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Growth, yield and silvicultural management of exotic timber species in Kenya 1983

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GROWTH, YIELD AND SILVICULTURAL MANAGEMENT OF EXOTIC TIMBER SPECIES IN KENYA by WINSTON JOSHUA KAMURU MATHU B.Sc.F. University of new Brunswick, Canada 1971 M.Sc.F. Uni v e r s i t y of Das-Es-Salaam, Tanzania 1977 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the FACULTY OF GRADUATE STUDIES Department of Forestry We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA ®March, 1983 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the l i b r a r y s h a l l make i t f r e e l y a v ailable for reference and study. I further agree that permission for extensive copying of the thesis for s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives. I t i s understood that copying or p u b l i c a t i o n of t h i s t h esis for f i n a n c i a l gains s h a l l not be allowed without my written permission. Department of Forestry The U n i v e r s i t y of B r i t i s h Columbia 2357 Main M a l l Vancouver, B.C. Canada V6T 1W5 March |g, 1983 i ABSTRACT Supervisor: D.D. MUNRO This study presents the growth, y i e l d and the s i l v i c u l t u r a l manage- ment of Cupressus l u s i t a n i c a , Pinus patula and Pinus ,radiata, the three most important timber species growing i n the Kenya highlands. The study Is based on 163, 176 and 164 permanent sample plots for the three species r e s p e c t i v e l y . The stand dominant height development was predicted as a function of stand age and s i t e index, defined as dominant height at reference age of 15 years. The Chapman-Richard's growth function was used for C_. l u s i t a n i c a and P_. radiata while a l i n e a r quadratic equation was used to describe dominant height development for P̂ . patula by geographical regions. Height development for the two pine species was found to be s i g n i f i c a n t l y d i f f e r e n t (up to age 20 years) i n the Shamba and grassland establishment s i t e s . Stand basal area before thinning was predicted as a function of stand age, dominant height and number of stems using a Weibull-type growth equation. In thinned stands basal area was predicted through a basal area increment nonlinear equation. For P_. r a d i a t a , basal area increment was predicted as a function of basal area at the beginning of the growth period (1 year) and age. For C_. l u s i t a n i c a and P. patula, a t h i r d term-stand density index, defined as the percent r a t i o of average spacing between trees to stand dominant height was included. The Weibull p r o b a b i l i t y density function was used to characterize stand i i diameter d i s t r i b u t i o n with the Weibull parameters predicted as a function of stand parameters. Stand volumes were determined from tree volume equations for the respective species while the mean DBH of stems removed i n thinnings was predicted from mean stand DBH before thinning and weight of thinning. Using the above functions, a growth and y i e l d simulation model EXOTICS was constructed. Written i n FORTRAN IV G-level which i s compatible with IBM System/360 and System/370, EXOTICS i s an i n t e r a c t i v e whole-stand/distance independent model with an added c a p a b i l i t y f or providing diameter d i s t r i b u t i o n (by 3 cm diameter classes) to give f i n a l main stand y i e l d by size c l a s s e s . The model i s intended to f a c i l i t a t e s i l v i c u l t u r a l management of the three species i n the Kenya highlands. On v a l i d a t i o n , EXOTICS was found to have no bias within the range of v a l i d a t i o n data, and 95% confidence l i m i t s of 16%, 20% and 17% for C_. l u s i t a n i c a , P. patula and P_. r a d i a t a respectively. Using EXOTICS, the current s i l v i c u l t u r a l management schedules i n Kenya were studied. The thinning regimes were found to have marked e f f e c t s on the current annual volume increment. It was therefore con- cluded that at the present l e v e l of s i l v i c u l t u r a l management, Moller's theory that thinning has no appreciable e f f e c t s on t o t a l volume y i e l d does not hold for the three species i n Kenya. The current thinning p o l i c y aimed at production of large-sized sawlog crop i n as short a r o t a t i o n as possible at the expense of some loss i n t o t a l y i e l d i s discussed and found to have been overtaken by events. A p o l i c y based on the concept of maximum volume production i s advocated. A thinning experiment (using C. l u s i t a n i c a ) demonstrated that t o t a l merchantable i i i volume could be increased by between 5 and 10% (using 20% thinning i n t e n s i t y ) depending on s i t e q u a l i t y c l a s s . Within the range of stockings maintained i n plantations i n Kenya, thinning i n t e n s i t y was found to be the most important consideration, with stocking before thinning having very l i t t l e e f f e c t on both mean annual volume increment and t o t a l merchantable volume y i e l d up to age 40 years. iv TABLE OF CONTENTS Page ABSTRACT i TABLE OF CONTENTS i v LIST OF TABLES v i i LIST OF FIGURES x i ACKNOWLEDGEMENT x i v DEDICATION xv INTRODUCTION 1 CHAPTER 1: BACKGROUND INFORMATION 7 1. Species Nomenclature and D i s t r i b u t i o n 7 2. Climate and S o i l s 10 3. S i l v i c u l t u r a l Forest Management i n Kenya 16 4. S i l v i c u l t u r a l Problems Related to E c o l o g i c a l Features of Exotic Plantations In Kenya 22 5. Permanent Sample Plots 27 5.1 The Permanent Sample Plot Program i n Kenya 27 5.2 Permanent Sample Plots Data as a Basis for Growth and Y i e l d Studies 32 5.3 Problems Associated with Permanent Sample Plots Data 32 6. Study Methods 35 CHAPTER 2: STAND DEVELOPMENT AND GROWTH FUNCTIONS 38 1. Height Development and S i t e Index Curve Construction .. 38 1.1 Introduction 38 1.2 Site Index Curve Construction 40 V Page 2. M o r t a l i t y , Stand Density Development and Thinning Practices i n Kenya Plantations 85 2.1 M o r t a l i t y 85 2.2 Stand Density Development 86 2.3 Thinning 93 3. Basal Area Growth Before F i r s t Thinning 104 4. Basal Area Development i n Thinned Stands 116 5. Stand Diameter D i s t r i b u t i o n 128 6. Stand Volume Determination 143 CHAPTER 3: YIELD MODEL CONSTRUCTION AND VALIDATION 147 1. General P r i n c i p l e 147 2. Simulation A p p l i c a t i o n to Growth and Y i e l d Models 149 2.1 Forest Stand Simulation Models 150 3. Y i e l d Model Construction 159 3.1 E s s e n t i a l Features for the Envisaged Growth and Y i e l d Model 159 3.2 Growth and Y i e l d Model Synthesis 160 4. Model V a l i d a t i o n 173 4.1 Introduction 173 4.2 V a l i d a t i n g EXOTICS 177 4.3 Conclusion 201 CHAPTER 4: SILVICULTURAL MANAGEMENT MODELS FOR KENYA 203 1. Introduction 203 2. Current Thinning Models for Sawtimber Regime i n Kenya 207 2.1 Summary on the Current Thinning Model for Kenya .. 217 v i Page 3. A l t e r n a t i v e Thinning Models for Sawtimber Crop i n Kenya 217 3.1 Thinning P o l i c y Considerations 217 3.2 Thinning Experiment f o r C_. l u s i t a n i c a 220 3.3 Results from the thinning experiment 224 3.4 Summary on the thinning experiment 242 4. Pulpwood Production Regime for Kenya 242 CHAPTER 5: SUMMARY: THEORETICAL AND PRACTICAL ASPECTS OF THIS STUDY, SUGGESTED FUTURE DEVELOPMENTS AND APPLICATION ., 249 1. Growth and Y i e l d Relationships 249 2. Construction of the Growth and Y i e l d Model 256 3. S i l v i c u l t u r a l Management Models for Kenya 259 4. Future Research and Development A r i s i n g from t h i s Study 262 5. A p p l i c a t i o n of the Results 263 6. Conclusion 265 BIBLIOGRAPHY 266 v i i LIST OF TABLES Table Page 1 D i s t r i b u t i o n of the species by area i n the major countries where the species are grown 8 2 Summary of r a i n f a l l and a l t i t u d e f o r weather stations representative of the highland zone 15 3 Seed per kilogram and germinative capacity for C_. l u s i t a n i c a , P_. patula and P_. radiata i n Kenya .... 17 4 Pruning schedules f or C_. l u s i t a n i c a , P_. patula and P_. radi a t a i n Kenya as per relevant t e c h n i c a l orders 21 5 Basic thinning schedules f o r sawtimber and plywood crops for the three major plantation species i n Kenya 23 6 Basic thinning schedules for pulpwood crops f o r the three major plantation species i n Kenya 24 7 Summary of the permanent sample plot data 31 8 C o e f f i c i e n t s for the dominant height over age l i n e a r equations 47 9 Comparison of the modified Weibull and Chapman-Richard models for height over age curves 54 10 Asymptotic standard deviations for the estimated c o e f f i c i e n t s of Table 9 57 11 C o e f f i c i e n t estimates and other s t a t i s t i c s for the height over age and s i t e index equation 2.13 59 12 Asymptotic standard deviations for the estimated c o e f f i c i e n t s of Table 11 59 13 D i s t r i b u t i o n of plot s showing s i t e index over age c o r r e l a t i o n at .05 p r o b a b i l i t y l e v e l f o r the three species 62 14 Covariance analysis f o r slope test f o r height over age equations f o r P_. patula and P. radiata for d i f f e r e n t establishment s i t e s 66 15 Covariance analysis for slope test for height over age equations f o r I?, patula i n d i f f e r e n t regions i n Kenya 71 v i i i Table Page 1 6 R a i n f a l l data and elevation f o r geographical regions recognized for separate s i t e index curves 7 3 1 7 Height and age data for P_. patula by geographical regions 7 5 1 8 Regression c o e f f i c i e n t s f o r s i t e index curves for P_. patula by geographical regions 7 7 1 9 D i s t r i b u t i o n of plot s showing s i g n i f i c a n t s i t e index over age co r r e l a t i o n s at . 0 5 l e v e l f o r P. patula by geographical regions 7 8 2 0 Summary of thinning data by species and relevant v a r i a b l e s 9 7 2 1 Mean DBH of thinning/mean DBH before thinning r e l a t i o n s h i p 9 8 2 2 Basal area of thinning/basal area before thinning r a t i o 1 0 1 2 3 Parameter estimates and other s t a t i s t i c s f o r the DBH of thinning equation 2 . 2 7 1 0 3 2 4 Parameter estimates and other s t a t i s t i c s for the DBH of thinning equation 2 . 2 8 1 0 4 2 5 Summary of the basal area before thinning data 1 0 6 2 6 Parameter estimates and relevant s t a t i s t i c s for basal area before thinning equation 2 . 2 9 1 0 9 2 7 Asymptotic standard deviations for the estimated c o e f f i c i e n t s of Table 2 6 1 1 0 2 8 Summary of basal area increment data 1 2 0 2 9 Parameter estimates and other relevant s t a t i s t i c s for the basal area increment equation for C_. l u s i t a n i c a , P_. patula and P. r a d i a t a (Kenya and New Zealand) .... 1 2 3 3 0 Asymptotic standard deviations for the parameter on Table 2 9 1 2 3 3 1 Summary of the DBH d i s t r i b u t i o n data 1 3 3 3 2 Linear and c u r v i l i n e a r c o r r e l a t i o n of the D L and estimated Weibull parameters with other stand variables 1 3 7 ix Table Page 33 Comparison of predicted and observed Weibull parameters f o r the test plots 141 34 Equations and c o e f f i c i e n t s for the tree volume equations f o r C_. l u s i t a n i c a , P_. patula and P. ra d i a t a i n Kenya 145 35 C o e f f i c i e n t s for R-factor equation (2.44) f o r the merchantable l i m i t s for the respective species 146 36 Domain of the y i e l d model EXOTICS with respect to input variables 168 37 To t a l volume y i e l d table for C_. l u s i t a n i c a s i t e Index 20 169 38 Merchantable volume y i e l d table for C_. l u s i t a n i c a s i t e index 20 170 39 Stand table at c l e a r f e l l f o r C_. l u s i t a n i c a s i t e index 20 171 40 Beta weights, mean bias, standard deviation and the 95% confidence l i m i t s of percentage differences between observed and simulated t o t a l volume overbark, and the Chi-square values for three hypothesized l e v e l s of accuracy: (2. l u s i t a n i c a 181 41 Beta weights, mean bias, standard deviation and the 95% confidence l i m i t s of percentage differences between observed and simulated t o t a l volume overbark, and the Chi-square values for three hypothesized l e v e l s of accuracy: P_. patula 182 42 Beta weights, mean bias, standard deviation and the 95% confidence l i m i t s of percentage differences between observed and simulated t o t a l volume overbark, and the Chi-square values for three hypothesized l e v e l s of accuracy: P. radi a t a 183 43 Bias percentage for dominant height and basal area f o r te s t permanent sample plots by species 193 44 Volume y i e l d and other relevant stand parameters under the current sawtimber thinning regime for C_. l u s i t a n i c a to a r o t a t i o n age of 40 years: Technical Order No. 42 of March 1969 211 X Table Page 45 Volume y i e l d and other relevant stand parameters under the current sawtimber thinning regime f or P_. patula (Nabkoi) to a r o t a t i o n age of 20 years: Technical Order No. 53 of May 1981 212 46 Volume y i e l d and other relevant stand parameters under the current sawtimber thinning regime f or P_. radiata to a r o t a t i o n age of 30 years: Technical Order No. 44 of March 1969 213 47 Basal area before thinning (m^/ha) f o r the a l t e r n a t i v e thinning regime 221 48 Maximum MAI (nr/ha) and b i o l o g i c a l r o t a t i o n age (culmination age) for d i f f e r e n t thinning regimes for C. l u s i t a n i c a S.I. 18 227 49 Volume y i e l d (m /ha), increase % ( r e l a t i v e to current thinning regime) and other stand parameters at 40 years r o t a t i o n age for d i f f e r e n t thinning regimes for C. l u s i t a n i c a S.I. 18 231 50 Volume pr o d u c t i v i t y (m^/ha) and stand mean DBH (cm) up to age 40 years for various thinning l e v e l s at 20% thinning i n t e n s i t y by s i t e index classes f o r C_. l u s i t a n i c a , r e l a t i v e to the current thinning regime 235 51 E f f e c t of i n i t i a l stocking on y i e l d under thinning regime C:20 for C. l u s i t a n i c a i . e . thinning based on proportion of basal area to remove when a c r i t i c a l stand basal area i s equalled or exceeded 241 52 Total volume y i e l d (V(l)(m 3/ha) f o r P. patula (Nabkoi) by s i t e index classes for various stocking l e v e l s and establishment s i t e s up to age 15 years ... 244 x i LIST OF FIGURES Figure Page 1 Kenya forest blocks and weather stations 11 2 Ombrothermic diagrams for weather stations representa- t i v e of the highland zone i n Kenya 13 3 Height/age r e l a t i o n s h i p f o r C_. l u s i t a n i c a plots 43 4 Height/age r e l a t i o n s h i p for P_. patula plots 44 5 Height/age r e l a t i o n s h i p f or P_. radiata plots 45 6 S i t e index estimation procedure for C_. l u s i t a n i c a ... 49 7 Height over age curves for d i f f e r e n t stand e s t a b l i s h - ment s i t e 67 8 Height/age r e l a t i o n s h i p f o r P_. patula by geographical regions 72 9 S i t e index curves for C_. l u s i t a n i c a i n Kenya ........ 81 10 Site index curves for P̂. r a d i a t a i n Kenya 82 11 S i t e index curves f o r P_. patula i n Kenya Nabkoi group 84 12 No. stems/height/S% r e l a t i o n s h i p by species and s i t e index classes i n Kenya 92 13 Basal area over age curves for various s i t e index classes at stand density of 1200 s.p.h I l l 14 Observed and predicted basal area for unthinned plots not used i n formulating the basal area equation 113 15 Basal area Increment curves (a) P_. radiata Kenya (b) P. radiata New Zealand 127 16 Basal area increment curves (a) C_. l u s i t a n i c a (b) P. patula 129 17 Diameter d i s t r i b u t i o n histogram and the frequency curve r e s u l t i n g from the f i t t e d Weibull p r o b a b i l i t y density function 135 18 Diameter d i s t r i b u t i o n histograms, f i t t e d Weibull p.d.f. and the predicted Weibull p.d.f. for the 8-test plots 142 x i i Figure Page 19 Ove r a l l forest planning system showing the in t e g r a t i o n of the y i e l d model 161 20 Flowchart of the y i e l d model EXOTICS 163 21 Two simulations of thinning experiment 345 i n Tanzania (P_. patula) from d i f f e r e n t s t a r t i n g conditions 178 22 Comparison of simulated and observed t o t a l volume (overbark) f o r two C_. l u s i t a n i c a t e s t plots 180 23 D i s t r i b u t i o n of £. l u s i t a n i c a test plots by age and volume bias % 195 24 D i s t r i b u t i o n of P_. patula test plots by age and volume bias % 196 25 D i s t r i b u t i o n of P_. ra d i a t a test plots by age and volume bias % 197 26 D i s t r i b u t i o n of £. l u s i t a n i c a test plots by s i t e index and volume bias % 198 27 D i s t r i b u t i o n of P_. patula test plots by s i t e index and volume bias % 199 28 D i s t r i b u t i o n of P_. radi a t a test p l o t s by s i t e index and volume bias % 200 29 The maximum size-density r e l a t i o n s h i p and the natural stand data used i n po s i t i o n i n g t h i s r e l a t i o n s h i p .... 206 30 Main stand basal area/age r e l a t i o n s h i p under the current sawtimber thinning regimes by species and s i t e index classes 208 31 Mean and current annual volume increment r e l a t i o n s h i p with age for the current sawtimber thinning regimes by species and s i t e Index classes 209 32 Number of stems and basal area at d i f f e r e n t ages for d i f f e r e n t thinning l e v e l s and thinning i n t e n s i t i e s for C. l u s i t a n i c a S.I. 18 222 33 MAI and GAI over age curves for d i f f e r e n t thinning l e v e l s and thinning i n t e n s i t i e s f o r C. l u s i t a n i c a S.I. 18 225 x i i i Figure Page 34 D i s t r i b u t i o n of merchantable volume (m^/ha) f o r d i f f e r e n t thinning regimes for C. l u s i t a n i c a S.I. 18 233 35 Merchantable volume increase (%) for d i f f e r e n t thinning regimes ( r e l a t i v e to current thinning regime) on d i f f e r e n t s i t e index classes 238 36 CAI and MAI curves for various stocking l e v e l s f or _P. patula s i t e index class 21 245 37 Diameter/age r e l a t i o n s h i p at various stocking l e v e l s f o r s i t e index 21 for P_. patula (Nabkoi) 246 xiv ACKNOWLEDGEMENTS The author wishes to thank his two major research supervisors Dr. J.P. Demaerschalk who supervised the i n i t i a l phase and Dr. D.D. Munro who supervised the f i n a l phase of t h i s study. Their guidance and encouragement are greatly appreciated. The author also thanks h i s supervisory committee members: Drs. N. Reid, A. Kozak, G. Weetman, J . Thirgood, C. Goulding and Mr. P. Sanders for t h e i r comments and constructive c r i t i c i s m on the research, also for reviewing the thesis and providing most welcome advice. Special thanks also go to the computer programming s t a f f of the Forestry Department, e s p e c i a l l y Mr. Barry Wong for h i s help with the programming phase of the study. Also to a l l s t a f f and students i n the Faculty of Forestry who i n one way or another helped make t h i s study a success. The author would also l i k e to thank the Chief Conservator of Kenya Forest Department Mr. O.M. Mburu for permission to use the permanent sample plo t s data and other information for t h i s study. Also to Mr. H.L. Wright of the Commonwealth Forestry I n s t i t u t e , Oxford for r e t r i e v i n g the data and other relevant information from the data bank at Oxford for the author. This study was made possible through f i n a n c i a l support from International Development Association fellowship (through University of N a i r o b i ) , McPhee fellowship and the Forest Products fellowship (through U.B.C. Faculty of F o r e s t r y ) . This assistance i s g r a t e f u l l y acknowledged. The author also thanks the University of Nairobi for the 3 year study leave. XV DEDICATION I dedicate t h i s thesis to my wife - N e l l i e Muthoni Mathu. N e l l i e gave up her career In teaching i n order to come and minister to me during my study here i n Canada. Her dedicated love and encouragement have been my main source of strength. Also to my children - Muthoni, Mwihaki and Mathu who provided most welcome d i s t r a c t i o n from the rigours of my study, e s p e c i a l l y i n the evenings. 1 INTRODUCTION The t o t a l forest land i n Kenya consists of about 2 m i l l i o n hectares or 3% of the t o t a l land area of the country, a l l of which i s p u b l i c l y owned and administered by the Kenya Forest Department. Of t h i s land, about two th i r d s i s designated protection f o r e s t , leaving about 660,000 hectares for timber production. To date, over 150,000 hectares have been converted to exotic softwood plantations, mainly Cupressus l u s i t a n i c a M i l l e r , Pinus patula Schlecht and Cham and Pinus radiata D. Don. These species are grown primarily for the supply of sawtimber and pulpwood to p r i v a t e l y owned forest i n d u s t r i e s . The demand for timber and timber products i n Kenya has been r i s i n g and w i l l continue to r i s e i n the foreseeable future. This i s as a re s u l t of two components: increasing population estimated at an annual rate of 4% and increasing per capita consumption of wood and wood products, a r e s u l t of a r i s i n g standard of l i v i n g . To meet th i s r i s i n g demand for wood and wood products within the constraint of a fixed forest land base, one of the options open to the government i s more intensive forest management to maximize y i e l d from the available forest land through s i l v i c u l t u r a l manipulation of the stand. To do t h i s e f f e c t i v e l y requires a good knowledge of the growth and y i e l d of the candidate species under the various physical, edaphic and s i l v i c u l t u r a l conditions p r e v a i l i n g i n the country. When av a i l a b l e , t h i s knowledge forms a basis for the formulation of a l t e r n a t i v e management strategies to meet the desired goals and objectives. This can best be achieved i f a r e l i a b l e means of forecasting growth and y i e l d under the relevant 2 phy s i c a l , edaphic and s i l v i c u l t u r a l constraints i s available to provide the necessary quantitative information. Since large scale plantation f o r e s t r y started i n Kenya around 1936 (C_. l u s i t a n i c a ) and 1946 (P_. patula and P_. r a d i a t a ) , several studies on growth and y i e l d of plan t a t i o n species have been undertaken. Some of these, i n c l u d i n g Wimbush (1945), G r i f f i t h and Howland (1961) and Paterson (1967) were li m i t e d i n scope, ei t h e r because they were based on lim i t e d data or were for s p e c i f i c regions. However, since the s e t t i n g up of a permanent sample plots program i n 1964, three important studies based on data from these plots have been undertaken: 1. Wanene (1975, 1976) and Wanene and Wachiuri (1975) constructed v a r i a b l e density y i e l d tables for P. patula, P_. ra d i a t a and C_. l u s i t a n i c a , r e s p e c t i v e l y . These tables were of p a r t i c u l a r s i g n i f i c a n c e as they were the f i r s t tables of t h e i r kind for y i e l d estimation. However, they were based on simple regression methods and so could not represent the dynamic processes of stand growth very w e l l . Shortcomings i n these y i e l d tables were discussed by Mathu (1977). 2. Mathu (1977) studied the growth and y i e l d of £. l u s i t a n i c a i n Kenya as part requirement of an M.Sc. study program. This study improved on the methodology used by Wanene and Wachiuri by casting the p r i n c i p a l growth function as a rate of change of the stand basal area. However, th i s study was lim i t e d i n that i t only considered growth for an average s i t e i n Kenya. Two major areas for future research were i d e n t i f i e d . 3 (a) Need f o r further i n v e s t i g a t i o n of basal area increment i n overstocked and understocked stands and i n d i f f e r e n t s i t e c l a s s e s . (b) Need to investigate further the height development i n the d i f f e r e n t regions i n the country. S 3. Alder (1977) developed a single stand y i e l d p r e d i c t i o n model - the VYTL-2 - as a subroutine i n the PYMOD forest management program as part of the requirements i n a Ph.D. study program. VYTL-2 presents the stand state as a l i s t of diameters (derived from a diameter p r o b a b i l i t y density function), stand growth i s driven by a s i t e index curve while a tree diameter increment equation (a function of s i t e index, stand basal area and dominance r a t i o - a measure of competitive stress) e f f e c t s d i a - meter growth for diameter classes as percentiles of a cumula- t i v e d i s t r i b u t i o n function. This, therefore, i s a single tree distance independent model although growth i s for diameter classes rather than the i n d i v i d u a l tree (see Chapter 3 Section 2), The VYTL-2 was designed to simulate y i e l d s for single stands of C_. l u s i t a n i c a , P_. patula and P_. radiata and, according to Alder (1977), i t i s also capable of simulating d i f f e r e n t thinning treatments and so can be used as a s i l v i c u l t u r a l t o o l . However, two main factors mitigate against use of t h i s model as a s i l v i c u l t u r a l t o o l : 1. On v a l i d a t i o n at 95% confidence l e v e l , the output from t h i s model ranged from 40% underestimate to 20% overestimate of t o t a l volume y i e l d (Alder 1978). According to the author, 4 these errors appeared to be associated with a v a r i e t y of f a c t o r s : genetic, b i o t i c and c l i m a t i c ; and not with any s t r u c t u r a l flaw i n the model. Whatever the cause, the e f f e c t s of these systematic errors i s to reduce the confidence of the model e s p e c i a l l y as a t o o l for s i l v i c u l t u r a l research. 2. The thinning algorithms i n the model are based on purely hypo- t h e t i c a l assumptions such as low thinning, mechanical thinning, etc. which imply a f i x e d d i s t r i b u t i o n of the removed stems i n the stand. These assumptions which amount to accepting the concept of an " i d e a l " thinning type are u n r e a l i s t i c i n p r a c t i c e . In addition, the model allows only one thinning option within a single simulation run. In practice however, d i f f e r e n t thinning c r i t e r i a may be required at d i f f e r e n t points i n the l i f e of the stand, necessitating use of more than one thinning option within a single simulation run. This f l e x i b i l i t y i s lacking i n the VYTL-2 model. The PYMOD model of which VYTL-2 i s a subroutine was intended as a long-term forecasting and f e a s i b i l i t y analysis system for e n t i r e f o r e s t s . For given i n i t i a l planting i n t e n s i t i e s , several plantations and t h e i r s i t e i n d i c e s , the model simulates species mix, product mix and y i e l d s using s a t i s f y i n g (as opposed to optimizing) c r i t e r i a . This model i s s i m i l a r to the A u s t r a l i a n model FORSIM which simulates the growth of many stands comprising a t o t a l forest estate at one time and summarizes y i e l d s by various s i z e classes from a l l compartments (Gibson et a l . 1970). However, unlike the A u s t r a l i a n s i t u a t i o n , the present East 5 A f r i c a n scene for which PYMOD was designed lacks forest planning systems capable of defining a l l the constraints - economic, product mix, species mix, e t c . that are required to u t i l i z e PYMOD c a p a b i l i t y . This may p a r t l y explain why the model has received so l i t t l e a ttention i n East A f r i c a . The growth conditions under which the three species are managed i n Kenya vary considerably i n terms of s i t e q u a l i t i e s (including s i t e factors such as r a i n f a l l d i s t r i b u t i o n and i n t e n s i t y , s o i l s , e levation above sea l e v e l , e t c . ) , establishment s i t e s (defined l a t e r ) and s i l v i - c u l t u r a l regimes. From a review of past studies, i t i s apparent that the growth and y i e l d of these species under the p r e v a i l i n g conditions has not been adequately addressed; neither has the p o s s i b i l i t y of the adoption of a l t e r n a t i v e s i l v i c u l t u r a l regimes been considered as a means of increasing growth and y i e l d of the stand. It i s also apparent that there i s no r e l i a b l e means of forecasting growth and y i e l d that can be used to f a c i l i t a t e s i l v i c u l t u r a l manipulation of the stand towards the desired goals and objectives. These problems gave r i s e to the three objectives of t h i s study: 1. To study growth and y i e l d of the three species: £ . l u s i t a n i c a , P_. patula and P_. radiata under the v a r i e t y of s i t e s , e s t a b l i s h - ment s i t e s , stand structures and s i l v i c u l t u r a l practices found i n Kenya. 2. To construct a stand growth and y i e l d model as a means for y i e l d p r e d i c t i o n and as an aid to s i l v i c u l t u r e for evaluation of d i f f e r e n t management strategies. To evaluate the Impact of the present and a l t e r n a t i v e manage- ment schedules on stand development for d i f f e r e n t management objectives. 7 CHAPTER 1 BACKGROUND INFORMATION 1. Species Nomenclature and D i s t r i b u t i o n Cypressus l u s i t a n i c a M i l l e r (1768) also c a l l e d Mexican cypress, Portuguese cypress, Cedar of Goa or just cypress i s of the family Cupressaceae and genus Cupressus. In nature, i t occurs n a t u r a l l y between l a t i t u d e 15-45° North and i s widespread i n Central and Southern Mexico, Guatemala, Honduras and E l Salvador, where i t grows to a height of 30 meters at 1800-2400 meters elevation (Dyson 1968). Outside i t s natural range, t h i s species i s planted extensively i n the South and West of France, Japan, Portugal, Spain and East A f r i c a . In New Zealand, South A f r i c a and Malawi, i t i s planted as a minor plantation species. Cupressus l u s i t a n i c a was introduced to Kenya i n 1905 but large scale planting did not begin u n t i l 1936 (Wimbush 1945). Figures for d i s t r i b u - t i o n by area are only a v a i l a b l e for Kenya as shown i n Table 1. Pinus patula Schlecht and Cham also c a l l e d Patula pine or spreading- leaved pine occurs n a t u r a l l y i n a comparatively r e s t r i c t e d range i n the States of Queretaro, Hildalgo, Pueblo, Mexico and Vera Cruz i n Central Mexico at an elevation of 1500-3000 meters elevation with annual average r a i n f a l l of 1200 mm (Mirov 1967). Outside i t s range, P. patula i s planted mostly i n A f r i c a where South A f r i c a pioneered i t s use. It was introduced to Kenya i n 1910 but large scale planting did not star t u n t i l 1946 (Pudden 1957). P_. patula i s grown mainly for pulpwood and saw- timber but the wood can be used for other purposes, e.g. veneer as i n 8 TABLE 1: D i s t r i b u t i o n of the species by area i n the major countries i n which the species are grown. Species Country Area i n hectare Source of data C. l u s i t a n i c a Kenya 66,000 Kenya Forest Department (1981) (Per. Comm.) P. patula South A f r i c a Kenya Malawi 174,000 Crowe (1967) 50,000 Kenya Forest Department (1981) (Per. Comm.) 23,000 Marshall and Foot (1969) P. radi a t a New Zealand Ch i l e A u s t r a l i a Spain South A f r i c a Kenya 307,000 261,000 185,000 173,000 35,000 18,000 Grut (1970) Kenya Forest Department (1981) (Per. Comm.) 9 Kenya. Table 1 shows the d i s t r i b u t i o n of the species i n South A f r i c a , Kenya and Malawi. Pinus ra d i a t a D. Don also known as radiata pine, i n s i g n i s pine, Monterey pine or remarkable pine belongs to a group of pines known as closed-cone pines, believed to have been widespread along the coast of C a l i f o r n i a In p r e h i s t o r i c times (Forde 1966). The present d i s t r i b u t i o n of P_. radi a t a i s the coastal region of central C a l i f o r n i a and Mexico, where i t grows n a t u r a l l y on some 4,050 hectares (Scott 1960). It exi s t s i n two v a r i e t i e s ; two needles and three needles. In i t s natural habitat, P_. radi a t a i s a small to medium sized bushy tree which grows quickly and bears cones at an early age - 6 years. Cones are hard and p e r s i s t for many years on the stems and branches and may remain closed for many years a f t e r maturity. In th i s form, the tree i s of l i t t l e economic value. The climate i n central C a l i f o r n i a and Mexico where ]?. ra d i a t a grows i s a s p e c i a l type of mediterranean climate with l i t t l e summer rain but adequate summer moisture from frequent sea fogs and mists. It grows on gentle to moderate slopes from the coastline to 10 kilometers inland, on deep sandy s o i l types (sandy loam s o i l s ) with r a i n f a l l ranging between 500 to 1050 mm per annum. 70 to 75% of the r a i n f a l l s i n winter. P. radi a t a i s the most widely planted species outside i t s natural habitat. As with P̂ . patula, i t was introduced to Kenya i n 1910 and large scale planting started i n 1946 (Pudden 1957). The d i s t r i b u t i o n of t h i s species by area i n the major countries where i t has been introduced i s shown i n Table 1. 10 2. Climate and S o i l s The bulk of softwood plantations i n Kenya are found i n the Kenya highlands between 1800 to 2750 meters above sea l e v e l . Within the high- land zone, v a r i a t i o n s i n r e l i e f do occur, of which the most s i g n i f i c a n t i s the Great R i f t V a l l e y which transects the central highlands i n a North/South d i r e c t i o n , as shown on Figure 1. This v a r i a t i o n i n r e l i e f r e s u l t s i n great v a r i a t i o n i n climate and vegetation. As shown on Figure 1, the highland zone l i e s right across the equator. One would therefore expect an equatorial type of climate characterized by high i n s o l a t i o n , high r a i n f a l l and high evaporation rates. However, the combination of a l t i t u d e and the r e l i e f v a r i a t i o n modifies the climate so that no single c l i m a t i c type can characterize the climate of t h i s zone. According to Gilead and Roseman (1958), the most important elements of climate for plant growth are warmth (temperature) and water ( r a i n - f a l l ) . Using these two elements, several procedures have been used to characterize the water loss from evapotranspiration and water gain from r a i n f a l l i n r e l a t i o n to plant growth. These include the Thornthwaite system of c l i m a t i c c l a s s i f i c a t i o n (Thornthwaite 1948), Penmans method (Penman 1948) and the Gaussens method (Gaussen 1954). Of these three, the Gaussens method has been adopted i n t h i s study mainly because i t u t i l i z e s the only two c l i m a t i c data available - r a i n f a l l and temperature measurements. B a s i c a l l y the Gaussen procedure consists of constructing ombrother- mic diagrams as follows: 11 FIGURE 1 KENYA FOREST BLOCKS AND WEATHER STATIONS 12 1. On the abscissa scale, plot the months of the year. 2. On the ordinate scale, l a b e l the right axis with monthly p r e c i p i t a t i o n i n mm and the l e f t axis with monthly average temperatures i n degrees centrigrade to a scale double that of p r e c i p i t a t i o n . 3. J o i n a l l the l e v e l s of monthly temperature to get the thermic curve, j o i n a l l the l e v e l s of monthly r a i n f a l l to get the ombrographic curve. When the ombrographic curve sinks below the thermic curve, p r e c i p i t a t i o n < 2 temperature. The space enclosed by the two curves indicates the duration and severity of the dry season. This c r i t e r i a for dry season i s based on a rule of thumb but the r e s u l t s are consistent with those obtained using other procedures (FAO 1974). Figure 2 shows the ombrothermic diagrams for weather stations representative of the highland zone (see also Figure 1) while Table 2 gives the r a i n f a l l summary, a l t i t u d e and an i n d i c a t i o n of the dry months for each s t a t i o n . Data was obtained from the East A f r i c a n Meteorological Department (1973, 1975). Figure 2 shows that of the three stations representing the high- lands west of the R i f t (Eldoret, Molo and Kericho), there i s no dry season as defined by the ombrothermic diagrams. In general the diagrams for these stations i n d i c a t e one long rainy season between March to September and one season with minimum r a i n f a l l between October to February. 13 FIGURE 2 OMBROTHERMIC DIAGRAMS FOR WEATHER STATIONS REPRESENTATIVE OF THE HIGHLAND ZONE IN KENYA Ombrographic Curve Thermic Curve J f m a m j j a s o n d j ( m » m j j » » o n d  15 TABLE 2: Summary of r a i n f a l l and a l t i t u d e for weather stations representative of the highland zone. Weather s t a t i o n Elevation i n R a i n f a l l i n mm Dry season meters above per annum sea l e a v e l Edoret Molo Kericho Kimakia Muguga Nanyuki 2084 2477 2134 2439 2096 1947 1124 (24) 1177 (28) 2081 (7) 2288 (14) 995 (20) 759 (32) no dry season M It September January & February Number i n bracket ind i c a t e number of years of record. 16 Of the three stations representing the highlands east of the R i f t (Kimakia, Muguga and Nanyuki), Kimakia has the highest t o t a l annual r a i n f a l l followed by Muguga, with Namyuki having the l e a s t . Muguga has a one month dry season (September) while Nanyuki has two months (January and February) of dry season. In general, the diagrams for these stations ind i c a t e one long r a i n season between March and May with a peak i n A p r i l and a short rainy season between October to December with a peak i n November. January/February and July/August/September are seasons of minimum r a i n f a l l . In general the highland s i t e s where plantations are established are o v e r l a i d with volcanic loam s o i l s , usually of great depth. They are usually well drained so that there i s r a r e l y any water table within reach of tree roots. However, there are exceptions and i n some places where drainage i s impeded, pans of l a t e r i t e may be found which i n some cases act as b a r r i e r to tree root penetration to lower s o i l s t r a t a . Volcanic loams are generally very suitable for tree growth. 3. S i l v i c u l t u r a l Forest Management i n Kenya In discussing forest management i n Kenya, there are two possible approaches: the t e c h n i c a l approach whereby the success of a given management a c t i v i t y i s measured on the basis of b i o l o g i c a l c r i t e r i a , e.g. highest volume production; and the economic approach whereby the c r i t e r i a for success i s based on economic evaluation. To date, the economic evaluation of management practices have never been attempted i n Kenya, mainly because of lack of a basis for assessing the quantitative 17 impact of the s i l v i c u l t u r a l p r a c t i c e s . Only technical aspects of forest management w i l l be discussed i n t h i s section. Nursery p r a c t i c e A r t i f i c i a l regeneration through planting i s the standard p r a c t i c e i n Kenya. Nursery practice i s a highly developed technology. Seed i s usually sown i n l e v e l , shaded beds which are usually netted to protect the seedlings from birds and small mammals. The s o i l mixture consists mostly of sand with no humus material. Seed pretreatment i s not required f o r any of the three species. Table 3 shows the number of seed per kilogram and the germinative capacity for each of the species under normal nursery p r a c t i c e . TABLE 3*: Seeds per kilogram and germinative capacity for £. l u s i t a n i c a P. patula and P_. radi a t a i n Kenya. Species Seeds per kg Germinative No. seedlings capacity % per kg C. l u s i t a n i c a 236,000 30% 70,000 to 75,000 P. patula 180,000 30% 50,000 to 60,000 P. r a d i a t a 35,000 40% 15,000 to 20,000 *Data obtained from the respective technical reports. As soon as the seeds have germinated and before any side roots are developed, the seedlings are pricked out into trays of siz e 38 x 40 cm and 10 cm high containing s o i l to 8 cm depth or r a r e l y i n t o transplant beds ra i s e d 8 cm above the ground l e v e l . For C_. l u s i t a n i c a , ordinary 18 . forest s o i l i s used, while pine s o i l i s required for the pine species, as the l a t e r contains a mycorrhiza species necessary for the s u r v i v a l of the pine seedlings. Each box contains 49 plants so that each seedling enjoys approximately 250 cu.cm of s o i l . Use of f e r t i l i z e r s i n the nursery i s standard p r a c t i c e . Seedlings are considered mature for planting out i n the f i e l d when they are approximately 20-30 cm high. Plantation establishment methods There are two systems of plantation establishment i n Kenya: 1. Shamba System: This i s a highly developed taungya system whereby the land earmarked for tree planting Is issued to permanent forest employees for crop c u l t i v a t i o n . A f t e r one or two years of crop growing, trees are planted and the employees continue growing t h e i r crops u n t i l the trees are too t a l l f o r a g r i c u l t u r a l crops. Under t h i s system, the employees gain from the a g r i c u l t u r a l crops while the trees are planted i n c u l t i - vated ground and get free weeding the f i r s t few seasons of t h e i r l i f e . 2. Grassland Planting: Forest glades and open grasslands are r a r e l y suitable f or a g r i c u l t u r a l crops. Tree planting i n these s i t e s i s preceded by minimum land preparation consisting of s t r i p ploughing or simply digging p i t s where the trees w i l l be planted. Experience with (2. l u s i t a n i c a has shown i t to be very i n t o l e r a n t of weed competition, e s p e c i a l l y from grass. As such i t i s never planted on grassland s i t e s . Pine species on the other hand can be established 19 under either system as t h e i r s u r v i v a l on grassland s i t e s i s quite acceptable. I n i t i a l spacing and early tending i n plantations The objective of stand establishment i n Kenya i s s i m i l a r to that of South A f r i c a : to grow trees to merchantable size i n as short a time as p o s s i b l e . Thus, the stand establishment i s characterized by wide spacing accompanied by heavy thinning and pruning to insure high q u a l i t y stems. Presently, the i n i t i a l spacing i s 2.5 x 2.5 meters (1600 stems per hectare) and, as indicated i n the 1981 revised Technical Order for P. patula there i s a move to even wider spacing (3.0 x 3.0 meters = 1110 stems per hectare). This i s very wide spacing compared to ce n t r a l Europe where spacing i s 1.4 x 1.4 meters = 5000 seedlings per hectare for 2+2 year old spruce seedlings or B r i t a i n where spacing ranges between 2.2 x 2.2 meters to 2.0 x 2.0 meters (2000 to 2300 seedlings per hectare) (Kuusela 1968). For the l a t t e r countries, the objectives of stand establishment are to obtain s u f f i c i e n t l y dense stands to u t i l i z e f u l l y the s i t e p r o d u c t i v i t y and to improve the q u a l i t y of the tree stems by s e l f pruning. These and the slower growth rates explain the closer i n i t i a l spacing. An important consideration i n adopting a s p e c i f i c i n i t i a l spacing p o l i c y i s the cost of r a i s i n g the seedling i n the nursery and planting out i n the f i e l d . Wide i n i t i a l spacing may be j u s t i f i e d i n Kenya, f i r s t l y because i t means fewer seedlings to be raised and therefore l e s s nursery and planting expenses and, secondly because the intensive ground 20 preparation and subsequent tending e s p e c i a l l y under the Shamba system, ensures higher s u r v i v a l of seedlings. Pruning By d e f i n i t i o n , pruning involves removal of l i v e branches so as to ensure production of timber free of dead knots. The wide i n i t i a l spacings accompanied by heavy thinnings as practiced i n Kenya imply an increase i n si z e of branches and delay i n natural pruning. Pruning i s therefore a standard p r a c t i c e . Table 4 shows the current pruning schedules for the three species. Thinning Under t r a d i t i o n a l f o r e s t r y practices as practiced i n Europe and B r i t a i n , the objective of thinning i s to harvest those trees which would be wasted as mortality and to better the q u a l i t y of standing stock. Thinnings are therefore very l i g h t so that the density of standing stock i s kept as high as i s necessary for maximum volume production. The thinning practices i n Kenya on the other hand are based on Craib's (1939, 1947) thinning p o l i c y which advocated very heavy thinnings so as to promote tree diameter growth. This p o l i c y i s not consistent with the objective of maximum volume production but aims at production of larger sized material i n as short a r o t a t i o n as possible at the expense of some loss i n t o t a l y i e l d . Craib's revolutionary ideas on thinning went against t r a d i t i o n a l thinning p o l i c i e s , e s p e c i a l l y with regard to the recommendation for more severe thinnings on poor s i t e s . As a r e s u l t t h i s p o l i c y has received varied comments, some against (Hawley and Smith 1954, Johnston 1962) but 21 TABLE 4: Pruning schedules f o r C_. l u s i t a n i c a , P_. patula and P_. radiata i n Kenya as per relevant technical orders. Species Age/dominant height i n Pruning height from ground Number of to be p stems/ha runed meters l e v e l Sawtimber/ plywood Pulp- wood c . l u s i t a n i c a 2 years 1/2 height but not over 2 meters A l l stems A l l stems 4 years 1/2 height but not over 4 meters A l l stems A l l stems 9.25 meters 2/3 tree height 533 stems A l l 11.25 meters 2/3 tree height 533 stems N/A 13.75 meters 2/3 tree height Minimum 9 meters Maximum 11 meters 533 stems N/A p. patula 3 years 1/2 height + 1 whorl A l l N/A 4 years 1/2 height + 1 whorl N/A A l l 8 meters 1/2 height + 1 whorl 600 N/A 12 meters 1/2 height + 1 whorl 600 N/A 16 meters 10 meters 600 N/A p. radi a t a 3 years 1/2 height + 1 whorl A l l A l l 12.0 meters 1/2 height + 1 whorl 426 A l l 17.5 meters 1/2 height + 1 whorl 426 N/A 24.5 meters 1/2 height + 1 whorl 213 N/A 22 mostly i n support (De V i l l i e r s et a l . , 1961, Hile y 1959, Fenton 1972, Lewis 1964). Due to changing management objectives, Craib's management recommendations have been revised i n South A f r i c a to s u i t s p e c i f i c objectives, such as production of pulpwood or to improve on timber q u a l i t y . For further discussion on these, the reader i s referred to Kotze (1960) and De V i l l i e r s et a l . (1961). The thinning regimes adopted i n Kenya compare c l o s e l y with those of South A f r i c a . However, while thinning regimes for the l a t t e r are t i e d down to s p e c i f i c s i t e q u a l i t y classes, those for Kenya are for an average s i t e consequent on the lack of a means of assessing s i t e q u a l i t y c l a s s e s . Thinning regimes for Kenya can therefore be considered as a compromise between the thinning regimes for the poorest and best s i t e q u a l i t y classes i n South A f r i c a , with some adjustments for the higher rates of growth i n Kenya. These are shown on Table 5 for sawtimber and plywood and Table 6 for pulpwood. In general, sawtimber crop i s considered mature for c l e a r f e l l i n g when the mean stand DBH i s 48 cm while for plywood, f i n a l mean stand DBH i s 51 cm. Rotation age for pulpwood plantations varies but i s usually between 15 to 20 years. 4. S i l v i c u l t u r a l Problems Related to E c o l o g i c a l Features of Exotic Plantations i n Kenya Before the s i l v i c u l t u r a l problems of stand management i n Kenya can be discussed, i t i s worthwhile to mention a few ec o l o g i c a l features which are peculiar to these plantations as these have a dir e c t bearing on the problems: 23 TABLE 5*: Basic thinning schedules f or sawtimber and plywood crops f or the three major plantation species i n Kenya. Species Treatment Dominant height Stem/ha a f t e r thinning or age at thinning No. % of planting (C. l u s i t a n i c a Planting 1st thinning 2nd thinning 3rd thinning 4th thinning P. patula Planting 1st thinning 2nd thinning 3rd thinning 4th thinning 11.25 meters but not before age 6 years 5 years a f t e r 1st thinning 10 years a f t e r 1st thinning 15 years a f t e r 1st thinning Before 1981 A f t e r 1981 16 meters 5 years a f t e r 1st thinning 10 years a f t e r 1st thinning 15 years a f t e r 1st thinning (plywood planta- tions only) 1,600 888 55.5 533 33.3 355 22.2 266 16.6 1,600 1,110 600 54.0 400 36.0 250 22.5 170 15.3 P_. radiata Planting 1st thinning 2nd thinning 3rd thinning 4th thinning 12 meters 17.5 meters 7 years a f t e r 2nd thinning 13 years a f t e r 2nd thinning 1,600 853 53.3 426 26.6 266 16.6 213 13.3 *Data from the respective technical orders. 24 TABLE 6: Basic thinning schedules for pulpwood crops for the three major plantation species i n Kenya. Species Treatment Age or dominant height at thinning Stems per hectare a f t e r thinning No. % of planting £. l u s i t a n i c a Planting c l e a r f e l l i n g or thinning 15-20 years 15 years 1,322 840 63.5 P. patula Planting: before 1981 a f t e r 1981 1st thinning (old plantations) New plantations 12 years 1,322 1,110 980 74.1 No thinning P. ra d i a t a Planting C l e a r f e l l i n g or Thinning 15-20 years 15 years 1,322 880 66.6 25 1. The plantations are monocultures, meaning there i s only one species i n a given stand. 2. The species have only recently been introduced to Kenya, so that they have not yet f u l l y adapted themselves to the new environment. 3. The stands are even-aged, so that there i s only one stratum i n terms of s p a t i a l d i s t r i b u t i o n . 4. The species are usually very fast growing so that the rate of nutrient impoverization through tree harvest may be very rapid. The above e c o l o g i c a l features implies that these man-made ecosystems are very f r a g i l e and e c o l o g i c a l l y immature so that they are very susceptible to pest outbreaks. In p a r t i c u l a r , the following problems have been i d e n t i f i e d i n Kenya. Diseases: Two important fungal diseases have been i d e n t i f i e d i n Kenya plantations: 1. Cypress canker disease, caused by a p a r a s i t i c fungus Monochaetia unicornis (Cook and E l l i s ) Sacc. It i s not known i f t h i s fungus was present i n Kenya or i f i t was introduced. The fungus causes lesions on the stem of cypress trees, e s p e c i a l l y Cupressus macrocarpa Hartw. but has been known to af f e c t C_. l u s i t a n i c a to a smaller degree. This disease was responsible for stoppage of any further planting of £. macrocarpa despite the fact that i t was the more superior species i n terms of tree growth. 26 2. Dothlstroma pint species, another p a r a s i t i c fungus, was responsi- ble f o r the cessation of a l l planting of P. ra d i a t a i n 1961 when the fungus was discovered, apparently having been introduced into Kenya from elsewhere. This disease i s known to weaken the trees and sometimes k i l l them at a young age between 5 to 15 years. A f t e r that age,most trees not already k i l l e d recover and s t a r t to grow normally again. Attempts to control the disease are s t i l l i n progress i n Kenya. Another fungus of minor economic importance i s the universal A r m i l l a r i a mellea i n both C_. l u s i t a n i c a and pine plantations. Insect: Among the important insect pests i s the newly introduced woolly aphid, a Pineus species which attacks mostly pines. This insect attacks young twigs and needles, weakening the trees and eventually k i l l i n g them. Another important insect i s the Oemida gahani Distant which enters heartwood of l i v i n g C_. l u s i t a n i c a trees through pruning or i n j u r y scars, thus degrading the q u a l i t y of the logs. Rodents and big game damage: Rodents, including moles, and rats f i n d the bark of the young softwood plants e s p e c i a l l y palatable. S i m i l a r l y , b i g game such as elephants, buffaloes and Sykes monkeys are a continuous problem i n forest plantations, e i t h e r by pushing over the trees or by feeding on the succulent bark. F i r e problems: As indicated on the ombrothermic diagrams of Figure 2, January and February are usually the d r i e s t months of the year i n most areas of the highlands. Numerous forest f i r e s do occur, mostly 27 o r i g i n a t i n g from honey hunters and causing considerable damage to forest plantations. S o i l degradation: Although no study has been done to determine i f s o i l f e r t i l i t y w i l l decrease i n successive r o t a t i o n s , evidence from Southern A u s t r a l i a on P_. radi a t a (Keeves 1965) and from Swaziland on P_. patula (Evans 1975) in d i c a t e that y i e l d In successive rotations can be expected to be lower. This i s as expected due to the fact that a large quantity of organic matter and mineral nutrients are removed when harvesting the trees. A l l the above problems presents a very d i f f i c u l t challenge to forest management, e s p e c i a l l y as new problems are continuously a r i s i n g . Nonetheless, the advantages of even-aged monoculture plantations mitigate, the problems. In view of the l i m i t e d forest area i n Kenya, the very high y i e l d obtainable from plantations makes the investments i n forest protection worthwhile. 5. Permanent Sample Plots 5.1 The permanent sample plot program i n Kenya Objective The permanent sample plots (hereafter refereed to as p.s.p's) establishment program was i n i t i a t e d i n Kenya i n 1964 for a l l three species: C_. l u s i t a n i c a , P_. patula and P_. ra d i a t a . The main objectives of the program were: 28 1. To obtain information on growth rates of these species under the c l i m a t i c and edaphic conditions p r e v a i l i n g i n Kenya. 2. To obtain information on the y i e l d of these species under the s i l v i c u l t u r a l p r actices obtaining i n Kenya. 3. To provide a basis for the development of plantation management guide. Plots d i s t r i b u t i o n and layout i n the f i e l d A l l the p.s.p's are located i n the major geographical regions where these species are planted i n Kenya. In order to give a clear picture of the growth pattern, the o r i g i n a l p.s.p's were established i n 5, 10, 15, 20, 25, 30 and 35 year old plantations. The number of plots i n each age class was proportional to the t o t a l area of that class so that the larger the area for a given age c l a s s , the higher the number of p l o t s . S i m i l a r l y , the larger the area f o r a given species, the higher the number of plo t s allocated to i t . The whole idea of d i s t r i b u t i n g plots i n 5 year age i n t e r v a l s was that a f t e r 5 years of continuous measure- ments, the growth pattern f o r the whole ro t a t i o n would be known. Within a given age c l a s s , the d i s t r i b u t i o n of the plots was based primarily on the basis of pla n t a t i o n s t r a t i f i c a t i o n i n t o s i t e types, geographical regions and c l i m a t i c f a c t o r s : mainly r a i n f a l l d i s t r i b u t i o n . Due to t e c h n i c a l and f i n a n c i a l problems, the i n i t i a l sampling i n t e n s i t y within a given age class was 2 plots per 50 hectares. Within the selected p l a n t a t i o n , the l o c a t i o n of the plot was subjective. The plots s i z e was 0.04 hectare (.25 acre) and c i r c u l a r i n shape. Since the 29 I n i t i a l establishment, many more plots have been established using the same procedures (see Table 7). I d e a l l y the p.s.p's should be marked as inconspicuously as possible so that they do not receive p r e f e r e n t i a l treatments during thinning or pruning operations. For p r a c t i c a l purposes however, and ease of r e l o c a - t i o n , the plots are c l e a r l y marked on the ground. This may point to a p o s s i b i l i t y that the p l o t s may be receiving d i f f e r e n t treatments from the rest of the plantation. However, t h i s p o s s i b i l i t y i s minimum since treatments i n the plantations are supervised by the forest o f f i c e r - i n - charge of the forest d i s t r i c t . Plot measurements and preliminary data analyses Two main plot parameters are measured annually: 1. Diameter at 1.3 meters above ground, referred to as diameter at breast height or DBH. This i s measured for a l l trees on the plot to the nearest 0.1 cm with a diameter tape. 2. Stand dominant height, defined as the mean height of the 100 l a r g e s t diameter trees per hectare, i s measured for each p l o t . This i s accomplished by measuring the height of the four largest diameter trees on the p l o t . In the past, such i n s t r u - ments as Haga and Blume-Leiss hyposometers have been used but currently, the Suunto clinometer i s i n use. The measuring of p.s.p's i s usually timed for just before the t h i n - ning operations s t a r t . This timing i s c r u c i a l since the measurements of trees removed i n a thinning are taken as those at t h e i r previous years measurement. 30 Under an agreement between the Kenya government and the Common- wealth Forestry I n s t i t u t e , Oxford University, a l l the basic p.s.p. data i s sent to the l a t t e r f o r further c a l c u l a t i o n , processing and storage on computer tapes. The following information on p.s.p's i s available f o r up to and including 1979: 1. For Main Crop: (a) Age at time of thinning (b) Number of stems per hectare (c) Mean DBH of the stand ( i n cm) (d) Dominant height of the stand (in meters) (e) Basal area per hectare (In square meters/hectare) ( f ) Stand density indices (g) Volume of the stand i n cubic meters per hectare. This i s obtained using the tree volume equations for the respective species (see Chapter 3). 2. For Thinnings: (a) Number of stems removed per hectare (b) Basal area removed ( i n square meters/hectare) (c) Volume removed ( i n cu.meters per hectare) calculated as fo r main crop. 3. For Tot a l Production: (a) T o t a l basal area production i n sq.meters per hectare (b) Total volume production i n cu.meters/hectare (c) Current annual volume increment (d) Mean annual volume increment. Table 7 shows a summary of the basic p.s.p. data for up to 1979. 31 TABLE 7: Summary of the permanent sample pl o t data S P E C I E S C_. l u s i t a n i c a P_. patula P_. radiata No. Pl o t s Remeasurements 163 1,413 176 1,452 164 1,625 Age i n Years Minimum 4.7 Mean 17.4 Maximum 43.6 Standard deviation 8.6 3.6 12.5 27.7 4.9 5.5 13.6 34.6 5.4 Dom. Height i n m. Minimum 4.3 Mean 20.2 Maximum 39.4 Standard deviation 7.2 3.8 19.0 37.3 6.7 5.3 24.2 51.2 8.8 Hart's Density Index Minimum 12.1 Mean 21.4 Maximum 74.9 Standard deviation 5.0 8.4 20.2 57.0 6.8 7.9 16.6 168.8 9.7 32 5.2 Permanent sample p l o t s data as a basis for growth and y i e l d studies In general, growth and y i e l d studies are based on data obtained from e i t h e r permanent sample plots ( c f . continuous forest inventory), temporary sample p l o t s , or on a combination of both. When temporary sample plots are used, greater error can be expected since the e f f e c t s of past stand treatments and v a r i a t i o n i n growth over time are not taken into account. In regions where annual growth rings are a standard feature of tree growth, the problems of temporary samples are overcome through stem analysis procedure. Growth rings i n the exotic species i n Kenya are usually a r e f l e c t i o n of seasonal fluctuations i n r a i n f a l l rather than annual growth and so are unreliable as a guide to past tree growth. Permanent sample plots are therefore the only suitable source of data for growth and y i e l d studies i n these plantations. 5.3 Problems associated with permanent sample pl o t data General problems The problems associated with p.s.p. data have been discussed by several authors, including Vuokila (1965) and Adlard (1974). These can be summarized as follows: 1. Mensurational errors (also common to a l l sample plot systems). These include sampling errors i n terms of plot d i s t r i b u t i o n by age classes, s i t e classes, etc. and measuring errors that may occur due to use of f a u l t y measuring equipment or techniques. 2. Problems associated with plot management. These include f a i l u r e to relocate the plots i n the f i e l d , to r e - i d e n t i f y i n d i v i d u a l trees i n the plot or d i f f e r e n t i a l treatment of the 33 plo t s In r e l a t i o n to the rest of the plantations (see 5.1 above). 3. Problems Inherent i n the p.s.p. system i t s e l f . The main problems inherent i n the p.s.p. system a r i s e s from accidental damage to the plot or elements i n the plots e.g., damage to i n d i v i d u a l trees from game, diseases or in s e c t s . These problems r e s u l t s i n errors of varying magnitude i n the f i n a l parameters estimated from the data. Barring the p o s s i b i l i t y of sampling e r r o r s , the r e s u l t s of measuring errors are i n general minimal,and i n the long run,of a random nature. S i m i l a r l y , p o s s i b i l i t y of f a i l u r e to relocate plots i n the f i e l d i s very remote i n Kenya as the plots are c l e a r l y mapped on the plantation maps, while d i f f e r e n t i a l treatment of p.s.p's i s minimized through s t a f f supervision. The problem of autocorrelation i n p.s.p. data One of the important assumption underlying the use of regression analysis - the p r i n c i p l e data analysis precedure i n th i s study - i s that the residuals are independent. For example i n the usual regression model: Y i a B0 + B l X i l + B 2 X i 2 + e p x i P + H 1.1 where Dependent variable X • • • X ip Independent variables e 0 , e 2 . . . B p Regression parameters to be estimated A normally d i s t r i b u t e d error with mean zero and variance = a*, 34 i t i s assumed that the i n d i v i d u a l error (e^) f o r a given observation i s independent and not predictable from the error of any other observa- t i o n . When t h i s assumption i s v i o l a t e d , autocorrelation i s said to e x i s t . This problem often occurs with permanent sample plots data due to repeated measurements being taken on the same sample p l o t . The general theory on the problem of autocorrelation has been discussed i n d e t a i l by Durbin and Watson (1950) and Johnston (1960). With respect to a p p l i c a t i o n i n f o r e s t r y , the problem has been recognized and handled i n d i f f e r e n t ways by d i f f e r e n t researchers working on growth and y i e l d studies. For example, Buckman (1962) working on growth and y i e l d of red pine (Pinus resinosa A i t ) i n Minnesota recognized the problem but went ahead and used ordinary least squares regression method i n the hope that the error involved was not large. S i m i l a r l y , C u r t i s (1967) recognised t h i s problem with Douglas-fir (Pseudotsuga menziesii (Mirb.) Franco) volume measurements from permanent sample p l o t s . He went ahead and used ordinary least-squares procedure but took care of the problem through an ad hoc procedure inv o l v i n g testing for s i g n i f i - cant c o r r e l a t i o n between any two contiguous observation on each p l o t . A more d e t a i l e d study on correlated errors was c a r r i e d out by S u l l i v a n and C l u t t e r (1972). Using permanent sample plot data for L o b l o l l y pine (Pinus taeda L . ) , they compared two y i e l d models, one developed by the ordinary least-squares procedure and the other by the maximum l i k e l i h o o d estimating procedure, the l a t t e r being one of the possible methods of overcoming the weaknesses of ordinary least squares procedure when autocorrelation e x i s t s . The r e s u l t s of t h i s comparison 35 indicated that f o r a l l p r a c t i c a l purposes, the two models were the same. Because of the d i f f i c u l t y involved i n estimating parameters using the maximum l i k e l i h o o d procedure (using i t e r a t i v e procedures), S u l l i v a n and C l u t t e r wondered i f i t was worth the e f f o r t . These sentiments agreed with those of Swindel (1968) who had addressed the same problem and come to the conclusion that ordinary least-squares should be given due cons i - deration f o r parameter estimation because of i t s s i m p l i c i t y , e s p e c i a l l y when parameter estimates rather than confidence i n t e r v a l s are the main i n t e r e s t . A more recent example of the a p p l i c a t i o n of the maximum l i k e l i h o o d estimators of parameters for l i n e a r models when the error components are corre l a t e d due to having repeated measurement on plots i s provided by Seagrist and Stanford (1980). This example serves to demonstrate the d i f f i c u l t y involved i n using t h i s procedure f o r parameter estimation. From a review of the l i t e r a t u r e , i t may be stated that, to date, there i s no simple procedure for handling the problem of autocorrelation i n l i n e a r regression models and that further research i s required i n t h i s area. The view adopted i n t h i s study i s that the problem of autocor- r e l a t i o n may ex i s t i n the permanent sample plots data but no e f f o r t w i l l be expended to resolve i t because of the s i m p l i c i t y of the ordinary least-squares procedure. 6. Study Methods The basic procedure i n growth and y i e l d studies involve derivation of the growth or y i e l d function. The p r i n c i p l e t o o l adopted f o r parameter estimation for the functions i n t h i s study i s the ordinary 36 least-squares procedure for both l i n e a r and nonlinear regressions, wherever possib l e , nonlinear models were preferred to l i n e a r models for reasons given l a t e r i n Chapter 2). The ordinary least-squares method for the estimation of the regres- sion parameters for the l i n e a r models are d e t a i l e d i n many books on mathematics and s t a t i s t i c s . In p a r t i c u l a r , Draper and Smith (1981, 2nd e d i t i o n ) have given d e t a i l e d procedures for f i t t i n g l i n e a r and nonlinear regression models, wherever l i n e a r models are used i n t h i s study, rigorous regression analysis and study of residuals have been performed to ensure that regression analysis assumptions are met and that the model was appropriate for the data. However, not a l l d e t a i l s of the analysis are shown and i n most cases, only the relevant s t a t i s t i c s are presented. In a d d i t i o n to regression a n a l y s i s , tests of regression bias were performed and i n d i v i d u a l functions validated using independent data (data not used i n deriving the functions) wherever possible. These independent data were created by s e t t i n g aside 20 randomly selected p l o t s for each species. F i n a l l y , i n nearly a l l cases, graphical or tabular presentation of the r e s u l t s from the derived function were studied to ensure that the function performance i s consistent with expectation. It should be mentioned here that the major c r i t e r i a i n the choice of variables entering a given function was t h e i r having a b i o l o g i c a l meaning i n the function. In t h i s study, nonlinear equations were f i t t e d using the nonlinear least-squares subroutine P:3R of the BMDP developed by the University of C a l i f o r n i a i n Los Angeles. The routine estimates the model parameters 37 i t e r a t i v e l y by minimizing the sums of the squared error of pred i c t i o n using the Gauss-Newton i t e r a t i v e procedure. The subroutine requires that the function and the f i r s t p a r t i a l d e r ivatives (with respect to the parameters to be estimated) be s p e c i f i e d and the i n i t i a l estimates of the parameters be supplied. Further d e t a i l s on BMDP are contained i n the BMDP manual. The simulation procedure used for model construction i s discussed i n Chapter 3. 38 CHAPTER 2 STAND DEVELOPMENT AND GROWTH FUNCTIONS 1. Height Development and Si t e Index Curve Construction 1.1 Introduction S i t e i n fo r e s t r y terminology re f e r s to the i n t e r a c t i o n of both the physic a l and b i o l o g i c a l factors determining the productive capacity of an area f o r a given tree species (or i t s provenance or v a r i e t y ) . Because of the large number of combinations of physical factors and b i o l o g i c a l factors of a species, there i s therefore almost an i n f i n i t e number of s i t e s . In prac t i c e however, s i t e s are recognised for a given tree species (and or i t s provenance or variety) within a given area within which the environmental conditions are considered more or less homogeneous. Over the years, several methods of quantifying s i t e have been developed. Spurr (1952), Husch et a l . (1972) and many other authors have d e t a i l e d the d i f f e r e n t methods, a l l of which can be grouped into two categories: 1. Those based on s i t e factors considered c l o s e l y associated with y i e l d : examples are the in d i c a t o r plant approach developed by Cajander i n Finland (Ilvessalo 1927) and the environmental factors approach, using c l i m a t i c f a c t o r s , s o i l f a c t o r s , fauna and vegetation (Spurr (1952), Husch et^ al^. (1972) and others). 2. Those using stand c h a r a c t e r i s t i c s as phytometers. Examples are use of volume y i e l d or the expected mean annual volume 39 increment at a predetermined reference age (Hamilton and C h r i s t i e 1971) and the use of stand dominant height with or without reference to a predetermined age. In general, the second category has been the most widely used. This can be a t t r i b u t e d to the widely accepted assumption that the various s i t e factors a f f e c t i n g the growth of a given tree species can be q u a n t i f i e d i n t h e i r influence on stand c h a r a c t e r i s t i c s . Between the two stand c h a r a c t e r i s t i c s most widely used to quantify s i t e , volume appears the most l o g i c a l i n that i t i s usually the primary i n t e r e s t i n forest management. Its main drawback i s that volume y i e l d can be influenced by s i l v i c u l t u r a l treatments such as thinning and i n i t i a l spacing. As a r e s u l t , the most widely used procedure i n growth and y i e l d studies has been to derive the height over age and age over t o t a l volume y i e l d r e l a t i o n s h i p s for various s i t e s . When t h i s r e l a t i o n - ship i s known, height may be used to estimate t o t a l volume y i e l d (Crawe 1967). The objective of t h i s section i s to investigate the height over age r e l a t i o n s h i p f or the three species: C_. l u s i t a n i c a , P_. patula and P_. radiata i n Kenya. The stand dominant height as used i n t h i s study i s defined as the mean height of the 100 largest diameter trees per hectare (Hummel 1953). This i s preferred to stand mean height as i t i s l i t t l e affected by stand treatment, e s p e c i a l l y low thinning. However, i t should be noted that t h i s assumption may not always hold, for example i n the case of high thinning (an exception i n Kenya thinning practice) and i n situations where the dominant trees might die of disease or insect attack or other f a c t o r s . 40 The basic assumption underlying use of dominant height over age re l a t i o n s h i p for s i t e c l a s s i f i c a t i o n i s that a stand of a given age and height w i l l always y i e l d the same t o t a l volume on a given s i t e i f the s i t e remains unchanged. This assumption w i l l be v a l i d provided that the supposition that t o t a l volume y i e l d i s not affected by the degree of thinning (Moller 1947) holds. However, i t i s not known i f t h i s supposi- t i o n holds for Kenya thinning p r a c t i c e s . 1.2 Si t e Index Curve Construction Procedure The dominant height attained by a given forest stand at a predeter- mined reference age i s the most widely used index of s i t e q u a l i t y and w i l l be used i n t h i s study. To f a c i l i t a t e objective a p p l i c a t i o n of t h i s approach, a system of height over age curves, c a l l e d the s i t e index curves i s developed. Over the years, two main methods for construction of height over age and s i t e index curves have evolved: 1. Anamorphic curves procedure: This procedure consists of f i t t i n g one guiding curve to the height over age data, either g r a p hical- l y or using s t a t i s t i c a l methods, and then f i t t i n g a family of anamorphic curves above and below the guide curves at a r b i t r a r i l y defined i n t e r v a l s . These curves assume: (a) Constant p r o p o r t i o n a l i t y between growth curves for a l l s i t e s and stand conditions. (b) S i t e q u a l i t y i s independent of age. 41 These assumptions are consistent with theory which suggests that a l l other factors being equal, stands on poor s i t e s develop at a slower rate than those on better s i t e s . Because of i t s s i m p l i c i t y , t h i s procedure was the f i r s t to be developed and nearly a l l e a r l i e r s i t e index curves were, based on i t . Nonetheless, these assumptions have been proven f a l s e for several species, e.g. Stage (1963), Beck and Trousdell (1973), Spurr (1955), Powers (1972), Carmean (1956), King (1966) and others. This has given r i s e to the second procedure: 2. Polymorphic curve procedure: According to Stage (1963), the polymorphic s i t e index curve approach i s a generalization of the assumption that s i t e index i s independent of age. I t recognises that height over age curves for a p a r t i c u l a r species may vary i n shape for d i f f e r e n t c l i m a t i c regions, vegetation types, s o i l s and other fa c t o r s . For example, according to Beck and Trousdell (1973), height growth for red pine (Pinus resinosa A i t ) on high-quality s i t e s i s rapid at f i r s t but the curve f l a t t e n s while stands are s t i l l young, while height growth on low-quality s i t e s i s sustained at a slower rate for a longer period, thus j u s t i f y i n g polymorphic curves. From the above discussion i t would appear that the discussion of anamorphic curve procedure i s purely academic since i t i s now a well established fact that s i t e index curves are e s s e n t i a l l y polymorphic, which vary i n shape from one s i t e to another (Rawat and Franz 1974). In p r a c t i c e however, the issue i s not so simple. For example, the proce- dure used to estimate the i n d i v i d u a l plot s i t e index i n t h i s study (discussed below) dictated that the r e s u l t i n g curves would be inherently anamorphic, thus n e c e s s i t a t i n g further t e s t i n g to support the f a c t . 42 This was e s p e c i a l l y necessary since two main factors may be expected to introduce polymorphism to height over age curves i n Kenya. 1. Differences i n r a i n f a l l ( d i s t r i b u t i o n and quantity) i n d i f f e r e n t parts of the country. As mentioned i n Chapter 1 Section 2, the western part of the country has one rainy season and one dry season; while the East of the R i f t v a l l e y has two rainy seasons and two dry seasons during the year. This may cause differences i n growth rates. 2. Differences i n s o i l types In the d i f f e r e n t parts of the country may cause differences i n growth rates. Estimating Plot S i t e Index The permanent sample plot data for s i t e index curve construction consisted of an n by 2 matrix for each plot where n was the number of plot remeasurements for the two variable e n t r i e s , age and stand dominant height. A t h i r d v a r i a b l e , s i t e index for each plot was therefore required for the f i n a l s i t e index curve equation. Figures 3, 4 and 5 shows the height over age data used i n t h i s study. By d e f i n i t i o n , s i t e index refers to the dominant height attained by a given stand at some a r b i t r a r i l y predetermined reference age. Several factors a f f e c t the choice of t h i s age. For example according to C u r t i s et^ a l . (1974), index age should approximate the r o t a t i o n age since the main i n t e r e s t i s usually the t o t a l production over the r o t a t i o n . According to Trousdell et a l . (1974) however, index age should be chosen such that the period of rapid growth i s completed, and should preferably FIGURE 3 HEIGHT/AGE RELATIONSHIP FOR C . LUS ITANICA PLOTS FIGURE 4 HEIGHT/AGE RELATIONSHIP FOR P, PATULA PLOTS F I G U R E 5 H E I G H T / A G E R E L A T I O N S H I P F O R P . R A D I A T A P L O T S ID a l 1 1 1 1 1 r 0 5 10 IS 20 25 30 A G E I N Y E A R S F R O M P L A N T I N G 46 be somewhat les s than the usual r o t a t i o n age for the species. They recommended an index age near the average age of the stands to be predicted so as to generate the most accurate p r e d i c t i o n s . In Kenya, the r o t a t i o n age for the species covered i n t h i s study ranges between 15. years for pulpwood plantations to about 30 years for sawlog and peelerlog plantations. From the point of view of management therefore, choice of 15 years as reference age i s l o g i c a l as i t includes the r o t a t i o n age for pulpwood plantations. Other factors i n favour of t h i s choice are that at t h i s age, the stands are high enough to have passed the juvenile stage and the age i s central enough to the range of data covered i n t h i s study (Alder 1977). Site index i n t h i s study therefore refers to the dominant height attained by the stand at age 15 years. The f i r s t step i n the estimation of the i n d i v i d u a l plot s i t e index was to f i t the best l i n e a r model to the dominant height over age data for each species, with the dominant height as the dependent va r i a b l e . Table 8 gives the equations, the estimated c o e f f i c i e n t s and other relevant s t a t i s t i c s for each species. The above equations provided the guide curve, representing the general growth trend for the respective species. Using these guide curves, the i n d i v i d u a l plot s i t e index was estimated as follows: 47 TABLE 8. C o e f f i c i e n t s f o r the dominant height (m) over age (yrs) l i n e a r equations C_. l u s i t a n i c a P. patula P_. radiata Hdom = b 0 + b l l o * 1 0 A Hdom = b 0 + b l A + b 2 A ' -16.0380 30.5460 2 r>2 r or R SEE H(15) 1413 .90 2.29 19.9 -1.0064 1.8909 -0.0205 1452 .88 2.35 22.8 -4.5323 2.6424 -0.03337 1625 .92 2.54 27.6 48 At any given age, the plot s i t e index, r e l a t i v e to the mean dominant height at age 15 years (H(15)) i s given by: H, x ft(15) S» - — 2.1 A. H i Estimated s i t e index corresponding to height H^. Plot dominant height at the given age i n meters. Dominant height i n meters (corresponding to H^) estimated from the guide curve equation. where S' = From these estimates, the average plot s i t e index S, i s estimated as follows: S = n i = l S 2.2 where n = number of plot remeasurements. The r e s u l t i n g dataset for each plot i s an n by 3 matrix: Age a l a 2 Height h, Site Index S *n n where column 3 i s the same number for each p l o t . Figure 6 i l l u s t r a t e s t h i s procedure diagrammatically using the equation for C_. l u s i t a n i c a . 49 50 Choice of the Height Over Age Model Forestry l i t e r a t u r e has numerous examples of height over age growth models, both l i n e a r and nonlinear. The l i n e a r models are of the general form: Y = 3 Q + 3 l X l + B 2 X 2 . . . . 3 p X p + E ± 2.3 where Y = dependent v a r i a b l e . , X2...Xp = Independent va r i a b l e s . B Q , B ^ . . . 3 p = Regression parameters to be estimated. = A normally d i s t r i b u t e d error with mean zero and variance cr2. The term l i n e a r implies that the model i s l i n e a r i n parameters. This family of models has been widely employed i n the past to describe height over age r e l a t i o n s h i p s . Examples are provided by Meyers (1940), Schumacher (1939), Trorey (1932) and others. Schumacher's equation was used for height over age rel a t i o n s h i p s for the exotic timber species i n East A f r i c a (Alder 1977). In general, the l i n e a r models are easy to develop and to apply and often f i t the data best. However, they are not always the most appro- p r i a t e . For example, the height over age curve of an i n d i v i d u a l tree or a forest stand i s known to be a t y p i c a l b i o l o g i c a l growth curve: Sigmoid i n shape, s t a r t i n g at the o r i g i n and increasing monotonically to an i n f l e c t i o n point, and then approaching the asymptote, determined by the genetic c o n s t i t u t i o n of the i n d i v i d u a l tree or stand and s i t e f a c t o r s . From a t h e o r e t i c a l standpoint therefore, l i n e a r models are inappropriate since they do not s a t i s f y the b i o l o g i c a l p r i n c i p l e s of the growth process. 51 One example of nonlinear models has general form: Y = P 2 ( e 0 + BjX + E ) 2.4 e where 3 Q » 1̂ a n c * 2̂ a r e t n e parameters to be estimated. X i s the predictor v a r i a b l e . The term nonlinear implies that the function (2.4) i s nonlinear i n parameters and cannot be l i n e a r i z e d through transformations. In general, most nonlinear models used i n growth and y i e l d studies conform to the t h e o r e t i c a l laws of b i o l o g i c a l growth and are f l e x i b l e , thus permitting changes i n shape, form and scale of the curves to f i t the data of i n t e r e s t . In addition, they provide a basis for formulation of general hypotheses that express the underlying laws of growth. This, according to Pienaar and Turnbull (1973) provides a j u s t i f i c a t i o n for extrapolation beyond the range of conditions represented by the data. For height over age curves, the most popular nonlinear model has been the Chapman-Richard's equation i n the form: H 2.5 where H Stand height. A Age of the stand. b^ and b 2 are the c o e f f i c i e n t s to be estimated. 52 This function i s a generalization of Von Bertalanffy's growth model (Richards 1959) where some of the parameters have a p h y s i o l o g i c a l i n t e r p r e t a t i o n such that bg represents the b i o l o g i c a l s i t e p o t e n t i a l or maximum height attainable on a given s i t e , while bj represents the stand growth rate, bn i s related to b, value such that = culmination age of the current annual increment. The a p p l i c a t i o n of t h i s model to forestry was popularized through the work of Pienaar and Turnbull (1973), when they demonstrated i t s a p p l i c a t i o n to basal area growth. It now forms the basis for most height over age curves i n North America. Examples are Hegyi et^ a l . (1979), Trousdell et a l (1974), Beck (1971), B r i c k e l (1968) and others. Another nonlinear model that i s increasingly a t t r a c t i n g i n t e r e s t i n growth and y i e l d studies i s the modified Weibull function i n the form: H = b Q ( l - e " b l A 2 ) 2.6 where H = Stand height. A = Age of the stand. bg, bj and b 2 are c o e f f i c i e n t s to be estimated. The c o e f f i c i e n t s bg, b^ and b 2 have the same i n t e r p r e t a t i o n as for Chapman-Richard's equation. This function was developed by Weibull (1939, 1951) as a p r o b a b i l i t y d i s t r i b u t i o n function of the form: 53 X F 1 2.7 where F = Frequency for a given class of i n t e r e s t . X - Class s i z e . a = A scale parameter. X = A shape parameter. As a p r o b a b i l i t y d i s t r i b u t i o n function, t h i s model has been used In f o r e s t r y to model tree diameter d i s t r i b u t i o n , basal area, surface area etc. By introducing an expanding factor (a) to the model, Yang et a l . (1978) demonstrated that the modified form: performed as a highly f l e x i b l e , monotomically increasing sigmoid curve with very desirable growth c h a r a c t e r i s t i c s such as passing through o r i g i n when age i s zero, having an i n f l e c t i o n point and tending to an asymptote as age increases. For height over age curves i n t h i s study, both the Chapman-Richard's equation 2.5 and the modified Weibull equation 2.6 were tested to determine the most appropriate. Comparing the Two Models Both equations 2.5 and 2.6 were applied to the height over age data for each of the three species. The r e s u l t s are given on Table 9a, b and c which shows: 1. The estimated parameters for each model. F = a (1 2.8 TABLE 9. Comparison of the Modified Weibull and Chapman-Richard models for height over age curves C o e f f i c i e n t s Mean bias i n meters by age classes Equation h r SEE bo D l D2 <10 years 10-20 years >20 years C. l u s i t a n i c a a) Chapman-Richards (2.5) 36.1960 -.06092 1.2005 .89 2.40 -.14 .11 -.05 Weibull (2.6) 36.4873 -.03905 1.0978 .89 2.40 -.23 .17 -.05 n = 1201 (802) (448) (451) P. patula b) Chapman-Ri chards (2.5) 54.9478 -.04249 1.1650 .89 2.36 -.07 -.02 .37 Weibull (2.6) 51.1519 -.02707 1.1399 .89 2.36 -.04 -.02 .41 n = 1231 (457) (680) (94) P. radiata c) Chapman-Richards (2.5) 55.2035 -.06871 1.5531 .92 2.56 .14 -.15 .27 Weibull (2.6) 52.3892 -.01997 1.3414 .92 2.56 .09 -.10 .23 n = (406) (843) (170) 55 2. Number of remeasurements on which the parameters are based. 3. The standard error of estimate defined as: SEE - , A ("* ' ̂  2'9 n - p where p = Number of parameters estimated i n the model, n = Total number of observations. and are the observed and predicted dominant heights. 4. C o e f f i c i e n t of determination for each model, estimated as: 2 TSS - RSS „ , n r * TSS 2 ' 1 0 where TSS = Total sums of squares " J l ( H i " 5 ) 2 RSS = Residual sums of squares = J U i - v 2 where , and H are the observed, predicted and the mean of observed stand dominant height. 5. The mean bias by a r b i t r a r i l y defined age classes: <10 years, 10-20 years and >20 years. The mean bias was calculated as: 56 n B = " 2 - 1 1 where tij = Number of observations i n age class j , and are as above. Number of remeasurements within each age class are shown i n brackets. As seen from Table 9, both models gave i d e n t i c a l r e s u l t s f or each species, based on c o e f f i c i e n t of determination or standard error of estimate. S i m i l a r l y , mean bias of age classes was n e g l i g i b l e f o r both models for a l l three species. However, a look at the c o e f f i c i e n t bg shows that for C_. l u s i t a n i c a , both models gave i d e n t i c a l values. This i s as expected since for both models t h i s parameter estimates the maximum attainable height for the species. For the two pine species however, the model gave d i f f e r e n t values of bg, that for Chapman- Richard being higher than that estimated by the Weibull function i n both cases. The explanation for t h i s i s that for these two species, the data did not cover the asymptotic phase of growth and so the estimated value of bg cannot be very r e l i a b l e . This i s further confirmed by Table 1 0 which shows the asymptotic standard deviation for the estimated c o e f f i - c i e n t s . For C. l u s i t a n i c a , the asymptotic standard deviation for bg i s much lower compared to that for P. patula and P_. ra d i a t a for both models. I t i s worth noting that between the two pine species, the asymptotic standard deviation for bg i s lower (for both models) for P_. r a d i a t a whose data cover higher age classes than that for P. patula (35 57 years JP. r a d i a t a , 25 years P_. patula upper age l i m i t ) ; see also Figures 3, 4 and 5. Table 10 also shows that for bg a n d b 2 ' t h e Chapman-Richards model gave lower asymptotic standard deviation than the Weibull model, while for bi» Chapman-Richards gave a s l i g h t l y higher value f o r a l l three species. Based on these r e s u l t s therefore, Chapman-Richards equation (2.5) was selected f o r the height over age re l a t i o n s h i p s i n t h i s study. TABLE 10. Asymptotic standard deviations for the estimated c o e f f i c i e n t s of Table 9 C o e f f i c i e n t function 2.5 2.6 2.5 2.6 2.5 2.6 C_. l u s i t a n i c a P_. patula P. rad i a t a 0.5737 0.9808 2.7884 4.2858 0.9137 2.2989 0.002100 0.001827 0.003013 0.001032 0.001697 0.000047 0.007812 0.02981 0.004198 0.03933 0.005463 0.01084 2.5 = Chapman-Richards function. 2.6 = Modified Weibull function. Introducing S i t e Index to Height Over Age Model Having decided on the general height over age model, the next step was to introduce s i t e index as the second independent variable i n the equation. From t h e o r e t i c a l considerations, one would expect that the better the s i t e q u a l i t y , the higher would be the height growth rate. Similarly, one would expect that the better the s i t e q u a l i t y , the higher 58 the expected maximum attainable dominant height. Site index was there- fore introduced as a l i n e a r function of the c o e f f i c i e n t s bg a n ^ b p the c o e f f i c i e n t s associated with maximum attainable height and growth rate r e s p e c t i v e l y i n equation 2.5. Three equations were tested: -biA b 0 H, = b n S ( l - e 1 ) 1 2.12 dom u -biAS b 0 H, = b n ( l - e 1 ) 1 2.13 dom 0 -b,AS b 0 Hdom * V < 1 " 6 > 2 ' 1 4 where S = plot s i t e index. A l l other symbols are as before. For a l l species, equation 2.13 gave the best r e s u l t s i n terms of mean bias by age classes and standard error of estimate, suggesting that s i t e q u a l i t y expressed i t s e l f best i n i t s e f f e c t on growth rate. This was unexpected since as mentioned above, one would expect that s i t e q u a l i t y should also be associated with c o e f f i c i e n t bg for maximum attainabl e dominant height, thus favouring equation 2.14. This could p a r t l y be explained as being due to lack of s u f f i c i e n t data i n the asymptotic phase of growth, more so for the two pine species, as shown on Figures 3, 4 and 5. Table 11 shows the parameter estimates and other relevant s t a t i s t i c s (calculated as for Table 9) from equation 2.13 for each of the species while Table 12 gives the asymptotic standard deviations f o r the estimated parameters. 59 TABLE 11. C o e f f i c i e n t estimates and other s t a t i s t i c s for the height over age and s i t e index equation 2.13 Co e f f i c i e n t s Mean bias by age classes Species b Q bl b„ R 2 SEE <10 10-20 >20 C. l u s i t a n i a 41.9764 -0.002153 1.0481 0.97 1.24 -0.04 0.26 -0.23 P. patula 52.6155 -0.002038 1.2048 0.97 1.14 -0.02 0.04 -0.31 P. radi a t a 61.6871 -0.001941 1.3583 0.98 1.22 -0.07 0.03 -0.15 TABLE 12. Asymptotic standard deviations for the estimated c o e f f i c i e n t s of Table 11 Species J0 Co e f f i c i e n t s b l C_. l u s i t a n i c a P_. patula P. rad i a t a 0.4891 1.0308 0.7352 0.000054 0.00010 0.000092 0.00253 0.00190 0.001753 The estimated parameters (Table 11) appear both l o g i c a l and consis- tent with expectation. For example, the value of c o e f f i c i e n t s bg and b^ are highest f o r radi a t a followed by JP. patula and C_. l u s i t a n i c a , i n that order as expected. S i m i l a r l y , the magnitude of estimated bg appear reasonable as indicated by the plot growth trends on Figures 3, 4 and 5. The estimated c o e f f i c i e n t s of determination are very high for the three species with a standard error of estimate i n the order of 1.2 60 meters. The mean bias by age classes i s n e g l i g i b l e for P. r a d i a t a , while the model f o r C_. l u s i t a n i c a gives a p o s i t i v e bias of .26 meters between ages 10-20 years and a negative biases of .23 meters above age 20 years. For p r a c t i c a l purposes, these biases can be considered i n s i g n i f i c a n t . For P̂ . patula, the bias up to age 20 years i s n e g l i g i b l e while above age 20 years, the model gives a bias of .31 meters, which could also be considered i n s i g n i f i c a n t . The o v e r a l l f i t of equation 2.13 to the data therefore appeared quite s a t i s f a c t o r y for a l l three species, except for the minor biases. V a l i d a t i n g the Si t e Index Model The s i t e index estimation procedure used i n t h i s study was based on a guide curve, thus presuming that height development on a given s i t e i s proportional to height development on other s i t e s . This implied there- fore that the dominant height development was inherently anamorphic. In theory, i f the s i t e index model i s correct and the p r i n c i p l e of anamorphic height development holds, one would expect the dominant height of a given stand to develop along the same s i t e index curve throughout the r o t a t i o n . In other words, for a given p l o t , i f the proposed growth model i s adequate, there should be no c o r r e l a t i o n be- tween s i t e index and age. In practice however, random s h i f t s i n s i t e index may be expected from year to year due to: 1. Cl i m a t i c fluctuations from year to year. 2. Measurement e r r o r s . 3. Interruptions i n dominant height development a r i s i n g from: 61 (a) rare cases of high thinning, r e s u l t i n g i n removal of dominant trees. (b) death of dominant trees due to diseases, insect attack or windthrow. In addition, i f a species i s composed of d i f f e r e n t provenances or v a r i e t i e s , one would expect some plots to have p o s i t i v e and others nega- t i v e c o r r e l a t i o n of s i t e index to age, depending on the growth rate of the p a r t i c u l a r provenance or v a r i e t y i n r e l a t i o n to the general growth model for the species. In general, we would expect these c o r r e l a t i o n trends to be random with respect to age and s i t e index, while systematic trends would in d i c a t e that the model was inadequate for the p a r t i c u l a r species. The l a t t e r would be indicated by: 1. Plots i n s p e c i f i c s i t e classes showing s p e c i f i c c o r r e l a t i o n trends. 2. Plots i n a given age group showing a p a r t i c u l a r c o r r e l a t i o n trend. 3. Plots from a given region or with a s p e c i f i c c h a r a c t e r i s t i c , e.g. establishment method or an i d e n t i f i e d provenance or v a r i e t y showing a p a r t i c u l a r c o r r e l a t i o n trend. To test for these trends, the following procedure was followed: 1. For each p l o t , predicted s i t e index at each remeasurement was calculated by solving for S i n equation 2.13 to get: S = l n ( l - (S. ) b 2)/-b,A 2.15 bo 1 where In r e f e r to natural logarithm. 62 2. For each p l o t , a simple l i n e a r regression of the calculated s i t e index on age (using standardized variables) was f i t t e d . The regression c o e f f i c i e n t s obtained at step 2 above for each plot are beta weights, which are measures of the c o r r e l a t i o n c o e f f i c i e n t be- tween s i t e index and age. These were studied as the index of f i t for the model for each species. Table 13 gives the t o t a l number of plots and the number and percentage of plots which showed s i g n i f i c a n t c o r r e l a - tions at .05 l e v e l . TABLE 13. D i s t r i b u t i o n of plots showing s i t e index over age c o r r e l a t i o n at .05 p r o b a b i l i t y l e v e l for the three species To t a l no. T o t a l no. plots with Correlation sign p l o t s s i g n i f i c a n t c o r r e l a - t i o n at .05 l e v e l No. % + - C. l u s i t a n i c a 139 41 29.5 22 19 P. patula 144 54 37.5 18 36 P. r a d i a t a 148 42 28.4 19 23 Table 13 shows that of the 41 (29.5%) £. l u s i t a n i c a plots with s i g n i f i c a n t c o r r e l a t i o n s at .05 p r o b a b i l i t y l e v e l , 22 had p o s i t i v e and 19 had negative c o r r e l a t i o n s . A study of the d i s t r i b u t i o n of these plots with respect to s i t e index and age showed no p a r t i c u l a r trend and therefore these s h i f t s could be considered random, a r i s i n g from any of 63 the causes mentioned above. The s i t e index model 2.13 was therefore; considered s a t i s f a c t o r y for C_. l u s i t a n i c a . S i m i l a r l y , of the 42 (28.4%) P_. r a d i a t a p l o t s with s i g n i f i c a n t c o r r e l a t i o n s , 19 had p o s i t i v e and 23 had negative c o r r e l a t i o n s , while a study of t h e i r d i s t r i b u t i o n showed no p a r t i c u l a r trend with respect to s i t e index or age. S i t e index model 2.13 was therefore also considered adequate for t h i s species. For P_. patula however, the r e s u l t s from the above test indicated problems. Not only did i t have the highest number of plots with s i g n i f i c a n t c o r r e l a t i o n s ; 54 (37.5%); but of these, 18 were p o s i t i v e and 36 negative. This raised doubts regarding the a p p l i c a b i l i t y of the model. Further i n v e s t i g a t i o n of these plots indicated the following discrepancies. 1. A l l the p l o t s with s i g n i f i c a n t correlations showed a d e f i n i t e regional bias. For example, a l l plots from Nabkoi group (Nabkoi, Buret, Cengalo and Timboroa) had p o s i t i v e c o r r e l a - t i o n s , while a l l plots from Elburgon and Kiandongoro groups had negative c o r r e l a t i o n s , etc. 2. Of a l l the 10 plots from grassland planting s i t e s that had s i g n i f i c a n t c o r r e l a t i o n s , 9 of them, a l l coming from the Nabkoi group had p o s i t i v e c o r r e l a t i o n s . The lone plot with a negative c o r r e l a t i o n came from the Nanyuki group. These discrepancies indicated the p o s s i b i l i t y of v a r i a b i l i t y of height over age r e l a t i o n s h i p s i n the d i f f e r e n t regions of the country and according to establishment methods. This c a l l e d for further i n v e s t i g a t i o n . 6 4 Height Development On D i f f e r e n t Establishment Sites As mentioned i n Chapter 1 Section 3 the two pine species are established on e i t h e r of the two s i t e types: 1. Grassland planting. Planting on grassland s i t e s i s done with minimum land preparation other than digging p i t s into which seedlings are planted. 2. Shamba planting, a highly developed taungya system which r e s u l t s i n well c u l t i v a t e d f i e l d s f o r tree planting and ensures care of the young trees i n t h e i r i n i t i a l one or two years of l i f e i n the f i e l d . On a l l s i t e s previously occupied by high f o r e s t , shamba planting i s the only method of plantation establishment. Grassland s i t e s on the other hand can e i t h e r be converted to shambas before tree planting or planted as grasslands. However, the current practice i s to plant grass- land s i t e s simply as grasslands. To date, the e f f e c t s of establishment s i t e on the growth and y i e l d of plantations are not known although the general opinion i s that they a f f e c t s growth only i n the f i r s t few years of the plantation l i f e . This section of the study examines the e f f e c t s of establishment s i t e on height development for the two pine species between ages 5 to 20 years (the age range covered by grassland p l o t s ) . Data Analysis and Procedure A l l the grassland p l o t s for P̂ . patula except one came from Nabkoi and Turbo regions. Analysis for t h i s species was therefore l i m i t e d to p l o t s from these two regions. Grassland plots for P_. radiata were 65 d i s t r i b u t e d over the whole range where the species i s grown. The number of pl o t s and t o t a l number of remeasurements for each species are shown on Table 14. For data from each establishment s i t e , the best height over age l i n e a r equation: Hdom " b0 + b l A + V 2 2 ' 1 6 where H, = Plot dominant height i n meters, dom e A = Stand age from planting i n years. bg, bj and b 2 are regression c o e f f i c i e n t s to be estimated. was computed and covariance analysis c a r r i e d out to determine whether the differences i n the regression c o e f f i c i e n t s between the two establishment s i t e s could be ascribed to sampling error or to r e a l differences between the establishment s i t e s . This was accomplished using the U.B.C. S:SLTEST routine (Chinh 1980) to test the hypothesis that the regression c o e f f i c i e n t s bi and b 2 are i d e n t i c a l among the two establishment s i t e s and i f not rejected, to test the hypothesis that a common equation can be used. The l a t t e r tests whether the intercepts (bg) are equal, given that the regression c o e f f i c i e n t s are equal. Table 14 gives the r e s u l t s of t h i s analysis while Figure 7 shows the height over age curves for each species by establishment s i t e . For s t a t i s t i c a l theory on the t e s t s , the reader i s referred to Chinh (1980) and Kozak (1970). 66 TABLE 14. Covariance analysis for slope test for height over age equations for P. patula and P. radiata for d i f f e r e n t establishment s i t e s P. patula (Nabkoi and Turbo) Shamba Grassland Common slope P. rad i a t a (whole country) Shamba Grassland Common slope 0.6275 1.7560 -0.01494 -0.9087 1.8400 -0.02365 1.6420 -0.012 -3.230 2.7140 -0.03779 -4.5390 2.5940 -0.03256 2.7130 -0.03800 No. plots 137 23 145 25 282 48 532 38 878 106 1410 144 Equation ^ i o m = b Q + bjA + b 2A TEST HYPOTHESIS OF A COMMON SLOPE F F(.05) DF(1) DF(2) P r o b a b i l i t y 1.92 3.00 2 276 0.1489 0.56 3.00 2 1404 0.5721 TEST HYPOTHESIS OF A COMMON EQUATION F F(.05) DF(1) DF(2) P r o b a b i l i t y 49.87 3.84 1 278 0.0000 171.37 3.84 1 1406 0.0000 67 F I G U R E 7 H E I G H T O V E R A G E C U R V E S F O R D I F F E R E N T S T A N D E S T A B L I S H M E N T S I T E S P . R A D I A T A S H A M B A P , R A D I A T A G R A S S L A N D P . P A T U L A S H A M B A P . P A T U L A G R A S S L A N D — i 1 1 1 — r 5 10 IS 20 25 A G E I N Y E A R S F R O M P L A N T I N G 68 Results and Discussion For both species, the common slope hypothesis (Table 14) i s not rejected at the .05 l e v e l since the calculated F-values are less than the c r i t i c a l F-values. Thus, the regression surfaces for the two establishment s i t e s may be assumed p a r a l l e l for the two species. Despite t h i s r e s u l t , Figure 7 shows that while the curves f o r I>. ra d i a t a are a constant distance from one another, the distance between the P_. patula curves tend to increase with age. The test for a common equation led to r e j e c t i o n of the hypothesis that the intercepts are the same for both establishment s i t e s for both species since the calculated F-values were much higher than the c r i t i c a l F-values. This meant that the c o e f f i c i e n t b 0 i n equation 2.16 cannot be assumed equal for the two establishment s i t e s , thus precluding use of a common equation. The above r e s u l t s suggest that the growth rate for the two establishment s i t e s can be assumed to be the same. For P_. radiata, the growth curve for grassland s i t e s remain about 2 meters below that for shambas for the whole period covered by the data i . e . up to age 20 years. For P_. patula, the curve for grassland s i t e s remains below that for shamba but the distance between them increases with age, though not s i g n i f i c a n t l y so at .05 l e v e l . This distance worked out to be 0.156 of age up to age 20 years. The above observations that the e f f e c t s of establishment s i t e s remain throughout the l i f e of the plantation (as suggested by the growth trend up to age 20 years) were unexpected. For example, while the i n i t i a l differences can be expected due to the competition for nutrients 69 and moisture between the young trees and grass on grassland s i t e s , i f a l l other factors remained the same, one would expect that the e f f e c t s would diminish with age. The unexpected r e s u l t s therefore point to two p o s s i b i l i t i e s both of which could be operative: 1. Grassland s i t e s may be i n t r i n s i c a l l y poorer than high forest s i t e s . 2. There may be a fa c t o r of growth, e.g. mycorrhiza, which may be lacking or i s less e f f e c t i v e under grassland conditions. Further i n v e s t i g a t i o n on the true cause of the differences i n growth between the two establishment s i t e s i s needed. For example i f the differences are e n t i r e l y due to the I n i t i a l competition, i t may in d i c a t e a need to c u l t i v a t e the grasslands before tree planting and subsequent weeding. Conclusion The most important f i n d i n g from t h i s i n v e s t i g a t i o n was that grass- land planting r e s u l t s i n slower height development compared to shamba plan t i n g f o r patula and P_. r a d i a t a . The e f f e c t s of establishment s i t e are continued up to age 20 years. The immediate p r a c t i c a l s i g n i f i - cance of t h i s f i n d i n g are: 1. Where the forest manager has a choice over establishment s i t e for the two pine species, shamba planting i s to be preferred. 2. Any growth and y i e l d model for the two pine species should have establishment s i t e as one of the input v a r i a b l e s . 70 Height Development by Geographical Regions f or P_. patula For t h i s i n v e s t i g a t i o n , the P̂ . patula plots were s t r a t i f i e d into 8 groups according to the geographical regions In which they are located. These regions conform to the inventory zones already recognized by the Kenya Forest Department. For data from each region, equation 2.16 was computed and the covariance analysis and F-tests performed to test the hypotheses of common slope and common equation, as i n the previous section. Table 15 gives the r e s u l t s of t h i s analysis while Figure 8 shows the r e s u l t i n g height over age curves for each region. The r e s u l t s on Table 15 indicated an F value of 8.65 which i s s i g n i f i c a n t at .05 l e v e l . This led to the r e j e c t i o n of the hypothesis of a common slope, which implied that at least one of the regression surface was not p a r a l l e l to one other. This r e s u l t could have been anti c i p a t e d from a study of Figure 8 which shows the curves from d i f f e r - ent geographical regions crossing each other and growing i n d i f f e r e n t d i r e c t i o n s . Use of a single set of s i t e index curves was therefore considered inappropriate for t h i s species. Because of the small number of p l o t s i n regions 4 and 8, these regions were eliminated from further analyses so that only s i x regions were considered. For the time being however, the equation for region 3 (Elburgon) can be used to approximate growth i n region 4 ( L i k i a ) while the equation for region 1 (Nabkoi region) can be used to approximate growth i n region 8 (Timboroa), based on geographical proximity of the regions. Table 16 shows the s i x regions, t h e i r mean annual r a i n f a l l and the elevation above sea l e v e l of the weather s t a t i o n . TABLE 15. Covariance analysis for slope test f o r height over age equations for P. patula i n d i f f e r e n t regions i n Kenya Geographical region Common slope b 0 b l b 2 n No. plots -7.3550 2.5250 -0.03391 124 15 -11.6800 3.9930 -0.1063 108 9 -1.395 2.0180 -0.0.3106 395 40 -2.4440 1.9440 -0.03127 16 2 -7.6360 3.0950 -0.06403 75 10 -1.8870 2.0870 -0.02598 175 22 -3.9060 2.9160 -0.08390 140 34 -5.9540 2.2330 -0.03160 35 4 1.9720 -0.0230 1072 136 Equation: H, = b~ + b.A + b„A dom 0 1 2 TEST HYPOTHESIS OF COMMON SLOPE F F(.05) DF(1) DF(2) P r o b a b i l i t y = 8.65 1.70 14 1048 - 0.000 where Region 1 2 3 4 5 6 7 8 Nabkoi Nanyuki Elburgon L i k i a Kiandongoro Kinale Turbo Timboroa 72 S i F I G U R E 8 H E I G H T / A G E R E L A T I O N S H I P F O R P . P A T U L A B Y G E O G R A P H I C A L R E G I O N in. or 10. rsi rsi LU ^ in. o o in N A B K O I N A N Y U K I E L B U R G O N L I K I A K I A N D O N G O R O K I N A L E T U R B O T I M B O R O A - i 1 1 — r 5 10 IS 20 A G E I N Y E A R S F R O M P L A N T I N G 73 TABLE 16. R a i n f a l l data and elevation f o r geographical regions recognized for separate s i t e index curves Zone Region Forest Mean annual Average Weather d i s t r i c t r a i n f a l l e levation s t a t i o n mm m Buret 1 Nabkoi Cengalo 1160.6 (26) 2592 Nabkoi FS Nabkoi O n t u l i l i 908.3 (23) 2256 O n t u l i l i FS 2 Nanyuki Nanyuki 1001.0 (25) 2317 Nanyuki FS Gathiuru 954.0 (10) 2287 Gathiuru FS 3 Elburgon Elburgon West 1093.6 (56) 2378 Elburgon FS Elburgon East 1094.6 (23) 2439 Nessuit FS 4 Kiandongoro Kiandongoro 1670.0 (24) 2378 Kiandongoro Kabage 1373.8 (14) 2287 Kabage FS Kamae 5 Kinale Kimakia 1465.5 (14) 2591 Kamae FS Kinale Kieni 6 Turbo Turbo 1170.0 (26) 1890 Turbo FS Number i n brackets r e f e r to record years. FS = Forest Station. 74 The recognition that height development trends d i f f e r s from one region to another i n Kenya i s a new f i n d i n g . For example Alder (1977) and Wanene (1975) developed a single system of s i t e index curves for the whole country. As can be seen from Table 16 the v a r i a b i l i t y i n height development i n the d i f f e r e n t regions cannot be wholly attributed to differences i n r a i n f a l l and elevation. For example Nanyuki region and Elburgon region are almost s i m i l a r i n terms of mean annual r a i n f a l l and elevation. Yet t h e i r height development (curves 2 and 3 on Figure 11) are quite d i f f e r e n t . This suggests that there may be other factors a f f e c t i n g growth, for example s o i l s , r a i n f a l l d i s t r i b u t i o n , i nteractions of these with r a i n f a l l (quantity) and a l t i t u d e and genetics. Further investigations are needed to determine what factors are most important i n respect to t h i s v a r i a b i l i t y . S i t e Index Curves for P_. patula For the permanent sample plots from each region, s i t e index was reca l c u l a t e d using equation 2.1 and 2.2 and the appropriate equation for the region (Table 15) to obtain H(15) and H^. However, an attempt to f i t the Chapman-Richards model (2.5) or the modified Weibull function (2.6) to the height over age data for each region separately f a i l e d f o r some, due to the short range of ages covered by the data, as shown on Table 17 which gives the range of data for each region. The best l i n e a r model for the data, equation 2.16 was therefore used and s i t e index introduced: 75 TABLE 17. Height and age data for ]?. patula by geographical regions n Age Region dom No. of No. of Min. Max. Min. Max. plots remeasurements 1. Nabkoi 5.2 34.2 5.5 23.6 15 124 2. Nanyuki 8.3 33.7 5.6 18.7 9 108 3. Elburgon 6.4 30.5 3.6 22.6 40 394 4. Kiandongoro 7.2 30.8 6.7 18.7 10 74 5. Kinale 5.9 32.2 4.7 21.6 22 179 6. Turbo 5.0 21.1 4.5 13.6 34 140 76 H = b Q + b xS + b 2AS + b 3A + b ^ + b^S 2.17 where variable labels are as before. For convenience and ease of regression analysis, a nested regres- sion equation was f i t t e d , using dummy variables to d i f f e r e n t i a t e between the regions. Table 18 gives the c o e f f i c i e n t s for the regression equa- t i o n 2.17 and the predicted heights at ages 10 and 15 years for each region. The nested regression 2.17 gave a lower standard error of estimate (.81 meters) compared to that given by the nonlinear equation 2.13 for P_. patula (1.14 meters). Thus, s t r a t i f i c a t i o n of data by geographical regions led to higher p r e c i s i o n . The predicted dominant height at age 15 years for s i t e index 20 was about 20 meters for a l l regions (Table 18) except for Turbo whose data did not cover t h i s age. This i s as expected from the d e f i n i t i o n of s i t e index. However, the predicted dominant height for the same s i t e index at age 10 years varies between 12.7 meters for Nabkoi to 16.9 meters for Turbo. This underscores the essence of polymorphic growth, that plots may be of the same s i t e index but have d i f f e r e n t growth curves, depending on the environmental conditions under which the plots are growing. It should be noted here that the quadratic model used for t h i s species shows very rapid decrease beyond the range covered by the data and so extrapolation should be avoided. TABLE 18. Regression C o e f f i c i e n t s for s i t e index curves for I_. patula by regions Predicted values S AS A A A S for S.I. 20 Region : b Q b x b 2 b 3 b 4 b 5 H 1 5 H 1 Q 1. Nabkoi 0.1831 -0.3322 0.1298 -0.3148 0.01663 -0.002545 20.0 12.7 1. Nanyuki -13.0639 -0.1388 0.01763 3.1635 -0.1616 0.003034 19.8 14.8 3. Elburgon -9.5594 0.2714 0.05252 1.2761 -0.03566 -0.0005576 20.2 14.4 4. Kiandongoro -21.0919 0.8502 -0.8114 4.2404 -0.1863 0.006048 20.5 15.6 5. Kinale -1.6231 -0.03267 0.07984 0.2728 -0.007494 -0.0009530 19.8 13.8 6. Turbo 2.2528 -0.2011 0.1762 -1.2576 0.1199 -0.007974 - 16.9 R^ = .98 SEE = .81 meters 78 Va l i d a t i n g S i t e Index Curves f or ]?. patula A s i m i l a r procedure as used to validate equation 2.13 was used to val i d a t e equation 2.17 by solving f o r S and regressing predicted s i t e index against age for each p l o t : S = ( H d o m - b Q - b 3A - b 4 A 2 ) / ( b 1 + b 2A + b 5A 2) 2.18 Table 19 gives the r e s u l t s . TABLE 19. D i s t r i b u t i o n of plots showing s i g n i f i c a n t s i t e index over age c o r r e l a t i o n at .05 l e v e l f o r P. patula by geographical regions Region To t a l No. Plots with s i g n i f i c a n t Correlation p l o t s c o r r e l a t i o n s i g n i f i c a n t No. % + - Nabkoi 15 3 20.0 1 2 Namyuki 9 1 11.1 0 1 Elburgon 40 17 42.5 10 7 Kiandongoro 10 1 10.0 1 0 Kinele 22 5 22.7 2 3 Turbo 34 2 5.9 1 1 130 29 18.7 15 14 The r e s u l t s of Table 19 indicates almost an equal number of plot s showing both negative and p o s i t i v e c o r r e l a t i o n for a l l regions. The 79 t o t a l number of pl o t s with s i g n i f i c a n t c orrelations i s also quite reasonable, the percentage for a l l regions being lower than that obtained on Table 13, except for Elburgon region. A plot of the s i t e index on age for the 17 plots from Elburgon with s i g n i f i c a n t c o r r e l a t i o n at .05 p r o b a b i l i t y l e v e l showed no apparent trend. Thus, these c o r r e l a - t i o n s appeared random and therefore could be a t t r i b u t e d e i t h e r to measurement errors or i n t e r r u p t i o n of dominant height development discussed e a r l i e r . Site Index curves f o r P. patula as defined by equation (2.17) f o r each region were therefore accepted. The F i n a l Curves As mentioned e a r l i e r , the d e f i n i t i o n of s i t e index requires that at reference age, s i t e index equals stand dominant height. However, due to the s t a t i s t i c a l nature of the curve f i t t i n g procedure, t h i s condition was found to be not s a t i s f i e d . For example Table 18 above shows that the predicted dominant height at reference age 15 years as used i n t h i s study varied between 19.8 to 20.5 meters for s i t e index 20 for _P. patula. Similar v a r i a t i o n existed f o r C. l u s i t a n i c a and P_. radi a t a . This r e s u l t was not s u r p r i s i n g since the least squares procedure used to f i t the curves assumed dominant height and s i t e index to be independent, which i s not the case. In other words, there was no b u i l t i n procedure to guarantee that dominant height w i l l equal s i t e index at reference age. To insure that t h i s condition was s a t i s f i e d , the following conditioning procedure was used: 80 For £. l u s i t a n i c a and ?. radiat a : At age 15 years we require that: - b i A S b o Hdom = S - b*0<! " e > 2 ' 1 9 where b*Q = by c o e f f i c i e n t that s a t i s f i e s the condition: Hdom = s a t age 15 years. g Thus b * n = v A O v.— 0 (1 - e " b l A S ) b 2 — . § — B — 2.20 (1 - C s) 2 _ 1 5 b l where c = e A Thus the f i n a l equation f o r £. l u s i t a n i c a and P_. radiata s i t e index curves was: c f btASNb, = L____ ( l - e 1 )  2 2.21 dom o Do \» J (1 - c S ) 2 where a l l variables are as before. This conditioning could only be done numerically since t h i s equation cannot be solved by least squares method. Figures 9 and 10 shows the f i n a l curves for C. l u s i t a n i c a and P. radiata using equation F I G U R E S I T E I N D E X C U R V E S F O R C , 9 L U S I T A N I C A IN K E N Y A FIGURE 10 INDEX CURVES FOR P. RADIATA IN KENYA 1 1 1 : r- 10 15 20 ^ 5 AGE IN YEARS FROM PLANTING 83 For JP. patula: The conditioning of equation 2.17 was accomplished through equation 2.18 as follows: Let S* be the s i t e index defined by the model. Let S be the s i t e index that w i l l insure that s i t e index = dominant height at age 15 years. Then at age 15 years: Hdom = S * = S ( b l + b 2 A + b 5 A 2 ) + ( b 0 + b 3 A + b 4 A 2 ) 2.22 S = S* - ( b 0 + b 3A + b 4A 2) 2.23 (b x + b 2A + b 5 A 2) S* - c x where cl = b Q + b 3(15) + b 4 ( 1 5 ) 2 c 2 = bl + b 2(15) + b 5 ( 1 5 ) 2 Thus, the conditioning of the s i t e index model for P. patula was through numerical adjustment of the s i t e index of i n t e r e s t . Figure 11 gives the s i t e index curves f o r P. patula for Nabkoi region only. F I G U R E 11 S I T E I N D E X C U R V E S F O R P , P A T U L A I N K E N Y A N A B K O I G R O U P 85 2.0 M o r t a l i t y , Stand Density Development, and Thinning Practices i n Kenya Plantations 2.1 M o r t a l i t y As defined by Husch et a l . , 1972, mortality i s the number or volume of trees p e r i o d i c a l l y rendered unusable through natural causes such as old age, competition, insect and diseases, wind, etc. From t h i s d e f i n i - t i o n , two types of stand mortality can be i d e n t i f i e d ; that a r i s i n g from natural mortality as a r e s u l t of competition and the i r r e g u l a r type arising- from unnatural causes; diseases and insect attack, f i r e , wind- throws, etc. The l a t t e r category can range from i n s i g n i f i c a n t to catas- trophic, occurring at intermittent i n t e r v a l s during the l i f e of a stand. Natural mortality i s c e n t r a l to stand dynamic and simulation studies procedures for dealing with t h i s type of mortality i n growth and y i e l d models have been discussed by Smith and Williams (1980), Hamilton (1980), Lee (1974) and others. Irregular mortality on the other hand i s very l i t t l e understood and very d i f f i c u l t to p r e d i c t . According to Lee (1974), s t a t i s t i c a l procedures are available for t r e a t i n g t h i s type of m o r t a l i t y , e.g. t r e a t i n g i t as an o v e r a l l p r o b a b i l i t y or stochastic process for a given area. The major problem however Is related to a v a i l a b i l i t y of adequate data needed to develop the models. For example, windthrow may be related not only to the regional weather patterns and the root systems of the species but also to the s i l v i c u l t u r a l treatments that the stands have received. 86 Because of the intensive l e v e l of plantation management i n Kenya, natural mortality a r i s i n g out of competition may be considered non- existant a f t e r f i r s t thinning. This i s because the f i r s t thinning i s designed to remove dead, dying, diseased, wolfed and suppressed stems so that the remaining stems are healthy and therefore able to withstand any competition that may occur before the next thinning. Before f i r s t thinning however, mortality might occur, e s p e c i a l l y i f thinning i s delayed beyond the prescribed stage. However, there was no evidence of mortality i n the a v a i l a b l e p.s.p. data. Irregular mortality on the other hand i s a constant threat to plantation management, as mentioned e a r l i e r i n Chapter 1. Although lack of data may be c i t e d as the reason why t h i s problem i s not addressed i n t h i s study, the view i s also taken that t h i s e f f o r t would not be worth- while. This i s because forest managers are constantly improving f o r e s t protection procedures such as f i r e protection, game damage and insect and disease control so that any mortality model may well be i n v a l i d before i t i s ready for use. 2.2 Stand Density Development D e f i n i t i o n and Importance of Stand Density The growth rate of an i n d i v i d u a l tree i s determined by i t s genetic c h a r a c t e r i s t i c s , the s i t e q u a l i t y and the amount of growing space a v a i l - able to i t . Stand density refers to the measure of the aggregate degree to which a given tree species u t i l i z e s the growing space. The genetic character of an i n d i v i d u a l tree and s i t e q u a l i t y can be manipulated 87 through tree breeding techniques and through use of f e r t i l i z e r s and s i t e preparation techniques r e s p e c t i v e l y . However, i t i s through density c o n t r o l ; i n i t i a l spacing, thinning and other s i l v i c u l t u r a l techniques that the forest manager has the best chance of d i r e c t i n g growth towards the desired goals and objectives. This i s one of the reasons why the e f f e c t s of stand density on stand development has been so extensively studied. Some of the important studies include: Braathe (1957), Beekhuis (1966), Marsh (1957), Cromer and Pawsey (1957), Adlard (1957), Newnham and Mucha (1971), Smith and Williams (1980), Hummel (1947) and Ba s k e r v i l l e (1965). Measures of Stand Densities Several measures of stand density have evolved over the years, the basic ones being volume per unit area, basal area per unit area and number of stems per unit area. These measures have been very widely used mainly because they are simple and e a s i l y understood. However, they are rela t e d to age and s i t e q u a l i t y , which i s a disadvantage when the stand density i s required to express the degree of s i t e occupancy. Stand density expressions that are independent of age and s i t e , referred to as stand density i n d i c e s ; have therefore come into general use. Examples of these are Reineke's stand density index (Reineke 1933), Chisman and Schumacher's tree area equation (Chisman and Schumacher 1940) and Schumacher and Co i l e ' s stocking per cent (Schumacher and Coi l e 1960). For f u l l d e t a i l s on these, see Husch, M i l l e r and Beers (1972) Chapter 17-2, C u r t i s (1970), Crowe (1966), etc. Among the stand density indic e s mentioned above i s the spacing between trees expressed i n terms 88 of stand dominant height, referred to i n some fore s t r y l i t e r a t u r e as Hart's density index and i n others as Hart-Becking stand density index (Crowe 1967, Wilson 1979). This stand density index has been very widely used i n thinning research i n Europe (Braathe 1957), South A f r i c a i n t e r e s t i n t h i s study. I t i s therefore discussed below i n greater d e t a i l s . Harts Stand Density Index This stand density index was f i r s t proposed by Hart (1928) but according to Wilson (1979), the concept of spacing had been used i n Denmark as f a r back as 1851 where I t may have been formulated as: Relative distance of trees = Height The present d e f i n i t i o n of t h i s index can take any of the two forms: 1. For square spacing: (Crowe 1967) and North America (Wilson 1979) and i s of p a r t i c u l a r 2.25 S% - 100 H dom 89 2. For t r i a n g u l a r spacing: f10.000 x Sin 60 c No. trees/ha S% - . 100 2.26 dom where S% = Stand density index. Between these two forms, the square spacing formula has received a much wider acceptance, mainly because i t i s free of the constant (Sin 60°) and therefore much easier to apply. However, one problem associated with use of t h i s index i s that i t assumes a regular spacing i n the stand, even a f t e r repeated thinning. This may not always be true. I t s main advantages are that i t i s la r g e l y Independent of stand age, s i t e q u a l i t y and species, besides being simple and easy to apply. One of the desirable c h a r a c t e r i s t i c s of a stand density index expression i s that i t should give an idea of the degree to which an area i s being u t i l i z e d by trees. As i t stands, Harts density index does not f u l l y account f o r the degree of s i t e u t i l i z a t i o n . For example, consider two stands of the same stand dominant height and the same spacing. According to t h i s index, the two would have the same density index. Yet i f one had been thinned to the present density e a r l i e r than the other, I t would have a considerably higher basal area. This shortcoming i s e s p e c i a l l y important i n i n t e n s i v e l y managed plantations where the hi s t o r y of stand treatment i s an important consideration i n growth and y i e l d determination. It should also be noted that stand dominant height i s used instead of stand mean height, because the former i s l i t t l e a f f e c t e d by stand treatments, e s p e c i a l l y thinning. 90 Stand Density Development f o r the Plantation Species i n Kenya (based on Harts Density Index) The opinion over whether stands should be managed at constant stand density index or not d i f f e r s among fo r e s t e r s . For example, Hummel and C h r i s t i e (1953) favoured maintenance of constant index throughout the . ro t a t i o n period, while Becking (according to Braathe 1957) recommended constant index f o r Douglas-fir and poplars and a decreasing (with age) index f o r beech and l a r c h i n Holland. Another strong proponent of thinning to a constant stand density index i s Wilson (1979) who recom- mended leaving a spacing of 20.6% of dominant height a f t e r each thinning at Star Lake, Minnesota. Although not e x p l i c i t l y defined i n terms of S%, thinning schedules f o r P_. patula and P_. radiata i n South A f r i c a and New Zealand r e s p e c t i v e l y are defined i n terms of number of stems per hectare to be l e f t at a given stand dominant height (Crowe 1967, Fenton 1972), which translates to thinning to a constant density index. In general, thinning to a constant density index i s favoured, i f only because i t provides an objective c r i t e r i a for deciding when a stand i s due f o r thinning. As shown on Table 5 of Chapter 1 on s i l v i c u l t u r e , the f i r s t t h i n - ning for a l l the three species i n Kenya i s defined i n terms of number of stems to be l e f t at a s p e c i f i e d stand dominant height. This translates to an S% of about 30%, 25.5% and 27.7% f o r C_. l u s i t a n i c a , P_. patula and P_. radi a t a r e s p e c t i v e l y a f t e r f i r s t thinning. For 1?. r a d i a t a , 2nd thinning i s defined s i m i l a r l y and translates to the same S% a f t e r 91 thinning. The 2nd and subsequent thinning f o r C_. l u s i t a n i c a and P_. patula and also subsequent thinnings for P_. radiata are defined i n terms of number of stems to be l e f t a f t e r each thinning while the thinnings are spaced at constant time i n t e r v a l s . The e f f e c t s of t h i s p r a c t i c e on S% development i n the d i f f e r e n t s i t e index classes are shown on Figure 12a,b and c for C_. l u s i t a n i c a , P_. patula and P_. radiata r e s p e c t i v e l y . The figure shows: 1. For C_. l u s i t a n i c a S% varies between 18-30% while for P. patula and P_. r a d i a t a , i t varies between 15-25%. 2. Stand density index (S%) varies for the d i f f e r e n t s i t e q u a l i t i e s for each species, with wide spacing developing on poor s i t e and overcrowing on good s i t e s . For example for C_. l u s i t a n i c a , S% for f i n a l thinning on s i t e index 12 i s 30% while that for s i t e index 24 i s 20%. The p r a c t i c a l implications of t h i s practice are that i t i s l i k e l y that the f u l l s i t e capacity on poor s i t e s i s not f u l l y exploited while overcrowding on very good s i t e s may be a f f e c t i n g growth. In addition, t h i s practice ignores the e f f e c t s of spacing (S%) on stand development, thus depriving the management of one of the basic tools for control of tree diameter development. This i s e s p e c i a l l y c r i t i c a l from a u t i l i z a - t i o n point of view. To date, the e f f e c t s of the present schedules on growth and y i e l d are not known and i t i s hoped that r e s u l t s from t h i s study w i l l provide some knowledge on the magnitude of these e f f e c t s . 92 FIGURE 12 NO. STEMS/HEIGHT/S% RELATIONSHIP BY SPECIES AND SITE INDEX CLASSES IN KENYA Ul z o s 5 o. in S.77 . 2 0 0 5 0 0 . 4 0 0 4 0 8 - 6 0 0 3 . 5 4 - 8 0 0 3.16-1000' 2 .89-1200 2 .67-1400 2 .5-1600 (A) C. LUSITANICA S I T E I N D E X 12 15 18 21 24 r I I 4 t h thinning 3 ' d S% T _ _ f f 30 _ T — 25 " 20 18 orrd 7 / / / 7 7 / / / '// • / - — — / 10 — I — 3 0 5.77. 200 5-00. 400 4 . 0 8 . 600\ 3 . 5 4 . 800 3 .16*1000 2 . 8 9 . 1 2 0 0 ^ 2 .67 .1400 1600 P. PATULA (Nabkoi) S I T E I N D E X IS 18 21 24 27 S ^ I 1 I 1 - l _ _ 3 0 — _**.—. 25 3 r d thinning 2nd • 1 , 4 / // / / / / / / / / ! ' 1 * 15 / 10 20 D O M I N A N T H E I G H T IN M E T E R 3 z z o 5 . 7 7 - 2 0 0 5 . 0 0 - 4 0 0 4 . 0 8 - 6 0 0 3 . 5 4 - 8 0 0 ' 3 .16-1000 2 .89-1200 2.67-1400' 2 . 5-1600 Cc ) P. RADIATA S I T E I N D E X 21 24 27 30 33 r T i l l 4 ' n thinning 3 rd «nd 1»« / / / n /A 7 sr. - 20 — 15 1 I I n 1 ; 10 20 >° D O M I N A N T H E I G H T IN M E T E R 4 0 93 2.3 Thinning D e f i n i t i o n : Braathe (1957) defined thinning as the act of removing some of the stems i n an immature stand of trees i n order to give the remaining trees better conditions f o r growing and producing wood of high q u a l i t y . However, t h i s d e f i n i t i o n i s inadequate without further q u a l i - f i c a t i o n as to: 1. Type of thinning This q u a l i f i c a t i o n describes the trees to be removed; based on the p o s i t i o n of the trees with respect to e i t h e r s p a t i a l d i s t r i b u - t i o n or tree s i z e s . For example low thinning which i d e a l l y implies removal of the smallest trees, s t a r t i n g with the suppressed ones or high thinning, which i d e a l l y removes trees from the dominant and codominant categories. The d e f i n i t i o n of types of thinnings i s well documented i n most texts on s i l v i c u l t u r e e.g. Smith (1962) pages 90-94. However, i t should be noted that i n p r a c t i c e , most thinnings are a combina- t i o n of two or three types at the same time or i n sequence. This i s because the thinning objectives change as the stand structure changes and the i n d i v i d u a l stand develops. 2. Thinning i n t e n s i t y This r e f e r s to the proportion of the stand removed i n thinning as a function of the stand before thinning. Several measures of thinning i n t e n s i t y have been used. 94 The r a t i o d/D a t t r i b u t e d to Eide and Langsaeter (Braathe 1957) where: d = Mean DBH of trees removed i n a thinning D = Mean DBH of trees before thinning. This r a t i o b a s i c a l l y measures the type of thinning c a r r i e d out. Thus according to the above authors, thinning types can be defined according to the value of the r a t i o d/D: £ 0 . 7 0 = Low thinning 0.70 to .85 = No d e f i n i t e low or crown thinning 0.85 to 1.0 = Crown thinning > 1 = Selection thinning This r a t i o has been used by Reukema and Bruce (1977) to define the recommended thinning type f or Douglas-fir. The number of trees to be removed at each thinning, expressed e i t h e r as a pure number or as a proportion of i n i t i a l stand density; such as i s presently practiced i n Kenya. Basal area (or volume) removed as a proportion of basal area (or volume) before thinning. For example Reukema and Bruce (1977) defined the i n t e n s i t y of thinning for Douglas-fir i n terms of the minimum recommended re s i d u a l basal area to be maintained i n the stand a f t e r each thinning. The Forest Management Tables (Metric) f o r Great B r i t a i n use the same p r i n c i p l e by defining the 95 marginal thinning i n t e n s i t y i n terms of the annual rate of volume removal equivalent to 70% of the maximum mean annual increment. This proportion represents the maximum thinning i n t e n s i t y which can be maintained without loss of volume production (Hamilton and C h r i s t i e 1971). 3. Thinning c y c l e : which refers to the p e r i o d i c i t y of thinning. Choice of thinning cycle i s a function of both economic and b i o - l o g i c a l considerations. For example, frequent but l i g h t thinnings are preferred from y i e l d and b i o l o g i c a l considerations while fewer (longer cycles) but heavier thinnings are a t t r a c t i v e from an economic point of view. The b i o l o g i c a l considerations often override the economic considerations mainly because the main concern i s the growth conditions of the remaining stand. Thus, although not i m p l i c i t l y stated, the thinning cycle for Douglas-fir (Reukema and Bruce 1977) i s defined by the growth rate of the stand, the time i t takes f o r the basal area to grow from the recom- mended r e s i d u a l l e v e l to the maximum basal area, where maximum i s defined as the approximate maximum basal area to which a given number of merchantable trees should be grown i n a managed stand. The concept of maximum size-density advocated by Drew and Fl e w e l l i n g (1979) f o r control of plantation density for Douglas-fir i n New Zealand reinforces the importance of b i o l o g i c a l considera- t i o n i n determining the thinning c y c l e . It should however be noted that the thinning type, thinning i n t e n s i t y and thinning cycle are very c l o s e l y i n t e r r e l a t e d and choice of one influences the other. 96 Thus the forest manager has to compromise between the b i o l o g i c a l d e s i r a b i l i t y of a thinning and the economic considerations when determining both the thinning i n t e n s i t y and thinning cycles, since choice of one a f f e c t s the other. Thinning Type and Intensity for Plantation Species i n Kenya So f a r , the thinning schedules for the three species have been described i n terms of thinning cycle (time i n t e r v a l s between thinnings) and i n t e n s i t y ( i n terms of number of stems and spacing (S%). I t remains to describe the thinning type and the thinning i n t e n s i t y i n terms that w i l l reveal both the size and the proportion of the stand removed i n terms of mean D B H and basal area of stand before thinning. The summary of data for t h i s study i s shown on Table 20 by species and basic variables as measured on the permanent sample p l o t s . From these v a r i a b l e s , other d e s c r i p t i v e v a r i a b l e s , the r a t i o s of the basal area, D B H and number of stems were derived. In addition, each thinning was categorized as precommercial (those which took place at ages < 13.5 years) or commercial. 1. Mean D B H of thinning/mean D B H before thinning r a t i o Table 21 shows the r a t i o s of the mean D B H of thinnings to mean D B H of stand before thinning for precommercial, commercial and combined thinnings for each species. The following observations can be noted from t h i s table. D B H ( T ) ^ , 1. That for a l l three species the r a t i o TJBH(BT) precommercial and commercial thinnings are almost equal although those for 97 TABLE 20. Summary of thinning data by species and relevant variables Variable Species Mean Mininum Standard Maximum deviation Age of C. l u s i t a n i c a 117 12.1 thinning P. patula 139 11.0 (years) P. radiata 121 11.5 5.6 5.7 5.5 30.4 24.5 30.4 5.6 4.2 4.8 Dominant height (m) £. l u s i t a n i c a P. patula P. r a d i a t a 117 16.0 7.6 30.2 5.6 139 17.2 8.0 35.4 6.5 121 20.6 5.9 44.2 8.3 No. stems C. l u s i t a n i c a 117 289 74 thinned P. patula 139 278 74 P. radiata 121 2706 49 No. stems C. l u s i t a n i c a 117 1007 198 before P. patula 139 908 222 thinning P. radia t a 121 959 99 DBH of C. l u s i t a n i c a 117 18.4 6.2 thinning P. patula 139 17.5 7.2 (cm) P. radiata 121 15.5 3.2 DBH C. l u s i t a n i c a 117 21.3 10.2 before P. patula 139 19.9 8.6 thinning P. radia t a 121 18.9 7.6 Basal C. l u s i t a n i c a 117 6.5 0.3 area of P. patula 139 6.3 0.4 thinning P. radia t a 121 4.6 0.1 791 1161 988 1631 1680 2001 51.0 35.7 45.8 45.1 37.0 47.9 22.5 21.5 15.1 195 185 196 400 347 422 8.4 5.8 8.1 8.4 5.9 7.9 4.2 4.2 3.4 Basal area before thinning £• l u s i t a n i c a P. patula P. rad i a t a 117 29.9 7.6 53.0 10.0 139 25.9 2.4 57.4 9.7 121 22.5 7.4 45.8 9.1 98 TABLE 21. Mean DBH of thinning/mean DBH before thinning r e l a t i o n s h i p Mean D B H m 5 5 5 ( B T ) Minimum Maximum Standard deviation C. l u s i t a n i c a .84 P_. patula .87 P. ra d i a t a .77 Precommercial ,53 1.0 .11 .59 1.1 .09 .30 1.0 .15 75 91 82 £. l u s i t a n i c a .88 P. patula .89 P. radi a t a .86 .54 .66 .45 Commercial 1.13 1.05 1.10 .11 .07 .14 42 48 39 C. l u s i t a n i c a P. patula P. rad i a t a .85 .88 .80 .53 .59 .30 Combined 1.13 1.10 1.10 .11 .09 .15 117 139 121 99 precommercial are s l i g h t l y lower. In general, a bigger difference was expected between these two types of thinnings since with commercial thinnings, there i s a tendency towards a systematic type of thinning so as to remove some stems of economic value to o f f s e t cost of thinning, while precommercial thinnings are aimed at stand hygiene, removing trees mostly from the lower diameter classes. This difference may have been p a r t l y reduced by the presence of wolfed trees which are usually removed during precommercial thinnings. The combined r a t i o f o r C_. l u s i t a n i c a and that for P. patula are almost equal while that for P_. radiata i s s l i g h t l y lower. This may r e f l e c t the e f f e c t s of dothistroma pine disease mentioned i n Chapter 1 since thinning, e s p e c i a l l y precommer- c i a l , removes diseased trees as a p r i o r i t y . Based on the c l a s s i f i c a t i o n of Eide and Langsaeter (according to Braathe 1 9 5 7 ) the r a t i o Q B H ( B T ) F O R - R A D I A T A ( ° « 8 ° ) corresponds to no d e f i n i t e low or crown thinning while that f o r C_. l u s i t a n i c a and P_. patula ( 0 . 8 5 and 0 . 8 8 respectively) borders on the lower side of crown thinning. 100 2. Basal area of thinning/basal area before thinning r a t i o and number of stems thinned/number of stems before thinning r a t i o Both of these r a t i o s followed exactly the same pattern as for the DBH(BT) T a t-^°> except f ° r the magnitude. Table 22 shows the magnitude f o r the basal area r a t i o f o r the precommercial, commercial and combined data by species. The table indicates that the average l e v e l of basal area removal i s between 20-25% of the basal area before thinning although i t can be as low as 1% and as high as 70%. 3. Mean DBH of thinning The mean DBH of stems removed i n a thinning can be expected to be a function of: (1) Mean diameter of stand before thinning. (2) Thinning type: For example, mean DBH of thinnings i n low- thinning can be expected to be lower than mean stand DBH before thinning, mean DBH of thinnings i n se l e c t i o n thinning would be higher than mean stand DBH before thinning. (3) Weight of thinnings: In low, crown or i n sel e c t i o n thinning, the heavier the thinning, the closer would t h e i r mean DBH approach the mean stand DBH before thinning. Other variables that can a f f e c t the mean DBH of thinnings include stand dominant height and age. Preliminary investigations however indicated the stand mean DBH before thinning as the variable best correlated with mean DBH of thinnings. 101 TABLE 22. Basal area of thinning/basal area before thinning r a t i o s Mean r a t i o Minimum Maximum Standard deviation Cypress P. patula P. radiate .22 .24 .19 Precommercial .02 .56 .12 .03 .70 .12 .01 .56 .13 75 91 82 Cypress P. patula P. radiate .22 .25. .23 .06 .05 .02 Commercial .61 .71 .68 .11 • 14 .13 42 48 39 Cypress P. patula P. r a d i a t a .22 .24 .20 .02 .03 .01 Combined .61 .71 .68 .12 .13 .13 117 139 121 102 B A ( T ) As a measure of weight of thinning the r a t i o g^g^) was better correlated to mean DBH of thinnings than the r a t i o ^jjjfy This i s as expected since basal area i s a better measure of stand density than number of stems. Both r a t i o s showed s i g n i f i c a n t c o r r e l a t i o n at .05 p r o b a b i l i t y l e v e l except for the r a t i o of number of stems f or C. l u s i t a n i c a . For t h i s study, one thinning option i n the envisaged y i e l d model required that thinnings be defined i n terms of number of stems to be l e f t a f t e r the thinning. For t h i s option therefore, the appropriate measure of weight of thinning i n the DBH of thinning equation was ^ g j ) • The following multiple l i n e a r equation was formulated: N(T) 9 97 DBH(T) = b Q + b 1 DBH(BT) + b 2 l'^> Table 23 shows the estimated parameters and other relevant s t a t i s t i c s f o r the respective equations for each species. A l l the independent variables were s i g n i f i c a n t at .05 p r o b a b i l i t y l e v e l , i n c l u d i n g those for £. l u s i t a n i c a while the study of residuals indicated no systematic trends. 103 TABLE 23. Parameter estimates and thinning equation 2.27 other s t a t i s t i c s for the.DBH of Species - . C. l u s i t a n i c a P. patula P. r a d i a t a b 0 b l b2 -3.3840 .9508 5.2153 -2.1596 .9262 4.0669 -4.5192 .9637 6.1003 n R 2 SEE 117 .92 2.37 cm (12.9%) 139 .92 1.64 cm (9.4%) 121 .91 2.42 cm (15.6%) The other thinning option i n the envisaged model required that thinning be defined i n terms of proportion of basal area to be removed at each thinning. Thus i f basal area before thinning i s known, basal area to be removed i s completely defined. The problem with t h i s procedure was how to determine the number of stems removed i n thinning since only basal area removed i s known. This was accomplished by f i r s t c a l c u l a t i n g the mean diameter of thinned stems using the equation: DBH(T) = b Q + bj DBH(BT) + b £ f f ^ y 2-28 Table 24 shows the estimated parameters and other s t a t i s t i c s f o r equation 2.28 f o r each species. 104 TABLE 24. Parameter estimates and other s t a t i s t i c s for the DBH of thinning equation 2.28 Species C. l u s i t a n i c a P. patula P. radiata b 0 °1 b 2 n -4.0099 -2.4008 -4.4723 .9316 .9129 .9274 11.6300 7.2044 11.9780 117 139 121 R2 .94 .94 .93 SEE 2.06 (11.2%) 1.46 (8.3%) 2.14 (13.8%) A l l the independent variables were s i g n i f i c a n t at .05 p r o b a b i l i t y l e v e l and the study of residuals indicated no systematic trends. 3. Basal Area Growth before F i r s t Thinning 3.1 Introduction Basal area i s one of the most important stand c h a r a c t e r i s t i c s i n that i t i s one of the d i r e c t l y measurable independent variable i n the stand volume equation. There i s also no doubt that basal area i s the simplest and most widely accepted measure of stocking density (Rawal and Franz 1973). Thus, i n i n t e n s i v e l y managed plantations, basal area i s often used as one of the c r i t e r i o n f o r determining when the stand i s due for f i r s t and subsequent thinnings. I t i s also one of the variables 105 often used for i n i t i a l i z i n g the stand f o r growth and y i e l d i n whole stand, diameter free models (Smith and Williams, 1980). Basal area development i n managed stands can be considered i n two phases, before and a f t e r f i r s t thinning. Before f i r s t thinning the development i s mostly a function of stand density, s i t e q u a lity and stand age. A f t e r f i r s t thinning, the basal area of the stand i s a function of not only these factors but also of the i n t e n s i t y and frequency of thinnings. This section of the study investigates the development of basal area before f i r s t thinning for the three species. 3.2 Data f o r Basal Area Development Data f o r basal area development were derived from p l o t s established when stands were 3 to 8 years old and had number of stems higher than 1,000 stems per hectare. Table 25 shows a summary of the data. A preliminary study of the basal area growth for P_. patula had shown that for Kinale region, basal area development was markedly d i f f e r e n t from that of the rest of the country. For example, the mean basal area at 2 2 age 7 years was 28.6 m compared to 19.7 m f o r the rest of the country. This explains why data for t h i s species were treated as two d i f f e r e n t groups. 3.3 Choice of the Regression Model The growth of the i n d i v i d u a l tree basal area i s described by Spurr (1952). He further stated that the growth curve for basal area of the 106 TABLE 25. Summary of the basal area before thinning data Species Variable n Mean Standard deviation Mininum Maximun Age (years) 112 7.1 1.29 4.7 11.5 C. l u s i t a n i c a No. stems/ha 112 1337 177.28 1013 1656 Height (dom) m 112 10.2 2.32 5.0 15.8 2 Basal area m 112 19.5 6.66 4.2 38.2 Age (years) 172 7.2 2.02 3.6 16.5 P. patula No. stem/ha 172 1300 162.22 1012 1631 rest of country Height (dom) m 172 11.9 3.47 3.8 26.3 Basal area m 172 19.7 8.52 1.1 49.1 Age (years) 45 7.0 .87 5.5 8.5 P. patula No. stem/ha 46 1296 119.71 1038 1532 Kinale Group Height (dom) m 46 12.0 1.86 8.2 16.4 o Basal area m 46 28.6 5.81 14.7 40.8 Age (years) 95 7.2 1.10 5.5 9.5 P. ra d i a t a No. stem/ha 95 1342 178.29 1038 1705 Height (dom) m 95 12.8 2.89 6.1 20.0 2 Basal area m 95 15.3 3.84 5.7 24.4 107 stand i s s i m i l a r , e s p e c i a l l y f o r managed stands where mortality i s n e g l i g i b l e . Spurr was r e f e r r i n g to the sigmoid growth curve. Pienaar and Turnbull (1973) used Chapman-Richards growth model for basal area growth of a stand for d i f f e r e n t stand d e n s i t i e s . In t h e i r study, the authors f i t t e d t h i s equation to data from d i f f e r e n t stocking d e n s i t i e s from c o n t r o l l e d experiments i n South A f r i c a (CCT experiments worked upon by Marsh (1957)). They demonstrated that the magnitude of the asymptote parameter may be density dependent, but within c e r t a i n l i m i t s could be considered constant for d i f f e r e n t d e n s i t i e s , thus supporting the hypothesis of constant y i e l d . The above study indicated that the basal area growth curve i s sigmoid, s t a r t i n g at the o r i g i n and tending to an asymptote as age increases. However, i n t h e i r study of asympotic growth curves, Rawat and Franz (1973) noticed that f o r some species l i k e pine (Pinus s i l v e s t r i s , L . ) , basal area per acre over age curves were not asymptotic but bend downwards a f t e r reaching the maximum value. They explained t h i s as due to the fact that the s i t e i s not capable of supporting optimum number of stems per unit area at higher ages. They therefore recommended de r i v a t i o n of stand basal area from basal area over age curve of mean tree and number of trees per unit area over age curves. For t h i s study however, the data did not cover the higher ages. Basal area per hectare was retained as the dependent variable as i t i s the va r i a b l e of d i r e c t i n t e r e s t . From experimentation with the Chapman-Richard's and modified Weibull function; the l a t t e r was selected as most suited to the data: 108 BA . „ 0 ( l - .-VW*) 2 . 2 , 2 Where BA = Stand basal area i n m /ha A = Stand age i n years H = Stand dominant height (includes e f f e c t s of s i t e q u a l i t y ) N = Number of stems per hectare bo; bj.-.b^ are c o e f f i c i e n t s to be estimated where bg = The asymptote. b^ = The rate at which basal area approaches the asymptote b2» b^ and b^ are scale parameters. This model assumed constant asymptote for a l l s i t e q u a l i t i e s and stand d e n s i t i e s . Both assumptions appear j u s t i f i e d because of the narrow range of de n s i t i e s covered by the data and because the data does not cover the asymptotic phase of growth. In other words, the estimated asymptotes are mainly a r e f l e c t i o n of the rate at which the plots are growing i n the e a r l y ages. Table 26 shows the estimated parameters and other relevant s t a t i s t i c s from equation 2.29 for a l l the species. SEE and R values were estimated using equations 2.9 and 2.10 r e s p e c t i v e l y with BA substituted f o r H. Table 27 gives the asymptotic standard deviations f o r the estimated c o e f f i c i e n t s of Table 26 while Figure 13 shows the basal area over age curves f o r various s i t e index classes at stand density of 1200 stems per hectare. 109 TABLE 26. Parameter estimates and relevant s t a t i s t i c s f o r basal area before thinning: Equation 2.29 Species C. l u s i t a n i c a P. patula* P. patula(k) P. radiata b 0 39.6342 53.1844 87.7894 22.1194 b x -0.00001426 -0.0000003329 -0.000006209 -0.00000727 b 2 1.2418 0.9381 1.2788 0.2540 b 3 1.1797 1.1033 0.5626 1.7142 b 4 0.7807 1.3367 n 112 172 46 95 R 2 .85 .91 .91 .63 SEE 2.61 m2 or 2.58 m2 or 1.79 m2 or 2.36 m2 or 13.35% 13.1% 6.26% 15.4% * P_. patula r e s t of the country, (k) P. patula Kinale region. n o TABLE 27. Asymptotic standard deviations of the estimated c o e f f i c i e n t s of Table 26 Species b 0 b l b 2 b 3 b4 c. l u s i t a n i c a 4.6935 0.000000 0.2540 0.1662 0.1657 p. patula* 4.6404 0.000000 0.1292 0.1082 0.1394 p. patula(k) 38.8152 0.000000 0.2091 0.1311 - p. ra d i a t a 2.1430 0.000000 0.4698 0.3400 * P. patula rest of the country, (k) P. patula from Kinale group. I l l FIGURE 13 BASAL AREA OVER AGE CURVES FOR VARIOUS SITE I HOEX CLASSES AT STAND DENSITY OF 1200 S.P.H. C . L U S I T A N I C A P R A D I A T A AGE IN YEARS FROM PLANTING 112 3.4 V a l i d a t i o n and Discussion of the Basal Area before Thinning Equation 2.29 Equation 2.29 was accepted on the basis of the high c o e f f i c i e n t of determination and the good f i t indicated by a study of the residuals for the i n d i v i d u a l species equations. However i t remained to see how well these equations predicted basal area development of permanent sample plo t s not used to formulate the equations. Unthinned pl o t s from the 20 permanent sample plo t s set aside for each species as test data were used for t h i s . Figure 14a,b,c,d shows the observed and predicted basal area for C_. l u s i t a n i c a , P_. radiata, P_. patula (rest of the country) and P_. patula (Kinale) p l o t s respectively. The figure indicates very accurate p r e d i c t i o n for some of the p l o t s , e.g. p l o t s 348 f o r C.. l u s i t a n i c a , and p l o t s 391 and 276 f o r P. patula from the rest of the country and from Kinale respectively. For the other p l o t s , the growth rate appears very accurately modeled but the predicted curves are s h i f t e d either above or below the observed curve. The s h i f t for C_. l u s i t a n i c a and P_. patula (both groups) i s within +2 sq. meters of basal area, with no i n d i c a t i o n of bias. The s h i f t f o r r a d i a t a plot 289 i s of the order of 4 sq. meters at age 8.5 years but t h i s reduces to 2.5 sq. meters at age 11.5 years. For plot 164, the s h i f t i s almost n i l at age 7.5 years but the curves diverges towards age 9.5 years. This trend i s reversed f o r plot 373. In general, the v a r i a b i l i t y i s expected to be higher for t h i s species as indicated by the lower c o e f f i c i e n t of determination (.63) compared to that of the other species (Table 26). This could be the e f f e c t s of 113 FIGURE 14 OBSERVED AND PREDICTED BASAL AREA FOR UNTHINNED PLOTS NOT USED IN FORMULATING THE BASAL AREA EQUATION C . LUSITANICA (a) - 4 0 3 0 25 2 0 < <! 1 0 pa 0«-i-̂ - 8 9 1 0 3 4 8 8 9 1 0 I I 1 2 Observed Predicted FIGURE 14 con't P . PATULA (Rest of the country) AGE IN YEARS FROM PLANTING 115 dothistroma pine disease which attacks t h i s species i n the e a r l y age, between 3 to 15 years. S u r p r i s i n g l y , t h i s v a r i a b i l i t y was not noticed f o r height development, which would suggest that probably the e f f e c t s of t h i s disease on stand development are expressed mainly on basal area development. There i s however no i n d i c a t i o n of bias or inconsistency i n these p r e d i c t i o n s . The higher rate of basal area development f o r P. patula plots from Kinale region was unexpected. For example from the study on height development, there was no i n d i c a t i o n that t h i s region had better s i t e q u a l i t y than the others (see Figure 8 curve No. 6) Also from Table 16 we see that t h i s region has the second highest r a i n f a l l to Kiandongoro region, while Table 25 f o r the basal area data does not i n d i c a t e differences i n stand density for plots from Kinale and those from the rest of the country. This leaves two possible explanation: 1. That there are s i t e factors possibly associated with s o i l s which expresses themselves through basal area development f o r P_. patula. This would be contrary to the general growth theory which holds that e f f e c t s of s i t e f actor are expressed mainly through stand height development. 2. That the P_. patula grown In Kinale region i s of a d i f f e r e n t provenance or v a r i e t y from that grown i n the rest of the country. To date, there has been no evidence of t h i s p o s s i b i l t y . The i m p l i c a t i o n here i s that P. patula trees from Kinale region are of d i f f e r e n t form from that of trees from the rest of the country. 116 This phenomenon needs further i n v e s t i g a t i o n as the findings are of both economic and b i o l o g i c a l importance. 4. Basal Area Development i n Thinned Stands 4.1 Introduction As mentioned i n the previous section, basal area development i n plantations can be viewed i n two phases, before and a f t e r f i r s t thinning. This section deals with the second phase i n which the e f f e c t s of thinning are a fac t o r . Thinning influences stand development through two changes i n stand conditions: 1. I t reduces the standing basal area by removing some of the trees, the basic units on which growth occur. 2. I t reduces competition for l i g h t and nutrients, thus creating conditions condusive to fas t e r rate of growth of the remaining trees i n the stand. The f i r s t e f f e c t r e s u l t s i n an instant reduction i n t o t a l basal area per unit area and an instant change i n mean stand DBH, whose magnitudes depend on type of thinning. For example thinning (as practiced i n Kenya) r e s u l t s i n the remaining stand having a higher mean DBH than before thinning. This e f f e c t has been termed mechanical or s t a t i s t i c a l growth (Grut 1970). The second e f f e c t a f f e c t s the subsequent rate of stand basal area growth or mean stand DBH increment. Most studies on basal area 117 development i n thinned stands have considered basal area increment as the variable of i n t e r e s t rather than basal area i t s e l f . Examples are Crowe (1967), Grut (1970), C u r t i s 91967), C l u t t e r (1963), S u l l i v a n and C l u t t e r (1972), C l u t t e r and A l l i s o n (1974) and others. In studying basal area increment of a stand, an important consi- deration i s the period of growth. In general, a longer period, e.g. 5 or 10 years has the advantage of evening out the year to year c l i m a t i c v a r i a t i o n s . This would r e s u l t i n an apparent higher p r e c i s i o n i n the p r e d i c t i o n equation. In p r e d i c t i n g p e r i o d i c growth however, an assump- t i o n i s made that within that period, annual growth rates are the same. This could r e s u l t i n serious errors e s p e c i a l l y for f a s t growing t r o p i c a l species such as are the subject of t h i s study. For t h i s reason and because permanent sample plots i n Kenya are measured annually, annual growth i n t e r v a l was used. 4.2 Choice of Dependent and Independent Variables As mentioned above, the variable of Interest i n basal area develop- ment a f t e r f i r s t thinning i s u s u a l l y basal area Increment. The p o s s i b i l i t y existed of using either basal area increment per hectare or basal area increment of the tree of mean basal area i n the stand. The former was selected as i t i s more d i r e c t l y r e l a t e d to stand basal area, one of the main d r i v i n g variables of the envisaged growth and y i e l d model. For independent variables i n the basal area increment equation, the following variables were considered: 118 1. Basal area i n square meters per hectare at the beginning of the growth period. This variable forms the basic unit from which basal area increment i s formed and on which i t i s accumulated. Basal area i s also one of the most e a s i l y obtainable and widely accepted measure of stocking density. It was therefore an obvious candidate as an independent variable i n the basal area increment equation. 2. Stand age: For a given standing basal area, one would expect that the older the stand, the lower the basal area increment. 3. Interaction of number of stems per hectare and stand dominant height. The e f f e c t s of number of stems on basal area Increment are accounted for l a r g e l y by basal area. S i m i l a r l y , some of the e f f e c t s of s i t e q u a l i t y are absorbed by the basal area since the better the s i t e q u a l i t y , the higher the basal area, provided competition i s not l i m i t i n g . However i t should be noted that the better the s i t e q u a l i t y , the higher the dominant height of the stand and therefore for a fixed number of stems per hectare, the higher the competition and therefore lower basal area increment. In the preliminary study of the basal area increment equation, the t h i r d independent v a r i a b l e ( i n t e r a c t i o n ) was introduced as the stand density index - S%, calculated from equation 2.25: 119 / 10,100 \/No. trees/ha S% = . 100 Hdom S = Harts stand density index (see also Section 2.2). Thus, t h i s stand density index acted as a measure of the i n t e r - a c t i o n between number of stems and stand dominant height, an equivalent of competition index measuring the e f f e c t of average spacing on basal area increment on a per unit area basis. For C_. l u s i t a n i c a and P̂ . patula, t h i s variable proved s i g n i f i c a n t at the .05 l e v e l but non- s i g n i f i c a n t f o r P_. radiata i n the preliminary l i n e a r equations. Table 28 gives the summary of data used for basal area increment study. 4.3 Basal Area Increment Equation In the preliminary study of the r e l a t i o n s h i p of the basal area Increment to the independent v a r i a b l e s , the following curve shapes were observed: 1. With age: an exponential decay curve; a l l three species. 2. With i n i t i a l basal area: also an exponential decay curve, approximating a hyperbola: a l l three species. 3. With S%: an approximate l i n e a r r e l a t i o n s h i p f o r £. l u s i t a n i c a and P_. patula; no r e l a t i o n s h i p indicated f o r P. r a d i a t a . From l i t e r a t u r e review and preliminary t r i a l with nonlinear models, the f i n a l equation adopted was an extension and more generalized model 120 TABLE 28. Summary of basal area increment data Species Variable Mean Standard deviation Minimum Maximum Age 20.3 8.43 7.5 43.6 BA 34.6 10.34 8.5 67.9 S 21.2 3.95 12.6 35.9 BAI 2.2 1.10 0.1 5.9 Age 14.3 4.56 7.4 27.7 BA 28.0 10.64 5.2 62.2 S 19.6 5.70 8.4 57.0 BAI 2.4 1.29 0.1 7.7 Age 15.2 5.44 7.5 34.6 BA 25.6 10.15 4.7 57.0 S - - - — BAI 2.1 0.79 0.2 4.7 n C. l u s i t a n i c a P. patula P. r a d i a t a 658 638 723 121 for basal area increment f o r P. radi a t a i n New Zealand (Clutter and A l l i s o n 1974): BAI . _ (MA b 2 + b 3B b4) 2 > 3 0 Where: A = I n i t i a l age B = I n i t i a l per acre basal area BAI = Predicted basal area increment during the next year b p b 2 , b 3 and b^ are regression c o e f f i c i e n t s . In t h e i r equation, C l u t t e r and A l l i s o n had b4=-l which was generalized i n t h i s study. The extension consisted of i n c l u s i o n of a t h i r d independent v a r i a b l e , S% (Hart's stand density index), to account for the i n t e r a c t i o n between stocking and stand dominant height. Preliminary investigations with t h i s variable however proved n o n s i g n i f i - cant f o r P_. radiata and so the extended form of the equation was used only for C_. l u s i t a n i c a and P_. patula: BAI = e + ^ + b 5 S ) 2.31 Where: BAI = Basal area increment i n nr/ha B = Basal area i n m2/ha at beginning of the growth period A = Age at end of growth period S = Harts stand density index at end of growth period b p b 2...bij are regression c o e f f i c i e n t s . Equation 2.31 (and s i m i l a r l y 2.30) can be rewritten as: 122 b 3B b 4 b 5S 2.32 BAI . e . e so that e b i i n d i c a t e the rate of growth of the basal area increment due to the respective independent v a r i a b l e , while b 2 and b 4 are scale parameters rel a t e d to the shape of the r e l a t i o n s h i p of the respec- t i v e v a r i a b l e sto basal area increment. Table 29 gives the estimated parameters from equation 2.31 (C_. l u s i t a n i c a and P_. patula) and equation New Zealand P. radi a t a equation are also given for comparison purposes. Table 30 gives the asymptotic standard deviations of the estimated para- meters from equations 2.30 and 2.31. 4.5 Results and Discussion The equations for a l l three species show moderate to low c o e f f i - cient of determination: 0.61, 0.69 and 0.31 for C_. l u s i t a n i c a , P_. patula and P_. radi a t a r e s p e c t i v e l y . The c o e f f i c i e n t for P. radiata i s e s p e c i a l l y low compared to that for New Zealand. However i t should be noted that the data f o r New Zealand came from a r e l a t i v e l y r e s t r i c t e d area - the New Zealand Forest Products Limited f o r e s t s , while that for Kenya came from the whole country. This and the dothistroma disease problem i n Kenya may p a r t l y explain the high v a r i a b i l i t y i n Kenya data. In general, high v a r i a b i l i t y i n basal area increment may be expected since the increment i s a function of several other factors not included i n the model: 2.30 (P. radiata) and t h e i r relevant s t a t i s t i c s . The parameters for the 1. Climatic v a r i a t i o n from year to year. 123 TABLE 29. Parameter estimates and other relevant s t a t i s t i c s f o r the basal area increment equation for £. l u s i t a n i c e , IP. patula and P. rad i a t a (Kenya and New Zealand) C o e f f i c i e n t s C. l u s i t a n i c a P. patula P. radiata P. radiata (Kenya) (N.Z.) b 2 b 3 b 4 ^5 7.7226 -0.3836 -4.7451 -0.2227 0.01653 11.1362 -0.2327 -7.1706 -0.07970 0.01282 5.9090 -0.6218 -11.1983 -1.0632 5.978 -0.2941 -21.663 -1 R SEE 658 .61 .68 m1 or 31% 638 .69 .72 m2 or 30% 723 .31 .66 m2 or 32% .76 18.2% TABLE 30. Asymptotic standard deviations for the parameters on Table 29 C o e f f i c i e n t s C. l u s i t a n i c a P. patula P_. radiata (Kenya) b 2 b 3 b 4 b5 0.2864 0.1362 0.8785 0.0948 0.0043 3.5863 0.1691 4.4411 0.0666 0.0036 0.5602 0.05722 5.3622 0.2440 1 2 4 2. Stand disturbances during c u l t u r a l operations. For example i n a study of basal area Increment for P_. r a d i a t a , Grut (1970) found that during approximately the f i r s t year a f t e r a thinning, the Increment was lower than that normal f o r the p a r t i c u l a r age, s i t e q u a l i t y and number of stems. This, he stated, could p a r t l y be explained by the fact that a portion of the growing stock had been removed and the remaining crop had not yet adjusted i t s e l f to the new conditions. A f t e r a year, the basal area increment rose above that which i s normal f o r the p a r t i c u l a r age, s i t e q u a l i t y and number of stems. This could p a r t l y be explained as due to the manurial e f f e c t s of the released slash and roots l e f t In the ground. Although c l i m a t i c factors can be measured f o r i n c l u s i o n i n the basal area increment equation, they are not useful for p r a c t i c a l pur- poses since i t i s not possible to project accurately what these factors w i l l be i n future. The e f f e c t s of s i l v i c u l t u r a l disturbances on the other hand can only be measured for controlled experiments: information that was not a v a i l a b l e for t h i s study. The signs for a l l the predicted parameters are consistent with theory. For example, the p o s i t i v e sign associated with by for age implies that basal area increment w i l l be decreasing as age increases because of the negative value of b£» while the negative sign associated with b 3 f o r basal area implies that basal area increment w i l l be increasing as basal area increases, a l l other factors being held constant, because of the negative value associated with b4« The 125 p o s i t i v e sign associated with c o e f f i c i e n t for density index for C. l u s i t a n i c a and P. patula means that basal area increment w i l l increase as density index of the stand increases. This can r e s u l t from: 1. Increasing spacing as a r e s u l t of heavier thinnings. 2. For fi x e d spacing, lower s i t e q u a l i t y means higher stand density index. In e i t h e r case, the r e s u l t i s less crowding of the remaining stand, l e s s competition and therefore higher basal area increment for the remaining stand. Simi l a r l y , stands with a higher number of stems per unit area or on high q u a l i t y s i t e s means higher crowding: more competi- t i o n and therefore lower basal area increment. Thus t h i s sign i s con- s i s t e n t with growth theory. An important observation i n Table 29 i s that the parameter bj which i s associated with growth rate with respect to age i s almost the same for both the Kenyan equation and the New Zealand equation for _?. ra d i a t a . The parameter b 4 associated with shape of the basal area increment to basal area r e l a t i o n s h i p i s also almost the same; thus confirming the hyperbolic r e l a t i o n s h i p , while the other parameters d i f f e r by almost twice each other. However, the comparison of the magnitude of these parameters i s in v a l i d a t e d by the fact that the equations are not based on the same u n i t s . I t i s also s i g n i f i c a n t that f o r P. radi a t a i n Kenya, the i n t e r - action between number of stems and stand dominant height as measured by stand density index (S%) f a i l e d to be s i g n i f i c a n t . As mentioned e a r l i e r , t h i s variable measures the e f f e c t s of competition on basal area 126 increment per unit area. Thus f or t h i s species, stand density index, within the range maintained i n plantations i n Kenya, has no s i g n i f i c a n t e f f e c t on basal area increment on a unit area basis. This may well explain why C l u t t e r and A l l i s o n (1973) used only basal area and age i n t h e i r basal area increment equation f o r t h i s species i n New Zealand. Figure 15a and b shows the predicted basal increment for various d e n s i t i e s f o r rad i a t a i n Kenya and New Zealand, using equation 2.30. The range of densities i s within the range covered by the data for Kenya. Two main observations are worth of note: 1. For any given basal area, basal area increment i s much higher for Kenya than for New Zealand. This could be a r e s u l t of s i t e factors i n Kenya being more favourable to the growth of t h i s species. 2. For Kenya, basal area increment i s much more affected by stand density (as measured by basal area) than i n New Zealand, e s p e c i a l l y for basal area under 30 m /ha. Thus, the theory that basal area increment i s not affected by changes i n stand density does not hold f o r Kenya at the present range of de n s i t i e s . For P. radiata i n New Zealand, the theory appears to hold c l o s e l y . This may be a r e f l e c t i o n of the difference i n s i t e q u a l i t y , suggesting that for Kenya, s i t e q u a l i t y i s much higher so that the present stand densities do not e n t i r e l y u t i l i z e the s i t e p o t e n t i a l , e s p e c i a l l y below 30 m per hectare. 1 2 7 FIGURE 1 5 BASAL AREA INCREMENT CURVES 128 Figure 16a and b shows the predicted basal area increment for various stand d e n s i t i e s f o r C_. l u s i t a n i c a and P. patula, using equation 2.31. These curves are f o r the average stand density index (21.2 and 19.6 r e s p e c t i v e l y ) within the range of densities covered by the data. For a given basal area, the curve would be higher f o r higher value of S% and vice versa. The figure indicates the same pattern as for F_. ra d i a t a . Thus, basal area increment for these species varies with stand density within the range of d e n s i t i e s maintained i n Kenya plantations. 5. Stand Diameter D i s t r i b u t i o n The information on stand structure and diameter d i s t r i b u t i o n i s ce n t r a l to forest stand management. Besides forming the basis f o r the stand table construction, t h i s knowledge i s e s p e c i a l l y important i n stands managed for sawtimber and veneer production since the f i n a l y i e l d i n these plantations i s c l o s e l y r e l a t e d to the size of the logs. Knowledge of the volume d i s t r i b u t i o n by siz e classes forms the basis for deci s i o n making as to when a stand can be economically harvested for a given end product. According to Hyink (1980), three basic approaches have been employed i n modelling growth and y i e l d a t t r i b u t e s by size classes: 1. Approaches employing Markov chains and systems of d i f f e r e n t i a l equations. According to Moser (1980), t h i s approach uses a square matrix of condit i o n a l p r o b a b i l i t i e s that correspond to the p r o b a b i l i t y of going from state i to state j a f t e r one FIGURE 16 BASAL AREA INCREMENT CURVES 130 step or t r a n s i t i o n . In p r a c t i c e , t h i s approach corresponds to the updating of a stand table. 2. I n d i v i d u a l tree model approach: By t h e i r nature, i n d i v i d u a l tree models, pioneered by Newnham (1964^ provide growth and y i e l d a t t r i b u t e s f o r i n d i v i d u a l trees. 3. Diameter d i s t r i b u t i o n approach based on p r o b a b i l i t y theory. The f i r s t two approaches place no r e s t r i c t i o n on the form or shape of the underlying diameter d i s t r i b u t i o n (Hyink 1980), an advantage over the t h i r d approach. I t should however be noted that these approaches are inherent to the modelling strategy and so are not a l t e r n a t i v e s for the t h i r d approach i n the whole stand, diameter free models. I t i s t h i s t h i r d approach which i s of i n t e r e s t i n t h i s study. 5.1 Th e o r e t i c a l Considerations The basic assumption underlying the p r o b a b i l i t y d i s t r i b u t i o n approach to stand diameter d i s t r i b u t i o n i s that the l a t t e r can be adequately characterized by a given p r o b a b i l i t y density function (pdf). In particular,even-aged stands tend to have unimodal shape and form, which has lead to the wide use of continuous unimodal p r o b a b i l i t y density function to characterize t h e i r diameter d i s t r i b u t i o n . 131 In general, the p r o b a b i l i t y density function has the property: f(x;6) dx = 1 2.3 Where 6 = a vector containing the parameters of the p a r t i c u l a r pdf. Put i n another form: x 2 p(x x < X <x 2) - f(x;6) dx 2.34 x l which reads that the p r o b a b i l i t y of the random variable X assuming a value between x^ and x 2 i s given by the i n t e g r a l of the p r o b a b i l i t y density function between the two values (x^ and x 2 ) . The problem therefore i n applying t h i s theory to f o r e s t r y i s one of fin d i n g the appropriate pdf and estimating i t s parameters. 5.2 Choosing the Appropriate pdf As mentioned above, even-aged stands i n general have diameter d i s - t r i b u t i o n with unimodal shape and form. Over the years, several mathe- matical functions have been used to model t h i s d i s t r i b u t i o n . Examples including the normal curve (e.g. Crowe 1967), Gram-charlier Series (e.g. Meyer 1930, Crowe 1967), Pearl-Reed population growth curve (Osborne and Schumacher 1935, Nelson 1964, Crowe 1967), Beta d i s t r i b u t i o n ( C l u t t e r and Bennett 1965, McGee and Delia Bianca 1967, Lenhart and C l u t t e r 1971), the Gamma d i s t r i b u t i o n (Nelson 1964) and the Weibull function (e.g. Bailey and D e l l 1973, C l u t t e r and A l l i s o n 1974, Alder 1977, 132 Rustagi 1978 and Hyink 1980, Schreuder and Swank 1974). In an analyses of the s u i t a b i l i t y of most of these function for characterizing diameter d i s t r i b u t i o n , B ailey and D e l l (1973) concluded that no other diameter d i s t r i b u t i o n function e x h i b i t s as many desirable features as the Weibull function. These include the s i m p l i c i t y of algebraic manipulation and i t s a b i l i t y to assume a va r i e t y of curve shapes. This explains why most of the recent work on diameter d i s t r i b u t i o n and stand structure has used t h i s function, as evidenced by the examples quoted above. This function was chosen for t h i s study for the above reasons and because preliminary investigations indicated that i t would adequately characterize the diameter d i s t r i b u t i o n of the permanent sample plots available for t h i s study. The Weibull p r o b a b i l i t y density function has already been discussed i n Section 1.2 of t h i s chapter i n connection with height over age growth function. For diameter d i s t r i b u t i o n , the usual procedure i s to work with the cumulative d i s t r i b u t i o n form of the function, defined as: x-x. - ( F(x) = 1 - e b 2.35 f o r x > x. 0 0 < F(x) < 1 where F(x) measures the area under the curve between X = x 0 and X = x. Thus i n terms of diameter d i s t r i b u t i o n , 133 F(x) = p (X < d) or p(d < X < d ) - F(d ) - F(d ) -.- 2.36 This equation i s equivalent to equation 2.34. 5.3 F i t t i n g the Weibull Model to Diameter D i s t r i b u t i o n Data The data for the diameter d i s t r i b u t i o n study consisted of 58 permanent sample plo t s made up of a l l three species: 30 for P_. patula, 18 of £. l u s i t a n i c a and 10 of P_. ra d i a t a . The number of stems per plot ranged between 4 to 70 stems on which tree DBH had been measured to one decimal place of a cm. For consistency with timber measurement practices i n Kenya, the tree diameters for each plot were s t r a t i f i e d into 3 cm diameter classes. Of the 58 p l o t s , 8 were selected at random and set aside as test data. Data summary for the 50 remaining plots are given on Table 31. TABLE 31. Summary of the DBH d i s t r i b u t i o n data Standard Variable n Minimum Maximum Mean deviation Age 50 9.700 41.700 18.568 6.5781 Height 50 16.100 39.000 24.754 6.0117 Stems 50 74.000 1160.0 523.08 278.37 The f i r s t step was to f i t the Weibull cumulative d i s t r i b u t i o n function to the 50 diameter d i s t r i b u t i o n histograms to obtain an estimate of the parameter: 134 F(D) = 1 - e-b*D*c 2.37 where b* = ^ c D* = D - (DL + 3) DL = Minimum diameter class on the histogram D = A given diameter c l a s s b and c are Weibull parameters F(D) = Observed p r o b a b i l i t y of trees having a DBH < D. The parameters b and c were estimated using BMDP:3R nonlinear sub- routine. Figure 17 shows the histogram for plot 135 and the curve r e s u l t i n g from the estimated parameters. The next step was to test how well the Weibull function character- ized the diameter frequency d i s t r i b u t i o n of the 50 p l o t s . For each p l o t , the expected frequency per diameter class was calculated using the estimated parameters and a Chi-square test of goodness of f i t performed using the following equation: x a . K W j - V 2 2.38 i = l N where = Observed number of trees i n a diameter class i = Expected number of trees i n a diameter class i predicted using the estimated parameters k = T o t a l number of diameter classes X 2 = Calculated Chi-square value. 135 FIGURE 17 DIAMETER DISTRIBUTION HISTOGRAM AND THE FREQUENCY CURVE RESULTING FROM THE FITTED WEIBULL PROBABILITY DENSITY FUNCTION Plot no 135 N = 716 Sph n s 39 D _ = 18 b= 10.6246 C si .9257 18 21 24 27 3 0 33 3 6 DIAMETER CLASSES 136 The calculated x values were then compared with the tabulated 9 X values for .05 p r o b a b i l i t y l e v e l of s i g n i f i c a n c e and k-3 degrees of freedom. Out of the 50 p l o t s , x values could not be calculated f or 8 pl o t s as they had 3 or fewer diameter classes. Of the remaining 42 p l o t s , only 4 showed a lack of f i t , e i t h e r because they had very long t a i l s or had sharply truncated data. On basis of t h i s test therefore, the hypothesis that the diameter d i s t r i b u t i o n follows the Weibull d i s t r i b u t i o n was accepted. 5.4 Estimating the Parameters and V a l i d a t i o n of the Estimating Model The Weibull parameters estimated for each i n d i v i d u a l plot define the shape and form of the frequency d i s t r i b u t i o n by diameter classes of the p a r t i c u l a r p l o t . As i s expected, these parameters d i f f e r from pl o t to plot due to d i f f e r i n g stand conditions. In order to be able to predict the appropriate I > L and Weibull function parameters for given stand conditions, the l i n e a r and c u r v i l i n e a r c o r r e l a t i o n of and the estimated parameters with stand variables was studied (Table 32). Table 32 indicates that the minimum diameter i n a stand i s best correlated with the stand mean dbh, age, stand dominant height and the log^Q number of stems per hectare i n that order. The high c o r r e l a - t i o n of with age and dominant height could be explained as l a r g e l y due to the high c o r r e l a t i o n between the stand mean dbh, age and stand dominant height and so any one of these could be used as the predictor va r i a b l e without loss of p r e c i s i o n i n the estimation of t h i s v a r i a b l e . 137 TABLE 32. Linear and c u r v i l i n e a r c o r r e l a t i o n of the D^ and estimated Weibull parameters with other stand variables n = 50 DF = 48 r @ .05 = 0.28 r @ .01 = 0.36 Stand variables Age Height No. stem DBH DL DBH - D L °L DBH 1 No. stem Log Age Log Height Log No. stems Log ( D B H - D L ) Log °L DBH Log D B H Log D L 0.92 0.80 -0.76 0.94 1.00 -0.02 0.66 0.77 0.91 0.76 -0.81 -0.14 0.63 0.93 0.95 0.34 0.28 -0.10 0.37 0.04 0.94 -0.54 -0.04 0.29 0.26 -0.07 0.90 -0.50 0.35 0.01 -0.46 -0.43 0.40 -0.43 -0.43 0.13 -0.44 -0.42 -0.50 -0.44 0.42 0.15 -0.42 -0.46 -0.52 DBH = Mean stand DBH i n cm (D) Height = Stand dominant height i n meters D L = Minimum stand DBH i n cm No. stems = Number of stems per hectare b & c = Estimated Weibull parameters. 138 On the other hand, the number of stems per hectare i s not so c l o s e l y correlated with the stand mean DBH. These two variables were therefore selected as predictors of the minimum stand diameter or the xg parameter. The Inverse of number of stems was used instead of the regression equation with a higher c o e f f i c i e n t of determination. The f i n a l equation was: D = Stand mean dbh N = Number of stem/ha n = 50 R 2 = .90 SEE = 3.1 cm = 15% of the mean a l l the predictor variables were s i g n i f i c a n t at .05 l e v e l . The scale parameter b was very strongly correlated with the d i f f e r - ence between the stand mean DBH and the minimum stand DBH, followed by the r a t i o of the two. This was not s u r p r i s i n g since t h i s parameter i s defined i n terms of i t s p o s i t i o n on the x-axis, on which both D_ and D are located (Bailey and D e l l 1973). A regression for t h i s para- meter with these two variables as predictors gave the following equation: of the number of stems because the former resulted i n a D = -3.2048 + .7472D + 691.14(i) L N 2.39 where D_ Minimum stand dbh 139 b = -3.6207 + 1.2263(5 - D. ) + 5.7910(—) 2.40 L D n = 50 R 2 = .91 SE = 1.24 = 11% of mean Both predictor variables were s i g n i f i c a n t at .05 p r o b a b i l i t y l e v e l . The parameter c on the other hand did not have any stand a t t r i b u t e with which i t was strongly correlated. T r i a l s with several combinations of predictor variables produced the following best equation: \ c = 4.6585 - 2.0635(—) - .02360D 2.41 D n = 50 R 2 = .27 SE = .68 = 27% of mean Both predictor variables were s i g n i f i c a n t at .05 p r o b a b i l i t y l e v e l . 5.5 Model V a l i d a t i o n Equations 2.39 and 2.40 proved quite s a t i s f a c t o r y based on t h e i r c o e f f i c i e n t of determination and the study of t h e i r residuals plotted against the predicted values. Equation 2.41 proved the best possible with s a t i s f a c t o r y d i s t r i b u t i o n of r e s i d u a l s . However, i t s t i l l remained to see how well these equations would predict the respective Weibull 140 parameters for independent data from the same population as used to derive the equations. Table 33 gives the predicted parameters and the observed parameter obtained by f i t t i n g the Weibull function to the frequency d i s t r i b u t i o n s of the 8 p l o t s set aside for t h i s purpose. In terms of mean bias the equations for i n d i c a t e no bias. S i m i l a r l y , the equation f o r c i n d i c a t e s , no o v e r a l l bias. However, the model i s only moderately s e n s i t i v e to changes i n stand a t t r i b u t e s , so that high values of c are associated with p o s i t i v e r e s i d u a l s and low values with negative r e s i d u a l s . Parameter b, on the other hand, has one large p o s i t i v e r e s i d u a l , that of p l o t 327, which has l a r g e l y contributed to the p o s i t i v e bias for t h i s parameter. Otherwise, the other residuals range between +3.6, which i s within the expected v a r i a b i l i t y as indicated by equation 2.40. Equation 2.40 can therefore be considered unbiased since, as seen from Figure 18, plot 327 represents an o u t l i e r . In terms of p r e c i s i o n , the standard errors of estimate for a l l parameters, are i n general higher than those obtained i n the d e r i v a t i o n of t h e i r equations. This can be explained p a r t l y by the lower degrees of freedom associated with the test sample. In addition the standard error f or parameters b and c can be expected to depart even farther from that of t h e i r equation since t h e i r v a r i a b i l i t y includes the v a r i a b i l i t y i n (as a predictor variable which i s i t s e l f a predicted v a r i a b l e ) . Figure 18 shows the histograms of the diameter frequency d i s t r i b u - tions of the test p l o t s , the observed d i s t r i b u t i o n ( s o l i d curve) obtain- ed by f i t t i n g the Weibull function to the i n d i v i d u a l plot cumulative p r o b a b i l i t y data, and the predicted d i s t r i b u t i o n (dotted curve) obtained from Weibull parameters predicted from equations 2.39, 2.40 and 2.41. TABLE 33. Comparison of predicted and observed Weibull parameters f o r the test plots Plot # Observed Predicted Residual Observed Predicted Residual Observed Predicted Residual 210 11.7 14.8 -3.1 13.6583 10.0550 3.6033 3.6882 2.7844 0.8538 327 26.6 33.3 -6.7 21.5330 14.1089 7.4241 3.3244 2.0608 1.2636 128 18.4 19.9 -1.5 14.2300 11.2948 2.9352 2.4216 2.5530 -0.1314 147 19.3 16.4 2.9 8.7052 10.4026 -1.6974 2.8789 2.7055 0.1734 189 26.8 23.2 3.6 9.0294 12.7995 -3.7701 1.7225 2.4350 -0.7125 202 14.8 15.1 -0.3 10.4126 9.7630 0.6496 4.0712 2.7574 1.3138 221 25.1 23.6 1.5 10.4964 11.1698 -0.6734 1.4104 2.3862 -0.9758 238 18.8 14.7 4.1 5.9215 9.5166 -3.5951 1.8959 2.7720 -0.8761 SE = 4.41 cm SE = 4.64 cm SE = 1.1248 or 21.8% or 39% or 42% Mean bias = 0.0625 Mean bias = 0.6095 Mean bias = 0.1136 142 FIGURE 18 DIAMETER DISTRIBUTION HISTOGRAMS, FITTED WEIBULL P.D.F. AND THE PREDICTED WEIBULL P.D.F. FOR THE EIGHT TEST PLOTS 10 8 6 4 2 .4 .3 .2 .1 PLOT 2I0 N = 34 (0) PLOT 3 2 7 PLOT I28 N = 8 N = 1 9 / ( b ) ( c ) pdf f i t t e d d i r e c t l y to plot data pdf predicted from equations 12 15 18 21 24 27 30 27 30 33 38 39 42 45 4B 51 54 16 21 24 27 30 33 36 39 10 8 .5 .2 PLOT I47 N = 24 (d) PLOT I 8 9 N = 23 (e) PLOT 202 N = 23 (f) 18 21 24 27 30 27 30 33 36 39 42 15 18 21 24 27 • s .4 .3 • 2 PLOT 221 N =12 i g ) 0 24 27 30 33 36 39 42 18 21 24 27 30 DIAMETER CLASSES IN Cm 143 In general, the histograms Indicate the range of diameter frequency d i s t r i b u t i o n s that can be expected i n plantations where thinning i s a standard p r a c t i c e . Thus, plot 327 represents an extreme case tending towards a uniform d i s t r i b u t i o n while most of the other plots with multimodal d i s t r i b u t i o n s can be expected as depicted by pl o t s 327, 128, 189 and 221. A l l these cases underline the problem associated with the use of any one p r o b a b i l i t y d i s t r i b u t i o n function to characterize a l l these shapes. Assuming the unimodal shape and form to be the predominant charac- t e r i s t i c , the f i t t e d Weibull function ( s o l i d curve) appears to charac- t e r i z e t h i s d i s t r i b u t i o n quite s a t i s f a c t o r i l y . S i m i l a r l y , the curves from the predicted parameters (dotted) follow s i m i l a r shape and form to that f i t t e d from the data, with no i n d i c a t i o n of bias or inconsistency. Thus, the Weibull parameter p r e d i c t i o n equations were accepted for the diameter frequency d i s t r i b u t i o n modeling i n t h i s study. 6.0 Volume Determination Tree volume equations for C_. l u s i t a n i c a , P. patula and P_. ra d i a t a i n Kenya were developed by Wright i n 1969 (Wright 1977). Since then, these volume equations have been very widely used i n management practices and form the basis for the tree volume tables for these species (Wright 1974) and the y i e l d tables (Wanene and Wachiuri 1975, Wanene 1975, 1976). They can therefore be said to have passed the f i e l d t e s t s . These equations were used to determine the stand volumes i n t h i s study. 144 The tree volume equations provide for the determination of tree volume from the tree DBH and e i t h e r the i n d i v i d u a l tree height or the tree dominant height, obtained from determining the dominant height of the stand. Only equations based on tree dominant height are of i n t e r e s t i n t h i s study since i n d i v i d u a l tree heights are not a v a i l a b l e . Table 34 gives the equations and the respective c o e f f i c i e n t s f o r each of the species. These equations give t o t a l overbark volume. For a l l species, the conversion of the t o t a l volume to merchantable volume i s achieved through m u l t i p l i c a t i o n by a factor R which i s dependent on the tree DBH: b 0D R = b 0 + b i e 2 2.44 Table 35 gives the relevant c o e f f i c i e n t s for each of the species for the merchantable standards i n Kenya: 15 cm and 20 cm top diameter l i m i t s f o r P. patula and P. r a d i a t a , 15 cm only for C;. l u s i t a n i c a . 145 TABLE 34. Equations and c o e f f i c i e n t s for the tree volume equations for C. l u s i t a n i c a , P. patula and P_. radiata i n Kenya 1. V - bo + b xD 2 + b 2DH + b 3D 2H 2. log 1 ( )V = bo + t ^ l o g ^ D + b 2logH C. l u s i t a n i c a P. patula P. radiata t>0 -0.01733 -0.0072 -4.2643 b l 0.0001937 0.00002887 2.0598 b 2 0.00005069 0.00002077 0.7875 b 3 0.00002296 0.00003276 Equation 2.47 i s for £. l u s i t a n i c a and P_. patula Equation 2.48 i s for P_. radiata V = Tree volume i n cu.meters D = Tree DBH i n cm H = Stand dominant height i n meters. 146 TABLE 35. C o e f f i c i e n t s f o r R-factor equation (2.44) f o r the merchantable l i m i t s for the respective species 15 cm b l 20 cm b. C. l u s i t a n i c a 0.9870 -11.1577 -0.1742 - P. patula 0.98471 -8.6658 -0.16135 0.97352 -21.9737 -0.15407 P. r a d i a t a 0.98348 -14.7231 -0.17505 0.97622 -18.7751 -0.13971 147 CHAPTER 3 YIELD MODEL CONSTRUCTION AND VALIDATION 1.0 General P r i n c i p l e The general p r i n c i p l e on which growth and y i e l d models are based can be derived d i r e c t l y from the d e f i n i t i o n of these two terms. For example i n the simple case of an even-aged stand with a standing volume, V, growth rate may be expressed as a function of age, A, i n terms of d i f f e r e n t i a l notation: = f(A) 3.1 dA where — = rate of change of stand volume with respect to d A stand age. f(A) = a function of stand age. When the age i s expressed i n years, the rate of change as given i n equation 3.1 i s referred to as annual growth or annual increment i n volume. From t h i s , volume y i e l d from an i n i t i a l age A Q to a future age AQ + t can be obtained as the summation of the growth rates within t h i s time period. A n a l y t i c a l l y , y i e l d i s obtained by integra t i n g the age function within the l i m i t s A Q to A Q + T ; : - Y = ^ ° + t f(A)dA = F(A) 3.2 148. where Y = Stand y i e l d or the summation of the annual growth increments between the i n i t i a l stand age A Q , and a future stand age, A Q ^ . F(A) = Y i e l d function obtained by mathematically Integrating the growth-rate equation, f ( A ) . The essence of t h i s p r i n c i p l e helps put the whole concept of growth and y i e l d into i t s h i s t o r i c a l perspective. Thus, the e a r l i e r y i e l d models represented by normal y i e l d tables and l a t e r by the variable density y i e l d tables presented stand y i e l d (derived using graphical and l a t e r by regression techniques) as a function of stand c h a r a c t e r i s t i c s as d i s c r e t e v a r i a b l e s . Forest l i t e r a t u r e has numerous examples of t h i s class of models: Plonski (1956), Barnes (1962), etc. This category of models i s described as S t a t i c models i n growth and y i e l d l i t e r a t u r e . As a r e s u l t of recent advances i n computational techniques and computer technology, recent y i e l d models presents stand y i e l d derived from stand parameters which are rates of change (growth r a t e s ) . This category of models i s termed Dynamic where one (or more) of the independent variable i s an i n t e g r a l of the dependent v a r i a b l e . The e a r l i e s t example of a dynamic model appears to be the growth and y i e l d model for ponderosa pine stands by Lemraon and Schumacher (1963) which estimated tree growth as a function of diameter growth a f t e r thinning. The t r a n s i t i o n between the s t a t i c models and dynamic models presented problems to the mensurationists, since most models tended to treat growth and y i e l d as e s s e n t i a l l y independent phenomena. The r e s u l t was that y i e l d derived from conventional ( s t a t i c ) models d i f f e r e d from 149 y i e l d derived from dynamic models. This contradicted the basic p r i n c i p l e of growth and y i e l d which required that t o t a l growth ( y i e l d ) be synonymous with the summation of continuous growth increments. This problem of i n c o m p a t i b i l i t y between the two categories of models was recognised by C l u t t e r (1963) when he developed compatible growth and y i e l d models for L o b l o l l y pine. This he did by d i f f e r e n t i a t i n g already accepted y i e l d models (with respect to age) to produce cubic-foot and basal area growth functions. He defined as compatible those y i e l d models whose algebraic form can be derived by mathematical i n t e g r a t i o n of t h e i r growth model. 2. Simulation A p p l i c a t i o n to Growth and Y i e l d Models By f a r , the most widely used method for growth and y i e l d modeling i s simulation technique (Lee 1967). B a s i c a l l y , the term ref e r s to any model that exhibits a behaviour s i m i l a r to the r e a l system. Because of the complex nature of a forest stand and the need f o r a f l e x i b l e and comprehensive technique to handle a l l the component interactions with as few r e s t r i c t i v e assumptions as possible, simulation has become almost the standard t o o l . I t s a p p l i c a t i o n has been stimulated by the advances i n programming languages and access to high speed computers. I t s r o l e i n simulating stand growth and the b i o l o g i c a l and economic assumptions underlying i t s a p p l i c a t i o n were discussed by Smith (1966), while Gould and O'regan (1965) discussed i t s role to better forest planning. Among i t s main advantages are the time compression e f f e c t s whereby the techni- que accomplishes i n seconds what might otherwise take several years of 150 actual experimentation and the f a c i l i t y to experiment with the simulated system rather than the actual system. I t should however be stated here that simulation i s not a perfect t o o l as i t only provides estimates of the model state r e s u l t i n g from predetermined decision v a r i a b l e s . Thus, according to H i l l i e r and Lieberman (1980), i t only compares a l t e r n a t i v e s rather than generating the optimal one. Besides this,one pays for the comprehensiveness and f l e x i b i l i t y of simulation i n terms of a n a l y t i c i n t r a c t i b i l i t y . The general approach to forest stand simulation Involves the development of the i n d i v i d u a l growth relationships and functions to describe the i n d i v i d u a l i n t e r a c t i o n s . These form the elements (blocks) of the system. These are then put together i n a systematic and l o g i c a l sequence (as a computer programme) to form the mathematical simulator of the forest stand system. This i s the general approach adopted to t h i s study. 2.1 Forest Stand Simulation Models Computer based forest stand simulation models f i r s t appeared i n the early 1960's, heralded by the pioneering work of Newnham (1964). Since then, t h e i r development has been very rapid so that as of 1980, Smith and Williams (1980) counted not less than 26 published models. Most of t h i s development has been mainly i n response to forest management needs for planning t o o l s , academic endeavours to broaden the knowledge of forest stand modeling,or both. The l e v e l of d e t a i l obtainable from 151 each model depends on the type of model and the simulation approach, which i t s e l f i s l a r g e l y governed by the available data within the p r e v a i l i n g f i n a n c i a l and technical constraints. Although several researchers have attempted to c l a s s i f y stand simulation models: M i t c h e l l (1980), Smith and Williams (1980), Munro (1974): the terminology by Munro (1974) w i l l be adopted here f i r s t l y because i t i s comprehensive and because i t encompasses the philosophy on which these models are based. Munro's c l a s s i f i c a t i o n recognises three categories of models, based on i n t e r - t r e e dependence status and primary unit parameter requirements: 1. Single tree - distance dependent forest stand models: Newnham's (1964) y i e l d model for Douglas-fir was not only the f i r s t computer based y i e l d model but also introduced a new generation of forest stand models based on the i n d i v i d u a l tree i n the stand as the basic unit, and characterized by the requirement that i n d i v i d u a l tree p o s i t i o n i n the stand be known. These models grow the physical a t t r i - butes of the i n d i v i d u a l tree by f i r s t obtaining the p o t e n t i a l growth for a free growing tree and then reducing t h i s by a factor dependent on the degree of competition to which the tree i s subjected. This procedure popularized the concept of a competition index which can be defined as a r e l a t i v e measure of the degree of competition. Different stand a t t r i - butes have been used as competition indices including mean distance of subject tree to a predetermined number of competing neighbours (Adlard 1974), angle count density (Lowe 1971), crown overlap (Newnham 1964, Lee 1967) size-distance of competing neighbours r e l a t i v e to subject tree, etc. For further discussion on competition in d i c e s , the reader i s 152 referred to Adlard (1974), Newnham and Mucha (1971), Gerrard (1968) and Opie (1968). Since Newnham's (1964) model, several models based on s i m i l a r concept but d i f f e r i n g i n complexity and l e v e l of a t t r i b u t e d e t a i l have been developed. Examples of some of those with unique features follow: Arney's (1972) Douglas-fir model improved on Newnham's model by simulating the tree diameter growth at each whorl up the tree so that' tree form was a r e f l e c t i o n of d i f f e r e n t i a l growth rates at d i f f e r e n t sections up the tree. This feature was p a r t i c u l a r l y s i g n i f i c a n t i n permiting modeling tree responses to such s i l v i c u l t u r a l treatments as pruning, thinning and f e r t i l i z a t i o n . Hegyi's (1974) BUSH model for jack pine (Pinus banksiana Lamb.) used Arney's approach. M i t c h e l l ' s (1969, 1975) TASS I (white spruce, Douglas-fir and hemlock) and TASS II (Douglas-fir) models resp e c t i v e l y represent the highest development In the singl e - t r e e distance dependent models based on the l e v e l of d e t a i l of stand a t t r i b u t e s modeled. The unique feature of these models i s that the crown of the simulated tree i s modelled e x p l i c i t l y with the growth of i n d i v i d u a l branches responding to such factors as competition from other branches either of the same subject tree or from adjacent trees. A f t e r c a l c u l a t i n g Individual branch size and f o l i a r volume, the net production of photosynthates i s then predicted and proportionately d i s t r i b u t e d to the stem bole which increases i n diameter accordingly. Thus, these models come closest to modeling the whole tree biomass based on the photosynthetic a c t i v i t y of the f o l i a g e . They are therefore t h e o r e t i c a l l y capable of modelling tree responses to c u l t u r a l treatments (pruning, thinning and f e r t i l i z e r s ) 153 and to insects and diseases which a f f e c t the quantity and q u a l i t y of tree f o l i a g e . Ek and Monserud's (1974) FOREST model i s also a s i n g l e - t r e e distance dependent model but i s unique i n that while most of the other models i n t h i s category are for even-aged stands of a single species, FOREST simulates growth and reproduction of even- or uneven-aged mixed species forest stands. This i s accomplished through modeling the reproduction (regeneration) and understory development e x p l i c i t l y , with the s p a t i a l pattern of the new stems determined by the p r i o r stand conditions. As indicated by the few examples quoted above, t h i s category of models i s capable of providing very d e t a i l e d information on i n d i v i d u a l tree growth i n response to s i l v i c u l t u r a l treatments and insect and disease attacks, depending on the l e v e l of complexity. They are also capable of simulating the development of heterogenous stands ( i n terms of age, species and s p a t i a l d i s t r i b u t i o n ) . However, t h e i r use i s l i m i t e d by t h e i r complexity and the requirement that the coordinates of each i n d i v i d u a l tree i n the stand be known. This, according to Munro (1974) makes them very expensive i n terms of time required to develop them and the excessive computer time required to execute the extra space for storage of tree p o s i t i o n records. As an example of how expensive these models can be, M i t c h e l l (1980) stated that TASS system had taken 17 years for development and testing at a cost of about $1,000,000 Canadian. Nonetheless, t h e i r p o t e n t i a l value as a research tool i n f o r e s t s i l v i c u l t u r e and economics of si n g l e forest stands may more than j u s t i f y the commitment i n time and funds. For t h i s study, however, tree 154 p o s i t i o n data were not ava i l a b l e and so t h i s category of models was not appl i c a b l e . 2. Single tree - distance independent stand models: Conceptually, t h i s term ref e r s to the class of stand simulation models which recognise the i n d i v i d u a l tree i n the stand as the basic production unit but do not require that the i n d i v i d u a l tree positions be provided. In a p p l i c a t i o n however, i t i s not clear what constitutes a single tree - distance independent model. For example, according to Munro (1974), i t Includes models where trees are grown i n dimensions i n d i v i d u a l l y or i n groupings of s i m i l a r diameters. If t h i s d e f i n i t i o n i s accepted, then according to Moser (1980), most of the models i n t h i s category could be considered as s i m i l a r to the t r a d i t i o n a l stand table p r o j e c t i o n approach presented i n most mensuration texts. However, i t should be noted that the concept of the single tree as the basic production unit i s contradicted when growth i s applied to a group of trees of s i m i l a r diameters. A common feature of a l l single tree - distance independent stand models i s that the i n i t i a l model state consists of a l i s t of tree diameters or tree diameter classes. This l i s t can be s p e c i f i e d either from inventory data or from diameter p r o b a b i l i t y density functions. Beyond t h i s stage however, two general strategies for modeling stand growth appear: 1. For some models, the pot e n t i a l growth i s computed for the aggregate stand, and then allocated among the trees i n the DBH l i s t . Stand l e v e l competitive stress i s thus incorporated i n 155 the growth equation. Growth i s then a l l o c a t e d among the i n d i v i d u a l trees or diameter classes according to t h e i r p o s i t i o n i n the ordered diameter l i s t . Examples of t h i s type of model are STANDSIM (Opie 1970), C l u t t e r and A l l i s o n s (1974) model f o r P. radiata i n New Zealand, FORSIM (Gibson et a l . , 1969, 1970), etc. 2. Other models predict the diameter increment of each tree or each diameter class e x p l i c i t l y as a function of other state v a r i a b l e s , including at least one that i s tree diameter (or class diameter) dependent to give d i f f e r e n t i a l growth rates between the classes. Examples of these are TOPSY (Goulding 1973), Prognosis (Stage 1973), PYMOD (Alder 1977), etc. In general, these models do have appeal to forest managers mainly because they are computationally e f f i c i e n t and use conventional inven- tory data as input. In addition the tree diameter l i s t i s an invaluable feature, e s p e c i a l l y i n evaluating the returns from management decisions i n terms of product size and diameter d i s t r i b u t i o n . This i s e s p e c i a l l y important for planning purposes. A serious shortcoming of single tree - distance independent models i s that they cannot predict the growth of a s p e c i f i c s i n g l e tree with any r e l i a b i l i t y . Thus, according to Munro (1974), they cannot be used to examine i n d i v i d u a l trees for crown shape and growth, bole shape changes or d e f o l i a t i o n . Nonetheless,models with c a p a b i l i t y to respond to thinning, spacing and i n some cases f e r t i l i z a t i o n interventions have been developed. These models are of i n t e r e s t In t h i s study and so a discussion of some of them i s germane. 156 STANDSIM (Opie 1970) i s a single tree - distance independent model developed to simulate preferred s i l v i c u l t u r a l treatments f o r Aust r a l i a n Mountain Ash (Eucalyptus regnans F. Muell). The model i s considered as a s i l v i c u l t u r a l rather than a planning model i n that i t simulates the growth of a single stand only (Alder 1977). However, i t has been i n c o r - porated into a planning and management system MARSH, for prediction of growth under a l t e r n a t i v e s i l v i c u l t u r a l schedules (Weir 1972). Model state consists of a l i s t of i n d i v i d u a l tree diameters, while growth i s accomplished by c a l c u l a t i n g the gross basal area increment per unit area. Thus competition i s i m p l i c i t l y included. Individual tree growth Is effected by d i s t r i b u t i n g the gross basal area increment propor- t i o n a t e l y according to the i n d i v i d u a l tree s i z e . C l u t t e r and A l l i s o n (1974) developed a single tree - distance independent model f or P_. radiata i n New Zealand. Unlike STANDISM, the i n i t i a l stand state consisted of a fixed number of diameter classes of equal p r o b a b i l i t y instead of the single tree diameter l i s t . Tree diameter d i s t r i b u t i o n from which the diameter classes are derived i s obtained from a Weibull p r o b a b i l i t y density function. Annual growth i s accomplished by p r e d i c t i n g gross annual basal area Increment and then d i s t r i b u t i n g t h i s to the diameter class medians as a function of stand age, current median diameter, current and projected basal area, and current and projected number of trees per acre. S i m i l a r l y , gross mortality i s predicted and d i s t r i b u t e d among the diameter class medians as a function of class basal area r e l a t i v e to stand basal area. This model i s designed f o r single species even-age stands. 157 PROGNOSIS model for stand development (Stage 1973) represents a very high l e v e l of development i n t h i s class of models i n that i t can simulate growth of mixed stands i n terms of species, age classes and size classes. The high l e v e l of r e s o l u t i o n i n the model i s accomplished by recording not only the DBH l i s t but also tree height and crown dimensions. The key growth component i s annual basal area increment computed from DBH, s i t e , habitat type, crown r a t i o , r e l a t i v e stand density, and the pe r c e n t i l e of the tree i n the basal area d i s t r i b u t i o n . In addition, the basal area growth function incorporates a stochastic element although the t o t a l growth process remains e s s e n t i a l l y deter- m i n i s t i c . I n dividual trees are incremented i n DBH (as a function of p o s i t i o n of the p a r t i c u l a r tree i n the basal area d i s t r i b u t i o n ) , height (as a function of DBH growth, habitat type, DBH and height) and crown length or clear bole length (as a function of r e l a t i v e stand density, basal area p e r c e n t i l e and DBH). Thus, the model has a va r i e t y of tree c h a r a c t e r i s t i c s which allow simulation of a wide range of s i l v i c u l t u r a l p r e s c r i p t i o n s . Stage (1973) has demonstrated use of t h i s model to prognose lodgepole pine stand development i n the presence of an i n f e s t a - t i o n of mountain pine beetle (Dendroctonus ponderosae Hopkins). VYTL-2 used as a subroutine i n PYMOD for e s t planning programme (Alder 1977) (discussed i n the Introduction) f a l l s i n t h i s c l a s s . 3. Whole stand - distance independent models: This category of models i s based on the same philosophy as the conventional y i e l d tables, normal or variable density, i n that the basic unit of production i s the whole stand. However, conventional y i e l d 158 tables d i f f e r from computer-based simulation models i n that the l a t e r are dynamic while the former are s t a t i c . From a philosophical point of view, t h i s class of models cannot be j u s t i f i e d as the concept of the whole stand as the basic production unit contradicts the bionomic (ec o l o g i c a l ) theory of i n d i v i d u a l i s t i c systems (such as the i n d i v i d u a l tree i n the stand) as stated by Boyce (1978): "Each l i v i n g organism and i t s environment forms an i n d i v i d u a l i s t i c system with negative feedback loops guiding behaviour i n accordance with the goal of s u r v i v a l . Behaviour i s directed by decision mechanisms. These mechanisms are g e n e t i c a l l y and environmentally determined and are p h y s i o l o g i c a l , anatomical and morphological structure of the i n d i v i d u a l - i s t i c system. Each i n d i v i d u a l i s t i c system senses and reacts to i t s own state. Past actions influence future actions to achieve the goal of s u r v i v a l . " The single tree approach to growth and y i e l d i s consistent with t h i s theory. On the other hand, the whole stand concept i s consistent with the management objective of ordering the i n d i v i d u a l i s t i c systems that make up the forest i n t o a forest community with the suitable structure to achieve s p e c i f i c goals and objectives. Models based on t h i s concept therefore have received attention since they are usually computationally l e s s expensive and e f f i c i e n t as computation of i n d i v i d u a l tree information i s eliminated. As with single tree - distance independent models, whole stand - distance independent models take conventional inventory data as input. The major difference between the two classes therefore appears to be i n the output information since the l a t t e r does not provide i n d i v i d u a l tree information. It should however be pointed out that whole stand 159 simulation models can recapture some of the i n d i v i d u a l tree information by incorporating diameter d i s t r i b u t i o n models. This w i l l provide frequency d i s t r i b u t i o n of the trees by diameter classes i n t o which growth or y i e l d can be d i s t r i b u t e d whenever t h i s information i s required. This approach i s more economical from the programming and computational point of view and therefore has been adopted for t h i s study. An example of the models i n t h i s class i s the Douglas-fir managed y i e l d simulator (DFIT) of Bruce, De Mars and Reukema (1977). Stand state i s represented as the number of trees per acre, stand basal area and the mean stand DBH. Height growth i s obtained from a s i t e index equation while stand volume growth (a function of state variables) i s modified by a density dependent f a c t o r , the r a t i o of average stand basal area to the maximum l i m i t i n g basal area of the stand. 3. Y i e l d Model Construction 3.1 E s s e n t i a l Features f o r the Envisaged Growth and Y i e l d Model The overriding objective of th i s study i s to extend the under- standing of the t h e o r e t i c a l and p r a c t i c a l aspects of growth and y i e l d of the three respective species under the c l i m a t i c , edaphic and s i l v i c u l - t u r a l conditions obtaining i n Kenya. Consequently, the three major questions that the envisioned model must answer are: 1. What i s the expected y i e l d under present s i l v i c u l t u r a l practices? 2. What i s the impact of the present s i l v i c u l t u r a l practices on stand development? 160 3. What i s the impact of adopting a l t e r n a t i v e s i l v i c u l t u r a l p ractices on the future development of the stand? Answers to these questions w i l l depend on the quantitative informa- t i o n provided by the growth and y i e l d model. To do t h i s e f f e c t i v e l y , the model must contain the following e s s e n t i a l features: 1. The model should permit s p e c i f i c a t i o n s of the management practices (decision variables) as input. 2. The model output should provide not only t o t a l volume but also volume to d i f f e r e n t merchantable l i m i t s and by size classes. 3. The model should be able to evaluate d i f f e r e n t l e v e l s of stand management. 4. The f i n a l model should be integratable into the o v e r a l l forest planning system. Within the data and resource l i m i t a t i o n s , i t was f e l t that a l l the above features could be b u i l t into an i n t e r a c t i v e stand l e v e l simulation model with s u f f i c i e n t d e t a i l for management, planning and s i l v i c u l t u r a l research purposes. Figure 19 shows the p o s i t i o n of the envisaged model i n the o v e r a l l planning system and also serves to show the sequence of the present research project ( s o l i d l i n e s ) . 3.2 Growth and Y i e l d Model Synthesis The growth and y i e l d functions developed i n Chapter 2 form the bui l d i n g blocks of the y i e l d model EXOTICS; an acronym f o r Exotic Species. These functions were coded as FORTRAN subroutines and then organized i n t o a l o g i c a l sequence (programme) capable of simulating the Figure 191. Overall forest planning system showing the i n t e g r a t i o n of the y i e l d model. PSP Data System K f I I IMPLEMENTATION -> Data Analysis Y i e l d Model Construction HYPOTHESES <- Model Testing & V a l i d a t i o n SIMULATION Y i e l d predications under various management schedules FIELD TRIALS W POLICY FORMULATION <r -ANALYSIS & EVALUATION Economic C r i t e r i a Technical) C r i t e r i a ) B i o l i o g i c a l Design model adopted from Munro (1974). Broken l i n e s indicate the stages of the o v e r a l l sequence not covered i n t h i s study. 162 growth and y i e l d of a si n g l e , even-aged monospecific stand. Figure 20 shows the flow chart of the f i n a l program configuration. Documentation and variable d e s c r i p t i o n for the main and a u x i l i a r y subroutines i s given at the beginning of each subroutine on the program l i s t i n g a v a i l a b l e from the author on request. The following i s a de t a i l e d d e s c r i p t i o n of the main fu n c t i o n a l subprogrammes: MAIN: This programme i n i t i a l i z e s and defines variables which are s p e c i f i c to i t or are common or shared with other sub- programme. I t i s also responsible for control of the sequence i n which various subprogrammes are c a l l e d and b a s i c a l l y d i r e c t s the whole programme including i n i t i a l i z a t i o n , input and output. HDSTDC: Calculates stand dominant height as a function of stand age and s i t e index using equation 2.21 for C_. l u s i t a n i c a and P_. radi a t a and equation 2.22 for P_. patula. Correction for establishment s i t e i s accomplished by deducting 2 meters and (0.156 x Age) meters from dominant height for P_. radi a t a and P_. patula respectively. BACALC: Calculates the stand basal area before f i r s t thinning using equation 2.29 for a l l three species. This subroutine i s only c a l l e d i f i n i t i a l stand basal area i s not provided (indicated by entering a negative value of basal area when using the model) and stand i n i t i a l i z a t i o n age i s 5 years. 163 FIGURE 20 FLOWCHART OF THI YIELD MODEL EXOTICS INPUT VARIABLES 1-C. LUSITANICA I.SPECIES 2'P. RADIATA 3-B-P. PATULA 2.ESTABLISHMENT 1-SHAMBA 2'GRASSLANO 3.A0E-5-40 YEARS 4.SITE INDEX-10-3S 3. STOCKING. 100-1«00 IF AQE>S YEARS OR -1000-1600 IF AGE'S YEARS -VE IF BA. IS UNKNOWN 6.BASAL AREA •VE IF BA. IS SIVEN (AGE MUST BE 9 YEARS IF BA. UNKNOWN) T.THINNING SPECIFICATIONS (OPTIONS) 0- NO THINNING 1- BY NO. OF STEMS TO BE REMOVED 2- BY BASAL AREA TO BE REMOVED 8.CLEARFELLING OPTIONS 1- IF DBH>SPECIFIEO VALUE 2- IF AGE>SPECIFIED VALUE GENERATE INITIAL STAND STATE 1. DOMINANT HEIGHT-F(A.SI,ESTAB.) 2. BA/HA.aF(A.H.N) 3.MEAN STAND 08H-F(BA..N) 4.STAND DENSITY INOEX-F(H.M) IS.STAND VOLUME/HA"F(DBH.H,N) (TOTAL AND MERCHANTABLE) _NO_ SIMULATE THINNING BY SPECIFYING- t.THINNING OPTION 2. NO.STEMS TO BE LEFT OR XBA.TO BE REMOVED 3. THINNING CRITERIA (WHEN THE SPECIFIED AGE.DOMINANT HEIGHT OR BASAL AREA ISREACHEO OREXCEEDEp) CALCULATE THINNING OUTPUT 1.NO.STEMS REMOVED 3.BA. REMOVEO 3.VOLUME REMOVEO TOTAL AND MERCHANTABLE < STORE THINNING OUTPUT CALCULATE RESIDUAL STAND STATE 1 NO. STEMS/HA 2.BA/HA 2 UPDATE STAND STATE 1.STAND DOMINANT HEIGHT 2.STAND DENSITY INDEX 3.STAND BA. VIA BASAL AREA INCREMENT EOUATION: BAI«F(A,BA.) FOR P.RADIATA BAI-F(A.BA.SX) FOR C.LUSITANICA AND P.PATULA 4.STAND DBH STANO VOLUME CALCULATE 1. WEIBULL PARAMETERS 1.X FROM DBH AND STOCKING 3.b FROM DBH AND X 3 C FROM DBH AND X 2. NO.STEMS BY OBH CLASSES USING WEIBULL DISTRIBUTION FUNCTION 3. TOTAL AND MERCHANTABLE VOLUME BY OBH CLASSES O U T P U T R E S U L T S 2.STI» VAlI ( " * " " > " « » T H I N N I N G S ) 164 BABAI: This subroutine updates the stand basal area i n thinned stands using basal area at the beginning of the growth period and age at the end of the growth period (equation 2.30 f o r P̂ . radiata) and S% (equation 2.31 for C_. l u s i t a n i c a and P_. patula) to calculate the basal area increment. This i s then added to the basal area at beginning of growth period to give basal area at end of the growth period. This subroutine i s c a l l e d i f i n i t i a l stand basal area i s known, for example from inventory data. I n i t i a l i z i n g stand simulation at an age >5 years automatically c a l l s t h i s routine and so i n i t i a l stand basal area must be given. It allows f o r stand simulation from any given age, an important feature when v a l i d a t i n g the model using f i e l d data ( c f . p.s.p.s'). VOLCAL: Calculates t o t a l overbark stand volume using tree volume equations 2.42 and 2.43 and the number of stems per hectare for the respective species. It also calculates the merchant- able volume by f i r s t c a l c u l a t i n g the relevant R-factor for the respective species and merchantable l i m i t (equation 2.44). THIN: Controls thinning operations of which there are three options: 0 = No thinning 1 = Thinning based on number of stems to be l e f t when a predetermined age i n t e r v a l or stand dominant height i s equalled or exeeded. 165 2 = Thinning based on proportion of basal area to be removed when a c r i t i c a l predetermined basal area i s equalled or exceeded. This subroutine allows for the use of any one option or a combination of two or a l l three options in one simulation run, thus allowing for a change of management decision (with regard to thinning criteria) at any age within the l i f e of the plantation. THNCAL: This subroutine calculates thinning variables: DBH(T) or DBH(T) N(T) or N(T) BA(T) or BA(T) V(T) Where f[DBH(BT), N(T), N(BT)] i f thinning option = 1 f[DBH(BT), BA(T), BA(BT)] i f thinning option = 2 f[N(BT), N(AT)] i f thinning option f[BA(T), DBH(T)] i f thinning option f[DBH(T), N(T)] i f thinning option a specified proportion of BA(BT) i f thinning option equation 2.27 equation 2.28 1 2 1 = 2 f[DBH(T), H, N(T)] N(T) = No. stems removed i n a thinning/ha N(BT) = No. stems before thinning/ha N(AT) = No. stems after thinning/ha DBH(T) = Mean DBH of thinnings in cm DBH(BT) = Mean DBH of stand before thinning in cm BA(T) = Basal area of thinnings in m2^ha BA(BT) = Basal area of stand before thinning in m2/ha V(T) = Volume of thinnings in m" 3. 166 CHKCLF: Checks i f stand i s due for c l e a r f e l l i n g by checking i f the predetermined c l e a r f e l l age or DBH i s equalled or exceeded. C l e a r f e l l i n g has p r i o r i t y over thinning operation ( c f . Figure 20). CLRFEL: Calculates t o t a l y i e l d at c l e a r f e l l by diameter classes: 1. Calculates Weibull parameters XQ = f (DBH, N) equation 2.39 b = f(DBH, D_) equation 2.40 c = f(DBH, D_) equation 2.41 2. Calculates number of trees per diameter class from t o t a l number of trees/ha and the cumulative d i s t r i b u t i o n function (equation 2.42) using the Weibull parameters calculated above. 3. Calculates volume y i e l d by diameter classes: V_ = f ( D 1 , H, N ±) V£ r e f e r to volume corresponding to diameter class (at breast height) D_» H Is stand dominant height at c l e a r - f e l l and N_ i s number of stems/ha at c l e a r f e l l i n diameter class D_« Input v a r i a b l e s f o r EXOTICS At the beginning of each simulation run, the programme c a l l s f o r several input v a r i a b l e s : State v a r i a b l e s : Species, establishment s i t e , age, s i t e index, stocking and basal area. 167 Decision v a r i a b l e s : Thinning option and control inputs (when to t h i n and how much to remove) and c l e a r f e l l i n g c r i t e r i a . These variables are given on Figure 20, while Table 36 gives t h e i r domain or the range within which t h e i r values should be. In e s t a b l i s h - ing the variable domain, the main guiding f a c t o r was the range of the data used i n developing the various growth and y i e l d functions. It should be noted that the thinning variables entered at the beginning of the simulation run apply to f i r s t thinning only. The programme c a l l s f o r further thinning structions at the end of each thinning. Output from EXOTICS In conformity with the basal area Increment equation whose i n c r e - ment period i s one year, the simulation cycle for EXOTICS Is one year. At each cycle, several main stand state parameters are calculated, namely age, No. stems, dominant height, mean stand DBH, basal area, volume and the stand density Index, S%. At each thinning, the para- meters No. stems, DBH, basal area and volume of thinnings are calcu- l a t e d . Both main stand y i e l d and y i e l d of thinning are combined to give t o t a l stand production: basal area, volume, current annual increment (CAI) and mean annual increment (MAI). A l l these variables are stored i n an array for output at the end of the simulation run. Table 37 (output as Table 1 i n the programme) i s an example using C. l u s i t a n i c a sawtimber regime for s i t e index 20. In addition to Table 1, two other tables are output: Table 2 which gives the merchantable volume to the respective merchantable l i m i t s and Table 3 which gives the main stand f i n a l volume ( t o t a l and merchantable) TABLE 36. Domain of the y i e l d model EXOTICS with respect to input variables Species code 1 2 3 4 5 6 7 8 Establishment code 1 only 1 or 2 1 or 2 1 or 2 1 or 2 1 or 2 1 or 2 1 or 2 Age i n year 5-40 5-35 5-20 5-20 5-20 5-20 5-20 5-12 S i t e index 12-24 21-33 15-27 15-27 15-27 15-27 15-27 21-30 I n i t i a l stocking 1000-1600 1000-1600 1000-1600 1000-1600 1000-1600 1000-1600 1000-1600 1000-1600 Basal area m 10-60 10-60 10-60 10-60 10-60 10-60 10-60 10-60 No. stems to remove 10-50% 10-50% 10-50% 10-50% 10-50% 10-50% 10-50% 10-50% Basal area to remove 10-50% 10-50% 10-60% 10-50% 10-50% 10-50% 10-50% 10-50% Species code: Establishment code: 1 = C. l u s i t a n i c a 1 = Shamba 2 = P. radiata 2 = Grassland 3 = P_. patula Nabkoi 4 = P. patula Nanyuki 5 = P_. patula Elburgon 6 = P. patula Kiandongoro 7 = P_. patula Kinale 8 = P_. patula Turbo No. stems and basal area to remove are given as percent of values before thinning. T a b l e 37 EXAMPLE OF OUTPUT FROM EXOTICS TABLE 1 TOTAL VOLUME YIELD TABLE SPECIES: CUPRESSUS LUSITANICA INITIAL SITE INDEX: 20.0 ESTABLISHMENT: SHAMBA STANDING CROP THINNING TOTAL Pf JODUCTIOf' •1 AGE NO. OF HDOM DBH(1) BA(1) V(1) S% NO. OF DBH(2) BA(2) V(2) BA(3) V(3) CAI MAI STEMS STEMS CU.M YEARS M CM SO M cu M CM SO.M CU.M SO M CU M cu M 5.0 1200 0 7 8 10 4 10 3 32 9 37 1 10 3 32 9 0 0 6.6 G.0 1200 0 9 2 12 5 14 6 61 6 31 3 14 6 61 6 28 7 10.3 7.0 1200 0 10 6 14 2 19 1 94 7 27 2 19 1 94 7 33 1 13.5 8.0 1200 0 12 0 15 8 23 4 130 1 24 2 312.0 13.0 4. 1 21.5 23 4 130 1 35 4 16.3 9.0 888 0 13 2 18 1 22 9 140 6 25 3 27 0 162 1 32 0 18.0 10.0 888 0 14 5 19 4 26 3 173 3 23 2 30 4 194 8 32 7 19.5 11 .0 888 0 15 7 20 5 29 4 206 3 21 4 33 5 227 9 33 1 20.7 12.0 888 0 16 8 21 5 32 3 239 6 20 0 36 4 261 1 33 2 21.8 13.0 888 0 17 9 22 4 35 1 272 9 18 7 355.0 20.0 11.2 86.4 39 2 294 5 33 3 22.7 14.0 533 0 19 0 25 0 26 2 213 7 22 8 41 5 321 7 27 2 23.0 15.0 533 0 20 0 26 1 28 4 241 2 21 7 43 7 349 2 27 5 23.3 16.0 533 0 21 0 27 0 30 6 268 9 20 6 45 9 376 9 27 7 23.6 17 .0 533 0 21 9 27 9 32 6 296 7 19 8 47 9 404 6 27 8 23.8 18.0 533 0 22 8 28 7 34 6 324 4 19 0 178.0 25.7 9.2 86.5 49 9 432 4 27 8 24.0 19.0 355 0 23 7 31 2 27 1 261 6 22 4 51 6 456 1 23 7 24.0 20.0 355 O 24 5 32 1 28 8 285 4 21 7 53 3 479 9 23 8 24.0 21 .0 355 0 25 3 33 1 30 5 309 3 21 0 55 0 503 8 23 9 24.0 22.0 355 0 26 1 33 9 32 1 333 3 20 4 56 6 527 7 23 9 24 .0 23.0 355 0 26 8 34 7 33 6 357 3 19 8 89.0 31.0 6 . 7 71.2 58 2 551 7 24 0 24.0 24.0 266 0 27 5 36 9 28 .4 307 5 22 3 59 6 573 1 21 4 23.9 25.0 266 0 28 2 37 8 29 8 329 0 21 8 61 0 594 7 21 5 23.8 26.0 266 0 28 8 38 7 31 2 350 6 21 3 62 4 616 3 21 6 . 23.7 27.0 266 0 29 5 39 5 32 6 372 2 20 8 63 8 637 8 21 6 23.6 28.0 266 0 30 0 40 3 34 0 393 8 20 4 65 2 659 4 21 6 23.6 29.0 266 0 30 6 41 1 35 3 415 3 20 0 66 5 681 0 21 5 23.5 30.0 266 0 31 2 41 9 36 6 436 8 19 7 67 8 702 5 21 5 23.4 31 .O 266 0 31 7 42 6 37 9 458 3 19 3 69 1 723 9 21 4 23.4 32.0 266 0 32 2 43 3 39 2 479 7 19 0 70 4 745 3 21 4 23.3 33.0 266 0 32 7 44 0 40 4 501 0 18 8 71 7 766 6 21 3 23.2 34 .0 266 0 33 1 44 7 41 7 522 2 18 5 72 9 787 8 21 2 23.2 35.0 266 o 33 6 45 3 42 9 543 3 18 3 74 1 808 9 21 1 23. 1 T a b l e 38 EXAMPLE OF OUTPUT FROM EXOTICS TABLE 2 MERCHANTABLE VOLUME YIELD TABLE SPECIES: CUPRESSUS LUSITANICA INITIAL SITE INDEX: 20 .0 ESTABLISHMENT: SHAMBA AGE V( 15) CU.M V(20) CU.M YEARS MAIN THINNING TOTAL MAIN THINNING TOTAL 5.0 0 O 0 0 6 . 0 0 0 0 0 7.0 4 91 4 91 8 .0 35 10 0 . 0 35 10 9 .0 72 OS 72 03 10.0 105 29 105 29 1 1 .0 139 28 139 28 12.0 173 63 173 63 13.0 208 1 1 55.82 208 1 1 14.0 180 44 236 27 15.0 209 37 265 20 16.0 238 31 294 13 17.0 267 20 323 02 18.0 295 99 74 .35 351 82 19.0 245 44 375 61 20 .0 269 94 400 1 1 2 1 . 0 294 42 424 59 22 .0 318 85 449 03 2 3 . 0 343 22 66 .65 473 39 2 4 . 0 297 93 494 76 25 .0 319 68 516 51 2 6 . 0 341 41 538 23 27 .0 363 10 559 93 28 .0 384 74 581 57 29 .0 406 33 603 15 30 .0 427 83 624 66 31 .0 449 26 646 09 32 .0 470 59 667 42 33 .0 491 83 688 66 34 .0 512 95 709 78 35 .0 533 97 730 80 T a b l e 39 EXAMPLE OF OUTPUT FROM EXOTICS TABLE 3 STAND TABLE AT CLEARFELL SPECIES: CUPRESSUS LUSITANICA INITIAL SITE INDEX: ESTABLISHMENT: SHAMBA DIAMETER NO. OF V(1) V( 15) V (20) CLASS TREES CM CU.M CU.M CU.M 33 .0 9 .00 9.81 9 .33 36 .0 26 .00 33 .66 32.51 39 .0 39 .00 59. 15 57 .64 42 .0 4 5 . 0 0 79.03 77.42 4 5 . 0 43 .00 86.57 85 .06 4 8 . 0 36 .00 82 .35 81 .06 5 1 . 0 27 .00 69.64 68 .62 54 .0 18.00 51 .99 51 .26 57 .0 1 1 .00 35.36 34 .88 6 0 . 0 6 . 0 0 21 . 35 2 1 .07 6 3 . 0 3 .00 1 1 .76 1 1 .60 6 6 . 0 1 .00 4 .30 4 .24 TOTALS 1 264.001 544.95 I 534.71 I I FINAL AGE AT CLEARFELL= 172 d i s t r i b u t e d by diameter classes. Table 38 and Table 39 are examples from the same simulation run as for Table 37. After these outputs, the programme returns to the beginning f o r further i n s t r u c t i o n : to stop or to begin another simulation. As expected, the f i n a l number of stems i n Table 37 i s the same as those on Table 39, except for a rounding-off error of +2 stems. S i m i l a r l y , we would expect that the f i n a l main stand volume on Table 37 should be the same as t o t a l volume ( V ( l ) ) i n Table 39. Also the f i n a l volume V(15) and V(20) of Table 38 should be the same as V(15) and V(20) r e s p e c t i v e l y of Table 39. However, as Tables 37 and 39 show, t h i s may not always be the case. The main cause of the discrepancy i s that the volume as calculated i n Table 37 i s based on the DBH of tree of mean basal area and stand dominant height and thus assumes a normal d i s t r i b u - t i o n of the trees by basal area. However, the volume as calculated for Table 39 w i l l depend very much on the d i s t r i b u t i o n of the trees by diameter classes. Thus, d i s t r i b u t i o n s skewed to the l e f t w i l l r e s u l t i n lower t o t a l volume and those skewed to the right w i l l r e s u l t i n higher t o t a l volume i n Table 39 compared to the f i n a l main stand volume ( V ( l ) ) of Table 37. One possible s o l u t i o n to t h i s problem would be to calculate volumes at each cycle as the sum of the volumes calculated through the diameter d i s t r i b u t i o n function. However, i t was f e l t that for t h i s study, the diameter d i s t r i b u t i o n data did not cover the young stand ages adequately, e s p e c i a l l y before f i r s t thinning and so the diameter d i s t r i b u t i o n function would not be v a l i d i n unthinned stands. At the moment, the discrepancy appears to be of the order of not more than +5% and so can be considered i n s i g n i f i c a n t for p r a c t i c a l purposes. 173 4.0 Model V a l i d a t i o n 4.1 Introduction Model v a l i d a t i o n i s the process of buil d i n g an acceptable l e v e l of confidence that an inference about a simulated process i s a correct or v a l i d inference about a simulated process (Van Horn 1968). It i s an e s s e n t i a l accompaniment for any simulation model as a test that both the component parts of the model and the performance of the model as a whole are i n agreement with the behaviour of the r e a l system. I f the model f a i l s to pass t h i s t e s t , then changes must be made i n either the va r i a b l e s , parameters estimates, or the structure of the model. This process serves two purposes: 1. It builds confidence of prospective model users i n the model. 2. It helps delineate the l i m i t s to model v a l i d i t y . The knowledge of the l i m i t s to model v a l i d i t y , e s p e c i a l l y with respect to accuracy and p r e c i s i o n i s very important to the user when choosing between a l t e r n a t i v e models. The problems and procedures for model v a l i d a t i o n have been discussed by several researchers Including Naylor and Finger (1967) and Van Horn (1968) with a d e t a i l e d summary by Goulding (1972). Naylor and Finger (1967) suggested a three stage approach: 1. Construct a set of hypotheses and postulates for the process using a l l available information: observations, general knowledge, relevant theory, and i n t u i t i o n . 2. Attempt to v e r i f y the assumptions of the model by subjecting them to empirical t e s t i n g . 174 3. Compare the input/output transformations generated by the model to those generated by the r e a l system. Steps one and two of t h i s approach e n t a i l d e t a i l e d t e s t i n g of i n d i v i d u a l model components and the underlying assumptions while the t h i r d step validates the performance of the whole model. The order of t e s t i n g was j u s t i f i e d on the grounds that t e s t i n g of assumptions and components (before synthesis of the model) i s cheaper than t e s t i n g the p r e d i c t i o n s . Van Horn (1968) summed up the problems of v a l i d a t i n g simulations as s i m i l a r to the standard problems of empirical research: 1. Small samples due to high cost of data. 2. Too aggregate data. 3. Data whose own v a l i d i t y i s questionable. He suggested the following possible v a l i d a t i o n actions - i n rough order of decreasing value-cost r a t i o : 1. Find models with high face v a l i d i t y . 2. Make use of e x i s t i n g research, experience, observations and any other available knowledge to supplement models. 3. Conduct simple empirical tests of means, variances, and d i s t r i b u t i o n s using available data. 4. Run "Turing" type t e s t s . 5. Apply complex s t a t i s t i c a l tests on available data e.g. spectral a n a l y s i s , T h e i l ' s i n e q u a l i t y c o e f f i c i e n t (Naylor and Finger (1967)). 175 6. Engage i n s p e c i a l data c o l l e c t i o n . 7. Run prototype and f i e l d t e s t s . 8. Implement the r e s u l t s with l i t t l e or no v a l i d a t i o n . The appropriate v a l i d a t i o n action w i l l depend on several factors including a v a i l a b i l i t y of data (including available time and funds) and the type of model. For example, complex s t a t i s t i c a l tests may be appropriate for stochastic models but not for deterministic models. I t should be noted here that a l l the s t a t i s t i c a l tests i n the v a l i d a t i o n process constitute n u l l hypotheses. Acceptance of the n u l l hypothesis w i l l not be a "proof" that the model i s correct but simply indicates acceptance of the model as an acceptable approximation of the simulated system at the required l e v e l of d e t a i l . V a l i d a t i o n of Forest Growth and Y i e l d Models The v a l i d a t i o n of most growth and y i e l d models i n for e s t r y appears to follow c l o s e l y the approach proposed by Naylor and Finger (1967). However, l e v e l of model accuracy and p r e c i s i o n has i n general received l i t t l e a t tention i n s p i t e of concern raised by some researchers including Munro (1974), Row and Norcross (1978) and Smith and Williams (1980). This may be a t t r i b u t e d to among others: 1. For management and planning purposes (the object of most growth and y i e l d models), the required l e v e l of accuracy i s often not very high so that modellers have tended to downplay the process of model v a l i d a t i o n . 176 2. Most growth and y i e l d models are deterministic so that v a l i d a t i o n has been r e s t r i c t e d to simple tests of comparison of outputs from the model with the r e a l system. 3. F i e l d t e s t s , which should be e s s e n t i a l for any given model, are u s u a l l y very expensive and time consuming and so are often not conducted. With respect to the t h i r d problem, the general trend has been to use long-terra study samples or permanent sample plots data not used i n the construction of the model to test how well the model performs. This approach was advocated by Munro (1974a) and has been e f f e c t i v e l y employed by Ek and Mouserud (1979), Moser et a l . (1979) and Alder (1977). Ek and Mouserud (1979) used remeasured plot data to compare the performance of two models, FOREST and SHAFT (both c a l i b r a t e d f o r northern hardwood stands i n Wisconsin) with respect to accuracy and p r e c i s i o n . They found the former to be more accurate. Moser et a l . (1979) used long-term study (cutting cycle) data to validate a simula- t i o n model of uneven-aged northern hardwoods f o r e s t . They observed that basal area and volume were more accurately predicted than number of stems although the accuracy of the l a t t e r was acceptable for management and planning purposes. Alder (1977) used psp data to v a l i d a t e PYMOD although i t was not c l e a r whether the data had been used i n the con- s t r u c t i o n of the model. A more pressing concern i n growth and y i e l d model v a l i d a t i o n i s the length of period over which the test plots are remeasured. Most simula- t i o n models are designed to "grow" the stand f or the whole r o t a t i o n . 177 Unfortunately, any one test plot u sually covers only a f r a c t i o n of the r o t a t i o n period. It i s therefore obvious that tests of accuracy and p r e c i s i o n based on data from these short period test p l o t s cannot be a true r e f l e c t i o n of the model accuracy. Put i n other words, the length of the simulation, run governs the p r e c i s i o n of estimates obtained by simulation j u s t as sample siz e determines the p r e c i s i o n of r e a l - l i f e experiment. This problem i s demonstrated on Figure 21 from Alder (1977), which shows two simulations of thinning experiment 345 i n Tanzania (P_. patula) from d i f f e r e n t s t a r t i n g conditions. The figure indicates that i n general, the t o t a l volume simulated from the actual diameter d i s t r i b u t i o n at age 8 years i s closer to the actual measured volume than the t o t a l volume simulated from age zero. Thus, v a l i d a t i o n of the model using the experimental plots between ages 8 to 15 does not t e l l the true picture of model v a l i d i t y between ages 0 to 15. This fi g u r e i l l u s t r a t e s i n general the problem of v a l i d a t i n g the growth and y i e l d model using permanent sample plot data or other data whose remeasurement period covers only a f r a c t i o n of the rotation period over which the model w i l l be used. Any quoted measures of accuracy or p r e c i s i o n w i l l appear much better than they a c t u a l l y are when the model i s used for long-term p r e d i c t i o n s . This problem w i l l be p a r t l y resolved when permanent sample plots covering the whole r o t a t i o n become a v a i l a b l e . 4.2 V a l i d a t i n g EXOTICS Using Independent Permanent Sample Plot Data The data available for model v a l i d a t i o n i n t h i s study consisted of 20 permanent sample plots for each species (see Chapter 1 Section 6) and F i g u r e 2 1 t Two S i m u l a t i o n s o f Th inn ing Experiment 3 U 5 i n Tanzan ia ( P . p a t u l a ) from d i f f e r e n t s t a r t i n g c o n d i t i o n s . (from Alder 1977) T o t a l Volume ( m 3 A a ) 7 0 0 n « A c t u a l measurements S imu la ted from age 0 600 i 5 0 0 H 3 0 0 i 2 0 0 H 1 0 0 S i m u l a t e d from a c t u a l d iameter d i s t r i b u t i o n a t age 8 Treatments A 1 3 9 0 s t e m s A a B 9 8 7 C 6 9 U D 3 U 7 E 1 2 0 T~ 8 ~ 1 — 1 0 —1— 1 2 1 1 * Age from P l a n t i n g ( years ) 179 therefore constituted an independent set of data coming from the same population as those used to construct the model. The data consisted of the basic v a r i a b l e s : age, dominant height, basal area and t o t a l over- bark volume, with some plots having received one or more thinnings during t h e i r remeasurement period. The i n i t i a l remeasurement age for most test plots was d i f f e r e n t from 5 years, the i n i t i a l i z a t i o n age for the model. This presented problems when simulating basal area since p r i o r stand state was not known. For these p l o t s , the i n i t i a l observed basal area was assumed to be the same as the predicted basal area and the simulation started on the second year of measurement. Where plot stocking was less than 1000 stems per hectare at age 5 years, thinning was assumed to have already been c a r r i e d out and simulation done as for plots i n i t i a l i z e d at ages other than 5 years. For each p l o t , the stand dominant height, basal area, and the basal area removed i n thinning ( i f any), a l l corresponding to the observed measurements were simulated and from these, the corres- ponding stand t o t a l volume overbark was calculated. Figure 22 shows the r e s u l t s of t h i s procedure f o r two C_. l u s i t a n i c a p l o t s , one i n i t i a t e d at age 5.5 years, the other at age 21.5 years. Of the many output variables from the simulation model, t o t a l volume overbark was selected as the p r i n c i p l e variable to be validated since i t i s the primary i n t e r e s t i n forest management. Several s t a t i s - t i c s were computed for comparing the observed and simulated volume and are presented on Tables 40, 41 and 42 for C_. l u s i t a n i c a , P_. patula and P. rad i a t a r e s p e c t i v e l y . These s t a t i s t i c s were computed as follows: FIGURE 22 i i i • i i i i i i i i 8 10 12 14 16 18 20 22 24 26 28 30 32 34 AGE IN YEARS FROM PLANTING TABLE 40: Beta weights, mean bias, standard deviation and the 95% confidence l i m i t s of percentage differences between observed and simulated t o t a l volume overbark, and the Chi-square values f o r three hypothesized le v e l s of accuracy. C. l u s i t a n i c a Plot No. N Beta-weight % Mean bias % Standard deviation % 95% confidence l i m i t s 15 X value 20 25 4 10 -0.84* -0.40 4.42 3.16 3.41 1.92 1.23 37 12 0.64* -6.42 5.03 3.20 17.09 9.61 6.15 54 12 0.27 -1.25 2.25 1.43 1.33 0.75 0.48 116 5 0.89* -0.28 4.16 5.17 1.27 0.72 0.46 117 11 0.95* -5.30 9.12 3.85 27.65* 15.55 9.95 121 11 -0.88* -7.65 8.10 5.37 30.86* 17.36 11.11 181 11 -9.60* 1.32 4.83 3.24 4.20 2.36 1.51 190 11 0.55 4.92 5.44 3.66 8.10 4.56 2.92 202 12 -0.94* -18.18 9.95 6.32 164.24* 92.38* 59.12* 233 13 -0.77* -9.39 8.57 5.18 56.53 31.80* 20.35 246 10 -0.85* -7.76 3.51 2.51 15.11 8.50 5.44 261 10 0.50 8.57 5.08 3.63 13.56 7.63 4.88 279 11 0.42 17.83 8.40 5.64 47.86 26.92* 17.23 288 11 0.95* -1.39 4.34 2.92 3.86 2.17 1.39 295 10 -0.36 3.03 8.05 5.76 7.87 4.43 2.84 331 9 -0.88* -4.65 5.31 4.08 8.83 4.97 3.18 336 9 -0.95* -4.21 2.40 1.85 3.97 2.23 1.43 348 10 0.88* 2.32 13.93 9.97 33.73* 18.97* 12.14 379 9 0.60 0.36 3.98 3.06 2.09 1.18 0.75 388 7 -0.68 -14.63 12.11 11.20 66.02* 37.14* 23.77* B = -2.21 S.D.= 7.93 6.36 291.1* 186.335 Chi-square value at the .05 p r o b a b i l i t y l e v e l of 204 degrees of freedom = 238.04. TABLE 41: Beta weights, mean bias, standard deviation and the 95% confidence l i m i t s of percentage differences between observed and simulated t o t a l volume overbark, and the Chi-square values for three hypothesized le v e l s of accuracy. P. patula Standard 95% X 2 value Plot No. N Beta-weight Mean bias deviation confidence % % % l i m i t s 15 20 25 2 9 0.91* 2.46 19.83 15.24 101.65 57.18* 36.60* 12 11 -0.02 -1.21 3.39 2.28 2.32 1.30 0.83 34 12 -0.72* 5.37 14.20 9.02 26.20* 14.74 9.43 59 11 0.77* 22.68 13.47 9.05 74.44* 41.87* 26.80* 123 10 -0.92* -15.42 14.87 10.63 106.95* 60.16* 38.50* 126 10 -0.02 -1.01 4.19 3.03 3.24 1.82 1.16 144 8 -0.45 0.54 6.86 5.74 5.13 2.89 1.85 147 12 -0.90* -5.40 15.22 9.67 58.93 33.15 21.22* 154 11 -0.18 -8.10 4.71 3.16 20.71 11.65 7.46 167 12 -0.86* -22.82 8.40 5.33 232.31* 130.67* 83.63* 203 12 -0.68* 0.04 11.84 7.52 25.30* 14.23 9.11 209 13 -0.86* -9.51 11.99 7.24 69.24 38.95* 24.93* 252 11 -0.82* 12.84 10.61 7.13 33.91* 19.08* 12.21 270 11 0.64 3.88 2.56 1.72 3.55 2.00 1.28 276 11 -0.25 3.31 6.53 4.39 7.87 4.42 2.83 312 10 -0.71* -8.53 4.25 3.04 19.63* 11.04 7.06 315 10 0.24 -0.88 4.46 3.19 3.39 1.91 1.22 324 9 -0.89* -13.20 3.96 3.05 40.00* 22.52* 14.41 342 10 -0.84* 4.68 3.97 2.84 4.20 2.36 1.51 391 8 -0.81* -0.28 11.43 9.56 15.05 8.46 5.41 B = -1.53 S.D.= 10.02 10.10 307^33* Chi-square value at the .05 p r o b a b i l i t y l e v e l of 207 degrees of freedom = 241.28. TABLE 42: Beta weights, mean bias, standard deviation and the 95% confidence l i m i t s of percentage differences between observed and simulated t o t a l volume overbark, and the Chi-square values for three hypothesized le v e l s of accuracy. P. radiata Standard 95% X 2 value Plot No. N Beta-weight Mean bias deviation confidence % % % l i m i t s 15 20 25 6 14 0.48 -7.17 11.24 6.49 57.91* 32.57* 20.84 18 11 0.44 2.58 3.55 2.38 3.07 1.73 1.11 31 11 0.63* 11.92 12.00 8.06 33.08* 18.61 11.91 91 6 -0.88* -7.67 5.62 5.89 11.13 6.26 4.01 96 13 -0.84* 14.29 18.50 11.18 61.36* 34.51* 22.09 99 13 -0.93* 6.88 20.16 12.18 64.69* 36.39* 23.29* 103 9 0.34 6.52 3.62 2.78 7.06 3.97 2.54 112 9 0.44 -0.10 2.22 1.71 0.70 0.40 0.26 134 8 0.72* -2.71 5.22 4.37 5.00 2.81 1.80 138 12 -0.48 20.22 6.45 4.10 60.29* 33.91* 21.70* 164 11 0.59 3.69 10.21 6.86 16.52 9.29 5.94 177 10 0.25 -8.19 3.10 2.22 16.06 9.03 5.78 238 13 -0.46 -8.43 10.56 6.38 53.75* 30.23* 19.35 256 10 -0.94* -1.31 5.39 3.85 5.30 2.98 1.91 289 8 0.36 -2.13 3.73 3.12 2.54 1.43 0.92 340 8 -0.62 -2.00 3.69 3.08 2.42 1.36 0.87 373 9 0.19 -1.22 3.40 2.62 1.99 1.12 0.72 383 9 0.44 16.07 8.19 6.30 34.50* 19.41* 12.42 400 7 0.77* 8.86 6.42 5.94 10.49 5.90 .3.78 402 8 -0.68 -3.32 3.84 3.21 3.77 2.12 1,36 B = 2.34 S.D. = 8.50 10.14 254.02* 162.58 Chi-square value at the .05 p r o b a b i l i t y l e v e l of 199 degrees of freedom = 232.64. 184 Beta-weight %: were obtained from the standardized regression of the r e s i d u a l s (express as percent of simulated value): E = -V i . 100 3.3 V s Where E = Residual corresponding to each remeasurement V Q = Observed volume V S = Simulated volume on remeasurement age. The p o s i t i v e values indicate p o s i t i v e c o r r e l a t i o n of residuals with respect to age and vice versa for negative values, while the magnitude of the beta weight indicate the degree of c o r r e l a - t i o n . I d e a l l y , the c o r r e l a t i o n should be zero. S i g n i f i c a n t c o r r e l a - tions at the .05 p r o b a b i l i t y l e v e l are shown i n asterisks on Tables 40, 41 and 42. Mean bias %: t h i s was calculated as the mean res i d u a l for each p l o t : B = • ? E ±/n 3.4 Where E = As i n 3.3 above n = T o t a l number of remeasurements per p l o t . Standard deviation %: computed as the standard deviation of the residuals i n percent: 185 s.d. = £ (E. - B) 2/n - 1 3.5 1=1 1 Where E = As i n 3.3 B = As i n 3.4 s.d. = Standard deviation. 95% confidence l i m i t s : computed as: C * L * = t(.05, n-1)' S d 3 , 6 Where C L . - Confidence l i m i t s S.j = Standard error of the residuals i n percent for each p l o t . Chi-square values: corresponding to three hypothesized l e v e l s of accuracy: 15, 20 and 25%. These were computed using Freese's 1960 test of accuracy: X2, . \ » • - V 2 3.7 ("'P) i-1 a2 2 E i p 2 V 0 2 Where a z = _ 1 = 2 2 1.96Z 196^ P = Hypothesized percent of the true value unless a l-in-20 chance has occurred E^ = Specified allowable error as percent of the true (observed) value V Q , and V G are as above. 186 The calculated Chi-square value for each plot was compared with the c r i t i c a l Chi-square value f o r .05 p r o b a b i l i t y l e v e l and n-degrees of freedom. I f the calculated Chi-square value exceeded the c r i t i c a l value, then the simulated volume did not meet our accuracy requirement and the n u l l hypothesis of a common d i s t r i b u t i o n of the observed and simulated volumes rejected. These are shown i n aster i s k s on Tables 40, 41 and 42. Results and Discussion 1. Beta-weight t e s t : The expected c o r r e l a t i o n between the residuals and age i s zero. However, due to both random and systematic e r r o r s , both p o s i t i v e and negative correlations are expected. The true value of t h i s test therefore i s to detect systematic trends i n the model by revealing i f there i s a preponderance of e i t h e r p o s i t i v e or negative signs and i f these are s i g n i f i c a n t at the .05 p r o b a b i l i t y l e v e l . Table 40 f o r C_. l u s i t a n i c a shows that there were 10 p o s i t i v e and 10 negative beta-weights, an i n d i c a t i o n of lack of bias. Of the 13 plots whose beta-weights were s i g n i f i c a n t , 5 were p o s i t i v e and 8 negative, which would i n d i c a t e a s l i g h t tendency to overestimate. Table 41 for P_. patula i n d i c a t e that 16 pl o t s had negative c o r r e l a t i o n s and only 4 were p o s i t i v e i n d i c a t i n g overestimation by the model. Of the 13 pl o t s with s i g n i f i c a n t c o r r e l a t i o n s , only 2 were p o s i t i v e , again pointing to overestimation. 187 Table 42 f o r P_. radi a t a indicates 8 negative and 12 p o s i t i v e beta- weights, a s l i g h t underestimate. Of the 7 plots with s i g n i f i c a n t c o r r e l a t i o n s , 3 were p o s i t i v e and 4 negative, i n d i c a t i n g lack of bias i n the model. From t h i s t e s t , the models f or C_. l u s i t a n i c a and P_. ra d i a t a i n d i - cate no apparent systematic trends while that f o r P_. patula point to a possible tendence f o r the residuals to increase with age. 2. Mean bias t e s t : The mean bias measured as percent of the predicted value i s i d e a l l y expected to be zero. For reasons mentioned above, however, t h i s s i t u a t i o n cannot be r e a l i z e d and the best one can hope f or i s that over a large number of simulations, the o v e r a l l mean bias w i l l be zero or close to zero. Table 40 f o r C_. l u s i t a n i c a indicates plot mean bias ranging from +0.3% to +18% with a mean of -2.21%. Table 41 f o r P̂ . patula shows a range of ±0.5 to ±23% with a mean of -1.53% while Table 42 for P. ra d i a t a shows a range of ±0.1% to ±20% with a mean of 2.34%. For a l l three species, the mean bias can be considered as n e g l i g i b l e . It i s worth noting that i n spite of the higher v a r i a b i l i t y i n the mean bias for P. patula (measured by the range), i t has a s l i g h t l y lower mean bias than the other two species. I t i s also worth noting that plot No. 59 and 167 (P. patula) represents o u t l i e r s . However, the two compensate each other so that t h e i r e f f e c t on the mean bias i s almost n i l . 188 3. Standard deviation and standard error of the residuals as percentage of simulated volume: The standard deviation f o r the £. l u s i t a n i c a plots i s f a i r l y homogeneous, ranging between 2 to 14% with a standard error of 7.93%. The v a r i a b i l i t y f o r V_. patula and P_. ra d i a t a i s higher, ranging between 1 to 20% with a standard error of 10% and 8.50% r e s p e c t i v e l y . Thus, the true difference between the observed and simulated volume w i l l i n general l i e within ±16% for C_. l u s i t a n i c a , +20% f o r P_. patula and +17% for _P. ra d i a t a unless l-in-20 chance occurs. This i s an improvement on VYTL2 whose 95% confidence l i m i t s ranged between 40% underestimation and 20% overestimation (Alder 1978). 4. The i n d i v i d u a l plot 95% confidence l i m i t s : These indi c a t e the range within which the true difference between the observed and simulated volume would be expected to be unless a l-in-20 chance occurred-just as the standard error i n 3 above. Thus, i f the r e s u l t s of 3 above are c o r r e c t , no more than one plot would be expected to have a 95% confidence l i m i t greater than twice the standard error calculated i n 3. As i t turned out none of the three species had any plot with 95% confidence l i m i t s greater than twice the standard e r r o r , which could be considered very s a t i s f a c t o r y r e s u l t considering only twenty sample plots were a v a i l a b l e . 5. Chi-square test of accuracy: Freese (1960) discussed the use of both the t - t e s t and the C h i - square test for tests of accuracy. In regard to the t-test he concluded that i t i s not suitable as i t uses one form of accuracy (precision) to 189 test for the other form (freedom from b i a s ) , frequently with anomalous r e s u l t s . He therefore recommended use of Chi-square test as i t w i l l r e j e c t inaccurate r e s u l t s , regardless of the source of inaccuracy. To use the test as proposed by Freese (1960), three statements are required: 1. Statement of accuracy required 2. A measure of accuracy attained 3. An objective method of deciding whether the accuracy attained i s equal to the accuracy required. For t h i s study, the statement of the required accuracy consisted of three hypothesized l e v e l s of accuracy: 15, 20 and 25% of the observed values, while the observed accuracy consisted of the calculated C h i - square values by equation 3.7. These are shown i n the respective table for each species. The a s t e r i s k s i n d i c a t e plots where the simulated volume did not meet the required l e v e l of accuracy. This conclusion was reached by comparing the calculated Chi-square value with the c r i t i c a l Chi-square value for n-degrees of freedom at .05 p r o b a b i l i t y l e v e l . For the o v e r a l l model, a l l the measurements for each species were considered together and the o v e r a l l Chi-square value calculated. This was compared with the c r i t i c a l Chi-square value approximated by: x 2 ( V ) d f = ° * 8 5 3 + V + 1.645 7 2 V - 1 3.8 (From Freese (1969)) Where V = Degrees of freedom. 190 Table 40 f o r C. l u s i t a n i c a indicates that at a 15% l e v e l of accuracy, the model gave 7 unacceptable r e s u l t s out of 20. This i s unacceptable at .05 p r o b a b i l i t y l e v e l . S i m i l a r l y , a 20% l e v e l of accuracy gave 5 out of 20 unacceptable r e s u l t s , which again i s unaccept- able. However, at 25% l e v e l of accuracy only two plots were unaccept- able and the o v e r a l l model i s accepted as meeting the stated l e v e l of accuracy. Thus, the true l e v e l of accuracy f or the C_. l u s i t a n i c a model l i e s somewhere between 20 and 25%. These r e s u l t s apply also to the P. r a d i a t a model as shown on Table 42. It can therefore be stated that barring a l-ln-20 chance, the models for these two species are accurate i f the required l e v e l of accuracy i s 20% or l e s s . The model for P. patula (Table 41) on the other hand had a s l i g h t l y lower l e v e l of accuracy compared to the other two species. At 15% l e v e l of accuracy, 11 plots out of 20 were unacceptable, 8 were unacceptable at 20% and 6 were unacceptable at 25%. The o v e r a l l model was unaccept- able at the required accuracy l e v e l of 25% but was acceptable at 30% (X 2 = 213.42 compared to c r i t i c a l X 2 = 241.28). Thus, the true accuracy l e v e l for t h i s species at .05 p r o b a b i l i t y l e v e l l i e s somewhere between 25 and 30%. This lower l e v e l of accuracy i s seen from the table to be associated with the larger biases and/or standard deviations. The above l e v e l s of accuracy are comparable with the accuracy of two models already i n operation - FOREST and SHAFT (Ek and Monserud 1979). For example both models were found to predict basal area with approximately a l-in-20 chance of a 20% or greater erro r . Number of stems for trees with DBH > 12.7 cm were predicted with 22 and 39% or 191 greater e r r o r by FOREST AND SHAFT res p e c t i v e l y . It should be noted here that FOREST i s a s i n g l e - t r e e , distance dependent model while SHAFT i s a whole stand, diameter d i s t r i b u t i o n model. EXOTICS i s a whole stand diameter free model although diameter d i s t r i b u t i o n i s av a i l a b l e i n the f i n a l output. It i s therefore of i n t e r e s t to note how d i f f e r e n t types of models can have claim to the same l e v e l of accuracy. For purposes of a p p l i c a t i o n i t should be noted that the l e v e l s of accuracy calculated above for EXOTICS are based on a .05 p r o b a b i l i t y l e v e l . I f a user i s prepared to tolerate lower l e v e l s , the models appear acceptable down to 15% acceptable error f o r £. l u s i t a n i c a and P_. r a d i a t a and 20% for P. patula. Indeed Ek and Monserud (1979) considered FOREST sui t a b l e for development of management guides and analysis of s i l v i c u l t u r a l a l t e r n a t i v e s i n d e t a i l at these l e v e l s of accuracy. It i s therefore conceivable that EXOTICS w i l l be a very u s e f u l t o o l for that purpose, i n addition to y i e l d predictions for management and planning purposes. 6. Sources of E r r o r s : The large bias and standard deviation exhibited by some of the plot s on Tables 40, 41 and 42 could have arisen from three possible sources: 1. From model components. 2. Bias from age class or s i t e index d i s t r i b u t i o n . 3. Errors from exogenous f a c t o r s . In EXOTICS, two major components could give r i s e to er r o r s : dominant height and basal area functions. The biases f o r 192 dominant height and basal area (as percent of predicted values) f o r a l l three species are shown on Table 43. In general, the o v e r a l l mean bias f o r both dominant height and basal area was almost n e g l i g i b l e f o r a l l species, except the mean bias f o r P_. radiata basal area with an under- estimate of 2.74%. This may be the cause of the volume underestimate of 2.34% on Table 42. It should be noted here that the mean bias from volume i n t h i s case i s lower than mean bias from basal area because the mean bias from dominant height i s negative. In general the error i n volume i s approximately the sum of the component errors. For a l l species, the plots height biases are very low, ranging from almost zero to 5%. This i s confirmed by the low standard error of estimate: 1.38, 3.07 and 1.04 for C_. l u s i t a n i c a , P. patula and P_. ra d i a t a r e s p e c t i v e l y . Plot No. 324 for P_. patula appears to be an o u t l i e r with a -10.45% bias. This was because the plot age extended to 28 years which i s beyond the range covered by the height over age data f o r P_. patula. Excluding t h i s plot gave a standard error of 2.12%, which i s s t i l l higher than that for the other two species. The p l o t basal area biases on the other hand are very variable as indicated by the standard error of estimate: 7.94, 6.66 and 8.52 f o r C_. l u s i t a n i c a , P_. patula and P. radiata r e s p e c t i v e l y . It therefore appears that nearly a l l of the v a r i a b i l i t y i n the _C. l u s i t a n i c a and P. radi a t a models i s a r e s u l t of t h i s component while v a r i a b i l i t y i n the P_. patula model can be apportioned to both dominant height and basal area functions i n the r a t i o of 1:2. In addition to the sources of v a r i a b i l i t y i n basal area estimate discussed i n Chapter 2, i t should be noted that i n the simulation model, errors could a r i s e from three TABLE 43: Bias percentage for dominant height and basal area for test permanent sample plots by species. £. l u s i t a n i c a P_. patula P. rad i a t a Basal Basal Basal Plot No. Height area Plot No. Height area Plot No. Height area 4 -0.08 -0.30 37 -1.12 -5.04 54 -1.91 0.26 116 -4.04 3.03 117 -1.06 -4.09 121 -1.36 -6.61 181 -1.81 2.85 190 -2.14 6.54 202 -0.29 -17.36 233 -1.20 -8.38 246 -3.41 -5.25 261 -2.98 11.48 279 -0.93 17.21 288 -0.64 -0.83 295 -0.34 3.19 331 -2.02 -3.21 336 -1.79 -2.83 348 0.79 1.47 379 -0.61 1.04 388 1.82 -14.72 Means -1.26 -1.08 S.D. 1.38 7.94 2 -1.87 5.23 12 0.30 -1.49 34 -.190 6.71 59 -1.04 23.78 123 1.40 -16.96 126 -0.53 -0.45 144 -0.42 0.87 157 -0.05 -6.14 154 -0.90 -7.01 167 -0.74 -21.92 203 0.55 -0.69 209 -0.42 -9.60 252 2.55 9.60 270 2.77 1.16 276 2.59 0.69 312 -5.81 -2.59 315 -3.14 2.25 324 -10.45 -3.35 342 -3.58 8.17 391 0.57 -1.34 -1.01 -0.65 3.07 6.66 6 -0.49 -6.52 18 0.10 2.41 31 -0.73 12.27 91 -2.55 -5.65 96 -1.55 15.08 99 -1.75 7.67 103 -0.33 6.68 112 -0.84 0.56 134 0.12 -2.72 138 -1.83 21.29 164 -0.98 4.41 177 -1.66 -6.59 238 2.33 -9.62 256 -0.14 -1.15 289 -0.45 -1.76 340 0.40 -2.18 373 -0.10 -1.11 383 -0.55 16.02 400 -0.10 8.69 402 -0.35 -2.93 -0.57 2.74 1.04 8.52 Excluding plot 348 for P. patula: Mean ) -0.48 S.D. ) 2.12 194 sub-components: basal area function before f i r s t thinning, basal area increment equation and the thinning function. In general however, the v a r i a b i l i t y appears to be of a random nature except f o r the tendency f o r the P_. radiata model to s l i g h t l y underestimate the basal area component. This may require further refinement i n future work. Figures 23, 24 and 25 show the d i s t r i b u t i o n of the test plots by age and plot mean bias % f o r C_. l u s i t a n i c a , P_. patula and P_. radi a t a r e s p e c t i v e l y . In a l l cases, there i s no evidence of bias i n the d i s t r i b u t i o n of the bias with respect to age. S i m i l a r l y , Figures 26, 27 and 28 show the d i s t r i b u t i o n of the test plots by s i t e index and plot mean bias and again there i s no evidence of bias with respect to s i t e index. Plot No. 238 f o r P_. radiata appears to be an o u t l i e r . It i s worth noting the range of the s i t e indices within which the test plots belonged: 17 to 23, 18 to 28 and 24 to 31 for C. l u s i t a n i c a , P. patula and P_. radi a t a r e s p e c t i v e l y . S i m i l a r l y , the age range for test plots was 5 to 43, 5 to 20 and 5 to 30 for C_. l u s i t a n i c a , P_. patula and P_. radi a t a respectively. F i n a l l y but not l e a s t , there are several exogenous factors that may introduce error i n the model. These include: 1. Measurement errors. 2. B i o t i c factors: including game damage, insect and disease, etc. 3. C l i m a t i c and edaphic factors some of which may not have been covered i n the study. These include annual weather f l u c t u a - tions and cumulative drought e f f e c t s , s o i l f a c t o r s , etc. FIGURE 23 DISTRIBUTION OF C. LUSITANICA TEST PLOTS 3Y AGE AND VOLUME BIAS Z , 279 • 116 54 -336 -117 •37 . 121 246 • 233 1 1 I I n « 14 16 18 20 22 24 26 28 AGE IN YEARS FROM PLANTING , • 1 . • 1 1 — 30 32 34 36 38 40 42 VO FIGURE 24 DISTRIBUTION OF P. PATULA TEST PLOTS BY AGE AND VOLUME BIAS % -34 270 -2 144 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 AGE IN YEARS FROM PLANTING VO FIGURE 25 DISTRIBUTION OF P^ RADIATA TEST PLOTS BY AGE AND VOLUME BIAS % 373- 238- • 400 99 -103 12 AG!' 16 20 24 IN YEARS FROM PLANTING FIGURE 26 DISTRIBUTION OF C. LUSITANICA TEST PLOTS BY SITE INDEX AND VOLUME BIAS % • 279 • 190 • 181 •116 * 4 2 8 8 » , S 4 • 336 • 117 • 37 • 2 4 6 »121 • 233 • 388 • 202 - I 1 r — — i • 1 1 1 1 • . . — . r - 10 12 14 16 1 8 20 22 24 PLOT SITE INDEX IN METERS FIGURE 27 DISTRIBUTION OF P. PATULA TEST PLOTS BY SITE INDEX AND VOLUME BIAS % 20' • 252 • 34 • 342 • 270 • 276 • 2 . 2 0 3 « 1 4 4 ' 391 * 1 2 6 « 1 2 • 312 • 209 • 324 • 123 167 " • ' 17 18 19 20 21 22 23 24 25 26 27 PLOT SITE INDEX IN METERS FIGURE 28 DISTRIBUTION OF P. RADIATA TEST PLOTS BY SITE INDEX AND VOLUME BIAS % • 138 • 383 • 98 • 31 • 400 • 103 • 18 • 164 . 3 7 3 " " 2 • 2 5 6 * 3 4 0 8 , 3 4 " \ 402 • 238 • • -91 » 1 7 7 y\ . .. (S3 ' l 8 19 20 21 22 23 24 25 26 27 28 29 30 31 O PLOT SITE INDEX IN METERS 201 The e f f e c t s of these factors i s to introduce e r r a t i c behaviour i n some plots so that t h e i r observed values w i l l d i f f e r markedly from the simulated values. The question of how to deal with these factors i s more ph i l o s o p h i c a l than p r a c t i c a l since these problems are for the most part there to stay. Thus, i n my view, p l o t s showing e r r a t i c behaviour ( o u t l i e r s ) c a l l for s p e c i a l attention to determine the cause but should not be eliminated unless there Is clear evidence that they are from outside the population of Interest. 4.3 Conclusion The o v e r a l l conclusion from the v a l i d a t i o n process i s that the model i s unbiased for a l l species except for a s l i g h t tendency to under- estimation for P. r a d i a t a . The 95% confidence l i m i t s for the difference between simulated and observed volume was ±16% for C_. l u s i t a n i c a +20% for _P. patula and +17% for P. r a d i a t a . The model i s acceptable at an error s p e c i f i c a t i o n between 20 and 25% (C_. l u s i t a n i c a and P_. radiata) and 25 to 30% for .P. patula, unless a l-in-20 chance has occurred. However, the model i s f a i r l y accurate for error s p e c i f i c a t i o n . up to 15% for C_. l u s i t a n i c a and P_. radiata and up to 20% for P_. patula If the lower p r o b a b i l i t y l e v e l i s acceptable. Basal area was i d e n t i f i e d as the main source of e r r o r for £. l u s i t a n i c a and P. radiata models. Both dominant height and basal area contributed to the error for P_. patula model i n the r a t i o of 1:2. Thus future refinement to the model should be directed at basal area components for a l l species. Dominant height function for P. patula may also need further refinement. 202 The test plots covered s i t e index ranges of 17 to 23, 18 to 28 and 24 to 31 f o r 0, l u s i t a n i c a , P_. patula and P. r a d i a t a and age ranges of 5 to 43, 5 to 20 and 5 to 30 years for the same species r e s p e c t i v e l y . The accuracy l e v e l s mentioned above apply to these ranges. It should also be noted that i n d i v i d u a l plot runs were l i m i t e d to an average of 10 years. It i s l i k e l y that longer simulations w i l l not r e s u l t i n the same l e v e l of accuracy. From the above discussion, i t i s clear that EXOTICS i s capable of accurately simulating stand growth for the three species within the l i m i t a t i o n s stated above. It s u t i l i t y for predicting stand y i e l d and as a management guide i n analyzing d i f f e r e n t s i l v i c u l t u r a l a l t e r n a t i v e s i s the subject of the next chapter. This w i l l serve as the s e n s i t i v i t y test with respect to input v a r i a b l e s . 203 CHAPTER 4 SILVICULTURAL MANAGEMENT MODELS FOR KENYA 1. Introduction As mentioned i n the introduction (Chapter 1) a thorough knowledge of the growth and y i e l d of the forest resources under d i f f e r e n t physical and b i o l o g i c a l conditions i s basic to formulation of sound forest management plans, including s i l v i c u l t u r a l management schedules. Forest inventory systems, of which the permanent sample plot programme for Kenya forms a part, are the main source of t h i s information, while growth and y i e l d models are invaluable tools for planning and experimen- t a t i o n with a l t e r n a t i v e schedules. This chapter Is devoted to the study of the growth and y i e l d of the three species under the present manage- ment schedules and the formulation of al t e r n a t i v e schedules using the y i e l d model EXOTICS developed i n the previous chapter. In t h i s regard, thinning i s singled out as the p r i n c i p a l s i l v i c u l t u r a l means for stand manipulation towards the desired goals and objectives. Fundamentally, thinning involves the periodic removal of some of the trees, with the main objective being to provide the remaining trees with adequate growing conditions. In p r i n c i p l e , therefore, the whole process amounts to stand density control to achieve the desired objec- t i v e s . The main concern to foresters has been to decide what measures of stand density to employ and the l e v e l of stand density control to apply. Since thinning consists of removal of some of the trees i n the stand, i t seems obvious that stem count should be the l o g i c a l means of 204 density c o n t r o l . However, as the number of trees diminishes over the r o t a t i o n , the s i z e of the i n d i v i d u a l trees increases. Thus, according to Wilson (1979), stem count must be q u a l i f i e d by some measure of tree s i z e i f i t i s to have meaning. Wilson proposed the use of spacing as a function of stand dominant height, the concept of which has already been discussed under the section on stand density. Wilson's (1946) proposal has been applied i n thinning research i n several other countries as Hart's density index. In addition to i t s disadvantage already discussed, i t i s demonstrated l a t e r i n t h i s section that for some species, stands of the same stand density index w i l l have d i f f e r e n t basal areas per hectare for d i f f e r e n t s i t e index classes. For these species t h i s index i s inadequate as a measure of the degree to which a given species i s u t i l i z i n g the s i t e . A more recent approach to stand density control has been proposed by Drew and F l e w e l l i n g (1977, 1979). B a s i c a l l y the approach employs the concept of maximum size-density as a general p r i n c i p l e of plant popula- t i o n biology: In pure stands, the maximum mean tree size attainable for any density can be determined by a r e l a t i o n s h i p known as the -3/2 power law: where v = a = P = mean tree volume a constant stand density expressed as number of trees. 205 The above law can be rewritten as: In v = a' - — lnp 4.2 2 So that -3/2 represents the slope of the maximum size-density r e l a t i o n s h i p as shown on Figure 29 for coastal Douglas-fir from Washington and Oregon (adopted from Drew and F l e w e l l i n g 1979). The empirical determination of t h i s law and i t s t h e o r e t i c a l derivation were developed by Yoda et al.(1963) while I t s a p p l i c a t i o n to f o r e s t r y has been demonstrated by Yoda et a l . (1963) and Drew and Flewelling (1977, 1979). According to Harper (1977), there i s evidence that t h i s -3/2 power law holds true for forest trees as well as for annual plants. The p r a c t i c a l i m p l i c a t i o n of t h i s law i s that a stand of a given i n i t i a l density ( i n terms of number of trees per unit area) w i l l main- t a i n volume growth u n t i l the mean tree volume (or size) reaches the maximum size for that density given by equation 4.1. This indicates the size-density at which s e l f - t h i n n i n g (competition induced) mortality sets i n and indicates the point at which the stand i s due for thinning. The problem for the forest manager then i s to determine that t h i s law applies to the species he i s dealing with and the r e l a t i o n s h i p (equation 4.1) which w i l l depend on the species and s i t e f a c t o r s . The a p p l i c a - b i l i t y of t h i s r e l a t i o n s h i p i s best determined from controlled experi- ments, e s p e c i a l l y the constant stocking t r i a l s . The approach of Reukema and Bruce (1977) u t i l i z e s a s i m i l a r p r i n c i p l e to that of Drew and F l e w e l l i n g (1977, 1979) except that they used maximum stocking l e v e l (basal area) per unit area as a guide to when a stand i s due for thinning. Besides providing the forest manager 206 Mean Tree Volume (m») (ff) ( t r e e s / a c r e ) 300 500 1000 1500 2000 3000 5000 ( t r e e s / h e c t a r e ) Dens i ty Figure 29 The maximum size-density relationship and the natural stand data used in positioning this relationship. (from Drew and Flewelling 1979) 207 with an objective guide (maximum basal area) to thinning, t h i s approach has the advantage of ensuring e f f i c i e n t u t i l i z a t i o n of s i t e by a species. In addition, i t provides a continuous mechanism for tree dimension con t r o l since mean stand DBH can be derived from the basal area at any time. I t s main drawback rests In defining q u a n t i t a t i v e l y the maximum stocking l e v e l . In addition to deciding when a stand i s due for a thinning, two other variables enter a thinning model: the i n t e n s i t y of thinning and the length of time between thinnings. As mentioned e a r l i e r i n Chapter 2, these two are i n t e r r e l a t e d , since the higher the i n t e n s i t y , the longer the thinning i n t e r v a l and vice versa. Thus, i f the basal area a f t e r thinning and basal area growth rate are known, then the time i t takes before the stand i s due for the next thinning i s known. Both these variables are functions of the economics of thinning and the b i o l o g i c a l f a c t o r s , as mentioned elsewhere. In t h i s study, the econo- mics of thinning Is not considered and therefore only b i o l o g i c a l factors w i l l be discussed. The problem reduces to one of determining the quantity of the stand to be removed at each thinning. In t h i s respect, the current thinning i n t e n s i t y used i n Kenya was used as a guide. 2. Current Thinning Models f o r Sawtimber Regimes i n Kenya Figure 30a,b, and c, i l l u s t r a t e the current basal area thinning model for £ . l u s i t a n i c a , P_. patula and P_. radiata r e s p e c t i v e l y while Figure 31a,b, and c, i l l u s t r a t e the MAI and CAI (smoothed) curves for the respective species under the current sawtimber thinning regimes. FIGURE 30 208 MAIN STAND BASAL AREA/AGE RELATIONSHIP UNDER THE CURRENT SAWTIMBER THINNING REGIMES BY SPECIES AND S.I. CLASSES 4 8 1 2 1 6 2 0 2 4 2 8 3 2 AGE IN YEARS FROM PLANTING 209 FIGURE 31 MEAN AND CURRENT ANNUAL VOLUME INCREMENT RELATIONSHIP WITH AGE FOR THE CURRENT SAWTIMBER THINNING REGIME BY SPECIES AND S.I. CLASSES 50 <40 P3 en 2 3 0 .20 oio > M.A.I, and C.A.I, m (smoothed out) a) C . LUSITANICA. 10 20 30 40 C.A.I, m (unsmoothed) 2 4 - 2 1 ' /v---*r _V.% 18'x 10 20 30 40 210 Unsmoothed CAI curves are also shown on Figure 31 f o r each species to i l l u s t r a t e the e f f e c t s of thinning on the current annual volume i n c r e - ment development. Tables 44, 45 and 46 give the volume y i e l d and other stand c h a r a c t e r i s t i c s for C_. l u s i t a n i c a , P_. patula and P_. radiata r e s p e c t i v e l y . The following features c h a r a c t e r i s t i c of the present sawtimber thinning model are worth noting: (a) For £ . l u s i t a n i c a thinning model 1. At f i r s t thinning, c a r r i e d out when stands on a l l s i t e classes have the same stand density index of 25%, stand basal areas are 32.2, 28.5, 25.3, 24.3 and 17.7 m2 p e r hectare f o r s i t e index classes 12, 15, 18, 21 and 24 re s p e c t i v l e y (see Table 44). For t h i s species therefore, thinning to a common stand density index r e s u l t s i n d i f f e r e n t l e v e l s of s i t e u t i l i z a t i o n , with the poor s i t e s carrying a much heavier basal area than the good s i t e s . This i s the opposite to what the s i t u a t i o n should be and demonstrates the weakness, already mentioned, inherent i n using Hart's stand density index as a guide to thinning for t h i s species. 2. Except f o r f i r s t thinning, the average maximum basal area before thinning i s 33 m per hectare, and the average basal area a f t e r thinning i s 26 m per hectare. Thus, there i s an implied maximum basal area which, judging from the basal area curve trends, appear to be well below the maximum consistent with maximum basal area y i e l d , e s p e c i a l l y for s i t e classes >18. 211 TABLE 44. Volume y i e l d and other relevant stand parameters under the current sawtimber thinning r e g i me for C. l u s i t a n i c a to a r o t a t i o n age of 40 years: Technical Order No. 42 of March 1969. S i t e Index 12 15 18 21 24 Hdom a t l s t thinning 11.25 11.25 11.25 11.25 11.25 Age at l s t thinning 14 11 9 8 6 2 BA (m ) before l s t thinning 32.2 28.5 25.3 24.3 17.7 Culmination age (approx.) 30 26 24 21 19 MAI max (m3) 16.0 19.0 22.0 25.2 28.1 CAI max (m ) 23(12) 27(11) 32(10) 37(9) 40(7) DBH at age 40 years 43.7 46.1 47.6 48.3 50.1 Tot a l V ( l ) mJ at age 40 years 637.5 749.3 849.7 943.0 1037.7 Thinning volume as % 37.3 33.0 30.3 29.7 28.1 To t a l V(15) nr at age 40 years 567.3 674.2 , 771.4 860.6 961.8 Thinning V(15) m3 as % 30.8 26.7 24.4 24.2 21.6 No. i n bracket ind i c a t e age of max. CAI. 212 TABLE 45. Volume y i e l d and other relevant stand parameters under the current sawtimber thinning regime f o r P_. patula (Nabkoi) to a r o t a t i o n of 20 years: Technical Order No. 53 of May 1981 P. patula (Nabkoi) Shamba S i t e Index 15 18 21 24 27 Hdom a t l s t thinning 16.9 16.8 16.8 17.5 17.7 Age at l s t thinning 17 14 12 11 10 BA (m2) before l s t thinning 40.1 36.6 33.7 33.0 31.5 MAI max (m ) at age 20 year 18 21 24 27 30 CAI max (m 3) 33(14) 40(13) 48(12) 55(11) 61(10) 3 Tot a l V ( l ) m at age 20 year 358.9 416.2 472.0 542.6 642.0 Thinning volume as % 35 45 39 35 32 To t a l V(15) m3 at age 20 year 261.0 312.8 364.3 428.8 520.7 Thinning V(15) m3 as % 28 37 29 26 22 To t a l V(20) m3 at age 20 year 101.9 160.5 218.0 277.0 374.3 Thinning volume as % 0 12.0 8.0 8.0 7.0 No. i n bracket i n d i c a t e age of max. CAI. 213 TABLE 46. Volume y i e l d and other relevant stand parameters under the current sawtimber thinning regime for P_. radi a t a to a ro t a t i o n age of 30 years: Technical Order No. 44 of March 1969. P. radiata Shamba Si t e Index 21 24 27 30 33 Hdom a t l s t t h i n n i n g 12.2 12.5 12.4 14.1 13.5 Age at l s t thinning 9 8 7 7 6 o BA nr at l s t thinning 14.9 14.9 14.5 16.3 15.4. 2 BA nr at 2nd thinning 21.0 19.5 19.8 19.1 18.9 3 MAI max m up to age 35 22 25 28 31 34 3 CAI max m 32(35) 35(35) . 38(35) 40(17) 44(16) DBH to age 30 years 41.6 44.3 46.2 47.7 49.5 Tot a l V ( l ) m at age 30 years 655.8 751.8 855.8 946.9 1052.8 Thinning volume as % 33.1 28.7 27.2 25.8 24.2 To t a l V(15) m3 a t age 30 years 542.8 642.6 743.4 831.0 939.1 Thinning V(15) as % 21.4 18.5 18.0 17.1 16.7 3 To t a l V(20) nr at age 30 years 471.3 573.6 713.4 760.9 869.5 Thinning volume as % 14.4 12.4 11.9 12.0 12.0 No. i n bracket i n d i c a t e age of max. CAI. 214 3. The CAI curves (unsmoothed) in d i c a t e that except for f i r s t thinning, a l l other thinnings have a marked e f f e c t on the current annual volume increment, indicated by the drop i n CAI a f t e r thinning. This i n e v i t a b l y has an e f f e c t on t o t a l volume y i e l d which indicates that for t h i s species, Moller's theory that thinning has no e f f e c t on t o t a l volume y i e l d does not hold under the present thinning regime i n Kenya. 4. Important figures to note i n Table 44 are the volume of t h i n - ning ( t o t a l and merchantable) as a percentage of the volume y i e l d ( t o t a l and merchantable) which on the average works out to 30% and 24% r e s p e c t i v e l y . These percentages, along with the DBH at age 40 years are important i n comparing outputs from a l t e r n a t i v e schedules. (b) For P_. patula thinning model 1. At f i r s t thinning which i s c a r r i e d out at the common stand density index of 17%, stand basal areas are 40.1, 36.6, 33.7, 33.0 and 31.5 m per hectare for s i t e index classes 15, 18, 21, 24 and 27 re s p e c t i v e l y . Thus, as for C_. l u s i t a n i c a , Hart's density index i s inappropriate as a measure of s i t e occupancy. 2. The f i r s t thinning appears very severe, removing an average of 42% of the basal area before thinning. This i s confirmed by Figure 31b which shows a very d r a s t i c drop i n current annual increment a f t e r t h i s thinning for a l l s i t e classes. Subsequent thinnings also appear to have an e f f e c t on CAI. It i s evident 215 therefore that for t h i s species, Moller's theory does not hold at the present l e v e l of thinning i n t e n s i t y . 3. Except for the f i r s t thinning, subsequent thinnings appear l i m i t e d to an average maximum basal area before thinning of o 30 xsr per hectare and an average minimum basal area a f t e r thinning of 22.5 m per hectare. The maximum appears well short of the maximum consistent with maximum basal area y i e l d f o r a l l s i t e classes. 4. Table 45 shows that thinning volume constitutes a very high percentage of the t o t a l volume y i e l d up to age 20 years, a re s u l t of the heavy thinnings for t h i s species. 5. At the present l e v e l of thinning i n t e n s i t y , culmination of growth does not occur before age 20 years on any s i t e c l a s s . From the smoothed CAI and MAI curves (Figure 31b) i t would appear from extrapolation that culmination would occur at the same age on a l l s i t e classes: between age 20 and 21 years. However i t should be noted that t h i s may be due to the heavy thinnings a f f e c t i n g the stand development or to the smoothing out of curves using subjective judgement. For _P. radiata thinning model 1. For t h i s species, basal area at f i r s t and second thinning are 2 an average 15.2 and 19.7 m per hectare respectively on a l l s i t e classes (see Table 46). These thinnings are car r i e d out at a common stand density index of 23% and 18% respectively on 216 a l l s i t e classes. I t therefore appears that for t h i s species, Hart's stand density index i s a good approximation of the degree to which the species i s u t i l i z i n g the s i t e . It would be an appropriate basis for timing when a stand i s due for a thinning for t h i s p a r t i c u l a r species. 2. In general the thinning model indicates that the l e v e l of basal area before thinning increases with age (Figure 30c). Figure 31c for CAI (unsraoothed) shows that the f i r s t thinning on a l l s i t e index classes has no marked e f f e c t on current annual volume increment. However, subsequent thinnings do have an appreciable e f f e c t as indicated by the drop i n CAI a f t e r each thinning. 3. From extrapolation, i t would appear that culmination age for P_. r a d i a t a would occur well beyond the range covered by the data i n t h i s study (see Figure 31c). Indeed the CAI appears to be s t i l l a c c elerating up to age 35 years for s i t e index classes 21, 24 and 27. This suggests that t h i s species has higher y i e l d p o t e n t i a l and that the r o t a t i o n (up to age 35 years) does not e x p l o i t t h i s p o t e n t i a l f u l l y . 4. At age 30 years, thinning volume on the average constitutes 27.2%, 18.0% and 11.9% of t o t a l volume y i e l d , merchantable volume to 15 cm top DBH and merchantable volume to 20 cm top DBH r e s p e c t i v e l y (Table 46). Thus, the thinnings are consider- ably l i g h t e r than those f o r P. patula and £. l u s i t a n i c a . 217 2.1 Summary on the Current Thinning Model f o r Kenya 1. Hart'8 stand density index i s Inadequate as a guide to thinning f o r C_. l u s i t a n i c a and P_. patula as i t re s u l t s i n higher s i t e occupancy on poor s i t e s than on good s i t e s , based on basal area. For P_. radi a t a the index appears quite s a t i s f a c t o r y as a measure of the degree of s i t e u t i l i z a t i o n by the species. 2. For £. l u s i t a n i c a and _P. r a d i a t a , f i r s t thinning appears to have no marked e f f e c t on CAI. However, subsequent thinnings do appear to have an appreciable e f f e c t , r e s u l t i n g i n lowering of the current annual volume increment. For P_. patula, both f i r s t and subsequent thinnings do have a d r a s t i c e f f e c t on CAI. 3. As a follow-up to the observations on 2 above, i t i s i n f e r r e d that under the current sawtimber thinning regimes, Moller's theory that thinning has no appreciable e f f e c t on t o t a l volume production does not hold for the three species i n Kenya. 3. A l t e r n a t i v e thinning model f or sawtimber crop i n Kenya 3.1 Thinning P o l i c y Considerations As mentioned i n Chapter 1 Section 3, the thinning p o l i c y for Kenya aims at production of large-sized material i n as short a ro t a t i o n as possible at the expense of some loss i n t o t a l y i e l d . At the time t h i s p o l i c y was adopted i n the f i f t i e s and early s i x t i e s and documented i n the relevant Technical Orders i n 1969, the predominant purpose of plantation management was production of sawlogs as quickly as possible 218 as the shortage of sawtimber from indigenous forests was already being f e l t . At that time, there were no other major wood-using i n d u s t r i e s , neither was the pressure on the l i m i t e d forest resource acute as the population was s t i l l very low with low per capita consumption of wood. Since then, several developments have occurred: 1. The population has increased from an estimated 6 m i l l i o n i n 1950 to 8 m i l l i o n i n 1960 and to 15 m i l l i o n i n 1980, with a population growth rate of 4%, estimated to be the highest i n the world (according to Kenya Bureau of S t a t i s t i c s 1982). This has put a l o t of pressure on the forests f or the supply of sawtimber, firewood, general purpose wood, pulp and paper products, etc. 2. There has been a very rapid increase i n forest i n d u s t r i e s , ranging from modern sawmills, p a r t i c l e board manufacturing i n d u s t r i e s , plywood industries and a modern p u l p m i l l , which came i n t o production i n 1972. The implication of t h i s development i s that while a few plantations may s t i l l be managed exclusively for supply of only one end product, the majority of plantations w i l l be managed for supply of multiple end products. Thus even i n a predominantly sawtimber management zone, there w i l l be a component for pulpwood, p a r t i c l e board and plywood. 3. There has been a rapid Increase i n standard of l i v i n g , r e s u l t i n g i n an increase i n consumption of wood and wood products. For example, the per c a p i t a l roundwood consumption 219 f o r Kenya i n 1950 was 0.1 m3. This had r i s e n to 1.8 m3 by 1979 (F.A.O. Yearbook of forest products s t a t i s t i c s 1947-1951, F.A.O. Yearbook of forest products 1979). Most of t h i s increase has been a r e s u l t of increased l i t e r a c y l e v e l , r e s u l t - ing i n higher consumption of pulp and paper products, and the change to modern st y l e s of b u i l d i n g which require more timber. A l l these factors point to a need for a change i n thinning p o l i c y i n favour of the objective of maximum volume production. The develop- ment of integrated forest i n d u s t r i e s which can u t i l i z e both small size logs from thinnings and large si z e logs from f i n a l f e l l i n g s favour t h i s p o l i c y . I t should also be noted that forests i n Kenya constitute only 3% of the land area and therefore the mounting population and demand for wood products can only favour the adoption of maximum volume y i e l d on any a v a i l a b l e forest land. The need for changing the thinning p o l i c y to accommodate the changes i n the f o r e s t r y i n d u s t r i a l sector appear already to have been appreciated by the Kenya Forestry Department. This i s evidenced by the 1981 r e v i s i o n of the management schedule f o r P. patula (Technical Order No. 53 of May 1981). The most s i g n i f i c a n t changes i n t h i s Technical Order are i n regard to the delay and heaviness of the f i r s t thinning. These changes appear to have been i n s t i t u t e d to provide a higher volume of l a r g e r - s i z e d thinnings to go to the p u l p m i l l . Unfortunately, as has already been demonstrated i n the previous section, t h i s thinning schedule i s not consistent with the concept of maximum volume y i e l d over the whole r o t a t i o n . 220 3.2 Thinning Experiment for C_. l u s i t a n i c a In order to investigate the p o s s i b i l i t y of a l t e r n a t i v e thinning regimes for the exotic timber species i n Kenya, a thinning experiment was designed with the following objectives: 1. To investigate the e f f e c t s of d i f f e r e n t thinning l e v e l s and thinning i n t e n s i t i e s on growth and y i e l d on d i f f e r e n t s i t e index classes. 2. Based on r e s u l t s from (1) above, to i d e n t i f y the appropriate thinning regime based on the c r i t e r i a of highest merchantable y i e l d . In order to draw reasonable bounds to the study, only one species, C_. l u s i t a n i c a was considered. This species was singled out on two counts; (1) i t i s the preferred species for sawtimber, and (2) i t s model domain covered the whole range of i t s rotation under the current management schedules. Experimental design: Five thinning l e v e l s A, B, C, D and E were a r b i t r a r i l y selected i n order of increasing basal area before thinning. These lev e l s are given on Table 47. Within each thinning l e v e l , four thinning i n t e n s i t i e s were selected based on the proportion of basal area to be removed as a percentage of basal area before thinning. These were 10, 20, 30 and 40%. These treatments were repeated over the f i v e s i t e index classes for C_. l u s i t a n i c a i . e . 12, 15, 18, 21 and 24 f o r a t o t a l of 100 treatment combinations. A l l the experiments were conducted using the y i e l d model EXOTICS to simulate r e s u l t s . Figure 32 shows how these thinning regimes translates i n terms of basal area before and a f t e r 221 TABLE 47. Basal area before thinning (Mz/ha) f o r the a l t e r n a t i v e thinning regimes Thinning l e v e l l s t thinning 2nd thinning 3rd thinning 4th thinning A . 2 5 35 35 35 B 25 35 40 40 C 25 35 45 45 D 25 40 45 45 E 25 40 50 50 thinning and number of stems ( i n i t i a l stocking of 1200 s.p.h. assumed) at d i f f e r e n t ages f o r s i t e index class 18. A major concern i n s e l e c t i n g the thinning l e v e l s was whether these exceeded the maximum basal area p o t e n t i a l f o r each s i t e q u a l ity c l a s s . A preliminary attempt to f i n d these maxima using the -3/2 power law (Drew and Fle w e l l i n g 1977, 1979) f a i l e d , suggesting that the plantations from which the data was drawn were managed below the maximum s i t e p o t e n t i a l . Faced with the problem of defining the maximum basal area for these stands, Alder (1977) had f i t t e d hand-drawn curves over the maximum basal area observed on the p.s.p.s. and then quantified these curves using a nonlinear equation: (-b,H).b, G - b n (1 - e 1 ) 2 4.3 max 0 v ' where = Maximum basal area i n m2/ha. H = Stand dominant height (represents e f f e c t s of age and s i t e ) . FIGURE 32 NUMBER OF STEMS AND BASAL AREA AT DIFFERENT AGES FOR DIFFERENT THINNING LEVELS AND THINNING INTENSITIES FOR C. LUSITANICA S.I. CLASS 18 40 % 400 O- 10 20 30 40 SO O 10 20 30 40 50 0 10 20 AGE I N YEARS FROM PLANTING 30 40 50 rO ho ho STEMS PER HA 224 bg, b^ and b 2 are the regression c o e f f i c i e n t s . For £. l u s i t a n i c a i n Kenya, he obtained the values: bg = 63.9, b^ = 0.1219 and b2 = 2.551. No c r i t e r i a for goodness of f i t was given for t h i s equation as i t was based upon a hand-drawn curve and hence such c r i t e r i a would be meaningless. For lack of better means of guidance i n t h i s study, t h i s equation was used to determine the G m a x curves shown on Figure 32. 3.3 Results from the Simulated Thinning Experiment (a) E f f e c t s of a l t e r n a t i v e thinning regimes on MAI and b i o l o g i c a l r o t a t i o n age: Figure 33 gives the MAI and CAI curves for the d i f f e r e n t thinning l e v e l s and thinning i n t e n s i t i e s for £. l u s i t a n i c a s i t e index 18 while Table 48 gives a summary of the maximum MAI and the age at which t h i s maximum i s obtained (culmination age) for each thinning l e v e l and thinning i n t e n s i t y for the same s i t e index c l a s s . The maximum MAI and the culmination age for the current thinning regime and some s i t e index cl a s s are also shown on Table 48 as control. Several observations can be noted from both Figure 33 and Table 48. 1. For a l l thinning l e v e l s , MAI decreases with increasing severity of thinning. For example under 10% thinning i n t e n s i t y , the e f f e c t s of thinning are minimal so that growth can almost be considered as for unthinned stands. MAI can therefore be expected to be at maximum and to decrease with increasing severity of thinning so that i t i s minimum at 40% i n t e n s i t y .   227 TABLE 48. Maximum MAI (mJ/ha) and b i o l o g i c a l r o t a t i o n age (culmination age) for d i f f e r e n t thinning regimes for C_. l u s i t a n i c a S.I. 18 Intensity 10 20 30 40 Control Level MAI Age MAI Age MAI Age MAI Age MAI Age A 24.2 36 22.6 22 22.0 22 21.0 26 B 24.2 34 22.9 25 22.2 25 21.0 26 C 24.3 32 23.3 26 22.4 28 21.1 32 D 24.4 32 23.5 27 22.6 28 21.7 24 E 24.6 30 23.8 28 22.7 32. 21.7 20 Control 22.0 24 i 228 This confirms the e a r l i e r observation (Chapter 4 Section 2) that Moller's theory with respect to e f f e c t s of thinning on volume y i e l d does not hold for C_. l u s i t a n i c a within the thinning i n t e n s i t i e s considered i n t h i s study. Within a given thinning i n t e n s i t y , MAI increases with increasing l e v e l of thinning i . e . increases from thinning l e v e l A to E. This i s as expected since MAI i s a function of basal area increment which i n turn i s a function of basal area before thinning. This increase however i s very small compared to the increase r e s u l t i n g from changes In thinning i n t e n s i t y and may be considered unimportant for p r a c t i c a l purposes. From the above observations, i t i s concluded that for C_. l u s i t a n i c a , thinning i n t e n s i t y rather than thinning l e v e l (measured by basal area before thinning) i s the more c r i t i c a l consideration with regard to MAI. For 10% thinning i n t e n s i t y , the culmination age decreases with increasing l e v e l of basal area before thinning. This indicates that t h i s thinning i n t e n s i t y i s so l i g h t that stand development i s as f o r unthinned stand. Increasing basal area before thinning therefore has same e f f e c t as improving s i t e q u a l i t y . For 20 and 30% thinning i n t e n s i t i e s , culmination age increases with increasing basal area before thinning i . e . from thinning l e v e l s A to E. This i s mainly a r e s u l t of the CAI curves being s h i f t e d further to the right as the basal area 229 before thinning i s raised ( r e s u l t i n g i n delay i n thinnings) while the MAI curves are l i t t l e affected (see Figure 33). Thus r a i s i n g l e v e l of basal area before thinning when thinning i n t e n s i t i e s are heavy has the same e f f e c t s as decreasing s i t e q u a l i t y . This e f f e c t i s also apparent for 40% thinning i n t e n - s i t y but i s reversed for thinning l e v e l s D and E as the e f f e c t s of the t h i r d thinning on MAI and CAI curves diminishes. 5. Within a given thinning l e v e l , the culmination age i s expected to increase with increasing thinning i n t e n s i t y . This i s manifest i n thinning l e v e l s C, D and E for thinning i n t e n s i t i e s 20, 30 and 40% ( l e v e l s A, B and C only). Based on MAI and culmination age, the most promising thinning regimes are those with 20 and 30% thinning i n t e n s i t y . The 10% thinning i n t e n s i t y gives high culmination age i n s p i t e of the higher MAI. Besides t h i s , the l i g h t thinnings are accompanied by short thinning cycles and therefore are unattractive economically. The 40% thinning i n t e n s i t y r e s u l t s i n low MAI compared to the current thinning schedule, suggesting that i t i s probably too severe. Between the 20 and 30% thinning i n t e n s i t i e s , the former has an edge i n both MAI and culmination age. Thinning regime A:20 appears to be the best with a higher MAI and lower culmination age than the current thinning regime. This however does not mean that t h i s i s the optimum regime. A l l the other thinning regimes under 20% thinning i n t e n s i t y have higher MAI but longer r o t a t i o n age than the current thinning regime. I t i s therefore not possible to determine the best regime without an economic a n a l y s i s . 230 The b i o l o g i c a l r o t a t i o n of a plantation (as discussed above) provides the r o t a t i o n of highest t o t a l volume y i e l d . For sawtimber production however, the main Interest i s the t o t a l merchantable volume production f o r a given end product. As a r e s u l t , b i o l o g i c a l r o t a t i o n i s hardly ever used i n sawtimber production regimes. As mentioned e a r l i e r , the current r o t a t i o n for sawtimber plantations i n Kenya i s the age at which a DBH of 48 cm i s attained. This however i s a gpoor c r i t e r i a for a r o t a t i o n since the 48 cm DBH can be attained i n a plantation at d i f f e r e n t ages depending on the i n i t i a l stocking and thinning i n t e n s i - t i e s . I t therefore does not r e l a t e to volume y i e l d . A commonly used method i n f o r e s t r y i s to calculate the economic r o t a t i o n , defined either as the r o t a t i o n of the highest economic land value or of the highest rate of return (Crowe 1967, Grut 1970, and others). This would require such information as the economics of plantation establishment, log class assortments and a c l e a r d e f i n i t i o n of product mix, a l l of which were not a v a i l a b l e to t h i s study. For purposes of y i e l d analysis under d i f f e r e n t thinning regimes, a common r o t a t i o n age of 40 years was adopted, mainly because i t i s the average age at which sawtimber crop attains 48 cm DBH under the current thinning regime i n Kenya. (b) E f f e c t s of a l t e r n a t i v e thinning regimes on p r o d u c t i v i t y Table 49 shows various measures of productivity up to age 40 years for C_. l u s i t a n i c a s i t e Index 18 for the various thinning regimes, including the current thinning regime as c o n t r o l . The table also gives the increase i n y i e l d of the a l t e r n a t i v e thinning regimes (expressed as percentage) r e l a t i v e to the y i e l d under the current thinning regime. TABLE 49. ll^TZ^t^li^' l n f r M 2 ? « ( r e l a " v e t 0 c u r r e n t thinning regime) and other stand parameters at 40 year rotation age for diff e r e n t thinning regimes for C. l u s i t a n i c a S.I. 18 V ( l ) t o t a l Increase V(15) t o t a l Increase V(15) Thinning Increase V(15) main Increase DBH(40) " * m3 X ra3 X m3 Stand parameter Thinning regime control 849.7 771.4 188.5 582.9 47.6 10 A 20 30 40 966.1 883.0 821.5 784.9 13.7 3.9 -3.3 -7.6 894.1 806.1 744.1 709.2 15.9 4.5 -3.5 -8.1 45.9 152.1 282.9 309.6 -75.6 -19.3 50.1 64.2 848.2 654.0 461.2 399.6 45.5 12.2 -20.9 -31.4 38.3 44.7 48.9 49.1 10 B 20 30 *40 966.8 888.2 839.6 801.0 13.8 4.5 -1.2 -5.7 895.0 812.3 763.2 726.1 16.0 5.3 -1.1 -5.9 67.0 177.3 332.4 289.8 -64.4 -5.9 76.3 53.7 834.0 635.0 430.8 436.3 43.1 8.9 -26.1 -25.2 37.9 44.0 47.2 45.7 10 C 20 30 *40 969.7 899.1 861.3 816.4 14.1 5.8 1.4 -3.9 898.2 824.3 785.7 741.9 16.4 6.8 1.8 -3.8 75.9 211.9 380.7 324.5 -59.7 12.4 102.0 72.1 822.3 612.4 405.0 417.4 41.1 5.1 -30.5 -28.4 37.5 43.1 45.8 44.7 10 D 20 *30 *40 969.9 903.9 869.3 836.8 14.1 6.4 2.3 -2.7 897.2 829.3 794.5 753.3 16.3 7.5 3.0 -2.3 83.4 230.7 236.3 356.9 -55.8 22.4 25.4 89.3 813.8 598.6 558.2 396.4 39.6 2.7 -4.2 -32.0 37.2 42.5 43.1 43.6 10 E 20 *30 **40 974.4 917.7 878.5 847.9 14.7 8.0 3.4 -0.2 902.7 844.0 804.1 774.9 17.0 9.4 4.2 0.4 98.7 263.8 257.7 147.9 -47.6 39.9 36.7 -21.5 804.0 580.2 546.4 627.0 37.9 -0.5 -6.3 7.6 37.0 41.8 42.7 40.9 * Received only three thinnings. **Received only two thinnings. 232 Figure 34 shows the d i s t r i b u t i o n of the merchantable volume for the same thinning regimes between thinnings and f i n a l crop. The following observations may be noted: 1. The difference i n y i e l d between the d i f f e r e n t thinning i n t e n s i - t i e s i s considerably greater than the difference between the d i f f e r e n t thinning l e v e l s within a given thinning i n t e n s i t y . This confirms the e a r l i e r observation that thinning i n t e n s i t y i s a more c r i t i c a l consideration i n choosing a thinning regime. It also confirms the observation that within the range of thinning i n t e n s i t i e s considered i n t h i s study, Mollers theory that thinning has l i t t l e e f f e c t on volume y i e l d does not hold for C_. l u s i t a n i c a i n Kenya. 2. The 10% thinning i n t e n s i t y has the highest t o t a l and merchant- able volume y i e l d up to age 40 years. Most of th i s y i e l d comes at f i n a l harvest, with only about 5-10% (depending on thinning l e v e l ) recovered as thinning volume. Besides the shortcomings already mentioned regarding t h i s regime, i t should also be noted on Table 49 that i t also r e s u l t s i n the lowest stand DBH, a r e s u l t of the large number of stems at rotation age (see Figure 32). 3. Of the rest of the thinning i n t e n s i t i e s , 20% resulted i n the highest percent increase i n both t o t a l volume and t o t a l merchantable volume. The t o t a l merchantable volume increase ranged from 4.5% for thinning regime A:20 to 9.4% for thinning regime E:20. 233 FIGURE 34 DISTRIBUTION OF MERCHANTABLE VOLUME (M3/HA) FOR DIFFERENT THINNING REGIMES FOR C. LUSITANICA S.I. 18 Thinning Volume Final Stand Merchantable Volume to 15 cm Thinning Level A top dbh Thinning Level B 1000- a E o > 600 ' 200 I C 10 20 30 40 Intensity 800- n E « E 3 O > 200 IZZ A A C 10 20 30 40 Intensity Thinning Leve l C 1000- m ^ 4 0 0 I 2 0 0 o > i C 10 20 30 40 Intensity 600 ' n E 0) E 200 3 "o > Thinning Level V?7 C 10 20 30 40 Intensity Thinn ing Leve l E 234 4. Within a given thinning i n t e n s i t y , mean stand DBH decreases with increasing l e v e l of basal area before thinning i . e . decreases from thinning l e v e l A to E. This i s as expected since r a i s i n g the basal area l e v e l has the e f f e c t of increasing the length of the thinning cycle and so the stand i s at a higher stocking l e v e l . A l l the thinning regimes under the 20% thinning i n t e n s i t y could be considered for adoption depending on the production p r i o r i t y . For example i f the f i n a l crop i s the p r i o r i t y , thinning regime A:20 with highest f i n a l crop merchantable volume increase f 12.2% and highest DBH (among those considered) would be preferred. If on the other hand the d i s t r i b u t i o n of y i e l d over the r o t a t i o n i s a p r i o r i t y , thinning regime E:20 with merchantable volume of thinning increase of 39.9% would be preferred. The optimum regime however cannot be i d e n t i f i e d without economic inputs, as already mentioned elsewhere. It should be noted here that the above observations apply only to pr o d u c t i v i t y on s i t e index 18. P o s s i b i l i t y therefore existed that the e f f e c t s of these a l t e r n a t i v e regimes on productivity may be d i f f e r e n t on d i f f e r e n t s i t e q u a l i t y classes. This p o s s i b i l i t y i s explored i n the following section, using thinning regimes under the 20% thinning i n t e n s i t y only. (c) E f f e c t s of the new thinning regimes on produ c t i v i t y on d i f f e r e n t s i t e q u a l i t y classes Table 50 gives the produ c t i v i t y and stand mean DBH up to age 40 years for the various s i t e index classes for C_. l u s i t a n i c a under the TABLE 50. Volume productivity (m3/ha) and stand mean DBH (cm) up to age 40 years for various thinning l e v e l s at 20% thinning i n t e n s i t y for various s i t e index classes for £. l u s i t a n i c a r e l a t i v e to the current thinning regime Thinning S i t e V ( l ) t o t a l Increase V(15) t o t a l Increase V(15) thinning Increase V(15) main Increase DBH(40) 3 3 3 3 regime index m % m % m % . m % c m 12 637.5 567.3 175.0 392.3 43.7 Current 15 749.3 674.2 180.2 494.0 46.1 regime 18 849.7 771.4 188.5 582.9 47.6 21 943.0 860.6 208.1 652.5 48.3 24 1037.7 961.8 207.3 754.5 50.1 12 671.0 5.2 604.3 6.5 122.7 -29.9 481.6 22.8 42.9 15 781.2 4.2 710.3 5.4 134.1 -25.6 576.2 16.6 44.1 A: 20 18 883.0 3.9 806.1 4.5 152.1 -19.3 654.0 12.2 44.7 21 977.9 3.7 891.1 3.5 166.5 -20.0 724.6 11.0 45.1 24 1075.6 3.6 983.8 2.3 178.9 -13.7 804.9 6.7 45.9 12 675.6 6.0 610.0 7.5 150.1 -14.2 459.9 17.2 41.3 15 787.1 5.0 717.4 6.4 165.6 -8.1 551.8 11.7 43.1 B:20 18 888.2 4.5 812.3 5.3 177.3 -5.9 635.0 8.9 44.0 21 984.3 4.4 898.6 4.4 194.3 -6.6 704.3 7.9 44.3 24 1083.1 4.4 992.6 3.2 209.7 1.2 782.9 3.8 45.1 tsJ Table 50 (cont'd) Thinning S i t e V ( l ) t o t a l Increase V(15) t o t a l Increase V(15) thinning Increase V(15) main Increase DBH(40) regime index m3 % m3 % m3 % m % cm C:20 D:20 E:20 12 15 18 21 24 12 15 18 21 24 12 15 18 21 24 684.0 797.1 899.1 966.7 1097.1 685.7 797.4 903.9 1002.4 1099.2 698.4 811.6 917.7 1017.5 1115.2 7.3 6.4 5.8 5.7 5.7 7.6 6.4 6.4 6.3 5.9 9.6 8.3 8.0 7.9 7.5 619.2 9.1 177.3 1.3 441.9 12.6 40.9 728.3 8.0 196.6 9.1 531.7 7.6 42.2 824.3 6.8 211.9 12.4 612.4 5.1 43.1 912.3 6.0 232.1 11.5 680.2 4.2 43.5 1008.0 4.8 251.2 21.2 756.8 0.3 44.3 621.0 9.5 187.7 7.2 433.3 10.4 40.5 728.7 8.1 204.4 13.4 524.3 6.1 41.9 829.3 7.1 230.7 22.4 598.6 2.7 42.5 918.2 6.7 253.0 21.6 665.2 1.9 42.9 1010.1 5.0 263.1 26.9 747.0 -1.0 43.9 634.3 11.8 218.1 24.6 416.2 6.1 39.6 743.7 10.3 238.8 32.5 504.9 2.2 41.0 844.0 9.4 263.8 39.9 580.2 -0.5 41.8 934.3 8.6 288.7 38.7 645.6 -1.1 42.2 1027.2 6.8 302.2 45.8 725.0 -3.9 43.2 C O 237 current and a l t e r n a t i v e thinning regimes at 20% thinning i n t e n s i t y . Figure 35a,b and c shows the percent increase i n p r o d u c t i v i t y ( v ( l ) , V(15) main stand and V(15) thinning r e s p e c t i v e l y ) r e l a t i v e to the current thinning regime on the d i f f e r e n t s i t e index classes for the same thinning regimes. The following observations may be noted: 1. Figure 35a shows that a l l the thinning regimes resulted i n an increase i n the t o t a l merchantable volume, with highest increase on regime E:20. The increase decreased with increasing s i t e index class ranging between 11.8% for s i t e index class 12 to 6.8% for s i t e index c l a s s 24. This suggests the highest response to the new thinning regimes i s on the poor s i t e s , which confirms the suspicion expressed i n Chapter 2 Section 2.3 that the f u l l s i t e capacity on poor s i t e s may not be getting f u l l y u t i l i z e d under the current thinning regimes i n Kenya. 2. Figure 35b shows the increase i n f i n a l crop merchantable volume decreasing with increasing s i t e index c l a s s . This i s a r e s u l t of the greater difference i n DBH between the current and a l t e r n a t i v e thinning regime as s i t e index class increases (see Table 50). The response i n t h i s case i s highest on thinning regime A:20 and lowest on E:20. This i s as expected since, as pointed above, the highest basal area l e v e l s imply delaying thinnings and consequently lower f i n a l crop mean DBH to produce lower f i n a l crop merchantable volume. This e f f e c t i s reversed for the merchantable volume of thinnings (Figure 34c) which shows thinning regime E:20 with highest response because of the FIGURE 35 MERCHANTABLE VOLUME INCREASE (%) FOR DIFFERENT THINNING REGIMES (RELATIVE TO CURRENT THINNING REGIME) ON DIFFERENT SITE INDEX CLASSES 239 l a r g e r - s i z e d thinnings r e s u l t i n g from the delay. Figure 35c also shows the response increasing with increasing s i t e index class within each thinning regime. Thus the highest response with respect to thinning volume occurs on the best s i t e s . The r e s u l t s from t h i s section complement the observations of the previous section that thinning model A:20 would be preferable i f the f i n a l crop i s the p r i o r i t y , while E:20 would be preferred i f d i s t r i b u - t i o n of y i e l d over the whole r o t a t i o n i s a major concern. On the average, thinning model C:20 would be a good compromise as i t r e s u l t s i n p o s i t i v e increase of both merchantable f i n a l crop volume and volume of thinnings (see Figure 35b and c ) . The response i s dependent on the s i t e q u a l i t y c l a s s , the poorest s i t e s responding best to the f i n a l crop merchantable volume (12.6% on S.I. 12 compared to almost zero on S.I. 24) while the best s i t e s responds best to the merchantable volume of thinning (21.2% on S.I. 24 compared to 1.3% on S.I. 12). (d) E f f e c t s of i n i t i a l stocking on y i e l d under a s p e c i f i c thinning regime Under the current thinning model for C. l u s i t a n i c a (and for P. patula and P_. radiata as w e l l ) , the emphasis i s on the number of stems to be l e f t a f t e r thinning. Increasing the i n i t i a l number of stems say from 1200 to 1600 stems per hectare (sph) w i l l have very l i t t l e e f f e c t on the y i e l d of the stand with respect to volume production. However, a s l i g h t decrease i n DBH may r e s u l t . For example for s i t e Index 18 under the current thinning regime, C. l u s i t a n i c a DBH (to 40 years) decreased from 47.6 cm to 46.6 cm when i n i t i a l number of stems were increased from 240 1200 to 1600 sph. This decrease i s a r e s u l t of the i n d i v i d u a l stems having a lower DBH at time of f i r s t thinning, an e f f e c t that i s c a r r i e d forward to age 40 years. Under the new thinning model proposed for C_. l u s i t a n i c a i n t h i s study, the number, of stems to be removed at each thinning i s a function of the basal area removed i n the thinning and the mean DBH of thinnings (see equation 2.28). Increasing the i n i t i a l stocking, while holding both the basal area before thinning and the proportion of basal area to be removed constant r e s u l t s i n an Increase i n the l e v e l of stocking to be maintained a f t e r each thinning. This i n e v i t a b l y r e s u l t s i n a fixed amount of basal area being a l l o c a t e d among the higher number of stems and therefore a lower mean stand DBH. The net r e s u l t i s a decrease i n DBH to age 40 years since the stand w i l l be at a higher stocking l e v e l . This e f f e c t i s demonstrated on Table 51, which shows the number of stems and basal area before and a f t e r each thinning, DBH and merchantable volume (to 15 cm top diameter) at age 40 years for C_. l u s i t a n i c a s i t e index 18 under the thinning regime C:20 at the two stockings of 1200 and 1600 sph. The difference i n merchantable volume to age 40 years does not appear much d i f f e r e n t under the two stockings, mainly because the decrease of 4.7 cm i n DBH under the 1600 sph stocking i s almost made up for by the larger number of stems removed i n thinnings. However, t h i s decrease i n DBH i s quite appreciable. Thus, i n using t h i s thinning model, t h i s e f f e c t must be taken into consideration. I t should be noted that t h i s e f f e c t can be minimized by adjusting the proportion of basal area to be removed at f i r s t thinning so as to leave a reasonably lower 241 TABLE 51. E f f e c t of i n i t i a l stocking on y i e l d under thinning regime C:20 for £. l u s i t a n i c a i . e . thinning based on proportion of basal area to remove when a c r i t i c a l stand basal area i s equalled or exceeded. Stocking 1200 1600 Before After Before After 1st thinning 1200 850 1600 1124 (25.0) (20.0) (25.0) (20.0) 2nd thinning 850 621 1124 813 (35.0) (28.0) (35.0) (28.0) 3rd thinning 621 459 813 597 (45.0) (36.0) . (45.0) (36.0) 4th thinning 459 342 597 442 (45.0) (36.0) (45.0) (36.0) DBH (40*) cm 43.1 38.4 V(15) (40*) m3 824.3 812.6 SI = 18. o No. i n brackets are basal area i n m . 40* = age 40 years. 242 number of stems. Lowering the i n i t i a l number of stems would r e s u l t i n higher f i n a l crop DBH. 3.4 Summary on the Simulated Thinning Experiment The preceding analysis of the pro d u c t i v i t y under the d i f f e r e n t thinning regimes served to demonstrate the a b i l i t y of the y i e l d model EXOTICS as a v e r s a t i l e t o o l for stand manipulation to study stand development under various s i l v i c u l t u r a l schedules. Thinning i n t e n s i t y was i d e n t i f i e d as the most c r i t i c a l consideration when formulating a thinning p o l i c y , with l e v e l of stocking before thinning having very l i t t l e e f f e c t on t o t a l and merchantable volume y i e l d f o r C. l u s i t a n i c a up to age 40 years. From a p r a c t i c a l perspective, the analysis demonstrated that by adopting thinning regime C:20 for C_. l u s i t a n i c a , t o t a l merchantable thinning volume could be increased by between 1 to 21% for s i t e index classes 12 to 24 r e s p e c t i v e l y while at the same time increasing the f i n a l crop merchantable volume by between zero and 12.6% for s i t e index classes 24 to 12 respectively. This thinning schedule i s not necessarily optimal and i s but one of several a l t e r n a t i v e s that can be formulated f o r various s i l v i c u l t u r a l and economic constraints, including product mix, minimum DBH at c l e a r f e l l age and the a v a i l a b i l i t y of f a c i l i t i e s to u t i l i z e thinning volume. 4. Pulpwood Production Regime f o r Kenya For primary pulpwood production plantations, the basic management p o l i c y i s not to t h i n but to manipulate the i n i t i a l stand density 243 to maximize t o t a l volume production. 1?. patula i s the favoured species f o r pulpwood plantations and so i t i s not s u r p r i s i n g that for t h i s species, data for unthinned stands covered up to age 16.5 years (see Table 25 - P_. patula, rest of the country which includes Nabkoi, the zone for pulpwood production). _P. patula (Nabkoi) was therefore used to study volume y i e l d f o r pulpwood production under various stocking l e v e l s and establishment s i t e s at a 15 year r o t a t i o n age. Table 52 gives the t o t a l y i e l d at age 15 years f o r the various stocking l e v e l s , s i t e index classes and establishment s i t e s . The percentage decreases i n volume under grassland establishment s i t e s are . also given. Figure 36a and b shows the CAI and MAI curves for the various stocking l e v e l s at the two establishment s i t e s for s i t e index class 21 while Figure 37 shows the DBH development under shamba establishment s i t e for s i t e index 21 (average s i t e index class for P. patula). As expected, Table 52 shows that volume y i e l d increases as s i t e index class increases and with increase i n number of stems for both Shamba s i t e s and grassland s i t e s , with the y i e l d of the l a t t e r estab- lishment s i t e being lower than the former for a given s i t e index c l a s s . The percentage decrease on grasslands also decreases with increasing s i t e index class and increasing stocking. This i s an e f f e c t of the constant reduction i n height being expressed as a percentage of an increasing height as s i t e index class increases or as basal area increases (as a r e s u l t of increase i n number of stems). The important point to note however i s that y i e l d under grassland establishment s i t e s i s on the average about 16% lower than on shamba establishment s i t e s f o r TABLE 52. Total volume y i e l d V ( l ) f or P. patula (Nabkoi) by s i t e index classes for various stocking levels and establishment s i t e s up to age 15 years. S.I. s.p.h. 15 18 21 24 27 1000 1200 1400 1600 201.8 233.0 258.7 279.4 150.6 (25.4) 176.3 (24.3) 198.5 (23.3) 217.1 (22.3) 272.8 309.5 338.3 360.1 217.0 (20.4) 249.5 (19.4) 276.0 (18.4) 297.1 (17.5) 347.8 388.5 418.7 440.4 289.0 (16.9) 326.8 (15.9) 355.9 (15.0) 377.9 (14.2) 425.4 468.6 498.8 519.3 364.7 (14.3) 406.1 (13.3) 436.4 (12.5) 457.8 (11.8) 504.5 548.6 577.9 596.6 442.7 (12.2) 486.2 (11.4) 516.3 (10.6) 536.4 (10.0) No. i n bracket indicate % decrease i n y i e l d under grassland r e l a t i v e to Shamba y i e l d . S = Shamba s i t e s . G = Grassland s i t e s . FIGURE 36 C.A.I. AND M.A.I. CURVES FOR VARIOUS STOCKING LEVELS FOR P. PATULA SITE INDEX 21 S 3 FIGURE 37 DIAMETER/AGE RELATIONSHIP AT VARIOUS STOCKING LEVELS FOR SITE INDEX 21 FOR P. PATULA (NABKOI) AGE IN YEARS FROM PLANTING 247 a stocking of 1200 sph. This i s an important f i n d i n g h i t herto not recognized by the forest managers i n Kenya. Figure 36a and b shows the CAI and MAI curves for higher stocking l e v e l s being higher than for the lower l e v e l s , with those for grassland s i t e s being lower than for Shamba s i t e s , again as expected. However, at some points i n time, the CAI curves for higher stocking le v e l s are seen to f a l l below those of lower curves. This indicates the e f f e c t s of competition. This i s confirmed on Figure 37 which shows the DBH development on a l l stocking l e v e l s being almost the same at the lower ages ( i n absence of competition) but with the curves for the higher stocking l e v e l s f a l l i n g below those of lower stocking l e v e l s much e a r l i e r : 9, 10 and 11 years for stockings 1600, 1400 and 1200 respec- t i v e l y . This i s as expected since competition i s expected to set i n e a r l i e r on stands with higher stocking. I t should however be noted that the age at which CAI curves of a given stocking l e v e l f a l l s below that of the next lower stocking l e v e l i s l a t e r than the age at which that stockings DBH curve f a l l s below the general growth trend curve. This i s because the CAI i s not only a frunction of DBH but also of the number of stems. Thus when DBH development s t a r t s slowing down due to competi- t i o n , the CAI i s maintained above that of the next lower stocking l e v e l due to the higher number of stems u n t i l a c r i t i c a l DBH i s reached below which the higher number of stems do not compensate the CAI s u f f i c i e n t l y . The true onset of competition therefore i s marked not by the age at which CAI s t a r t s f a l l i n g below that of the lower stocking l e v e l but by the age at which DBH development s t a r t s slowing down due to competition. 2 4 8 The conceptual basis for the use of DBH development as a c r i t e r i a for stand density control i s s i m i l a r to that of the maximum size-density proposed by Drew and Fl e w e l l i n g (1977, 1979). It therefore r e l i e s on constant stocking experiments to provide the c r i t i c a l size-density values at which stands should be thinned. As has just been demon- stra t e d , EXOTICS provides a us e f u l t o o l for conducting constant stocking experiments, provided data from unthinned stands i s a v a i l a b l e , both for c a l i b r a t i n g the model and f o r v a l i d a t i o n . As i t is, the model i s v a l i d f or constant stocking experiments for up to 10 years ( f o r C_. l u s i t a n i c a and P_. radiata) and 15 years ( f o r P_. patula) and for stocking between 1000 to 1600 sph for a l l species. 249 CHAPTER 5 SUMMARY: THEORETICAL AND PRACTICAL ASPECTS OF THIS STUDY, SUGGESTED FUTURE DEVELOPMENTS AND APPLICATION As stated i n the introductory chapter, the main objective of t h i s study was to advance our knowledge of the growth and y i e l d of the three species: C_. l u s i t a n i c a , P_. patula and P_. rad i a t a under the p r e v a i l i n g c l i m a t i c , edaphic and s i l v i c u l t u r a l regimes i n Kenya. This objective was pursued i n three phases: 1. A study of the growth and y i e l d r e l a t i o n s h i p and derivation of the appropriate growth functions. 2. Construction of a growth and y i e l d model as a means of pr e d i c t i n g growth and y i e l d under various p h y s i c a l , b i o l o g i c a l and management constraints. 3. An a n l y s i s of the growth and y i e l d of these species under the present and a l t e r n a t i v e s i l v i c u l t u r a l management regimes. This chapter summarizes the accomplishments of the study, i t s t h e o r e t i c a l and p r a c t i c a l implications and suggests areas for future development and a p p l i c a t i o n . The summary i s presented by phases as they occur i n the study. 1. Growth and Y i e l d Relationships D i f f e r e n t aspects of stand development were studied and relevant growth functions derived as follows: 250 Dominant height development: Stand dominant height (defined as the mean height of the 100 largest diameter tres per hectare) was studied as a function of age from planting and s i t e index (defined as dominant height of the stand at age 15 years). Two nonlinear functions, the Chapman- Richards (equation 2.5) and the modified Weibull function (equation 2.6) were considered and the former found to be more appropriate, based on the asymptotic standard deviation. The f i n a l equation was of the form: Hdom - bod - e - b l A S I ) b 2 where A = Age of stand from planting SI = S i t e index Hdom = ^ t a n c * dominant height i n meters bQ» bj and b£ are regression constants. On v a l i d a t i o n , t h i s function was accepted for C_. l u s i t a n i c a and P_. ra d i a t a . For P. patula, height development was found to d i f f e r from one region to another, suggesting that height develop- ment was polymorphic. Covariance analysis for the development curves for each region indicated that for at least some of the regions, the growth curves were s i g n i f i c a n t l y d i f f e r e n t at the .05 si g n i f i c a n c e l e v e l . This phenomenon was suspected to be due to edaphic differences i n the d i f f e r e n t regions but more research i s required i n t h i s regard. Data f or t h i s species was s t r a t i f i e d by geographical regions and a l i n e a r quadratic model (equation 2.17) f i t t e d : 251 Hdom = b 0 + b l S I + b 2 A + b 3 A 2 + b 4 A S I + b 5A 2SI where va r i a b l e names are as above. The f i n d i n g that height development for P_. patula i s poly- morphic i s unique to th i s study. A l l previous studies on t h i s species have used one set of s i t e index curves for the whole country. The p r a c t i c a l i m p l i c a t ion of t h i s f i n d i n g i s that planta- tions may be of the same s i t e index class at a given point i n time but that the development curves may be d i f f e r e n t . Hence the need for d i f f e r e n t s i t e index curves f o r d i f f e r e n t geographical regions. For both P. patula and P_. r a d i a t a , two types of establishment s i t e s are used - Shamba and grassland. Dominant height under each type of establishment was studied, again using covariance a n a l y s i s . The conclusion was that up to age 20 years, height development under grassland was s i g n i f i c a n t l y (at .05 l e v e l ) lower than under Shamba. This f i n d i n g , which hitherto had not been recognised, has two important p r a c t i c a l s i g n i f i c a n c e : (1) Where the forest manager has a choice, Shamba planting i s to be preferred. (2) Any growth and y i e l d model for the two pine species should have establishment s i t e as one of the input v a r i a b l e s . (b) M o r t a l i t y , stand density development and thinning practices i n Kenya: Due to the intensive nature of stand management, including thinning, m o rtality was not considered i n t h i s study. Stand density 2 5 2 development and thinning practices were i n t e n s i v e l y studied and the following important findings noted: (1) Under the present s i l v i c u l t u r a l p r e s c r i p t i o n s , the stand density Index (S%) varies between 18-30% for C_. l u s i t a n i c a and 15-25% for JP. patula and P_. r a d i a t a . (2) Stand density index varies for d i f f e r e n t s i t e classes for a l l species, with wide spacing developing on poor s i t e s and over- crowding on good s i t e s . The second observation was suspected to have an influence on pro d u c t i v i t y . This p o s s i b i l i t y was explored further i n Chapter 4. Thinning types for Kenya, based on the c l a s s i f i c a t i o n of Eide DBH(T) and Langsaeter (Braathe 1957) (based on DBTJ('B'J.) r a t i o ) was found to be 0.80 for P_. radiata (no d e f i n i t e low or crown thinning), 0.85 and 0.88 f o r C_. l u s i t a n i c a and P_. patula respectively, which borders on the lower side of crown thinning. DBH of thinning was expressed as a function of DBH before thinning and i n t e n s i t y of thinning, measured as a r a t i o e i t h e r of the number of stems thinned over number of stems before thinning or as basal area thinned over basal area before thinning. These functions were necessary f or the simulation of thinnings i n the l a t t e r part of the study. 253 Basal area development before f i r s t thinning: Basal area before thinning was described as a function of stand age, stand dominant height (which includes e f f e c t s of s i t e q u a l i t y ) and number of stems using nonlinear equation 2.29 as follows: where BA = Basal area i n m^/ha A = Age i n years from planting H = Stand dominant height i n meters N = No. stems/ha-. An unexpected fin d i n g from t h i s study was that basal area development f or P_. patula from Kinale region d i f f e r e d from that for the rest of the country, a phenomenon that could not be explained from any of the b i o l o g i c a l or c l i m a t i c factors available to t h i s study. More i n v e s t i g a t i o n w i l l be required to determine the under- l y i n g factor or fact o r s : Basal area development i n thinned stands: Basal area increment rather than basal area per se was studied i n thinned stands as i t i s r e l a t i v e l y independent of the e f f e c t s of thinning. Nonlinear equation 2.31 which i s an extension and more generalized form of the basal area increment equation f o r P_. radiata f o r New Zealand (Clutter and A l l i s o n 1974) was used i n t h i s study: BAI = e ( b l A b 3BA _ b4 + b5s) 254 where A = Age i n years at the end of growth period BA = Stand basal area at beginning of growth period i n m2 per ha S = Stand density index, calculated as: S = §_ x 100 Hdorn /l0,000 where a = / — 1 V N N = No. stems per hectare For P_. r a d i a t a , the term S% was found non-significant at .05 l e v e l . This term was also not included i n the New Zealand equation which suggests that i t s e f f e c t s on basal area increment may be species s p e c i f i c . Diameter d i s t r i b u t i o n : The Weibull p r o b a b i l i t y density function was selected to model diameter d i s t r i b u t i o n for several reasons: (1) I t i s simple and mathematically handy. (2) I t s a b i l i t y to assume a v a r i e t y of curve shapes. (3) On f i t t i n g t h i s model to 58 diameter d i s t r i b u t i o n histograms, only four pl o t s (with a multimodal histogram) were rejected as not having a Weibull d i s t r i b u t i o n , based on a goodness of f i t t e s t . The cumulative form of t h i s d i s t r i b u t i o n (equation 2.35) was f i t t e d to the data: 255 F(x) = 1 - for x > xg 0 < F(x) < 1 where x = observed diameter class x 0 = minimum observed diameter b and c are the Weibull constants. The estimated parameters for each plot were then correlated with stand parameters. F i n a l equations for the prediction of these parameters were: where x = Mean stand DBH i n cm. XQ = Mininum stand DBH i n cm. b and c are predicted Weibull parameters. Stand volume c a l c u l a t i o n s : Tree volume equations for the three species discussed i n t h i s study have been i n existence i n Kenya since 1969. These have been the basis for the volume tables and the y i e l d tables and so were deemed to have passed the test of time i n the f i e l d . Therefore they were used i n t h i s study to derive stand volume. x Q = b Q + bx5 + b 2 I b = b Q + b x ( x - x Q ) + b 2 J i c = b 0 + b 1 256 In the study of the dominant height development, the procedure adopted i m p l i c i t l y implied that the s i t e index curves for the three species were anamorphic. Therefore the v a l i d a t i o n procedure amounted to a test of the n u l l hypothesis that dominant height development i s anamorphic. This hypothesis was rejected for P. patula only. Another important t h e o r e t i c a l consideration i n t h i s study re l a t e d to basal area development i n thinned stands. A generally accepted supposition i n f o r e s t r y l i t e r a t u r e i s that basal area increment i s not a f f e c t e d by changes i n stand density over a wide range of d e n s i t i e s , a theory that has come to be known as Moller's theory ( B a s k e r v i l l e 1965). This theory i s consistent with i n t u i t i o n since a decrease i n number of stems on a unit area basis i s compensated for by an increase i n diameter growth of the remaining trees and thus increment i n basal area should vary very l i t t l e . In t h i s study however, basal area increment was found to be a function of the basal of the stand at the beginning of the growth period for a l l three species. Inasmuch as basal area i s a measure of stand density, i t was concluded that for these species and at the present l e v e l of stand d e n s i t i e s , Moller's theory does not hold. 2. Construction of the Growth and Y i e l d Model A growth and y i e l d model EXOTICS was constructed with the main objective being to provide a planning and s i l v i c u l t u r a l management t o o l that would allow manipulation of a single stand to meet forest manage- ment objectives. Written i n FORTRAN IV G l e v e l which Is compatible with IBM System/360 and System/370, EXOTICS i s an i n t e r a c t i v e whole-stand/ distance independent model designed to handle a single even-aged 257 monospecific stand at a time. The model also provides diameter d i s t r i b u t i o n by 3 cm diameter classes which allows output of the f i n a l main stand y i e l d by size classes. The following features make t h i s model unique i n the Kenya scene: 1. Refinement of the s i t e index curves through i n c l u s i o n of establishment s i t e as an input v a r i a b l e , and the polymorphic growth pattern f o r P_. patula. These are new findings from t h i s study. 2. The model allows for three thinning options, a l l of which can be addressed i n the same simulation run: 0 = No thinning. 1 = Thinning based on number of stems to leave a f t e r thinning when a predetermined age or stand dominant height i s equalled or exceeded. 2 = Thinning based on a proportion of basal area to remove when a predetermined basal area i s equalled or exceeded. This feature allows f o r f l e x i b i l i t y i n thinning decision and for use of d i f f e r e n t options at d i f f e r e n t stages of stand development. 3. The i n t e r a c t i v e aspect of the model makes i t a very handy t o o l f o r s i l v i c u l t u r a l research. On v a l i d a t i o n , EXOTICS was found to be acceptable within the following l i m i t a t i o n s : 258 1. The model was found to have no apparent bias for a l l three species. 2. 95% confidence l i m i t s f o r the difference between observed and simulated volumes were C. l u s i t a n i c a : ± 16% P. patula : ± 20% P. radiata : ± 1 7 % This was a l o t of improvement on VYTL-2 which had an average 95% confidence l i m i t s of ±30% (Alder 1977). 3. The model was found acceptable f o r error s p e c i f i c a t i o n of between 20-25% (C. l u s i t a n i c a and P. radiata) and 25-30% for P_. patula unless a l-in-20 chance has occurred. This compared well with the accuracy of some models already i n operation: FOREST and SHAFT (Ek and Monserud 1979). 4. Nearly a l l v a r i a b i l i t y for C_. l u s i t a n i c a and P_. radiata was from the basal area component. Thus, for these two species, future refinement of the model should be directed to t h i s component. For P_. patula, t o t a l v a r i a b i l i t y i n the predicted volume was contributed to by dominant height and basal area components i n the r a t i o s of 1:2. For t h i s species future refinement should be addressed to both components. As noted e a r l i e r i n the study, the v a l i d a t i o n process constituted a test of the n u l l hypothesis that the model i s an acceptable approxima- t i o n of the r e a l system. In t h i s case the n u l l hypothesis was accepted within the above stated l i m i t a t i o n s . 259 3. S i l v i c u l t u r a l Management Models f o r Kenya The current thinning p o l i c y f or Kenya aimed at production of l a r g e - sized sawlog crop i n as short a r o t a t i o n as possible at the expense of some loss i n t o t a l y i e l d was discussed and found to have been overtaken by events. A new thinning p o l i c y based on the concept of maximum volume production was proposed as more appropriate i n the presence of l i m i t e d f o r e s t land, increasing demand for wood products and increased i n t e g r a - t i o n of the f o r e s t r y i n d u s t r i a l sector. Using the y i e l d model EXOTICS, growth and y i e l d under the current thinning p r e s c r i p t i o n s was studied and the following main observations noted: 1. Hart's stand density Index was found to be inadequate as a guide to when a stand i s due for thinning for C_. l u s i t a n i c a and P_. patula as t h i s led to development of d i f f e r e n t l e v e l s of s i t e occupancy (with respect to basal area) on d i f f e r e n t s i t e q u a l i t y classes. However t h i s index was found to be appro- p r i a t e for P_. r a d i a t a . 2. For C_. l u s i t a n i c a and P_. r a d i a t a , f i r s t thinning was found to have no apparent e f f e c t on volume CAI. Subsequent thinnings were found to have marked e f f e c t s on CAI. For P_. patula, a l l thinning had marked e f f e c t s on volume CAI. 3. Consequent to 2 above, i t was concluded that Moller's theory (Moller 1947) that thinning has no appreciable e f f e c t s on t o t a l volume y i e l d does not hold for these species at the present l e v e l of s i l v i c u l t u r a l management. 260 Using C_. l u s i t a n i c a , a thinning experiment was designed to i n v e s t i - gate the e f f e c t s of a l t e r n a t i v e thinning regimes on growth and y i e l d . Five thinning l e v e l s (based on basal area l e v e l s before thinning) were a r b i t r a r i l y selected f or study. Within each l e v e l , four thinning i n t e n - s i t i e s : 10, 20, 30 and 40% of the stand basal area to be removed were investigated over the range of s i t e index classes for C_. l u s i t a n i c a . Several important findings were observed: 1. Within the range of thinning l e v e l s and thinning i n t e n s i t i e s considered, thinning Intensity i s the most important considera- t i o n with respect to volume y i e l d . Thinning l e v e l has very l i t t l e e f f e c t on both the MAI and the t o t a l y i e l d of merchant- able volume up to age 40 year. 2. The thinning i n t e n s i t y of 20% was the most appropriate, based on b i o l o g i c a l r o t a t i o n age. Without economic data however, i t was not possible to i d e n t i f y the optimum thinning l e v e l under t h i s thinning i n t e n s i t y . B i o l o g i c a l r o t a t i o n age ranged be- tween 22 to 28 years for s i t e index class 18 depending on thinning l e v e l , shortest r o t a t i o n associated with the lower l e v e l s of basal area before thinning. 3. By adopting thinning model C:20, i t was possible to increase the t o t a l merchantable y i e l d f o r C_. l u s i t a n i c a (up to age 40 years) by between 5 to 10% depending on the s i t e q u a l i t y cla s s . The poorest s i t e q u a l i t y c l a s s (12) responded best to f i n a l crop merchantable volume (12.6% increase compared to 0.3% for 261 S.I. 24) while the best s i t e q u a l i t y class (24) responded best to merchantable volume of thinning (21.2% compared to 1.3% for S.I. 12). This model was not n e c e s s a r i l y the optimum but one of the possible a l t e r n a t i v e s depending on both the economic and b i o l o g i c a l constraints that may be imposed. 4. Using the thinning model C:20, i t was observed that increasing the i n i t i a l stocking resulted i n lowering the f i n a l stand DBH but had very l i t t l e e f f e c t on t o t a l merchantable volume up to age 40 years. Using P_. patula (Nabkoi), y i e l d f o r pulpwood production regimes under d i f f e r e n t stand d e n s i t i e s was studied under the two establishment s i t e s . Y i e l d under grassland s i t e s was found to be between 10-25% lower than that on shamba s i t e s depending on s i t e index class and the stocking. High stocking and better s i t e q u a l i t i e s were associated with lower percentage decrease. E f f e c t s of competition on DBH and CAI development were also studied using simulated r e s u l t s from these pulpwood regimes. Besides i n d i c a t i n g possible a l t e r n a t i v e management strategies for increasing y i e l d , r e s u l t s from t h i s phase of the study served to demonstrate the use of EXOTICS as a s i l v i c u l t u r a l research t o o l with the s i l v i c u l t u r a l models as the framework within which the y i e l d model must operate. The model appears to provide both r e a l i s t i c and r e l i a b l e r e s u l t s with respect to stand development, thus providing the framework on which economic analysis can be based. 262 4. Future Research and Development A r i s i n g from t h i s Study One of the c h a r a c t e r i s t i c s of most research undertakings i s that they tend to unearth areas that require further research and/or r e f i n e - ment. Since most research i s proscribed within s p e c i f i e d time, f i n a n - c i a l or other l i m i t a t i o n s , i t i s not always possible to address these areas i n the p a r t i c u l a r study. The following areas were i d e n t i f i e d or are anticipated for improvement on the present study: 1. More data are required on the mature phase of plantation development for a l l species and, i n p a r t i c u l a r , those for P_. patula are urgently required. Immediately, these data could be obtained from temporary sample plots but i n the long run, permanent sample plot data w i l l be more appropriate. This information i s necessary to provide a basis for estima- t i o n of the asymptotes i n the s i t e index curve and basal area development functions. 2. Constant stocking experiments are required covering a wide range of densities and re p l i c a t e d over a wide range of s i t e q u a l i t i e s . This w i l l provide a basis f o r the study of e f f e c t s of density and s i t e q u a l i t y on basal area development, maximum basal area and maximum size-density concept: a l l of which are necessary for stand density c o n t r o l . 263 3. More detai l e d studies on factors a f f e c t i n g basal area develop- ment are needed. In p a r t i c u l a r , c l i m a t i c factors and e f f e c t s of thinning need to be investigated for possible i n c l u s i o n i n the basal area increment function. 4. Refinement of the diameter d i s t r i b u t i o n model through i n c l u s i o n of data covering the early phase of plantation development. An a l t e r n a t i v e to the procedure used i n t h i s study may also include modeling basal area frequency by diameter classes rather than stem frequency. This requires d e t a i l e d study of basal area development over the whole r o t a t i o n . 5. Inclusion of an economic analysis model to provide economic analysis of any chosen s i l v i c u l t u r a l treatment. In t h i s connection, optimization techniques such as network analysis or c r i t i c a l pathway analysis should be investigated to allow for optimization of s i l v i c u l t u r a l treatments for given economic and b i o l o g i c a l constraints. This requires more involved research and therefore may not be Immediately tenable. 6. Future studies should be addressed to the e f f e c t s of subsequent rotations on stand growth and y i e l d . 5, A p p l i c a t i o n of the Results The r e s u l t s from t h i s study w i l l be immediately applicable i n three main areas: 264 Production of up-to-date y i e l d tables for any s p e c i f i e d s i l v i - c u l t u r a l treatment, s i t e index and establishment s i t e : These tables are required by the Kenya Forest Department for day-to- day planning and management purposes. For research i n formulation of a l t e r n a t i v e s i l v i c u l t u r a l treatments. The r o l e of the y i e l d model EXOTICS w i l l be to provide quantitative information on stand response to various treatments. This a p p l i c a t i o n w i l l be useful to the Forest Research Section of the Kenya Forest Department, the Kenya A g r i c u l t u r a l Research I n s t i t u t e and the s t a f f at the Forest Department of the University of Nairobi. For teaching purposes at the University of Nairobi Forestry Department: The r e s u l t s from t h i s study and the author's experience on growth and y i e l d w i l l be transferred to the University of Nairobi to the benefit of the students and other interested p a r t i e s . For the f i r s t time, r e l i a b l e information on stand development for the three species w i l l be available as a basis for teaching and further research i n the f i e l d of growth and y i e l d . In addition, EXOTICS could form a basis for student's experiments, p a r t i c u l a r l y i n s i l v i c u l t u r e and mensuration experiments. 265 6. Conclusion A growth and y i e l d study on (3. l u s i t a n i c a , P_. patula and P. radiata under the p r e v a i l i n g c l i m a t i c , edaphic and s i l v i c u l t u r a l management regimes i n Kenya has been presented. This study represents a s i g n i f i - cant extension of our knowledge of the development of these species, e s p e c i a l l y with respect to y i e l d under d i f f e r e n t s i t e q u a l i t i e s and s i l v i c u l t u r a l regimes, in c l u d i n g establishment s i t e . A growth and y i e l d model EXOTICS was developed as a planning and s i l v i c u l t u r a l research t o o l . This model represents an improvement on an e a r l i e r model VYTL-2, both i n terms of p r e c i s i o n and i n f l e x i b i l i t y i n handling thinning decision models. It i s therefore hoped that t h i s model w i l l be an invaluable aid to forest management (producing y i e l d t a b l e s ) , s i l v i c u l t u r a l research ( i n formulating and monitoring stand response to various treatments) and as a teaching a i d . The present thinning p o l i c y i s discussed and found to be i n c o n s i s - tent with the current conditions i n the country. 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