- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Yield-density responses in monocultures and mixtures...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Yield-density responses in monocultures and mixtures of Beans (Phaseolus vulgaris L.) and Beets (Beta.. 1991
pdf
Page Metadata
Item Metadata
Title | Yield-density responses in monocultures and mixtures of Beans (Phaseolus vulgaris L.) and Beets (Beta vulgaris L.) |
Creator |
Mchaina, Grace Masala |
Publisher | University of British Columbia |
Date Created | 2011-02-03T16:53:16Z |
Date Issued | 2011-02-03T16:53:16Z |
Date | 1991 |
Description | Interference among neighbouring plants, often due to competition for limited resources, is central to subjects such as yield-density relationships, intercropping, self-thinning in dense plant stands and low reproductive yield in certain crops. An experiment was conducted to investigate plant interference in associated populations of beans {Phaseolus vulgaris L.) and beets [Beta vulgaris L.). Plants of the two species were grown at different total densities and at different mixture proportions in a randomized complete block design. Several analytical procedures were used to interpret and define treatment effects. The analysis of variance indicated that yield was significantly reduced with either increasing total population density or increasing bean proportions in mixtures. The interactions of total population density and mixture proportions were only occasionally significant. Parameters of non-linear models used to define yield-density relationships indicated that beans were the superior competitor, both against themselves and against beets. The model parameters were also used to determine differential yield responses on total dry weight, leaf dry weight, leaf number and leaf area in the bean-beet mixtures. Yield advantage was observed in leaf dry weight and leaf number when model parameters were used in calculating land equivalent ratios whereas total dry weight and leaf area showed yield disadvantage. Using observed values to calculate land equivalent ratios indicated yield advantage in all four variables. Plant size inequalities, as determined by the Gini coefficient tended to decrease in beet monocultures with increasing population density. In monocultures of beans and in the bean-beet mixtures, plant size distribution was not systematically changed by density and mixture treatments. Yield component analysis indicated that the variation in total yield due, to either population density or mixture treatments increased with age; the variation due to the population density by mixture proportions Interaction remained relatively constant throughout the growing season. Leaf number per plant was the yield component which was most frequently a significant source of yield variation both in the forward and backward yield component analysis. Plant growth analysis indicated that leaf area ratio and specific leaf weight were higher at higher population densities and at higher bean proportions. Harvest index decreased with increasing population density and with increasing proportions of the competing species in beets. Absolute growth, relative growth and unit leaf rates increased with time and declined after reaching a peak at about 68 days after planting. Both the lowest population density of 16 plants m ̄² and the mixture treatment with the least proportion of beans had the greatest increase in absolute growth, relative growth and unit leaf rates. Allometric relationships between total plant dry weight and any secondary measure per plant were influenced in different ways by density and mixture treatments and by time of harvest. The composition of models also varied considerably. The interpretation of plant interference, therefore is strongly influenced by the choice of plant characteristics which are measured, and by the time of measurement. |
Subject |
Plant Interference |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project [http://www.library.ubc.ca/archives/retro_theses/] |
Date Available | 2011-02-03T16:53:16Z |
DOI | 10.14288/1.0100349 |
Degree |
Doctor of Philosophy - PhD |
Program |
Plant Science |
Affiliation |
Land and Food Systems, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/31038 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0100349/source |
Download
- Media
- UBC_1991_A1 M33_5.pdf [ 13.98MB ]
- Metadata
- JSON: 1.0100349.json
- JSON-LD: 1.0100349+ld.json
- RDF/XML (Pretty): 1.0100349.xml
- RDF/JSON: 1.0100349+rdf.json
- Turtle: 1.0100349+rdf-turtle.txt
- N-Triples: 1.0100349+rdf-ntriples.txt
- Citation
- 1.0100349.ris
Full Text
YIELD-DENSITY RESPONSES IN MONOCULTURES AND MIXTURES OF BEANS [PHASEOLUS VULGARIS L.) AND BEETS {BETA VULGARIS L.) by GRACE MASALA MCHAINA B.Sc, University of Zambia, M.Sc, University of Manitoba A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE DEPARTMENT OF PLANT SCIENCE We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1991 ©GRACE MASALA MCHAINA, 1991 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of K f t ^ 1 S c ^ ^ c f - The University of British Columbia Vancouver, Canada Date 9-1- - QCo - ° \ y DE-6 (2/88) i i ABSTRACT Interference among neighbouring plants, often due to competition for limited resources, is central to subjects s u c h as yield-density relationships, intercropping, self-thuiriing i n dense plant stands a n d low reproductive yield i n certain crops. A n experiment was conducted to Investigate plant interference i n associated populations of beans {Phaseolus vulgaris L.) and beets [Beta vulgaris L.). Plants of the two species were grown at different total densities a n d at different mixture proportions i n a randomized complete block design. Several analytical procedures were used to interpret and define treatment effects. Th e analysis of variance indicated that yield was significantly reduced with either increasing total population density or increasing bean proportions i n mixtures. T he interactions of total population density a n d mixture proportions were only occasionally significant. Parameters of non-l inear models used to define yield-density relationships indicated that beans were the superior competitor, both against themselves a n d against beets. Th e model parameters were also used to deterrnine differential yield responses on total dry weight, leaf dry weight, leaf n u m b e r a n d leaf area i n the bean-beet mixtures. Yield advantage was observed i n leaf dry weight a n d leaf n u m b e r when model parameters were used i n calculating land equivalent ratios whereas total dry weight a n d leaf area showed yield disadvantage. U s i n g observed values to calculate land equivalent ratios indicated yield advantage i n al l four variables. Plant size inequalities, as detennined by the G i n i coefficient tended to decrease i n beet monocultures with increasing population density. In monocultures of beans and i n the bean-beet mixtures, plant size distri- but ion was not systematically changed by density and mixture treatments. Yield component analysis indicated that the variation i n total yield due, to either population density or mixture treatments increased with age; the variation due to the population density by mixture proportions Interaction remained relatively constant throughout the growing season. Leaf number per plant was the yield component which was most frequently a significant source of yield variation both i n the forward and backward yield component analysis. Plant growth analysis indicated that leaf area ratio and specific leaf weight were higher at higher population densities and at higher bean proportions. Harvest index decreased with increasing population density a n d with Increasing proportions of the competing species i n beets. Absolute growth, relative growth a n d unit leaf rates increased with time a n d declined after reaching a peak at about 68 days after planting. Both the lowest population density of 16 plants n r 2 and the mixture treatment with the least proportion of beans h a d the greatest increase i n absolute growth, relative growth a n d uni t leaf rates. Allometric relationships between total plant dry weight and any secondary measure per plant were influenced i n different ways b y density and mixture treatments and by time of harvest. T h e composition of models also varied considerably. T he interpretation of plant interference, therefore is strongly influenced b y the choice of plant characteristics which are measured, a n d by the time of measurement. i v TABLE OF CONTENTS ABSTRACT il TABLE OF CONTENTS iv LIST OF TABLES ix LIST OF FIGURES xiii LIST OF APPENDICES xvii SYMBOLS AND TERMINOLOGY xxii ACKNOWLEDGEMENTS xxv 1. INTRODUCTION 1 2. LITERATURE REVIEW 3 2.1 The Crops Used in This Study 3 2.1.1 Beans 3 2.1.2 Beets 5 2.2 Crop Mixtures 7 2.3 Plant Interference 10 2.3.1 Nature of Interference 10 2.3.2 Some Experimental Approaches to the Study of Plant Interference 12 2.3.3 Yield-density Relationships 16 2.3.4 Monoculture Models 16 2.3.5 Mixture Models 20 2.4 Differential Yield Responses 21 2.5 Size Hierarchies 23 2.6 Plant Growth Analysis 25 2.7 Allometry 28 2.8 Objectives of the Thesis 31 3. MATERIALS AND METHODS 32 3.1 Experimental Layout and Crop Production Procedures 32 3.2 Harvests and Primary Data Collection 34 3.3 Grading Procedures for Bean Pods 35 3.4 Analytical Procedures 36 3.4.1 An Overview of the Data Analysis 36 3.4.2 Analysis of Variance 37 3.4.3 Yield-density Relationships 37 3.4.3.1 Intraspecific Interaction 37 3.4.3.2 Interspecific Interaction 37 3.4.4 Differential Yield Response 38 3.4.5 Plant Hierarchies 39 3.4.6 Yield Component Analysis 39 3.4.7 Plant Growth Analysis 41 3.4.8 Plant Allometric Relationships 41 4. RESULTS 43 4.1 An Overview of the Results 43 4.2 Visual Observations 43 4.3 Analysis of Variance 44 4.3.1 Homogeneity of Variance Test 44 4.3.2 General Results from the Analysis of Variance 44 4.3.2.1 Beans 1984 44 4.3.2.2 Beans 1987 54 4.3.2.3 Beets 1984 57 4.3.2.4 Beets 1987 57 4.3.3 Summary of Analysis of Variance Results 64 v i 4.4 Yield-density Relationships 64 4.4.1 Yield-density Regressions 68 4.4.2 Summary of Yield-density Relationships Results 68 4.5 Differential Yield Responses of Mixtures 77 4.5.1 Summary of Differential Yield Responses Results 83 4.6 Size Hierarchies 83 4.6.1 Summary of Size Hierarchies Results 92 4.7 Yield Component Analysis 92 4.7.1 Beans 1984 92 4.7.2 Beans 1987 96 4.7.3 Beets 1984 96 4.7.4 Beets 1987 ~ 101 4.7.5 Summary of Yield Component Analysis Results 101 4.8 Plant Growth Analysis 103 4.8.1 Beans 1984: Growth Indices (Leaf Area Ratio, Specific Leaf Area, Leaf Weight Ratio and Harvest Index): Analysis of Variance Results 103 4.8.2 Beans 1987: Growth Indices (Leaf Area Ratio, Specific Leaf Area, Leaf Weight Ratio and Harvest Index): Analysis of Variance Results 105 4.8.3 Beets 1984: Growth Indices (Leaf Area Ratio, Specific Leaf Area, Leaf Weight Ratio and Harvest Index): Analysis of Variance Results 109 4.8.4 Beets 1987: Growth Indices (Leaf Area Ratio, Specific Leaf Area, Leaf Weight Ratio and Harvest Index): Analysis of Variance Results 117 v i i 4.8.5 Beans 1984: Primary Variables (Total Dry Weight, Leaf Dry Weight and Leaf Area): Regression Results 120 4.8.6 Beets 1984: Primary Variables (Total Dry Weight, Leaf Dry Weight and Leaf Area): Regression Results 123 4.8.7 Beans 1984 Growth Indices (Leaf Area Ratio, Leaf Weight Ratio, Harvest Index and Specific Leaf Area): Regression Results 123 4.8.8 Beets 1984 Growth Indices (Leaf Area Ratio, Leaf Weight Ratios Harvest Index and Specific Leaf Area): Regression Results 127 4.8.9 Beans 1984: Growth Indices (Absolute Growth Rate, Relative Growth Rate and Unit Leaf Rate): Regression Results 131 4.8.10 Beets 1984: Growth Indices (Absolute Growth Rate, Relative Growth Rate and Unit Leaf Rate): Regression Results 131 4.8.11 Summary of Plant Growth Analysis Results 136 4.9 Plant Allometric Relationships 136 4.9.1 Beans 1984 137 4.9.2 Beans 1987 . 141 4.9.3 Beets 1984 143 4.9.4 Beets 1987 146 4.9.5 Summary of Allometry Results 148 5. DISCUSSION 149 5.1 An Overview 149 5.2 Visual Observations 149 5.3 Analysis of Variance 150 v i i i 5.4 Yield-density Relationships 150 5.5 Differential Yield Response of Mixtures 152 5.6 Size Hierarchies 153 5.7 Yield component Analysis 154 5.8 Plant Growth Analysis 156 5.9 Plant Allometric Relationships 158 5.10 A Summary of the Discussion 160 6. CONCLUSIONS 163 7. LITERATURE CITED 166 8. APPENDICES 178 i x LIST OF TABLES 3.1 Treatment combinations of beans and beets 33 3.2 Population densities and plot sizes in the 1987 experiment 33 4.1 Summary of homogeneity of variance test of the raw data; percentage of variates homogeneous at the 5% level of significance as influenced by experimental treatments 45 4.2 Summary of homogeneity of variance test of the trans- formed data (log10 scale); percentage of variates homogeneous at the 5% level of significance as influenced by experimental treatments 46 4.3 Analysis of variance results for the 1984 bean data: Variance ratios for the effects of population density and mixture proportions on primary variables tested at different stages of growth 47 4.4 Analysis of variance results for the 1987 bean data: Variance ratios for the effects of population density and mixture proportions on primary variables tested at the the final harvest 55 4.5 Analysis of variance results for the 1984 beet data: Variance ratios for the effects of population density and mixture proportions on primary variables tested at different stages of growth 58 4.6 Analysis of variance results for the 1987 beet data: Variance ratios for the effects of population density and mixture proportions on primary variables tested at the final harvest 65 4.7a Estimates of parameter values for the response of total dry weight per plant to plant population densities 69 4.7b Estimates of parameter values for the response of leaf dry weight per plant to plant population densities 70 4.7c Estimates of parameter values for the response of leaf number per plant to plant population densities 71 X 4.7d Estimates of parameter values for the response of leaf area per plant to plant population densities 72 4.8a Standard deviations and error mean squares for the response of total dry weight per plant to population densities 73 4.8b Standard deviations and error mean squares for the response of leaf dry weight per plant to population densities 74 4.8c Standard deviations and error mean squares for the response of leaf number per plant to population densities 75 4.8d Standard deviations and error mean squares for the response of leaf area per plant to population densities 76 4.9a Gini coefficients for total dry weight distribution of beans grown in monocultures and mixtures 86 4.9b Gini coefficients for leaf dry weight distribution of beans grown in monocultures and mixtures 86 4.9c Gini coefficients for leaf number distribution of beans grown in monocultures and mixtures 87 4.9d Gini coefficients for leaf area distribution of beans grown in monocultures and mixtures 87 4.10a Gini coefficients for total dry weight distribution of beets grown in monocultures and mixtures 88 4.10b Gini coefficients for leaf dry weight distribution of beets grown in monocultures and mixtures 88 4.10c Gini coefficients for live leaf number distribution of beets grown in monocultures and mixtures 89 4. lOd Gini coefficients for leaf area distribution of beets grown in monocultures and mixtures 89 4.1 la Gini coefficients for bean yield variables in ascending order 91 Glnl coefficients for beet yield variables in ascending order Two dimensional partitioning of yield in beans: 1984 data (forward analysis) Two dimensional partitioning of yield in beans: 1984 data (backward analysis) Two dimensional partitioning of yield in beans: 1987 data (forward analysis) Two dimensional partitioning of yield in beans: 1987 data (backward analysis) Two dimensional partitioning of yield in beets: 1984 data (forward analysis) Two dimensional partitioning of yield in beets: 1984 data (backward analysis) Two dimensional partitioning of yield in beets: 1987 data (forward analysis) Two dimensional partitioning of yield in beets: 1987 data (backward analysis) Analysis of variance results for the 1984 bean data: Variance ratios for the effects of population density and mixture proportions on growth indices tested at different stages of growth Analysis of variance results for the 1987 bean data: Variance ratios for the effects of population density and mixture proportions on growth indices tested at the final harvest Analysis of variance results for the 1984 beet data: Variance ratios for the effects of population density and mixture proportions on growth indices tested at different stages of growth X l l 4.19 Analysis of variance results for the 1987 beet data: Variance ratios for the effects of population density and mixture proportions on growth indices tested at the final harvest 118 4.20 Summary of the allometric analysis for the 1984 bean data. Standard partial regression coefficients for allometric relationships of secondary variables with ln W (=y) 138 4.21 Summary of the allometric analysis for the 1987 bean data. Standard partial regression coefficients for allometric relationships of secondary variables with ln W (=y) 142 4.22 Summary of the allometric analysis for the 1984 beet data. Standard partial regression coefficients for allometric relationships of secondary variables with ln W (=y) 144 4.23 Summary of the allometric analysis for the 1987 beet data. Standard partial regression coefficients for allometric relationships of secondary variables with ln W (=y) 147 x i i i LIST OF FIGURES 4.1 The effect of population density on bean yield variables at different stages of growth: (1984 experiment) 50 4.2 The effect of mixture proportions on bean yield variables at different stages of growth: (1984 experiment) 52 4.3 The effect of population density on bean yield variables: (1987 experiment) (a) Marketable pod number and seed number (b) Pod fresh weight, marketable pod dry weight and seed dry weight 56 4.4 The effect of population density on beet yield variables at different stages of growth: (1984 experiment) 60 4.5 The effect of mixture proportions on beet yield variables at different stages of growth: (1984 experiment) 62 4.6 The effect of population density on beet live leaf number total dry weight, root fresh weight, leaf dry weight, petiole dry weight, and root dry weight :(1987 experiment) 66 4.7 The effect of mixture proportions on beet yield variables: (1987 experiment) (a) Live leaf number, leaf area and root diameter (b) Root fresh weight, leaf dry weight, petiole dry weight, root dry weight and total dry weight 67 4.8 Land equivalent ratio for total dry weight per unit land area at total population density of 66 plants nr2: (1984 experiment) (a) Predicted (b) Observed 78 4.9 Land equivalent ratio for total dry weight per unit land area at total population density of 16 plants nr2: (1984 experiment) (a) Predicted (b) Observed 79 4.10 Land equivalent ratio for leaf dry weight per unit land area at total population density of 66 plants nr2: (1984 experiment) (a) Predicted (b) Observed 80 4.11 Land equivalent ratio for leaf number per unit land area at total population density of 66 plants nr2: (1984 experiment) (a) Predicted (b) Observed 81 Land equivalent ratio for leaf area per unit land area at total population density of 66 plants nr2: (1984 experiment) (a) Predicted (b) Observed Observed land equivalent ratio for marketable yield per unit land area at total population density of 66 plants nr2: (1984 experiment) The effect of population density and mixture proportions on bean specific leaf area at different stages of growth: (1984 experiment) (a) Population density (b) Mixture proportions The effect of population density and mixture proportions on bean leaf weight ratio at different stages of growth: (1984 experiment) (a) Population density (b) Mixture proportions The effect of population density, mixture proportions and population density by mixture proportions interaction on bean leaf area ratio at different stages of growth: (1984 experiment) (a) Population density (b) Mixture proportions (c) Population density by mixture proportions interactions (92 days from planting) The effect of population density and mixture proportions on beet specific leaf area at different stages of growth: (1984 experiment) (a) Population density (b) Mixture proportions The effect of population density and mixture proportions interactions on beet specific leaf area: (1984 experiment) (a) Harvest 5 (b) Harvest 6 (c) Harvest 7 The effect of population density and mixture proportions on beet leaf weight ratio at different stages of growth: (1984 experiment) (a) Population density (b) Mixture proportions The effect of population density and mixture proportions on beet harvest index at different stages of growth: (1984 experiment) (a) Population density (b) Mixture proportions The effect of mixture proportions on beet leaf weight ratio: (1987 experiment) X V 4.22 Changes In total dry weight, leaf dry weight and leaf area per plant i n beans during growth resulting from increasing mixture proportions of beets at total population density of 66 plants n r 2 : (1984 experiment) (a) Total dry weight per plant, (b) Leaf dry weight per plant and (c) Leaf area per plant 121 4.23 Changes i n total dry weight, leaf dry weight and leaf area per plant i n beans dur ing growth resulting from increasing total population density at 2:2 bean:beet mixture proportion: (1984 experiment) (a) Total dry weight per plant, (b) Leaf dry weight per plant and (c) Leaf area per plant 122 4.24 Changes i n total dry weight, leaf dry weight a n d leaf area per plant i n beets dur ing growth resulting from increasing mixture proportions of beans at total population density of 66 plants n r 2 : (1984 experiment) (a) Total dry weight per plant, (b) Leaf dry weight per plant a n d (c) Leaf area per plant 124 4.25 Changes i n total dry weight, leaf dry weight a n d leaf area per plant i n beets dur ing growth resulting from increasing total population density at 2:2 bean:beet mixture proportion: (1984 experiment) (a) Total dry weight per plant, (b) Leaf dry weight per plant a n d (c) Leaf area per plant 125 4.26 Changes i n leaf area ratio, leaf weight ratio, specific leaf area a n d harvest index per plant i n beans dur ing growth resulting from increasing mixture proportions of beets at total population density of 66 plants m" 2 : (1984 experiment) (a) Leaf area ratio (b) Leaf weight ratio (c) Specific leaf area (d) Harvest index 126 4.27 Changes i n leaf area ratio, leaf weight ratio, specific leaf area a n d harvest index per plant i n beans dur ing growth resulting from increasing total population density at 2:2 bean:beet mixture proportion: (1984 experiment) (a) Leaf area ratio (b) Leaf weight ratio (c) Specific leaf area (d) Harvest index 127 x v i 4.28 Changes In leaf area ratio, leaf weight ratio, specific leaf area and harvest index per plant in beets during growth resulting from increasing mixture proportions of beans at total population density of 66 plants nr2: (1984 experiment) (a) Leaf area ratio (b) Leaf weight ratio (c) Specific leaf area (d) Harvest index 129 4.29 Changes in leaf area ratio, leaf weight ratio, specific leaf area and harvest index per plant in beets during growth resulting from increasing total population density at 2:2 bean:beet mixture proportion: (1984 experiment) (a) Leaf area ratio (b) Leaf weight ratio (c) Specific leaf area (d) Harvest index 130 4.30 Changes in absolute growth rate, relative growth rate and unit leaf rate per plant in beans during growth resulting from increasing mixture proportions of beets at total population density of 66 plants nr2: (1984 experiment) (a) Absolute growth rate, (b) Relative growth rate and (c) Unit leaf rate ~ 132 4.31 Changes in absolute growth rate, relative growth rate and unit leaf rate per plant in beans during growth resulting from increasing total population density at 2:2 bean:beet mixture proportion: (1984 experiment) (a) Absolute growth rate, (b) Relative growth rate and (c) Unit leaf rate 133 4.32 Changes in absolute growth rate, relative growth rate and unit leaf rate per plant in beets during growth resulting from increasing mixture proportions of beans at total population density of 66 plants nr2: (1984 experiment) (a) Absolute growth rate, (b) Relative growth rate and (c) Unit leaf rate 134 4.33 Changes in absolute growth rate, relative growth rate and unit leaf rate per plant in beets during growth resulting from increasing total population density at 2:2 bean:beet mixture proportion: (1984 experiment) (a) Absolute growth rate, (b) Relative growth rate and (c) Unit leaf rate 135 X V I 1 LIST OF APPENDICES 8.1.1.1 The effect of population density and mixture proportions interactions on bean leaf number harvested at 69 days after planting: (1984 experiment) 179 8.1.1.2 The effect of population density and mixture proportions interactions on bean pod dry weight harvested at 69 days after planting: (1984 experiment) 180 8.1.1.3 The effect of population density and mixture proportions interactions on bean pod fresh weight harvested at 69 days after planting: (1984 experiment) 181 8.1.2.1 The effect of population density and mixture proportions interactions on bean leaf number harvested at 75 days after planting: (1984 experiment) 182 8.1.2.2 The effect of population density and mixture proportions interactions on bean branch number harvested at 75 days after planting: (1984 experiment) 183 8.1.2.3 The effect of population density and mixture proportions interactions on bean pod number harvested at 75 days after planting: (1984 experiment) 184 8.1.2.4 The effect of population density and mixture proportions interactions on bean leaf dry weight harvested at 75 days after planting: (1984 experiment) 185 8.1.2.5 The effect of population density and mixture proportions interactions on bean pod dry weight harvested at 75 days after planting: (1984 experiment) 186 8.1.2.6 The effect of population density and mixture proportions interactions on bean total dry weight harvested at 75 days after planting: (1984 experiment) 187 8.2 Lorenz curve showing size inequality in a yield variable (hypothetical data) 188 8.3.1 Gini coefficients for stem dry weight distribution of beans grown in monocultures and mixtures 189 X V 1 1 1 8.3.2 G i n i coefficients for marketable pod number distribution of beans grown i n monocultures and mixtures 189 8.3.3 G i n i coefficients for unmarketable pod number distribu- tion of beans grown in monocultures and mixtures 190 8.3.4 G i n i coefficients for pod fresh weight distribution of beans grown i n monocultures and mixtures 190 8.3.5 G i n i coefficients for marketable pod dry weight distri- but ion of beans grown i n monocultures and mixtures 191 8.3.6 G i n i coefficients for unmarketable pod dry weight distri- but ion of beans grown i n monocultures and mixtures 191 8.3.7 G i n i coefficients for seed n u m b e r distribution of beans grown i n monocultures a n d mixtures 192 8.3.8 G i n i coefficients for seed dry weight distribution of beans grown i n monocultures a n d mixtures 192 8.4.1 G i n i coefficients for dead leaf n u m b e r distribution of beets grown i n monocultures and mixtures 193 8.4.2 G i n i coefficients for petiole dry weight distribution of beets grown i n monocultures and mixtures 193 8.4.3 G i n i coefficients for root diameter distribution of beets grown i n monocultures a n d mixtures 194 8.4.4 G i n i coefficients for root fresh weight distribution of beets grown i n monocultures and mixtures 194 8.4.5 G i n i coefficients for root dry weight distribution of beets grown i n monocultures and mixtures 195 8.5.1 Parameters a n d statistics for the best subset multiple regression models of allometric relationship between In (W) and l n (T) i n beans 1984 experiment 196 8.5.2 Parameters and statistics for the best subset multiple regression models of allometric relationship between l n (W) and l n (LN) i n beans 1984 experiment 198 X I X 8.5.3 Parameters and statistics for the best subset multiple regression models of allometric relationship between In (W) and In (LA) in beans 1984 experiment 200 8.5.4 Parameters and statistics for the best subset multiple regression models of allometric relationship between In (W) and In (WL) in beans 1984 experiment 202 8.5.5 Parameters and statistics for the best subset multiple regression models of allometric relationship between In (W) and In (WST) in beans 1984 experiment 204 8.5.6 Parameters and statistics for the best subset multiple regression models of allometric relationship between In (W) and In (BN) in beans 1984 experiment 206 8.5.7 Parameters and statistics for the best subset multiple regression models of allometric relationship between In (W) and In (PN) in beans 1984 experiment 208 8.5.8 Parameters and statistics for the best subset multiple regression models of allometric relationship between In (W) and In (FWPD) in beans 1984 experiment 210 8.5.9 Parameters and statistics for the best subset multiple regression models of allometric relationship between In (W) and In (WPD) in beans 1984 experiment 212 8.6.1 Parameters and statistics for the best subset multiple regression models of allometric relationship between In (W) and In (LN) in beans 1987 experiment 214 8.6.2 Parameters and statistics for the best subset multiple regression models of allometric relationship between In (W) and In (LA) in beans 1987 experiment 215 8.6.3 Parameters and statistics for the best subset multiple regression models of allometric relationship between In (W) and In (WL) in beans 1987 experiment 216 8.6.4 Parameters and statistics for the best subset multiple regression models of allometric relationship between In (W) and In (WST) in beans 1987 experiment 217 X X 8.6.5 Parameters and statistics for the best subset multiple regression models of allometric relationship between In (W) and In (MPN) in beans 1987 experiment 218 8.6.6 Parameters and statistics for the best subset multiple regression models of allometric relationship between In fW) and In (UPN) in beans 1987 experiment 219 8.6.7 Parameters and statistics for the best subset multiple regression models of allometric relationship between In (W) and In (FWPD) beans 1987 experiment 220 8.6.8 Parameters and statistics for the best subset multiple regression models of allometric relationship between In (W) and In (WUPD) beans 1987 experiment 221 8.6.9 Parameters and statistics for the best subset multiple regression models of allometric relationship between In (W) and In (WMPD) beans 1987 experiment 222 8.6.10 Parameters and statistics for the best subset multiple regression models of allometric relationship between In (W) and In (SN) beans 1987 experiment 223 8.6.11 Parameters and statistics for the best subset multiple regression models of allometric relationship between In (W) and In (WS) beans 1987 experiment 224 8.7.1 Parameters and statistics for the best subset multiple regression models of allometric relationship between In (W) and In (T) beets 1984 experiment 225 8.7.2 Parameters and statistics for the best subset multiple regression models of allometric relationship between In (W) and In (LN) beets 1984 experiment 227 8.7.3 Parameters and statistics for the best subset multiple regression models of allometric relationship between In (W) and In (LA) beets 1984 experiment 229 8.7.4 Parameters and statistics for the best subset multiple regression models of allometric relationship between In (W) and In (WL) beets 1984 experiment 231 Parameters and statistics for the best subset multiple regression models of allometric relationship between In (W) and ln (WP) beets 1984 experiment Parameters and statistics for the best subset multiple regression models of allometric relationship between ln (W) and ln (DR) beets 1984 experiment Parameters and statistics for the best subset multiple regression models of allometric relationship between ln (W) and ln (FWR) beets 1984 experiment Parameters and statistics for the best subset multiple regression models of allometric relationship between ln (W) and ln (WR) beets 1984 experiment Parameters and statistics for the best subset multiple regression models of allometric relationship between ln (W) and ln (LN) beets 1987 experiment Parameters and statistics for the best subset multiple regression models of allometric relationship between ln (W) and ln (LA) beets 1987 experiment Parameters and statistics for the best subset multiple regression models of allometric relationship between ln (W) and ln (WL) beets 1987 experiment Parameters and statistics for the best subset multiple regression models of allometric relationship between ln (W) and ln (WP) beets 1987 experiment Parameters and statistics for the best subset multiple regression models of allometric relationship between ln (W) and ln (DR) beets 1987 experiment Parameters and statistics for the best subset multiple regression models of allometric relationship between ln (W) and ln (FWR) beets 1987 experiment Parameters and statistics for the best subset multiple regression models of allometric relationship between ln (W) and In (WR) beets 1987 experiment SYMBOLS AND TERMINOLOGY Symbols Page Definition (First Appearance) a 29 Allometric coefficient a 17 Intercept i n yield-density models A G R 103 Absolute growth rate A N O V A 36 Analys is of variance p 29 Allometric exponent b 17 Regression coefficient i n yield-density models B N 47 B r a n c h n u m b e r c 21 Regression coefficient i n yield-density models C P 196 Mallows C P statistic 8 30 Non-allometric parameter D 22 Total mixture population density d.f. 47 Degrees of freedom D R 40 Root diameter D R / W L 40 Root diameter per leaf dry weight e 29 Base of natural logarithms E 103 Uni t leaf rate e' 29 Residual variation i n yield i n allometric models F 103 Leaf area ratio F W P D 47 Pod fresh weight F W R 58 Root fresh weight G 39 G i n i coefficient Y 30 Non-allometric parameter G' 39 Unbiased G i n i coefficient X X Symbols Page Definition (First Appearance) H 103 Harvest index H N 45 Harvest number i , j 17 Subscripts indicating species. Subscript i denotes the test species. Subscript j , when present, denotes the companion species k 30 Subscript indicating k * allometric and n o n allometric exponent L A 39 Leaf area L A / L N 40 Leaf area per leaf number L E R 22 L a n d equivalent ratio for a mixture L L 55 Live leaf number L N 39 Leaf n u m b e r l o g 1 0 37 Logarithms base 10 logg (same as ln) 29 Natural logarithm L W R 103 Leaf weight ratio M P N 55 Marketable pod n u m b e r n 39 Sample size i n G i n i coefficient computation p, q 39 Subscripts denoting p * and q * yield per plant i n G i n i coefficient computation P N 39 Pod n u m b e r P N / W S T 40 Pod n u m b e r per stem dry weight R 103 Relative growth rate R 2 140 Coefficient of determination s 39 G r a n d m e a n yield per plant of a variable i n G i n i coefficient computation s 39 Yield per plant i n G i n i coefficient computation xxiv Symbols Page Definition (First Appearance) SLA 103 Specific leaf area <E>, O' 18 Exponents controlling the form of the yield-density models SN 55 Seed number T 47 Plant height TDP 36 Two dimensional partioning UPN 55 Unmarketable pod number W 47 Total dry weight (total shoot dry weight for beans, and total shoot and storage root dry weight for beets) WL 39 Leaf dry weight W L / L A 40 Leaf dry weight per leaf area WMPD 55 Marketable pod dry weight WP 58 Petiole dry weight WPD 40 Pod dry weight WPD/PN 40. Pod dry weight per pod number WR 40 Root dry weight W R / D R 40 Root dry weight per root diameter WS 55 Seed dry weight WST 39 Stem dry weight W S T / W L 40 Stem dry weight per leaf dry weight WUPD 55 Unmarketable pod dry weight \ 30 Non-allometric parameter X 17 Population density of a species (plants per land area) y 17 Mean yield of the test species per plant Y 17 Mean yield of the test species per unit land area z 29 Yield of secondary measure of a plant in allometric relationships X X V ACKNOWLEDGEMENTS I would like to express my sincere thanks to Dr. P.A. Jolliffe for his guidance and encouragement throughout this project. His contribution in preparing this thesis is greatly appreciated. I would also like to thank Dr. G.W. Eaton and Dr. M.D. Pitt for their advice in statistical procedures. I have learnt a lot from them during the entire course of my program. I am very grateful to the members of my supervisory committee, Dr. V . C . Runeckles, Dr. R.A. Turkington and M.K. Upadhyaya for their willingness to assist me at all times. I am indebted to Madukar Potdar for allowing me to use part of his data and to Paul Lui and Elaine Wright, for their help, encouragement and support during my studies. Thanks to Peter, Tino, Derek, Andrew, Semion, Ingrid, Karen and Caline for their help with the field work. The assistance offered by the Maranatha Christian Church during the weeding period and the advice in technical matters of Ashley Herath, Al Neighbour, Derek White, Christa Roberts and Sean Treheame are greatly appreciated. I am also grateful to Eleanor for letting me use her computer even during the most awkward hours of the night. The assistance and patience of Patsy in helping me with the typing of the thesis is sincerely appreciated. The help of Owalabi, Titsesto, Patrice, and Gladys in child caring is highly appreciated. I acknowledge the Canadian International Development Agency and the University of Zambia for their financial support. Last but not least, I wish to thank my husband David, my daughter Nemhina and my son Hiza to whom I dedicate this thesis. And thanks mom, you made this study possible. 1 1. INTRODUCTION Interference among neighbouring plants, often due to competition for limited resources, is central to subjects such as yield-density relationships, intercropping, mortality in dense plant stands, and low reproductive yields in crops. Despite its importance, the nature and mechanisms of competition among plants are not well understood. There are many reasons for this lack of understanding. Among them are the inconsistent usage of, or different meanings attributed to, the term competition when it is used in relation to plants. Also, the experimental designs used to study competition have often had shortcomings, and methods of analyzing results from competition experiments have needed improvement. Further complications arise from the potentially complex nature of competitive relationships, because different environmental resources might be the cause of competition at different times during growth, and because non- competitive interferences (e.g. allelochemical interference) may take place among associated plants. The research described in this thesis pertains to the interference between intercropped bean {Phaseolus vulgaris L.) and beet {Beta vulgaris L.) plants. In addition to considering the effects of interference on final crop yields, this work was intended to advance our knowledge of the mutual influences of intercropped species upon each other. For this reason, experiments were undertaken to detail the timing and sites within plants of their responses to species population densities and mixture proportions, and to detail the effects of such treatments on the distribution of plant size among members of each species. Specific objectives of the research were: 2 (i) to develop mathematical relationships defining the response of shoot biomass, and certain other measures of plant growth, to population densities of beans (Phaseolus vulgaris L.) and beets {Beta vulgaris) in monocultures and mixtures; (ii) to use those mathematical relationships to determine whether intercrops of bean and beet plants are more productive than corresponding monocultures (le. to quantify the 'differential yield response' of the intercrops); (iii) to assess how different population densities and mixture proportions affect the frequency distribution of plant size within each species; (iv) to evaluate the effects of population density treatments on quantitative relationships among different measures of growth {le. allometric relationships) in each species; (v) to analyze the dynamics of plant growth, using procedures of plant growth analysis to assess the performance of plants and plant parts, as it was affected by experimental treatments; and (vi) to detennine the contributions of morphological yield components to variation in final agricultural yield, and to quantify the effects of population density and mixture treatments on those components. 3 2. LITERATURE REVIEW 2.1 The Crops Used i n This Study 2.1.1 Beans Out of about 150 species in the genus Phaseolus, the snap or common bean, Phaseolus vulgaris L. , is the species most widely cultivated (Yamaguchi 1983, Singh 1989). A member of the Legumlnosae, the plant is an annual which reproduces solely by seed. The plant can be a determinate (bush) type or ^determinate (climbing or pole) type (Peirce 1987). After the primary leaves, the plant produces alternate trifoliate leaves. P. vulgaris may self-pollinate, but there is large variation in plant characteristics within and among cultivars (Singh 1989). Pods are long and narrow, varying in length from 8 to 20 cm, and are 1 to 1.5 cm wide. The number of seeds in a filled pod also varies between 4 and 12 depending on cultivar and conditions during growth. Also varying are characteristics such as seed length (0.7 to 1.5 cm), seed weight (< 15 to > 60 g/100 seeds), seed form (globular to kidney shaped), and seed coat color (white, yellow, greenish pink, reddish purple, brown or black). Seed color can be solid, striped or mottled (Singh 1989). "String" (te. fibrous pod tissue) formation Is controlled genetically, and is influenced by cultivar and temperature (Drijfhout 1978). P. vulgaris is a warm season crop which is thought to have originated in central America (Gepts et al 1988). It was distributed to other parts of the world soon after European contact with central America (Ware and McCollum 1975). Now it is grown in temperate zones during the warm months, and in tropical and subtropical fanning regions (Wallace 4 1978), P. vulgaris is a mesophyte, requiring about 500-800 mm of water per growing season for optimum growth. In areas with low rainfall, irrigation is recommended for higher yields (Ware and McCollum 1975). The optimum temperature for germination is about 30C. Germination is greatly reduced above 35C, and below IOC germination does not occur (Peirce 1987). Vegetatively, the plant grows best between 21 and 26.7C. High temperatures during flowering can cause embryo abortion (Ware and McCollum 1975). P. vulgaris is usually planted in rows with intra-row spacing varying between 5 and 10 cm and inter-row spacing of 45 to 90 cm (Lorenz and Maynard 1988). Population densities therefore, are usually in the range of 11 to 39 plants per square meter. A planting depth of 2 to 5 cm is typical, and well aerated soils with good drainage are recommended (Pierce 1987). Beans form a symbiotic association with nitrogen fixing bacteria, and such soil conditions are excellent for nitrogen fixation, thus reducing beans requirement for nitrogen fertilizer. P. vulgaris is mostly grown for fleshy pods and immature seeds (Peirce 1987). In some parts of the world, the tender shoots are used as a pot herb (Singh 1989). Pod harvesting occurs between about 45 and 90 days after planting (Shoemaker 1953). In a bush type, frequent harvesting is recommended in order to enhance greater pod formation before the plant reaches maturity (Yamaguchi 1983). P. vulgaris can be susceptible to a number of fungal diseases (e.g. anthracnose [Colletotrichum lindemuthianum Sacc. & Magn.), angular leaf spot (Isariopsis griseola Sacc), rust [Uromyces phaseoU typica Arth.), ascochyta leaf spot [Ascochyta phaseolorum Sacc), downy mildew [Phytophthora phaseoU Thaxt), viral diseases (e.g. bean curly top virus, 5 bean dwarf mosaic virus, bean yellow mosaic virus, bean golden mosaic virus) and bacterial diseases (e.g. common bacterial blight [Xanthornonas phaseoli E .F . Smith), halo blight (Pseudomonas phaseolicola Burk.). Insect pests, such as bean leaf beetle [Cerotoma trijurcata), can also be a major problem. 2.1.2 Beets Beets, Beta vulgaris L. , belong to the family Chenopodiaceae, the goosefoot family. The species B. vulgaris has four types: sugar beet, table beet, fodder beet and swiss chard (Whitney and Duffus 1986). In this thesis, beets or B. vulgaris will refer to only the table beet type. The plant originated from the Mediterranean region of north Africa, Europe, and west Asia (Ware and McCollum 1975). Beets have been cultivated since at least the third century, AD. B. vulgaris is a biennial, and is grown in cultivation for storage roots and tops (Peirce 1987). The plant has a short and platelike stem, the crown. The leaves are simple in form and are arranged on the crown in a closed spiral (Ware and McCollum 1975). The color of the leaves may vary from dark red to light green (Ware and McCollum 1975). The "seed", which is in fact a fruit, may contain 2 to 6 true seeds (Shoemaker 1953, Ware and McCollum 1975). A monogerm type in which each fruit has only one seed, has also been released (Peirce 1987). The storage root is the result of swelling of the hypocotyl plus a small portion of the tap root. The swelling is caused by growth of several concentric vascular cambia which are visible as 'rings' when the roots are sectioned. Storage roots are usually red in color, but golden cultivars are not uncommon. The red color in B. vulgaris is due to betacyanin pigment, but roots also contain a yellow pigment, 6 betaxanthin (Peirce 1987). The main root system is a taproot that can grow to a depth of 3 m. A few lateral roots also tend to develop at the base of the swollen edible structure (Peirce 1987). B. vulgaris is a cool season crop. It can tolerate both cold and hot temperatures, but the plant cannot withstand severe freezing (Shoemaker 1953). Gennination occurs at soil temperatures between 10 and 29C, while exposure for 14 days or more to temperatures between 4 and IOC induces bolting at the expense of fleshy root development (Peirce 1987). Flower induction is also accelerated if the plants are exposed to long days (Peirce 1987). Beets can thrive on a range of soils but for optimum growth, slightly acid soils with pH between 6.0 and 7.0 are recommended (Shoemaker 1953, McCollum 1975). The water requirement for B. vulgaris varies depending on soil type, but adequate soil moisture during the entire growing season is required to maintain tender root tissues (Shoemaker 1953). B. vulgaris is normally planted at a depth of 1.5 cm to 2.5 cm in rows 30 to 75 cm apart which are thinned after crop emergence to achieve a plant spacing of 5 to 10 cm within rows (Lorenz and Maynard 1988). This corresponds to population densities in the range of 13 to 67 plants per square meter. Fertilizer requirements vary with soil type and fertility. If soils are deficient in boron, application of boric acid at a rate of 9 to 36 kilograms per hectare is recommended to prevent internal black spot. Harvesting of B. vulgaris depends on its intended use. Early in the growing season, the ttdnnings can be used as greens, or the crop can be harvested for bunching purposes when the roots are between 3 and 4 cm in diameter. The 'baby beet' is harvested when the roots are between 4 and 5 cm diameter. Roots are harvested for pickling and canning when 7 they are about 7 cm in diameter. The mature harvesting stage is when roots are between 7 and 10 cm in diameter. These roots have their shoots removed, and may be stored for several months after harvest (Shoemaker 1953). While small sized beet storage roots are marketed intact, larger sizes are used for sliced or diced products (Peirce 1987). B. vulgaris is rarely attacked by disease but can have cercospora leaf spot [Cercospora beticola Sacc.) phoma leaf spot [Phoma betae), downy mildew {Peronospora schachtii) (Ruppel 1986a), and leaf curly top, a virus disease transmitted by beet leafhoppers (Circulifer tenellus Baker) (Ruppel 1986b). Insect pests such as beet leafhopper, webworm [Loxostege sticticalis L.), spinach leaf miner [Pegomya hyoseyarni Panzer), cutworms and wireworms can be a problem too (Shoemaker 1953). Lack of boron may also cause black pitting, surface cankers, heart rot, or dry rot (Shoemaker 1953). 2.2 Crop Mixtures Systems of agricultural plant production include both monospecific plant associations, referred to as monocultures, and associations of different plant genotypes, referred to as mixtures. Trenbath (1974) suggested that combinations of different cultivars, or different age classes of the same cultivar, also can be considered to be crop mixtures. In agriculture, the use of monocultures offers certain advantages for crop management, including ease of mechanization, and relative simplicity of pest control practices (Beets 1982). Species mixtures, however, have the potential to exploit a greater range of environmental resources than can be utilized by a single species. Mixed pastures and forages are of worldwide importance, and are the best example of the use of crop mixtures in 8 developed countries. Otherwise, the use of crop mixtures is now prominent mainly in tropical and subtropical regions (Kass 1978). This type of agriculture has persisted in those regions because of advantages such as better utilization of environmental resources (Baker and Norman 1975), greater yield stability in variable environments (Beets 1982), reduced soil erosion (by rapidly providing vegetative cover, Beets 1982, Gomez and Gomez 1983), greater tolerance to disease and pests (where the severity of attack is proportional to host plant population density, Andrews and Kassam 1976), easier pest control in some crop mixtures (Andrews and Kassam 1976), better weed control, and sometimes better labour utilization (Baker and Norman 1975). It should be noted that intercropping may have disadvantages, such as: yield reduction due to adverse competition and allelopathic effects, and complexities in management, especially in cases where a high level of mechanization is essential (Willey 1979a & b). Two systems exist for growing crop mixtures: mixed cropping and intercropping. Both involve the simultaneous growing of two or more crop species on the same piece of land (Mead 1979, Yunusa 1989). In intercropping, plants are grown in rows, and various arrangements of the species within and between the rows are possible. Mixed cropping, however, entails less organization; it involves species randomly mixed within rows or broadcasted together (Mead 1979). In some cases the terms intercropping and mixed cropping have been used interchangeably (Willey 1979a & b). The distinction between them may be significant, however, because the spatial arrangement of coexisting species within mixtures is sometimes of primary importance in determining mixture performance (Andrews 1972, Yunusa 1989, Mead 1979). 9 Plants in mixed cropping or intercropping systems need not be sown or harvested at the same time, but must grow together for a significant part of their growing period (Willey 1979a & b, Ofori and Stern 1987). Several types of mixed cropping/intercropping exist: row intercropping, te. the growing of mixtures in rows (Andrews and Kassam 1976); strip cropping, te. growing of mixtures in alternating strips or blocks on the same piece of land (Beets 1982, Trenbath 1974); relay intercropping, te. the coexisting species are not sown or harvested at the same time (Andrews and Kassam 1976); and patch intercropping, te. the growing of mixtures in patches (Papendick et al 1976). Crop rnixtures are believed to have the potential to yield more than monocultures on an equivalent land area basis (Trenbath 1974, Andrews and Kassam 1976). This could be due to reduced interspecific competition compared to intraspecific competition, or it could be a result of the fuller exploitation of environmental resources due to niche differentiation among different species (Trenbath 1974). As a result, many different plant species have been grown in mixtures. Common mixture combinations include maize [Zea mays) associated with some members of the Leguminosae family including Phaseolus vulgaris (Harwood and Price 1976), maize with millet (Setaria italica) or cassava [Manihot esculenta) (Harwood and Price 1976), many forage/legume crop associations (Drolsom and Smith 1976), and sorghum [Sorghum spp.) in mixture with a legume or with millet (Kassam and Stockinger 1973, Norman 1974). Tree/tree and tree/annual crop associations are also common (Harwood and Price 1976). Other than the binary mixtures mentioned above, multi-species crop mixtures are also used (Norman 1974). Previous research on bean/beet intercropping, however, seems to be absent. 10 2.3 Plant Interference 2.3.1 Nature of Interference Competition among organisms has been an important issue in biology. Darwinian expressions such as "struggle for existence" and "survival of the fittest" highlight the importance of competition and other types of interactions among organisms. Competition is important in both plant and animal associations, but it has proved difficult to advance a generally accepted definition for plant competition (Clements et al 1929, Bleasdale 1960, Grime 1979, Begon and Mortimer 1981). Grime (1979) emphasized the importance of environmental resources, defining competition as: "the tendency of neighbouring plants to utilize the same quantum of light, ion of mineral nutrient, molecule of water or volume of space." On the other hand, Bleasdale (1960) indicated that competition between two plants occurs when "the growth of either or both plants is reduced or modified as compared with their growth or form in isolation." Harper (1961) criticized the use of the term competition to describe the overall aspects of interactions among neighbours, and suggested that the term should be abandoned and be replaced by other terms. Grime (1977), however, argued that competition is too useful a word and competition for resources is too important a mechanism to be discarded. Hall (1974a) pointed out that competition tends to be used to describe ecological and agronomic processes in a rather loose manner. Silvertown (1987) divided the different versions of definitions of the word competition into two categories: those that define the interaction among species on the basis of the mechanisms involved (e.g. Grime 1979), and those that define 11 the interaction in terms of the outcome between two competing species (e.g. Begon and Mortimer 1981, Bleasdale 1960). Partly due to the difficulty in providing an acceptable definition for competition, some workers (Harper 1961, Hall 1974a & b, Trenbath 1974, Penney 1986) have preferred to use the term interference. The definition of interference was given by Harper (1961) as: "all responses of an individual plant or plant species to its total environment as this is modified by the presence and/or growth of other individuals." Thus, interference is a comprehensive term which encompasses both competitive and noncompetitive interactions among neighbours. A plant might interfere with its neighbours through competition for resources or in other ways, e.g. through allelochemistry or through influences on herbivores or pests. Harper's (1961) definition permits the possibility of beneficial interference, such as the promotion of pollinators, or sheltering of a plant from environmental stresses. Harper's (1961) concept of interference has two main advantages: (I) it is a term that does not imply a specific mechanism by which neighbours affect a plant's growth, and (ii) it directs attention to plant responses, which are the means by which the effects of neighbours can be assessed experimentally. Different mechanisms of interference may occur together and interact flrenbath 1976, Harper 1977). Intraspecific interference can occur in monocultures, and among members of the same species within mixed crop associations. Interspecific interference occurs between plants of different species in mixtures. Although competition for resources is only one potential component of interference, there is reason to believe that competition may often be of considerable importance. Many agronomic studies, for example, have 12 demonstrated strong improvements in crop growth with additional resource supply when population densities are high. The effectiveness with which a particular plant competes for essential growth resources depends on a number of factors such as plant population density (Wiener 1984), plant arrangement (Yunusa 1989) and the ability of the particular plant or plant species to access resources in its vicinity (Watkinson 1980, 1984). At very low population density, plants may be too widely spaced to compete for resources and may grow as if they are in isolation (Trenbath 1974). This state can be approximated in a young crop, before plant size is sufficient to cause interference. As growth continues, however, expanding root and shoot systems lead to interference, and possibly competition for resources, among neighbours. Despite competition and other detrimental components of interference among neighbours, coexistence among plant species is a common phenomenon in both natural and agricultural plant communities. Grubb (1977) has reviewed the different mechanisms by which plants coexist. Variation in competitive ability with age (Watt 1955), balanced mixtures (Marshall and Jain 1969), differences in life forms (Turkington 1975), phenologies! separation (Bratton 1976), and local variations in the environment (Thomas and Dale 1976) are some of the mechanisms known to play a part in plant coexistence. 2.3.2 Some Experimental Approaches to the Study of Plant Interference Much of our present knowledge concerning plant interference has come from agricultural cropping systems and laboratory studies. This is because agricultural systems offer several experimental advantages 13 compared to natural plant communities: they contain simple plant populations, have a quick turnover through the use of annual crops, and they allow appropriate control treatments to be used. Also, in agricultural systems, the supply of resources can be partly controlled and/or managed, the populations under study are relatively uniform, the environment can be defined, and other experimental circumstances such as plant population density, plant arrangement, and timing of association are under some control of the experimenter (Radosevich and Holt 1984, Radosevich 1987, Snaydon 1980). Moreover, relevance to agricultural yield is another motivation for studying interference in agricultural cropping systems. There are limitations, however, of using agricultural systems for studies of interference. These include: the simplicity of agricultural studies limits the direct application of their results to complex natural situations; agricultural research has concentrated on annuals, biennials and other short lived perennials, and this may not relate well to long term natural associations; the scale of agricultural experiments is limited in extent; and, agricultural experiments have concentrated on yield rather than other biologically important outputs. Many different approaches to studying plant interference and competition have been developed. Techniques of neighbourhood analysis have been used by Levin and Kerster (1971), Bella (1971), Trenbath and Harper (1973), Yeaton and Cody (1976), Mack and Harper (1979) Ford and Diggle (1981), Weiner (1984), and Cannell et al (1984). These techniques take into account the importance of the pattern and arrangement of individuals in a population. This is important because it has been shown that the ability of an individual plant to exploit its environment depends on its position within the area defined by its neighbours, its time of 14 emergence, stage of development and its size relative to its neighbours (Ross and Harper 1972, Ford 1975). The focus of neighbourhood analysis is on the individual plant and its immediate surroundings, and it can be applied in relatively complex circumstances. It is insufficient, however, simply to define a plant's neighbourhood. The quality of the target plant must also be considered because it is the balance between the target plant and its neighbours that will determine the fate of a competitive interaction. Other approaches concentrate more on the collective performance of plant populations. Two experimental forms which commonly have been used to investigate interference in crop systems are the replacement series (de Wit 1960) and the additive series (Donald 1963). In binary replacement series experiments, proportions of two species in mixture are varied, but total density is held constant. In additive experiments, a constant density of one species is established with a variety of densities of another species. The additive series has been useful for the study of weed-crop associations because it can assess how different weed population densities affect a crop at fixed population density (Dew 1972, Cousens 1985). It is a simple approach, but the effects of total population density and weed population density on the crop are confounded since both factors change together. Replacement series have been used widely in competition studies. Results are often presented graphically as replacement diagrams in which the yield of each species is plotted against its proportion in mixture. Replacement diagrams have been used to indicate the stronger competitor and the degree of niche overlap between species (Khan et at 1975). Replacement series have been used to study competition for specific 15 nutrients (Hall 1974b). The replacement series approach is not preferred in crop-weed studies because high weed proportions are not usual in crops. A drawback with replacement series is that the choice of total population density is arbitrary and could condition species performance, especially when species of different sizes are mixed. Results from replacement series have proved difficult to interpret. This is because the performance of species in mixtures has been interpreted on the basis of their performance in monocultures at the arbitrary chosen density. Jolliffe et al (1984) criticized the original method of interpreting replacement series, developed by de Wit (1960), which is no longer used. Some studies have incorporated both replacement and additive series together by repeating a replacement series experiment at a range of total population densities, a structure termed as an addition series by Spitters (1983). Addition series experiments embody the exploration of a range of proportions and densities of the mixed species, and they are favoured by contemporary researchers (Radosevich 1987, Radosevich and Roush 1990, Rejmanek et al 1989). Some other experimental forms used to study interference have controlled the spatial arrangement of plants. These include the honeycomb layout in which a test plant is surrounded by six equidistant plants (Martin 1973,). The neighbouring plants could either be of the same species or different species. Nelder (1962) and Bleasdale (1967) described experimental arrangements in which plant spacing was systematically varied, although these have not been used widely in recent years. In his 1979 review, Mead (1979) indicated that new approaches need to be developed for this field of study. Replacement and additive series, as well as systematic designs, are forms of yield-density experiments. During 16 the past decade, general yield-density studies have proved to be useful in investigating interference, as discussed in the following section. 2.3.3 Tield-density Relationships Defining the relationships that exist between population density and yield has been of great concern to plant scientists (Kira et al 1956, Shinozaki and Kira 1956, Holliday 1960, Bleasdale and Nelder 1960, Bleasdale and Thompson 1966, Mead 1966, Gillis and Ratkowsky 1978, Vandermeer 1984). In addition to the agronomic need to define appropriate densities for crop production, mathematical models of yield- density relationships can be used to express plant interference. Models of yield-density relationships can be divided into two groups: those'that describe plants in monocultures and those that describe plants growing in mixtures. Willey and Heath (1969) thoroughly reviewed early attempts to construct yield-density models, which form the basis of current models. Yield-density models are of greatest value when their parameters possess meaning relevant to the biology of plant growth and interference. 2.3.4 Monoculture Models Monoculture yield-density data have often been successfully described using mathematical equations (Kira et al 1953, Holliday 1960, Mutsaers 1989). Many of these yield-density models have proved to be asymptotic in that an increase in population density leads to an increase in yield per unit land area until an upper limit is reached at high population densities. Typical of asymptotic relationships are data of total shoot biomass or other measures of vegetative parts. Parabolic yield- 17 density relationships have also been encountered, particularly with data from reproductive yield such as grain or seed. It is preferable to model the relationship between population density (X) and yield per plant (y), rather than yield per land area (Y), because the latter combines dependent and independent variables (Y = yX). Many yield density models were reviewed by Willey and Heath (1969). Among these are reciprocal yield density equations which can describe both asymptotic and parabolic yield-density relationships. These models seem to have been widely accepted because they can be derived from basic concepts of Interference (Jolliffe 1988), they offer a potentially powerful approach to data interpretation, and they contain parameters which seem to have biological relevance (Jolliffe 1988, Rejmanek et al 1989). Reciprocal yield-density models were first applied empirically by Kira et al (1953) and Shinozaki and Kira (1956). A simple functional relationship between mean shoot dry weight per plant and population density was expressed as: y t-i =a1 + b11X1 (2.1) where y{ is the mean yield per plant and ^ is plant population density. Parameter a expresses the reciprocal mean yield of an isolated plant while the parameter b expresses the strength of intraspecific competition. This equation can only describe an asymptotic yield-density relationship. Holliday (1960) also developed a reciprocal model and extended the above relationship to include parabolic yield-density responses by adding a quadratic term. 18 yf^ai + buXi + bu'X2 (2.2) Again parameter a represents the inverse of mean yield of an isolated plant and bu and bu' are expressing intraspecific interference. When bu = 0, then there is an asymptotic relationship; when bu is greater than zero the relationship is parabolic. Bleasdale and Nelder (1960) modified Kira et al's (1953) equation to give a more general fit to yield-density data: the ratio of <D and <&' determine the form of the yield-density relationship. The relationship is parabolic when <& is less than and it is asymptotic when <& = <&'. Since the ratio <t» to <&' is the main factor influencing Bleasdale and Nelder's equation, Bleasdale (1966) simplified the equation by setting o' equal to unity. Thus the equation becomes: Vandermeer (1984) argued that Bleasdale (1966) could equally have suggested setting * equal to unity in which case the equation: (2.3) yf* = aj + buXi or (yt = (aj + b^-V*) (2.4) y4-i = ai + buXj*. or (yt = (̂ + h&v)-*) (2.5) emerges, which according to Vandermeer (1984) has a satisfying biological interpretation. The parameter b u relates to the area and intensity of the competitive interaction, while <t>' relates to the rate at which the intensity of 19 competition decays as a function of interplant distance. He achieved this by altering the assumption of equal competition within a specified region for the classical yield-density relationship. He replaced this assumption with that of variable competition. Another reciprocal model equivalent to equation 2.4 was put forward by Watkinson (1980, 1984): ^ y m a x U + a u - X i K (2.6) where a '̂ = 9^-%, b t' = and y m a x = af 1/*. Watkinson (1980, 1984) attached biological meanings to model parameters: parameter y m a x is the yield of an isolated plant, a^ is the population density at which interference among neighbouring plants begins to be present, and b' is a measure of the efficiency of resource acquisition from the area surrounding the plant. Bleasdale and Nelder's (1960) equation (equation 2.4) has been widely used in analyzing yield-density data (Gillis and Ratkowsky 1975, 1978), but it has been found to produce biased estimations in cases where the data were non-normally distributed (Gillis and Ratkowsky 1978). It has also been criticized for being difficult to give a simple biological interpretation of parameters a and b when the model describes a parabolic relationship (Vandermeer 1984, Watkinson 1980). In an asymptotic situation, as population density approaches zero, the value of y approaches 1/a; thus, the reciprocal of a can be used as a measure of a species' genetic potential in a certain environment. Similarly, as the population density approaches mfinity, yield per plant approaches the asymptotic value of 1/b, and the inverse of b can be used as a measure of 20 environmental potential. Gillis and Ratkowsky (1978) indicated that in the parabolic relationship where <& is less than 1, the biological meaning for the parameters a and b are confounded with the effects of o. They pointed out that a"*1/*) measures genetic potential in a parabolic situation. Gillis and Ratkowsky (1978) reparameterized Bleasdale and Nelder's (1960) equation. A simple biological interpretation to all of their new parameters, however, is not clear. 2.3.5 Mixture Models Wright (1981) and Spitters (1983) extended the inverse monoculture yield density model to a two species system. They expressed a pair of equations as: yy-i = ai + b ^ + byX, (2.7) y-jfi = a, + bjjX, + bjjXj (2.8) where the first subscript corresponds to the species whose biomass is represented as the dependent variable (Le. the test species) and the second subscript identifies the associated species. The coefficients b u and bjj measure effects of intraspecific competition. The coefficients by and bj t measure the effects of the associated species on the test species. Thus, the coefficients formally separate intra- and interspecific competition. Watkinson's (1981) reparameterized equation was also extended to include a binary mixture situation (Firbank and Watkinson 1985) that corresponds to: yy-^i = ai +b11X1 + byXj (2.9) 21 yjf*l = a, + byX, + bjjXi (2.10) Jolliffe (1988) indicated that the model could potentially be extended to include interaction terms and higher order polynominals, for example yy-^ = aj + buXi + byX, + CuX^ + CyXiX, + CJ JXJXJ (2.11)" Potentially, these same models can be extended to multiple species systems where more than two species are intercropped though so far few experiments with more than two species have been done (Jolliffe 1988, Rejmanek et al 1989). The above equations have been fitted by normal multiple regression procedures. When population density treatments Xi and Xj are correlated, Jolliffe (1988) proposed an alternative approach of fitting the regression in stages. Jolliffe (1988) also showed the applicability of such models to the interpretation of differential yield responses. 2.4 Differential Yield Responses Overyielding in intercrops in relation to the corresponding monocultures is of central importance in mixed cropping. Several indices to determine the performance of crop mixtures have been suggested and were reviewed by Potdar (1986). Jolliffe (1988) demonstrated the use of yield-density models for the interpretation of differential yield responses. Land equivalent ratio (LER) is a useful index of the combined performance of species in binary mixtures (Willey and Osiru 1972). This index is calculated from: 22 LER = (Yy/Yu) + CYji/Yj,) (2.12) where Y represents yield per unit land area, the first subscript designates the species providing the data for Y and the second subscript indicates the companion species, le. Yy indicates the yield per unit land area of species i grown in mixture with species j . Similarly Yy is the yield per unit land area of species j in monoculture. In this evaluation, the mixtures and monocultures are assessed at the same total population density. There are three possible outcomes from a mixed crop (Willey 1979a & b): mutual inhibition, (LER < 1) mutual cooperation (LER > 1) and mutual compensation (LER =1). In equation 2.12 Y u , Yy, YJJ and YJJ are expressed on a per unit land area basis. Since Y u = y ^ , equation 2.4 can be converted to a unit land area basis as: Y u = yuXi = Xtfaj + buXj)-!/*! (2.13) for monoculture yield per unit land area of species I. Similarly, Yy = yyXt = X f o + buXi + byXj)-i/*i (2.14) expresses yield of species i per unit land area in the presence of species j . The same relationships can be formed for species j to express Yy and Yj|. Let D = X 1 + Xj, le. D represents the total population density of the mixture. Then: 23 X 1 = D - X J (2.15) It follows that (Jolliffe 1988): = VyXj = X^at + b u(D - X,) + byXjJ-l/^i = X ^ + buD-fbu - bjjJX,)-!/*! (2.16) Similarly: Yji = yjjX, = Xj(a, +bf> - (bu - bjJXJ-i/*! (2.17) Combined mixture yield will therefore be modelled by the sum of equation 2.16 and 2.17. From equation 2.13 monoculture yield per unit land area at X^ = D is given by Comparison with equation 2.16 indicates that the differential yield intraspecific interference is greater than interspecific interference, te. b u - by is positive, then this is subtracted from (aj + b uD), which reduces the denominator of equation 2.14, hence increasing yield per unit land area. 2.5 Size Hierarchies Plant populations often contain a few large individuals and many suppressed small individuals (Weiner 1984), te. the size distribution is positively skewed. This can develop in even-aged monoculture populations Y u = Xj/fau + b uD)-i/*i (2.18) response for one of the species is controlled by the difference: b u - by. If 24 which begin with a normal frequency distribution of seedlings. In such a skewed distribution, a few dominant individuals represent most of the biomass, while the numerous suppressed individuals contribute only a small portion of the biomass (Stem 1965, Weiner 1985, Weiner and Thomas 1986). Weiner and Solbrig (1984) have referred to such distributions as size hierarchies or size inequalities. Variations in size have ecological and evolutionary significance because small individuals are likely to suffer density-dependent mortality while large plants are more likely to contribute to future generations (Harper 1977). Also, size variations have commercial implications in cases where uniformity of produce is desirable. Plant size inequalities are the result of variation in plant growth rates which are caused by competition (Rejmanek et at 1989) and other factors, such as: genetic variation (Bonan 1988), seed size, order of seedling emergence, age differences (Ross and Harper 1972), environmental non-uniformity (Hara 1984a & b), neighbourhood effects (Hara 1984a & b), parasites, herbivores, pathogens and interactions among these factors (Weiner and Thomas 1986). Competition seems to be a major contributor, and efforts have been made to classify its effects. Two types of effects have been distinguished: symmetric effects in which competition is thought to act on all individuals in proportion to their sizes, thus reducing variation in growth rate and size inequalities (Weiner 1985, Connolly 1986, Weiner and Thomas 1986); and asymmetric effects where differences in growth rate are amplified by disproportionate sharing of available resources so that large plants utilize more resources and deprive the small individuals (Ford 1975, Bonan 1988, Weiner 1985). 25 The importance of competition in causing size hierarchies was demonstrated by Bonan (1988). He found that size structure was partly due to spatial distribution and availability of resources within a stand. On the other hand Weiner (1985), working with Lolium and Trtfoltum, found that spatial pattern was of minor importance in causing inequalities. Weiner (1985) also found that size inequalities always increased with increasing population density. In a mixture of Lolium and Trijolium, he found that the dominant species, LoUum, exhibited less size Inequality than in monocultures, whereas the subordinate species, Trijdtixsm, had greater size inequality in mixtures. The inequalities were greater for reproductive dry matter than for overall shoot weight, and these findings represent an asymmetric effect. Thus, competition is one-sided, operating more strongly on small than on large individuals (Weiner and Thomas 1986). In Weiner's (1985) study, interference also caused decreased mean plant mass, increased relative variation in plant mass and increased concentration of mass within a small fraction of the population. 2.6 Plant Growth Analysis If interference is the response of plants to their neighbours (Harper 1961), then progress in understanding interference may occur through the documentation of those responses. Measurement of final crop yield or seasonal production can Indicate the agronomic results of interference. Inferences concerning processes of interference, however, could be formed from knowledge of the timing and sites of responses within plants. Methods of plant growth analysis offer one means to provide such knowledge. 26 Formal methods of plant growth analysis date from the early work of Gregory (1918), Blackman (1919), Briggs et al (1920a, 1920b) and Engeldow and Wadham (1923). Modem reviews of the subject were done by Evans (1972), Causton and Venus (1981) Hunt (1982) and Fraser and Eaton (1983). Methods of growth analysis are an aid in the quantitative interpretation of growth variation. One advantage they possess is their requirement for simple input data which can be collected in field experiments, such as leaf areas and component dry weights. Several variants of plant growth analysis exist. Conventional plant growth analysis involves growth indices such as relative growth rate, leaf area ratio and unit leaf rate. Those indices are obtained from a series of observations of leaf areas and dry weights during the course of growth. The 'classical' form of conventional plant growth analysis computes growth indices from observations made at pairs of harvests; the 'functional' approach to conventional plant growth analysis computes growth indices from growth curves fitted to data from a set of harvests (Hunt 1982). Conventional plant growth analysis is helpful in assessing how plant performance is dependent on growth rates, persistence of growth, and dry matter partitioning (Jolliffe et al 1982). Much attention has been given to methods of fitting growth curves; polynomials, splined polynomials and the Richards function are growth functions commonly used today (Hunt 1982, Causton and Venus 1981). Sub-organismal demographic analysis is a second major variant of plant growth analysis. It was introduced by Bazzaz and Harper (1977), and it applies demographic concepts to the population of components which exist within individual plants. Issues such as the appearance ('births'), abundance, disappearance ('deaths'), lifetimes, functional 27 histories ('fates') and prominence of plant components are assessed. Sub- organismal demographic analysis has not yet been widely applied to plants but is becoming widely used by zoologists working with sessile organisms, and it is most useful for plants which produce large numbers of similar components (Hunt and Bazzaz 1980). In a statistical sense, 'yield' can be used to indicate any particular dependent variable. In the agronomic sense, which will be the predominant usage in this thesis, 'yield' denotes some particular output of plant growth, such as seed or fruit. The third major variant of plant growth analysis, yield component analysis, takes yield to be the mathematical product of a set of yield components (Engeldow and Wadham 1923). Yield components are in turn formed as ratios from measures of morphological constituents of the plant (Fraser and Eaton 1983). Thus, yield component analysis is concerned with the contributions of yield components to variation in yield, and the relationships among yield components. Yield component analysis has been used extensively in improving the grain yield in rice (Matsushima et al 1964 Matsushima 1966, 1976, 1980, Ishizuka 1971, Yoshida 1972, Yoshida and Parao 1972, Cock and Yoshida 1973, Yoshida 1973a & b, Murata and Matsushima 1974, Murayama 1979). Complex relationships can exist among yield components (Siefker and Hancock 1986). Matsushima (1966) reported that the components can act in parallel, in opposition, or sometimes may control each other, thereby compensating for either increases or decreases in other components. Many statistical procedures have been used to explore such relationships (Fraser and Eaton 1983). A relatively new procedure, the analysis of yield component relationships by two dimensional partitioning 28 (TDP), combines multiple regression procedures with the analysis of variance (Eaton et al 1986). Other workers (Siefker and Hancok 1986, Hancock et al 1984, 1983) have analysed yield component relationships using path coefficient analysis (Li 1956). This procedure involves standardization of the regression coefficients so that the degree of influence of the independent variable on the dependent variable is unrelated to physical units. Thus, conventional plant growth analysis, suborganismal demographic analysis, and yield component analysis treat different aspects of plant growth. The three approaches were developed independently, but in recent years they have been linked with each other and can be considered to form different branches of the general field of plant growth analysis (Jolliffe and Courtney 1984). Hunt (1980) and Hunt and Bazzaz (1981) showed that parallel analyses could be performed at the sub- organismal level using demography and conventional growth analysis. Jolliffe et al (1982) linked conventional plant growth analysis and yield component analysis. Further connections among all three approaches were demonstrated by Jolliffe and Courtney (1984). The application of plant growth analysis in studying yield-density relationships in mixtures has been rninimal. Roush and Radosevich (1985) used plant growth analysis to characterize the competitiveness of four annual weed species. Few other workers have exploited these analytical procedures in intercropping studies (Potdar 1986, Jolliffe et al 1988). 2.7 Allometry Allometric relationships can be defined as quantitative relationships that exist among different features of an organism as growth proceeds 29 (Jolliffe et al 1988). Mathematical models have been used to describe such allometric relationships, particularly the power function for bivariate allometry popularized by Huxley (1932): where y and z are two measures of an organism, or part of an organism, parameter a is the allometric coefficient and parameter P is the allometric exponent. It should be noted that a and p express the proportionality between y and z, as can be seen by dividing the above equation by z. Also, the power function is an empirical relationship, although the allometric exponent is related to the relative growth rates of y and z (Whitehead and Myerscough 1962). Hence, concepts of allometry are relevant to plant growth analysis. Parameter a is dependent on the arbitrary choice of scale of measurement for z since it is the value of y when z equals 1.0. The power function equation is linearized by transforming it to logg scale: where loggfe) has been added to account for residual variation in loggfy) not accounted for by log^a), p and loggfz). Jolliffe et al (1988) expanded this model to detail allometric responses to experimental treatments le. experimental treatments can affect loge(y) through allometric adjustments, via changes in a and/or p, or through non-allometric adjustments, via e. y = cczP (2.19) l0ge(y) = loge(a) + pl0ge(z) + loggfe) (2.20) 30 For two experimental treatments ( e.g. Xj and Xj), their effect on a, P and e can be expressed as: loge(cc) = loge(oo) + S^og^a )̂ + S^og^X,) + 83loge((X3XiXj) (2.21) p = p0 + PiXi + P2X, + P3X1XJ (2.22) loge(e) = loge(e0) + SilogefeiXJ + ^log^X,) + l̂ logefEaXiX,) (2.23) The treatment effects are expressed through values of 8k, P k £;k where k > 0. Terms that are difficult to separate can be grouped as follows: loge(a') = loge(ao) + Ŝ Ogê ) + S ôgê ) + 8 3 ^ ( 0 3 ) (2.24) loge(e') = loge(eo) + ^log^) + yogeGfc) + S3loge(e3) (2.25) Yk = sk + ^ k ( w h e r e k = L 2 - o r 3 ) (2.26) Treatment effects on terms having P k have been separated from other treatment effects which have been expressed by terms containg yk. Thus equation 2.20 expands to: lofefcr) = loge(a') + p0loge(z) + PjXilogetz) + PaXjlogJz) + p3X1XJloge(z) + YilogefXi) + Y2loge(Xj) + YslogefXpc,) + logje') (2.27) In their work with orchardgrass [Dacytylis glomerata L.) and timothy [Phleum pratense L.), Jolliffe et al (1988) found that population density 31 treatments changed allometric exponents. This adjustment changed as growth proceeded and each species responded differently. 2.8 Objectives of the Thesis Interference among associated plants, in monocultures and mixtures of different plant species is an active area of interest and study to both agronomists and ecologists. Methodologies for measuring interference and for accounting for yield responses have advanced rapidly during the past decade. As specified in the introduction, this study was done to measure interference in defined populations of beans and beets, and to explore how interference arose by detailing the timing and sites of plant responses. These' crops were chosen because: (1) both crops grow well in British Columbia, (2) recommended plant spacings for both crops overlap, (3) both crops mature at about the same time, (4) they provide an interesting contrast between a crop whose yield depends on above ground reproductive development (bean) and one which depends on vegetative growth (beet), and (5) they provide a contrast between a nitrogen fixing crop (bean) and a non-fixing crop (beet). This study has the potential, therefore to address questions about plant interference in terms of both practical consideration of growing beans and beets, and the academic purpose of trying to better understand interference in simple plant associations. 32 3. MATERIALS AND M E T H O D S 3.1 Experimental Layout and Crop Production Procedures Field experiments were performed at the Totem Park Field Station of the University of British Columbia, Vancouver, Canada in the summers of 1984 and 1987 on a sandy loam soil with pH 6.1. The 1984 experiment was designed and conducted by Potdar (1986). Two plant species, Phaseolus vulgaris L. cv 'Topcrop' and Beta vulgaris L. cv 'Ruby Queen' were used in both years. The experimental design was a randomized complete block with 20 treatments randomized within each block. The 20 treatments were made up of 4 planting densities (66, 50, 33, 16 plants nr2) of each species, and 5 mixtures at each density. The mixture proportions varied between 0 and 100% in uniform steps. Table 3.1 shows the treatment combinations in both replacement series (de Wit 1960) and additive series (Donald 1963) present within each block. In 1984, there were 3 blocks, and in 1987 there were 2 blocks with treatments replicated twice in each block. In 1984, individual plots were 14.6 m long and 3.2 m wide, while in 1987 plots varied in size between 3.2 m and 14.6 m in length and were all 3.2 m wide. The less dense plots were longer than the highly populated plots as shown in Table 3.2. Before seeding, the experimental area was sprayed with Dazomet (BASF Basamid Granular 90%) at a rate of 400 kg/ha in 1983 and with glyphosate at the rate of 5 L/ha in 1987 for weed control. In both years, the experimental areas were also fertilized before seeding with ammonium phosphate (11:50:0) at the rate of 224 kg/ha. Seeding was done by hand between May 30 and June 3 in 1984 and between June 1 and June 4 in 1987. Both beans and beets were planted at 0.45 m between row spacing in both years with varying within row Table 3.1 Treatment combinations of beans and beets 33 Mixture Proportions Total Population Density 0:4 1:3 2:2 3:1 4:0 (plants nr2) Species Population Densities (plants nr2) 16 00:16* 04:12 08:08 12:04 16:00 33 00:33 08:25 16:16 25:08 33:00 50 00:50 12:37 25:25 37:12 50:00 66 00:66 16:50 33:33 50:16 66:00 *Species population densities, Beans:Beets. Table 3.2 Population densities and plot sizes in the 1987 experiment Plants m of Row Length Plot Size nr2 per Plant* m 4.17 0.600 3.2 x 14.6 8.34 0.300 3.2 x 10.1 12.50 0.200 3.2 x 10.1 16.67 0.150 3.2 x 3.2 25.00 0.100 3.2 x 3.2 33.33 0.075 3.2 x 3.2 37.50 0.067 3.2 x 3.2 50.00 0.050 3.2 x 3.2 66.67 0.038 3.2 x 3.2 *Interrow spacing was 0.45 m in all plots 34 spacing depending on plant density and mixture proportion (Table 3.2). To facilitate planting, the same seeding depth (2 cm) was used for both beets and beans. Thinning was done when shoots were about 5 cm tall. In both years, experimental areas were weeded by hand throughout the growing period. Occasional irrigation was done as required to supplement natural rainfall. 3.2 Harvests and Primary Data Collection In 1984, six harvests, performed at 40, 51, 63, 69, 75, and 92 days after planting were done for both beans and beets. A seventh harvest was included at 107 days after planting for beets. In 1987, only one harvest at the end of the season was done for both plant species: beans were harvested between August 21 and August 23, and beets were harvested between August 27 and August 31. In 1984, 5 plants per species per harvest were collected from each plot, and in 1987 15 plants were taken. For beans, shoots were cut to ground level, while for beets, the shoots and major roots were pulled from the ground. The harvested material was taken to the laboratory where measurements were begun before plant material wilted. Heights of individual plants for each species were determined using a ruler. Measurements from the base of the stem to the apex of the longest leaf were recorded. Each plant was then subdivided into components: leaves, stem, pods, and flowers for beans, and leaves, petioles and storage roots for beets. Live and dead leaves were counted, and the area of live leaves per plant was measured using an LI-COR LI- 3000 leaf area meter. Pods were graded into marketable and unmarketable grades, as described in section 3.3, and the numbers in each category were recorded. Fresh weights per plant of marketable pods 35 and storage root were recorded for beans and beets respectively. In beets, storage root diameters were also determined. Dry weights per plant of all components were obtained after drying the material in a forced air oven at 75C. Duration of drying was 4 days, except for beet storage roots which were left for 7 days or more until constant weights were approached. After obtaining the bean pod dry weights, seeds were removed and their numbers and dry weights were determined. 3.3 Grading Procedure for Bean Pods Pods were graded into marketable and unmarketable categories by using holes cut through a Plexiglas template as guidelines. Pods from individual plants were sieved through holes of different diameters, and the following grades were obtained: Grade I - Pods which passed through 6.6 mm diameter hole (< 6.6 mm) Grade II - Pods which passed through 9.1 mm diameter hole (6.6- 9.05 mm) Grade III- Pods which do not pass through 9.1 mm diameter hole (>9.1 mm) These grades were condensed from Canadian government standards. The pod size range for the above grades are as follows: Range Average Sieve Size 5.8-7.4 mm 6.6 mm 2 7.4-8.4 mm 7.9 mm 3 8.4-9.7 mm 9.1mm 4 9.7-10.7 mm 10.2 mm 5 >10.2mm >10.2 mm 6 36 Sieve sizes 2 to 4 are considered to be marketable pods in British Columbia. 3.4 Analytical Procedures 3.4.1 A n Overview of the Data Analysis In outline, the data analysis included several steps: (i) Analysis of variance (ANOVA), which was carried out to determine the occurrence of significant effects due to density and rnixture treatments. (ii) Overall yield was analyzed next using non-linear regression models developed to describe yield-density relationships in both monocultures and mixtures. The models were then used to interpret competitive performance of beets and beans as well as the differential yield responses of the mixtures. (iii) The Gini coefficient (Weiner and Solbrig 1984) was computed on the 1987 data to determine the degree of size inequality among individuals of different characteristics within treatments. (iv) The contributions of yield components to total yield variation, and the relationships among yield components, were determined using two dimensional partitioning (TDP), a procedure involving a combination of the analysis of variance and stepwise multiple regression analysis. (v) Conventional plant growth analysis was done on the 1984 data to detect the timing and sites of treatment effects and the responses of physiologically relevant measures of plant performance. (vi) The effects of population density and mixture proportions treatments on quantitative relationships between different measures of plant growth were explored by allometric analysis. 3 7 Through these procedures, the overall effects of experimental treatments on each species are defined, and some of the relationships underlying yield of each species are detailed. 3.4.2 Analysis of Variance A partitioned Layard's homogeneity of variance test was done on the raw data before conducting an analysis of variance. Based on the homogeneity of variance test results, the data were transformed to a log 1 0 scale, and the analysis of variance was then done on the transformed data to test for significant differences among treatments. 3.4.3 Yield-density Relationships 3.4.3.1 Intraspecific Interaction In the first stage in developing these relationships, total dry weight, leaf dry weight, leaf area and live leaf number data for each species in pure stand (monoculture) were analyzed by fitting a simple reciprocal model (equivalent to equation 2.4): (y11)-*i = a 1 + b 1 1 X 1 (3.1) The P:9R BMDP statistical program (Dixon 1985) was used to fit the monoculture regressions. 3.4.3.2 Interspecific Interaction Parameter estimates obtained in the monoculture yield-density regressions were used in a second stage to determine interspecific 38 interference in the bean-beet mixtures (Jolliffe 1988). This additional, interspecific, interference is measured by the mixture yield-density model: (y1J)-*i = a 1 + b l i X 1 + b i JX J (3.2) This model is an extension of the monoculture model (3.1). In the mixture yield-density model, b u and « t were fixed at values obtained in the monoculture yield-density model using the monoculture yield data. Values for by were then obtained from a linear regression of residuals obtained after fitting the monoculture model, te the difference between y^-* and the value of y u-* predicted from model 3.1 was regressed against Xj, with no constant being formed in that regression. The assumption used in this second stage in developing the model for yy is that there will be no difference between the yield in mixtures, yy-*, and the yield of that species in monocultures, y u - ° , when interspecific interference, by, is zero. 3.4.4 Differential Yield Response The predicted combined yield per unit land area for bean-beet mixtures for total dry weight, leaf dry weight, leaf number and leaf area at harvest 6 was described by using the equation (Jolliffe 1988): LER = {(X^ + b u D - (bu - b y J X j H / o O A D ^ + buD)-i/*i} + {Xjfa, + byD - (bjj - bjiJXtH/^/Dfay + bjjDH/Oj (3.3) L E R stands for land equivalent ratio, and D represents total population density in mixtures te. (Xt + Xj). The predicted values of LER were compared with the observed values which were calculated from equation: 39 LER = (Yy/Yu) + (Yji/Yj,) (3.4) which is identical to equation 2.12 3.4.5 Plant Hierarchies Plant inequalities were determined using the Gini coefficient (Weiner and Solbrig 1984), a measure of inequality. This was calculated using the equation: G = I Z I sp - sq I / 2n2 S (3.5) p=l q=l H H where G, the Gini coefficient is the arithmetic average of the absolute values of the differences between all possible pairs of individuals, s represents the size of individuals in the sample, s is the sample mean, p and q are subscripts denoting all pairs of individual observations and n is the sample size. Since the calculated G for a small sample is a biased estimator of the population's G, sample G's were multiplied by n/(n-l) to give unbiased estimates of the population Gini coefficient G'. Standard error estimates were obtained from a bootstrapping procedure (Efron 1981, 1982). 3.4.6 Yield Component Analysis Primary variates for beans were leaf number (LN), leaf area (LA), leaf dry weight (WL), stem dry weight (WST), pod number (PN), and pod dry weight (WPD). For beets, the primary variates were LN, LA, WL, root j diameter (DR), and root dry weight (WR). Ratios of the primary variates le. yield components, were constructed, after arranging the variates in an 40 assumed chronological sequence of development. The sequence was based on the order of development of the components during plant growth te. leaves appear and expand in area, then they accumulate dry matter which is translocated to the stem and pods in beans or to the storage roots in beets. The overall yield component models were as follows: Beans: Y = LN X (LA/LN) X (WL/LA) X (WST/WL) X (PN/WST) X (WPD/PN) (3.6) where Y = WPD. Beets: Y = LN X (LA/LN) X (WL/LA) X (DR/WL) X (WR/DR) (3.7) where Y = WR. To measure the contribution of each yield component to variation in yield, and to assess the effects of treatments on these variates, two dimensional partitioning was done on the data. This procedure involves a combination of stepwise multiple regression analysis and analysis of variance (Eaton et al 1986). Both the forward analysis, in which the components are entered into the regression the way they appear in equations 3.6 and 3.7, and the backwards analysis, in which the components are entered into the regression in the reverse sequence, were performed on the data. 41 3.4.7 Plant Growth Analysis Plant growth analysis was done to measure the quantitative performance of plants, or plant parts, as affected by treatments during the entire growing period. Plant growth curves were fitted to the 1984 data over time for leaf area per plant, leaf dry weight per plant, leaf area ratio, leaf weight ratio, and harvest index using a cubic spline regression technique (Jolliffe and Courtney 1984). Growth indices commonly used in functional plant growth analysis were then computed from the fitted curves. This study involved 16 combinations of population density and mixture proportions (Table 3.1). In order to simplify the presentation of the results of the growth analysis, a subset of the overall data was selected for intensive analysis. Because interference among neighbouring plants was assumed to be most intense at high population densities, the data from the highest population density of 66 plants n r 2 were analysed to determine growth responses to mixture proportions. Also the 2:2 mixture treatment (Table 3.1) represents equality of opportunity for interference between species and that mixture was selected for analysis to determine growth responses to population density. 3.4.8 Plant Allometric Relationships The 1984 and 1987 data were fitted separately to the model: lOgefo) = l O g e M + Pologe(Zl) + h^O^) + pa^lo&fet) + ^X^lo^) + YilogeOq) + Y2loge(Xj) + YalogJXjXj) + logje1) (3.8) to determine the effect of population densities and mixture proportions on allometric relationships. In the model, y t represents shoot dry weight per 42 plant and Zj represents a secondary plant variable being evaluated in relation to shoot dry weight. In beans z t was taken to be leaf area, leaf dry weight, seed number, live leaf number, total pod dry weight, unmarketable pod dry weight, plant height, marketable pod dry weight, or marketable pod fresh weight. Similarly in beets, z t represented leaf area, leaf dry weight, petiole dry weight, live leaf number, storage root diameter, storage root fresh weight or storage root dry weight. As before, the Xj represents the population density of the test species while Xj represents the population density of the competing species. A best subset multiple regression analysis using the P:9R BMDP statistical package (Dixon 1985) was used in the analysis as described by Jolliffe et al (1988). 43 4. R E S U L T S 4.1 A n Overview of the Results Presentation of the results will be as follows: Visual observations made on plants during growth will be described first followed by analysis of variance (ANOVA) results on all primary variables. Yield-density relationships and differential yield responses will be presented next. Yield component analysis, plant growth analysis and allometric relationships will be reported last. 4.2 Visual Observations Plants started to emerge about 10 days to 21 days after planting. Beans were early to emerge, while beets were slow. Generally, germination was very good for beans, while in beets the sowing of more than 3 seeds per hill helped to produce the required planting densities, except in one replicate of the 66 plants n r 2 monoculture of the 1987 beets where fewer plants grew than intended. The possible reason for the slow and poor germination in beets could have been due to a deeper seeding depth (about 2 cm) than was warranted by the small seed size. Visually, both beans and beets seemed to be healthy throughout the growing seasons. Plants were noticeably smaller in the high population density treatments. Bean stalks from the less dense treatments were thicker than those from densely populated plots, which will be evident in the ANOVA results for bean stem dry weight presented in section 4.3.2. 44 4.3 Analysis of Variance 4.3.1 Homogeneity of Variance Test Homogeneity of variance is one of the assumptions for many statistical tests including ANOVA. This was tested for all primary variables and ratio indices using the partitioned Layard's homogeneity of variance test. The raw data indicated heteroscedasticity at the 5% level of significance for both the 1984 and 1987 data (Table 4.1). After transformation of the data to a log 1 0 scale, the homogeneity of variance assumption was largely satisfied as most variables had homogeneous variances at the 5% level of significance for both crop species in both years (Table 4.2). 4.3.2 General Results from the Analysis of Variance As detailed below, the results for both species both in 1984 and 1987 indicated a significant reduction in yield per plant of all primary variables with increasing total population density. The effects due to increasing mixture proportions of the competing species were also a decrease in yield per plant of the test species in beets and an increase in yield per plant with increasing mixture proportions of the competing species in beans. The effects due to interactions between population density and mixture proportions were infrequently significant. 4.3.2.1 Beans 1984 In the 1984 study, the results of the ANOVA for beans varied from harvest date to harvest date and from variable to variable (Table 4.3). Data from the last 5 harvests were analyzed for all variables. Pod number, pod fresh weight and pod dry weight data from the first harvest, Table 4.1 Summary of homogeneity of variance test of the raw data; percentage of variates homogeneous at the 5% level of significance as influenced by experimental treatments 1984 1987 Harvest Dates Source of variation HNlt HN2 HN3 HN4 HN5 HN6 BEANS Population density (D) 70 33 53 43 10 55 85 Mixture proportions (M) 50 70 58 75 58 60 85 D x M 70 73 48 80 95 100 72 Overall mean 63 88 53 66 66 72 81 1984 1987 Harvest Dates Source of variation HNlt HN2 HN3 HN4 HN5 HN6 HN7 B E E T S Population density (D) 60 38 55 45 53 48 55 75 Mixture proportions (M) 50 25 10 28 63 45 23 25 D x M 40 68 65 70 100 83 63 25 Overall mean 50 44 43 48 72 59 47 42 "•"Harvest number (1-6 in beans and 1-7 in beets) Table 4.2 Summary of homogeneity of variance test of the transformed data (log10 scale); percentage of variates homogenous at the 5% level of signifince as influenced by experimental treatments 1984 1987 Harvest Dates Source of variation HNlt HN2 HN3 HN4 HN5 HN6 BEANS Population density (D) 80 87 88 91 28 91 100 Mixture proportions (M) 90 66 62 82 34 74 100 D x M 90 82 71 91 100 100 75 Overall mean 87 78 74 88 54 88 92 1984 1987 Harvest Dates Source of variation HNlt HN2 HN3 HN4 HN5 HN6 HN7 BEETS Population density (D) 85 72 84 88 100 82 100 100 Mixture proportions (M) 90 42 47 59 84 36 47 100 D x M 95 60 54 60 84 71 82 75 Overall mean 90 58 62 69 89 63 76 92 ^Harvest number (1-6 in beans and 1-7 in beets) 47 Table 4.3 Analysis of variance results for the 1984 bean data: Variance ratios for the effects of population density and mixture proportions on primary variables tested at different stages of growth Age at Harvest Source of Variables (days) Variation d.f. T LN BN PN FWPD LA WL WST WPD W 40 Blocks 2 6* 4* 4* 3 5* 4* 5' (HNl)t Density 3 1 2 2 - - 5** 7** 1 - 4' Mixture 3 2 2 3 - - 1 3** 2 - 2 D x M 9 1 1 1 - - 1 1 1 - 1 Exp. Err. 30 3** 2* 1 - - 3** 2* 2** - 2' Samp. Err. 192 - - - - - - - - - - Total 239 - - - - - - - - - - 51 Blocks 2 4* 1 2 3 2 3 2 1 2 - 2 (HN2) Density 3 0 23** 11** 0 0 17** 33** 15** 0 24' Mixture 3 2 4* 2 0 1 1 5** 1 1 3 D x M 9 1 1 1 1 1 1 2 2 1 2 Exp. Err. 30 4** 1 2* 2** 2* • 3** 2* 2** 1 2' Samp. En. 192 - - - - - - •- - - - Total 239 - - - - - - - - - - 63 Blocks 2 1 2 4* 3 21** 2 1 1 23** 1 (HN 3) Density 3 0 39** 13** 17** 10** 19** 31** 15** 7** 22' Mixture 3 2 14** 6** 5** 2 3 10** 3* 1 6' D x M 9 0 1 0 0 1 0- 1 1 2 1 Exp. Err. 30 6** 1 1 1 1 4** 2** 3** 1 2 Samp. Err. 192 Total 239 t Harvest number (1-6 in beans and 1-7 in beets) *SignificantatP = 0.05 *SignificantatP = 0.01 48 Table 4.3 (cont'd) Analysis of variance results for the 1984 bean data: Variance ratios for the effects of population density and mixture proportions on primary variables tested at different stages of growth Age at Harvest Source of Variables (days) Variation d.f. T LN BN PN FWPD LA WL WST WPD W 69 Blocks 2 2 1 5* 4* 9** 4* 1 1 17** 1 (HSU) Density 3 3 54** 28** 36** 60** 18** 32** 10** 75** 17** Mixture 3 0 34** 21** 22** 45** 10** 21** 7** 47** 11** D x M 9 0 3* 1 2 2* 1 1 1 3* 1 Exp. Err. 30 8 1 1 1 0 3** 2 4** 0 3** Samp. Err. 192 - - - - - - - - - - Total 239 - * • - - • - - - - 75 Blocks 2 1 4* 5* 3 2 1 2 3 1 3 (HN5) Density 3 0 78** 29** 53** 25** 35** 49** 18** 64** 27** Mixture 3 4 35** 12** 17** 19** 9** 19** 4** 19** 9** D x M 9 1 4** 3** 2* 1 2 3** 2 5** 2* Exp. Err. 30 5 0 1 1 2 2 2* 3** 1 2** Samp. Err. 192 - - - - - - - - - - Total 239 - - - - - - - - - - 92 Blocks 2 0 3 0 1 2 1 1 0 0 0 (HN6) Density 3 1 47** 28** 53** 50** 20** 31** 25** 45** 13** Mixture 3 1 26** 8** 18** 24** 10** 18** 6** 18** 12** D x M 9 0 2 1 1 1 0 1 1 1 1 Exp. Err. 30 5 1 1 2** 1 2 3 3 2** 3** Samp. Err. 192 Total 239 *SignificantatP = 0.05 *SignificantatP = 0.01 49 done 40 days after planting, were not analyzed as the majority of observations for these variables were zero. The ANOVA indicated significant treatment responses for most variables tested. Total population density was found to significantly affect total dry weight, leaf dry weight, and leaf area at all harvest dates. From harvest 2 to harvest 6, all variables were significantly affected by population density except plant height which was significantly affected only at harvest 4. Branch number was also not significantly affected by population density at harvest 2 (Table 4.3). When density effects were significant, increasing population density reduced the mean yield per plant of the variable in question (Fig. 4.1). The effects due to mixture proportions were significant much later in the growing season on most of the variables, except for leaf dry weight which was influenced at all harvest dates. Leaf number was significantly affected from harvest 2 onwards, pod number and stem dry weight from harvest 3 and pod dry weight, pod fresh weight and branch number from harvest 4 onwards. Plant height was never significantly affected by mixture proportions throughout the study (Table 4.3). In all significantly affected cases, increasing mixture proportions of the competing species increased yield per plant (Fig. 4.2). Unlike the main effects, population density by mixture proportions interactions were seldom significant. Only three variables, leaf number, pod fresh weight and pod dry weight were significantly affected at harvest 4 and six variables, leaf number, branch number, pod number, leaf dry weight, pod dry weight and total dry weight at harvest 5 (Table 4.3). The trend was a decrease in yield per plant with increasing total population density and increasing mixture proportions of beets. The effect of 50 (a) (b) 35 16 3 3 5 0 Population density (plants rn 2 ) 6 6 16 3 3 6 0 -2 6 6 Population density (plants m ) Harvaet number —— Harvest 2 —t— Harvest 3 - * - Harvaet 4 - ° - Harvaet 6 Harvest 8 Fig. 4.1 The effect of population density on bean yield variables at different stages of growth (1984 experiment) 51 Harvest number Harvest 2 Harvest 3 Harvest 4 - B " Harvest 6 -*~ Harvest 8 Fig. 4.1 (cont'd) The effect of population density on bean yield variables at different stages of growth (1984 experiment) 52 1:3 Mixture proport ions (Beans:Beets) 2:2 3:1 4 : 0 Mixture proport ions (Beans:Beets) (d) 2:2 3:1 Mixture proport ions (Beans:Beets) 4:0 (f) 2:2 3:1 Mixture proport ions (Beans:Beets) Herveet 2 -a- Herveet B 4:0 Harvest number +- Herveet 3 Herveet 6 2:2 3:1 Mixture propor t ions (Beans:Beets) Herveet 4 4:0 Fig. 4.2 The effect of mixture proportions on bean yield variables at different stages of growth (1984 experiment) 53 *~ 2 0 h ' 1 2:2 3:1 * : 0 Mixture proportions (Bean»:Be»ts) Harvest number Harvest 2 Harvest 3 Harvest 4 -a- Harvest 6 Harvest 6 Fig. 4.2 (cont'd) The effect of mixture proportions on bean yield variables at different stages of growth (1984 experiment) 54 increasing mixture proportions of beets seemed to be greater at higher than at lower population densities (Appendix 8.1). Block effects were significant only early in the growing season (Table 4.3). 4.3.2.2 Beans 1987 The ANOVA results for the 1987 experiment showed significant population density effects on number of seeds, seed dry weight, marketable pod fresh and dry weights, and marketable pod number (Table 4.4). In all these variables, the trend was an increase in yield per plant from the first population density level of 16 plants n r 2 to the second population density level of 33 plants n r 2 , followed by a decrease at the third and fourth density levels of 50 plants n r 2 and 66 plants n r 2 respectively (Fig. 4.3). Other variables (live leaf number, dead leaf number, pod number, leaf area, leaf dry weight, stem dry weight, unmarketable pod dry weight, and total dry weight) did not show significant effects due to population density, but also tended to have similar responses (data not shown). This pattern would suggest that at the lowest population densities there was little or no interference. Above the second density level, plant interference tended to result in a reduction in mean yield of the variables tested. Large variations observed among individuals also made it difficult to detect significant differences between treatments. The same problem could account for lack of significant differences due to mixture proportions in any of the variables tested (Table 4.4). Total population density by mixture proportions interactions also were not statistically significant at the 5% level. 55 Table 4.4 Analysis of variance results for the 1987 bean data: Variance ratios for the effects of population density and mixture proportions on primary variables of beans tested at die final harvest. Source of Variation d.f. LL MPN UPN FWPD Variables WS SN LA WL WST WMPD WUDP W Blocks 1 0 0 8** 0 2 3 0 0 2 0 17** 0 Density 3 3 3 1 4* 5** 4* 2 2 1 4* 0 3 Mixture 3 2 1 1 2 2 2 1 1 0 2 1 1 D x M 9 0 0 0 1 1 2 0 0 0 1 1 0 Exp. Err. 15 4** 5** 4** 4** 2* 1 5** r* 3** 4** 2* 6** Samp. Err. 32 6** 5** 2** 4** 2** 2** 6** 5** 7** 4** 2* 6** Sub Samp. Err. 896 - Total 959 - *Significant at P •Significant at P = 0.05 = 0.01 56 Population density (plant per m 12 c « Q. 10 i_ <D a C 8 as a t_ 01 a CL >. 6 •o «~ T5 od w ei 4 .Q >. w a TJ «5 •a 2 (0 2 co Population density (plants m Yield variable MPN -4— SN c ra a> a a> tt) o a. Yield variables FWPD •-+- WMPD WS ng. 4 .3 The effect of population density on bean yield variables (1987 experiment) (a) Marketable pod number and seed number (b) Pod fresh weight, marketable pod dry weight and seed dry weight 57 4.3.2.3 Beets 1984 Compared to the bean data for 1984, the beet data were much more consistent from harvest date to harvest date and from variable to variable. As in beans, only some of the variables (plant height, leaf number, leaf dry weight, petiole dry weight, leaf area and total dry weight) were analyzed for the first harvest done 40 days after planting. Root diameter, root fresh weight and root dry weight were not analysed as the storage roots had not yet begun to enlarge. Other than at harvest 1, where no •significant differences due to population density were detected in any of the variables tested, results for harvests 2 to 7 showed significant effects due to total population density on all variables except plant height at harvest 6 (Table 4.5). Mixture proportions exerted significant effects on all variables tested, except on plant height, at harvest 2 and 3, leaf area at harvests 4, 5 and 6, and petiole dry weight at harvest 6. The total population density by mixture proportions interactions were significant only for leaf number and leaf dry weight at harvest 2 (Table 4.5). The responses in all these harvests were reductions in yield per plant with increasing total population density (Fig. 4.4). Similar trends were also observed due to increasing mixture proportions of beans (Fig. 4.5) except for beet plant height where an increase in height due to increasing bean proportions was observed (Fig. 4.5i). As for beans, block effects on the beet results were only significant in a few cases (Table 4.5). 4.3.2.4 Beets 1987 The ANOVA for the 1987 results for beets indicated significant effects of both population density and mixture proportions on all primary 58 Table 4.5 Analysis of variance for the 1984 beet data: Variance ratios for effects of population density and mixture proportions of primary variables tested at different stages of growth Age at Source of Variables Harvest Variation d.f. T LN DR FWR LA W WL WP WR 40 Blocks 2 1 1 0 0 0 0 (HNl)t Density 3 0 0 - - 0 1 1 1 - Mixture 3 0 2 - - 1 1 2 1 - D x M 9 1 1 - - 1 1 1 1 - Exp. Err. 30 7" 4" - - 6** 7** 7** 7** - Samp. Err. 192 - - - - - - - - - Total 239 - - - - - - - - - 51 Blocks 2 4* 0 0 1 2 1 1 1 0 (HN2) Density 3 r* 15** 8** 15** 11** 7** 12** 9** 3* Mixture 3 i 10** 11** 12** 6** 10** 15** 5* 8** D x M 9 i 3* 2 1 2 1 2* 1 1 Exp. Err. 30 3** 3** 3** 2* 3** 3** 2** 3** 4** Samp. Err. 192 - - - - - - - - - Total 239 - - - - - - - - - 63 Blocks 2 1 0 0 0 1 00 0 1 (HN3) Density 3 6** 15** 15** 16** 15** 17** 18** 15** 19** Mixture 3 0 12** 17** 18** 5** 15** 16** 7** 23** D x M 9 0 1 1 0 0 1 1 0 1 Exp. En. 30 7** 3** 2** 3** 4** 3** 4** 4** 3** Samp. Err. 192 Total 239 tHarvest number (1-6 in beans and 1-7 in beets) *SignificantatP = 0.05 "Significant at P = 0.01 59 Table 4.5 (cont'd) Analysis of variance for the 1984 beet data: Variance ratios for effects of population density and mixture proportions of primary variables tested at different stages of growth Age at Source of Variables Harvest Variation d.f. T LN DR FWR LA W WL WP WR 69 Blocks 2 2 2 4* 2 1 1 1 0 2 (HN 4) Density 3 1 8" 3* 5" 4" 5** 6" 3* 5** Mixture 3 2 8" 5" 10" 3* 9** 10** 5** 13" D x M 9 0 1 1 1 0 1 0 0 0 Exp. Err. 30 7" 3" 4" 5" 7** 5** 6** 7" 4** Samp. Err. 192 - - - - - - - - - Total 239 - - - - - - - - - 75 Blocks 2 2 1 1 0 2 0 0 0 0 (HNS) Density 3 2 9" 11" 12" 6** 10" 10" 5** 13" Mixture 3 0 7" 14" 14" 2 11" 10** 3* 17" D x M 9 1 0 0 0 0 0 0 0 1 Exp. En. 30 12" 6" 7" 7" 12** 8** 9** 10" 7** Samp. En. 192 - - - - - - - - - Total 239 - - - - - - - 92 Blocks 2 3* 0 1 1 3 1 2 2 0 (HN 6) Density 3 2 10" 8 10" 6** 8** 8** 4** 10** Mixture 3 1 .9" 13 15** 2 10" 9** 2 16** D x M 9 0 0 0 0 0 0 0 0 0 Exp. Err. 30 10" 3" 6 5" 10** 7** 9** 9** 5" Samp. Err. 192 - - - - - - - - - Total 239 - - - - - - - - - 107 Blocks 2 9" 4* 1 1 5 2 3 4* 1 (HN 7) Density 3 3" 6" 11 10** 8** 10" 12** 7** 10** Mixture 3 2 7" 23" 23** 4" 17" 15" 4** 23** D x M 9 2 1 1 1 1 0 l 1 1 Exp. Err. 30 6" 3 3" 3** 5** 3" 4** 3** 3** Samp. Err. 192 Total 239 *SignificantatP = 0.05 *SignificantatP = 0.01 60 16 16 0.06 16 Fig. 4.4 33 50 66 Population density (plants m 2 ) (c) 16 33 50 66 Population density (plants m ) 33 50 66 Population density (plants m"2 ) (e) 16 33 50 66 Population density (plant m - 2 ) ( f ) 33 60 Population density (plants m* 2) Herveet 2 -a- Herveet 8 33 50 Population density (plants m~ 2 ) Harvest number -t— Herveet 3 Herveet 8 * Herveet « Herveet 7 -a- Herveet 8 — n«r»»«i » The enect of population density on beet yield variables at different stages of growth (1984 experiment) 61 16 3 3 50 6 6 16 Population density (plants m" 2 ) 33 60 Population density (plants m"2) 6 6 (i) 33 60 Population density (plants m ^ 6 6 Harvest 2 -a- Harvest S Harvest number - * - Harveet 3 * Harvest 4 -x- Herveet 6 - * - Harvest 7 Fig. 4.4 (cont'd) The effect of population density on beet yield variables at different stages of growth (1984 experiment) 62 0:4 1:3 2:2 3:1 Mixture proportions (Beans:Beets) (c) 0 0:4 0 0:4 Fig. 4.5 0:4 1:3 2:2 3:1 Mixture proportions (Beans:Beets) (Q) 1:3 2:2 Mixture proportions (Beans:Beets) 3:1 (e) 0:4 1:3 2:2 3:1 Mixture proportions (Beans:Beets) ( f ) 1:3 2:2 Mixture proportions (Beans:Beets) Herveet 2 •-a- Herveet 6 0:4 Harvest number -•— Herveet 3 * - Herveet 8 1:3 2:2 Mixture proportions (Beans:Beets) Herveet 4 Herveet 7 3:1 The effect of mixture proportions on beet yield variables at different stages of growth (1984 experiment) 63 1:3 2:2 Mixture proport ions (Beans:Beets) 1:3 2:2 Mixture proport ions (Beans:Beets) 3:1 Plant height (cm per plant) 0:4 1:3 2:2 Mixture proport ions (Beans:Beets) Harvest 2 Harvest S Harvest number — Harvest 3 **— Harvest 8 Harvest 4 Harvest 7 Fig. 4.5 (cont'd) The effect of mixture proportions on beet yield variables at different stages of growth (1984 experiment) 64 variables tested (Table 4.6). Increasing plant population density generally reduced total dry weight, number of live leaves, tuber fresh weight, leaf dry weight, petiole dry weight and root dry weight per plant (Fig. 4.6). Similarly, increasing the mixture proportions of beans reduced total dry weight, number of live leaves, tuber diameter, leaf area, tuber fresh weight, leaf dry weight, petiole dry weight and root dry weight per plant (Fig. 4.7). The population density by mixture proportions interaction was not statistically significant at the 5% level of significance (Table 4.6). 4.3.3 Summary of Analysis of Variance Results In summary, both population density and mixture proportions significantly affected yield of almost all variables in both species in 1984 and in beets in 1987. The response was a reduction in yield per plant with increasing population density for both species and with increasing proportions of the competing species in beets. In beans, increasing mixture proportions of beets increased yield per plant. Population density by mixture proportions interactions were rarely significant. 4.4 Tield-density Relationships Non-linear models were used to define yield-density relationships in both monocultures and mixtures using equations 3.1 and 3.2 respectively. For each species, total dry weight, leaf number, leaf weight and leaf area were considered as the dependent variables in separate models. 65 Table 4.6 Analysis of variance results for the 1987 beet data: Variance ratios for the effects of population density and mixture proportions on primary variables tested at the final harvest Source of Variation d.f. LL DR Variables FWR LA WL WP WR W Blocks 1 2 0 0 0 0 0 0 0 Density 3 13** 7** 13** 8** 13** 7** 11** 3* Mixture 3 23** 15** 21** 9** 21** 7** 21** 5* D x M 9 1 1 1 1 1 2 1 1 Exp. Err. 15 1 1 1 1 1 1 1 2 Samp. Err. 32 2** 4** 4** 4** 3** 4** 4** 4** Sub Samp. Err. 192 Total 255 *SigniHcantatP = 0.05 *Significant at P = 0.01 66 4.6 The effect of population density on beet live leaf number, total dry weight, root fresh weight, leaf dry weight, petiole dry weight and root dry weight (1987 experiment) 67 (a) Live leaf number, leaf area and root diameter (b) Root fresh weight, leaf dry weight, petiole dry weight, root dry weight and total dry weight 68 4.4.1 Yield-density Regressions Table 4.7 shows the parameters and statistics while Table 4.8 shows standard deviations and error mean squares of the reciprocal yield-density model 3.2. As indicated earlier, parameters a and b serve to express different aspects of species performance and interrelationships. Parameter b u expresses responsiveness of y to intraspecific interference and parameter by is a measure of plant responsiveness to interspecific interference. Hence, the ratio b u /by estimates the relative influence of intra- to interspecific interference. Parameter a which expresses the reciprocal mean yield of an isolated plant as scaled by the parameter «D was found to be negative in some cases (Table 4.7) implying negative reciprocal mean yield. This could partly be due to the fact that the model best describes interference at high plant population densities and thus would not provide a good estimate of yield of an isolated plant. Under the conditions of this experiment, the results indicate that beans were the stronger competitor, both against themselves and against beets (Table 4.7). The exponent, (-<D), which may be related to the acquisition and utilization of resources within the space accessible by a plant (Watkinson 1984), was negative in all cases; large variations in o among harvests were noticed in both species. This variation might be a result of the shifts in plant development or changes in the relative importance of competition at different stages of plant development. 4.4.2 Summary of Yield-density Relationships Results Interference among associated plants of beans and beets grown in monocultures and mixtures were found to be quite complex. The Table 4.7a Estimates of parameter values for the response of total dry weight per plant to population densities* Estimates of Model Parameter Values Age at Harvest <J>j aj by tyi/by (days) BEANS 40 -0.630 0.0426 51 -2.35 0.352 63 -8.58 0.599 69 -0.544 -0.00028 75 -0.784 -0.00097 92 -0.819 -0.00184 BEETS 40 -0.559 3.99 51 -0.286 0.132 63 -51.2 0.956 69 -23.8 0.908 75 -0.525 0.0148 92 -6.40 0.576 107 -32.10 0.904 0.0041 0.0036 1.11 0.00431 0.0030 1.43 0.0021 0.0017 1.24 0.00003 0.000013 2.31 0.00016 0.000082 1.95 0.00024 0.000095 2.53 0.0081 5.77 0.001 0.00001 342.000 <0.001 0.00042 0.002 0.21 0.00098 0.003 0.28 0.0012 0.851 0.001 0.00037 0.014 026 0.0006 0.003 0.20 •Equation (3.2): (yy)"461 = â + b ^ + byXj 70 Table 4.7b Estimates of parameter values for the response of leaf dry weight per plant to population densities Estimates of Model Parameter Values Age at Harvest Oj aj b /̂bi (days) BEANS 40 -0.64 -0.0064 51 -1.69 0.2772 63 -2.47 0.3880 69 -0.41 -0.0429 75 -0.89 - :0.0012 92. -1.25 -0.0551 BEETS 40 -0.30 49.294 51 -0.40 -5.663 63 -18.32 0.9354 69 -0.54 0.4070 75 -0.46 -0.1863 92 -2.17 0.4326 107 -0.54 -0.045 0.0123 0.0099 1.24 0.0076 0.0050 1.52 0.0055 0.0020 2.75 0.0031 0.0010 3.26 0.0070 0.0273 0.26 0.0181 0.0039 4.64 0.3272 5563.000 0.0001 0.0326 94.290 0.0003 0.0010 0.0051 0.2000 0.0028 1.919 0.001 0.0145 7.207 0.002 0.0079 0.0481 0.16 0.0083 0.7358 0.01 •Equation (3.2): (vy)-** = £4 + bjPq + byXj Table 4.7c Estimates of parameter values for the response of leaf number per plant to population densities Estimates of Model Parameter Values Age at Harvest <J>j aj by îi/by (days) BEANS 40 -0.21 0.00002 51 -0.27 -0.00003 63 -1.06 0.0522 69 -0.63 0.0016 75 -1.51 0.1081 92 -1.62 0.0873 BEETS 40 -0.24 0.0006 51 -0.65 0.0377 63 -0.56 0.0117 69 -0.61 0.0291 75 -0.27 0.00001 92 -2.02 0.3357 107 -2.41 0.3865 0.00004 0.00002 0.20 0.00001 0.00001 0.70 0.0013 0.0005 2.60 0.00064 0.0001 6.40 0.0029 0.0007 4.14 0.0064 0.0013 4.92 0.000001 0.0001 0.01 0.000123 0.0017 0.07 0.0003 0.0016 0.19 0.00012 0.0015 0.08 0.00001 0.0002 0.05 0.0005 0.0032 0.16 0.00005 0.0029 0.02 •Equation (3.2): (yy)"*4 = aj + b̂ Xj + byXj Table 4.7d Estimates of parameter values for the response of leaf area per plant to population densities Estimates of Model Parameter Values Age at Harvest Oj aj by by ^v/^ij (days) BEANS 40 -1.12 -552.600 67.76 -44.431 -1.52 51 -0.95 3.650 0.3126 0.303 1.03 63 -20.99 1.067 0.0011 0.0006 1.67 69 -0.26 -8535.000 534.9000 463.813 1.15 75 -0.84 0.7503 0.3548 0.2141 1.66 92 -1.22 -0.5917 0.2984 0.2187 1.36 :ETS 40 -0.91 300.55 0.3648 -12.382 -0.03 51 -0.60 483.76 13.48 1233.04 0.01 63 -10.50 1.3107 0.0027 0.0101 0.27 69 -2.47 3.409 0.0345 0.1285 0.27 75 -0.39 -11533.000 770.100 153967.000 0.01 92 -0.73 24.653 5.065 73.964 0.16 107 -0.41 -2480.000 289.900 48920.100 0.01 •Equation (3.2): (yy)"01 = aj + bjPq + byXj 73 Table 4.8a Standard deviations and error mean squares for the response of total dry weight per plant to population densities STAGE 1 STAGE 2 Age at Asymptotic standard deviations RMS Std. error RMS harvest Oj aj bjj by BEANS 40 0.545 0.055 1.09 3.09 0.001 0.043 51 0.699 0.003 4.93 4.17 0.0005 0.014 63 3.896 0.023 449.72 444.6 0.0003 0.004 69 4.998 0.199 0.384 0.076 0.000001 0.000002 75 0.949 0.045 0.49 0.1310 0.0001 0.0003 92 1.164 0.054 0.492 0.124 0.0004 0.0008- BEETS 40 1885.84 5.85 178.62 0.131 1.26 100027.9 51 1107.30 2.61 425.83 0.004 118.99 885655352 63 12.08 0.11 83.50 0.002 0.0001 0.0005 69 19.08 0.13 183.59 0.002 0.0001 9000.07 75 0.047 0.005 0.693 84.37 0.187 2196.35 92 3.26 0.055 34.41 0.003 0.165 187.22 107 5.94 0.072 103.11 0.014 0.123 2459.05 74 Table 4.8b Standard deviations and error mean squares for the response of leaf dry weight per plant to population densities STAGE 1 STAGE 2 Age at Asymptotic standard deviations RMS Std. error RMS harvest Oj aj bjj by BEANS 40 0.49 0.008 9.21 1.42 0.002 0.28 51 0.52 0.004 2.10 1.30 0.001 0.030 63 0.76 0.005 4.78 2.81 0.0004 0.012 69 0.55 0.002 2.22 1.40 0.0003 0.006 75 0.62 0.001 4.96 1.28 0.061 230.62 92 0.61 0.003 0.53 3.73 0.001 3.40 BEETS 40 11140.68 240.81 6.42 5.64 1674.86 51 0.07 0.001 8.67 2.71 44.58 124330936.07 63 2.87 0.048 621.77 1.28 0.0002 0.0034 69 10.85 0.029 7.85 1.37 0.294 5407.66 75 1.11 0.014 5.85 2.33 1.84 211728.66 92 0.36 0.004 0.98 2.92 0.004 0.77 107 0.25 0.061 0.90 2.36 0.140 1234.73 75 Table 4.8c Standard deviations and error mean squares for the response of leaf number per plant to population densities STAGE 1 STAGE 2 Age at Asymptotic standard deviations RMS Std. error RMS harvest Oj aj bj| by BEANS 40 122.21 3.27 0.46 0.001 0.000004 0.000001 51 8.09 2.33 0.09 0.002 0.000002 0.0000 63 4.18 0.033 0.08 0.001 0.0001 0.001 69 7.44 0.125 0.053 0.001 0.0001 0.0002 75 9.02 0.176 1.76 0.001 0.0002 0.002 92 8.15 0.137 1.59 0.002 0.0004 0.011 BEETS- 40 1568.65 5.68 0.065 0.003 0.00002 0.00002 51 8.52 3.19 0.179 0.002 0.0001 0.002 63 1.89 0.25 60.120 0.002 0.0001 0.001 69 3.05 0.13 5.63 0.002 0.0001 0.001 75 2.19 0.11 0.88 0.001 0.00003 0.0001 92 4.07 0.08 0.76 0.001 0.0002 0.004 107 1.09 0.081 0.94 0.023 0.0003 0.004 76 Table 4.8d Standard deviations and error mean squares for the response of leaf area per plant to population densities STAGE 1 STAGE 2 Age at Asymptotic standard deviations RMS Std. error RMS harvest <t>j aj bjj by BEANS 40 6200.28 631.05 0.48 0.0007 4.23 1134141.99 51 20.37 1.007 0.83 0.0808 0.04 100.09 63 1.51 0.019 332.75 0.002 0.0001 0.001 69 13948.79 3703.62 0.18 0.002 177.84 197845779 75 11.36 0.83 0.57 0.002 0.037 86.82 92 10.11 0.98 1.29 0.003 0.022 29.93 BEETS 40 742688.53 1033.13 401.77 0.00001 0.69 29369.34 51 23588.49 599.63 4.31 0.0001 262.79 4320283383 63 7.53 0.06 172.63 0.0004 0.001 0.21 69 38.78 0.66 10.68 0.0003 0.011 8.27 75 38807.85 8558.93 0.45 0.0007 42231.2 10562564.91 92 405.91 35.12 0.87 0.0005 14.92 13922821.8 107 40576.54 4259.63 0.611 0.0005 15669.8 Too large 77 mathematical relationships as defined by non-linear inverse models have indicated strong intra- and interspecific interferences. In this study, the model parameters consistently suggested that beans were better competitors both in monoculture populations and in mixtures. 4.5 Differential Yield Responses of Mixtures The predicted combined yield per land area for bean-beet mixtures was evaluated using equation 3.3 while the observed combined yield was done using equation 3.4 at 92 days after planting. The predicted land equivalent ratio (LER) for total dry weight indicated yield disadvantage while the observed figures indicated yield advantages for total population density of 66 plants per m" 2 (Fig. 4.8). Similar results were obtained when other total population densities of 33 and 50 plants m" 2 were considered (data not shown), but the total population density of 16 plants per n r 2 predicted overyielding while the observed results indicated yield disadvantage (Fig 4.9). The predicted and the observed LER for leaf dry weight and leaf number at population density of 66 plants n r 2 were found to be greater than 1 indicating yield advantage in these two variables (Figs. 4.10 and 4.11). The other three total population densities of 16, 33 and 50 plants n r 2 also indicated yield advantage (data not shown). In case of leaf area, the predicted land equivalent ratio at total population density of 66 plants n r 2 indicated yield disadvantage while the observed figures indicated yield advantage (Fig. 4.12). Other total population densities also indicated similar results (data not shown). Marketable yield land equivalent ratios were also calculated but model 3.4 was not totally successful in predicting combined yield per land area Mixture proportions (Beans:6eets) (b) 1.6 i 4:0 3:1 2:2 1:3 0:4 Mixture proportions (Beans:Beets) Beans Beets Combined Fig. 4.8 Land equivalent ratio for total dry weight per unit land area at total population density of 66 plants n r 2 (1984 experiment) (a) Predicted (b) Observed 79 4:0 1.2 5 0.6 0.4 0.2 Hg. 4.9 Mixture proportions (Beans:Beets) / / / / / / " „ / - / .+- —K""" I 1 3:1 2:2 1:3 0:4 Mixture proportions (Beans:Beets) — Beans —t- Beets • • * - Combined Land equivalent ratio for total dry weight per unit land area at total population density of 16 plants nr 2 (1984 experiment) (a) Predicted fb) Observed 80 Mixture proportions (Beans:Beets) —~~ Beans H ~ Beets Combined Fig. 4.10 Land equivalent ratio for leaf dry weight per unit land area at total population density of 66 plants n r 2 (1984 experiment) (a) Predicted (b) Observed Fig. 4.11 Land equivalent ratio for leaf number per unit land area at total population density of 66 plants n r 2 (1984 experiment) (a) Predicted (b) Observed 82 (a) 1.2 i Fig. 4.12 Land equivalent ratio for leaf area per unit land area at total population density of 66 plants n r 2 (1984 experiment) (a) Predicted (b) Observed 83 in this variable. The problem was in beets where although the non-linear regressions converge, the parameters estimated were associated with error mean squares and coefficient of variation of over 1000. Thus the parameter estimates obtained could not produce reasonable LER when fitted to equation 3.4, but observed LER were greater than 1 indicating yield advantage at 66 plants n r 2 (Fig. 4.13) and at other population densities as well (data not shown). 4.5.1 Summary of Differential Yield Responses Results Applying the mathematical models to determine differential yield responses indicated yield advantage for leaf number and leaf dry weight and a yield disadvantage for total dry weight and leaf area in mixtures as compared to their corresponding monocultures while observed values indicated yield advantage in all the variables tasted. 4.6 Size Hierarchies Size inequality among individuals in bean and beet populations was evaluated using the Gini coefficient. The Gini coefficient has a minimum value of 0, when all individuals are equal, and it has a maximum of 1.0 in an infinite population in which all individuals except one have a value of zero. Mathematically, the Gini coefficient is calculated from equation 3.5 and graphically, after ranking individuals according to biomass in ascending order, the cumulative percentage of biomass is plotted against the cumulative percentage of the population. A perfect equality will produce a diagonal line from the origin to the right corner (Appendix 8.2). The degree of deviation from the diagonal line, the Lorenz curve, is a measure of inequality. It can be expressed as the ratio 84 Beans -+ - Beets Combined Fig. 4.13 Observed land equivalent ratio for marketable yield per unit land area at total population density of 66 plants n r 2 (1984 experiment) 85 of the area between the diagonal line and the curve over the triangular area below the diagonal. This ratio is the Gini coefficient (Weiner and Solbrig 1984). Tables 4.9 and Appendix 8.3 contain the results for the plant size distribution in beans while Table 4.10 and Appendix 8.4 contain those for beets. Size distribution of almost all the bean variables (Tables 4.9 and Appendix 8.3) seemed not to have followed any particular pattern in response to treatments except in the case of leaf number where in monoculture populations a steady increase in Gini coefficients with increasing population density was observed (Table 4.9c). When the mixture proportions were pooled, no pattern was again observed, but pooling densities consistently indicated that the bean monoculture treatment had the highest G' (unbaised Gini coefficient) in all variables except seed number and seed weight, and the 3:1 bean:beet ratio had the lowest G' except for marketable pod number, unmarketable pod number, seed number and seed weight. The ranking of the yield variables in order of increasing G', was as follows: leaf number (0.269) < total dry weight (0.291) < pod fresh weight (0.307) < stem dry weight (0.320) < marketable pod number (0.324) < marketable pod dry weight (0.336) < leaf area (0.338) < leaf dry weight (0.373) = seed dry weight (0.373) < seed number (0.374) < unmarketable pod number (0.384) < unmarketable pod weight (0.549) (Table 4.11a). When 95% confidence intervals were obtained from the bootstrapping procedure, it was found that leaf number had significantly lower size inequality compared to all variables tested. Total dry weight was not significantly different from pod dry weight which was in turn not significantly different from stem dry weight and marketable pod number. Stem dry weight was not significantly 86 Table 4.9a Gini coefficients for total dry weight distribution of beans grown in monocultures and mixtures Total Population Density Statistic 4:0* Mixture Proportions 3:1 2:2 1:3 0:4 Mixture Pooled 16 G' 0.165 0.241 0.290 0.205 0.286 33 G" 0.275 0.185 0.202 0.173 - 0.259 50 G' 0.218 0.208 0.172 0.255 - 0.262 66 G' 0.286 0.190 0.236 0.239 - 0.344 Density Pooled G' 0.287 0.208 0.242 0.237 - 0.291t Table 4.9b Gini coefficients for leaf dry weight distribution of beans grown in monocultures and mixtures Total Population Density Statistic 4:0* Mixture Proportions 3:1 2:2 1:3 0:4 Mixture Pooled 16 G' 0.232 0.321 0.366 0.250 0.380 33 G' 0.395 0.285 0.228 0.220 - 0.399 50 G' 0.273 0.284 0.208 0.359 - 0.342 66 G' 0.363 0.234 0.315 0.363 - 0.419 Density Pooled G' 0.363 0.286 0.299 0.309 - 0.373t *Beans:Beets tGini coefficient for the whole data set 87 Table 4.9c Gini coefficients for leaf number distribution of beans grown in monocultures and mixtures Total Population Mixture Proportions Mixture Density Statistic 4:0* 3:1 2:2 1:3 0:4 Pooled 16 G' 0.177 0.218 0.236 0.281 0.272 33 G' 0.190 0.169 0.218 0.361 0.211 50 G' 0.222 0.171 0.165 0.270 0.245 66 G" 0.323 0.133 0.259 0.231 0.333 Density Pooled G' 0.281 0.175 0.231 0.257 0.269t Table 4.9d Gini coefficients for leaf area distribution of beans grown in monocultures and mixtures Total Population Mixture Proportions Mixture Density Statistic 4:0* 3:1 2:2 1:3 0:4 Pooled 16 G' 0.208 0.316 0.336 0.272 0.338 33 G' 0.349 0.256 0.243 0.216 0.300 50 G' 0.227 0.299 0.186 0.333 0.316 66 G' 0.379 0.232 0.326 0.298 0.389 Density Pooled G' 0.331 0.280 0.289 0.289 0.338t *Beans:Beets +Gini coefficient for the whole data set 88 Table 4.10a Gini coefficients for total dry weight distribution of beets grown in monocultures and mixtures Total Population Density Statistic 4:0* Mixture Proportions 3:1 2:2 1:3 0:4 Mixture Pooled 16 G' 0.394 0.505 0.385 0.606 0.519 33 G' 0.397 0.595 0.448 0.343 0.548 50 G' 0.426 0.463 0.529 0.431 0.525 66 G' 0.530 0.499 0.772 0.376 0.757 Density Pooled G' 0.472 0.604 0.800 0.500 0.625t Table 4.10b Gini coefficients for leaf dry weight distribution of beets grown in monocultures and mixtures Total Population Mixture Proportions Mixture Density Statistic 4:0* 3:1 2:2 1:3 0:4 Pooled 16 G' 0.315 0.410 0.316 0.409 0.388 33 G* 0.337 0.460 0.331 0.263 0.434 50 G' 0.300 0.340 0.430 0.321 0.398 66 G' 0.383 0.326 0.487 0.305 0.427 Density Pooled G' 0.365 0.470 0.440 0.356 0.438t *Beans:Beets ^Gini coefficient for the whole data set 89 Table 4.10c Gini coefficients for live leaves distribution of beets grown in monocultures and mixtures Total Population Mixture Proportions Mixture Density Statistic 4:0* 3:1 2:2 1:3 0:4 Pooled 16 G' 0.141 0.152 0.172 0.219 0.179 33 G' 0.135 0.167 0.110 0.135 0.184 50 G' 0.128 0.143 0.150 0.125 0.163 66 G' 0.138 0.132 0.176 0.120 0.164 Density Pooled G' 0.154 0.177 0.181 0.163 0.185t Table 4. lOd Gini coefficients for leaf area distribution of beets grown in monocultures and mixtures Total Population Mixture Proportions Mixture Density Statistic 4:0* 3:1 2:2 1:3 0:4 Pooled 16 G" 0.308 0.362 0.296 0.501 0.391 33 G' 0.272 0.405 0.294 0.237 0.384 50 G' 0.293 0.324 0.391 0.363 0.384 66 G' 0.348 0.330 0.443 0.271 0.365 Density Pooled G' 0.333 0.434 0.403 0.401 0.413t Beans:Beets f Gini coefficient for the whole data set 90 different from marketable pod number, marketable pod dry weight, and leaf area. No significant differences were observed between leaf dry weight, seed dry weight, seed number and unmarketable pod number, but unmarketable pod dry weight had significantly the highest G' among all variables tested (Table 4.1 la). In beet monocultures, all the variables tested showed a general decline in the Gini coefficients with increasing population density. In all cases, the highest Gini coefficient was obtained at a population density of 16 plants n r 2 , while the lowest value was at 33 plants n r 2 with the other two density treatments lying in between (Table 4.10 and Appendix 8.4). The beet plants grown in mixture with beans showed no clear pattern in their Gini coefficients among treatments. Whereas pooling the densities did not show any pattern in G 1 , when mixture proportions were pooled, there was a general increase in G' with increasing population density for all variables except leaf area and number of dead leaves. For beets, the G* results were as follows in an ascending order: leaf number (0.185) < root diameter (0.350) < leaf area (0.413) < dead leaf number (0.436) < leaf weight (0.438) < petiole dry weight (0.449) < total dry weight (0.625) < root dry weight (0.641) < root fresh weight (0.661) (Table 4.11b). Based upon the confidence interval obtained from the bootstrapping procedure, leaf number had significantly the lowest while root diameter had the second lowest size inequality compared to the other variables tested. Leaf area and number of dead leaves did not differ significantly from each other so were number of dead leaves, leaf dry weight and petiole dry weight. Total dry weight, root dry weight and root fresh weight had significantly the highest Gini coefficients indicating high size inequality (Table 4.1 lb). Table 4.1 la Gini coefficients for bean yield variables in ascending order 91 Variable Gini coefficient 95% (G') confidence interval LN 0.269a* 0.257-0.281 W 0.291b 0.282 - 0.300 FWPD 0.307bc 0.296-0.318 WST 0.320cd 0.312-0.328 MPN 0.324cd 0.314-0.334 WMPD 0.336d 0.325 - 0.347 LA 0.338d 0.328 - 0.348 WL 0.373e 0.361 - 0.385 WS 0.373e 0.362 - 0.384 SN 0.374e 0.365 - 0.383 UPN 0,384e 0.375 - 0.393 WUPD 0.549f 0.542 - 0.556 Table 4.1 lb Gini coefficients for beet yield variables in ascending order Variable Gini coefficient 95% (G') confidence interval LN 0.185a 0.181-0.189 DR 0.350b 0.338 - 0.362 LA 0.413c 0.404 - 0.422 DLN 0.436cd 0.421-0.451 WL 0.438d 0.427 - 0.449 WP 0.449d 0.439 - 0.459 W 0.625e 0.598 - 0.652 WR 0.641e 0.627 - 0.655 FWR 0.661e 0.647 - 0.675 Gini coefficients with the same letter are not significantly different from each other based upon 95% confidence interval (P <0.05). 92 4.6.1 Summary of Size Distribution Results In summary, high Gini coefficients were generally obtained in most variables in both species. Responses due to treatments were not detected but in beets, a general decline in Gini coefficients with increasing population density was observed. Comparisons within species but between variables indicated that leaf number had the lowest Gini coefficient in both species but comparisons between species seemed to indicate that beets had higher G's than beans. 4.7 Yield Component Analysis The contribution of yield components to the total yield variation, and the relationships among yield components were determined. The two dimensional partitioning (TDP) procedure (Eaton et at 1985) was used in the analysis. 4.7.1 Beans 1984 The data for the 1984 bean experiment collected at 40 days from planting were not analyzed as most of the plants at this stage had no pod dry weight, the component considered to be yield. The results for later harvest dates indicated that some effects of treatment were significant from 63 days through to 92 days after planting (Table 4.12). The effects of population density on total yield variation rose from 0% at 51 days to 42% at the final harvest. For mixture proportions, the percent of total yield variation rose from 1% at 51 days from planting to 16% at the final harvest. The population density by mixture proportions interactions were significant, but remained relatively constant at about 5% throughout the growing period (Table 4.12). 93 Table 4.12a Two dimensional partitioning of yield variation in beans: 1984 data (forward analysis) Age at Harvest Source of Yield Components Sum of Yield (days) Variation d.f. LN LA/LN WL/LA WST/WL PN/WST WPD/PN Product WPD 51 Blocks 2 0 0 0 0 3 0 1 5* (HN2)t Density 3 5" 0 0 1 3 0 -8 0 Mixture 3 1 0 0 0 1 0 -1 1 D x M 9 1 0 0 0 4* 0 -1 5* Exp. Err. 30 2* 0 0 1 16" 0 2 22** Samp. Err. 192 10" 0 0 4" 45" 2 7 67** Total 239 18" 0 0 6" 73" 3 - 100 63 Blocks 2 0 0 0 0 1 6" 10 17** (HN3) Density 3 8" 0 0 0 0 1 -2 8** Mixture 3 3 0 0 1 0 0 -3 1 D x M 9 1 0 0 0 1 4* -1 5* Exp. Err. 30 2 1 0 1 5" 6" -5 11** Samp. Err. 192 12" 1 0 1 20" 22" 2 58** Total 239 27" 3" 0 3" 27" 39" - 100 69 Blocks 2 0 0 0 0 0 1 2 4 (HN4) Density 3 20" 0 0 0 0 0 6 26** Mixture 3 13" 0 0 0 0 0 4 17** D x M 9 3 0 0 0 0 1 -1 3 Exp. Err. 30 4" 2" 0 0 0 4 -6 4** Samp. Err. 192 27" 3" 1 0 2 16" -3 46** Total 239 66" 7" 2 0 2 23" - 100 75 Blocks 2 1 1 0 0 0 0 -2 0 (HN5) Density 3 21" 0 0 0 0 0 15 37** Mixture 3 10" 0 0 0 0 1 0 11** D x M 9 3 1 0 0 0 2 3 9** Exp. Err. 30 3" 3 0 0 1 7" -7 6** Samp. Err. 192 22" 4" 1 0 6" 13" -8 38** Total 239 60" 8" 1 0 8" 22" - 100 92 Blocks 2 0 0 0 0 0 0 -1 0 (HN6) Density 3 8" 1 2 1 0 0 30 42** Mixture 3 5" 0 2 0 0 0 9 16** D x M 9 2 1 0 1 1 0 -2 3 Exp. Err. 30 4" 9" 2 2 2 2 -10 9** Samp. Err. 192 16" 13" 9" 5" 5" 8" -25 30** Total 239 35" 24". 15" 8" 8" 10" — 100 tHarvest number (1 -6 for beans and 1 -7 for beets) Note: Numbers within the table are expressed as a percentage of total sum of squares for yield at each harvest *SignificantatP = 0.05 **SignificantatP = 0.01 94 Table 4.12b Two dimensional partitioning of yield variation in beans: 1984 data (backward analysis) Age at Harvest Source of Yield Components Sum of Yield (days) Variation d.f. WPD/PN PN/WST WST/WL WL/LA LA/LN LN Product WPD 51 Blocks 2 3 0 0 0 0 0 1 5* (HN2)t Density 3 1 1 0 0 1 0 -2 0 Mixture 3 1 0 0 0 0 0 0 1 D x M 9 3 1 0 0 0 0 0 5* Exp. Err. 30 16** 3** 0 0 1 0 2 22** Samp. Err. 192 58** 10** 0 0 1 1 -2 68** Total 239 81** 15** 0 1 3** 1 - 100 63 Blocks 2 21** 0 0 0 0 1 -5 17** (HN3) Density 3 2 0 0 0 1 0 5 8** Mixture 3 0 0 0 1 0 0 0 1 D x M 9 7** 0 0 0 0 0 -2 5* Exp. Err. 30 12** 0 0 1 1 1 -4 11** Samp. Err. 192 46** 0 0 1 2** 3** 6 58** Total 239 88** 0 0 3** 4** 4** - 100 69 Blocks 2 1 0 0 0 1 0 2 4 (HN4) Density 3 1 0 0 0 4* 4* 17 26** Mixture 3 2 0 0 0 2 3 10 17** D x M 9 2 0 0 0 1 2 -1 3 Exp. Err. 30 8** • 0 0 0 6** 3 -13 4 Samp. Err. 192 29** 0 0 1 13** 18** -15 46** Total 239 43** 0 0 2** 26** 29** - 100 75 Blocks 2 1 0 0 0 1 1 -3 0 (HN 5) Density 3 1 0 0 0 2 9** 24 37** Mixture 3 1 0 0 0 0 3 6 11** D x M 9 3 0 0 1 1 2 2 9** Exp. Err. 30 8** 0 0 2 5** 4** -13 6** Samp. Err. 192 17** 0 1 9** 10** 18** -17 38** Total 239 31** 0 1 13** 19** 36** - 100 92 Blocks 2 0 0 0 0 0 0 -1 0 (HN6) Density 3 0 0 0 1 3 12** 26 42** Mixture 3 0 0 1 1 1 1 13 16** D x M 9 0 0 0 0 2 2 -1 3 Exp. Err. 30 2** 0 1 1 9** 5** -8 9** Samp. Err. 192 10** 0 3 5** 15** 25** -29 30** Total 239 13** 1 6** 8** 29** 44** - 100 ^Harvest number (1-6 for beans and 1-7 for beets) Note: Numbers within the table are expressed as a percentage of total sum of squares for yield at each harvest *SignificantatP = 0.05 **SignificantatP = 0.01 95 Block effects were larger in the early stages of growth and declined to 0% in the later stages. The experimental error was significant, indicating large variations among plants within the same treatment. In the forward analysis, the yield components were included in the stepwise regression in chronological order of their development. This analysis indicated that treatments had strong effects on yield late in the growing season (Table 4.12a) The yield components, LN, L A / L N , PN/WST and WPD/PN made significant contributions to total yield variation at all harvest dates, except L A / L N did not make a significant contribution at harvest 2 or PN/WST at harvest 4. Other yield components, namely W S T / W L and W L / L A , were significant at harvests, 2, 3 and harvest 6 respectively (Table" 4.12a). The yield component LN was almost always the major component source of the treatment effects. It was affected by population density throughout the growing season. Mixture proportions effected LN starting at harvest 4, and population density by mixture proportions interactions were not significantly effective. The component PN/WST was a source of treatment effects at harvest 2, and WPD/PN contributed significantly at harvest 3. Both of those components were significantly affected by population density by mixture proportions interactions. Other yield components contributed to yield variation, especially at the final harvest, but not through direct treatment effects. In the backward analysis, yield components were fitted in the stepwise regression equation in the inverse order of their chronological development. The analysis indicated that WPD/PN and L A / L N always had significant contributions to total yield at all harvest dates (Table 4.12b). The yield component PN/WST had a significant effect only at the 96 second, harvest in contrast to WST/WL which made significant contributions to yield variation only at the last harvest. W L / L A and LN made significant contributions to yield variation at all harvests except at 51 days from planting. Significant effects due to population density were observed for L A / L N at 69 days and for LN at 69, 75 and 92 days. Significant effects due to population density by mixture proportions interactions, were observed with the yield component WPD/PN at 63 after planting 4.7.2 Beans 1987 In the 1987 study, where only one final harvest was done at about 83 days from planting, the TDP for bean indicated that yield was significantly affected by population density, mixture proportions and population density by mixture proportions interactions (Table 4.13). Due to large variations observed among plants within the same treatment, none of the components showed significant effects due to treatments. Overall, the yield components LN, W L / L A , PN/WST and WPD/PN made significant contributions to yield variation in the forward analysis, and the yield components WPD/PN, PN/WST, L A / L N and LN had significant effects in the backward analysis (Table 4.13a and b). 4.7.3 Beets 1984 Similar to the bean results for the 1984 growing season, the beet data collected at 40 days from planting were not analyzed because the storage root had not began to grow. The results obtained in the forward analysis at later harvests indicated that both population density and mixture proportions treatments had significant effects at harvest dates 97 Table 4.13a Two dimensional partitioning of yield variation in beans: 1987 data (forward analysis) Source of Yield Components Sum of Yield Variation d.f. LN LA/LN WL/LA WST/WL PN/WST WPD/PN Product WPD Blocks 1 0 0 0 0 1 0 0 1 Density 3 1 0 0 0 1 0 3 4* Mixture 3 0 0 0 0 0 0 2 3 D x M 9 0 0 0 0 2 0 2 4* Exp. Err. 15 1 0 1 0 2 0 4 7** Samp. Err. 32 1 0 0 0 3 0 2 6** Sub Samp. Err. 896 5" 0 3 0 72** 5** -11 74** Total 959 8" 0 4** 0 81" 6** _ 100 Table 4.13b Two dimensional partitioning of yield variation in beans: 1987 data (backward analysis) Source of Variation df. WPD/PN Yield Components PN/WST WST/WL WL/LA LA/LN LN Sum of Product Yield WPD Blocks 1 0 2 0 0 0 0 -1 1 Density 3 0 2 0 0 0 0 2 4* Mixture 3 0 1 0 0 0 0 1 3 D x M 9 0 2 0 0 0 0 1 4* Exp. Err. 15 0 2 0 0 1 1 3 8** Samp. Err. 32 0 5** 0 0 1 1 -1 6** Sub Samp. Err. 896 7" 62** 0 0 3" 7** -5 74** Total 959 8** 77** 0 0 5** 10** - 100 Note: Numbers within the table are expressed as a percentage of total sum of squares for yield at each harvest *SignificantatP = 0.05 **SignificantatP = 0.01 98 (Table. 4.14a). Population density by mixture interactions were significant except at 63 and 92 days after planting. The effect of population density rose from 8% at 51 days from planting to 33% at both 75 and 92, days and then declined to 15% at 107 days after planting. The mixture effect also rose from 20% at 51 days from planting to 34% at 107 days from planting. The population density by mixture proportions interaction remained below 10% throughout the growing period (Table 4.14a and b). Blocks did not make any significant contributions to variation in yield for beets. The forward analysis (Table 4.14a) indicated that the yield components LN, L A / L N , and W L / L A made significant contributions to yield variation at all harvest dates. The yield component D R / W L was significant at all harvests except at 69 days from planting. W R / D R was significant earlier in the growing season, but was not significant at both 92 and 107 days from planting. At all harvest dates, the main source of treatment effects was LN which was significantly affected by both population density and mixture proportions. The population density by mixture proportions interaction was also significant at 51 and 69 days from planting. In the backward analysis (Table 4.14b), the yield component W R / D R accounted for almost all the yield variation. D R / W L also had a significant contribution at 69 days from planting and L A / L N contributed significantly to yield variation at 51, 63, and 69 days from planting. The source of treatment effects was WR/DR, with population density and mixture proportions affecting that component significantly at all harvests from 51 days onwards. Population density by mixture proportions 99 Table 4.14a Two dimensional partitioning of yield variation in beets: 1984 data (forward analysis) Age at Harvest Source of Yield Components Sum of Yield (days) Variation d.f. LN LA/LN WL/LA DR/WL WR/DR Product WR 5 1 + Blocks 2 0 1 0 0 0 -1 1 (HN2)' Density 3 14** 0 0 0 1 -7 8** Mixture 3 9** 0 1 0 0 9 20** D x M 9 7** 0 0 1 1 9 18** Exp. Err. 30 9** 2** 1 1 6** 3 23** Samp. Err. 192 23** 10** 2** 3** 6** •4 39 Total 239 62** 14** 5** 5** 14** - 100 63 Blocks 2 0 0 0 0 0 -1 1 (HN3) Density 3 16** 1 0 0 0 6 24** Mixture 3 13** 0 4 0 0 12 29** D x M 9 2** 1 1 0 0 -1 3 Exp. En. 30 11** 2** 3** 2** 1 -5 13** Samp. Err. 192 24** 4** 7** 4** 4** -12 31** Total 239 66** 7** 15** 7** 6** - 100 69 Blocks 2 2 0 2 0 0 0 3 (HSU) Density 3 9** 0 1 0 0 1 11** Mixture 3 10** 1 6** 0 0 12 29** D x M 9 4** 1 1 0 1 0 5* Exp. Err. 30 12** 4** 1 0 2** 3 22** Samp. Err. 192 28** 5** 6** 1 5** -15 30** Total 239 64** 10** 17** 1 8" - 100 75 Blocks 2 h* 0 1 0 0 -3 1 (HN 5) Density 3 16** 0 0 0 0 8 24** Mixture 3 13** 0 4** 0 0 14 33** D x M 9 3 1 1 0 0 1 7** Exp. Err. 30 18** 2** 3** 1 1 -6 19** Samp. Err. 192 18** 2** 5** 4** 2** -14 17** Total 239 70** 6** 15** 6** 3** - 100 92 Blocks 2 0 2 0 1 0 -2 1 (HN6) Density 3 12** 0 0 0 0 7 20** Mixture 3 11** 0 5** 1 0 15 33** D x M 9 1 1 2 0 0 -2 2 Exp. Err. 30 12** 4** 3** 2** 1 -2 20** Samp. Err. 192 23** 5** 6** 7** 1 -16 25** Total 239 59** 13** 16** 10** 2 - 100 107 Blocks 2 3 1 1 0 0 -4 1 (HN 7) Density 3 6** 1 1 0 0 7 15** Mixture 3 7** 0 4 1 0 20 34** D x M 9 3 1 1 1 0 -2 5* Exp. Err. 30 11** 3** 1 2** 0 -3 15** Samp. Err. 192 26** 5** 3** 8** r -17 30** Total 239 57** 10** 11** 13** 2 100 tHarvest number (1-6 for beans and 1-7 for beets) Note: Numbers within the table are expressed as a percentage of total sum of squares for yield at each harvest. Significant at P = 0.05 Significant at P = 0.01 100 Table 4.14b Two dimensional partitioning of yield variation in beets: 1984 data (backward analysis) Age at Harvest Source of Yield Components Sum of Yield (days) Variation d.f. WR/DR DDR/WL WL/LA LA/LN LN Product WR 51 Blocks 2 1 0 0 0 0 0 1 (HN2)t Density 3 4* 0 0 0 0 4 8" Mixture 3 15" 0 0 0 0 5 20" D x M 9 7" 0 0 0 0 2 9" Exp. Err. 30 29" 0 0 1 0 -7 23" Samp. Err. 192 37" 1 0 3" 1 -2 39" Total 239 92" 1 0 5" 1 - 100 63 Blocks 2 1 0 0 0 0 0 1 (HN3) Density 3 22" 0 0 0 0 2 24" Mixture 3 28" 0 0 0 0 0 29" D x M 9 3 0 0 0 0 0 3 Exp. Err. 30 12" 0 0 1 0 0 13" Samp. Err. 192 30" 0 0 2" 1 -2 31" Total 239 96" 0 0 3" 1 - 100 69 Blocks 2 O 1 0 0 0 1 3 (HN4) Density 3 6" 1 0 1 0 4 11" .Mixture 3 18" 1 0 0 0 9 29" D x M 9 6" 1 0 1 0 -2 5* Exp. Err. 30 19" 5" 0 3" 0 -5 22" Samp. Err. 192 23" 9" 0 4" 1 -7 30" Total 239 72" 18" 0 9" 1 - 100 75 Blocks 2 0 0 0 0 0 0 1 (HN5) Density 3 22" 0 0 0 0 -2 24" Mixture 3 34" 0 0 0 0 -1 33" D x M 9 7" 0 0 0 0 0 7" Exp. Err. 30 18" 0 0 0 0 1 19" Samp. Err. 192 17" 0 0 1 1 -1 17" Total 239 98" 0 0 1 1 - 100 92 Blocks 2 0 1 0 0 0 0 1 (HN6) Density 3 19" 0 0 0 0 0 20" Mixture 3 33" 0 0 0 0 -1 33" D x M 9 3 0 0 0 0 -1 2 Exp. Err. 30 18" 0 0 0 0 1 20" Samp. Err. 192 25" 0 0 0 0 -3 25" Total 239 98" 0 0 1 1 - 100 107 Blocks 2 1 0 0 0 0 0 1 (HN7) Density 3 14" 0 0 0 0 1 15" Mixture 3 33" 0 0 0 0 1 34" D x M 9 4* 0 0 0 0 1 5* Exp. Err. 30 15" 0 0 0 0 -1 15" Samp. Err. 192 31" 0 0 0 0 -2 30" Total 239 99" 0 0 0 1 - 100 tHarvest number (1-6 for beans and 1-7 for beets) Note: Numbers within the table are expressed as a percentage of total sum of squares for yield at each harvest *SignificantatP = 0.05 "Significant at P = 0.01 101 Interactions was also significant at 51, 69, 75 and 107 days from planting. 4.7.4 Beets 1987 The forward analysis performed on beet data from the final harvest done at about 90 days after planting in 1987 showed that yield was influenced by population density, mixture proportions and population density by mixture proportions interactions. The last line in Table 4.15a indicates that all five yield components made significant contributions to variation in yield. The source of treatment effects was LN, which was significantly affected by population density.and mixture proportions. Blocks did not make any significant contributions to variation in total yield. In the backward analysis, the yield components W R / D R and L A / L N made significant contributions to variation in yield. The treatment effects came from W R / D R which was significantly affected by population density, mixture proportions and their interactions (Table 4.15b). 4.7.5 Summary of Yield Component Analysis Results Treatments were found to affect final agricultural yield of both beans and beets later in the growing season in 1984. Treatment effects were more drastic for main effects than their interactions in both years. In both the forward and backwards analysis, the yield component that entered into the equation first contributed the most to total yield variation. The yield component LN was generally found to be the source of treatment effects. 102 Table 4.15a Two dimensional partitioning of yield variation in beets: 1987 data (forward analysis) Source of Variation d.f. LN Yield Components LA/LN WL/LA DR/WL WR/DR Sum of Product Yield WR Blocks 1 0 0 0 0 0 0 0 Density 3 9** 0 0 0 0 5 15' Mixture 3 16" 0 0 0 0 10 27' DxM 9 3 1 0 1 1 0 5' Exp. Err. 15 3" 1 0 0 1 0 7' Samp. Err. 32 8" 4" 1 1 1 4 18' Sub Samp. Err. 192 25" 10" 3" 4" 5" -18 28' Total 255 64" 16" 4" 6" 9" - 100 Table 4.15b Two dimensional partitioning of yield variation in beets: 1987 data (backward analysis) Source of Variation d.f. WR/DR Yield Components DDR/WL WL/LA LA/LN LN Sum of Product Yield WR Blocks 1 0 0 0 0 0 0 0 Density 3 14** 0 0 0 0 1 15" Mixture 3 23** 0 0 0 0 4 27" DxM 9 5* 0 0 0 0 -1 5* Exp. Err. 15 5** 0 0 0 0 1 7" Samp. Err. 32 15** 0 0 1 0 2 18" Sub Samp. Err. 192 30** 1 0 3** 1 -6 28** Total 255 93** 1 0 4** 2 100 Note: Numbers within the table are expressed as a percentage of total sum of squares for yield at each harvest. *Significant at P = 0.05 "Significant at P = 0.01 103 4.8 Plant Growth Analysis Conventional plant growth analysis was used to determine the quantitative effect of experimental treatments over time using growth indices but before discussing plant growth analysis results, ANOVA results for ratio indices (leaf area ratio (F), specific leaf area (SLA), leaf weight ratio (LWR) and harvest index (H)) will be dealt with first. Analysis of variance results for total dry weight, leaf dry weight, and leaf area were already stated in section 4.2.2. Regression results for the primary variables (total dry weight, leaf dry weight and leaf area) will then be described, followed by those for ratio indices (leaf area ratio, specific leaf area, leaf weight ratio, and harvest index). Results from indices computed from first derivatives (absolute growth rate (AGR), relative growth rate (R) and unit leaf rate(E)) will be discussed last. Since fitted curves were not done on the 1987 data, where only one harvest was performed on each species, only growth curves for the 1984 data will be presented. 4.8.1 Beans 1984: Growth Indices (Leaf Area Ratio, Specific Leaf Area, Leaf Weight Ratio and Harvest Index): Analysis of Variance Results Analysis of variance done on growth indices for beans in 1984 indicated that leaf area ratio and harvest index were not significantly affected by treatments at any of the harvests except for leaf area ratio at harvests 5 and 1 where population density and mixture proportions had a significant effect respectively. On the other hand, specific leaf area responded significantly different to both treatments at all harvest dates (Table 4.16). Specific leaf area increased with increasing population Table 4.16 Analysis of variance results for the 1984 bean data: Variance ratios for the effects of population density and mixture proportions on growth indices tested at different stages of growth Age at Harvest Source of Variables (days) Variation d.f. F SLA LWR H 40 (HNl)t 51 (HN2) 63 (HN3) 69 (HN4) 75 (HN5) 92 (HN6) Blocks 2 2.83" 3.12 1.13 Density 3 1.86 3.17* 24.12" - Mixture 3 3.78* 9.90** 6.42** - D x M 9 1.26 1.17 2.28* - Exp. Err. 30 2.11" 1.68** 1.68** - Samp. Err. 192 - - - - Total 239 - - - - Blocks 2 7.51** 16.02** 2.71 2.31 Density 3 1.26 5.66** 8.17** 0.24 Mixture 3 1.68 7.72** 6.86** 0.46 D x M 9 0.36 0.36 0.60 0.81 Exp. Err. 30 3.44** 3.87** 2.87** 1.34 Samp. En. 192 - - - - Total 239 - - - - Blocks 2 11.14** 10.79" 19.02** 22.32" Density 3 0.77 3.05* 1.14 0.15 Mixture 3 0.36 13.14** 0.61 0.45 D x M 9 1.57 0.39 1.52 1.70 Exp. Err. 30 1.74** 3.55** 2.02** 1.25 Samp. Err. 192 - - - - Total 239 - - - - Blocks 2 0.24 17.09** 7.23" 2.98 Density 3 1.97 3.46* 1.64 1.93 Mixture 3 1.68 5.54** 1.21 1.52 D x M 9 0.63 0.67 0.58 0.54 Exp. Err. 30 2.26** 1.50* 2.06** 2.31" Samp. Err. 192 - - - - Total 239 - - - - Blocks 2 0.11, 8.33** 0.68 0.43 Density 3 2.93 7.67** 0.57 0.33 Mixture 3 2.21 12.38** 0.96 0.83 D x M 9 1.41 0.89 0.81 0.80 Exp. Err. 30 2.51** 2.27** 2.74** 4.68** Samp. Err. 192 - - - - Total 239 - - - - Blocks 2 3.30* 1.44 0.85 1.54 Density 3 1.30 4.33** 1.97 0.51 Mixture 3 0.18 4.29** 3.34* 0.03 D x M 9 0.29 1.24 0.91 0.94 Exp. En. 30 0.02 1.24 0.13 0.49 Samp. En. 192 - - - - Total 239 - - - - "'"Harvest number (1-6 in beans and 1-7 in beets) *SignificantatP = 0.05 *SignificantatP = 0.01 105 density and with decreasing beet proportions in mixtures (Fig 4.14). Leaf weight ratio was not affected by treatments except at harvests 1 and 2 where both population density and mixture proportions significantly reduced leaf weight ratio with increasing population density and decreasing beets proportions in mixtures (Fig. 4.15). Leaf weight ratio was also significantly reduced with decreasing beets proportions at harvest 6 (Fig. 4.15b). No differences were observed due to population density by mixture proportions interactions in any of the indices throughout the growing period. 4.8.2 Beans 1987: Growth Indices (Leaf Area Ratio, Specific Leaf Area, Leaf Weight Ratio and Harvest Index): Analysis of Variance Results Leaf area ratio, specific leaf area and leaf weight ratio results of beans in 1987 did not show any significant response to both main treatment factors (Table 4.17). Whereas no particular trends were noticed for leaf area ratio and leaf weight ratio, specific leaf area tended to be lowest for plants grown at 33 plants nr 2 and was highest for plants grown at 66 plants nr 2 . Plants at population density of 50 plants nr 2 treatment had the second highest while those at 16 plants nr 2 had the second lowest (data not shown). Regarding mixture proportions treatments, the 3:1 bean:beet ratio tended to have the lowest leaf area ratio and it increased with increasing beets proportions (data not shown). Similarly, the ANOVA results for harvest index indicated that both population density and mixture proportions did not significantly affect the proportions of the marketable yield (Table 4.17). Qualitatively, there was a non-significant increase from 51% at 16 plants nr 2 to 53% at 33 106 0.04 0.035 ~ 0.03 CM 0.025 S 0.02 w (0 ? 0.015 tt o = 0.01 o « W 0.005 16 33 50 66 0.04 0.035 Population density (plants m ) (b) 2:2 3:1 Mixture proportions (Beans:Beets) 4:0 Harvest 1 - B - Harvest 4 Harvest number - t - Harvest 2 * Harvest 3 Harvest 6 - • - Harvest 6 Fig. 4.14 The effect of population density and mixture proportions on bean specific leaf area at different stages of growth (1984 experiment) (a) Population density (b) Mixture proportions 107 O) C» O eg 0.7 0.6 0.5 0.4 (a) £ o. J n 0.1 1:3 16 33 50 66 Population density (plants m"^) (b) 2:2 3:1 Mixture proportions (Beans:Beets) Harvest number Harvest 1 Harvest 2 * Harvest 3 -Q- Harvest 4 Harveat 6 — * - Harvest 8 Fig. 4.15 The effect of population density and mixture proportions on bean leaf weight ratio at different stages of growth (1984 experiment) (a) Population density (b) Mixture proportions Table 4.17 Analysis of variance results for the 1987 bean data: Variance ratios for the effects of population density and mixture proportions on growth indices of beans tested at the final harvest Source of Variation d.f. F Variables SLA LWR H Blocks 1 0.31 3.38 0.32 9.08** Density 3 0.27 2.46 1.38 0.74 Mixture 3 1.06 1.56 0.85 0.18 D x M 9 0.56 0.26 0.33 1.07 Exp. Err. 15 1.52 2.15 4.93** 1.34 Samp. Err. 32 2.20** 2.40** 1.48** 1.92 Sub Samp. Err. 896 - - - Total . 959 _ _ *SignificantatP = 0.05 *Significant at P = 0.01 109 plants n r 2 followed by a decline to 50% at 50 plants n r 2 , and a further decrease to 48% at 66 plants n r 2 . A similar (non-significant) trend due to increasing beet proportions was also observed. An increase in marketable yield proportions from 51% at 4:0 to 53% at 3:1 followed by a decrease to 51% and 49% at 2:2 and 1:3 bean:beet proportions respectively. 4.8.3 Beets 1984: Growth Indices (Leaf Area Ratio, Specific Leaf Area, Leaf Weight Ratio and Harvest Index): Analysis of Variance Results Leaf area ratio, was significantly affected by both population density and mixture proportions at all harvest dates except at harvest 2 where only mixture proportions treatment effects were significant. The population density by mixture proportions interactions were significant at harvest 7 (Table 4.18). Leaf area ratio was found to increase with increasing population density and with increasing proportions of beans, the competing species (Fig. 4.16). This trend was easily seen at harvest 7 where the population density by mixture proportions interactions were detected. Leaf area ratio was highest at high population density and at high bean proportions in mixtures (Fig. 4.16c). Specific leaf area was significantly affected by both treatments at all harvest dates except at harvest 2 and 3 where only effects due to mixture proportions were significant. Interactions between population density and mixture proportions were found to be significant later in the growing season from harvest 4 to harvest 7 (Table 4.18). Specific leaf area increased with increasing population density and with increasing beans proportions in mixtures (Fig. 4.17a and b). At higher densities, Table 4.18 Analysis of variance for the 1984 beet data: Variance ratios for effects of population density and mixture proportions on growth indices tested at different stages of growth Age at . Harvest Source of Variables (days) Variation d.f. F SLA LWR H 40 (HN1)T 51 (HN2) 63 (HN3) 69 (HN4) 75 (HN5) 92 (HN6) Blocks Density Mixture D x M Exp. Err. Samp. Err. Total Blocks Density Mixture D x M Exp. En. Samp. Err. Total Blocks Density Mixture D x M Exp. Err. Samp. Err. Total Blocks Density Mixture D x M Exp. Err. Samp. Err. Total Blocks Density Mixture D x M Exp. Err. Samp. Err. Total 2 3 3 9 30 192 239 2 3 3 9 30 192 239 2 3 3 9 30 192 239 2 3 3 9 30 192 239 2 3 3 9 30 192 239 0.48 0.29 5.98" 7.14** 4.36" 10.64** 1.87 2.22 5.41" 5.85** 0.42 5.54** 1.29 2.75 11.63" 24.61** 0.92 0.80 5.00" 2.93** 3.78 3.27* 3.60* 1.76 32.27** 28.22** 1.67 1.21 1.63* 2.01 12.56** 26.15" 4.36** 8.66** 41.07** 81.88** 1.90 2.25 2.22 1.72* 0.84 12.49** 11.02" 8.06** 44.12** 54.37** 1.67 3.41** 3.77** 3.41" 1.13 24.12** 6.42** 2.23 1.69* 1.89 0.79 0.30 0.52 9.66** 1.28 2.78 3.13* 0.63 2.39** 0.61 0.70 3.50* 1.13 2.81** 2.82 6.61** 9.32** 0.65 3.63** 1.07 0.20 2.54 0.43 3.98 5.21** 5.46** 14.02** 0.82 2.92** 5.87** 3.13* 16.91** 1.48 3.68** 0.06 9.66** 22.81** 1.77 5.61** Blocks 2 1.49 2.98 0.03 0.83 Density 3 7.11" 2.33 6.21** 7.78** Mixture 3 39.81** 36.13" 10.07** 23.07** D x M 9 0.81 2.17* 0.85 0.89 Exp. Err. 30 3.71** 4.08** 2.13 3.09** Samp. Err. 192 - - - - Total 239 - - - - 107 Blocks 2 7.38** 12.50** 0.95 3.06* 1.87 (HN7) Density 3 8.52** 11.84" 5.53** Mixture 3 55.77** 61.95" 11.56** 21.64** D x M 9 2.85** 1.63* 3.51" 1.40 1.50 Exp. Err. 30 2.14** 1.69 2.66** Samp. Err. 192 - _ Total 239 - - - - ^Harvest number (1-6 in beans and 1-7 in beets) Significant at P = 0.05 *SignificantatP = 0.01 I l l 0.025 0.02 0.015 0.01 • 0.005 16 33 50 Population density (plants m"2) 0 0:4 1:3 2:2 Mixture proportions (Beans:Beets) — Harvest 1 - o - Harvss t 4 Harvest 7 Harvss t number Hsrves t 2 " * " Harvest 3 Harvest 6 Harvest 6 0.012 Population density (plants m ) Mixture proportions - — 0 : 4 ~+-1:3 -*-2:2 • - Q ' 3:1 Fig. 4.16 The effect of population density, mixture proportions and population density by mixture proportions interaction on bean leaf area ratio at different stages of growth (1984 experiment) (a) Population density (b) Mixture proportions (c) Population density by mixture proportions interaction (at 92 days from p l a n t i n g ) 0:4 1:3 2:2 3:1 Mixture proportions (Beans:Beets) Harvsst number Harves t l . —+— Harvest 2 -*~ Harvest 3 -a- Harvest 4 Harvest 6 Harvest 6 Fig. 4.17 The effect of population density and mixture proportions on beet specific leaf area at different stages of growth (1984 experiment) (a) Population density (b) Mixture proportions 113 the Increase in specific leaf area was more with increasing bean proportions than at lower densities (Fig. 4.18). Leaf weight ratio was also significantly affected by treatments later in the growing season from harvest 3 to harvest 7 though at harvest 4 only mixture proportions were significant (Table 4.18). Leaf weight ratio increased with increasing population density and with increasing beans proportions in mixtures (Fig. 4.19). The population density by mixture interactions were not significant at any time during the experiment. An ANOVA for harvest index was also not done for the beet data from the harvest done at 40 days after planting since most plants had not developed storage roots by that time. The results from harvest 2 indicated a significant effect due to mixture proportions while those of other "harvest dates indicated significant responses due to both population density and mixture proportions (Table 4.18). The trend was a decrease in marketable yield proportions with increasing population density (Fig. 4.20a). The largest difference between the lowest and the highest population densities was observed at harvest 5 where a 24% reduction in marketable yield was recorded across the range of densities. In mixtures, however, the decrease in marketable yield proportions was observed with increasing proportions of beans (Fig. 4.20b). The population density by mixture proportions interactions did not show significant responses at any of the harvests (Table 4.18). 114 (a) 0.06 16 33 60 Population density (plants m ^ 66 (b) 0:4 1:3 2:2 3:1 Population density (plants m ^ 0.025 Population density (plants m ) Mixture proportions 0:4 —r-1:3 - * 2:2 "3-3:1 Fig. 4.18 The effect of population density and mixture proportions interactions on beet specific leaf area (1984 experiment) (a) Harvest 5 (b) Harvest 6 (c) Harvest 7 115' 16 33 60 Population density (plants m"2) 66 0.8 (b) 0.7 - 0.1 - 0:4 1:3 2:2 Mixture proportions (Beans:Beets) Harvest number Harvest 1 ~~*~ Harvest 2 * Harvest 3 • Q - Harvest 4 Harvest 6 —*— Harvest 6 Harvest 7 3:1 Fig. 4.19 The effect of population density and mixture proportions on beet leaf weight ratio at different stages of growth (1984 experiment) (a) Population density (b) Mixture proportions 116 1:3 2:2 3:1 Mixture proportions (Beans:Beets) Harvest number Harvest 2 — H a r v e s t 3 ~ * Harvest 4 Harvest 6 - * * - Harvest 6 Harvest 7 The effect of population density and mixture proportions on beet harvest index at different growth stages (1984 experiment) (a) Population density (b) Mixture proportions 117 4.8.4 Beets 1987: Growth Indices (Leaf Area Ratio, Specific Leaf Area, Leaf Weight Ratio and Harvest Index): Analysis of Variance Results The response of leaf area ratio, and specific leaf area was not significantly different between treatments. Leaf weight ratio was also not significantly affected by population density but the effect due to mixture proportions was highly significant (Table 4.19). The mixture proportion 1:3 bean:beet ratio had the lowest leaf weight ratio while 0:4 had the second lowest followed by 2:2. The mixture proportion 3:1 had the highest leaf weight ratio of the 4 treatments (Fig.4.21). Leaf area ratio though not significant, tended to increase with increasing population density and with increasing proportions of beans in bean:beet mixture treatments. Leaf area ratio tended to be high for the population density of 33 plants n r 2 but had no clear pattern in rnixture proportions (data not shown). The population density by mixture proportions interaction was neither significant nor did it follow any particular pattern. Population density significantly affected harvest index in beets (Table 4.19). The percentage increase in marketable yield proportion was from 62% at 16 plants n r 2 to 112% at 33 plants n r 2 . Both high population density treatments of 50 and 66 plants n r 2 had lower proportions of 51% and 45% respectively. Mixture effects were not significant but a similar trend to the beans 1987 data was observed (i.e. 64% at 0:4, 117% at 3:1, 50% at 2:2, and 39% at 1:3 mixture proportions of beans :beets). The population density by mixture proportions interaction was not significant, but quantitative declines in marketable yield proportion with increasing total population density and bean proportions in the mixtures were observed (data not shown). Table 4.19 Analysis of variance results for the 1987 beet data: Variance ratios for the effects of population density and mixture proportions on growth indices tested at the final harvest Source of Variables Variation d.f. F SLA LWR H Blocks 1 2.58 0.61 0.20 0.20 Density 3 2.58 1.15 0.53 3.72* Mixture 3 1.01 3.05 8.19** 1.22 D x M 9 0.71 0.48 0.76 2.16 Exp. Err. 15 0.90 1.29 0.68 0.58 Samp. Err. 32 0.95 2.13** 1.76 1.89*' Sub Samp. Err. 192 Total 255 - - - *Significant at P = 0.05 "Significant at P = 0.01 119 0.35 1:3 2:2 Mixture proportions (Beans:Beets) 3:1 Fig. 4.21 The effect of mixture proportions on beet leaf weight ratio (1987 experiment) 120 4.8.5 Beans 1984: Primary Variables (Total Dry Weight, Leaf Dry Weight and Leaf Area): Regression Results Regressions for total dry weight indicated an increase in total dry weight per plant with time for all mixture proportions at the total population density treatment of 66 plants nr2. The increase was highest at mixture proportion 1:3 bean:beet ratio and decreased with increasing mixture proportions of beans. This response was evident after 51 days from planting though at the final harvest (92 days from planting), a slight decline was observed (Fig. 4.22a). Leaf dry weight and leaf area per plant underwent an early response to different mixture proportions at 66 plants nr 2 population density. The 1:3 bean:beet mixture proportion had again the highest increase in leaf dry weight and leaf area per plant. In the other mixture proportions treatments, the increase in leaf dry weight and leaf area per plant decreased with increasing beans proportions. The decline in leaf dry weight and leaf area was also observed at the final harvest (Fig 4.22b and c). The effect of increasing plant population density on total dry weight at 2:2 mixture proportions was evident after 51 days from planting whereas leaf dry weight and leaf area responded to treatments early in the growing period. The response curves were similar to those stated above. Total dry weight, leaf dry weight and leaf area per plant were seen to increase with time. The increase was highest at 16 plant nr 2 and decreased with increasing population density. These changes remained in effect throughout growth though a slight decline in yield at the final harvest was noticed at all population densities in all three variables tested (Fig. 4.23). This slight decrease in yield per plant seen at Mixture proportions (Beans:Beets) — 4:0 3:1 2:2 1:3 Fig 4.22 Changes in total dry weight, leaf dry weight and leaf area per plant in beans during growth resulting from increasing mixture proportions of beets at total population density of 66 plants n r 2 (1984 experiment) (a) Total dry weight per plant (b) Leaf dry weight per plant (c) Leaf area per plant (b) riMC A f T E R P L A N T I N G , d o y s U *t tl It TIME A f T E R P L A N T I N G , d a y s ( C ) Population density (plants m"2) — 16 33 50 66 Changes in total dry weight, leaf dry weight and leaf area per plant in beans during growth resulting from increasing total population density at 2:2 bean:beet mixture proportion (1984 experiment) (a) Total dry weight per plant fb) Leaf dry weight per plant (c) Leaf area per plant 123 the final harvest could be due to over fitting of the data by plant growth analysis procedure. 4.8.6 Beets 1984: Primary Variables (Total Dry Weight, Leaf Dry Weight and Leaf Area): Regression Results Fitted growth curves for total dry weight for beets were similar to those of beans. The effect due to increasing bean mixture proportions at 66 plants nr 2 started to show at 51 days from planting and remained in effect throughout the growing season. Treatment effects on leaf dry weight were earlier than on total dry weight. In both yield variables, the increase in yield decreased with increasing proportions of beans. Leaf area did not seem to be very drastically affected by treatments (Fig. 4.24). The effect of increasing population density beyond 16 plants nr 2 at 2:2 mixture proportion did not cause drastic increases in yield throughout the growing season. A big increase in total dry weight per plant took place only at the lowest population density of 16 plants nr2. A similar pattern was also observed for leaf dry weight and leaf area per plant (Fig 4.25). 4.8.7 Beans 1984: Growth Indices (Leaf Area Ratio, Leaf Weight Ratio, Harvest Index and Specific Leaf Area): Regression Results Leaf area ratio, leaf weight ratio and harvest index did not seem to be affected by different mixture proportions when examined at the total population density of 66 plants nr 2 (Fig. 4.26a-c). A general decline in leaf area ratio and leaf weight ratio was observed with time while harvest index generally increased. Although a general increase in specific leaf 124 Mixture proportions (Beans:Beets) — 0:4 1:3 2:2—3:1 Fig. 4.24 Changes in total dry weight, leaf dry weight and leaf area per plant in beets during growth resulting from increasing rrilxture proportions of beans at total population density of 66 plants nr 2 (1984 experiment) (a) Total dry weight per plant (b) Leaf dry weight per plant (c) Leaf area per plant 125 0.06 - TIME AfTER PLANTING, Coys Population density (plants m"2) — 16 33 50 66 Fig. 4.25 Changes in total dry weight, leaf dry weight and leaf area per plant in beets during growth resulting from increasing total population density at 2:2 bean:beet mixture proportion (1984 experiment) (a) Total dry weight per plant (b) Leaf dry weight per plant (c) Leaf area per plant 126 (b) 0 .020 c.on- er < 0 . 0 0 5 - 0 .0 JS 0.025 0.020 0 .0 U 0 . 0 0 5 - 0 . 0 0 0 - V U tl tt TIME A F T C R P L A N T I N G , d a y s id tt tl tt TIME AFTER PLANTING, doys (C) X * * « 8 S2 *6 TIME AFTER PLANTING, doys n « a 82 tt TIME AFTER PLANTING, doys Mixture proportions (Beans:Beets) — 4:0 3:1 2:2 —1:3 Fig. 4.26 Changes ln leaf area ratio, leaf weight ratio, specific leaf area and harvest Index per plant in beans during growth resulting from increasing mixture proportions of beets at total population density of 66 plants n r 2 (1984 experiment) (a) Leaf area ratio per plant fb) Leaf weight ratio per plant (c) Specific leaf area per plant (d) Harvest index per plant 127 area was also observed with time, the changes in rnixture proportions did affect this index (Fig. 4.26d). The effect was in the reverse order of what has been described for the three primary variables in sections 4.3.2.1 and 4.3.2.3. The increase in specific leaf area decreased with decreasing beets proportions. These results are similar to those of changing total population density at 2:2 mixture proportions (Fig. 4.27a-d). 4.8.8 Beets 1984: Growth Indices (Leaf Area Ratio, Leaf Weight Ratio, Harvest Index and Specific Leaf Area): Regression Results In beets, leaf area ratio, leaf weight ratio, harvest index and specific leaf area were all affected by changing the rnixture proportions at 66 plants nr 2 total population density treatment. Leaf area ratio, specific leaf weight, and leaf weight ratio, were reduced with time but the extent of reduction differed with rnixture proportions treatment. The decrease was highest for the mixture proportion treatment with more bean and decreased with decreasing beans proportions (Fig. 4.28a-c). The increase in harvest index also differed with different mixture proportions within the 66 plants nr 2 population density treatment. The increase decreased with increasing bean proportions (Fig. 4.28d). The results due to increasing population density at the 2:2 mixture proportions treatment were similar to those described above. Leaf area ratio, specific leaf area and leaf weight ratio decreased with time although a lot of fluctuations within treatments were seen (Fig. 4.29a-c). The increase in harvest index with time was also not systematic between treatments (Fig 4.29d). 128 V> *« 12 M TIME AFTER PLANTING, days b* tt 82 tt TIME AFTER PLANTING, doys Population density (plants m"2) — 16 33 50—66 Fig. 4.27 Changes In leaf area ratio, leaf weight ratio, specific leaf area and harvest index per plant in beans during growth resulting from increasing total population density at 2:2 bean:beet mixture proportion (1984 experiment) (a) Leaf area ratio per plant (b) Leaf weight ratio per plant (c) Specific leaf area per plant (d) Harvest index per plant 129 0.00 -t , , i o.a4 i — * ° ** M »* HO «0 V . U « t( 110 TIMC AfTER PLANTING, days TIME AfTER PLANTING, days Mixture proportions (Beans:Beets) — 0:4 1 : 3 2 : 2 — 3 : 1 Fig. 4.28 Changes in leaf area ratio, leaf weight ratio, specific leaf area and harvest index per plant in beets during growth resulting from increasing mixture proportions of beans at total population density of 66 plants nr2 (1984 experiment) (a) Leaf area ratio per plant (b) Leaf weight ratio per plant (c) Specific leaf area per plant (d) Harvest index per plant 130 (a) "E Hg. 4.29 te a: it TIME AFTER PLANTING, doys (c) a. (b) u te a: tt TIME AFTER PLANTING, doys (d) *» « : M TIME AFTER PLANTING, doys Population density (plants m"2) — 16 33 5 0 — 6 6 Changes In leaf area ratio, leaf weight ratio, specific leaf area and harvest index per plant in beets during growth resulting from increasing total population density at 2:2 bean:beet mixture proportion (1984 experiment) (a) Leaf area ratio per plant lb) Leaf weight ratio per plant (c) Specific leaf area per plant (d) Harvest index per plant 131 4.8.9 Beans 1984: Growth Indices (Absolute Growth Rate, Relative Growth Rate and Unit Leaf Rate): Regression Results All three growth indices (absolute growth rate, relative growth rate and unit leaf rate) were influenced by treatments. They all increased with time and reached a maximum in the middle of the growing season and thereafter declined. For absolute growth rate and unit leaf rate, the mixture proportion 1:3 bean:beet ratio had the most effect whereas the differences among other treatments was not so drastic (Fig. 4.30a and c). For relative growth rate, all treatments seemed to have the same effect (Fig 4.30b). The effect of increasing population density at the 2:2 mixture proportions treatments had similar responses as described above (Fig. 4.31a-c). 4.8.10 Beets 1984: Growth Indices (Absolute Growth Rate, Relative Growth Rate and Unit Leaf Rate): Regression Results Fig. 4.32a-c shows the effect of changing mixture proportions at 66 plants n r 2 population density treatment. No clear pattern was observed on treatment effect with time. Due partly to overfitting of the data, Fig. 4.32a-c indicate that all three indices fluctuated so much during growth (Fig. 4.32a-c). Similar patterns were seen for different total population densities at mixture proportion 2:2 treatment except the fluctuations were drastic only for 16 plants n r 2 population density (Fig. 4.33). 132 (b) - JO.O H 1 1 1 1 1 40 Vt M w M no TIME AFTER PLANTING, doyi Mixture proportions (Beans:Beets) — 4:0 3:1 2:2 1:3 Fig. 4.30 Changes in absolute growth rate, relative growth rate and unit leaf rate per plant in beans during growth resulting from increasing mixture proportions of beets at total population density of 66 plants n r 2 (1984 experiment) (a) lb) (c) Absolute growth rate per plant Relative growth rate per plant Unit leaf rate per plant 133 -*0.0 I i „ 4 0 »* M »2 * t TIME AFTER PLANTING, doys Population density (plants m"2) — 16 33 50 66 Fig. 4.31 Changes in absolute growth rate, relative growth rate and unit leaf rate per plant in beans during growth resulting from increasing total population density at 2:2 bean:beet inixture proportion (1984 experiment) (a) Absolute growth rate per plant (b) Relative growth rate per plant (c) Unit leaf rate per plant 134 (a) (b) 0.8 f 0.J-, 0.6- 0.2- - s . o - -10.0 -I i t *0 U 88 82 18 no TIME AFTER PLANTING, doys Mixture proportions (BeansrBeets) — 0:4 1:3 2:2—3:1 Fig. 4.32 Changes in absolute growth rate, relative growth rate and unit leaf rate per plant in beets during growth resulting from increasing mixture proportions of beans at total population density of 66 plants n r 2 (1984 experiment) (a) Absolute growth rate per plant (b) Relative growth rate per plant (c) Unit leaf rate per plant 135 (a) • (b) T I M E AFTER PLANTING. Ooys Population density (plants m*2) — 16 33 50 66 Fig. 4.33 Changes in absolute growth rate, relative growth rate and unit leaf rate per plant in beets during growth resulting from increasing total population density at 2:2 bean:beet mixture proportion (1984 experiment) (a) Absolute growth rate per plant fb) Relative growth rate per plant (c) Unit leaf rate per plant 136 4.8.11 Summary of Plant Growth Analysis Results Specific leaf area was found to increase with increasing population density in both species and with increasing mixture proportions of the competing species in beets. In beans, increasing mixture proportions of the competing species reduced specific leaf area. Leaf area ratio and specific leaf weight were significantly higher at higher population densities in beets monocultures and at high bean proportions in rnixtures but generally not significant for beans although both ratios tended to increase with increasing densities in bean monocultures and with decreasing beets proportions in mixtures. Derived ratios increased with time and declined after reaching a peak at about 68 days. Both the lowest population density of 16 plants n r 2 in both beans and beets, and the rnixture proportions treatment with the least proportion of the competing species in beets had the greatest increase in absolute growth, relative growth and unit leaf rates. In beans, the mixture proportion with the highest proportion of beets had the greatest increasing in derived ratios. This pattern was also observed in the primary variables; total dry weight, leaf dry weight, leaf number and leaf area. 4.9 Plant Allometric Relationships The relationships that exist between one part of the organism and another, or between part of the organism and the whole, are allometric relationships. The effect of treatments on allometric relationships were analysed using equation 3.8, and the results for each species are described in the following sections. 137 4.9.1 Beans 1984 Parameters and statistics for the allometric relationship of log,, shoot dry weight (W) (=y4) against either logg plant height (T), leaf number (LN), leaf area (LA), leaf dry weight (WL), stem dry weight (WST), branch number (BN), pod number (PN), pod fresh weight (FWPD), or pod dry weight (WPD) for the 1984 bean data are contained in Appendix 8.5, and Table 4.20 has the summary of the standard partial regression coefficients. The size and sign of the standard partial regression coefficient can be used to indicate the relative magnitude and direction of the relationships between W and independent variables. Significant terms containing logjz), le. those containing parameter (3, are indications of significant allometric relationships with logefW). Significant terms containing parameter a indicate treatment effects on loggfW) which are non-allometric. Generally, the allometric relationships between log^) and loggfzj) variables were influenced by different factors both between different z4 variables and between harvests within each z t. The models as determined by the best subset regressions varied considerably. Results from some harvest dates had as many as 7 terms (including the allometric parameter a) explaining the variation in loge(W) while the results in other harvest dates had as few as two terms, the allometric parameter a and the independent variable containing the (J0 exponent. The latter case indicates that treatments had no effect in influencing the allometric relationship between logefW) and loggtej) variable (e.g. Appendix 8.5.2 the harvest at 63 days from planting). In situations where the allometric relationship was dependant on more than the 2 terms mentioned above, treatment effects explain the presence of the other terms in those models. The treatments could have either 138 Table 4.20 Summary of the allometric analysis for the 1984 bean data. Standard partial regression coefficients for allometric relationships of secondary variables with In W (= y) Age at Potential Parameter Secondary variables (= z) harvest (days) independent variable T LN LA WL WST BN PN FWPD WPD 40 Intercept ln(a') -4.032" -0.166" 0.059" -0.039" -0.152" -0.178" -0.120** 0BSfl)T ln(z) P o 0.849" 0.055** 0.017" 0.010" 0.139" 0.019" 0.083" - - Xjln(z) P i 0.424" - - - - 0.0003* -0.001" - - Xjln(z) P 2 - - - - - - -0.003" - - XiXjln(z) P 3 -0.106 - - - - - 0.0001" - - ln(Xi) Ti -0.588" -0.035" - -0.010" -0.013" -0.046* - - - ln(Xj) Y2 -1.251" -0.071" 0.007" 0.004 0.006** 0.007" - - - ln(XiXj) Y3 0.977" 0.093" - - - - - 51 Intercept ln(a') -3.372" -0.153" 0.060" -0.039" -0.062" -0.196" -0.250* 0.232" -0.427** (HN2) ln(z) P o 0.692" 0.078** 0.028" 0.010** 0.016" 0.024" 0.140** 0.001* 0.187" X-ln(z) P i -0.130 -0.001" 0.0002" * - - -0.001" 0.003" 0.004 Xjln(z) P 2 - - - - - - -0.001" - - XjXjln(z) P 3 0.139 - - - - - - - - ln(Xi) Yl -0.560" - - -0.009" -0.014" -0.019" - -0.073" - ln(Xj) Y2 -0.178* -0.064* - - - - - -0.155" - ln(XiXj) Y3 - 0.084' - 0.178" 63 Intercept ln(a') -0.168 -0.245" 0.370" 0.206** 0.236" -0.447" 0.36" -0.181 -0.136" (HN3) ln(z) P o 0.337" 0.097" 0.100" 0.028** 0.032" 0.063" 0.200** 0.055" 0.038" Xjln(z) P i -0.530" - 0.002* - - - -0.002** -0.001 0.001" Xjln(z) P 2 - - - - - - - - - XiXjln(z) P3 - - - - - - - - ln(Xi) Yl - - 0.120* -0.043** -0.049** - - - - ln(Xj) Y2 - - - -0.081* -0.093** - -0.036 - - ln(XiXj) Y3 - - - -0.106* 0.121" - - -0.031 - tHarvest number (1-6 in beans and 1-7 in beets) *SignificantatP = 0.05 "Significant at P = 0.01 Note: In is synonymous for loge 139 Table 4.20 (cont'd) Summary of the allometric analysis for the 1984 bean data. Standard partial regression coefficients for allometric relationships of secondary variables with ln W (= y) Age at Potential harvest independent Parameter Secondary variables (= z) (days) variable T LN LA WL WST BN PN FWPD WPD 69 Intercept ln(a') 0.609** 0.311" 0.103" -0.248 -0.132" -0.319" 0.367" 0.106" -0.056 (HN4) ln(z) Po 0.135" 0.074" 0.057" 0.034" 0.033" 0.041" 0.124" 0.040" 0.008 Xjln(z) Pi -0.001" - 0.001" - - -0.0002" - - Xjln(z) P2 - - 0.0001* - - - 0.005* - - XiXjln(z) P3 - - - 0.00002* - - - -0.00004" ln(Xj) Yi -0.127* -0.483" - -0.039** - - -0.064** - 0.009 ln(Xj) Y2 - - - - 0.069" - -0.207** - -0.015 ln(XjXj) Y3 - - 0.040* 0.340* -0.093** - 0.209** 0.036" 0.019 75 Intercept ln(a") 0.905** 0.440** 0.099" -0.254** -0.143" -0.368** 0.141 0.111" -0.082 (HN 5) ln(z) Po 0.144" 0.086** 0.060" 0.036" 0.033** 0.047" 0.078 0.035" 0.009 Xjln(z) Pi 0.002 - 0.001" - - -0.0003** -0.001 -0.001" -0.0002 Xjln(z) P2 0.003* 0.003** - - - - - - - XjXjln(z) P3 - - - - - - - - - ln(Xj) Yi -0.220** -0.077** - -0.037" - - - - 0.022 ln(Xj) Y2 -0.306* -0.211" - - - - - - - lnOCjXj) Y3 0.305* 0.216" - - - - -0.027 - - 92 Intercept ln(a') 0.438" 0.095" 0.117" -0.435** -0.176" 0.221" 0.485" 0.175" -0.025" (HN 6) ln(z) Po 0.097" 0.037" 0.035" 0.053" 0.032" 0.084** 0.141** 0.033" 0.004** Xjln(z) Pi -0.001* -0.001" - - -0.001* -0.0003" -0.003** - 0.0001" Xjln(z) P2 - - - -0.001 -0.001" - - - 0.0002* XjXjln(z) P3 - - 0.00002** 0.00004** 0.00004" - - - -0.0001** ln(Xi) Yi 0.108" - -0.050" -0.076** - - -0.129 -0.037" -- ln(Xj) Y2 - - - - - - - - -0.022 ln(XiXj) Y3 - - - - -0.041" -0.023" - - 0.023* *SignificantatP = 0.05 **Significant at P = 0.01 Note: ln is synonymous for loge 140 influenced the allometry via those terms with B l f B 2 and B 3 , and/or the treatments could have influenced the non-allometric variation (yk terms) which account for some of the variation due to logje) in equation 3.8 and/or the treatment might have well influenced the allometric parameter a. It is possible that some of the treatment effects on the allometric parameter a could be allocated to the Yk terms and not log ĉx) so that a significant Yk term would either be due to direct treatment effects on loggfW) not related to allometry and/or may express treatment effects on the parameter a. The parameters B k and Yk acquire the units of measurement of the related independent variable. Except for a few cases (e.g. logJLA) for harvests at 40, 63 and 69 days from planting), population density, mixture proportions or population density by mixture proportions interaction generally reduced loggfW) (Appendix 8.5). All terms containing B 0 were positive and most of them were significant showing a direct allometric relationship aside from the effects of treatments. Significant interactions between treatments and the allometric exponents B l t B 2 , and B 3 were also detected in most regressions. Some of the independent terms in the models did not make significant contributions to explaining the variation in loggfW) but were included in the models because of the best subset regressions technique used (Appendix 8.5). The removal of the non-significant terms from the models may cause a non significant reduction in R 2 , but this may invalidate the best subset regression. The direction of response of the allometric parameters a, B k , and Yk on the allometric relationships followed no particular pattern. A factor would respond positively in one harvest and negatively in other harvest 141 within the same zi variable. Sirnilarly, no obvious pattern was observed between Zj variables at the same harvest date. The coefficients of determination (R2) which resulted from fitting equation 3.8 also varied from harvest date to harvest date within each z t variable. In almost all cases, the data harvested at 63 days from planting had the lowest R 2 value except in FWPD and WPD where the plants harvested at 51 days from planting had the lowest R 2 values. In most relationships the R 2 values were quite high, ranging between 0.70 to 0.99, but R 2 values of as low as 0.29 were observed. The higher R 2 values indicate an obvious allometric relationship (Appendix 8.5). 4.9.2 Beans 1987 As described by model 3.8, the results for the allometric relationships that exist between loggly,) and logjz,) are shown in Appendix 8.6, while Table 4.21 contains the standard partial regression coefficients. Similar to the 1984 bean data, each loggfet) variable related to loggtyj) had a different model. The number of independent variables explaining the variation in loggfW) varied from 5 to 7, again counting the allometric parameter a. All relationships indicated a strong treatment effect, either on the terms containing the allometric exponent B k or the terms containing non-allometric coefficients Yk- The allometric parameter a could also have been influenced by treatment factors which can be expressed in the Yk terms. Sirnilar to the 1984 bean data was also the presence of non-significant terms in some models because of the best subset regression technique used in the analysis. Unlike the 1984 bean data, some pattern was observed in the direction of response of the allometric exponent B 0 in that it was found to 142 Table 4.21 Summary of the allometric analysis for the 1984 bean data. Standard partial regression coefficients for allometric relationships of secondary variables with In W (= y) Potential Parameter Secondary variables (= z) independent variable LN LA WL WST MPN UPN FWPD WUPD WMPD SN WS Intercept ln(a') -0.305 2.910" 3.539" 4.042" 2.234" 4.745" -1.387" 6.505" 1.804" 1.450" 4.613" ln(z) Po 0.817" 0.817" 1.004" 0.781" 0.961" 0.489** 0.937" 0.164" 0.927" 0.729" 0.610" Xiln(z) Pi -0.256" - -0.111" - -0.115" -0.097 -0.228" 0.161" -0.116**-0.382" - Xjln(z) h -0.217" - -0.159" - -0.125" - -0.152" - -0.105"-0.305" - XjXjln(z) h -0.116* -0.123" 0.090** -0.071* -0.073 -0.183" -0.142" -0.075 -0.143"-0.149" -0.238" ln(Xj) Yl 0.269" -0.073* - -0.171" 0.087" -0.175" 0.280" -0.324" 0.127" 0.318" -0.125" ln(Xj) Y2 0.202" -0.451" - -0.849" - -1.152" 0.225** -1.560" - -0.869" ln(XjXj) Y3 - 0.491" 0.825" 0.130" 1.144" - 1.367" 0.186" 0.306" 0.937** 'Significant at P = 0.05 "Significant at P = 0.01 143 be positive in all the z t variables tested indicating a positive influence on the direct allometric relationship between logefW) and loge(zj) variable in question. The parameter a was also positive in all cases except in logg(FWPD) and logg(LN) z t variables. The allometric exponents f31( (32 and P 3 , and the non-allometric parameters Yi. Y2» Y3 did not follow any pattern. These parameters were found to respond positively In one z, variable and negatively in another. The R 2 values obtained also differed among z t variables. In general, the R 2 values were lower compared to those for the 1984 bean data. Most zi variables had R 2 values ranging between 0.58 to 0.79 except variables UPN, WUP, SN and WS who had R 2 values below 0.40. Although significant at 5% level of significance, these low R 2 values do not indicate an obvious allometric relationships between logefW) and the corresponding logjzi) variables. 4.9.3 Beets 1984 The allometric relationships between loge(yi) and loge^) varied greatly depending on the logjzt) variable being evaluated in relation to logglyt) and time at harvest (Table 4.22, and Appendix 8.7). Thus, a different model for each loggCŷ at each harvest date within and between loge(Z}) was the common occurrence. The largest model in the 1984 beets data had all 8 terms accounting for the variation in logefW) while the smallest models had 3 terms indicating treatment effects in all cases. Based on the same reasoning as in the bean data, the non-significant terms were retained in the equations. Just as in the bean data, the treatments could have influenced the allometry by influencing the allometric parameter a, the allometric terms 144 Table 4.22 Summary of the allometric analysis for the 1984 beet data. Standard partial regression coefficients for allometric relationships of secondary variables with ln W (= y) Age at Potential Parameter Secondary variable (= z) harvest independent (days) variable T LN LA WL WP DR WR FWR 40 Intercept ln(a") -7.136" -8.419" 5.211" -0.444" -2.037" -4.623" -1.870" (HNl)t ln(z) Po 0.886" 0.885" 0.923" 0.986** 0.932" 1.005" 0.964** - Xjln(z) Pi -0.115" -0.398* - - 0.087" -0.162" - - Xjln(z) P2 -0.337" -0.672** - - - - -0.058* - XjXjln(z) P3 - - 0.150" -0.017 - -0.038 - - ln(Xj) Yi - 0.439* - - 0.080" 0.175" -0.050* - ln(Xj) Y2 - 1.446** - -0.132** 0.318" - - - InPqXj) Y3 - -0.969* - 0.083** -0.240" - - - 51 Intercept ln(a-) -5.372" -5.377" 5.200" -0.202** -1.358" -2.150" -1.316" -0.389" (HN2) ln(z) Po 0.769" 0.818" 1.025" 1.099" 1.139" 0.840" 0.912" 0.859" Xjln(z) Pi - - -0.240" -0.199" -0.143" 0.174" 0.133" -0.089 Xjln(z) P2 -0.289" -0.126" -0.161" -0.100** -0.149" - 0.103 - XiXjln(z) P3 - - - 0.144" 0.180" - -0.210 0.170" ln(Xi) Yi - 0.051 -0.205" -0.051** -0.039* -1.171" 0.004* 0.079** ln(Xj) Y2 -0.112 - -0.776" -0.168** 0.143" -0.050 - 0.335" In(XiXj) Y3 - - 0.538" 0.122" - - - -0.365** 63 Intercept ln(cO -6.058" -5.122" 5.209" -0.500** -1.230" -1.995" -0.814** -0.003 (HN 3) ln(z) Po 0.755" 0.810" 0.970" 0.960** 1.030" 0.917" 1.006" 0.943** Xjln(z) Pi -0.311" -0.241" -0.274" -0.059" -0.069* -0.107" - -0.071* Xjln(z) P2 -0.298" - -0.195" - - . 0.167" -0.145" - XtXjln(z) P3 -0.174* -0.265" - 0.096" 0.093" - - 0.113" ln(Xi) Yl 0.337" 0.256" -0.271" - - - - - ln(Xj) Y2 - - -0.835" -0.018* 0.252" -0.442" -0.330" - ln(XiXj) Y3 -0.135* 0.465** -0.127 0.184" 0.234** Ĥarvest number (1-6 in beans and 1-7 in beets) *Significant at P = 0.05 ""Significant at P = 0.01 145 Table 4.22 (cont'd) Summary of the allometric analysis for the 1984 beet data. Standard partial regression coefficients for allometric relationships of secondary variables with In W (= y) Age at Potential Parameter Secondary variable (= z) harvest independent (days) variable T LN LA WL WP DR WR FWR 69 Intercept ln(a') (HN4) ln(z) Po Xfln(z) p! Xjln(z) P2 XJXjbKz) p3 ln(XJi) Yi ln(Xj) Y2 ln(XJXj) Y3 75 Intercept ln(a") (HN5) ln(z) Po Xiln(z) p! X:ln(z) p 2 XJiXjln(z) P3 ln(Xi) Yi ln(X:) Y2 lnQCJXj) 73 92 Intercept ln(a") (HN6) ln(z) Po Xjln(z) Pi X;ln(z) p 2 XJXjlnfz) p 3 Intfj) Y l ln(Xj) Y2 ln(XJXj) Y3 107 Intercept ln(oO (HN7) ln(z) Po Xiln(z) Pi Xjln(z) P2 xJXjln^) p 3 ln^j) Yi ln(Xj) y 2 ln(x]Xj) Y3 -4.631" -3.898" 5.589" -0.404** -1.443" -3.729" -0.730" -0.014 0.724" 0.775" 0.960" 0.921" 0.940" 1.320" 0.959** 0.935" - -0.123* -0.179" - - -0.265* - --0.207" -0.212* - - 0.138" -0.074** 0.062 -0.110 -0.195" -0.173" 0.042** - - -0.058** - - - -0.200" -0.047" 0.027 0.279* - --0.366" -0.188 -0.378" -0.173" 0.182" 1.365" -1.101" - - - - 0.107* - -0.687" - - -3.728" -2.663" 4.427" -0.678** -2.107" -3.076** -0.190 -0.182" 0.667" 0.748" 0.786" 0.943** 1.142" 1.066" 0.815" 1.012" -0.137" - - -0.045** -0.153" -0.105" 0.078" - - -0.171" - 0.081" - -0.118* 0.073* - -0.146" - - - 0.147" 0.071 0.078* - - -0.085 -0.050" - 0.146" 0.082 -0.065** - -1.007 - -0.337" - 0.660" 0.406" -0.409" 0.048" 0.408" -0.095 - -0.035* -0.396" -0.329" 0.263** - -2.984" -2.244" 4.785" -0.553** -1.367" -2.492" -0.608" -1.130" 0.643" 0.628** 0.812" 0.877** 0.896" 0.978" 0.955** 0.958" -0.209" -0.163" 0.044 - - -0.050" - - -0.223" 0.345* - 0.090** 0.205" 0.038 -0.037* 0.080** - -0.144 -0.088" - - - - -0.042 - - - -0.032* - - - --1.694" -1.248" -0.389" -0.071** 0.187" - -0.080** - -0.738" 0.712" - - - -0.058* - - -3.133" -3.817" 5.499" -0.335** -0.500" -3.680" -0.601" -0.032 0.599" 0.761" 0.819" 0.881" 0.906" 1.096" 0.900" 1.055" -0.127" -0.304" - - - -0.142* - - -0.222" - - 0.172" -0.179" - -0.061" - - -0.114" 0.052** - - - -- 0.206* -0.061* -0.092** - 0.141* - -0.003 -1.011" - -0.417" -0.273" 0.407" 0.686" -0.251" 0.079" 0.278 -0.178" - 0.190* -0.236* -0.464* 0.145* 0.145* * Significant at P = Significant at P = 0.05 0.01 146 containing the B k exponent with the exception of the B 0 term, and/or the non-allometric terms containing the y k parameters. The effect of the population density of beets in monocultures or of beans and beet-bean interactions in mixtures did not follow any pattern. Some specific patterns in the direction of response were observed in the case of the allometric parameter a. This parameter was found to be negative in all harvest dates in all the z t variables except in the case of loge(LA) where it was positive at all harvest dates. This would indicate a negative influence of the allometric parameter a in the other Zj variables and a positive influence on the relationship between loggfW) and loge(LA). The allometric exponent B 0 also followed a particular pattern in that it was found be have a positive influence on all the allometric relationships determined for the 1984 beets data. The other B k and all the y k terms did not follow any particular pattern. The R 2 values for almost all relationships in the 1984 beets data were above 0.70 indicating strong allometric relationships between logefW) and log^) variables. A lower R 2 value of 0.58 was observed for the allometric relationship between loggfW) and loge(LN) at 40 days from planting. Lower R 2 values indicate weaker allometric relationships. 4.9.4 Beets 1987 Appendix 8.8 contain the parameters and statistics while Table 4.23 has the standard partial regression coefficients for the 1987 beets data. Similar to the other cases above, the models have been found to be different for each log^) variable being tested in relation to log^) . The largest models had 8 independent terms explaining the variation in loggfW) and the smallest models had 4 independent terms indicating 147 Table 4.23 Summary of the allometric analysis for the 1987 beet data. Standard partial regression coefficients for allometric relationships of secondary variables with ln W (=y) Potential independent variable Parameter LN LA Secondary variables (= z) WL WP DR FWR WR Intercept ln(a') -0.620 0.029 1.073" 2.086" -1.442" -1.494" 0.315" ln(z) Po 0.564" 0.825" 0.889" 0.924" 0.786" 1.098" 1.075" Xjln(z) Pi 0.340" - - -0.157" 0.163" -0.186" -0.118" Xjln(z) P2 - 0.003" 0.281" 0.165" -0.172" - 0.152" XjXjln(z) Ps 0.248" -0.176" -0.375" -0.294" 0.155" -0.113" -0.246" ln(Xi) Yi -0.521" 0.079" - -0.069" -0.235" 0.247" 0.067" ln(Xj) Y2 -0.249" - - 0.321" - 0.097" -0.171" lnCXjXj) Y3 - -0.062" - -0.510" - - 0.187" "Significant at P = 0.01 148 treatment effects in all cases. Non-significant terms were again retained in the models. Just as in the 1984 beets data, the treatments influenced the allometric relationship either through the allometric parameter a and/or through the P k terms excluding the P 0 term and/or through the non allometric terms containing the parameters. The presence of the terms in the models did not follow any particular pattern. The allometric exponent J30 influenced the allometric relationships positively in all the loggfzi) variables, whereas the other parameters were inconsistent in their response. They would respond positively in one logjzi) variable and negatively in another variable, thus presenting no particular pattern The R 2 values obtained were all about 0.70 and above, except in the logefW) vs loge(LN) relationship where a lower R 2 value of 0.58 was observed. Again a lower R 2 value indicates a weaker relationship. 4.9.5 Summary of Allometry Results Allometric relationships in each species were found to be quite complex. The allometric relationships between logefW) and loggfz) were influenced by different factors within the same zi variable at different harvest dates in both species in 1984 and were influenced by different factors among z1 variables in both species in both years. There was no pattern in the direction of responses but most variables indicated strong relationships as indicated by high R 2 values. Strong treatment influences were also detected for each species in both years. 149 5. DISCUSSION 5.1 A n Overview Interference among neighbouring plants in monocultures and mixtures of different plant species has been widely examined, although this seems to be the first study to look at bean-beet associations. The general aims of this study were to quantify the interference between beans and beets, and to reach some understanding about the nature of those interferences by detailing the timing and sites within plants of interference effects. Thus, different analytical approaches were applied to the data to meet these specific aims. 5.2 Visual Observations Growth and development of both beans and beets was monitored throughout the growing season. Early in the growing season, beets were found to be quite slow to germinate. This could have been due to the excessive seeding depth in relation to seed size. Visually, plants in both monocultures and mixtures appeared to be quite healthy throughout the growing season. In both species, plants from densely seeded plots looked thinner and were taller than plants in less dense plots. The analysis of variance (ANOVA) results also indicated a reduction in per plant yield with increasing plant population density for most of the variables tested in both species in both years. The reductions in per plant yield reported in this study have commonly been observed by other researchers who have studied plant interference (e.g. Gaye 1990, Weiner 1984, Potdar 1986, Carmi 1986,). 150 5.3 Analysis of Variance In 1984, treatment effects [te. population density and mixture proportions) on most variables were large later in the growing season. For total dry weight, leaf dry weight and leaf area in beans the effects due to population density were detected as early as 40 days after planting, and for leaf dry weight the effects due to mixture proportions were also detected early during growth. In beets, the effects due to rnixture proportions on all variables except plant height were also detected early in the growing season. In 1987, there were fewer significant treatment effects especially in beans. This could have been due to the large variations among plants within the same treatment. In both 1984 and 1987, effects due to population density and mixture proportions interactions were not common. Hence, in both species and both years there were significant and diverse responses to treatments, and these responses were further explored by several lines of data analysis. 5.4 Tield-density Relationships Plants grown at low population densities grow as if they are in isolation, and thus experience no interference from each other (Spitters 1983, Jolliffe et al 1984). Interference among neighbouring plants intensifies with increasing proximity (Firbank and Watlunson 1985). Non-linear inverse models were used in this study to define yield-density responses because they have a biological basis (Jolliffe 1988) and also because their parameters serve as useful indices Of plant interference. Thus, they offer a potentially powerful approach to data interpretation. Four variables, total dry weight, leaf dry weight, leaf number and leaf area all of which were expressed on a per plant basis, were evaluated 151 In both beans and beets. Comparisons of the yield-density models formed for the different species and variables are made difficult by the effect of parameter <1> in scaling the yield variate. However, in this study all the models indicated that beans were better competitors both in monocultures and mixtures. This is indicated by the ratio by/by (Table 4.7), which was always greater than 1.0 for bean plants responding to themselves and beets, and less than 1.0 for beet plants responding to themselves and beans. Given the error in estimating parameters b u and by, the competitive balances between beans and beets were approximately the same for all four variables and all harvests. Inverse yield-density models have been found to be inadequate for certain yield variables, such as shoot height (Jolliffe 1988). Gillis and Ratkowsky (1978) also found the Bleasdale-Nelder equation to produce biased parameter estimates, especially in positively skewed data distributions. Large variations in the data could also impair the adequacy of the models, and these data were quite variable as could be seen from the high Gini coefficient values (indicators of size heterogeneity) for both monoculture and mixture data. The partitioned Layard homogeneity of variance test, and the significant F test between subsamples within a treatment were other Indications of high variation in the data. Gaye (1990) also found the Bleasdale-Nelder model unsatisfactory for describing yield for culled and undersized graded fruit of bell pepper (Capsium sativum) and attributed the difficulty to the small proportions of these grades relative to total fruit yield. This could not have been a contributing factor in this present study as all four variables chosen were major components of the total yield. 152 5.5 Differential Yield Responses of Mixtures The differential yield response of mixtures is based on a comparison of performance of crop mixtures in relation to their corresponding monocultures; overyielding in mixtures is the central issue in mixed cropping. Yield-density models (Jolliffe 1988) and direct computation of land equivalent ratio (LER) from the data were used to evaluate differential yield responses. When observed yields were used in calculating LER, results for both beans and beets usually indicated overyielding for the four variables (le. total dry weight, leaf dry weight, leaf number and leaf area) at all total population densities used in this study. The exception to this was with total dry weight at the total population density of 16 plants n r 2 , where a yield disadvantage was obtained. The results obtained at 16 plants n r 2 total population density might not be representative of interference effects, as the plants may not interfere strongly at low total population densities. When equation 3.3 was used to predict LER, two variables, leaf weight and leaf number indicated yield advantage, while total dry weight and leaf area indicated yield disadvantage. Two causes for the discrepancy between the observed and the modelled results for LER of leaf area and total dry weight can be advanced. First, the yield-density models were developed using data from all the population densities; they might not necessarily predict the results at a particular density successfully, given the high variability in the observations. Second, the models were developed for yield per plant, while differential yields are on a per land area basis. Multiplication of yield per plant by population density to produce yield per land area (Y = yX) magnifies errors of estimation. This could explain also why model 3.3 could not predict LER 153 for marketable yield in beets. The observed LER for all the four variables ranged between 1.0 and 2.5. 5.6 Size Hierarchies Plant size distribution, as measured by the Gini coefficient, was found to be highly and positively skewed for most variables at all four planting densities used in this study. That is, the results agree with the statement that plant populations often contain a few large individuals and many suppressed small neighbours (Weiner 1984). The results indicate that interference among bean, among beets and between bean and beet plants was "one-sided" [te. in a pairwise interaction there seemed to be a winner and a loser) (Weiner 1986, Weiner and Thomas 1986). Systematic changes in Gini coefficients in relation to experimental treatments appeared to occur in a few cases (e.g. increased Gini coefficient with increasing population density for leaf number of bean plants in monocultures). These patterns occurred more commonly in beets than In beans, but it is difficult to test the statistical significance of such apparent trends. Bonan (1988) found that the development of size hierarchies depends on a complex interaction of factors such as growth rate, spatial distribution and the degree to which competition for resources is symmetric or asymmetric. Pooling the mixture proportions to study the effect of population density did not reveal any pattern, but when population densities were pooled to evaluate the effect of rnixture proportions, it was consistently found that the monoculture bean treatments had higher Gini coefficients than mixed treatments. This may reflect that in beans intraspecific competition was more intense than interspecific competition. 154 In beet monoculture treatments, the 33 plant n r 2 population density treatment had the lowest Gini coefficient indicating less variation in plant size. Under normal circumstances, one would expect the lowest population density treatment to have the lowest Gini coefficient because of less or no interference among neighbouring plants (Weiner 1986). Whereas pooling population density treatments did not show any pattern, mixture proportions pooled indicated an increase in Gini coefficients with increasing population density. These findings are also in agreement with those of Wiener (1985), and Wiener and Thomas (1986) who found an increase in the Gini coefficient with increasing population densities. Rice (1990), working with two Erodium species, also found an increase in reproductive hierarchies with increasing population density. The high Gini coefficients for the whole data set for many variables are a clear indication of large variations in the data. Beets were more variable than beans, judging from the Gini coefficients values. Increasing the sample size by pooling population densities and mixture proportions tended to produce clearer treatment effects suggesting that a sample size of more than 60 individuals might have led to a clearer perception of treatment effects. 5.7 Yield Component Analysis Three approaches were used to analyze treatment effects on growth: yield component analysis, conventional plant growth analysis and allometric analysis. These procedures were intended to examine relationships underlying the overall growth responses among different plant attributes as they are influenced by experimental treatments. 1 5 5 Yield component analysis is a procedure that expresses yield as the product of a set of morphological yield components. The procedure has been used extensively in improving grain yield in rice (e.g. Matsushima 1966. 1980). The two dimensional partitioning (TDP) technique attempts to identify how the variation in yield is associated with variation in yield components, and how yield component variation depends on experimental sources of variation (Eaton et al 1985). The yield components are entered in the regression both in the forward manner (chronological sequence) and in the reverse order (le. the backward analysis). The effects of treatments on the yield and on individual yield components were more prominent later in the growing season. This is not surprising as interference among neighbouring plants is intensified as plants grow bigger. Expanding shoot and root systems most likely lead to competition as the resources start to be Urniting (Trenbath 1974). Yield component analysis indicated that the source of treatment effects was in most cases the yield component leaf number (LN), regardless of whether the forward or backward TDP analysis was done. Both population density and mixture proportions treatments, and their interaction, affected the yield components, though not necessarily at the same harvest time. Eaton et al (1985) and Jolliffe et al (1990) also found strong treatment effects on some yield components. Yield component analysis was therefore able to single out the yield components which were directly and indirectly affected by treatments and also quantified the contribution of each yield component to the total yield. The contribution of each yield component varied among harvests and differed depending on whether the forward or backward analysis was 156 used. In both cases the yield component that entered the regression first seemed to have contributed the most to the total yield variation. In part, this may be because the first component entering the model has the opportunity to account for all of the variation in yield; later components can only account for residual variation not explained by earlier ones in the model. Significant yield components which enter the model last can be considered to act directly on yield variation, since effects of all other components in the model have been taken into account. For example, this was the case with the yield component LN in the backwards analysis in beans at harvests 4 and 6 (Table 4.12b) 5.8 Plant Growth Analysis Conventional plant growth analysis is a time based analytical model which uses growth indices such as relative growth rate and leaf area ratio to express the presence of assimilatory structures and their performance as growth proceeds. Fitted growth curves are used to compute the growth indices and to indicate the timing and extent of changes in plant performance during growth. The ANOVA showed that the primary variables from which the growth indices were computed, were significantly affected by treatments. Similarly, ANOVA performed on the growth indices (leaf area ratio, leaf weight ratio, specific leaf weight) and the results usually indicated strong response of these indices to both population density and rnixture proportions. In both beans and beets, specific leaf area increased with increasing population density and in beets with increasing density of the competing species in mixtures. For beans increasing rnixture proportions of beets reduced specific leaf area. This could reflect a 157 mechanism by plants which serve to enhance the commitment of dry matter produced to leaf production as a plastic response to overcrowding. Whereas leaf area ratio and specific leaf weight did not generally respond to treatments in beans, both leaf area ratio and specific leaf weight were found to increase with increasing population density in beets monocultures, and with increasing proportions of beans in mixtures. The proportion of marketable yield produced within plants (Le. the harvest index) responded significantly to treatments in beets, but not in beans. An increase in population density or in the proportion of the competing species in mixtures caused a decrease in the proportion of beet storage root to total dry weight. This response is again a direct influence of interference in plant communities. The non-significant response of beans to treatment effects could be again due to large variations in the data set making the experimental error too large to be able to detect any significant differences. Growth curves were generated from a subset of the overall results, involving the 66 plants n r 2 population density and the 1:1 bean beet mixture proportions. Thus, the effects of increasing mixture proportions of the competing species was evaluated at 66 plants n r 2 total population density, and the effects of increasing population density was evaluated at mixture proportion of 1:1 bean:beet ratio. The choice of conditions for evaluating plant responses by conventional plant growth analysis was arbitrary but it was based on the belief that interference among plants is intense at high population density and the 1:1 bean:beet ratio would give each species an equal opportunity to cause interference. Fitted curves 158 were not done on the 1987 data as only a final harvest was done during that year. Fitted growth curves for the 1984 data showed that absolute growth rate, relative growth rate and unit leaf rate increased with time and reached a maximum at about 68 days from planting before declining. The monoculture beets or the mixture proportion with the most beets in beans had the highest of all three growth rates. Similarly the 16 plants n r 2 density treatment had the highest absolute, relative and unit leaf growth rates. Thus, both increasing population density at 1:1 bean:beet mixture proportions in and increasing mixture proportions of beans at 66 plants n r 2 , reduced absolute growth rate, relative growth rate and unit leaf rate of the test species. This is due to increasing intra- and interspecific interference with increasing total population density and increasing rnixture proportions of beans. 5.9 Plant Allometric Relationships Quantitative relationships that exist among different features of an organism as growth proceeds are referred to as allometric relationships (Jolliffe et al 1988). The procedure of Jolliffe et al (1988) was used to assess the effects of population density and mixture proportions on yield. The variation in yield (y) was partitioned to direct treatment effects, allometry on z, treatment effects with z, treatment effects on allometry with z, and residual variation. Since the allometric power equation (2.19) is equivalent to y /z = azP'l, this approach can thus be used to analyze the effects of treatments on plant proportions. The results obtained indicated that allometric relationships between logg^) and logjzi) variables were influenced by different factors. 159 The effects of treatments were evident in situations where the variation in loggfyi) was explained by more than two terms including the allometric parameter a and the independent variable containing the P 0 exponent. In most cases, it was found that treatments influenced the allometric relationship either through the allometric coefficient a or through the allometric exponent Pk, k>0 or the non-allometric adjustments expressed through with the y k terms. The y k terms explain the direct effects of treatment on y, independent of allometry with z (Jolliffe et al 1988). The relationship between W and LA in beans indicated a strong positive contribution from logJLA). The other allometric relationships were not consistent in the manner they responded. It is not clear why the direction of response of most of these parameters did not follow any particular pattern. One would be made to think that this was due to large variations which were observed in the data or it could be that the nature of response varies depending on the environmental factors not accounted for during the experiment. Such complexes in allometric responses were also shown by Jolliffe et al (1988). The keeping of non- significant terms in the model when the best subset regression technique is used is necessary or the method may not be valid. Few linear allometric relationships, in which only b 0 was significant, were observed in this study. Examples include W vs LN (Appendix 8.6.2), W vs LA (Appendix 8.6.3), W vs WL (Appendix 8.6.4). Linear allometries were mostly noted in beans at 40, 63 and 69 days from planting for LN, at 40 days for LA, at 40, 51, 63, and 75 days for WL. In beets only W vs LA (Appendix 8.8.3) at 75 days, W vs DR (Appendix 8.8.8) at 75 days, and W vs WR (Appendix 8.8.7) at 107 days 160 from planting were linear. All other relationships were curvelinear (i.e. at least one term containing B l t B2 or B3 was significant). High R 2 values in the allometric relationships are a good indication of strong relationships. Some of the relationships were significant but had very low R 2 values. The strength of these relationships is questionable. Significance could be a result of the many degrees of freedom which were available for the denominator component of the variance ratios tested. 5.10 A Summary of the Discussion In summary, the yield responses of beans and beets to both population density and mixture proportions in both 1984 and 1987 were generally detected to be significant regardless of the analytical procedure used to access the effects. Each method was able to demonstrate additional information to the general finding of significant treatment effects on yield although in some cases discrepancies between the different analytical methods were detected. Using the ANOVA indicated that high population densities reduced total plant yield in either species whether grown in monocultures or mixtures. Yield reduction was a result of intensified interference among plants with increasing population density and rnixture proportions of bean. In 1984, the effects were seen to take place later in the growing season usually after 51 days from planting and onwards in both species, though variables like total dry weight, leaf dry weight and leaf area in beans due to population density and for leaf dry weight the effects due to rnixture proportions were also detected early during growth. In beets, the effects due to rnixture proportions on all variables except plant height 161 were also detected at 40 days after planting. Yield-density models were able to show which of the two species was a good competitor both in monocultures and in mixtures. Yield-density models thus showed that beans were better competitors when grown in monocultures and mixtures. The analysis of differential yield responses on the other hand were able to show that growing beans and beets together might be beneficial as the observed LER indicated yield advantage in mixtures as compared to their corresponding monocultures in all the variables tested. Yield component analysis was able to show that interference is a major source of yield variation. This procedure was able to single out the variables which were directly or indirectly affected by treatments. Both the forward and the backward analysis indicated that leaf number (LN) was the component which was directly influenced by treatments in both species. Just as the ANOVA, this analysis also indicated that in 1984, the effects of treatments were significant later in the growing season. Treatment effects were difficult to determine in the 1987 data because of the large variation between plants within treatments. These large variations among plants were demonstrated by high Gini coefficient values which were obtained when size frequency distribution was determined. The Gini coefficient was also able to show which yield components were more variable compared to others. Similar to ANOVA and yield component analysis results, conventional plant growth analysis results also indicated that yield reduced with increasing population density and with increasing mixture proportions of beans. The reduction in yield was also evident 51 days after planting and onwards indicating intense plant interference as 162 plants grew larger. Both the efficiency and extent of assimilatory systems were affected by interference in beets. The allometric relationship between logefW) (total yield) and yield components (loge(z)) were accessed and it was found that treatments affected the allometry. The effects due to treatment were as early as 40 days after planting. The direction of response was difficult to determine as no pattern was obtained but it seems that each plant part was affected differently by the different treatments. 163 6. CONCLUSIONS 1. Visually, plants in high population densities were smaller than those in less dense treatments. These observations were confirmed by the analysis of variance (ANOVA) which indicated that per plant yield in almost all variables was reduced with increasing population density and with increasing rnixture proportions of beans. 2. Interference among associated plants of beans and beets was found to be quite complex. Non-linear inverse yield-density models, have indicated strong intra- and interspecific interferences. In this study, the model parameters consistently suggested that beans were better competitors than beets, both in monocultures and in mixtures. 3. Land equivalent ratios (LER) computed from observed yield values usually showed yield advantage for all variables. Applying the yield-density models to determine the differential yield responses indicated yield advantages for leaf number and leaf dry weight, and yield disadvantages for total dry weight and leaf area in mixtures as compared to their corresponding monocultures. 4. The effect of population density and mixture proportions on the frequency distribution of plant size, as expressed by the Gini coefficient, did not have obvious trends in beans monocultures and in beans or beets mixtures. In monocultures of beets, the plants were found to differ in size more at the lowest density of 16 plants nr 2 and was least variable at 33 plants nr 2 . The two population densities of 50 and 66 plants nr 2 164 were intermediate. This pattern was found to be true in all the yield variables tested. Pooling mixture proportions to study the effect of population density indicated a general increase in the Gini coefficient with increasing population density. When the yield variables were compared within species, leaf number was found to be least variable in either species but a comparison between species indicated that beets yield variables had higher Gini coefficient values than beans. 5. Treatments were found to affect agricultural yield of both beans and beets later in the growing season regardless of whether the forward and backward yield component analysis was done. Main effects were stronger than their interactions. In both the forward and backward analysis, the yield component that entered into the equation first contributed most to total yield variation. The yield component leaf number (LN) was often found to be the source of treatment effects. 6. Specific leaf area was found to increase with increasing population density and with increasing mixture proportion of beans in both beans and beets as time progressed. Leaf area ratio and specific leaf weight were significantly higher at higher population densities in beets monocultures and at high bean proportions in mixtures but generally not significant for beans although both ratios tended to increase with increasing densities in bean monocultures and with increasing beans proportions in mixtures. Derived ratios increased with time and declined after reaching a peak at about 68 days. Both the lowest population density of 16 plants nr 2 and the mixture treatment with the least proportion of beans had the greatest increase in absolute 165 growth, relative growth and unit leaf rates. This pattern was also observed in the primary variables; total dry weight, leaf dry weight, leaf number and leaf area. 7. Bivariate allometric relationships in each species were also found to be quite complex. Strong treatment influences were detected for each species and influences varied at different harvest dates. Different allometric relationships and different treatment effects on allometry, were determined. 166 7. LITERATURE CITED Andrews, D.J. 1972. Intercropping with sorghum in Nigeria. Expt. Agric. 8:139-150. Andrews, D.J. and A.H. Kassam. 1976. The importance of multiple cropping in increasing world food supplies, p. 1-10. In Papendick R.I., P.A. Sanchez, and G.B. Triplett (ed.) Multiple cropping. Spec. Pub. No 27. Am. Soc. of Agron., Madison, Wis. Baker, E.F.I., and D.W. Norman. 1975. Cropping system in Northern Nigeria. In Proceedings of the cropping systems workshop, IRRI, Los Banos, Philippines. Bazzaz, F.A. and J.L. Harper. 1977. The biology of Ambrosia trtflda L. IV. Demography of plants and leaves. New Phytol. 48:637-654. Beets, W.C. 1982. Multiple cropping and tropical farming systems. Gower Westview Press. Begon, M. and Mortimer, M. 1981. Population ecology. Blackwell, Oxford, Sinauer, Sunderland, MA. Bella, I.E. 1971. A new competition model for individual trees. Forest Sc. 17:364-372. Blackman, V.H. 1919. The compound interest law and plant growth. Ann. Bot. 33:353-360. Bleasdale, J.K.A. 1960. Studies on plant competition, in the biology of weeds. Ed. J.L. Harper, Symposium of the British Ecological Society, Oxford. April 2-4, 1959. Blackwell Scientific Publications, Oxford. Bleasdale, J.K.A. 1966. The effect of plant spacing on the yield of bulb onions {Allium cepaL.) grown from seed. J . Hort. Sci. 41:145-153. Bleasdale, J.K.A. 1967. Systematic designs for spacing experiments. Expt. Agric. 3:73-85. Bleasdale, J.K.A. and J.A. Nelder. 1960. Plant population and crop yield. Nature 188:342. 167 Bleasdale, J.K.A. and R. Thompson. 1966. The effect of plant density and the pattern of plant arrangement on the yield of parsnips. J. Hort. Sci. 41:371-378. Bonan, G.B. 1988. The size structure of theoretical plant populations: spatial pattern and neighbourhood effects. Ecology 69:1721-1730. Bratton, S.P. 1976. Resource division in an understory herb community: Responses to temporal and microtopographic gradients. Amer. Nat. 110:679-693. Briggs, G.E., R. Kidd, and C. West 1920a. A quantitative analysis of plant growth. Parti. Ann. Appl. Biol. 7:103-123. Briggs, G.E., R. Kidd, and C. West 1920b. A quantitative analysis of plant growth. Part II. Ann. Appl. Biol. 7:202-223. Cannell, M.G.R., P. Rothery and E.D. Ford. 1984. Competition within stands of Picea sitchensis and Pinus contorta. Ann. Bot. 53: 349- 362. Carmi, A. 1986. Effects of root zone volume and plant density on the vegetative and reproductive development of cotton. Field Crop Research 13:25-32. Causton, D.R. and J.C. Venus. 1981. The biometry of plant growth. Edward Arnold, London. Clements, F.E., J.E. Weaver and H.C. Hanson. 1929. Plant competition - an analysis of community function. Carnegie Inst, of Washington Publication No. 398. Cock, J.H. and S. Yoshida. 1973. Changing sink and source relations in rice [Oryza sativa L.) using carbon dioxide enrichment in the field. Soil Sci. Plant Nutr. 19:229-234. Connolly, J. 1986. On difficulties with replacement series methodology in rnixture experiments. J. Appl. Ecol. 23:125-137. Cousens, R. 1985. A simple model relating to weed density. Ann. Appl. Biol. 107:239-252. de Wit, CT. 1960. On competition. Verslagen van Landbouwkundige Onderzoekigen. 66:1-82. 168 Dew, D.A. 1972. An index of competition for estimating crop loss due to weeds. Can. J. Plant Sci. 52:921-927. Dixon, W.J. 1985. BMDP statistical software, 725 pp. University of California, Berkeley. Donald, CM. 1963. Competition among crop and pasture plants. Adv. Agron. 15:1-118. Drijfhout, E. 1978. Inheritance of tempeture-dependent string formation in common bean (Phaseolus vulgaris L.). Netherlands J. Agric. Sci. 26:99-105. Drolsom, P.N. and D. Smith. 1976. Adapting species for forage mixtures, p. 223-234. In Papendick RL, P.A. Sanchez and G.B. Triplett (ed.). Multiple cropping. Spec. Pub. No 27. Am. Soc. Agron., Madison, Wis. Eaton, G.W., P.A. Bowen and P.A. Jolliffe. 1985. Two-dimensional partitioning of yield variation. HortScience. 21:1052-1053. Efron, B. 1981. Non parametric estimates of standard error: The jackknife, the bootstrap and other methods. Biometrika 68:589- 599. Efron, B. 1982. The jackknife, the bootstrap and other resampling plants. SIAM Monograph No 38. Engledow, F.L. and S.M. Wadham. 1923. Investigations on yield in cereals: Part I. J. Agric. Sci. 13:390-439. Evans, G.C. 1972. The Quantitative Analysis of Plant Growth. Blackwell Scientific Publications, Oxford. Firbank, L.G. and A.R. Watkinson. 1985. On the analysis of competition within two-speices mixtures of plants. J. Appl. Ecol. 22:503-517. Firbank, L.G. and A.R. Watkinson. 1986. Modelling the population dynamics of an arable weed and its effects upon crop yield. J. Appl. Ecol. 23:147-159. Ford, E.D. 1975. Competition and stand structure in some even-aged plant monocultures. J. Ecol. 63:311-333. 169 Ford, E.D. and P.J. Diggle. 1981. Competition for light in a plant monoculture modelled as a spatial stochastic process. Ann. Bot. 48:481-500. Fraser, J. and G.W. Eaton. 1983. Applications of yield component analysis to crop research. Field Crop Abstr. 36:787-797. Gaye, M.M. 1990. The influence of row covers and plant population density on the growth and productivity of bell peppers [Capsicum annuum L.). M.Sc. Thesis, University of British Columbia. Gepts, P., K. Kimeicki, P. Pereira and F.A. Bliss. 1988. Dissemination pathways of common beans (Phaseolus vulgaris, Fabaceace) deduced from phaseolin electrophoretic variability. I. The Americas. Econ. Bot. 42:73-85. Gillis, P.R. and D.A. Ratkowsky. 1975. the use of non-linear functions in vegetable research working papers, Australian vegetable research conference, Port Sorell 4.1-4.13 Gillis, P.R and Ratkowsky. 1978. The behavior of estimators of the parameters of various yield-density relationships. Biometrics 34:191-198. Gomez, A.A. and K.A. Gomez. 1983. Multiple cropping in the Humid Tropics of Asia. Ottawa, Ont., 248p. Gregory, F.G. 1918. Physiological conditions in cucumber houses. Annual Report, Experimental Research Station, Chestnut, pp. 19- 28. Grime, J.P. 1977. Evidence for the existence of three primary strategies in plants and its relevance to ecological and evolutionary theory. Amer. Nat. 111:1169-1194. Grime, J.P. 1979. Plant Strategies and Vegetation Processes J. Wiley, New York. Grubb, P.J. 1977. The maintenance of species-richness in plant communities: The importance of the generation niche. Biol. Rev. 52:107-145. Hall, R.L. 1974a. Analysis of nature of interference between plants of different species. I. Concepts and extension of the de Wit analysis to examine effects. Aust. J. Agric. Res. 25:739-747. 170 Hall, R.L. 1974b. Analysis of nature of interference between plants of different species. II. Nutrients relations In a Nandi Setaria and Greenleaf Desmodium associations with particular reference to potassium. Aust. J. Agric. Res. 25:749-756. Hancock, J.F., J.H. Sielker, and N.L. Schulte. 1983. Cultivar variation in yield components of strawberries. HortScience 18:312-313. Hancock, J.F., M.P. Pritts, and J.H. Siefker. 1984. Yield components of strawberries maintained in ribbons and matted rows. Crop Res. (Hort. Res.) 24:37-43. Hara, T. 1984a. A stochastic model and the moment dynamics of the growth and size distribution in plant populations. J. Theor. Biol. 109:173-190. Hara, T. 1984b. Dynamics of stand structure in plant monocultures. J. Theor. Biol. 110:223-239. Harper, J.L. 1961. Approaches to the study of plant competition. In: Mechanisms in Biological Competition. F.L. Milthorpe (Ed.) Symp. Soc. Exp. Biol. 15:1.39. Harper, J.L. 1977. Plant Population Biology. Academic Press, New York. Harwood, R.R. and E.C. Price. 1976. Multiple cropping in Tropical Asia, p. 11-40. In Papendick R.I., P.A. Sanchez, and G.B. Triplett (ed.) Multiple cropping. Spec. Pub. No 27. Am. Soc. of Agron., Madison, Wis. Holliday, R. 1960. Plant population and crop yield. Nature 186:22-24. Hunt, R. 1982. Plant growth curves: the functional approach to plant growth analysis. Edward Arnold, London. Hunt, R. and F.A. Bazzaz. 1980. The biology of Ambrosia trifida L. V. Response to fertilizer, with growth analysis at the organismal and sub-organismal level. The New Phytol. 84:13-121. Huxley, J.S. 1932. Problems of relative growth, 307pp. Methuen, London Ishizuka, Y. 1971. Physiology of rice plant. Adv. Agron. 23:241-315. 171 Jolliffe, P.A. 1988. Evaluating the effects of competitive interference on plant performance. J. Theor. Biol. 130:447-459. Jolliffe, P.A. and W.H. Courtney. 1984. Plant growth analysis: Additive and multiplicative components of growth. Ann. Bot. 54:243-254. Jolliffe, P.A., A.J.P. Tarimo and G.W. Eaton. 1990. Plant growth analysis: Growth and yield component responses to population density in forage maize. Ann. Bot. 65:139-147. Jolliffe, P.A., A.N. Minjas, and V.C Runeckles. 1984. A reinterpretation of yield relationships in replacement series experiments. J . Appl. Ecol. 21:227-243. Jolliffe, P.A., G.W. Eaton and J. Lovett Doust. 1982. Sequential analysis of plant growth. New Phytol. 92:287-296. Jolliffe, P.A., G.W. Eaton and M.V. Potdar. 1988. Plant growth analysis: Allometry, growth and interference in orchardgrass and timothy. Ann. Bot. 62:31-42. Kass, D.C.L. 1978. Polyculture cropping systems: review and analysis. Cornell Int. Agric. Bull. 32:1-69. Kassam, A.H. and K. Stockinger. 1973. Growth and nitrogen uptake of sorghum and millet in mixed cropping. Samaru Agric. Newsl. 15:28-33. Khan, M.A., P.D. Putwain, A.D. Bradshaw. 1975. Population interrelationships. 2. Frequency-dependent fitness in Linum. Hereditary 34:145-163. Kira, T., H. Ogawa and N. Sakazaki. 1953. Intraspecific competition among higher plants I. Competition-yield-density interrelationship in regularly dispersed populations. J . Inst. Polytech., Osaka City Univ. D4:l-16. Kira, T., H. Ogawa, K. Hozumi, H. Koyama and K. Yoda. 1956. Intraspecific competition among higher plants. V. Supplementary notes on the C-D effect. J . Inst. Polytech., Osaka City Univ. D 7 : l - 14. Levin, D. and H.W. Kerster. 1971. Neighbourhood structure in plants under diverse reproductive methods. The Amer. Nat. 105:345-354. 172 Li, CC. 1956. The concept of path coefficients and its impact on population genetics. Biometrics 12:190-210. Lorenz, O.A. and D.N. Maynard. 1988. Knott's handbook for vegetable growers. 3rd ed. John Wiley and Sons. New York, Chichester, Brisbane, Toronto, Singapore. Mack, R.N. and J.L. Harper. 1977. Interference in dune annuals: spatial pattern and neighbourhood effects. J. Ecol. 65:345-363. Marshall, D.R. and S.K. Jain. 1969. Interference in pure and mixed populations of Avenafatua and A. barbata. J. Ecol. 57:251-270. Martin, F.B. 1973. Beehive designs for observing variety competition. Biometrics 29:397-402. Matsushima, S. 1966. Crop science in rice. Fuji Publishing Co., Tokyo. Matsushima, S. 1976. High yielding rice cultivation, a method for maximizing rice yield through "Ideal Plants". University of Tokyo Press. Matsushima, S. 1980. Rice cultivation for the millions. Diagnosis of rice cultivation and techniques of yield increase. Pub. by Japan Scientific Soc. Press 2-10 Hongo, 6-chome, Bunkyo-ku Tokyo, 113 Japan. Matsushima, S., G. Wada, and T. Tanaka. 1964. Analysis of yield deterrnining process and its application to yield-prediction and culture improvement of low land rice. LXXII. Effects of top- dressing with ammonium sulfate on grain yield and its components of directly sown rice plants at critical stages relating to determination of each yield-component. Proc. Crop Sci. Soc. Jap. 33:144. Mead, R. 1966. A relationship between individual plant spacing and yield. Ann. Bot. 30:301-309 Mead, R. 1979. Competition experiments. Biometrics 35:41-54. Murata, Y. and S. Matsushima S. 1974. Rice. In Crop physiology. Ed. Evans L.T. Murayama, N. 1979. Nitrogen and rice. Inter. Rice Res. Inst. The importance of nitrogen for rice production. 173 Mutsaers, H.J.W. 1989. A dynamic equation for plant interaction and application to yield-density-time relations. Ann. Bot. 64:521-531. Nelder, J.A. 1962. New kinds of systematic design for spacing experiments. Biometrics 18:283-307. Norman, D.W. 1974. Crop rnixtures under indigenous conditions in the northern parts of Nigeria. Samaru Research Bulletin No. 205, pp 130-144. Institute for Agricultural Research Samaru Nigeria. Ofori, F. and W.R. Stern. 1987. Relative sowing time and density of component crops in a maize/cowpea intercrop system. Expt. Agric. 23:41-52. Papendick, R.L., P.A. Sanchez and G.B. Triplett (Eds.). 1976. Multiple cropping. American Society of Agronomy, Special Publication No. 27. Peirce, C.L. 1987. Vegetables characteristics, production, and , marketing. University of New Hampshire. John Wiley and son Inc. Penney, K. 1986. The effect of ozone air pollution on plant competition. M.Sc. Thesis, University of British Columbia. Potdar, M.V. 1986. Interference in monocultures and rnixtures of orchardgrass {Dactylis glomerata L.) and timothy (Phleum pratense L.) Ph. D. Thesis, University of British Columbia. Radosevich, S.R. 1987. Methods to study Interactions among crops and weeds. WeedTechnol. 1:190-198. Radosevich, S.R. and J.S. Holt. 1984. Weed ecology: Implications for vegetation management. Willey, New York. Radosevich, S.R. and M.L. Roush. 1990. The role of competition in agriculture. Pg 341-363. In Perspectives on plant competition. Ed. J.B. Grace and D. Tilman. Academic Press, Inc. New York. Rejmanek, M., G.R. Robinson, and E. Rejmankova. 1989. Weed-crop competition: Experimental designs and model for data analysis. Weed Sci. 37:276-284. 174 Rice, K.J. 1990. Reproductive hierarchies in Erodium: Effects of variation in plant density and rainfall distribution. Ecology 71:1316-1322. Ross, M.A. and Harper, J.L. 1972. Occupation of biological space during seedling establishment. J. Ecol. 68:919-927. Roush, M.L. and Radosevich, S.R. 1985. Relationships between growth and competitioness of four annual weeds. J. Appl. Ecol. 22:895- 905. Ruppel, E.G. 1986a. Foliar diseases caused by fungi. In Whitney, E.D. and E. Duffus (ed.). Compendium of beet diseases and insects, pp 8-17. Ruppel, E.G. 1986b. Beet culy top. In Whitney, E.D. and Duffus (ed.). Compendium of beet diseases and insects, pp 31-32. Shinozaki, K. and Kira. 1956. Intraspecific competition among higher plants. VII. Logistic theory of the C-D effect. J. Inst. Polytech., Osaka City Univ. D7:35-72. Shoemaker, J.S. 1953. Vegetable growing. 2nd ed. New York. John Wiley and sons Inc. London, Chapman and Hall, Limited. Siefker, J.H. and J.F. Hancock. 1986. Yield components interactions in cultivars of the highbush blueberry. J. Amer. Soc. Hort. Sci. 111:606-608. Silvertown, J.W. 1987. Introduction to plant population ecology. Longman Icn., New York. Singh, S.P. 1989. Variation in cultivated common beans {Phaseohis vulgaris, Fabaceae). Econ. Bot. 43:39-57. Snaydon, RW. 1980. Plant demography in agricultural systems. In Demography and evolution in plant populations. Ed. O.T. Solbring University of California. Press, Berkeley, California. Spitters, C.J.T. 1983. An alternative approach to the analysis of mixed cropping experiments. I. Estimation of competition coefficients. Netherlands J. Agric. Sci. 31:1-11. 175 Stern, W.R. 1965. The effect of density on the performance of individual plants in subterranean clover swards. Aust. J. Agric. Res. 16:541- 555. Thomas, A.G. and H.M. Dale. 1976. Cohabitation of three Hieracium species in relation to the spatial heterogeneity in an old pasture. Can. J. Bot. 54:2517-2529. Trenbath B.R. and J.L. Harper. 1973. Neighbour effect in the genus Avena I. Comparison of crop species. J. Appl. Ecol. 10:379-400. Trenbath, B.R. 1974. Biomass productivity of mixtures. Adv. Agron. 26:177-210. Trenbath, B.R. 1976. Plant interaction in mixed crop communities, p. 129-169. In Papendick R.I., P.A. Sanchez, and G.B. Triplett (ed.). Multiple cropping. Spec. Pub. No 27. Am. Soc. of Agron., Madison, Wis. Turkington, R.A. 1975. Relationships between neighbours among species of permanent grassland (especially Tnfolxum repens L.). Ph.D. dissertation, University College of North Wales, Bangor. Vandermeer, J.H. 1984. Plant competition and the yield-density relationship. J. Theor. Biol. 109:393-399. Wallace, D.H. 1978. Adaptation of Phaseohis to different environments, pg 349-357. In advances in legume science. Ed. R.J. Summerfield and A.H. Bunting. Ware, G.W. and J.P. McCollum. 1975. Producing vegetable crops. The Interstate Printers & Publishers, Inc. Danville Illinois. Watkinson, A.R. 1980. Density dependence in single-species populations of plants. J. Theor. Biol. 83:345-357. Watkinson, A.R. 1981. Interference in pure and mixed populations of Agrostemma githago. J. Appl. Ecol. 18:967-976. Watkinson, A.R. 1984. Yield-density relationships: The influence of resource availability on growth and self-tJiinning in populations of Vidpiafasciculata Ann. Bot. 53:469-482. Watt, A.S. 1955. Bracken versus heather:study in plant sociology. J. Ecol. 43:490-506. 176 Weiner, J. 1984. Neighbourhood interference amongst Pinus rigida individuals. J.Ecol. 72:183-195. Weiner, J. 1985. Size hierarchies in experimental populations of annual plants. Ecology 66:743-752. Weiner, J. 1986. How competition for light and nutrients affects size variability in Ipomoea tricolor populations. Ecol. 67:1425-1427. Weiner, J. and C. Thomas. 1986. Size variability and competition in plant monocultures. Oikos 47:211-222. Weiner, J. and O.T. Solbrig. 1984. The meaning and measurement of size hierarchies in plant populations. Oecologia (Berlin) 61:334- 336. Whitehead, F.H. and P.J. Myerscough. 1962. Growth analysis of plants: the ratio of mean relative growth rate to mean relative of leaf area increase. New Phytol.. 61:314-321. Whitney, E.D. and E. Duffus. 1986. Compendium of beet diseases and insects. APS Press American Phytopathological Society p2. Willey, R.W. 1979a. Intercropping - its importance and research needs. II. Field Crop Abstr. 32:2-10. Willey, R.W. 1979b. Intercropping - its importance and research needs. I. Held Crop Abstr. 32:73-85. Willey, R.W. and D.S.O. Osiru. 1972. Studies on mixtures of maize and beans [Phaseolus vulgaris) with special reference to plant populations. J. Agric. Sci. 79:517-529. Willey, R.W. and S.B. Heath. 1969. The quantitative relationships between plant population and crop yield. Adv. Agron. 21:281-321. Wright, A.J. 1981. The analysis of yield-density relationships in binary rnixtures using inverse polynomials. J. Agric. Sci. 96:561-567. Yamaguchi, M. 1983. World vegetables principles, production and nutritive values. Avi pubUshing company Inc. Westport Connecticut. Yeaton, Rl. and M.L. Cody. 1976. Competition and spacing in plant communities: the northern mohave desert. 64:689-696. 177 Yoshida, S. 1972. Physiological aspects of grain yield. Ann. Rev. Plant Physiol. 23:437-464. Yoshida, S. 1973a. Effects of temperature on growth of the rice plant [Oryza saitva L.) in a controlled environment. Soil Sci. Plant Nutr. 19:299-310. Yoshida, S. 1973b. Effects of CO2 enrichment at different stages of panicle development on yield components and yield of rice [Oryza sativa L.). Soil Sci. Plant Nutr. 19:311-316. Yoshida, S. and F.T. Parao. 1972. Performance of improved rice varieties in the tropics with special reference to tillering capacity. Expt. Agric. 8:203-212. Yunusa, I.A.M. 1989. Effects of planting density on plant arrangement pattern on growth and yields of maize ( Zea mays L.) and soya beans [Glycine max (L.) Merr.) grown in mixtures. J . Agric. Sci. 112:1-8. 178 8. A P P E N D I C E S 179 Appendix 8.1.1.1 25 c 0" ' 1 1 16 33 50 . 66 -2 Population density (plants m ) Mixture proportions (Beans:Beets) -1:3 - 1 - 2 : 2 - * ~ 3:1 "B- 4:0 The effect of population density and mixture proportions interactions on bean leaf number harvested at 69 days after planting (1984 experiment) 180 Appendix 8.1.1.2 Population density (plants m. ) Mixture proportions (Beans:Beets) -1:3 -+~2:2 * 3:1 --Q- 4.0 The effect of population density and mixture proportions interactions on bean pod dry weight harvested at 69 days after planting (1984 experiment) 181 Appendix 8.1.1.3 140 0' ' 1 I • "•6 33 50 66 Population density (plants m - 2 ) Mixture proportions (Beans:Beets) 1:3 -4-2:2 * 3:1 " G - 4:0 The effect of population density and rnixture proportions interactions on bean pod fresh weight harvested at 69 days after planting (1984 experiment) 182 Appendix 8.1.2.1 25 3 (0 • 5 - 16 33 50 „ 66 -2 Population density (plants m ) Mixture proportions (Beans:Beets) — 1:3 - » - 2 : 2 * 3:1 "13-4:0 The effect of population density and mixture proportions interactions on bean leaf number harvested at 75 days after planting (1984 experiment) Appendix 8.1.2.2 183 1 - 16 33 50 _ 2 66 Population density (plants m ) Mixture proportions (Beans:Beets) "1:3 -4—2:2 * 3:1 " Q " 4:0 The effect of population density and mixture proportions interactions on bean branch number harvested at 75 days after planting (1984 experiment) 184 Appendix 8.1.2.3 30 5 - 0 i 1 1 1 16 33 50 „ 66 -2 Population density (plants m ) Mixture proportions (Beans:Beets) 1:3 —^2:2 * 3:1 4:0 The effect of population density and mixture proportions interactions on bean pod number harvested at 75 days after planting (1984 experiment) Appendix 8.1.2.4 12 16 33 50 _ 2 66 Population density (plants m ) Mixture proportions (Beans:Beets) ——1:3 -+-2:2 - * 3:1 ' "O" 4:0 The effect of population density and mixture proportions interactions on bean leaf dry weight harvested at 75 days after planting (1984 experiment) Appendix 8.1.2.5 186 200 16 33 50 _ 2 66 Population density (plants m ) Mixture proportions (Beans:Beets) — 1:3 "+-2:2 3:1 " H - 4:0 The effect of population density and mixture proportions interactions on bean pod dry weight harvested at 75 days after planting (1984 experiment) Appendix 8.1.2.6 200 16 33 SO _ 2 66 Population density (plants m ) Mixture proportions (Beans:Beets) — 1:3 -+-2:2 ~ * 3:1 4:0 The effect of population density and mixture proportions interactions bean total dry weight harvested at 75 days after planting (1984 experiment) Appendix 8.2 188 0 10 20 30 40 50 60 70 80 90 100 Cumulative percentage of population perfect equality Deviation from equal i t y Lorenz curve showing size inequality in a yield variable (hypothetical data) 189 Appendix 8.3.1 Gini coefficients for stem dry weight distribution of beans grown in monocultures and mixtures Total Population Density Statistic 4:0* Mixture Proportions 3:1 2:2 1:3 0:4 Mixture Pooled 16 G' 0.220 0.339 0.376 0.185 0.324 33 G" 0.409 0.261 0.257 0.232 0.310 50 G' 0.242 0.307 0.199 0.299 0.309 66 G' 0.289 0.369 0.304 0.324 0.327 Density Pooled G' 0.308 0.301 0.288 0.264 0.320t Appendix 8.3.2 Gini coefficients for marketable pod number distribution of beans grown in monocultures and mixtures Total Population Density Statistic 4:0* Mixture Proportions 3:1 2:2 1:3 0:4 Mixture Pooled 16 G' 0.198 0.270 0.319 0.276 0.313 33 G' 0.232 0.234 0.220 0.206 0.277 50 G' 0.278 0.263 0.204 0.309 0.301 66 G' 0.350 0.221 0.278 0.285 0.395 Density Pooled G' 0.308 0.249 0.278 0.293 0.324t *Beans:Beets tGini coefficient for the whole data set 190 Appendix 8.3.3 Gini coefficients for unmarketable pod number distribution of beans grown in monocultures and mixtures Total Population Mixture Proportions Mixture Density Statistic 4:0* 3:1 2:2 1:3 0:4 Pooled 16 G' 0.353 0.368 0.352 0.408 0.453 33 G' 0.339 0.356 0.380 0.286 0.369 50 G' 0.319 0.320 0.303 0.333 0.356 66 G* 0.326 0.304 0.259 0.305 0.339 Density Pooled G" 0.386 0.407 0.336 0.346 0.384t Appendix 8.3.4 Gini coefficients for pod fresh weight distribution of beans grown in monocultures and mixtures Total Population Density Statistic 4:0* Mixture Proportions 3:1 2:2 1:3 0:4 Mixture Pooled 16 G' 0.235 0.308 0.310 0.271 0.324 33 G' 0.244 0.314 0.267 0.212 0.299 50 G' 0.276 0.297 0.256 0.321 0.311 66 G' 0.400 0.365 0.289 0.290 0.421 Density Pooled G' 0.326 0.259 0.295 0.307 . 0.307t *Beans:Beets +Gini coefficient for the whole data set 191 Appendix 8.3.5 Gini coefficients for marketable pod dry weight distribution of beans grown in monocultures and mixtures Total Population Density Statistic 4:0* Mixture Proportions 3:1 2:2 1:3 0:4 Mixture Pooled 16 G' 0.214 0.289 0.354 0.283 0.326 33 G* 0.227 0.241 0.251 0.225 0.288 50 G* 0.292 0.258 0.225 0.322 0.302 66 G' 0.417 0.252 0.246 0.300 0.420 Density Pooled G' 0.338 0.267 0.290 0.313 0.336t Appendix 8.3.6 Gini coefficients for unmarketable pod dry weight distribution of beans grown in monocultures and rnixtures Total Population Density Statistic 4:0* Mixture Proportions 3:1 2:2 1:3 0:4 Mixture Pooled 16 G' 0.570 0.517 0.510 0.615 0.567 33 G' 0.588 0.558 0.557 0.521 0.556 50 G' 0.579 0.515 0.497 0.548 0.537 66 G' 0.549 0.476 0.462 0.525 0.503 Density Pooled G' 0.571 0.521 0.510 0.549 0.549t *Beans:Beets tGini coefficient for the whole data set 192 Appendix 8.3.7 Gini coefficients for seed number distribution of beans grown in monocultures and mixtures Total Population Mixture Proportions Mixture Density Statistic 4:0* 3:1 2:2 1:3 0:4 Pooled 16 G' 0.391 0.484 0.417 0.417 0.371 33 G" 0.349 0.385 0.425 0.413 0.320 50 G' 0.373 0.424 0.332 0.414 0.348 66 G' 0.432 0.491 0.344 0.477 0.450 Density Pooled G' 0.354 0.360 0.335 0.372 0.374t Appendix 8.3.8 Gini coefficients for seed dry weight distribution of beans grown in monocultures and mixtures Total Population Density Statistic 4:0* Mixture Proportions 3:1 2:2 1:3 0:4 Mixture Pooled 16 G' 0.254 0.406 0.382 0.336 0.377 33 G' 0.249 0.301 0.284 0.294 0.318 50 G' 0.335 0.320 0.316 0.368 0.346 66 G' 0.463 0.305 0.291 0.417 0.456 Density Pooled G 0.379 0.342 0.333 0.381 0.373t *Beans:Beets +Gini coefficient of the whole data set 193 Appendix 8.4.1 Gini coefficients for dead leaves distribution of beets grown in monocultures and mixtures Total Population Density Statistic 4:0* Mixture Proportions 3:1 2:2 1:3 0:4 Mixture Pooled 16 G' 0.256 0.397 0.374 0.460 0.379 33 G' 0.323 0.527 0.553 0.305 0.449 50 G' 0.459 0.547 0.460 0.390 0.478 66 G' 0.385 0.391 0.601 0.379 0.436 Density Pooled G' 0.360 0.471 0.496 0.393 0.436t Appendix 8.4.2 Gini coefficients for petiole dry weight distribution of beets grown in monocultures and mixtures Total Population Density Statistic 4:0* Mixture Proportions 3:1 2:2 1:3 0:4 Mixture Pooled 16 G' 0.315 0.411 0.328 0.519 0.410 33 G* 0.485 0.488 0.363 0.265 0.464 50 G' 0.303 0.421 0.475 0.390 0.418 66 G' 0.453 0.358 0.521 0.318 0.429 Density Pooled G' 0.413 0.475 0.450 0.406 0.449t *Beans:Beets tGini coefficient for the whole data set 194 Appendix 8.4.3 Gini coefficients for root diameter distribution of beets grown in monocultures and mixtures Total Population Mixture Proportions Mixture Density Statistic 4:0* 3:1 2:2 1:3 0:4 Pooled 16 G' - 0.207 0.356 0.264 0.358 0.313 33 G' - 0.204 0.398 0.273 0.156 0.342 50 G' - 0.222 0.303 0.332 0.290 0.336 66 G' 0.310 0.318 0.382 0.216 0.336 Density Pooled G' - 0.260 0.400 0.365 0.282 0.350t Appendix 8.4.4 Gini coefficients for root fresh weight distribution of beets grown in monocultures and mixtures Total Population Density Statistic 4:0* Mixture Proportions 3:1 2:2 1:3 0:4 Mixture Pooled 16 G' 0.458 0.527 0.474 0.732 0.613 33 G' 0.464 0.717 0.588 0.389 0.625 50 G' 0.509 0.584 0.658 0.526 0.630 66 G' 0.611 0.627 0.671 0.456 0.618 Density Pooled G' 0.546 0.694 0.654 0.622 0.661t *Beans .Beets tGini coefficient for the whole data set 195 Appendix 8.4.5 Gini coefficients for root dry weight distribution of beets grown in monocultures and rnixtures Total Population Density Statistic 4:0* Mixture Proportions 3:1 2:2 1:3 0:4 Mixture Pooled 16 G" 0.438 0.558 0.460 0.670 0.582 33 G' 0.479 0.689 0.585 0.389 0.625 50 G' 0.498 0.569 0.642 0.499 0.615 66 G' 0.612 0.621 0.671 0.440 0.619 Density Pooled G" 0.540 0.688 0.643 0.567 0.641t *Beans:Beets tGini coefficient for the whole data set 196 Appendix 8.5.1 Parameters and statistics for the best subset multiple regression models of the allometric relationship between In W and In T in beans - 1984 experiment Age at Potential Parameter Regression Std. partial Mallows R2 RMS d.f. harvest independent coefficient regression CP (days) variable coefficient 40 Intercept ln(a') -2.398** -4.032 6.32 0.85 0.05 6,233 (HNl)t ln(T) XilnCD Xjln(T) XiXjln(T) ln(Xj) ln(Xj) ln(XiXj) 51 Intercept ln(a') -2.014** -3.372 4.86 0.78 0.81 5,234 (HN2) ln(T) XjlnCT) Xjln(T) Po 1.824** 0.849 Pi 0.007** 0.424 P2 - - Ps -0.0001 -0.106 Yl -0.520** -0.588 Y2 -0.531** -1.251 Y3 -0.561** 0.977 Po 1.596** 0.692 Pi -0.001 -0.130 h - - P3 0.0001 0.139 Yi -0.496** -0.560 Y2 -0.076* -0.178 Y3 - ln(a') -0.168 -0.168 Po 1.157** 0.337 Pi -0.007** -0.530 XiXjlnCT) ln(Xi) ln(Xj) ln(XiXj) 63 Intercept l ( ') - . - . 2.48 0.29 0.54 2,237 (HN3) ln(T) XjlnCT) XjlnCT) k XiXjlnCT) P3 lnCXj) Yi ln(Xj) Y2 InCXiXj) Y3 Ĥarvest number (1-6 in beans and 1-7 in beets) *SignificantatP = 0.05 **Significant at P = 0.01 Note: In is synonymous for loge 197 Appendix 8.5.1 (cont'd) Parameters and statistics for the best subset multiple regression models of the allometric relationship between In W and ln T in beans - 1984 experiment Age at Potential Parameter Regression Std. partial Mallows harvest independent coefficient regression CP (days) variable coefficient 69 Intercept ln(a') 2.431** 0.609 4.20 (HN4) ln(T) Po 0.909** 0.135 XjlnCD Pi -0.005** -0.001 XjlnfT) P2 - - XiXj(T) Ps - - ln(Xi) Yi -0.314* -0.127 ln(Xj) Y2 - - ln(XiXj) Y3 - - R2 RMS d.f. 0.54 0.21 3,236 75 (HN 5) Intercept ln(a') 4.574** 0.905 ln(T) Po 0.663** 0.144 XjlnCT) Pi -0.004 0.002 Xjln(T) XiXjfD P2 0.008* 0.003 P3 - - ln(Xi) Yi -0.654** -0.220 ln(Xj) ln(XiXj) Y2 -0.761* -0.306 Y3 0.694* 0.305 6.02 0.59 0.21 6,233 92 (HN 6) Intercept ln(T) XilnCT) XjlnfT) XiXjCT) ln(X{) ln(Xj) ln(XiXj) ln(a') Po Pi P2 Ps Yi Y2 Y3 2.047** 1.375** -0.003* -0.705* 0.438 0.097 -0.001 0.108 2.80 0.77 0.15 3,236 *SignificantatP = 0.05 **Significant at P = 0.01 Note: ln is synonymous for loge 198 Appendix 8.5.2 Parameters and statistics for the best subset multiple regression models of the allometric relationship between In W and In LN in beans - 1984 experiment Age at Potential Parameter Regression Std. partial Mallows R2 RMS d.f. harvest independent coefficient regression CP (days) variable coefficient 40 Intercept ln(a') -0.670** -0.166 5.47 0.75 0.09 4,235 (HN l)t ln(LN) Xiln(LN) Xjln(LN) XiXjln(LN) ln(Xj) ln(Xj) lnOCjXj) Po 1.305** 0.055 Pi - ' - P2 - - P3 - - Yi -0.193** -0.035 Y2 -0.214** -0.071 Y3 0.293** 0.093 ln(a') -1.687** -0.153 Po 1.499** 0.078 Pi -0.003** -0.001 P2 - - P3 - - Yi - - Y2 -0.162* -0.064 Y3 0.186* 0.084 51 Intercept l ( ') - . 87" - . 2.78 0.77 0.08 4,235 (HN 2) ln(LN) Xiln(LN) Xjln(LN) XiXjIn(LN) ln(Xi) ln(Xj) lnOCjXj) 63 Intercept ln(a') -0.779** -0.245 0.54 0.53 0.37 1,238 (HN 3) ln(LN) p0 1.576** 0.097 XilnOJvO p! Xjln(LN) p2 XiXjlnOJSO P3 ln(Xi) Y l ln(Xj) Y 2 hKXjXj) Y3 Ĥarvest number (1-6 in beans and 1-7 in beets) *SignificantatP = 0.05 **Significant at P = 0.01 Note: In is synonymous for loge 199 Appendix 8.5.2 (cont'd) Parameters and statistics for the best subset multiple regression models of the allometric relationship between ln W and ln LN in beans - 1984 experiment Age at Potential Parameter Regression Std. partial Mallows R2 RMS d.f. harvest independent coefficient regression CP (days) variable coefficient 69 Intercept ln(a') 1.894** 0.311 2.15 0.72 0.13 2,237 (HN4) ln(LN) Xiln(LN) Xjln(LN) XiXj(LN) InCXj) In(Xj) ln(XiXj) 75 Intercept ln(a') 4.112** 0.440 5.57 0.70 0.16 5,234 (HN 5) ln(LN) XjlnOJSQ Xjln(LN) XiXj(LN) ln(Xi) ln(Xj) ln(XiXj) 92 Intercept ln(a') 3.139** 0.095 4.23 0.69 0.20 2,235 (HN 6) ln(LN) Xiln(LN) Xjln(LN) XjXjCLN) ln(Xj) Y l ln(Xj) Y 2 ln(XiXj) Y3 Po 1.124** 0.074 Pi - - P2 - - P3 - - Yi -0.173** -0.483 Y2 - - Y3 Po 0.767** 0.086 Pi - - P2 0.010** 0.003 Ps - - Yi -0.481** -0.077 Y2 -0.856** -0.211 Y3 0.825** 0.216 . . Po 0.809** 0.037 Pi -0.009** -0.001 *SignificantatP = 0.05 **Significant at P = 0.01 Note: ln is synonymous for loge 200 Appendix 8.5.3 Parameters and statistics for the best subset multiple regression models of the allometric relationship between ln W and ln LA in beans - 1984 experiment Age at Potential Parameter Regression Std. partial Mallows R 2 RMS d.f. harvest independent coefficient regression CP (days) variable coefficient 40 Intercept ln(a') 4.200** 0.059 3.91 0.94 0.02 2,237 (HNl)t ln(LA) po 0.980** 0.017 Xjln(LA) p! Xjln(LA) P2 XjXjln(LA) P3 ln(Xi) Yx ln(Xj) Y2 0.030** 0.007 ln(XiXj) Y3 51 Intercept ln(a') 4.148** 0.060 5.46 0.92 0.03 2,237 (HN2) ln(LA) p0 0.970** 0.028 XjhKLA) p! 0.001** 0.0002 Xjln(LA) p2 XiXjln(LA) P3 ln(Xi) Yi ln(Xj) Y2 hKXiXj) Y3 63 Intercept ln(a') 4.820** 0.370 5.21 0.60 0.31 3,236 (HN 3) ln(LA) p0 1.010** 0.100 XjlnfLA) px 0.004* 0.002 Xjln(LA) p2 XiXjln(LA) P3 ln(Xi) Yi 0-260* 0.120 ln(Xj) Y 2 ln(XiXj) Y3 ^Harvest number (1-6 in beans and 1-7 in beets) *SignificantatP = 0.05 **Significant at P = 0.01 Note: In is synonymous for loge 201 Appendix 8.5.3 (cont'd) Parameters and statistics for the best subset multiple regression models of the allometric relationship between In W and In LA in beans - 1984 experiment Age at Potential Parameter Regression Std. partial Mallows R2 RMS d.f. harvest independent coefficient regression CP (days) variable coefficient 69 Intercept ln(a') 5.720** 0.103 4.56 0.76 0.11 4,235 (HN4) ln(LA) XjlnOLA) Xjln(LA) XtXj(LA) ln(Xj) ln(Xj) ln(XjXj) 75 Intercept ln(a') 6.173** 0.099 6.89 0.79 0.11 2,237 (HN 5) ln(LA) Xiln(LA) Xjln(LA) X iXj(LA) ln(Xi) ln(Xj) ln(XiXj) 92 Intercept ln(a') 6.780** 0.117 1.87 0.80 0.13 3,234 (HN 6) ln(LA) Xjln(LA) p2 Xjln(LA) P2 XiXj(LA) P3 0.0001** 0.00002 Po 0.674** 0.057 Pi 0.003** 0.001 P2 0.0001* 0.0001 Ps - - Yi - - Y2 - - Y3 0.089* 0.040 Po 0.709** 0.060 Pi 0.005** 0.001 P V - - P3 ' - - Yi - - Y2 - - Y3 . . . Po 0.549** 0.035 ln(Xj) Yi -0.310** -0.050 ln(Xj) Y 2 InCXjXj) Y3 *SignificantatP = 0.05 **Significant at P = 0.01 Note: In is synonymous for loge 202 Appendix 8.5.4 Parameters and statistics for the best subset multiple regression models of the allometric relationship between ln W and ln WL in beans - 1984 experiment Age at Potential Parameter Regression Std. partial Mallows R 2 RMS d.f. harvest independent coefficient regression CP (days) variable coefficient 40 Intercept ln(a') -0.252** -0.039 0.46 0.98 0.09 3,236 (HNl)t InfWL) Po 1.041** 0.010 Xjln(WL) pj Xjln(WL) P2 XiXjln(WL) P3 ln(Xi) Yi -0.093** -0.010 ln(Xj) Y2 °-007 0 0 0 4 ln(XiXj) Y3 51 Intercept ln(a') -0.129** -0.039 1.62 0.98 0.08 2,237 (HN2) InfWL) p 0 0.935** 0.010 X ^ W L ) , p! Xjln(WL) P2 XjXjln(VvX) P3 ln(Xj) Yi -0.108** -0.009 ln(Xj) Y2 hKXjXj) Y3 63 Intercept ln(a') 1.533** 0.206 3.03 0.72 0.12 4,235 (HN 3) InfWL) Po 1.413** 0.028 Xiln(WL) Pi - - Xjln(WL) XiXjln(WL) P2 - - P3 - - ln(Xi) Yi -0.450** -0.043 ln(Xj) ln(XiXj) Y2 -0.163* -0.081 Y3 0.207* 0.106 ^Harvest number (1-6 in beans and 1-7 in beets) *SignificantatP = 0.05 "Significant at P = 0.01 Note: ln is synonymous for loge 203 Appendix 8.5.4 (cont'd) Parameters and statistics for the best subset multiple regression models of the allometric relationship between In W and In WL in beans - 1984 experiment Age at Potential Parameter Regression Std. partial Mallows R2 RMS d.f. harvest independent coefficient regression CP (days) variable coefficient 69 (HN4) Intercept ln(WL) XjhXWL) Xjln(WL) XiXj(WL) ln(Xi) ln(Xj) ln(XiXj) ln(a') Po Pi P2 Ps Yi Y2 Y3 -0.417 0.654** -0.268* -0.074* -0.248 0.034 2.49 0.83 0.071 4,235 0.0001* 0.00002 -0.039 0.340 75 (HN 5) Intercept ln(WL) Xiln(WL) Xjln(WL) XjXjOVL) ln(Xj) ln(Xj) ln(XiXj) ln(a') Po Pi P2 P3 Yi Y2 Y3 -1.096** 0.707** -0.190* -0.254 0.036 -0.037 -0.86 0.82 0.08 2,237 92 (HN6) Intercept ln(WL) Xiln(WL) Xjln(WL) XjXjOVL) ln(Xi) ln(Xj) ln(XiXj) ln(a') Po Pi P2 P3 Yi 72 Y3 -1.668** 0.882** -0.002 0.0001** -0.392** -0.435 0.053 -0.001 0.00004 -0.076 2.45 0.82 0.18 4,233 *SignificantatP = 0.05 "Significant at P = 0.01 Note: In is synonymous for loge 204 Appendix 8.5.5 Parameters and statistics for the best subset multiple regression models of the allometric relationship between in W and ln WST in beans - 1984 experiment Age at Potential Parameter Regression Std. partial Mallows R2 RMS d.f. harvest independent coefficient regression CP (days) variable coefficient 40 Intercept ln(cc') -0.387** -0.152 1.60 0.95 0.01 3,236 (HNl)t ln(WST) Xiln(WST) Xjln(WST) XiXjln(WST) ln(Xi) ln(Xj) ln(XiXj) 51 Intercept In(a') -0.346** -0.062 1.68 0.96 0.01 2,237 (HN 2) lnfWST) Xiln(WST) Xjln(WST) XiXjuXWST) ln(Xi) lnfXj) ln(XjXj) a Po 1.945** 0.139 Pi - - P2 - - Ps - - 71 -0.152** -0.013 72 0.017** 0.006 Y3 ~ ~ Po 1.058** 0.016 Pi - - P2 - - P3 - - Yi -0.114** -0.014 Y2 - - Y3 ln(a') 1.011** 0.236 Po 1.479** 0.032 Pi - - P2 - - Ps - - Yi -0.330** -0.049 Y2 -0.286** -0.093 Y3 0.356** 0.121 63 Intercept l ( ') . 1" . 4.28 0.65 0.15 4,235 (HN 3) lnfWST) XihXWST) Xjln(WST) XiXjln(WST) ln(X{) ln(Xj) ln(XiXj) Ĥarvest number (1-6 in beans and 1-7 in beets) *SignificantatP = 0.05 "Significant at P = 0.01 Note: ln is synonymous for loge 205 Appendix 8.5.5 (cont'd) Parameters and statistics for the best subset multiple regression models of the allometric relationship between In W and In WST in beans - 1984 experiment Age at Potential Parameter Regression Std. partial Mallows R2 RMS d.f. harvest independent coefficient regression CP (days) variable coefficient 69 Intercept ln(oc') -1.515** -0.132 1.38 0.73 0.11 3,236 (HN4) ln(WST) B0 0.752** 0.033 XihXWST) p! Xjln(WST) p2 XiXjCWST) 03 ln(Xj) Yi ln(Xj) Y2 0.269** 0.069 InpqXj) y3 -0.406** -0.093 75 Intercept ln(oc') -1.788** -0.143 0.00 0.69 0.13 1,237 (HN 5) lnCWST) p0 0.760** 0.033 X^WST) p! Xjln(WST) P2 - - XiXjCWST) P3 ln(Xj) Yi ln(Xj) Y2 m(XiXj) Y3 92 Intercept ln(a') -2.472** -0.176 7.66 0.84 0.10 5,234 (HN 6) ln(WST) Po 0.922** 0.032 Xiln(WST) Pi -0.001* -0.001 Xjln(WST) XiXj(WST) P2 -0.002** -0.001 Ps 0.0001** 0.00004 InCXj) Yi - - ln(Xj) InCXjXj) Y2 - - Y3 -0.128** -0.041 *Significant at P = 0.05 "Significant at P = 0.01 Note: In is synonymous for loge 206 Appendix 8.5.6 Parameters and statistics for the best subset multiple regression models of the allometric relationship between ln W and ln BN in beans - 1984 experiment Age at Potential Parameter Regression Std. partial Mallows R 2 RMS d.f. harvest independent coefficient regression CP (days) variable coefficient 40 Intercept ln(a') -4.589** -0.178 2.06 0.94 0.02 4,235 (HN l)t ln(BN) Po 0.970** 0.019 Xiln(BN) Pi 0.001* 0.0003 Xjln(BN) P2 - - XiXjln(BN) Ps - - lnfXj) Yl -0.102* -0.046 ln(Xj) lnPqXj) Y2 0.035** 0.007 Y3 - - 51 Intercept ln(a') -4.868** -0.196 0.85 0.92 0.30 2,237 (UN 2) ln(BN) p0 1.998** 0.024 Xjln(BN) px Xjln(BN) p2 XiXjln(BN) P3 lnfXj) Yj -0.055** -0.019 ln(Xj) Y2 ln(XiXj) Y3 - 63 Intercept ln(a') -5.018** -0.447 -1.24 0.59 0.32 1,238 (HN 3) ln(BN) p0 1.158** 0.063 X^BN) pi Xjln(BN) p2 XiXjln(BN) P3 lnCXj) Yi ln(Xj) Y2 ln(XiXj) Y3 ^Harvest number (1-6 in beans and 1-7 in beets) *SignificantatP = 0.05 "Significant at P = 0.01 Note: ln is synonymous for loge 207 Appendix 8.5.6 (cont'd) Parameters and statistics for the best subset multiple regression models of the allometric relationship between In W and In BN in beans - 1984 experiment Age at Potential Parameter Regression Std. partial Mallows R2 RMS d.f. harvest independent coefficient regression CP (days) variable coefficient 69 Intercept ln(a') -1.420** -0.319 (HN4) ln(BN) Po 0.819** 0.041 XihXBN) Pi -0.001** -0.0002 Xjln(BN) P2 - - XiXj(BN) Ps - - ln(Xi) Yl - - ln(Xj) Y2 - - lnQqXj) Y3 - - 1.87 0.76 0.11 2,237 75 Intercept ln(a') -1.751** -0.368 2.52 0.77 0.12 2,237 (HN 5)' ln(BN) XihXBN) Xjln(BN) XiXj(BN) ln(Xi) ln(Xj) InCXjXj) Po 0.900** 0.047 Pi -0.001** -0.0003 P2 - - Ps - . - Yi - - Y2 - - Y3 ln(a') 0.739** 0.221 Po 0.614** 0.084 Pi -0.002** -0.0003 P2 - - Ps - - Yi - - Y2 - - Y3 -0.061** -0.023 92 Intercept l ( ') . 39" . 2.40 0.80 0.12 3,234 (HN 6) ln(BN) Xiln(BN) Xjln(BN) XiXj(BN) lnCXj) ln(Xj) InCXjXj) *SignificantatP = 0.05 "Significant at P = 0.01 Note: In is synonymous for loge 208 Appendix 8.5.7 Parameters and statistics for the best subset multiple regression models of the allometric relationship between ln W and ln PN in beans - 1984 experiment Age at Potential Parameter Regression Std. partial Mallows R 2 RMS d.f. harvest independent coefficient regression CP (days) variable coefficient -1.320** -0.120 1.780** 0.083 -0.005** -0.001 -0.007** -0.003 0.0003** 0.0001 40 Intercept ln(a') 2.86 0.67 0.12 4,235 (HN l)t ln(PN) Po XjlnfPN) p! Xjln(PN) P2 XiXjln(PN) P3 ln(Xj) Yi InfXj) Y2 lnfXjXj) Ys 51 Intercept ln(a') -0.638* -0.250 1.68 0.48 0.19 3,236 (HN 2) ln(PN) Po 1.645** 0.140 XjlntPN) Pi -0.008** -0.001 Xjln(PN) XiXjln(PN) P2 -0.004** -0.001 Ps - - InfXj) Yi - - InfXj) Y2 - - lnfXjXj) Y3 - - 63 Intercept ln(a') 1.190** 0.36 2.00 0.33 0.52 3,236 (HN 3) ln(PN) Xiln^N) Xjln(PN) XiXjln(PN) InfXi) ln(Xj) lnfXiXj) Po 1.630** 0.200 Pi -0.011** -0.002 P2 - - P3 - - Yi - - Y2 -0.065 -0.036 Y3 - - "̂ Harvest number (1-6 in beans and 1-7 in beets) *SignificantatP = 0.05 "Significant at P = 0.01 Note: ln is synonymous for loge 209 Appendix 8.5.7 (cont'd) Parameters and statistics for the best subset multiple regression models of the allometric relationship between In W and In PN in beans - 1984 experiment Age at Potential Parameter Regression Std. partial Mallows R2 RMS d.f. harvest independent coefficient regression CP (days) variable coefficient 69 Intercept ln(a') 3.537** 0.367 4.73 0.68 0.15 5,234 (HN 4) ln(PN) Po 1.358** 0.124 Xiln(PN) Pi - - Xjln(PN) P2 0.013** 0.005 XjXjtPN) Ps - - ln(Xi) Yi -0.472** -0.064 ln(Xj) inpqxp Y2 -0.622** -0.207 Y3 0.624** 0.209 75 (HNS) Intercept ln(a') 2.800 0.141 ln(PN) Po 1.378 0.078 Xiln(PN) Pi -0.012 -0.001 Xjln(PN) XjXjOPN) P2 - - P3 - - ln(Xi) Yi - - ln(Xj) Y2 - - ln(XiXj) Y3 -0.056 -0.027 4.76 0.66 0.17 3,236 92 (HN 6) Intercept ln(PN) Xjln(PN) Xjln(PN) XiXj(PN) ln(Xi) ln(Xj) InPqXj) ln(a') Po Pi P2 P3 Yi Y2 Y3 3.209** 1.431** -0.010** -0.186 0.485 0.141 -0.003 -0.129 2.29 0.75 0.16 3,236 *SigificantatP = 0.05 "Significant at P = 0.01 Note: In is synonymous for loge 210 Appendix 8.5.8 Parameters and statistics for the best subset multiple regression models of the allometric relationship between ln W and ln FWPD in beans - 1984 experiment Age at Potential Parameter Regression Std. partial Mallows R 2 RMS d.f. harvest independent coefficient regression CP (days) variable coefficient 40 Intercept ln(a') (HN l)t ln(FWPD) p 0 XjlnfPWPD) pi Xjln(FWPD) p2 XiXjhKPWPD) P3 ln(Xi) Y l InfXj) Y2 lnfXjXj) Y3 51 Intercept ln(a') 3.905** 0.232 4.52 0.51 0.13 5,112 (HN 2) ln(FWPD) XjlncTWPD) Xjln(FWPD) XiXjln(FWPD) ln(Xi) InfXj) ln(XiXj) Po 0.003* 0.001 Pi 0.009** 0.003 P2 - - P3 - - Yi -0.676** -0.073 72 -0.585** -0.155 73 0.646** 0.178 63 Intercept In(a') -0.205 -0.181 0.82 0.70 0.23 3,236 (HN 3) ln(FWPD) Po 1.270** 0.055 Xiln(FWPD) Pi -0.001 -0.001 Xjln(FWPD) XjXjtafPWPD) P2 - - P3 - - InfXj) Yi - - InfXj) Y2 - - InfXiXj) Y3 -0.048 -0.031 ^Harvest number (1-6 in beans and 1-7 in beets) *SignificantatP = 0.05 "Significant at P = 0.01 Note: ln is synonymous for loge 211 Appendix 8.5.8 (cont'd) Parameters and statistics for the best subset multiple regression models of the allometric relationship between ln W and ln FWPD in beans - 1984 experiment Age at Potential Parameter Regression Std. partial Mallows R2 RMS d.f. harvest independent coefficient regression CP (days) variable coefficient 60 Intercept (HN4) ln(FWPD) X iln(FWPD) Xjln(FWPD) XjXjvTWPD) InfXj) InfXj) lnfXjXj) 75 Intercept (HN 5) ln(FWPD) Xiln(FWPD) Xjln(FWPD) XiXj(FWPD) lnCXj) InfXj) InCXiXj) 92 Intercept (HN 6) ln(FWPD) XjlnCFWPD) Xjln(FWPD) XjXjfPWPD) lnCXj) InfXj) Y 2 InfXiXj) 73 ln(a') 1.184** 0.106 5.28 0.77 0.11 3,236 Po 1.058** 0.040 Pi P2 -0.0001** -0.00004 Yi - - Y2 Y3 0.131** 0.036 ln(a') 2.090** 0.111 1.79 0.80 0.10 2,237 Po 1.980** 0.035 Pi -0.003** -0.001 P2 P3 - _ Yi - - Y2 - - Y3 ln(a') 2.811** 0.175 -0.22 0.90 0.06 2,237 Po 0.931** 0.033 Pi P2 P3 Yi -0.114** -0.037 •Significant at P = 0.05 "Significant at P = 0.01 Note: In is synonymous for loge 212 Appendix 8.5.9 Parameters and statistics for the best subset multiple regression models of the allometric relationship between In W and In WPD in beans - 1984 experiment Age at Potential Parameter Regression Std. partial Mallows R2 RMS d.f. harvest independent coefficient regression CP (days) variable coefficient 40 Intercept In(oc') (HNl)t ln(WPD) p0 X^WPD) p! Xjln(WPD) p2 XiXJln(WPD) P3 ln(Xj) Yi ln(Xj) Y2 ln(XiXj) Y3 51 Intercept ln(a') -3.325** -0.427 0.24 0.11 1.01 2,115 (HN2). ln(WPD) Po 0.668** 0.187 Xjln(WPD) Pi 0.007 0.004 Xjln(WPD) XiXjln(WPD) P2 - - P3 - - ln(Xi) Yl - - ln(Xj) Y2 - - InQCiXj) Y3 - - 63 Intercept ln(a') -2.700** -0.136 2.33 0.87 0.27 2,237 (HN 3) ln(WPD) Po 1.502** 0.038 Xiln(WPD) Pi 0.005** 0.001 Xjln(WPD) XiXjln(WPD) P2 - - Ps - - - ln(Xj) Yi - - ln(Xj) Y2 - - lnCXjXj) Y3 - - Ĥarvest number (1-6 in beans and 1-7 in beets) *SignificantatP = 0.05 "Significant at P = 0.01 Note: In is synonymous for loge 213 Appendix 8.5.9 (cont'd) Parameters and statistics for the best subset multiple regression models of the allometric relationship between In W and In WPD in beans - 1984 experiment Age at Potential Parameter Regression Std. partial Mallows R2 RMS d.f. harvest independent coefficient regression CP (days) variable coefficient 69 Intercept In(oc') -0.524 -0.056 2.45 0.99 0.004 4,235 (HN4) i ( ' ln(WPD) Po 1.066 0.008 XilnCWPD) Pi - - Xjln(WPD) P2 - - XjXjOVPD) Ps - - ln(Xj) Yi 0.026 0.009 ln(Xj) lnoqxp Y2 -0.026 -0.015 Y3 0.039 0.019 75 Intercept ln(a') -0.636 -0.082 0.59 0.99 0.004 3236 (HN 5) ln(WPD) p 0 1.071 0.009 XjlnCWPD) pj -0.0003 -0.0002 Xjln(WPD) P2 XiXj(WPD) P3 ln(Xj) Y l 0.076 0.022 ln(Xj) Y2 - - ln(XiXj) Ys 92 Intercept ln(a') -0.155** -0.025 6.25 0.99 0.002 6,233 (HN 6) ln(WPD) Po 1.008** 0.004 XjlnCWPD) Pi 0.0002** 0.0001 XjlnCWPD) XiXjCWPD) P2 0.001* 0.0002 Ps -0.00001** -0.0001 hKXj) Yi - - ln(Xj) Y2 -0.039 -0.022 lnCXjXj) T3 0.049* 0.023 *SignificantatP = 0.05 "Significant at P = 0.01 Note: In is synonymous for loge 214 Appendix 8.6.1 Parameters and statistics for the best subset multiple regression model of the allometric relationship between In W and ln LN in beans - 1987 experiment Potential independent variable Parameter Regression coefficient Standard partial regression coefficient Intercept ln(a') -0.168 -0.305 ln(LN) Po 0.799** 0.817 XjlnCLN) Pi -0.003** -0.256 Xjln(LN) P2 -0.003** -0.217 XiXjlnCLN) Ps -0.0001* -0.116 ln(Xj) Yi 0.202** 0.269 ln(Xj) Y2 0.082** 0.202 InCXiXj) Y3 - - Statistics Mallows CP 6.15 R2 0.58 Residue mean square 0.13 d.f. 6,950 •Significant at P = 0.05 "Significant at P = 0.01 215 Appendix 8.6.2 Parameters and statistics for the best subset multiple regression models of the allometric relationship between ln W and ln LA in beans - 1987 experiment Potential independent variable Parameter Regression coefficient Standard partial regression coefficient Intercept ln(a') 1.601** 2.910 ln(LA) Po 0.668** 0.817 XjlnCLA) Pi - - Xjln(LA) P2 - - XiXjlnfLA) Ps -o.ooo r* -0.123 ln(Xi) Yi -0.055* -0.073 ln(Xj) Y2 -0.183** -0.451 In(XiXj) Y3 0.287** 0.491 Statistics Mallows CP 5.26 R 2 0.70 Residue mean square 0.09 d.f. 5,951 *SignificantatP = 0.05 •Significant at P = 0.01 216 Appendix 8.6.3 Parameters and statistics for the best subset multiple regression models of the allometric relationship between In W and In WL in beans - 1987 experiment Potential independent variable Parameter Regression coefficient Standard partial regression coefficient Intercept ln(oc') 1.960** 3.539 ln(WL) Po 0.765** 1.004 XilnCWL) Pi -0.003** -0.111 Xjln(WL) P2 -0.005** -0.159 XiXjln(WL) Ps 0.0001** 0.090 ln(Xi) Yi - - ln(Xj) Y2 - - ln(XiXj) Y3 - - Statistics Mallows CP 2.96 R2 0.79 Residue mean square 0.06 d.f. 4,955 •Significant at P = 0.05 *SignificantatP = 0.01 217 Appendix 8.6.4 Parameters and statistics for the best subset multiple regression models of the allometric relationship between ln W and ln WST in beans-1987 experiment Potential independent Parameter Regression Standard partial variable coefficient regression coefficient Intercept ln(a*) 2.239** 4.042 ln(WST) Po 0.737** 0.781 XjlnCWST) Pi - - XjlnfWST) P2 - - XiXjinfWST) Ps -0.0001* -0.071 lnCXj) Yi -0.129** -0.171 ln(Xj) Y2 -0.346** -0.849 InCXiXj) Y3 0.486** 0.825 Statistics Mallows CP 5.26 R2 0.69 Residue mean square 0.10 d.f. 5,954 •Significant at P *Signiflcant at P = 0.05 = 0.01 218 Appendix 8.6.5 Parameters and statistics for the best subset multiple regression models of the allometric relationship between In W and In MPN in beans - 1987 experiment Potential independent variable Parameter Regression coefficient Standard partial regression coefficient Intercept ln(a') 1.195** 2.234 ln(MPN) Po 0.865** 0.961 XilnCMPN) Pi -0.002** -0.115 Xjln(MPN) P2 -0.003** -0.125 XiXjlnCMPN) Ps -0.0001 -0.073 ln(Xi) Yi 0.064** 0.087 ln(Xj) Y2 - ..•> - In(XiXj) Ys 0.074** 0.130 Statistics Mallows CP 6.36 R2 0.75 Residue mean square 0.07 d.f. 6,930 *Significant at P = 0.05 *SignificantatP = 0.01 219 Appendix 8.6.6 Parameters and statistics for the best subset multiple regression models of the allometric relationship between ln W and ln UPN in beans - 1987 experiment Potential independent variable Parameter Regression coefficient Standard partial regression coefficient Intercept ln(a') 2.607** 4.745 ln(UPN) Po 0.406** 0.489 XilnCUPN) Pi -0.002 -0.097 Xjln(UPN) P2 - - XiXjlnCUPN) Pa -0.0002** -0.183 ln(Xi) Yi -0.132** -0.175 ln(Xj) 72 -0.465** -1.152 l n o q x j ) 73 0.667** 1.144 Statistics Mallows CP 6.17 R2 0.29 Residue mean square 0.47 d.f. 6,911 *SignificantatP = 0.01 •Significant at P = 0.01 220 Appendix 8.6.7 Parameters and statistics for the best subset multiple regression models of the allometric relationship between In W and In FWPD in beans - 1987 experiment Potential independent variable Parameter Regression coefficient Standard partial regression coefficient Intercept ln(a') -0.740** -1.387 ln(FWPD) Po 0.749** 0.937 XilntFWPD) Pi -0.002** -0.228 Xjln(FWPD) P 2 -0.002** -0.152 XiXjlnCFWPD) P 3 -o.ooor* -0.142 ln(Xj) Yi 0.206** 0.280 ln(Xj) 0.089** 0.225 ln(XiXj) Y3 - - Statistics Mallows CP 7.11 R2 0.77 Residue mean square 0.07 d.f. 6,930 •Significant at P = 0.05 ^Significant at P = 0.02 Appendix 8.6.8 Parameters and statistics for the best subset multiple regression models of the allometric relationship between In W and In WUPD in beans - 1987 experiment Potential independent Parameter Regression Standard partial variable coefficient regression coefficient Intercept ln(a') 3.576** 6.505 ln(WUPD) Po 0.075** 0.164 XilnfWUPD) Pi 0.002** 0.161 Xjln(WUPD) P2 - - XiXjlnOVTJPD) Ps -0.0001 -0.075 ln(Xi) gl -0.243** -0.324 ln(Xj) 72 -0.631** -1.560 ln(XiXj) Ys 0.797** 1.367 Statistics Mallows CP 6.00 R2 0.21 Residue mean square 0.24 d.f. 6,910 *SignificantatP = 0.05 •Significant at P = 0.01 222 Appendix 8.6.9 Parameters and statistics for the best subset multiple regression models of the allometric relationship between In W and ln WMPD in beans - 1987 experiment Potential independent variable Parameter Regression coefficient Standard partial regression coefficient Intercept ln(a') 0.965** 1.804 ln(WMPD) Po 0.750** 0.927 XilnCWMPD) Pi -0.002** -0.116 Xjln(WMPD) P2 -0.002** -0.105 XiXjlnCWMPD) P3 -0.0001** -0.143 ln(Xi) Yi 0.093** 0.127 ln(Xj) - - InCXiXj) 73 0.106** 0.186 Statistics Mallows CP 6.01 R2 0.70 Residue mean square 0.09 d.f. 6,931 *SignificantatP = 0.05 *SignificantatP = 0.01 223 Appendix 8.6.10 Parameters and statistics for the best subset multiple regression models of the allometric relationship between ln W and ln SN in beans - 1987 experiment Potential independent variable Parameter Regression coefficient Standard partial regression coefficient Intercept ln(a') 0.768** 1.450 ln(SN) Po 0 . 5 3 9 * * 0.729 Xiln(SN) Pi -0.004** -0.382 Xjln(SN) P2 -0.004** -0.305 XjXjlnCSN) Ps -0.0001* -0.149 i n c q ) Yi 0 . 2 3 2 * * 0.318 ln(Xj) Y2 - - InCXiXj) Y3 0.173** 0.306 Statistics Mallows CP 6.96 R2 0.35 Residue mean square 0.18 d.f. 6,903 *Significant at P = 0.05 *SignificantatP = 0.01 224 Appendix 8.6.11 Parameters and statistics for the best subset multiple regression models of the allometric relationship between In W and In WS in beans - 1987 experiment Potential independent variable Parameter Regression coefficient Standard partial regression coefficient Intercept ln(cc') 2.443** 4.613 InfWS) Po 0.419** 0.610 XilnfWS) Pi - - Xjln(WS) P2 - - XjXjlnCWS) Ps -0.0003** -0.238 ln(Xi) Yi -0.091** -0.125 ln(Xj) §2 -0.340** -0.869 InCXjXj) Y3 0.529** 0.937 Statistics Mallows CP 4.62 R2 0.37 Residue mean square 0.18 d.f. 5,904 •Significant at P *Significant at P = 0.05 = 0.01 225 Appendix 8.7.1 Parameters and statistics for the best subset multiple regression models of the allometric relationship between ln W and ln T in beets - 1984 experiment Age at Potential Parameter Regression Std. partial Mallows R2 RMS d.f. harvest independent coefficient regression CP (days) variable coefficient 40 Intercept ln(oc') -6.571** -7.136 4.41 0.79 0.18 3,235 (HNl)t a' ln(T) Po 2.374** 0.886 Xiln(T) P i -0.002** -0.115 XjlnfT) XjXjlnfD P 2 -0.011** -0.337 P3 - - InfXj) Yi - - InfXj) Y2 - - InCXjXj) Y3 - - 51 Intercept ln(a') -6.563** -5.372 2.58 0.81 0.29 3,236 (HN2) ln(T) Po 2.427** 0.769 XilnCT) P i XjlnfT) P 2 -0.011** -0.289 XiXjlnfT) P3 InfXj) Yi InfXj) Y2 - 0 - 0 9 8 - ° 1 1 2 lnfXjXj) Y3 63 Intercept ln(a') -8.356** -6.058 6.03 0.83 0.33 6,233 (HN 3) lnfT) Po 2.715** 0.755 XjlnfT) P i -0.009** -0.311 XjlnfT) p 2 -0.012** -0.298 XiXjlnfT) P3 -0.0003* -0.174 lnCXj) Yi 0.689** 0.337 ln(Xj) Y 2 lnfXjXj) Y3 -0.179* -0.135 Ĥarvest number (1-6 in beans and 1-7 in beets) *Signiricant at P = 0.05 •Significant at P = 0.01 226 Appendix 8.7.1 (cont'd) Parameters and statistics for the best subset multiple regression models of the allometric relationship between In W and In T in beets - 1984 experiment Age at harvest (days) Potential independent variable Parameter Regression coefficient Std. partial regression coefficient Mallows CP R2 RMS d.f. -6.087** -4.631 2.471** 0.724 -0.007** 0̂.207 -0.0001 -0.110 -0.344** -0.366 -5.389** -3.728 2.548** 0.667 -0.004** -0.137 -0.0002** -0.146 -1.0420 -1.007 0.569** 0.408 -4.097** -2.984 2.240** 0.643 -0.006** -0.209 0.008** 0.223 -1.665** -1.694 0.979** 0.738 -4.036** -3.133 2.174** 0.599 -0.003** -0.127 -0.932** -1.011 69 (HN 4) 75 (HN 5) 92 (HN 6) 107 (HN7) Intercept InCD XjlnCT) Xjln(T) XiXjfT) ln(Xj) ln(Xj) lnQqXj) Intercept lnCT) XjlnCD XjlnCT) XiXjCT) lnCXj) ln(Xj) inQqxp Intercept ln(T) XjlnCT) XjlnCT) XjXjCT) ln(Xj) ln(Xj) moqXj) Intercept InCD XjlnCT) XjlnCT) XiXjlnCT) hi(Xi) ln(Xj) lnpCjXj) ln(a') Po P i P 2 P3 Yi Y2 Y3 ln(a') Po P i P 2 Ps Yi Y2 Y3 ln(a') Po P i P 2 P3 Yi Y2 Y3 ln(a') Po P i P3 P 4 Yi Y2 Y3 3.24 0.75 0.44 4,235 4.36 0.85 0.32 5,234 5.41 0.82 0.35 5,234 2.24 0.73 0.45 4,235 0.346 0.278 *SignificantatP = 0.05 "Significant at P = 0.01 227 Appendix 8.7.2 Parameters and statistics for the best subset multiple regression models of the allometric relationship between In W and In LN in beets -1984 experiment Age at Potential Parameter Regression Std. partial Mallows R2 RMS d.f. harvest independent coefficient regression CP (days) variable coefficient 40 Intercept ln(cc') -7.753** -8.419 6.03 0.58 0.36 6,232 (HN l)t ln(LN) Po 3.083** 0.885 XjlntLN) Pi -0.011* -0.398 Xjln(LN) XiXjln(LN) P2 -0.033** -0.672 P3 - - InfXj) Yi 0.599* 0.439 InfXj) InOqXj) Y2 0.957** 1.446 Y3 -0.862* -0.969 51 Intercept ln(a') -6.569** -5.377 2.64 0.76 0.37 3,236 (HN2)' ln(LN) p0 3.140** 0.818 XjlnaJSO p! Xjln(LN) p2 -0.008** -0.126 XiXjlnCLN) P3 InfXj) Yi 0.093 0.051 InfXj) Y2 InfXiXj) Y3 - - 63 Intercept ln(a') -7.065** -5.122 5.79 0.80 0.39 4,235 (HN 3) ln(LN) p0 Xiln(LN) p! Xjln(LN) p2 XiXjln(LN) P3 ln(Xi) Yi InfXj) Y 2 InfXiXj) Y3 3.347** 0.810 -0.009** -0.241 -0.001** -0.265 0.524** 0.256 Ĥarvest number (1-6 in beans and 1-7 in beets) •Significant at P = 0.05 "Significant at P = 0.01 2 2 8 Appendix 8.7.2 (cont'd) Parameters and statistics for the best subset multiple regression models of the allometric relationship between In W and In LN in beets - 1984 experiment Age at harvest (days) Potential independent variable Parameter Regression coefficient Std. partial regression coefficient Mallows CP R2 RMS d.f. 69 (HN4) 75 (HN 5) 92 (HN 6) 107 (HN 7) Intercept ln(LN) Xjln(LN) Xjln(LN) XiXj(LN) ln(Xi) ln(X-) ln(XiXj) Intercept ln(LN) Xjln(LN) Xjln(LN) XiXjCLN) ln(Xj) ln(Xj) ln(XiXj) Intercept ln(LN) Xjln(LN) Xjln(LN) XjXjCLN) ln(Xi) ln(Xj) ln(XiXj) Intercept ln(LN) XjlnCLN) X:ln(LN) XiXjlnCLN) ln(Xi) ln(Xj) lnQqXj) ln(a') -5.123** -3.898 Po 3.219** 0.775 Pi -0.004* -0.123 P2 -0.014* -0.212 Ps -0.0005** -0.195 Yl - - Y2 -0.177 -0.188 Y3 - - In(a') -3.851** -2.663 Po 2.933** 0.748 Pi - - P2 -0.013** -0.171 Ps - - Yi -0.182 -0.085 Y2 - - Ys -0.132 -0.095 ln(a') -3.081** -2.244 Po 2.569** 0.628 Pi -0.006** -0.163 P2 0.023* 0.345 Ps -0.0004 -0.144 Yi - - Y2 -1.226** -1.248 Y3 0.944** 0.712 ln(a') .̂919** -3.817 Po 2.980** 0.761 Pi -0.010** -0.304 P2 -0.013** -0.222 Ps - - Yi 0.394* 0.206 Y2 - - Ys -0.221** -0.178 5.57 0.74 0.46 5,234 3.77 0.81 0.63 4,235 6.06 0.73 0.72 6,233 5.38 0.77 0.40 5,234 *SignificantatP = 0.05 •Significant at P = 0.01 229 Appendix 8.7.3 Parameters and statistics for the best subset multiple regression models of the allometric relationship between ln W and In LA in beets -1984 experiment Age at Potential Parameter Regression Std. partial Mallows R2 RMS d.f. harvest independent coefficient regression CP (days) variable coefficient 40 Intercept ln(a') 4.798** 5.211 5.08 0.93 0.06 2,236 (HNl)t ln(LA) Xjln(LA) Xjln(LA) XjXjln(LA) lnCXj) InfXj) lnfXjXj) 51 Intercept ln(a') 6.352** 5.200 6.55 0.94 0.09 6,233 (HN 2) ln(LA) Xjln(LA) Xjln(LA) XiXjln(LA) ln(Xi) ln(Xj) InfXiXj) Po 1.075** 0.923 Pi - - P2 - - P3 0.0001** 0.150 Yi - - 72 - - 73 Po 1.231** 1.025 Pi -0.003** -0.240 P2 - -0.003** -0.161 P3 - - Yi -0.372** -0.205 Y2 -0.678** -0.776 Y3 0.635** 0.538 ln(a') 7.184** 5.209 Po 1.222** 0.970 Pi -0.004** -0.274 P2 -0.004** -0.195 P3 - - Yi -0.554** -0.271 Y2 -0.825** -0.835 Y3 0.619** 0.465 63 Intercept l ' . 4" . 6.08 0.91 0.17 6,233 (HN 3) ln(LA) Xjln(LA) Xjln(LA) XiXjln(LA) lnCXj) ln(Xj) InfXiXj) Ĥarvest number (1-6 in beans and 1-7 in beets) *SignificantatP = 0.05 *SignificantatP = 0.01 230 Appendix 8.7.3 (cont'd) Parameters and statistics for the best subset multiple regression models of the allometric relationship between ln W and ln LA in beets - 1984 experiment Age at harvest (days) Potential independent variable Parameter Regression coefficient Std. partial regression coefficient Mallows CP R2 RMS d.f. 69 (HN4) 75 (HN 5) 92 (HN 6) 107 (HN 7) Intercept ln(LA) Xjln(LA) Xjln(LA) XiXj(LA) InfXj) ln(Xj) InfXiXj) Intercept ln(LA) X{ln(LA) Xjln(LA) XjXj(LA) InfXj) ln(Xj) InfXiXj) Intercept ln(LA) Xiln(LA) Xjln(LA) XiXj(LA) InfXi) ln(Xj) InfXiXj) Intercept ln(LA) Xiln(LA) Xjln(LA) XiXjln(LA) InfXi) InfXj) InfXjXj) ln(a') 7.346** 5.589 Po 1.307** 0.960 Pi -0.003** -0.179 P2 - - P3 -0.0001** -0.173 Yi -0.390** -0.200 Y2 -0.356** -0.378 Y3 - - ln(a') 6.401** 4.427 Po 1.124** 0.786 Pi - - P2 - - P3 - - - Yi -0.106** -0.050 72 -0.348** -0.337 Y3 - - ln(a') 6.569** 4.785 Po 1.189** 0.812 Pi 0.001 0.044 P2 - - Ps -0.0001** -0.088 Yi - - 72 -0.382** -0.389 Y3 - - ln(a') 7.085** 5.499 Po 1.186** 0.819 Pi - - P2 - - P3 -0.0001** -0.114 Yi -0.117* -0.061 Y2 -0.384** -0.417 Y3 - - 7.16 0.93 0.13 5,234 0.40 0.94 0.13 3,236 2.73 0.92 0.15 4,235 3.18 0.90 0.17 4,235 *Significant at P = 0.05 "Significant at P = 0.01 231 Appendix 8.7.4 Parameters and statistics for the best subset multiple regression models of the allometric relationship between In W and In WL in beets -1984 experiment Age at Potential Parameter Regression Std. partial Mallows R2 RMS d.f. harvest independent coefficient regression CP (days) variable coefficient 40 Intercept In(a') -0.408** -0.444 5.32 0.99 0.01 4,234 (HNl)t ln(WL) Po 0.984** 0.986 Xjln(WL) Pi - - Xjln(WL) p2 XjXjln(WL) P3 -0.00002 -0.017 ln(Xi) Yi ln(Xj) Y2 -0.087** -0.132 ln(XiXj) Y3 0.074** 0.083 51 Intercept ln(a') -0.238** -0.202 8.00 0.98 0.03 7,232 (HN20 ln(WL) Po 1.057** 1.099 Xjln(WL) Pi -0.005** -0.199 Xjln(WL) XjXjlnCWL) P2 -0.005** -0.100 Ps 0.0002** 0.144 ln(Xi) Yi -0.089** -0.051 ln(Xj) lnQqXj) Y2 -0.142** -0.168 Y3 0.138** 0.122 63 Intercept ln(a') -0.625** -0.500 2.19 0.99 0.02 4,235 (HN 3) ln(WL) p0 0.869** 0.960 Xiln(WL) p! -0.002** -0.059 Xjln(WL) p2 XiXjlnCWL) P3 0.0002** 0.096 ln(Xi) Yi ln(Xj) Y2 -0.016* -0.018 InCXiXj) Ys tHarvest number (1-6 in beans and 1-7 in beets) •Significant at P = 0.05 *Significant at P = 0.01 232 Appendix 8.7.4 (cont'd) Parameters and statistics for the best subset multiple regression models of the allometric relationship between ln W and ln WL in beets - 1984 experiment Age at Potential Parameter Regression Std. partial Mallows R 2 RMS d.f. harvest independent coefficient regression CP (days) variable coefficient 69 Intercept ln(a') (HN 4) InfWL) p 0 Xiln(WL) Xjln(WL) p2 XiXjfWL) P3 ln(Xi) Yi InfXj) y2 lnfXjXj) 73 75 Intercept ln(a') (HN 5) InfWL) p 0 Xiln(WL) p! . XjInfWL) P2 XiXjfWL) P3 InfXj) Y l InfXj) 7 2 InfXiXj) 73 92 Intercept ln(a') (HN 6) InfWL) p 0 Xiln(WL) p! Xjln(WL) p2 XiXjfWL) P3 InfXi) 7i InfXj) 7 2 InfXiXj) 73 107 Intercept ln(a') (HN7) InfWL) p 0 Xjln(WL) pi Xjln(WL) P2 XiXjln(WL) P3 lnCXj) Y l InfXj) 7 2 InfXiXj) 73 -0.451" -0.404 5.69 0.98 0.03 5,234 0.782" 0.921 0.0001" 0.042 -0.077" -0.047 -0.138" -0.173 0.116* 0.107 -0.804** -0.678 3.42 0.98 0.03 4,235 0.774** 0.943 -0.001** -0.045 0.004** 0.081 -0.040* -0.035 -0.615** -0.553 3.79 0.96 0.05 4,235 0.710** 0.877 0.004** 0.090 -0.052* -0.032 -0.056** -0.071 -0.340** -0.335 5.14 0.93 0.07 5,234 0.694** 0.881 0.0001** 0.052 -0.138** -0.092 -0.199** -0.273 0.187* 0.190 •Significant at P = 0.05 •Significant at P = 0.01 233 Appendix 8.7.5 Parameters and statistics for the best subset multiple regression models of the allometric relationship between In W and In WP in beets -1984 experiment Age at Potential Parameter Regression Std. partial Mallows R2 RMS d.f. harvest independent coefficient regression CP (days) variable coefficient 40 Intercept ln(a') -1.963** -2.037 4.33 0.94 0.06 5,232 (HNl)t ln(WP) Xjln(WP) Xjln(WP) XtXjln(WP) ln(Xi) ln(Xj) lnOqXj) 51 Intercept ln(a*) -1.538** -1.358 7.30 0.94 0.08 6,233 (HN 2) ln(WP) Xjln(WP) Xjln(WP) XiXjln(WP) ln(Xi) ln(Xj) ln(XiXj) 63 Intercept ln(a') -1.545** -1.230 4.45 0.94 0.10 5,234 (HN 3) ln(WP) Xiln(WP) Xjln(WP) XiXjln(WP) InCXj) ln(Xj) ln(XiXj) Po 0.990** 0.932 Pi 0.002** 0.087 P2 - - Ps - - Yl 0.114** 0.080 Y2 0.221** 0.318 Y3 -0.224** -0.240 ' Po 1.056** 1.139 Pi -0.003** -0.143 P2 -0.007** -0.149 Ps 0.0003** 0.180 Yi -0.066* -0.039 Y2 0.116** 0.143 T3 — l ' - . - . Po 0.938** 1.030 Pi -0.002* -0.069 P2 - - P3 0.0002** 0.093 Yi - - Y2 0.227** 0.252 Y3 -0.155 -0.127 Ĥarvest number (1-6 in beans and 1-7 in beets) *SigmficantatP = 0.05 *SignificantatP = 0.01 234 Appendix 8.7.5 (cont'd) Parameters and statistics for the best subset multiple regression models of the allometric relationship between ln W and ln WP in beets - 1984 experiment Age at Potential Parameter Regression Std. partial Mallows R2 harvest independent coefficient regression CP (days) variable coefficient RMS d.f. 69 (HN 4) 75 (HN 5) 92 (HN 6) 107 (HN 7) Intercept lnfWP) Xiln(WP) Xjln(WP) XiXjfWP) ln(Xi) InfXj) tafXjXj) Intercept ln(WP) Xiln(WP) Xjln(WP) XJXJCWP) InfXj) ln(Xj) InfXiXj) Intercept lnfWP) Xjln(WP) Xjln(WP) XiXjfWP) InfXi) InfXj) InfXiXj) Intercept lnfWP) Xjln(WP) Xjln(WP) XjXjln(WP) W$ InfXj) InfXjXj) ln(a') -1.654** -1.443 Po 0.820** 0.940 Pi - - P2 0.007** 0.138 Ps - - Yi 0.046 0.027 Y2 0.149** 0.182 Y3 - - ln(a') -2.533** -2.107 Po 0.949** 1.142 Pi -0.004** -0.153 P2 - - Ps 0.0003** 0.147 Yi 0.261** 0.146 72 0.568** 0.660 Y3 -0.460** -0.396 ln(a') -1.544** -1.367 Po 0.737** 0.896 Pi - - P2 0.010** 0.205 Ps - - Yi - - Y2 0.151** 0.187 Ys - - ln(a') -1.432** -0.500 Po 0.672** 0.906 P i - - P 2 0.011** 0.172 Ps - - Yi - - Y2 0.278** 0.407 Y3 -0.217* -0.236 3.20 0.93 0.09 4,235 6.00 0.93 0.11 6̂ 33 3.00 0.91 0.112 3,236 2.49 0.86 0.13 4,235 •Significant at P = 0.05 'Significant at P = 0.01 235 Appendix 8.7.6 Parameters and statistics for the best subset multiple regression models of the allometric relationship between In W and In DR in beets -1984 experiment Age at Potential Parameter Regression Std. partial Mallows R2 RMS d.f. harvest independent coefficient regression CP (days) variable coefficient 40 Intercept ln(a') -4.257** -4.623 3.37 0.85 0.13 45,234 (HN l)tln(DR) po 1 - 7 0 6 * * l-°°5 Xjln(DR) p! -0.005** -0.162 Xjln(DR) p2 XjXjlnCDR) P3 -0.0001 -0.038 ln(Xj) Yi 0.239** 0.175 ln(Xj) Y2 InOqXj) Y3 51 Intercept ln(a") -2.626** -2.150 3.06 0.91 0.14 4,235 (HN 2) ln(DR) Po 1.490** 0.840 Xjln(DR) Pi 0.005** 0.174 Xjln(DR) XiXjln(DR) P2 - - P3 - - ln(Xi) Yi -0.310** -1.171 ln(Xj) ln(XiXj) 72 -0.044 -0.050 Y3 - - 63 Intercept In(a') -2.752** -1.995 5.41 0.95 0.09 5,234 (HN 3) ln(DR) Po 1.459** 0.917 Xiln(DR) Pi -0.003** -0.107 Xjln(DR) XiXjln(DR) P2 0.012** 0.167 P3 - - hKXj) Yi - - ln(Xj) lnCXjXj) Y2 -0.436** -0.442 Y3 0.244** 0.184 Ĥarvest number (1-6 in beans and 1-7 in beets) •Significant at P = 0.05 "Significant at P = 0.01 236 Appendix 8.7.6 (cont'd) Parameters and statistics for the best subset multiple regression models of the allometric relationship between In W and In DR in beets - 1984 experiment Age at harvest (days) Potential independent variable Parameter Regression coefficient Std. partial regression coefficient Mallows CP R2 RMS d.f. 69 Intercept ln(a') -4.901** -3.729 (HN 4) ln(DR) Po 1.663** 1.320 Xiln(DR) Pi -0.006* -0.265 Xjln(DR) P2 -0.050** -0.074 XiX;(DR) Ps - - ln(Xi) Yl 0.545* 0.279 ln(Xj) 72 1.284** 1.365 lntXJXj) 73 -0.871** -0.687 75 Intercept ln(a') -4.447** -3.076 (HN 5) ln(DR) Po 1.767** 1.066 Xjln(DR) Pi -0.003** -0.105 Xjln(DR) P2 -0.007* -0.118 X]Xj(DR) P3 0.0002 0.071 lnQq) Yi 0.176 0.082 ln(Xj) 72 0.420** 0.406 ln(XiXj) 73 -0.459** -0.329 92 Intercept In(a') -3.422** -2.492 (HN 6) ln(DR) Po 1.585** 0.978 Xiln(DR) Pi -0.001** -0.050 X:ln(DR) P2 0.002 0.038 XJX|(DR) P3 - - InPCj) Yi - - ln(Xj) Y2 - - InCXiXj) Y3 -0.077* -0.058 107 Intercept ln(a') -4.742** -3.680 (HN 7) ln(DR) Po 1.762** 1.096 Xiln(DR) Pi -0.003* -0.142 Xjln(DR) P2 -0.009** -0.179 XjXjlnCDR) P3 - - ln(Xj) Yi 0.270* 0.141 ln(Xj) Y2 0.633** 0.686 lnCXjXj) 73 -0.577* -0.464 6.39 0.73 0.48 6,233 8.00 0.96 0.08 7,232 3.13 0.96 0.07 4,235 6.81 0.94 0.10 6,233 *Significant at P = 0.05 *Significant at P = 0.01 237 Appendix 8.7.7 Parameters and statistics for the best subset multiple regression models of the allometric relationship between ln W and ln WR in beets -1984 experiment Age at Potential Parameter Regression Std. partial Mallows R2 RMS d.f. harvest independent coefficient regression CP (days) variable coefficient 40 Intercept ln(oc') -1.821** -1.870 1.38 0.87 0.13 3,234 (HNl)t InfWR) po 1.034** 0.964 Xjln(WR) p2 Xjln(WR) p2 -0.002* -0.058 XjXjln(WR) P3 ln(Xj) Yi -0.072* -0.050 InfXj) 7 2 InCX )̂ 73 51 Intercept In(a') -2.050** -1.316 9.93 0.92 0.19 5,234 (HN2)% InfWR) Po 1.162** 0.912 Xiln(WR) Pi 0.004** 0.133 Xjln(WR) XjXj^WR) P2 0.006 0.103 P3 -0.0004 -0.210 InfXi) Yi 0.102* 0.004 InfXj) Y2 - - InfXiXj) Y3 - - 63 Intercept ln(a') -1.542** -0.814 5.27 0.94 0.21 4,235 (HN 3) InfWR) p0 1.381** 1.006 Xiln(WR) px Xjln(WR) P2 -0.010** -0.145 XiXjlnfWR) P3 taf^) Yi InfXj) 7 2 -0.448** -0.330 InfXiXj) 73 0.427** 0.234 Ĥarvest number (1-6 in beans and 1-7 in beets) *SignificantatP = 0.05 *Significant at P = 0.01 238 Appendix 8.7.7 (cont'd) Parameters and statistics for the best subset multiple regression models of the allometric relationship between ln W and ln WR in beets - 1984 experiment Age at Potential harvest independent (days) variable Parameter Regression coefficient Std. partial regression coefficient Mallows CP R2 RMS d.f. 69 (HN4) 75 (HN5) 92 (HN 6) 107 (HN7) Intercept InfWR) XjhXWR) Xjln(WR) XiXjfWR) InfXj) ln(X:) lnfXjXj) Intercept InfWR) Xjln(WR) Xjln(WR) XiXjfWR) ln(Xj) InfXjXj) Intercept InfWR) Xtln(WR) Xjln(WR) XjXjCWR) lnCXj) InfXj) InfXiXj) Intercept InfWR) Xiln(WR) Xjln(WR) X|Xjln(WR) InfX-) InfXj) IntXiXj) ln(a') -1.394** -0.730 Po 1.393** 0.959 Pi - - P2 - . - P3 -0.0002** -0.058 Yi - - Y2 -0.138** -1.101 73 - - ln(a') -0.385 -0.190 Po 1.138** 0.815 Pi 0.004** 0.078 P2 0.007* 0.073 Ps -0.0003* 0.078 Yi -0.196** -0.065 Y2 -0.591** -0.409 Y3 0.513** 0.263 ln(a') -1.147** -0.608 Po 1.312** 0.955 Pi - - P2 -0.003* -0.037 P3 - - Yi - - Y2 -0.108** -0.080 73 - - ln(a') -1.105** -0.601 Po 1.284** 0.900 Pi - - P2 - - Ps - - Yi - - Y2 -0.330** -0.251 Y3 0.257* 0.145 2.93 0.95 0.17 3,236 8.00 0.97 0.14 7,232 1.92 0.96 0.14 3,236 3.94 0.94 0.20 3,236 •Significant at P = 0.05 •Significant at P = 0.01 239 Appendix 8.7.8 Parameters and statistics for the best subset multiple regression models of the allometric relationship between In W and In FWR in beets -1984 experiment Age at Potential Parameter Regression Std. partial Mallows R2 RMS d.f. harvest independent coefficient regression CP (days) variable coefficient 40 Intercept ln(a') (HN l)t ln(FWR) p0 XjlnvTWR) pt Xjln(FWR) &2 XiXjln(FWR) P3 hKXj) Yi ln(Xj) Y2 ln(XiXj) Y3 51 Intercept ln(a') -0.475** -0.389 6.55 0.82 0.27 6,233 (HN2) ln(FWR) Po 0.738** 0.859 Xiln(FWR) Pi -0.002 -0.089 Xjln(FWR) XiXjln(FWR) P2 - - P3 0.0003** 0.170 ln(Xi) Yl 0.144** 0.079 ln(Xj) lnfXjXj) T2 0.293** 0.335 Y3 -0.431** -0.365 63 Intercept ln(a') -0.004 -0.003 3.16 0.93 0.13 3,234 (HN 3) ln(FWR) B0 0.712** 0.943 Xjln(FWR) px -0.002* -0.071 Xjln(FWR) &2 XiXjlnQTWR) P3 0.0002** 0.113 ln(Xi) Yi - - ln(Xj) Y2 InCXiXj) Ys "̂Harvest number (1-6 in beans and 1-7 in beets) *Significant at P = 0.05 "Significant at P = 0.01 240 Appendix 8.7.8 (cont'd) Parameters and statistics for the best subset multiple regression models of the allometric relationship between In W and In FWR in beets - 1984 experiment Age at Potential Parameter Regression Std. partial Mallows R2 RMS d.f. harvest independent coefficient regression CP (days) variable coefficient 69 Intercept ln(a') -0.019 -0.014 (HN4) ln(FWR) Po 0.650** 0.935 XilnflFWR) Pi - - Xjln(FWR) P2 0.003 0.062 XiXjCFWR) P3 - - ln(Xi) Yl - - ln(Xj) Y2 - - moqxj) Y3 - - 75 Intercept ln(a') -0.264** -0.182 (HN 5) ln(FWR) Po 0.727** 1.012 Xiln(FWR) Pi - - Xjln(FWR) P2 - - XJXJO^WR) P3 - - ln(Xi) Yi - - ln(Xj) 72 0.049** 0.048 ln(XiXj) 73 - - 92 Intercept ln(a') -0.179** -1.130 (HD6) ln(FWR) Po 0.690** 0.958 Xiln(FWR) Pi - - Xjln(FWR) P2 0.004** 0.080 XiXjCFWR) P3 -0.0001 -0.042 ln(Xi) Yi - - hi(Xj) 72 - - ln(XiXj) 73 - - 107 Intercept ln(a') -0.041 -0.032 (HD7) ln(FWR) Po 0.723** 1.055 XilnCFWR) Pi - - Xjln(FWR) P2 -0.004** -0.061 XJXJUKPWR) P3 - - ln(Xi) Yi -0.063 -0.003 ln(Xj) Y2 0.073** 0.079 hKXjXj) Y3 0.257* 0.145 0.93 0.96 0.07 2,237 4.98 0.96 0.08 2,237 0.80 0.96 0.07 3̂ 36 3.87 0.94 0.09 4,235 *Significant at P = 0.05 *Significant at P = 0.01 241 Appendix 8.8.1 Parameters and statistics for the best subset multiple regression models of the allometric relationship between ln W and ln LN in beets - 1987 experiment Potential independent variable Parameter Regression coefficient Standard partial regression coefficient Intercept ln(a') -0.797 -0.620 ln(LN) Po 2.083** 0.564 XjlnCLN) Pi 0.011** 0.340 Xjln(LN) P2 - - XiXftsjlnCLN) P3 0.0005** 0.248 ln(Xi) Yi -0.911** -0.521 ln(Xj) Y2 -0.235** -0.249 lncqxp Y3 - - Statistics Mallows CP 4.07 R2 0.56 Residue mean square 0.75 d.f. 5,942 *Significant at P = *Significant at P = 0.05 0.01 Appendix 8.8.2 Parameters and statistics for the best subset multiple regression models of the allometric relationship between ln W and ln LA in beets - 1987 experiment Potential independent Parameter Regression Standard partial variable coefficient regression coefficient Intercept ln(a') 0.038 0.029 ln(LA) Po 1.254** 0.825 XjlnCLA) Pi - - Xjln(LA) P2 0.013** 0.003 XiXjlnCLA) Ps -0.0006** -0.176 InfXi) Yi 0.137** 0.079 ln(Xj) Y2 - - ln(XiXj) Y3 -0.085** -0.062 Statistics Mallows CP 7.24 R2 0.70 Residue mean square 0.49 d.f. 5,943 *SignificantatP = 0.05 *SignificantatP = 0.01 Appendix 8.8.3 Parameters and statistics for the best subset multiple regression models of the allometric relationship between in W and ln WL in beets - 1987 experiment Potential independent Parameter Regression Standard partial variable coefficient regression coefficient Intercept ln(a') 1.378** 1.073 InfWL) Po 1.233** 0.889 XilnfWL) P i - - Xjln(WL) P 2 0.020** 0.281 XiXjlnCWL) P3 -o.oor* -0.375 ln(Xi) Yi - - ln(Xj) Y2 - - lnQqXj) Y3 Statistics Mallows CP 2.40 R2 0.71 Residue mean square 0.47 d.f. 3,945 *SignificantatP = 0.05 •Significant at P = 0.01 244 Appendix 8.8.4 Parameters and statistics for the best subset multiple regression models of the allometric relationship between In W and In WP in beets - 1987 experiment Potential independent variable Parameter Regression coefficient Standard partial regression coefficient Intercept ln(a') 2.677** 2.086 ln(WP) Po 1.250** 0.924 XilnCWP) Pi -0.006** -0.157 Xjln(WP) \h 0.010** 0.165 XiXjln(WP) Ps -0.001** -0.294 ln(Xi) Yi -0.121** -0.069 ln(Xj) Y2 0.302** 0.321 In(XiXj) Ys -0.696** -0.510 Statistics Mallows CP 8.00 R2 0.69 Residue mean square 0.52 d.f. 7,941 *Significant at P = 0.05 *SignificantatP = 0.01 - 245 Appendix 8.8.5 Parameters and statistics for the best subset multiple regression models of the allometric relationship between ln W and in DR in beets - 1987 experiment Potential independent variable Parameter Regression coefficient Standard partial regression coefficient Intercept ln(a') -2.852** -1.443 in(DR) Po 1.329** 0.786 XilnCDR) Pi 0.004** 0.163 Xjln(DR) P2 -0.005** -0.172 XjXjlnCDR) Ps -0.0002** 0.155 ln(Xi) Yi -0.411** -0.235 ln(Xj) Y2 - - lnOqXj) Y3 - - Statistics Mallows CP 5.35 R2 0.71 Residue mean square 0.48 d.f. 5,943 *SignificantatP = 0.05 *SignificantatP = 0.01 246 Appendix 8.8.6 Parameters and statistics for the best subset multiple regression models of the allometric relationship between In W and In FWR in beets - 1987 experiment Potential independent variable Parameter Regression coefficient Standard partial regression coefficient Intercept ln(a') -1.918** -1.494 ln(FWR) Po 0.857** 1.098 Xjln(FWR) Pi -0.004** -0.187 Xjln(FWR) P2 - - XiXjlnfFWR) Ps -0.0002** -0.113 ln(Xi) Yi 0.432** 0.247 ln(Xj) Y2 0.091** 0.097 ln(X i X j) Y3 - - Statistics Mallows CP 4.75 R2 0.74 Residue mean square 0.42 d.f. 5,943 *Significant at P *Significant at P = 0.05 = 0.01 247 Appendix 8.8.7 Parameters and statistics for the best subset multiple regression models of the allometric relationship between ln W and ln WR in beets - 1987 experiment Potential independent variable Parameter Regression coefficient Standard partial regression coefficient Intercept ln(a) 0.405** 0.315 InfWR) Po 0.856** 1.075 XilnfWR) Pi -0.003** -0.118 XjlnfWR) P2 0.006** 0.152 XiXjlnfWR) Ps -0.0004** -0.246 ln(Xf) Yi 0.118** 0.067 ln(Xj) ~Y2 -0.166** -0.171 lnCXiXp Y3 0.255** 0.187 Statistics Mallows CP 8.00 R2 0.78 Residue mean square 0.37 d.f. 7,941 *Significant atP = *Significant at P = 0.05 0.01
Cite
Citation Scheme:
Usage Statistics
Country | Views | Downloads |
---|---|---|
India | 4 | 0 |
Japan | 2 | 0 |
China | 1 | 0 |
City | Views | Downloads |
---|---|---|
Unknown | 4 | 0 |
Tokyo | 2 | 0 |
Beijing | 1 | 0 |
{[{ mDataHeader[type] }]} | {[{ month[type] }]} | {[{ tData[type] }]} |
Share
Share to: