 0 1 4 3 H ( I ) = - l . O 0 1 4 4 n ( I l = - l . o 0 1 4 5 G O Tf) 1 5 0 1 4 6 1 6 ! c** C O N T I N U E L I V I N G T R F E S 0 1 4 7 L = L » 1 0 1 4 8 H I L ( L T = P H ! N ( 1 1 0 1 4 9 D I L I L ) = O I N ( I 1 0 1 5 0 H L ( L 1 =H( I 1 0 1 5 1 D L ( L ) =1 ( 1 ) 0 1 5 2 I F ( K R N . F Q . K R B ) G O T O 1 5 0 1 5 3 BA 1 =01.1 1 * * 2 0 1 5 4 B A = R A + B A l 0 1 5 5 V T = BA1 / l - O . 3 1 2 + 1 4 3 6 . 6 8 3 / H ! I I I 1 0 1 5 6 v=v+vr 0 1 5 7 I F ( 0 ( I I . L T . 3 . 6 1 G O T O 1 5 0 1 5 8 L 4 = L 4 + 1 0 1 5 9 BA 4 = 0 A 4 +B A 1 0 1 6 0 Z= ! ( 3 . 0 / 0 ( ! ) ) * * 2 ) * ( 1 . 0 + ( 0 . 5 / H ( [ ) > l 0 1 6 1 V 4 = V 4 + V T * ( C . 9 6 0 4 + Z * < - 0 . 1 6 6 - 0 . 7 8 6 B * Z ) ) 0 1 6 2 1 5 C O N T I N U F C * * O B T A I N : L I V I N G T R E E S T A T I S T I C S F O R ( O R AT T H E F N O O F ) T H E C U R R E N T c ** I N C R E M E N T P E R I O D , V I Z . , H E I G H T I N C . . D I A M E T E R I N C . , T O T A L H E I G H T c** A N D D I A M E T E R C 1 6 3 C A L L S T A T ( H I L , L , A H I L , S D H I L , C V F I L , H I L N , H I L X , N F H I L , C L H I L 1 0 1 6 4 C A L L S T A T i n l L . L . A D I L , S O D I L , C V O I L , 0 I L N , 0 [ L X , N F O I L , C L O 1 L ) 0 1 6 5 C A L L S T A T ( H L , L , » H L , S D H L , C V K , H . N , H L X , N F H L , C L H L ) 0 1 6 6 C A L L S T A T ( D L , L , A n L , S D O L , C V D L , C L N , n L X , N F D L , C L O L 1 0 1 6 7 WRIT E ( 6 , 9 2 2 1 0 1 6 H 9 2 2 F 0 R M A K / 6 X ' L I V I N G T R E E C H A R A C T E R I S T I C S ' I 0 1 6 9 W R 1 T E ( 6 , 9 2 4 ) A H I L , S D H I L , C V H I L , H I L N , H I L X , C L H I L , I N F H I L I I ) . 1 = 1 , 1 0 ) 0 1 7 0 9 2 4 F O R M A T ( / 6 X ' H E I G H T I N C 5 F 7 . 2 , F 7 . 3 , 81 4 , 2 I 3 I 0 1 7 1 W R I T E ( 6 , 9 2 5 1 A D ! L , S D D I L , C V D I L , D I L N , O I L X , C L D I L , ( N F D I L ( 1 ) , 1=1, 10) 0 1 7 2 9 2 5 F O R M A T ! 5 X , U H O B H I N C . . . 5F 7 . 2 , F 7 . 3 , 8 I 4 , 2 1 3 ) C 1 7 3 W k 1 T E I 6 , 9 2 6 ) A H L . S D F L . C V H L , H L N , H L X . C L H L , 1 N F H L I11,1=1,101 0 1 7 4 9 2 6 F O R M A T ( 5 X , I I H T O T H E I G H T 5 F 7 . 2 , F 7 . 3 , 8 1 4 , 2 I 3 1 0 1 7 5 W R I T E ( 6 , 9 2 7 ) A O L , S U D L , C V O L , O L N , D L X , C L D L , ( N F D L I I ) , I = I , 1 0 1 0 1 7 6 9 2 7 0 1 7 7 I F I M - l l 2 7 , 2 8 , 2 9 0 1 7 8 2 7 WR I T E ( 6 , 9 2 8 I O l 7 9 9 2 8 F O R M A T 1 h>< * N C M O R T A L I T Y I N T H E L A S T P E R I O D ' ) 0 1 8 0 G O TO 2 0 O l f l l 2 8 W R [ T E ( 6 , 9 2 9 ) 0 1 8 2 9 2 9 F 0 R M A M / 6 X ' M O R T A L I T Y T R E E C H A R A C T E R I S T I C S ' ) 0 1 8 3 .WR I T f ( 6 , 9 2 4 1 HIM( 1) 0 1 8 4 W R I T E ( 6 , 9 ? 5 I DI V ( 1 1 0 1 8 5 W f t | f E ( 6 , 9 2 6 ) HMI1 I 0 1 8 6 W H | 1 F<6 , 9 2 7 ) 0 M ( 1 I 0 1 8 7 GO TO 20 0 1 8 8 29 W R I T E ( 6 , 9 2 9 I C * * O R T A I N l C U P R E N T M O R T A L I T Y T R E E S T A T I S T I C S FOR THE I N C R E M E N T P E R I O D , c** V I Z . . H E I G H T I N C . , D I A M E T E R I N C . , F I N A L H E I G H T AND D I A M F T E R . 0 1 8 9 CALL SrAr (H lM,4 ,AH[M,Sf tH lM,CVHlM,H IMN,H IMX,NFHIM,CLHIM I 0 1 9 0 C A L L ST AT 1 D I M . M , A C I M , S 0 C 1 M . C V D I M , D I M N , D I M X , N F D I M . C L D I M ) 0 1 9 1 C A L L S T A T ( H M , M , A H M , S D H M , C V H M , H M N , H M X , N F H M , C L H M 1 0 1 9 2 C A L L ST AT ( D M , M, A C M , S O O M . C VOM ,DMN ,l)MX , N F 0 M , C L 0 M I 0 1 9 3 HR1TF16 , 9 2 4 1 A H I M , S D H I M . C V H I M . H I M N , H I M X , C L H I M , (NFH1 Ml 1 1 , 1 = 1 , 1 0 ) 0 1 9 4 W R I T E ( 6 , 9 2 5 ) A O I M . S D O I M . C V D I M . n l M N . O l M X , C L D ( M , ( N F 0 I M ( I I . I = 1 , 1 0 ) 0195 WRITF! (0 ,926) AHM,SOHM,CVHM,HMN,HMX,ClHM, lNFMM( 1 1 , 1 = 1 ,101 0 1 9 6 W R I T E ( 6 , ' ) 2 7 > A O M . S D D M . C V D M , O M N , O M X , CL CM, (NFOMI I 1, 1=1 , 10 ) 0 1 9 7 20 I F I K R N . E O . K P B 1 GO TO 21 c** IF NO Y I F L O T A B L E T Y P E I N F O R M A T I O N WANTED B Y P A S S TO 2 1 . 0 1 9 8 KPB =KR N c** S E L E C T THF N T T H T A L L E S T T R E E S FOR C A L C U L A T I N G TOP H E I G H T . 0 199 I=C 0 2 0 0 J= 1 0 2 0 1 25 1=1 + 1 0 2 0 2 24 J = J+ l 0 2 0 3 I K H L I I I . G E . H L ( J ) ) GO TO 23 0 2 0 4 DM Y= H L ( I ) 0 2 0 5 H L ( I I = HI. ( J ) 0 2 0 6 H L ( J ) = 0 M Y 0 2 0 7 23 I F ( J . L T . L ) GO TO 24 0 2 0 8 .1= 1 +1 0 2 0 9 I F ( I . L T . N T T H ) GO TO 25 0 2 1 0 S T H = 0 . 0 0 2 1 1 00 26 1 = 1 , N T T H 0 2 1 2 26 STH=STH+ H L ( I ) 0 2 1 3 A T F = S T H / F L O A T ( N T T H ) i O * C P . T M N Y I E L D T A B L E T Y P E O U T P U T . 0 2 1 4 F L = L 0 2 1 5 A O = S Q R T ( B A / F L 1 0 2 1 6 B A = B A * B A F * A C 0 2 1 7 V = V * A C 0 2 1 8 VPM=VPM4AC !0219 SVPM=SVPM+VPM 0 2 2 0 VGS = SVPM+V 0 221 J A C = A C + 0 . 2 0 2 2 2 L = L * J A C .022 J M = M K * J A C 0 2 2 4 I F ( L 4 ) 3 5 , 3 5 , 3 4 0 2 2 5 F L 4 = L 4 0 2 2 6 A D 4 = S 0 R T ( B A 4 / F L 4 I 0 2 2 7 B A 4 = H A 4 * B A F * A C 0 2 2 8 C 4 = L 4 « J A C 0 2 2 9 V 4 = V 4 * A C 0 2 3 0 35 WRITE(1 , 9 9 9 I K T , A T H . L . L 4 , M , A O , A D 4 , B A , B A 4 , V , V 4 , V P M , VGS 0 2 3 1 9 9 9 F 0 R M A T ( 5 X , 1 4 , F 8 . 1 , 1 7 , 2 1 5 , 2 X , 2 F 5 . 1 , 2 X , 2 F 5 . 0 , 2 X , 2 F 7 . 0 , 2 F 9 . 0 ) 0 2 3 2 MM=0 . 0 2 3 3 V P M = 0 . 0 0 2 3 4 21 K=K+1 0 2 3 5 1 F ( ( I S A + I ( K - 1 ) * L G P I ) . L E . I H A 1 GO TO 51 0 2 3 6 W R I T E ( 6 . 9 3 1 ) C * * W R I T E : O U T P U T H E A O I N G S . A C T U A L O U T P U T WILL COME FROM DISK S T O R A G E . C 2 3 7 W R I T E I 6 , 9 3 2 ) C 2 3 8 931 F O R M A T ! I H 1 1 7 X ' A G E TOP NUMBER OF T R E E S A V E . D B H B A S A L AREA 1 V . Y I E L D V . M O R T G R O S S V . ' 1 0 2 3 9 9 3 2 F O R M A T ( 2 6 X «H L . T O T . L . 4 + MORT A L L 4+ T U T . 4+ 1 T O T . 4+ C U R R E N T Y I E L D ' ) 0 2 4 0 W R I T E ! 2 , 9 9 8 ) 0 2 4 1 99R F O R M A T ( • A C T U A L S T A N D CI D I S T R . C H A R A C T E R I S T I C S : CIMI I N , C I M X . A C 1 , C V f t , N O . T R E E S , C L A S S W . . A N O N F C I I ) ' 1 0 2 4 2 C A L L E X I T 0 2 4 3 END STATEMENTS REPLACING "HGRO" SUBPROGRAM (IN PLACE OF STATEMENTS 0066 AND 0067) A H = A * ( 1 . 0 - 1 B 4 / 2 . 7 1 8 2 8 * * I B 5 * T K ) I ) I F ! ( K - 1 ) . G T . 0 1 GO TO 10 AHB=AH GO TO 11 _10 A HI N = A H - A H B  AHB= AH DO 1 2 1 = 1 , N T I F ( H ( I I . L E . 0 . 0 1 GO TO 12 P H I N ( I ) = A H I N 12 C O N T I N U E GO TO 36 183 0 0 0 1 S U R R O U T I N K H O R O ( K . A , 8 4 , B 5 . I S A , 1 X Y . H R N 0 , S 0 H , S O C , I H C ) , A H 1 N , K 11 C « * G E N E R A T E S H E I G H T S A N O H F I G H T I N C R l ' M F N T B Y D I F F E R E N C E A T R E Q U I R E D C * » I N T E R V A L S . 0 C C 2 D I M E N S I O N X( 1 0 0 0 1 , Y ( 1 0 0 0 1 , D ( 1 0 0 0 ) , I C ( 1 0 0 0 1 , P H 1 N ( 1 0 0 0 1 , C P R < 2 0 1 1 . H I 1 0 O 0 I 0 0 0 3 C O M M O N P S , N T , N C , N C C , N G P , X , Y , H , D , I C , P H I N , C P R 0 0 0 4 T K = K T C « * T K I S A G E 0 0 0 5 A H = A * < 1 . 0 - 1 B 4 / 2 . 7 1 8 Z B * * ( B 5 * T K I ) ) 0 0 C 6 I F I 1 K - 1 1 . C T . 0 ) A H I N = A H - A H H 0 0 0 7 A H B = A H 0 0 0 8 I F ( ( K - 1 1 . G T . O ) G O T O 4 0 0 0 9 A H 1 N = AH c«* G E N E R A T E ( H E I G H T ) G R O W T H P E R F O R M A N C E F O R E A C H C L O N E A T Y E A R I S A , c** A N O C A L C U L A T E C P R ( J I = C L O N E P E R F O R M A N C E R A T I O . 0 0 1 0 1 F I M C . G T . 1 1 G C T O 2 0 0 1 1 C P R I 1 1 = 1 . 0 0 0 ) 2 R E T U R N • o * N C = 1 , O N L Y O N E C L O N E , T H E N I H 0 = 1 A L S O . 0 0 1 3 2 H R = R A N D ( H R N O ) 0 0 1 4 C P R M = 9 9 9 9 9 9 . 0 0 1 5 J = l 0 0 1 6 3 H R = R A N O N ( 0 . 0 ) 0 0 1 - 7 H I N C = H R * A H I N / S D C + A H I N O O t f l I F ( H I N C . L E . 0 . 0 ) G O T O 3 0 0 1 9 C P R I J ) = H I N C / A H I N 0 0 2 0 I F ( C P R ( J ) . L T . C P R M ) C P R M = C P R ( J I 0 0 2 1 J = J * 1 0 0 2 2 I F I J . L E . N C ) G O T O 3 C * * G E N E R A T E I N D I V I C U A L T R E E H E I G H T I N C . . I T S D I S P E R S I O N M A Y B E R E L A T E O T U c * * A V E . C L O N E P E R F O R M A N C E I H I G H E R G R O W T H R A T E , G R E A T E R D I S P E R S I O N , I H O * 1 ) , c * * O R M A Y B E C O N S T A N T F O R A L L C L C N E S A T A G I V E N A G E ( P R O P O R T I O N A L c * * T O H I N C M I N , I H 0 = 2 ) . 0 0 2 3 4 J = l 0 0 2 4 H I N C = C P R ( J 1 * A H I N 0 0 2 5 1 0 1 = 1 c * * I I N D I V I D U A L T R E E , J C L O N E N U M B E R 0 C 2 6 5 I F ( I C ( I ) . N E . J ) G C T O 1 1 0 0 2 7 9 H R = R A N D N ( 0 . 0 ) 0 0 2 8 1 F ( 1 H O - 2 ) 6 , 7 , 7 0 0 2 9 6 P H I N I 1 ( = H R * H I N C / S D H + H I N C c * * P H I N P O T E N T I A L H E I G H T I N C R E M E N T O F T H E I - T H T R E E . 0 0 3 0 G O T O 8 0 0 3 1 7 H I N C M = C P R M * A H I N 0 0 3 2 P H I N I I ) = H R * H I N C M / S D H + H I N C 0 0 3 3 8 I F ( P H I N ( I I . L E . O . O I G O T O 9 0 0 3 4 I F ( P H I N I I ) . G T . ( H 1 N C * H I N C I 1 G O T O 9 0 0 3 5 1 1 1 = 1 * 1 0 0 3 6 I F ( I . L E . N T ) G O T O 5 0 0 3 7 J = J + l 0 0 3 8 I F U . L E . N C I G O T O 1 0 0 0 3 9 R E T U R N 0 0 4 0 . E N D Soooi S U B R O U T I N E C O M P E ( A I , B 1 , C 1 1 , E X , C C A S , T O I S , C I M N , C I M X , A C 1 , C I V R , N F C , A N ) C * * E V A L U A T E C O M P E T I T I V E S T A T U S ( C I I I ) l O F E A C H T R E E O N T H E S T U O Y P L O T . ! 0 0 0 2 D I M E N S I O N X( 1 0 0 0 ) , Y ( 1 0 0 0 ) , D ( 1 0 0 0 ) , I C C 1 0 0 0 ) , P H I N I 1 0 0 0 ) , C P R I 2 0 ) I , C I ( 1 0 0 0 ) , C M N M ( 5 0 1 , N F C ( 1 0 0 1 , C R I 1 0 0 0 ) , T O I S ( 4 ) , H ( 1 0 0 0 1 0 0 0 3 C n M M O N P S , N T , N C , N C C , N G P , X , Y , H , D . I C , P H I N , C P R , C I 0 0 0 4 P I = 3 . 1 4 1 5 9 0 0 0 5 0 0 1 1 = 1 , N T 0 0 0 6 ! F ( H ( I ) ) 1 , 1 , 2 c * * H = - 1 . 0 F O R D E A D T R E E S . c * * C R ( I ) = C O M P E T I T I V E R A O I U S . 0 0 0 7 2 C R ( I ) = ( A l + D I I ) * B l I * 0 . 5 * C 1 Z 0 0 0 8 1 C O N T I N U E 0 0 0 9 0 0 3 1 = 1 , N T 0 0 1 0 0 0 3 J = I , N T c » * I C O M P E T I N G T R E E A N D J C O M P E T I T O R . F O R E A C H I A L L J - S A R E T E S T E D . 0 0 1 1 I F ( H ( I I . L E . 0 . 0 . O R . H I J l . L E . O . O I G O T O 3 0 0 1 2 1 F I I . E 0 . J 1 G O T O 3 0 0 1 3 . C R S = ( C R ( I ( • C R ( J M * * 2 r . » « C R S - S U M O F C R - S S Q U A R E O 0 0 1 4 0 0 5 0 L = l , 4 0 0 1 5 5 0 T O I S I L 1 = 0 . 0 . C * * T D I S ( L ) = D I S T A N C E S B E T W E E N C O M P E T I N G T R E E A N D C O M P E T I T O R S C * » 1 1 1 O I R E C T D I S T A N C E B E T W E E N I A N D J , c * « ( 2 ) A O J A C E N T - T O - P C O T - B O U N O A R Y T R E E S B Y M I R R O R I N G A L O N G X , c * « ( 3 1 M I R R O R I N G A L O N G Y , c * * ( 4 ) F I L L I N G I N T H E F O U R E M P T Y C O R N E R S B Y M I R R O R I N G A L O N G X . 0 0 1 6 T 0 I S I 1 1 = 1 ( X I I ) - X ( J ) ) * * 2 I * ( < Y ( I l - Y ( J ) ) * * 2 ) 0 0 1 7 I F ( X ( I ) - X ( J I ) 5 , 7 , 6 0 0 1 8 5 T D I S ( 2 t = ( ( x l l ) - X ( J I + P S ) * * 2 l + ( I V H ) - Y ( J H * * 2 ' ) 182 0 0 1 < ) T P I S I M M ( X I 1 1-X( J 1 + P S ) * * 2 1 • ( I C S - A P S I V l l l-Y 1 J l ) »» *2 ) 0 0 2 0 00 T O 7 0 0 2 1 6 T D 1 S ( 2 1 = ( < X ( I ) - X ( J ) - P S I * * 2 I M ( Y ( I I - Y ( J I ) * * 2 > 0 0 2 2 r O I S ( 4 l = < ( X ( I I - X ( J ) - P S I * * 2 ) » ( ( P S - A B S ( Y ( l l - Y ( J I ) ) * * Z > 0 0 2 . 1 7 I F 1 Y I 1 l - Y ( J ) ) 8 , 1 0 , 9 0 0 2 4 8 r o i s i ? l = ( < x l 1 i - x u i 1 * * 2 ) • 1 ( v i i i-vu) VPS ) * * 2 ) 0 0 2 5 G O T O 1 0 C 0 2 6 9 T n i S ( 3 ) = ( ( X ( l ) - X ( J ) ) * * 2 ) + l ( Y ( l l - Y ( J ) - P S ) * * 2 1 0 0 2 7 1 0 0 0 300 1=1.4 0 0 2 8 1 F I T C I S ( L 1 . G E . C H S 1 G O T O 3 0 0 c * » C A L C U L A T E C I I 1 I 1 F C O M P E T I N G T R E E A N D C O M P E T I T O R I N F L U E C E Z O N E O V E R L A P . 0 0 2 9 4 4 l F I C R I I I - C R ( J ) 1 1 1 , 1 1 , 1 2 C C 3 0 1 1 R I = C R ( I ) 0 0 3 1 R 2 = C R ( J ) 0 0 3 2 G O T O 1 3 0 0 3 3 1 2 R I = C R ( J l 0 0 3 4 R 2 = C R I 1 I C * * R I i s L E S S T H A N R 2 . 0 C 3 5 1 3 T 0 = S O R T ! T D I S ( L 1 ) 0 0 3 6 S U = R 1 + T 0 - R 2 0 0 3 7 C * * I F I S U . L E . 0.0) G O T O 1 8 I F 1 8 , T H E N C O M P L E T E I N F L U E N C E Z O N E O V E R L A P . 0 0 3 B 1 7 S = 0 . 5 * I R H - R 2 * T n ) C C 3 9 S S = S * ( S - R 1 I * ( S - R 2 ) * ( S - T D > C 0 4 0 I F I S S ) 4 5 , 4 6 , 4 6 0 0 4 1 4 5 W R I T E ( 6 , 9 7 1 S S , R 1 , T D . P 2 , S 0 0 4 2 9 7 F O R M A T I 6 H - S O R T 5 F 1 0 . 3 0 C 4 3 G C T O 3 0 0 0 0 4 4 46 A 9 = S O R T ! S S I 0 0 4 5 V = 2 . C * A 9 / T D c * * S I N ( Q ) = V / R 1 , S t N ( Z ) = V / P . 2 0 0 4 6 V R 1 = V / R 1 - ' 1 0 C 4 7 V R 2 = V / R 2 0 0 4 8 U = A R S I N ( V R 1 I 0 0 4 9 Z = A R S I N ( V R 2 1 0 0 5 0 R 0 S = R 2 * * 2 - V * * 2 0 0 5 1 T D 2 = T D * * 2 0 0 5 2 I F ( T D 2 - R C S 1 1 4 , 1 5 , 1 5 C 0 5 3 1 4 A 2 = ( R 1 * * 2 ) * ( P I - C ) 0 0 5 4 G O T O 1 6 0 0 5 5 1 5 A 2 = ( R 1 * * 2 ) * Q 0 0 5 6 16 A 3 = I R 2 * * 2 ) * Z 0 0 5 7 A T = A 2 + A 3 - 2 . 0 * A 9 0 0 5 8 0 0 T O 1 9 0 0 5 9 lfi A T = P I * ( R 1 * * 2 ) C * * I N F L U E N C E Z O N E O V E R L A P I S M O D I F I E D BY E X P O N E N T I A L L Y W E I G H T E D C O M P E T I T O R -c ** C O M P E T I N G T R E E S I Z E R A T I O . 0 0 6 0 1 9 R J I = ( 0 ( J I / 0 ( 1 1 ) * * E X 0 0 6 1 C l ( I l = C 1 ( l l + ( A T / I P I * ( C R ( 1 1 * * 2 ) ) * R J 1 1 0 0 6 2 C I ( J ) = C I ( J ) + ( A T / ( P I * I C R ( J ) * * 2 ) ) * ( l . O / R J I ) ) 0 0 6 3 3 0 0 C O N T I N U E 0 0 6 4 3 C O N T I N U E C * * O B T A I N C I M E A N , M I N . , M A X . , V A R . A N D F R E O . O I S T R . F O R T R E E S N O T O P E N G R O W I N G 0 0 6 5 C 1 M N = 9 9 < ? 9 9 9 9 . 0 0 6 6 C 1 M X = 0 . 0 0 0 6 7 SC. 1 = 0 . 0 0 0 6 8 S S C I = 0 . 0 0 0 6 9 AN=0.0 0 0 7 0 D O 3 1 1 = 1 , N T C 0 7 1 I F I H ( I ) . L T . l . O ) G C T O 31 0 0 7 2 A N ' = A N + 1 . 0 0 0 7 3 I F I C K 1 I . L T . f . I M N ) C I M N = C I ( I I 0 0 7 4 I F ( C M I 1 . G T . C I H X 1 C I M X = C I ( I ) 0 0 7 5 S C I = SC. 1 +C I 1 1 ) 0 0 7 6 S S C I = S S C ( + C ! ( 1 1 * * 2 0 C 7 7 3 1 C O N T I N U E 0 0 7 8 R N G = C I M X - C 1 M N 0 0 7 9 C L A S = R N G / F L C A T ( N C C ) 0 0 8 0 J J = 0 0 0 8 1 00 4 2 J = l , N C C 0 0 8 2 N F C ( J ) = 0 0 0 8 3 P = C I M N < - F L 0 A T I J - 1 ) * C L A S 0 0 8 4 ( 0 = P t t L A S 0 0 8 5 D O 4 3 1 = 1 , N T 0 0 8 6 1 F ( H 1 I . L T . . 1 . 0 1 G O T O 4 3 0 0 8 7 I F I d ( I 1 . L T . P . G R . C l ( 1 1 . G E . O I G O T O 4 3 0 0 8 8 N F C ( J ) = N ' F C ( J ) + l 0 0 8 9 J J = J J + 1 0 0 9 0 4 3 C O N T I N U E 0 0 9 1 4 2 C O N T I N U E 0 0 9 2 ! F ( I F I . r i A T ( J J I » 0 . 2 ) . l T . A N > N F C I N C C ) = N F C I N C C 1 • 1 , 0 0 9 3 " A C 1 = S C 1 / A N 0 0 9 4 C 1 V R = 1 I S S C l - S C I * * 2 / A N 1 / ( A N - 1 . 1 ) 0 0 9 5 R E T U R N ' 0 0 9 6 E N D 183 ' 0 0 0 1 S U M R O U T I N E S T A T ( X V , N V , A V X , S O X , C O V X , X M N , X M X , I F C . C S 1 C * * C A L C U L A T E D E S C R I P T I V E T R E E S T A T I S T I C T I C S 0 0 0 2 O l f E N S I C N X V I N V I , I F C ( 1 0 1 0 0 0 3 X M N = 9 9 ° . 9 0 0 0 4 X M X = 0 . 0 C 0 0 5 S D X = o . n 0 0 0 6 C C V X = 0 . 0 0 0 0 7 A V X = 0 . 0 o o o n C S = 0 . 0 O O O n o 4 j = i , i o 0 0 1 0 4 I F C ( J ) = 0 0 C 1 1 I F I N V . L T . 2 ) R E T U R N 0 0 1 2 S X = 0 . 0 0 0 1 3 S X 2 = 0 . 0 0 0 1 4 o n i I = I , N V 0 0 1 5 X V I = X V ( I I 0 0 1 6 1 H X K N . C T . X V I ) X M N = X V 1 0 0 1 7 I F 1 X M X . L T . X V I 1 X M X = X V I 0 0 1 8 S X = S X + X V I 0 0 1 9 1 S X 2 = S X 2 * X V 1 * * 2 0 0 2 0 VN = N V 0 0 2 1 A V X = S X / V N 0 C 2 2 I F ( I X H N + X M X ) . L T . 0 . 0 1 ) R E T U R N 0 0 2 3 S O X = S O R T ! ( S X 2 - ( S X * * 2 > / V N 1 / ( V N - l . 0 1 1 0 0 2 4 C O V X = 1 0 0 . 0 * S D X / A V X 0 0 2 5 R N G = X M X - X M N • 0 0 2 6 C S = R N G / 1 0 . 0 C 2 7 K K = 0 0 0 2 8 0 0 2 J = l , 1 0 0 0 2 9 P = X M N + F L O A T 1 J - 1 ) * C S 0 0 3 0 0 = P + C S 0 0 3 1 D O 3 1 = 1 , N V 0 0 3 2 I F ( X V I I ) . L T . P . O R . X V I I ) . G E . 0 1 G O T O 3 0 0 3 3 I F C ( J ) = I F C ( J H 1 0 0 3 4 K K = K K « 1 . 0 0 3 5 3 C O N T I N U E 0 0 3 6 2 C O M 1 N U E 0 0 3 7 I F I K K . L T . N V I I F C ( 1 0 ) = I F C ( 1 0 1 + 1 C C 3 8 R E T U R N 0 0 3 9 E N O 184 APPENDIX I I I SIMULATED STAND TREE STATISTICS Abreviations: AVE Average SD Standard Deviation COV C o e f f i c i e n t of Varion MIN Minimum Value MAX Maximum Value CLN Class Width 1 2 3 10 10 20 30 Frequency D i s t r i b u t i o n Classes 185 S T A N O G R O W T H S I M U L A T I O N M O D E L P L O T D I M E N S I O N 6 6 . 0 A G E S T A R T I N G . . . . 1 4 X-Y C C - G R O . O P T I O N 1 I N I T I A L N O . O F T R E E S 4 0 9 A G E A T H A R V E S T 4 9 N U M O E R O F I N C R . P E R . . . B N U M B E R O F C L O N E S • • . . L E N G T H O F I N C R . P E R . T R E E S U M M A R Y S T A T I S T I C S A N O D I S T R I B U T I O N S F O R I N C R . P E R I O D . . 1, A G E . . . 1 4 * « * * * * * * « * * » « « N A M E A V E S O C O V M I N M A X C L W 1 2 3 4 . 5 6 a 9 1 0 L I V I N G T R E E C H A R A C T E R I S T I C S H E I G H T I N C 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 0 0 0 0 0 0 0 0 0 O B H I N C . . . 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 0 0 0 0 0 0 0 0 0 TUT H E I G H T 2 0 . 8 7 2 . 6 0 1 3 . 4 3 1 4 . 8 5 2 8 . 5 8 1 . 4 1 4 2 1 a 53 8 4 II 6 1 5 4 1 8 6 6 1 . 5 8 0 . 4 6 2 8 . 8 4 0 . 6 0 2 . 9 0 0 . 2 3 0 2 1 3 3 5 3 B 4 7 2 6 1 5 4 1 8 6 6 N C M O R T A L I T Y I N T H E L A S T P E R I C D T R E E S U M M A R Y S T A T I S T I C S A N D D I S T R I B U T I O N S F O R I N C R . P E R I O D . . 2 , A G E . 1 9 * * « « * « * * « « * * * * N A M E A V E S O C O V M I N M A X C L W 1 2 3 4 5 6 7 8 9 1 0 L I V I N G T R E E C H A R A C T E R I S T I C S H E I G H T I N C 4 . 3 3 0 . 7 8 1 8 . 0 9 1 . 9 6 6 . 2 6 0 . 4 3 C 4 7 2 6 3 2 4 8 7 4 8 0 4 S 1 9 4 D B H I N C . . . 0 . 4 1 0 . 1 5 3 7 . 5 1 0 . 0 7 0 . 9 1 0 . 0 8 4 1 3 4 0 5 2 7 4 6 7 5 4 2 5 1 0 4 3 T O T H E I G H T 2 5 . 9 1 3 . 2 t 1 2 . 3 8 1 7 . 4 3 3 5 . 2 5 1 . 7 8 2 3 2 2 3 9 5 0 7 9 6 7 4 7 2 2 9 4 0 . 7 7 3 . 8 1 0 . 3 0 4 6 2 8 4 3 6 2 7 7 6 9 3 0 1 5 1 0 2 M O R T A L I T Y T R E E C H A R A C T E R I S T I C S H E I G H T I N C 2 . 5 6 0 . 6 1 2 3 . 8 6 1 . 2 5 3 . 7 6 0 . 2 5 0 4 4 6 5 1 0 1 0 1 3 6 7 2 D B H I N C . . . 0 . 1 4 0 . 0 6 4 5 . 5 8 0 . 0 3 0 . 2 8 0 . 0 2 5 8 7 6 1 0 1 0 1 1 4 8 1 2 T C T H E I G H T 1 9 . 8 4 1 . 7 6 8 . 8 9 1 6 . 1 0 2 2 . 9 1 0 . 6 8 1 3 3 1 0 4 6 1 2 8 8 7 6 0 . 6 3 1 . 5 8 0 . 0 9 5 2 5 1 2 4 3 1 4 9 5 7 6 T R E E S U M M A R Y S T A T I S T I C S A N D D I S T R I B U T I O N S F O R I N C R . 1 P E R I O D . . 3 , A G E 2 4 6******4****** N A M E A V E S D C O V M I N M A X C L W 1 2 3 4 5 6 7 8 9 1 0 L I V I N G T R E E C H A R A C T E R I S T I C S H E I G H T I N C 3 . 9 1 0 . 6 5 1 6 . 5 9 2 . 1 8 5 . 6 6 0 . 3 4 8 6 1 4 2 5 4 2 5 4 6 4 5 3 2 0 " 1 2 3 O B H I N C . . . 0 . 4 0 0 . 1 4 3 6 . 1 2 0 . 1 1 0 . 9 0 0 . 0 7 9 1 8 3 9 5 4 6 4 5 7 3 5 1 4 6 3 3 T G T H E I G H T 3 0 . 5 0 3 . 4 4 1 1 . 2 8 2 2 . 3 3 4 0 . 9 1 1 . 8 5 8 7 2 3 3 9 5 3 6 4 5 3 3 3 1 0 9 2 1 . 2 6 4 . 7 1 0 . 3 4 5 1 1 ii 4<) 6 / 6 0 il 2 2 4 8 2 M O R T A L I T Y T R E E C H A R A C T E R I S T I C S H E I G H T I N C 2 . 5 6 0 . 6 2 2 4 . 3 7 1 . 1 3 3 . 7 9 0 . Z 6 5 2 0 4 8 9 1 2 0 5 5 4 D B H I N C . . . 0 . 1 6 0 . 0 8 5 0 . 9 6 0 . 0 3 0 . 3 6 0 . 0 3 3 2 7 1 1 1 3 2 3 2 3 4 2 T O T H E I G H T 2 4 . 3 7 2 . 8 1 1 1 . 5 2 1 8 . 5 6 ~ " 3 0 7 4 T ' " T . ~ 1 8 5 2 ~ 1 " " 6 1 2 1 0 4 3 3 5 3 0 . 8 0 2 . 5 6 0 . 1 7 6 2 3 a 1 5 7 1 2 7 1 3 T R E E S U M M A R Y S T A T I S T I C S A N D D I S T R I B U T I O N S F O R I N C R . P E R I O D . . 4 , A G E . . . 2 9 * * * * * * * * * * * * * * N A M E A V E S O C O V M I N M A X C L W . 1 2 3 4 5 6 7 8 9 1 0 L I V I N G T R E E C H A R A C T E R I S T I C S H E I G H T I N C 3 . 2 6 0 . 6 6 2 0 . 2 8 1 . 5 7 5 . 1 2 0 . 3 5 5 6 1 3 2 3 4 0 5 3 5 2 3 5 1 5 9 3 D B H I N C . . . 0 . 3 3 0 . 1 4 4 3 . 6 2 0 . 0 6 0 . 8 7 0 . 0 8 1 1 9 4 1 5 2 6 0 4 0 1 9 / u 0 3 T O T f - E I G H T 3 4 . 4 8 3 . 7 7 1 0 . 9 3 2 5 . 4 0 4 6 . 0 3 2 . 0 6 3 5 1 8 2 9 6 1 4 7 4 3 3 0 5 9 2 3 . 0 7 0 . 7 3 2 3 . 6 7 1 . 5 3 5 . 5 9 0 . 4 0 5 1 0 2 6 5 3 5 3 5 4 2 4 1 8 3 6 2 M O R T A L I T Y T R E E C H A R A C T E R I S T I C S H E I G H T I N C 2 . 1 4 0 . 5 6 2 5 . 9 8 1 . 0 2 3 . 2 3 0 . 2 2 1 i 3 6 2 1 / / 4 i 2 O B H I N C . . . 0 . 1 4 0 . 0 7 5 1 . 9 2 0 . 0 3 0 . 3 1 0 . 0 2 8 6 6 6 4 9 5 3 1 2 2 T O T H E I G H T 2 8 . 6 1 2 . 6 5 9 . 2 8 2 3 . 3 5 3 4 . 5 2 1 . 1 1 6 3 2 7 8 3 1 0 4 4 0 3 2 . 0 3 0 . 4 1 2 0 . 3 5 1 . 2 9 3 . 0 3 0 . 1 7 4 3 3 8 7 7 8 2 3 0 3 186 T R E E S U M M A R Y S T A T I S T I C S A N D D I S T R I B U T I O N S F O R I N C R . P E R I O D . . 5 , A G E . . . 3 4 * * * * • » * * « * * « * » N A M E A V E S D C C V M I N M A X C L W 1 2 3 4 5 6 7 8 9 1 0 L I V I N G T R E E C H A R A C T E R I S T I C S H E I G H T I N C 2 . 8 7 0 . 6 1 2 1 . 1 1 1 . 0 3 4 . 6 3 0 . 3 6 0 1 5 1 6 3 2 4 3 4 8 3 1 1 5 7 3 D B H I N C . . . 0 . 3 0 0 . 1 4 4 6 . 4 4 0 . 0 3 0 . 8 4 0 . 0 8 1 9 3 8 4 8 4 9 2 9 1 4 5 6 0 3 T O T H E I G H T 3 8 . 2 3 4 . 0 4 1 0 . 5 8 2 6 . 9 4 5 0 . 6 6 2 . 3 7 2 1 8 1 9 4 6 4 3 3 9 3 0 4 8 3 3 . 5 3 0 . 8 2 2 3 . 3 0 1 . 6 2 6 . 4 3 0 . 4 8 0 3 1 7 4 2 4 7 4 2 2 6 1 3 6 3 2 M O R T A L I T Y T R E E C H A R A C T E R I S T I C S "I H E I G H T I N C 1 . 9 8 0 . 5 5 2 7 . 7 5 0 . 9 3 2 . 9 2 0 . 1 9 9 3 4 4 4 7 6 7 4 3 6 D B H I N C . . . 0 . 1 4 0 . 0 8 5 5 . 1 8 0 . 0 3 0 . 2 9 0 . 0 2 7 6 8 5 7 5 4 4 1 4 4 T O T H E I G H T 3 2 . 7 8 3 . 1 8 9 . 7 1 2 6 . 3 3 3 8 . 8 4 1 . 2 5 1 1 5 5 7 4 6 7 5 5 3 2 . 5 3 0 . 5 2 2 0 . 6 3 1 . 5 6 3 . 6 0 0 . 2 0 4 2 7 6 7 3 7 7 1 6 2 T R E E S U M M A R Y S T A T I S T I C S A N D D I S T R I B U T I O N S F O R I N C R . 1 P E R I O D . . 6 , A G E 3 9 * * * * * * * * * * * * * * N A M E A V E S O C C V M I N M A X C L W 1 2 3 4 5 b 7 8 9 1 0 L I V I N G T R E E C H A R A C T E R I S T I C S H E I G H T I N C 2 . 6 7 0 . 5 2 1 9 . 6 2 1 . 3 8 4 . 1 8 0 . 2 8 0 3 8 2 3 3 2 3 7 3 2 1 8 7 6 3 D B H I N C . . . 0 . 3 0 0 . 1 3 4 4 . 7 4 0 . 0 7 0 . 8 0 0 . 0 7 3 1 1 3 4 4 6 3 6 1 7 1 2 4 6 0 3 T O T H E I G H T 4 1 . 6 7 4 . 2 6 1 0 . 2 3 3 2 . 2 8 5 4 . 8 4 2 . 2 5 6 3 1 9 2 8 3 5 3 4 2 2 1 7 2 7 2 3 . 9 8 0 . 9 2 2 3 . 0 9 2 . 2 6 1.23 0 . 4 9 1 7 i i i l 4 1 2 1 1 6 4 8 1 2 M O R T A L I T Y T R E E C H A R A C T E R I S T I C S H E I G H T I N C 1 . 9 2 0 . 4 5 2 3 . 2 3 0 . 8 4 2 . 7 6 0 . 1 9 2 1 0 3 5 5 4 5 4 3 2 O B H I N C . . . . 0 . 1 5 0 . 0 7 4 7 . 4 4 0 . 0 2 0 . 3 1 0 . 0 2 8 1 4 6 5 3 6 2 2 1 2 T O T H E I G H T 3 6 . 0 6 3 . 3 8 9 . 3 9 2 7 . 7 7 4 2 . 2 3 1 . 4 4 6 1 0 2 6 4 5 5 4 2 3 0 . 5 8 1 9 . 9 6 1 . 6 5 4 . 0 4 0 . 2 3 9 1 1 4 4 . 6 5 3 4 1 3 T R E E S U M M A R Y S T A T I S T I C S A N D D I S T R I B U T I O N S F O R I N C R . P E R I O D . . 7 , A G E 4 4 N A M E A V E S D C O V M I N M A X C L W 1 2 3 4 5 6 7 8 9 1 0 L I V I N G T R E E C H A R A C T E R I S T I C S H E I G H T I N C 2 . 2 5 0 . 5 3 2 3 . 5 9 1 . 0 6 3 . 7 8 0 . 2 7 3 5 9 2 1 2 2 3 1 2 0 1 2 7 4 3 D B H I N C . . . 0 . 2 5 0 . 1 3 5 2 . 8 9 0.05 0 . 7 6 0 . 0 7 1 1 6 3 1 3 4 2 6 1 3 4 4 i 1 2 T O T H E I G H T 4 4 . 8 9 4 . 6 2 1 0 . 3 0 3 5 . 1 1 5 8 . 6 2 2 . 3 5 2 6 1 0 2 0 3 2 2 7 1 9 9 4 5 2 1 . 0 4 2 3 . 4 2 2 . 5 6 7 . 9 9 0 . 5 4 3 9 2 0 3 2 3 1 1 8 1 2 3 6 1 2 M O R T A L I T Y T R E E C H A R A C T E R I S T I C S H E I G H T I N C 1 . 6 3 0 . 3 6 2 2 . 1 5 C . 7 6 2 . 2 0 0 . 1 4 5 1 2 1 2 5 7 i 6 i b O B H I N C . . . 0 . 1 2 0 . 0 5 4 3 . 6 0 0 . 0 2 0 . 2 2 0 . 0 2 C 2 3 4 6 4 3 5 1 S 2 T O T H E I G H T 3 9 . 6 0 2 . 8 5 7 . 2 0 3 3 . 0 4 4 5 . 2 7 1 . 2 2 3 1 1 3 6 4 7 3 5 3 2 0 . 5 1 1 5 . 2 7 2 . 2 8 4 . 4 2 0 . 2 1 3 2 I 4 7 4 6 2 5 2 2 T R E E S U M M A R Y S T A T I S T I C S , A N D D I S T R I B U T I O N S F O R I N C R . P E R I O D . . 8 , A G E 4 9 ' ************** N A M E A V E S D C O V M I N M A X C L W 1 2 3 4 5 6 7 8 9 1 0 L I V I N G T R E E C H A R A C T E R I S T I C S H E I G H T I N C 2 . 0 2 0 . 5 0 2 4 . 9 3 0 . 6 8 3 . 4 2 0 . 2 7 4 3 1 1 2 1 2 3 1 2 0 1 1 7 4 3 , O B H I N C . . . 0 . 2 3 0 . 1 3 5 5 . 3 2 0 . 0 2 0 . 7 1 0 . 0 6 9 7 2 0 3 4 1 8 1 1 5 4 2 1 2 T O T h E I G H T 4 8 . 0 3 4 . 8 8 1 0 . 1 7 3 5 . 7 9 6 2 . 0 4 2 . 6 2 5 3 1 1 1 2 4 2 6 1 7 1 1 4 5 2 1 . 1 4 2 3 . 2 8 2 . 5 8 8 . 7 0 0 . 6 1 2 3 1 0 2 6 2 6 1 6 1 2 2 6 1 2 M O R T A L I T Y T R E E C H A R A C T E R I S T I C S H E I G H T I N C 1 . 3 8 0 . 3 4 2 4 . 3 9 0 . 8 7 2 . 2 5 0 . 1 3 8 3 6 3 5 3 5 2 2 0 1 D B H I N C . . . 0 . 1 0 0 . 0 6 5 3 . 9 4 0 . 0 4 0 . 2 8 0 . 0 2 4 8 4 8 4 2 1 2 0 0 1 T C T H E I G H T 4 2 . 3 7 3 . 3 5 7 . 9 1 3 7 . 6 3 5 0 . 7 3 1 . 3 1 0 5 5 4 4 5 2 1 2 1 I 0 . 6 4 1 7 . 3 5 2 . 8 4 5 . 4 4 0 . 2 6 0 6 5 5 4 4 1 3 1 0 I T R E E C O M P E T I T I O N I N D E X S T A T I S T I C S 5 3 . 7 1 9 1 . 5 4 6 3 3 6 2 . 5 1 . 4 6 7 1 . 6 0 3 0 0 9 2 5 . 0 1 7 0 . 7 0 0 4 2 8 4 7 . 7 5 1 0 . 8 8 9 3 4 4 2 4 0 4 . 0 6 1 2 . 6 0 6 4 4 6 3 3 5 6 . 2 9 9 2 . 5 5 1 8 1 4 1 4 4 . 7 9 3 9 . 5 8 9 1 6 3 . 0 6 8 1 . 5 8 9 5 4 4 3 0 . 9 8 1 7 7 7 4 2 . 2 1 2 3 6 7 1 4 2 . 3 6 5 1 0 . 4 8 2 1 . 5 8 0 8 6 4 9 1 . 0 2 0 4 4 7 7 3 8 . 9 7 8 1 . 5 5 2 2 6 1 4 1 0 . 7 3 7 1 . 0 3 0 8 9 8 1 1 2 4 . 4 8 5 2 . 0 9 5 1 1 7 6 1 2 5 . 3 5 5 2 . 0 9 8 1 3 9 8 3 7 . 0 3 4 1 0 . 6 2 4 9 0 . 3 1 8 1 . 5 3 6 2 3 3 9 1 . 0 2 6 2 6 9 9 1 . 9 5 5 7 7 5 3 3 3 . 8 0 4 1 0 . 1 0 9 7 5 . 1 4 7 1 . 4 9 6 7 3 0 8 1 . 0 0 4 7 0 2 6 1 . 8 7 5 9 1 3 6 CLASS NUMBER WIDTH FREQUENCIES PER CLASS \ 10 2 0 3 0 4 0 9 . 9 7 6 1 6 6 9 8 6 7 3 8 18 16 21 7 9 17 11 5 3 2 7 2 1 2 2 1 0 2 2 3 1 ' 0 0 3 0 0 1 0 1 1 0 0 0 0 0 2 4 0 9 4 0 0 . 0 4 7 6 5 0 4 3 0 1 1 6 6 6 5 19 19 17 26 21 2 4 2 5 16 2 9 9 18 18 9 9 7 19 14 5 1 5 14 1 0 5 6 6 2 3 3 5 3 1 2 2 4 0 9 4 0 8 . 7 1 3 7 2 2 66 6 4 4 8 34 2 3 14 13 6 6 11 5 4 4 4 5 4 0 1 2 1 1 0 0 1 1 0 0 0 0 0 0 . 0 0 0 1 0 0 0 1 3 4 2 4 0 0 . 0 4 1 5 6 1 7 3 0 0 3 2 6 4 7 7 17 14 2 0 18 2 2 2 4 13 2 8 16 15 16 11 2 3 3 12 5 4 10 9 5 6 9 1 3 2 1 I 0 0 1 1 3 4 2 4 0 3 . 8 3 7 0 6 9 16 2 6 2 8 2 9 3 0 14 2 5 15 10 12 12 6 11 7 4 1 5 6 2 1 1 2 . 3 2 2 1 1 1 1 1 0 0 1 1 0 0 0 1 2 9 3 4 0 0 . 0 3 0 7 6 4 7 1 2 0 0 4 2 3 3 3 5 8 7 14 1 0 15 16 16 2 0 1 7 9 2 2 11 14 7 12 13 6 14 6 5 7 2 3 4 4 2 3 0 2 1 2 9 3 4 0 2 . 8 5 0 1 3 6 5 13 10 19 2 3 16 21 22 11 12 13 12 7 6 6 7 3 6 3 5 2 1 3 2 2 2 0 0 2 0 1 0 1 1 2 0 0 1 2 4 9 4 0 0 . 0 2 6 8 6 6 7 1 2 0 0 0 4 2 2 3 2 5 6 4 6 15 6 18 12 16 12 17 14 8 12 14 8 9 7 8 8 5 5 2 6 2 1 2 1 3 1 2 4 9 4 0 2 . 8 6 5 4 3 6 5 1 2 10 18 17 18 16 14 1 5 8 12 7 7 5 6 1 3 6 2 1 2 2 1 2 0 0 0 1 0 0 0 0 0 0 0 0 0 1 2 0 1 4 0 0 . 0 2 6 6 8 1 0 1 2 0 0 2 4 0 3 1 2 6 5 5 6 15 4 13 14 9 13 10 I S 7 9 10 8 6 6 3 7 3 2 2 3 0 1 0 0 0 1 2 0 1 4 0 1 . 9 9 2 4 3 0 6 3 6 6 6 7 15 9 1 5 7 8 11 6 8 13 3 3 5 6 1 6 2 3 2 3 1 0 1 0 1 2 0 0 0 0 0 0 1 169 4 0 0 . 0 2 3 2 3 7 6 1 1 1 0 0 0 3 2 2 1 3 0 7 2 4 6 4 9 8 7 17 5 7 U 7 9 12 6 5 6 5 5 4 3 1 1 3 0 0 1 169 4 0 1 . 6 2 6 0 3 0 4 3 3 2 7 4 5 9 8 8 6 15 4 4 4 7 2 5 7 3 4 1 1 1 2 1 2 2 2 0 1 0 3 0 0 0 0 1 1 3 4 4 0 0 . 0 2 1 7 8 0 3 1 1 1 0 0 0 2 2 1 2 1 3 0 3 5 3 3 5 6 7 6 8 12 7 4 3 8 4 8 7 4 2 1 3 4 2 1 3 0 1 1 3 4 ORIGINAL; T - TRANSFORMED COMPETITION INDICES C D APPENDIX IV ALBERTA NORMAL YIELD TABLES FOR ASPEN (FROM MACLEOD 1952) TOTAL HEIGHT OF AVERAGE DOMINANT ASPEN Age Total . ht. by i s i t e : index*" Age Total ht. by s i t e index 50 60 70 80 90 50 60 70 80 90 Feet Feet 10 10 14 17 20 23 70 47 56 66 75 85 20 20 26 31 36 42 80 50 60 70 80 90 30 28 35 41 48 55 90 52 63 73 84 94 40 34 42 49 57 65 100 54 65 76 86 97 50 39 47 56 64 73 110 56 67 78 89 90 60 43 52 61 70 79 120 58 69 80 91 102 ''"Reference age 80 years. TOTAL NUMBER OF TREES PER ACRE OVER i 0.5-INCH D.B.H. Age Trees per acre by s i t e index 50 60 70 80 90 Number 10 10,000 8,500 7,700 6,700 6,050 20 6,900 5,800 5,000 4,400 3,750 30 4,700 3,900 3,300 2,800 2,500 40 3,250 2,650 2,240 1,900 1,680 50 2,200 1,840 1,550 1,350 1,180 60 1,550 1,300 1,100 960 850 70 1,100 930 800 705 630 80 790 690 600 525 470 90 610 535 465 410 370 100 495 435 380 330 295 110 420 365 320 280 245 120 370 315 276 240 210 AVERAGE DIAMETER BY AGE AND SITE INDEX CLASS Age D.b.h. by s i t e index 50 60 70 80 90 Inches 10 0.6 0.7 0.8 1.0 1.3 20 1.3 1.5 1.7 2.0 2.3 30 1.9 2.2 2.5 2.9 3.2 40 2.5 2.9 3.3 3.8 4.2 50 3.2 3.6 4.1 4.6 5.1 60 3.8 4.4 4.9 5.5 6.1 70 4.6 5.2 5.9 6.5 7.1 80 5.4 6.1 6.8 7.6 8.3 90 6.2 6.9 7.8 8.6 9.4 100 6.9 7.7 8.6 9.6 10.5 110 7.5 8.4 9.4 10.4 11.5 120 8.0 9.0 10.1 11.2 12.4 ,TOTAL BASAL AREA PER ACRE FOR TREES OVER 0.5-INCH D.B.H. iAge Basal area per acre by s i t e index ,50 60 70 , 80 90 - Square feet  10 14 26 ,38 50 61 20 66 78 1 91 103 114 30 94 106 - 119 130 141 40 111 122 136 147 158 50 118 130 143 155 168 60 122 134 148 160 172 70 124 137 150 162 176 80 126 139 152 164 177 90 128 140 154 165 178 100 128 140 154 166 178 110 128 140 154 166 178 120 128 140 154 166 178 STEM WOOD VOLUME YIELD PER ACRE IN CUBIC FEET FOR TREES OVER 0.5-INCH D.B.H. Age 50 Y i e l d 60 per acre by s i t e 70 Cubic feet index 80 90 10 80 180 280 420 570 20 620 855 1120 1415 1715 30 1145 1480 1890 2315 2775 40 1605 2030 2575 3145 3775 50 1970 2480 3115 3800 4555 60 2270 2850 3560 4345 5155 70 2525 3180 3925 4790 5685 80 2735 3440 4240 5160 6115 90 2910 3650 4510 5460 6475 100 3050 3805 4730 5700 6760 110 3165 3920 4900 5900 6980 120 3255 4030 5035 6080 7170 \ 

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