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Interference in monocultures and mixtures of orchardgrass (Dactylis glomerata L.) and timothy (Phleum… Potdar, Madhukar Vishwanathrao 1986

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INTERFERENCE IN MONOCULTURES AND MIXTURES OF ORCHARDGRASS (DACTYLIS GLOMERATA L.) AND TIMOTHY (PHLEUM PRATENSE L.) by MADHUKAR VISHWANATHRAO POTDAR B.Sc, Marathwada Agricultural University, M.Sc, Marathwada Agricultural University A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE DEPARTMENT OF PLANT SCIENCE We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December 1986 © MADHUKAR VISHWANATHRAO POTDAR, 1986 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 P 1 at) \r S C^l The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date Q^C^mb e^ r /9-;/^^4 ABSTRACT Interference among plants involves responses of plants to their environment as modified by the presence and/or growth of neighbouring plants. An important theme of research on plant interference is the relationship between plant population density and measures of plant growth or agricultural yield. An experiment on plant interference was performed in which plots of two important forage species, orchardgrass (Dactylis glomerata L.) and timothy (Phleum pratense L.), were grown at different total population densities and mixture proportions. Measures of plant growth and yield were taken at five separate harvests during one growing season. The analysis of variance indicated that primary and derived measures of growth and yield generally were strongly affected by the three main experimental factors: time, total population density and mixture proportions. Best subset multiple regression analysis, using Mallow's CP criterion, helped to define which experimental factors and interactions were related closely to plant responses. Both of these analytical methods indicated that the main effects of experimental factors were often significant, while interactions among factors were less prominent. The best subset models were different in structure for different response variates, however, indicating that plant responses varied when different measures of growth were considered. Models were developed which provided an effective description of yield-density responses in monocultures and mixtures when interference was strong. Model parameters were used to compare the relative strengths of intraspecific and interspecific interference in each species. The higher-yielding species, timothy, exerted stronger interference, both within and between species than orchardgrass. Interference was significant early in growth and intensified with increasing population density. The parameters of the yield-density models were also used to assess differential yield responses in the mixtures. Net overyielding occurred in most mixtures because overyielding in timothy was not fully offset by underyielding in orchardgrass. The greatest yield advantage occurred in mixtures containing orchardgrass and timothy in proportions of 2:1. The dj'namics of plant growth were followed using methods of plant growth analysis. Absolute growth rate, relative growth rate, unit leaf rate and crop growth rate were among the growth indices showing strong responses to interference. Interference seemed to disturb the time course of growth in a complex way. Allometric relationships between dry weight per plant and either leaf area per plant or tiller number per plant were also affected by interference. The species differed in their allometric responses, and the relationships between allometry and different treatment factors also varied. TABLE OF CONTENTS ABSTRACT TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES LIST OF APPENDICES LIST OF SYMBOLS ACKNOWLEDGEMENTS 1. INTRODUCTION 1 2. LITERATURE REVIEW AND THEORY 3 2.1 Significance of Crop Mixtures 3 2.2 Plant Interference and Competition 6 2.2.1 Terminology: Competition vs. Interference 6 2.2.2 Some Experimental Approaches to Study Interference 8 2.2.3 Indices of Interference and Combined Yield 12 2.3 Quantitative Yield-density Relationships 19 2.3.1 Monoculture Models 20 2.3.2 Mixture Models 24 2.4 Plant Growth Analysis 27 2.5 Allometric Plant Relationships 29 3. MATERIALS AND METHODS 31 3.1. Experimental Design 31 3.2. Stand Establishment and Crop Management 31 3.2.1 Seeds 31 3.2.2 Land Preparation and Soil Fumigation 33 ii iv ix xii xv xvii xx V 3.2.3 Fertilizer Application 33 3.2.4 Layout and Planting 35 3.2.5 Post-emergence Crop Management 35 3.3 Harvesting and Data Collection 35 3.4 Data Analysis 37 3.4.1 Analysis of Variance 37 3.4.2 Simple and Multiple Regression Analysis 38 3.4.2.1 The Best Subset Multiple Regression 38 3.4.2.2. Yield-density Relationships 40 3.4.2.2.1 Monoculture Yield-density Model 40 3.4.2.2.2 Mixture Yield-density Model 42 3.4.2.2.2.1 Regression in One Stage 42 3.4.2.2.2.2 Regression in Two Stages 42 3.4.2.3' Differential Yield Responses 43 3.4.2.4 Plant Growth Analysis 43 3.4.2.5 Allometric Plant Relationships 44 3.4.2.5.1 Monoculture Allometry 44 3.4.2.5.2 Mixture Allometry 45 3.5 Statistical and Biological Considerations in Modelling 45 3.5.1 Statistical Considerations 45 3.5.2 Biological Considerations 51 4. RESULTS 52 4.1 Analysis of Variance 52 4.1.1 Homogeneity of Variance Test 52 4.1.2 Univariate Analysis of Variance 52 4.1.2.1 The Overall ANOVA Results 54 4.1.2.2 Orchardgrass Yield Responses 64 vi 4.1.2.2.1 Orchardgrass Shoot Dry Weight per Plant 64 4.1.2.2.2 Orchardgrass Shoot Biomass Density 64 4.1.2.3 Timothy Yield Responses 67 4.1.2.3.1 Timothy Shoot Dry Weight per Plant 67 4.1.2.3.2 Timothy Shoot Biomass Density 67 4.1.2.4 Total Mixture Shoot Biomass Density 72 4.1.2.5 Orchardgrass: Other Variates 72 4.1.2.6 Timothy: Other Variates 88 4.2 Best Subset Multiple Regression Analysis 104 4.2.1 Orchardgrass Best Subset Models 104 4.2.2 Timothy Best Subset Models 108 4.2.3 Total Mixture Best Subset Models 112 4.3 Yield-density Relationships 114 4.3.1 Monoculture Yield-density Relationships 114 4.3.1.1 Role of the Exponent in Describing Monoculture Yield-density Relationships 114 4.3.1.2 Orchardgrass Monoculture 117 4.3.1.3 Timothy Monoculture 119 4.3.2 Mixture Yield-density Relationships 125 4.3.2.1 Variation in Model Parameters and Mixture Yield-density Relationships 125 4.3.2.2 Comparison between Regresssion Approaches Describing Mixture Yield-density Relationships 128 4.3.2.3 Mixture Yield-density Relationships: Orchardgrass 131 4.3.2.4 Mixture Yield-density Relationships: Timothy 132 4.3.2.5 Species Relative Competitive Abilities 143 4.3.2.6 Differential Yield Responses 145 v i i 4.4 Plant Growth Analysis 158 4.4.1 Orchardgrass: Growth Indices 158 4.4.1.1 Orchardgrass: Leaf Area Index 158 4.4.1.2 Orchardgrass: Leaf Area Ratio 159 4.4.1.3 Orchardgrass: Absolute Growth Rate 159 4.4.1.4 Orchardgrass: Relative Growth Rate 166 4.4.1.5 Orchardgrass: Crop Growth Rate 166 4.4.1.6 Orchardgrass: Unit Leaf Rate 171 4.4.2 Timothy: Growth Indices 171 4.4.2.1 Timothy: Leaf Area Index 171 4.4.2.2 Timothy: Leaf Area Ratio 176 4.4.2.3 Timothy: Harvest Index 176 4.4.2.4 Timothy: Absolute Growth Rate 181 4.4.2.5 Timothy: Relative Growth Rate 181 4.4.2.6 Timothy: Crop Growth Rate 181 4.4.2.7 Timothy: Unit Leaf Rate 188 4.5 Allometric Plant Relationships 188 4.5.1 Allometric relationships: Monocultures 191 4.5.2 Allometric relationships: Mixtures 196 5. DISCUSSION 6. CONCLUSIONS 7. LITERATURE CITED 8. APPENDICES v i i i 202 217 220 227 LIST OF T A B L E S 3.1 Experimental design 3.2 Soil chemical analysis 3.3 Primary data: Measured and derived quantities 3.4 Data transformations for the best subset multiple regression analysis 3.5 Growth indices: symbols and definitions 4.1 Summary of homogeneity of variance test: Percentage of variates non-significant (P<0.05) as influenced by treatment factors 4.2 Summary of the overall ANOVA results: Percent frequency of significant variance ratios (P<0.05) for primary variates and ratio indices in orchardgrass 4.3 Summary of the overall ANOVA results: Percent frequency of significant variance ratios (P<0.05) for primary variates and ratio indices in timothy 4.4 Summary of ANOVA results: Variance ratios for the effects of total population density and mixture proportions on orchardgrass shoot dry weight per plant 4.5 Summary of ANOVA results: Variance ratios for the effects of total population density and mixture proportions on orchardgrass shoot biomass density 4.6 Summary of ANOVA results: Variance ratios for the effects of total population density and mixture proportions on timothy shoot dry weight per plant 4.7 Summary of ANOVA results: Variance ratios for the effects of total population density and mixture proportions on timothy shoot biomass density 4.8 Summary of ANOVA results: Percent sum of squares for the effects of total population density and mixture proportions on total mixture shoot biomass density 4.9 Summary of ANOVA results: Variance ratios for the effect of total population density and mixture proportions on other orchardgrass variates 4.10 Summary of ANOVA results: Variance ratios for the effects of total population density and mixture proportions on other timothy variates 4.11a Parameters and regression statistics for the best subset models in orchardgrass 4.1 lb Parameters and regression statistics for the best subset models in orchardgrass 106 4.12a Parameters and regression statistics for the best subset models in timothy 109 4.12b Parameters and regression statistics for the best subset models in timothy 110 4.13 Parameters and regressionstatistics for the best subset model for total mixture shoot biomass density 113 4.14 Parameters and regression statistics for orchardgrass monoculture yield-density models 118 4.15 Parameters and regression statistics for timothy monoculture yield-density models 124 4.16 Parameters and regression statistics for orchardgrass mixture yield-density models based on regression in stages 129 4.17 Parameters and regression statistics for timothy mixture yield-density models based on regression in stages 130 4.18 Relative sensitivity of orchardgrass and timothy to intraspecific and interspecific interference 144 4.19a Parameters and regression statistics for the best subset allometric model for orchardgrass in monoculture (leaf area per plant) 192 4.19b Parameters and regression statistics for the best subset allometric model for orchardgrass in monoculture (tiller number per plant) 193 4.20a Parameters and regression statistics for the best subset allometric model for timothy in monoculture (leaf area per plant) 194 4.20b Parameters and regression statistics for the best subset allometric model for timothy in monoculture (tiller number per plant) 195 4.21a Parameters and regression statistics for the best subset allometric model for orchardgrass in mixture (leaf area per plant) 197 4.21b Parameters and regression statistics for the best subset allometric model for orchardgrass in mixture (tiller number per plant) 198 4.22a Parameters and regression statistics for the best subset allometric model for timothy in mixture (leaf area per plant) 199 4.22b Parameters and regression statistics for the best subset allometric model for timothy in mixture (tiller number per plant) x i i LIST OF FIGURES 3.1 Residual us. predicted plot of orchardgrass shoot dry weight per plant (untransformed y) 47 3.2 Residual us. predicted plot of orchardgrass shoot dry weight per plant (log e y) 49 4.1 Time course of orchardgrass shoot dry weight per plant 65 4.2 Time course of orchardgrass shoot biomass density 68 4.3 Time course of timothy shoot dry weight per plant 70 4.4 Time course of timothy shoot biomass density 73 4.5 Time course of total shoot biomass density in the mixtures 75 4.6 Time course of orchardgrass dry weight per tiller 78 4.7 Time course of orchardgrass tiller number per unit land area 80 4.8 Time course of orchardgrass tiller number per plant 82 4.9 Time course of orchardgrass leaf area per plant 84 4.10 Time course of orchardgrass leaf area per tiller 86 4.11 Time course of timothy dry weight per tiller 89 4.12 Time course of timothy panicle dry weight per plant 91 4.13 Time course of timothy panicle dry weight per tiller 93 4.14 Time course of timothy tiller number per unit land area 95 4.15 Time course of timothy tiller number per plant 97 4.16 Time course of timothy leaf area per plant 99 4.17 Time course of timothy leaf area per tiller 101 4.18 Effect of different exponent values on the form of monoculture yield-density relationships 115 4.19 Fitted curves for monoculture shoot dry weight per plant in orchardgrass and timothy 120 4.20 Fitted curves for monoculture shoot biomass density in orchardgrass and timothy 122 x i i i 4.21 Effects of variation in model coefficients (bjj/bjj) on the form of mixture yield-density relationships in replacement diagrams 126 4.22 Replacement diagrams for shoot biomass density of each species and for the combined mixtures at 37 days after planting 133 4.23 Replacement diagrams for shoot biomass density of each species and for the combined mixtures at 48 days after planting 135 4.24 Replacement diagrams for shoot biomass density of each species and for the combined mixtures at 63 days after planting 137 4.25 Replacement diagrams for shoot biomass density of each species and for the combined mixtures at 77 days after planting 139 4.26 Replacement diagrams for shoot biomass density of each species and for the combined mixtures at 90 days after planting 141 4.27 Differential yield responses for component species and combined mixtures at 1/6 relative total population density 146 4.28 Differential yield responses for component species and combined mixtures at 2/6 relative total population density 148 4.29 Differential yield responses for component species and combined mixtures at 3/6 relative total population density 150 4.30 Differential yield responses for component species and combined mixtures at 4/6 relative total population density 152 4.31 Differential yield responses for component species and comined mixtures at 5/6 relative total population density 154 4.32 Differential yield responses for component species and combined mixtures at 6/6 relative total population density 156 4.33 Time course of leaf area index in orchardgrass 160 4.34 Time course of leaf area ratio in orchardgrass 162 4.35 Time course of absolute growth rate in orchardgrass 164 4.36 Time course of relative growth rate in orchardgrass 167 4.37 Time course of crop growth rate in orchardgrass 169 4.38 Time course of unit leaf rate in orchardgrass 172 4.39 Time course of leaf area index in timothy 174 4.40 Time course of leaf area ratio in timothy 177 4.41 Time course of harvest index in timothy 179 4.42 Time course of absolute growth rate in timothy 4.43 Time course of relative growth rate in timothy 4.44 Time course of crop growth growth rate in timothy 4.45 Time course of unit leaf rate in timothy X V LIST OF APPENDICES 8.1 Allometry and treatment responses 227 8.1.1 Allometry in monocultures 227 8.1.2 Allometry in mixtures 229 8.2 Summary of ANOVA results: Primary variates 8.2.1 Variance ratios for the effects of total population density and mixture proportions on orchardgrass dry weight per tiller 232 8.2.2 Variance ratios for the effects of total population density and mixture proportions on orchardgrass tiller number per unit land area 233 8.2.3 Variance ratios for the effects of total population density and mixture proportions on orchardgrass tiller number per plant 234 8.2.4 Variance ratios for the effects of total population density and mixture proportions on orchardgrass leaf area per plant 235 8.2.5 Variance ratios for the effects of total population density and mixture proportions on orchardgrass leaf area per tiller 236 8.2.6 Variance ratios for the effects of total population density and mixture proportions on timothy dry weight per tiller 237 8.2.7 Variance ratios for the effects of total population density and mixture proportions on timothy panicle dry weight per plant 238 8.2.8 Variance ratios for the effects of total population density and mixture proportions on timothy panicle dry weight per tiller 239 8.2.9 Variance ratios for the effects of total population density and mixture proportions on timothy tiller number per unit land area 240 8.2.10 Variance ratios for the effects of total population density and mixture proportions on timothy tiller number per plant 241 8.2.11 Variance ratios for the effects of total population density and mixture proportions on timothy leaf area per plant 242 xvi 8.2.12 Variance ratios for the effects of total population density and mixture proportions on timothy leaf area per tiller 243 cy 8.3 Goodness of fit (I ) and statistical significance for mixture models based on multiple regression analysis in one stage at individual total population density 244 8.4 Summary of ANOVA results: Derived quantities 8.4.1 Variance ratios for the effects of total population density and mixture proportions on Leaf Area Index in orchardgrass 245 8.4.2 Variance ratios for the effects of total population density and mixture proportions on Leaf Area Ratio in orchardgrass - 246 8.4.3 Variance ratios for the effects of total population density and mixture proportions on Leaf Area Index in timothy 247 8.4.4 Variance ratios for the effects of total population density and mixture proportions on Leaf Area Ratio in timothy 248 8.4.5 Variance ratios for the effects of total population density and mixture proportions on Harvest Index in timothy 249 x v i i LIST OF SYMBOLS Symbol Name or description a Intercept in yield-density regressions a' Allometric coefficient A Aggressivity coefficient AGR Absolute growth rate ANOVA Analysis of variance b Coefficient in yield-density regressions b Allometric exponent, the ratio of the relative growth rates of two plant characteristics BD Shoot biomass density c Coefficient in best subset multiple regressions used to analyze allometry CGR Crop growth rate C l Competition index C L . Confidence limit CP Mallows's CP statistic CR Competition ratio d days d In the definition formulae for growth indices, d denotes a first derivative (Table 3.5). d Coefficient in best subset multiple regressions used to analyze allometry D Total mixture population density DAP Days after planting d. f. Degrees of freedom e base of natural logarithms e, c' accounts for residual variation in yield in the allometric regressions x v i i i g Coefficient in best subset multiple regressions used to analyze allometry H Harvest index i Subscript indicating plant species (e.g. orchardgrass) IC Index of competition I Index of multiple determination j Subscript indicating plant species (e.g. timothy) k Crowding coefficient k Yield advantage (product of crowding coefficients) L Crop yield loss L Species land equivalent ratio LAI Leaf area index LAR Leaf area ratio LA/PL Leaf area per plant LA/TL Leaf area per tiller LER Land equivalent ratio for a mixture loge Natural logarithm m Subscript indicating monoculture M Subscript indicating the particular monoculture which has the same population density as the total population density in a replacement series N Number of plants O Orchardgrass o Subscript indicating no interference within or between species, p Subscript indicating a plant part P Plot size Q Exponent controlling the form of the yield-density regression r Relative yield R Relative growth rate R Coefficient of determination RM Relative monoculture response RMS Residual mean square RSS Residual sum of square RX Relative mixture response RYT Relative yield total t Time after planting T Timothy TLN/A Tiller number per land area TLN/PL Tiller number per plant ULR Unit leaf rate VR Variance ratio W Shoot dry weight per plant WPAN Panicle dry weight per plant W/PL Shoot dry weight per plant W/TL Dry weight per tiller WPAN/PL Panicle dry weight per plant WPAN/TL Panicle dry weight per tiller x Subscript indicating mixture X Population density, plants per land area y Yield per plant Y Total yield per land area, or a yield variable XX ACKNOWLEDGEMENTS I wish to thank my supervisor, Dr. Peter A. Joliiffe for his encouragement, moral support and helpful criticism throughout this program. This thesis owes a great debt to his enthusiasm, understanding, patience and willingness to spend long hours working with me on this project. I would also like to express my sincere thanks to the members of my supervisory committee, Dr. V. C. Runeckles, Dr. F. B. Holl and Dr. R. A. Turkington for their willingness to assist me at all times. I wish to acknowledge Dr. F. D. Moore, Department of Horticulture, Colorado State University, who suggested the use of the best subset multiple regression procedure. I also wish to express my special thanks to Dr. G. W. Eaton for his valuable advice on statistical analysis and many hours of stimulating discussion on design and analysis of experiments. I would like to thank Dave McArthur, Andrew Chow, Helen Evans for their help in collecting data. The assistance provided by Peter Garnett, Brian McMillan, Ashley Herath, Al Neighbour, Leo Pelletier, and Dave Armstrong is highly appreciated. I am grateful to my collegues Dr. David Ehret, Dr. David Kristie, Akwilin Tarimo, Paul Liu and Grace Mchaina for their support during my studies. The Canadian Commonwealth Scholarship Committee, the Natural Sciences and Engineering Research Council of Canada, the Marathwada Agricultural University, India, and the Government of India are gratefully acknowledged for their financial assistance. Finally, I wish to acknowledge the support and forebearance of my wife, Neeta and our children, Rashmi, Anjali and Monica which made this study possible. 1 1. I N T R O D U C T I O N It is a common observation that the growth of a plant can be influenced strongly by the presence of neighbouring plants. Such interference has widespread ramifications in biology. Competition among neighbours has long been considered to be a major force directing evolutionary change. Competition also contributes to successional relationships in plant populations. The main focus of this thesis, however, is on relatively short-term responses of plants to their neighbours. In particular, it deals with the effects of interference within and between species on the course of plant growth and on the agricultural productivity of plant populations. In addition to such physiological and agricultural issues, a major theme in this work has been the effort to advance analytical techniques which can be used to explore the origins of yield responses in plant populations. My specific experimental work followed growth and productivity in monocultures and mixtures of orchardgrass (Dactylis glomerata L.) and timothy (Phleum pratense L.). It was an intensive study carried out during one growing season, and the specific objectives of the research were: (i) to determine the relative strengths of intraspecific and interspecific interference on the yield (shoot dry weight per land area) of orchardgrass and timothy; (ii) to determine whether other growth characteristics of orchardgrass and timothy responded to intraspecific and interspecific interference in the same way as yield response; (iii) to determine the relative importance of treatment factors (time after planting, total population density, species proportions in the populations), and interactions among factors, on the growth and yield of each species; (iv) to evaluate the contributions of each species to differential yield responses (e.g. 'overyielding' or 'underyielding') in mixtures; 2 (v) to assess in detail the timing and physiological significance of growth responses in each species to experimental treatments using modern techniques of plant growth analysis; (vi) to determine the effects of experimental treatments on allometric relationships in each species During the time when this research was carried out, considerable attention was given by many scientists to the problems of how to study and interpret plant interrelationships in mixtures. As part of this effort, a final aim of the research was: (vii) to improve some of the methods by which plant responses to interference can be analyzed. 3 2. LITERATURE REVIEW AND THEORY 2.1 Significance of Crop Mixtures A crop mixture exists where two or more crops are grown simultaneously on the same land area. Crop mixtures are grown extensively throughout the world, although sole crops (monocultures) predominate where advanced crop production techniques are widely available. In temperate zones, mixed pastures and forage crops are the most common types of mixed crops. Two systems for growing crop mixtures, 'mixed cropping' and 'intercropping', are traditional practices of subsistence farmers in many subtropical and tropical countries. Intercropping refers to the practice of growing two or more different crops simultaneously in rows on the same land area. Mixed cropping is similar to intercropping, but it lacks a distinct row arrangement (Andrews and Kassam 1976; Freyman and Venkateswarlu 1977; Willey 1979a). Although the terms have sometimes been used synonymously, the distinction between intercropping and mixed cropping is important because the spatial arrangement of component species in mixtures may have a large influence on mixture yield responses (Andrews 1972; Freyman and Venkateswarlu 1977; Willey 1979b). In intercropping and mixed cropping the component species are not necessarily sown at the same time, and their harvest times may be different, but they are 'simultaneous' for a significant part of their growing season (Willey 1979a). Also, intercropping patterns vary according to the temporal and spatial arrangements of component species. Accordingly, several classes of intercropping can be distinguished: row intercropping, strip intercropping and relay intercropping (Andrews and Kassam 1976), and patch intercropping (Papendick et al. 1976). Crop mixtures usually involve different crop species (occasionally different cultivars), but sometimes mixtures are formed from different age classes of the same genotype (Trenbath 1974). There are large differences 4 among different crop mixtures with respect to the intimacy of individual plants or plant groups, the relative times of planting of the mixture components, the duration of mixing and the spatial arrangement of component species (Okigbo 1979). Although intercropping predominates in regions which are technologically less advanced than North America or Europe, it is not necessarily a primitive agricultural practice. The management of mixed crops can require considerable expertise because they are inherently more complex than monocultures. Crop mixtures are believed to have the potential to yield substantially more than monocultures on an equivalent land area basis (Trenbath 1974; Andrews and Kassam 1976). Among the reasons for this potential are the possibility that competition between different species may sometimes be less than within species. A similar idea is that a crop mixture may have the ability to more fully exploit environmental resources, because of niche differentiation among different species, than can a single species. The advantages and disadvantages of mixing crop species have been reviewed in detail by several researchers (Aiyer 1949; Webster and Wilson 1966; Wrigley 1969; Norman 1974; Okigbo 1978, 1979). Several reviews have emphasized the potential benefits of using mixtures in agriculture (Trenbath 1974, 1976; Willey 1979a, 1979b). Depending on context, I will use the term 'yield' in different ways. In discussions of agricultural productivity 'yield' will be used to describe some measure of agriculturally significant output. For example, in the studies of orchardgrass and timothy, 'yield' will specify shoot dry weight per unit land area. When regression models are being discussed, 'yield' will sometimes also be used in a more general way to refer to the dependent variable. I will use the expression 'differential yield response' to describe cases where yield per land area in a mixed crop differs from the combined yield obtained from component crops grown in monocultures with the same land area allocated per species. 'Overyielding' will indicate a differential yield response in which the mixed crop yield 5 surpasses the corresponding monoculture yields, and 'underyielding' will indicate the reverse. The literature reflects a rapidly growing research interest in several aspects of crop mixtures during the past decade. Studies of crop mixtures have become a major focus of research on crop production systems (Mead 1979; Mead and Riley 1981) and this trend is evident through recent annual reports of international agricultural research centres: The International Crops Research Institute for the Semi-Arid Tropics (ICRISAT), India; The International Rice Research Institute, Philippines; Centro Internacional de Agricultura Tropical, Colombia; International Institute of Tropical Agriculture, Nigeria. In addition, there have been several major conferences on intercropping: the Multiple Cropping Symposium at the American Society of Agronomy Annual Meeting, Tennessee, 1975; the Symposium on Intercropping in Semi-Arid Areas, Tanzania, 1976; the Symposium on Intercropping with Cassava, Philippines, 1978; the International Workshop on Intercropping, ICRISAT, India, 1979 (Mead and Riley, 1981). Despite such attention, little is known about the specific causes of differential yield responses of mixtures and how the beneficial effects of growing crops in mixtures could be improved. Some attempts have been made to provide a deeper understanding of mechanisms of plant interactions which result in underyielding and overyielding in mixtures (e.g. Trenbath 1974, 1976). The plant species used for my particular research, orchardgrass {Dactylis glomerata L.) and timothy (Phleum pratense L.) were chosen arbitrarily. They are both widespread temperate grasses, and both have the Cg pathway of photosynthetic C O 2 assimilation. They commonly coexist in pastures and meadows. Orchardgrass is relatively deep-rooted and is relatively shade-tolerant. Timothy is shallow-rooted and may be uprooted by grazing or by frost heaving. Both are considered to be desirable forage species. 6 2.2 Plant Interference and Competition 2.2.1 Terminology: Competition vs. Interference 'Competition' is a common but often misused term, and this has led to some confusion in different fields of biology. In plant ecology the term competition has had different meanings to different plant scientists. The term has been criticised because of its association with day to day human activities and because of its lack of an independent scientific meaning (Harper 1961). Hall (1974a) criticised the term on the basis that it is often used to describe ecological and agronomic phenomena in a rather loose manner with little scientific foundation. A few attempts have been made to review and formalize the definition of competition for plant scientists. This was discussed in detail by Clements et al. (1929) in relation to plant ecology and by Milne (1961) in the context of animal studies. An early and clear definition of plant competition was given by Clements et al. (1929). He wrote: "Competition is purely a physical process. With few exceptions, such as the crowding of tuberous plants when grown too closely, an actual struggle between competing plants never occurs. Competition arises from the reaction of one plant upon the physical factors about it and the effect of the modified factors upon its competitors. In the exact sense, two plants, no matter how close, do not compete with each other so long as the water content, the nutrient material, the light and the heat are in excess of the needs of both. When the immediate supply of a single necessary factor falls below the combined demands of the plants, competition begins". While Clements's definition of competition is straightforward, it should be noted that it is not intended to embrace all the mutual influences of plants growing together. Milne (1961) considered the concept of animal competition given by Clements and Shelford (1939) as being more acceptable. However, he slightly modified their original definition as: "Competition among animals is the endeavour of two (or more) animals to gain the same particular thing, or to gain the measure each wants from the supply of a thing when that supply is not sufficient for both (or all)". 7 There is a clear similiarity between these concepts of plant competition and animal competition. Accordingly, these concepts were fused to encompass competition among living organisms including both plants and animals (Donald 1963): "Competition occurs when each of two or more organisms seeks the measure it wants of any particular factor or thing and when the immediate supply of the factor or thing is below the combined demand of the organisms". All these definitions convey the sense that organisms compete for something, and the thing which is competed for might be termed a 'resource'. Again, the definitions of competition do not attempt to account for all mutual relationships among neighbouring organisms. Non-competitive relationships among plants, for example, include the direct stimulation of one species by another (e.g. the nitrogen fixed by a legume becoming available to non-legume) or the protection of one species by another (e.g. a coarse, unpalatable species in a pasture may prevent neighbouring species from being grazed). Having used the term competition in the specific sense of competition for resources, it is useful to have a more general term to describe the overall set of mutual relationships among neighbours, including both competitive and non-competitive interactions. Harper (1961) proposed the term 'interference' for this purpose. He defined plant interference as: "the response of an individual plant or plant species to its total environment as this is modified by the presence and/or growth of other individuals or species". Thus, competition for resources is one component of interference which may exist in plant mixtures, although it often seems to be the predominant component. In recent years, the term 'interference' has gained wide recognition in the scientific community (Hall 1974a and 1974b; Trenbath 1974; Radosevich and Holt 1984; Jolliffe et al. 1984). De Wit (1960) considered plant competition in terms of competition for the same or different 'space'. He used the term 'space' to express the region in the vicinity of plants in which all biological resources are embedded. He suggested that the description of space in terms of individual components is not necessary, always inaccurate and therefore inadvisable. The notion of competition for different space, however, is contradictory to the 8 above definition of competition, and De Wit's use of the term competition is considered inappropriate (Donald 1963; Hall 1974a). Hall (1974a) suggested that attempts to partition 'space' into particular growth factors may be fruitful because they may lead to a greater understanding of processes governing mutual influences between plants. He distinguished 'competitive' and 'non-competitive' interference, the latter type being often overlooked in experimental studies. Another term, 'interaction', is also used in the literature to mean all mutual influences of one species on another (Trenbath 1976; Minjas 1982). This term, therefore, closely corresponds to the term interference. Because 'interaction' is commonly used in statistical interpretations to mean interactions among treatment factors, I will not use the term interaction when interference is suitable. 2.2.2 Some Experimental Approaches to Study Interference Considering the diversity of biological issues pertaining to plant interference, it is not surprising that many approaches have been used in experiments on interference. Here, I will concentrate on approaches which provide insight on the relative strengths of intraspecific and interspecific interferences, on the relationship between interference and differential yield responses, and on the impact of interference on plant performance. In 'additive series' (Donald 1958) experiments, an 'indicator' or 'test' species grown at a fixed plant population density is supplemented with a 'competitor' species (usually a weed species) at a range of population densities. The additive series approach is widely used for evaluating crop-weed interference and crop losses due to weeds (Zimdahl 1980; Silvertown 1982; Radosevich and Holt 1984). In that context, the additive series approach is quite appropriate. It has the advantage of simulating the situation of a crop at fixed population density infested with weeds under agronomically realistic circumstances of relative population densities (i.e. crop population density > weed population density). Results from additive series experiments can be used to estimate threshold population 9 densities at which crop losses justify weed control measures. The interpretation of competitive relationships using results from the additive series approach, however, has difficulties. This is because the changes in species proportions and total population density are confounded in an additive design (Harper 1977; Firbank and Watkinson 1985). Another widely used experimental approach for research on plant interference is the 'replacement series' approach (de Wit 1960). In a replacement series, the overall population density of experimental plots is kept constant while the proportion of the total density allocated to component species is varied from 0 to 100%. , Confounding effects of changing total population density and species frequency are largely avoided in replacement series experiments. Perhaps for this main reason the replacement series approach has been utilized by many plant scientists over the last 25 years. It has been extensively employed for the interpretation of interference responses through replacement diagrams and mathematically derived indices such as the relative crowding coefficient, and relative yield total (Willey 1979a; Snaydon 1979; Mead and Riley 1981). This experimental approach has also been used in animal ecology (Ayala 1971) and in animal production (Nolan and Connolly 1977). However, many questions have been raised recently about the utility of this experimental approach. Mead (1979) claimed that most of the experiments designed to investigate crop responses to intercropping are complex and the effects of population densities and the spatial arrangements of component species in a mixtures are usually confounded. This problem applies to both additive and replacement series experiments. Conventional methods of interpreting results from replacement experiments have been criticised, on the basis that the comparisons between actual and expected yield fails to properly assess the relative contributions of intraspecific and interspecific interference in the determination of mixture yield (Jolliffe et al. 1984). Data from monoculture control plots, grown at a range of population densities, were considered essential for the interpretation of interference in replacement series experiments (Jolliffe et al. 1984). An important shortcoming of single replacement series is their assessment of 10 interference at an arbitrarily chosen total population density. The importance of varying both total population density and frequency in mixtures has been emphasized recently by Firbank and Watkinson (1985). Single replacement series, in which total population density is held constant, cannot be used to draw inferences regarding the nature of competitive relationships between species, nor can they predict the outcome of competition when densities are unconstrained (Inouye and Schaffer 1981). Connolly (1986) observed a wide variation in outcome when several commonly used indices of interference were applied to results from four mixtures grown in replacement series. He concluded that the replacement series approach is frequently a misleading tool for research on mixtures. Clearly, where replacement or additive series designs are used, they must be used with caution. In this light, the experimental design I used in my study with orchardgrass and timothy incorporated both replacement and additive series approaches. The treatments were organized in a complete factorial experiment, however, as advocated by Mead (1979). This made it possible to undertake a broader range of comparisons than allowed by either replacement or additive series alone. A simplified version of the replacement series approach, known as 'mixture diallels' (Trenbath 1974) has been widely used by population geneticists for interference studies. In this case a set of genotypes is grown in 1:1 mixtures of all possible pairs, including all monocultures. The analysis of mixture diallels gives information on both the aggressiveness and the productivity of genotypes in mixtures compared with corresponding monocultures. The interpretation of mixture diallel experiments has been reviewed by Trenbath (1978). In recent years 'systematic experiments', originally devised by Nelder (1962) and modified by Bleasdale (1967a), have also been introduced into intercropping research. These experiments have the potential advantage of efficient use of experimental material, through the reduction of non-experimental areas (Mead 1979). Systematic experiments lack randomization, however, which is one of the important assumptions of ANOVA. The 11 systematic approach has been applied in intercropping research by several researchers (Huxley and Maingu 1978; Wahua and Miller 1978). The 'fan' (Nelder 1962) and the 'parallel row' approaches of Bleasdale (1967a) have good potential for intercropping research as they can accomodate a wide range of population densities without changing the pattern of arrangement. Population density and row spacing can also be varied by using the two way systematic approach of Putnam et al. (1985). Another systematic approach, the 'beehive' approach (Martin 1973), uses the honeycomb form of hexagonal lattices and maintains equal distance between neighbours. Interference can be further partitioned into above-ground and below-ground components. Donald (1958) and Aspinall (1960) made early attempts to separate root and shoot interference for light and nutrients using separating panels in pot cultures. Subsequent modifications of Donald's technique (e.g., Schreiber 1967; Snaydon 1979) allow either addition or substitution of the components. Hall (1974a; 1974b) has made a first attempt to integrate the complementary approaches to interference developed by Donald and de Wit. Subsequent modifications made it possible to vary different experimental factors separately including the size of the stand, the overall population density, the relative density of component species, and the relative density of species in soil and in aerial space (Snaydon 1979). Thus, Snaydon's (1979) row technique is a comprehensive approach which combines the complementary additive and replacement experiments used for the study of interference. The approach has not been found to be very practicable under field conditions, although it was extended to field research on intercropping in a study of root and shoot components of interference (Willey and Reddy 1981). Such experiments may provide physiological insight into the mechanism and sites of interference. The main drawbacks of the approach involve the microenvironmentaJ effects of partitioning aerial and soil zones and restricting interference to two dimensions (i.e., between either horizontal or vertical rows). 12 2.2.3 Indices of Interference and Combined Yield Several indices have been proposed for the study of interference and combined yield in mixtures. In describing these indices I will used notation defined in the List of Symbols. My notation sometimes differs from conventional notation because of the limited size of the available character set and because I attempted to avoid overlap among symbols. A widely used interference index in ecological studies involving a replacement series is the 'relative crowding coefficient' of de Wit (1960). For each species, the relative crowding coefficient can be calculated from an 'actual yield' per plant in a mixture compared to the 'expected yield' that would be achieved if the species experienced the same degree of interference as in monoculture (at the same population density as the total mixture population density). The relative crowding coefficient (k^) of the test species (i) in the presence of a competing species (j) in a replacement series is defined by: where Y denotes yield of a species per unit land area and X denotes species population density. In Eq. 2.1, and elsewhere in this thesis where multiple subscripts occur, they should be read separately. Also, for subscripts i and j which indicate the species involved, the first subscript denotes the species being evaluated (i.e. the test species) and the second subscript denotes the species which is the potential source of interference in the test species. Thus, Yjj is the yield of species i in mixture with species j while is its yield in monoculture (= Yj if only monocultures are being discussed). The absolute sizes of relative crowding coefficients can be used to infer the relative yield performance of species in a mixtures. If is less than 1.0 the test species has produced less yield than 'expected', and if it is more than 1.0 it has exceeded the 'expected' yield (Willey 1979a). The relative crowding coefficient was further examined in detail by Hall (1974a, 1974b). The index was found to be useful for interpreting competitive (2.1) 13 relationships among species in number of studies on plant mixtures. Jolliffe et al. (1984), however, indicated that the comparison between actual and 'expected' yields implicit in the relative crowding coefficient was inappropriate for the interpretation of the relative responses of plants to interference. Their criticisms, however, do not invalidate the use of 'expected' yields as a basis for the interpretation of differential yield responses of The yield advantage (k) of a binary mixture, which is an index of differential yield response, can be determined by the product of the relative crowding coefficients: If k is greater than 1.0 there is yield advantage, and if it is less than 1.0 there is yield disadvantage (Willey, 1979a). Mclntyre (in Donald 1963) suggested a simple approach to formulate the 'competition index' (CI) for competing species in a mixture. The basic process is the calculation of 'equivalence factors' for each species. For species i, the equivalence factor is the number of plants of species i which is equally competitive to one plant of species j. If a species has an equivalence factor of less than 1.0 it is more competitive than the other species on a per plant basis. The competition index is then calculated based on equivalence factors and the actual density of species in the mixtures as: CI = (Product of species equivalence factors)/(Product of species frequency) (2.3) When CI is less than 1.0 there is minimal competitive interference and the mixtures overyield. This index has been used in a number of cases to determine beneficial or detrimental effects of intercropping (Willey and Osiru 1972; Osiru and Willey 1972). However, it requires that the monocultures have to be present at a range of plant mixtures. (2.2) 14 population densities so that equivalent plant numbers can be estimated (Willey, 1979a). Although the concept is good, the competition index has limited practical utility and has given conflicting results (Donald 1963). 'Relative yield total' (RYT) is an index which assesses the relative performance of mixtures compared with the corresponding monocultures (de Wit and van den Bergh 1965). In a binary replacement series, the relative yield total is defined as sum of the relative yields (r) of the two species: RYT = rj + rj = (Yy / Yjj) + (Yjj / Yjj) (2.4) where Y again denotes yield per land area and the subscripts denotes the species. The relative yield total has been used to describe the nature of mutual interference in a mixed population. If RYT equals 1.0, this has been taken to imply that the species utilize the same 'space' (sensu de Wit), and this situation seems to be common in practice. If RYT is greater than 1.0, overyielding has occurred. This can result when species occupy different niches in time or space or exhibit some kind of mutually beneficial relationship. Thirdly, if RYT is less than 1.0, one or both species are seriously affected by interspecific interference and this could indicate the occurrence of adverse interference (e.g. allelopathic effects). McGilchrist (1965) introduced two measures of interspecific interference between species in a 1:1 mixture. One is the increase in species i when grown in the presence of species j compared with its yield when grown in monoculture. The second is the depression interspecific interference causes in the yield of species j compared with its yield in monoculture. The two indices are defined by: 0.5 ( y i j - yjj) + 0.5 ( y j i - yg) (2.5) 1 5 which is an index of 'competitive advantage' of species i over species j, and: 0.5 (y h + y~) - 0.5 (yy + y j i) (2.6) which express 'competitive depression'. Here, y denotes yield per plant. McGilchrist and Trenbath (1971) subsequently revised those ideas and placed the index on a relative basis to express 'aggressivity' (A): Ay = 0.5 (y y / y i i) - (yjj/ y j j) (2.7) Aggressivity is related to relative yield total since: Ay = 0.5 (q - rj) (2.8) where rj and rj refers to the relative yields of species i and j grown in an equal mixture proportion. This equation can be generalized for any binary replacement series as: Ay = [yy /(yji x Xy)] - [ y j i / (j- x X^)] (2.9) where X denotes species population density. An aggressivity value of zero indicates that the component species are equally competitive. For other situations, both species have the same numerical value of A but the sign of the dominant species will be positive and that of the dominated negative. The greater the numerical value, the bigger the difference in competitive species ability (Willey 1979a). In practice, this index has seemed to be very unstable and did not necessarily give conclusive results in situations that gave different yield advantages (Hall 1974b; Willey and Rao 1980). A simple 'competitive ratio' (CR), as an index for relative competitive ability of species in a replacement series, was proposed by Willey and Rao (1980). This index is a further modification of the aggressivity index of McGilchrist and Trenbath (1971) and is defined as the ratio between the relative yields of component species: They claimed that the ratio provides an immediate measure of the degree of competition, in terms of the number of times that one species is more (or less) competitive than the other. In addition, they demonstrated that the competitive ratio varies with crop species, different intercropping systems and the population density at which they are grown. It was also suggested that the CR index could be useful in identifying plant characteristics which are associated with competitive ability and determining the competitive balance of a mixture that would give a maximum yield advantages (Willey and Rao 1980). An 'index of competition' (IC) was developed for wild oats (Avena fatua L.) competing with barley, wheat and flax in an additive series, using a simple regression procedure (Dew 1972). This index can be used for estimating the crop loss due to weeds when the population density of weed species and the expected weed-free yields are known. The method involves a simple regression of crop yield on the square root of the population density of an associated weed species: parameter b^ is the value of the regression coefficient of Y on the square root of the population density of the competing species. The coefficient fla;;) is of particular interest C R y ^ / r j (2.10) (2.11) where parameter aj is the value of the Y intercept (i.e., the weed-free crop yield) and because it is a direct index of yield response to the population density of the competitor. Using the regression parameters, the index of competition for a crop-weed mixtures was expressed as a simple ratio: I C ^ b y / a i (2.12) This index was found to be useful for studying the competitive abilities of crops against weed species in several studies (e.g. Dew and Keys 1976; Sibuga 1978; Minjas 1982). Once the basic yield-density relationship is known the predicted yield loss can be calculated as: where Ly denotes the yield loss per unit land area in species i due to the presence of species j. This information may be utilized to determine the economics of chemical weed control, provided the efficacy of the herbicide is known (Dew 1972). In recent years, the most widely used index for interpreting the impact of interference on the effectiveness of intercropping systems, is the 'land equivalent ratio (LER). Willey and Osiru (1972) defined LER as an index of combined yield: where L in this case denotes the land equivalent ratios for the component species and LER denotes the total land equivalent ratio for a particular intercropping combination. Thus, the LER is the relative land area required to produce the same yields in monocultures as are achieved by intercropping. Willey (1979a) summarized three major patterns of results L y - a j I C y X j (2.13) LER = Ly + L j i = (Yy /YH) + (Yji /Yjj) (2.14) 18 which are possible in binary replacement mixtures i.e. mutual inhibition, mutual cooperation and mutual compensation. Usually, these different patterns are distinguished by comparing the actual yields with 'expected' yields in replacement diagrams. However, Willey (1979a) argued that when the dominant species is yielding more than the expected, the comparison of intercrops with monocultures by other indices biased in favour of mixing. On the other hand if the dominant species is lower-yielding, the comparison is biased in favour of pure stands. He further argued that comparisons of intercrops with sole crops, if based on the sown proportions of the species, are biased in terms of the farmer's practical benefits. This problem is partly avoided in the LER concept (Willey and Osiru 1972). For a replacement series, the LER is identical to the Relative Yield Total. However, the LER concept can be applied to both an additive and replacement situations. In addition, LER is also more informative by incorporating the land equivalent ratios for component species, which is relevant to the central objectives of research on intercropping. Mead and Riley (1981) reviewed different methods of calculating the LER and stressed the need for research on statistical properties of using various different standardization methods and modifications of the LER index. As noted earlier, Jolliffe et al. (1984) criticized the conventional analysis of data from replacement series experiments. They suggested an alternative approach for assessing the relative contributions of intraspecific and interspecific interference to yield responses. The approach involves computing the 'projected' yield from the slope of the monoculture yield-density curve at = 0 as an initial reference point for yield comparisions. Two indices, the relative monoculture response (RM) and the relative mixture response (RX) were proposed as measures of plant response to intraspecific and interspecific interference, respectively: RMj = ( y i 0 - y i i ) / y i , (2.15) (2.16) where y j 0 denotes the projected yield per plant of species i in the absence of intraspecific interference, y^ denotes the actual monoculture yield, and yy denotes the actual mixture yield of species i in association with species j. All of the yield variates (y-1Q, y^ and yy) are determined at the same value of Xj. In Eq. 2.15 and 2.16 yields per land area can be substituted for yields per plant without affecting the values obtained. So far, these indices have not been extensively applied. In addition to the above indices, other measures of interference can be derived from a consideration of yield-density relationships, which will be discussed below. 2.3 Q u a n t i t a t i v e Y i e l d - d e n s i t y R e l a t i o n s h i p s As plants develop in a population they tend to interfere with each other's activities according to their age, size, functional capabilities and distance apart (Harper 1977). The distance between plants in a population is correlated with plant population density (e.g. in monoculture, the mean distance between plant centres will be 1/Xj^"^). For a given concentration of environmental resources, competition for resources will tend to intensify with increasing population density. Such interference usually results in either a reduced growth rate of individuals or an increased death rate of parts and/or whole individuals. The intensity and nature of the interference also may vary with plant age with changes in environmental factors. In practice, it has proved easier to measure how plants respond to interference than to determine the specific causes of such responses. By definition, the effects of plant population density on plant performance consititute interference. Hence the study of yield-density relationships is innately connected with the study of interference. From another viewpoint, the existence of large yield-density responses makes the selection of crop population density an important agricultural input. Because of such concerns, plant scientists have long been interested in 20 defining the quantitative relationships which exist between plant population density and crop yield, and early attempts were reviewed by Willey and Heath (1969). Quantitative yield-density relationships may be useful for determining optimum crop population density, comparing different cropping systems and for more generalized purposes in modelling crop production. In addition to describing the form of yield-density relationships, it is desirable that a mathematical model should also reflect the biological processes of interference and plant growth. 2.3.1 Monoculture Models Considerable success has been achieved in finding mathematical models that fit monoculture data from variety of experimental situations (Willey and Heath 1969; Thornley 1983). Attempts have also been made to provide a biological or mechanistic basis of these relationships. Holliday (1960b) first recognized that two basic types of monoculture yield-density relationships needed to be described by mathematical models. Many monoculture yield-density relationships seemed to be 'asymptotic' where, with increase in population density, yield per land area rose to a maximum and remained relatively constant at high population densities. In other cases a 'parabolic' relationship had been observed where yield per land area rose to a maximum but then declined at higher population densities. Total shoot biomass density or other measures of the yield of vegetative parts usually followed asymptotic relationships, while measures of total reproductive yield (e.g, grains and seeds) often followed parabolic relationships. However, such a correlation is not absolute (Willey and Heath 1969). The present review will emphasize one class of yield-density models, namely reciprocal equations, because these appear to have a good biological foundation as well as practical utility. Information on other classes of yield-density models can be obtained from the review by Willey and Heath (1969). 21 Japanese researchers (Kira et al. 1953, 1954, 1956; Ikusima et al. 1955; Hozumi et al. 1955; Hozumi et al. 1956; Shinozaki and Kira 1956; Koyama and Kira 1956; and Yoda et al. 1957) published a series of papers dealing with intraspecific interference and yield-density relationships of several crop species. A simple functional relationship between mean shoot dry weight per plant and population density was first recognized by Kira et al (1953). Shinozaki and Kira (1956) ultimately expressed the relationship in the form of a reciprocal model: y f 1 = aj + bjjXi (2.17) where a and b are model parameters defining the relationship between yj and Xj. The biological meaning of the parameters can be readily inferred. Parameter aj is the the inverse of the size of a plant at zero population density (i.e. in the absence of interference). Parameter b^ expresses the responsiveness of yj" * of a plant to changes in its own population density (i.e. it is an index of intraspecific interference). Eq. 2.17 was derived from a simple logistic curve and the law of constant final yield. This equation, however, can only describe an asymptotic yield-density relationship and is not applicable to a parabolic relationship. Holliday (1960a) also found a linear relationship between the reciprocal of yield per plant and population density in studies on a number of crop species. Holliday's equation is similar in structure to Shinozaki and Kira's (1956) reciprocal equation (Eq. 2.17), but it was based more directly on experimental results. Holliday also pointed out that this equation does not allow for a constant yield per plant at population densities too low for interference to occur. He interpreted yield per plant at zero population density to be the 'apparent maximum' yield per plant, which is equal to 1/a in Eq. 2.17. Jolliffe (1986) has pointed out that Eq. 2.17 can be derived directly from concepts of plant competition. Consider a plant whose growth is dependent on competition (i.e. y = 22 ffresources], where f denotes 'function of). The resources available to the plant will be related to the volume of the environment accessible to the plant (i.e. resources = f[height x area]). The mean area per plant is the inverse of plant population density (i.e. area = f[l/Xj]). Hence, growth per plant will be inversely related to plant population density (i.e. y = f[l/Xj]) and Eq. 2.17 is a sensible first approximation for such a relationship. It should be noted that this line of argument only considers competitive interference. If other types of interference are important, or if no competition is occurring, this type of model could be ineffective. Holliday (1960b) extended the above model to account for a parabolic yield-density situation by adding a quadratic term: y f 1 = aj + b ^ + bjjXj 2 (2.18) Although this equation is purely an empirical extension of the asymptotic model it often gives a good fit to experimental data (Willey 1982). Bleasdale and Nelder (1960) proposed a modification of Eq. 2.17, based on a generalization of Richard's growth curve (which is a superset of the logistic equation (Hunt 1982)). This equation gives a more universal fit to yield-density data: y{-Q = aj + b-xfi' (2.19) where Q and Q' are additional model parameters. This model can describe both the asymptotic and parabolic yield-density relationships. Where yield is asymptotic Q = Q' and where it is parabolic Q < Q'. The form of the yield-density relationship is determined by the ratio of Q to Q' rather than their absolute values. Bleasdale (1966b, 1967b), therefore, concluded that in practice the model could be simplified by taking the value of Q' as unity, so that Eq. 2.19 becomes: = «i + Vq (2.20) or Subsequently, this equation will be referred to as Bleasdale's simplified equation. The size of the exponent (-Q) determines the form of the yield-density model and this equation has proved very satisfactory in the practice. When Q = 1, Eq. 2.20 is the same as Eq. 2.17, the commonly found law of constant final yield is satisfied, and the model describes the asymptotic curves. When Q < 1 the equation models parabolic curves. Yield would theoretically increase monotonically with population density when Q > 1 (Watkinson, 1980). The biological meaning of parameter Q has proved difficult to express clearly. It has been related to the way in which plants utilize resources in their vicinity (Watkinson o 1980, 1984), and to the effectiveness in which 1/X measures space available per plant (Jolliffe 1986). It has been suggested that the parameters of Eq. 2.20 are difficult to interpret biologically except when Q = 1 (Willey and Heath 1969; Gillis and Ratkowsky 1978; Watkinson 1980). Accordingly, Eq. 2.20 was re.parameterized by Watkinson (1980, 1984) 7 i = y j ' U + X ^ 1 ' (2.21) In this form, a' = a"*b, b' = Q"1, and y' = a"*^. A biological meaning for these model parameters has been claimed (Watkinson 1980, 1984). Watkinson interpreted y' as the yield of an isolated plant, a' is the population density at which interference among individuals becomes appreciable and b' measures the effectiveness with which resources are taken up from the area around the plant. It is not yet clear whether this reparameterization offers significant statistical or interpretive advantages. I chose to use 24 the reciprocal model in the form of Eq. 2.20 because I found it convenient for interpretation and because that form can be extended simply to deal with interference in mixtures and with the evaluation of differential yield responses. 2.3.2 Mixture Models Wright (1981) and Spitters (1983a) extended Eq. 2.17 to model yield-density responses in binary mixtures. A model for mixture yield per plant of species i in association with species j was expressed by them as: yij"1 = a i + V i + bijXj <2-22) (Similarly, the model for mixture yield per plant of species j can be expressed as: yji"* = aj + byXj + bjjXj.) This model gave a good accounting of yield variation in binary mixtures (e.g. in grass mixtures, Wright (1981); in an intercropping situation, Spitters (1983a, 1983b)). Once these models were fitted, the values of the model parameters were used to interpret competitive relationships and to measure the degree of 'niche differentiation' (Spitters 1983a). Spitters (1983a) emphasized that the ratio of the model coefficients (e.g. bjj/bjj) offerred a direct assessment of the relative strengths of intraspecific and interspecific interference. The reparameterized version of the monoculture model (Watkinson (1981), Eq. 2.21) was also subsequently extended to account for interference by plants of a second species (Watkinson, 1981). The application of such models for the combined analysis of yield-density relationships and density-dependent mortality in mixtures was recently done by Firbank and Watkinson (1985). The model introduced by Firbank and Watkinson (1985) was a reparameterized version of Bleasdale's simplified model (Eq. 2.20) extended to account for interference in mixtures. Without reparameterization, that model is: 7f* = a i + biiXi + bijXj (2.23) As Jolliffe (1986) has noted, such a model could be extended further. For example, interaction terms such as bX^Xj, or higher order polynomial terms such as bXj , could be included. It is not yet clear whether such elaboration will be necessary, or whether Eq. 2.23 can by itself account for the majority of yield variation in mixtures. The conventional way of fitting Eq. 2.22 or 2.23 to experimental data has been the use of a normal multiple regression procedure. In that case, all of the monoculture and mixture data are incorporated in the analysis in one step. An alternative approach has been developed by Jolliffe (1986). He suggested that the Eq. 2.20 could first be established by regressing y m j against X^ to obtain Q, a-j and bjj. A second stage could then be performed using the mixture data by regressing the residual (y^"^-- yxj"^) against Xj (forcing the regression through zero), where in this case y m j"^ is the value predicted from Eq. 2.20 and y xj"^ is the transformed mixture yield per plant. The first stage in this approach, therefore, accounts for intraspecific interference. The second stage accounts for interspecific interference as a departure from the monoculture state (which presumably would exist if by were zero. Two potential advantages of the approach in stages are its utility in replacement series where there is matrix singularity between Xj and Xj, and its natural default to the monoculture condition when bjj or Xj go to zero. In my data analysis, both the conventional one-step multiple regression approach and the two stage approach were used and compared.-Jolliffe (1986) has recently demonstrated the applicability of such models to the interpretation of differential yield responses. Yield per unit land area is the product of plant population density and yield per plant: Y i = yft (2.24) 26 From Eq. 2.19 and 2.22 the models for yield per unit land area will be: Yj = Xj (aj + b ^ ) " 1 ^ (2.25) for monocultures and: Yy = Xj (aj + bjjXj + byXj)" 1^ (2.26) in mixtures. Let D be the total population density of a binary mixture (D = Xj + Xj). The evaluation of differential yield responses in mixtures is based on a comparison with the yield of each species in monoculture at density D, in terms of its proportion in the mixture (Xj/D). Hence the 'expected' yield, which provides the frame of reference for evaluating differential yield responses, is given by: Yj = (Xj/D) D (&i + b^D)" 1^ = Xj (aj + bjjD)- 1^ (2.27) Because in a mixture Xj equals D - Xj, Eq. 2.26 can be rearranged as: Yjj = Xj (aj + b u D + [by - bjjJXj)-1^ (2.28) Comparison of Eq. 2.27 and 2.28 indicates that deviations from the 'expected' yields is accounted for by the expression [bjj - bjj]Xj. This derivation therefore suggests that differential yield responses are driven directly by differences in the relative strengths of intraspecific and interspecific interference. 27 2.4 P l a n t G r o w t h A n a l y s i s Methods of plant growth analysis can help to assess the timing and sites of plant responses to experimental treatments during long-term experiments under field conditions. In addition, the general theory of plant growth analysis offers a conceptual framework for the interpretation of the structural and physiological origins of crop yield variations. Because of this, such methods have considerable potential application in studies of yield responses to population density and mixture treatments. The history and methodology of 'conventional' approaches to plant growth analysis, that is approaches revolving around concepts such as unit leaf rate, have been reviewed in the past decade by Causton and Venus (1981) and by Hunt (1982). Considerable attention has been given to problems and methods of fitting growth curves (Causton and Venus 1981; Hunt 1982). Parsons and Hunt (1981) introduced splined cubic polynomials, which were used to fit growth curves in my research. Cubic spline regressions provide a highly flexible, but entirely empirical, description of growth. Another important recent advance in 'conventional' plant growth analysis was the revised expression of relationships underlying crop growth rate by Warren Wilson (1981). That revision improved the framework for interrelating different growth responses, although Hardwick (1984) has noted that tautological difficulties can arise. Two other major approaches to plant growth analysis exist. The application of demographic analysis at the sub-organismal level of organization was introduced by Bazzaz and Harper (1977). Sub-organismal demographic analysis deals with the 'births' and 'deaths' of morphological components. Sub-organismal demographic analysis and 'conventional' plant growth were later linked by Hunt and Bazzaz (1980) and by Jolliffe and Courtney (1984). The final approach, yield component analysis, has been practiced for about the same length of time as 'conventional' plant growth analysis (more than 60 years). Yield component analysis deals with relationships between yield variation and variation in morphological features. Methods of yield component analysis were reviewed by Fraser and Eaton (1983) and continue to be refined (Eaton et al. 1986). A conceptual link between yield component analysis and conventional plant growth analysis was first established in 1982 (Jolliffe et al.). The subsequent work of Jolliffe and Courtney (1984) formally united all three areas of growth analysis in a common conceptual framework. Hence, the choice of which approach to use is not a choice between qualitatively different approaches. The methods used in my research fell within the 'conventional' approach because it was most relevant to my research objectives. Although plant growth analysis has been used extensively in studies on monocultures (ignoring the occurrence of weeds), its application to investigations of mixtures has been uncommon. Most studies on mixtures have emphasized the end result of the effects of interference, and not the morphological and physiological sequence of changes leading to the end result. Methods of plant growth analysis have recently been considered as a research tool in studies of crop-weed relationships (Radosevich and Holt 1984; Roush and Radosevitch 1985). Also, methods of plant growth analysis have begun to find application in studies of intercropping (Clark and Francis 1985; Gardiner and Craker 1981; Herbert and Litchfield 1984). 2.5 Allometric Plant Relationships Allometric relationships are growth relationships among different features of an organism. The particular form of any organism is, therefore, established through allometry. Allometry is often considered in terms of the growth relationship which exists between any part and the whole organism. In that sense, allometry provides a valuable bridge between different levels of organization. A general model for an allometric relationship between a part (ypj) and the whole (yj) was provided by Huxley (1932): y; = a y p i 6 (2.29) The allometric coefficient, a measures yj when y pj equals 1.0, and so it is highly dependent on the scales of measurement of yj and ypj. The allometric exponent, 6, is of greater interest, because it is the ratio of relative growth rates of yj and y pj (Whitehead and Myerscough 1962): b = Rj / Rpj (2.30) Jolliffe and Courtney (1984) demonstrated the involvement of b in growth relationships in additive systems. As a complex system grows, the contribution of each part to the whole is related to the fraction invested in each part and the allometric exponent relating each part to the whole. It may be noted that Eq. 2.20 and 2.29 provide two models for yj. There may be a tendency, therefore, to merge the two relationships and define a yield-density relationship for y pj from the parameters of the two models (Bleasdale 1966a, 1967b; Willey and Heath 1969). This approach has recently been extended to mixtures (Watkinson 1981). Some 30 consideration, however, suggests the need for caution. Eq. 2.19 and 2.28 could well be merged if they accounted for yield variation in the same way. Normally, however, there is unaccounted residual variation for each model, and the residuals would probably be different. In this thesis, a different approach was used for evaluating allometric responses to population densities and mixture proportions (Appendix 8.1). 31 3. MATERIALS AND METHODS 3.1 Experimental Design A 6 by 7 factorial experiment was conducted at the Totem Park Field Laboratory of the University of British Columbia during summer 1983. Plants were grown at six total population densities, ranging from approximately 650 to 4000 plants m (Table 3.1). At each density, seven mixtures of orchardgrass (Dactylis glomerata L.) and timothy (Phleum pratense L.) were established. Proportions of each species were varied from 0 to 100% in regular steps (Table 3.1). These 42 treatment combinations were arranged in a randomized complete block design in four blocks. This experimental approach encompasses both the replacement series (de Wit 1960) and the additive series (Donald 1963), which are standard experimental approaches used for research on plant interference. The gross plot dimensions for each replicated treatment were 3.9 m by 2.7 m. Within each plot, five sub-plots (0.30m by 2.7 m in dimension) were established and randomly selected for use at different harvests. 3.2 Stand Establishment and Crop Management 3.2.1 Seeds Seeds of orchardgrass cv. 'Sterling' and timothy cv. 'Climax' were obtained from Buckerfield's Ltd., Vancouver, British Columbia. Seed number per unit seed dry weight was determined from 10 random samples weighing one gram each. The mean values obtained were 1291 — 15 (SE) seeds g"* for orchardgrass and 2652 — 22 seeds g"* for timothy. A germination test was carried out under laboratory conditions on 10 random samples of 100 seeds per sample of each species. Seeds were placed on moist filter papers in Petri dishes at 22°C, and germination counts were made 5 days later when seeds had 3 2 Table 3.1 Experimental design R e l a t i v e Species proportions (orchardgrass: timothy) population d e n s i t y * 0:6 1:5 2:4 3:3 4:2 5:1 6:0 Population d e n s i t i e s ( p l a n t s m"^ ) 1/6 o»* 0 111 223 334 445 557 668 T 653 544 436 327 218 109 . 0 2/6 0 0 223 445 668 890 1113 1336 T 1307 1089 871 653 436 218 0 3/6 0 0 334 668 1002 1336 1670 2004 T 1960 1633 1307 980 653 327 0 4/6 0 0 445 890 1336 1781 2226 2671 T 2613 2178 1742 1307 871 436 0 5/6 0 0 557 1113 1670 2226 2783 3339 T 3267 2722 2178 1633 1089 544 0 6/6 0 0 668 1336 2004 2671 3339 4007 T 3920 3267 2613 1960 1370 653 0 * Maximum popu l a t i o n d e n s i t y (6/6) = 4007 p l a n t s m"^  f o r orchardgrass and 3920 p l a n t s m"2 f o r timothy ** 0 denotes orchardgrass cv. S t e r l i n g T denotes timothy cv. Climax Randomized Complete Block Design: 4 Blocks 42 Treatments (6 d e n s i t i e s X 7 mixtures) 5 Harvests (37, 48, 63, 77, and 90 days a f t e r p l a n t i n g ) reached maximum germination. The average per cent germination was 86 — 5% for orchardgrass and 82 — 9% for timothy. Seeding rates of 38 kg ha"''- for orchardgrass and 18 kg ha"''" for timothy were therefore used for the maximum population densities of the respective species in these studies. 3.2.2 Land Preparation and Soil Fumigation Land was prepared by disc ploughing followed by several harrowings in April to achieve a good tilth. On May 2, 1983, soil was fumigated with Dazomet (BASF Basamid Granular 90% ) at a rate of 400 kg ha"* to control weeds. Dazomet was evenly broadcast by hand on the experimental area followed by an immediate discing and soil compaction. Soil temperature on the day of application changed from 13.8°C (8 AM) to 15.7°C (4 PM) at 10 cm depth. Mean air relative humidity changed from 72% to 84% during the same period. On May 14, the land was cultivated to facilitate release of toxic herbicide fumes from the soil. To verify the disappearance of Dazomet, a germination test was carried out with cress {Lepidium sativum L. , a Dazomet-sensitive species) seed on composite samples collected from treated and untreated soil areas. When this test was performed in the last week of May, there was no evidence of an effect due to residual herbicide. 3.2.3 Fertilizer Application Soil samples were taken from the experimental area prior to planting. The soil was a sandy loam with pH of 6.08. A composite soil sample was analyzed by the Department of Soil Science, University of British Columbia. The results of the soil chemical analysis are presented in Table 3.2. Accordingly, a fertilizer application of 224 kg ha' 1 of Ammonium Phosphate (11:55:0), which corresponds to normal fertilizer recommendations, was administered to the plots and disced into the soil on May 30. Table 3.2 S o i l chemical analysis N u t r i e n t Approx. values in ppm (soil depth = 0 - 1 5 cm) N (as N0 3 ) 100 P 15 K 200 S0k 130 % total N 0 .21 35 3.2.4 Layout and Planting The field plots were laid out in four compact blocks, each block containing all 42 treatment plots. Pathways, 0.75 m wide between plots and 1.0 m wide between blocks, were kept to facilitate crop management. Each plot was identified with a labelled wooden stake. Planting was done on June 1, 1983, by broadcasting the requisite number of seeds on each plot followed by a gentle raking to mix seeds into the soil. Soil moisture at the time of planting was near field capacity. 3.2.5 Post-emergence Crop Management The crop approached full emergence by June 12, and emergence was uniform for both species. Due to the soil fumigation, there was minimal weed growth during the initial month of crop growth. After planting, weeds were controlled by hand pulling. The crop proved to be practically free from diseases and pests throughout the study. Rainfall was well distributed during the growing period; however, rainfall was supplemented with three irrigations provided by sprinkler during June (6th) and August (8th and 22nd). 3.3 Harvesting and Data Collection Shoot material was harvested from single quadrats, 0.3 m x 0.15 m in the center of each sub-plot, at 5 times during crop growth. Harvests were performed at 37, 48, 63, 77 and 90 days after planting (DAP). Shoots were cut to ground level and the harvested material was collected in labelled paper bags. The bags were transferred to the laboratory and stored in a cold room (4°C) until measurements were started. The shoot samples were separated into component species, and all measurements except dry weights were performed on the same day as the harvest. The primary data (Table 3.3) for each species were recorded on a per plot basis as well as a sub-sample basis. Leaf areas were obtained 36 Table 3.3 Primary data: Measured and derived quantities a) Measured q u a n t i t i e s : v a r l a t e sample s i z e Shoot dry weight, g per p lo t T i l l e r number " T i l l e r dry weight, g f i v e t i l l e r s Pan ic le dry weight, g " Leaf a rea , cm 2 " b) Derived q u a n t i t i e s : va r ia te symbol Shoot dry weight, g p l a n t - 1 W/PL Biomass Dens i ty , g m"2 BD T i l l e r dry weight, g t i l l e r - 1 W/TL Pan ic le dry weight, g p l a n t - 1 WPAN/PL Panic le dry weight, g t i l l e r - 1 WPAN/TL T i l l e r number m - ^ TLN/A T i l l e r number p l a n t - 1 TLN/PL Leaf area , p l a n t - 1 LA/PL Leaf a r e a , m2 t i l l e r - 1 LA/TL 37 using a LI-COR LI 3000 leaf area meter. Dry weights were obtained after samples were dried in a forced air oven at 70°C for five days. Primary data, also termed 'measured quantities' (Warren Wilson et al. 1986) or 'primary values' (Hunt 1982; Jolliffe et al. 1982), were transformed into 'derived quantities' (Warren Wilson et al. 1986) for subsequent growth analysis (Table 3.3) 3.4 Data Analyses The data were subjected to a series of different analyses to assess different aspects of growth and interference. Some of the analytical techniques had been applied to studies of growth and interference in previous research and are reviewed in detail by Mead and Riley (1981). Some of the following techniques, however, including the analysis of differential yield responses and the analysis of allometric changes, represent new approaches. 3.4.1 Analysis of Variance Univariate analysis of variance (ANOVA) was applied to the results for all primary data and all derived ratio quantities for each species. ANOVA was used to partition total sum of squares into components associated with different sources of variation using the multifactorial analysis of variance program package (MFAV) available through the Computing Center of the University of British Columbia. Separate ANOVAs were performed on data from each harvest. With quantitative factors, such as population density, the use of multiple range tests is invalid (Chew, 1976; Maindonald and Cox, 1984), and so such tests were not applied. In the ANOVA, the degrees of freedom and sum of squares were partitioned into linear, quadratic, and higher order components including interaction terms. This was used to provide a preliminary indication of quantitative responses to treatment factors. 38 ANOVA is based on several assumptions (Steel and Torrie 1980), including homogeneity of variance. In the ANOVA, data were subjected to Partitioned Layard's homogeneity of variance test. As detailed in the Results, most variates were found to be homogeneous in their variance in relation to both treatment factors (population density and mixture proportions). Where homogeneity was not found, data were transformed (Table 3.4) to improve homogeneity before the best subset multiple regression analyses, described below, were performed. 3.4.2 Simple and Multiple Regression Analysis Regression analyses were used to deepen the interpretation of the results by defining functional relationships among variables. Several different types of regression analyses were performed for different purposes. Best subset multiple regressions were developed for two purposes: (i) to assess the relative importance of different treatment factors, and (ii) to define allometric and treatment relationships underlying yield. Simple and multiple regression models were developed to define yield-density relationships of monocultures and mixtures. Finally, cubic spline regressions were developed to define the chronology of growth of different measured and derived quantities. 3.4.2.1 The Best Subset Multiple Regression In multiple regression, several independent variables are used to model a single response variable. Following ANOVA, the regression technique known as 'best subset multiple regression analysis' (Daniel and Wood 1971), was applied. This technique is based on 2 n possible equations from n candidate independent variables, and the BMDP9R statistical package program (Dixon 1983) was used to perform the regression analysis. The objective of the best subset multiple regression Table 3.4 Data transformations for the best subset multiple regression analysis Transformation Var iate Orchardgrass Timothy BD W/PL W/TL WPAN/PL WPAN/TL TLN/A TLN/PL LA/PL LA/TL LAI LAR log e (Y) log e (Y) log e (Y) log e (Y) log e (Y) log e (Y) Y log e (Y) 40 analysis in the present context was to evaluate the relative importance of different treatment factors in causing yield responses. The dependent variates (Y) included primary variates and derived quantities for each species (Table 3.3), the ratio growth indices for each species (BD, LAI, LAR, H defined in Table 3.5), plus the combined biomass density of mixtures. The potential independent variates included the time after planting (t, in days), orchardgrass population density (0, plants m ), timothy population density (T, plants m" o ), and the possible interactions between these treatment factors up to quadratic polynomials. Thus, the full regression model for each yield variate was: Y = bg + b-jt + b 2 0 + b 3T + b 4tO + b 5tT + bgOT +b ?tOT + b g t 2 + b g t 2 0 + b 1 Q t 2 T + b n t 2 O T + b 1 2 0 2 + b 1 3 t 0 2 + b 1 4 0 2 T + b 1 5 t 0 2 T + b 1 6 T 2 + b 1 ? t T 2 + b 1 8 O T 2 + b 1 9 t O T 2 (3.1) where bg.-.b^g denote the regression coefficients. 3.4.2.2 Yield-density Relationships 3.4.2.2.1 Monoculture Yield-density Model The monoculture yield-density relationship was modelled using the simple reciprocal model of Bleasdale (1966b, 1967b): i " Q = Km!*?* = «i + b i i X i (2.20) ' mi Bleasdale's simple model was fitted to mean shoot dry weights per plant from each harvest. Shoot dry weights ( y m j ) were initially transformed to twenty sets of ( y m j ) f o r values of Q ranging from 0.1 to 2.0 with the increment of 0.1. Twenty regressions (Eq. 2.20) were then performed for the transformed shoot dry weights using the Michigan Interactive Data Analysis System (MIDAS) program package available through the 4 1 Table 3.5 Growth indices: symbols and definitions Symbol Name D e f i n i t i o n f o r m u l a * U n i t s AGR A b s o l u t e growth r a t e AGR = (dW/cit) g d a y " 1 BD Shoot b iomass d e n s i t y BD = (NxW)/P) g m~2 CGR Crop growth r a t e CGR = (N/P)(dW/cit) g m"2 day - 1 H H a r v e s t index H = WPAN/W g g - 1 LAI L e a f a r e a Index LAI = N(LA/P) 2 -2 LAR L e a f a r e a r a t i o LAR = (LA/W) m 2 g "1 R R e l a t i v e growth r a t e R = ( i/W)(dW/dt) g g _ 1 day - 1 ULR U n i t l e a f r a t e ULR = (i/LA)(dw7dt) g n r 2 day' -2 * W d e n o t e s shoot dry we ight per p l a n t ; t d e n o t e s t i m e ; N d e n o t e s number o f p l a n t s ; P d e n o t e s p l o t s i z e ; LA d e n o t e s l e a f a r e a per p l a n t ; WPAN d e n o t e s p a n i c l e dry we ight per p l a n t 42 University of British Columbia Computing Centre (Fox and Guire 1976). The regression which gave the highest I (Ezekiel and Fox 1959) value was selected as an initial approximation of the best model. The model was further refined following twenty more (ymj)"^ transformations for values of the exponent in steps of 0.01 starting 0.1 below the Q obtained as the first approximation. The best model was then chosen on the basis of which regression gave the highest I value. 3.4.2.2.2 Mixture Yield-density Model 3.4.2.2.2.1 Regression in One Stage The additional interference which may occur from the associated species in a binary mixture can be taken into account by extending the monoculture model: Ey x i3-Q = n w Q = aj + byXj + byXj (2.23) As described above, twenty transformed sets of y xj"^ were established using values of Q from 0.1 to 2.0 in steps of 0.1. Those values were regressed in an ordinary multiple regression (Eq. 2.23) using the MIDAS package. Again, the model having the highest I 2 value was selected as an initial approximation of the best model. The model was further refined following twenty more (ymj) transformations for values of the exponent in steps of 0.01 starting 0.1 below the Q obtained as the first approximation. The best model was then chosen on the basis of which regression gave the highest I value. 3.4.2.2.2.2 Regression in Two Stages Data also were analyzed using an approach which develops the regression in two stages (Jolliffe 1986). The final model has the same form as Eq. 2.23, but Q, aj and bjj were fixed at values derived by fitting Eq. 2.20 to the monoculture yield-density data. Values for b- were then obtained from a linear regression of the residual: ' - y x i - ' " ^ m i - ' a S a m s t Xj. The relative effectiveness of the regression in two stages vs. the regression in one stage will be compared in the Results (Part 4.3.2.2). The regression in stages was found to be more successful on both statistical and conceptual grounds, and parameters Q, aj, b-j and by obtained using that model were applied in subsequent analyses. 3.4.2.3 Differential Yield Responses The differential yield response, that is the net loss or benefit obtained from growing species in mixtures compared to corresponding monocultures, is the most important consideration in intercropping research. The differential yield responses of each species and the combined mixture were assessed following Jolliffe's (1986) approach. According to Eq. 2.27, the expected yield per land area (where the differential yield response is zero) will be: Yj = X{ [aj + bjjD]- 1^ (2.27) However, the actual mixture yield per land area was modelled by: Y x i = Xj [aj + b uD + (b u - bij)X j]-1 /Q (2.28) The differential yield response of a species in mixture is therefore related to non-zero values of (bjj - by-). Differential yields for each species and for the combined species in mixtures were evaluated from the model parameters on this basis. 3.4.2.4 Plant Growth Analysis To provide more insight into the physiological basis for plant responses to treatments, the data were subjected to further analysis using modern techniques of plant 44 growth analysis. Relevant symbols and growth indices-are defined in Table 3.5. Curves describing growth over time were fitted to shoot dry weight per plant (W/PL) and leaf area per plant (LA/PL) using a cubic spline regression procedure (Jolliffe and Courtney 1984). The fitted curves were then used to compute the growth indices commonly used in conventional plant growth analysis (Hunt 1982).. For four growth indices [leaf area index (LAI), leaf area ratio (LAR), harvest index (H), and shoot biomass density (BD)], however, the results were expressed as mean ratios and progress curves were not computed. 3.4.2.5 Allometric Plant Relationships 3.4.2.5.1 Monoculture* Allometry The rationale and derivation of the following approach to the analysis of the effects of treatment factors on plant allometry is given in the Literature Review and in Appendix 8.1. The model evaluated for allometric relationships in monocultures was: log e(ym i) = loge(o') + b0loge{ym?i) + & 1tlog e(ym p i) + b 2 Xilog e (y m p i ) + 6 3 tX i log e (y m p i ) + C lloge(t) + c2log e(X i) + CgloggttXj) + loge(<?') (8.8) The best subset multiple regression approach, described in Part 3.4.2.1 above, was used to construct models in the form of Eq. (8.8) using the monoculture data from all harvests and population densities for each species. As before, the regression approach retained only those terms which made significant contributions to the final model. As detailed in Results, y m j was always taken to be shoot dry weight per plant; y p m j , however, varied for different models to investigate how different aspects of plant allometry were affected by time and population density. For both the orchardgrass and timothy results, y p m j was taken to be either leaf area per plant or tiller number per plant, in separate analyses. 45 3.4.2.5.2 Mixture Allometry As derived in Appendix 8.1, an extension of the above approach was applied to the analysis of allometric yield responses in mixtures. In this case, the treatment factors were taken to be time and the population densities of the two species: loge(yxi) = loge(o') + *>nloSefrxpi) + & 1 * ^ 0 ^ ) + 62xi loge^xpi) + 63Xj l 0MyXpi) + 64tXiloge(yXpi) + V X j ^ x p i ) + W ^ f r x p i ) + feytXjXjloggCyxpi) + CjloggOO + c2loge(Xi) + c3loge(Xj) + c4loge(tXi) + c5loge(tXj) + CgloggC^ Xj) + cylogettXjXj) + loge(e*) (8.9) Again, the best subset multiple regression method was used to develop models from the data. In all cases, yx^  was shoot dry weight per plant, and ypXj were the mixture equivalents of the variates analyzed in the previous section for the monocultures. These are new procedures for analyzing allometric changes in monocultures and mixtures, as will be reviewed in the Discussion. 3.5 Statistical and Biological Considerations in Modelling It will be evident from the previous sections that a prominent approach used in my analysis of experimental data involved the development of mathematical relationships between dependent and one or more independent variates. Where this has been done, I have tried to use both statistical and biological considerations while developing the various models. The following sections briefly outline those considerations. 3.4.1 Statistical Considerations The following statistical aspects were considered while fitting regression models: 1) The method of fitting regression models was based on the principle of the linear least squares. 46 2) The residual variation left after a model was fitted provided information about the appropriateness of the model. Hence, a detailed analysis of residuals was done at various stages of model development. Plots of residuals vs. other quantities were used to detect failures of the assumptions of the analysis (Fig. 3.1). When necessary, data transformations were performed to achieve random distribution of the residuals (Fig. 3.2). 3) Several statistics, including the-coefficient of determination (R ), residual mean square (RMS), Mallows's CP, and the index of multiple determination (I ), were used for model evaluations; however, the application of those statistics varied with the regression approach under consideration. Because Mallow's CP and the index of multiple determination have not been widely used, a brief description of them is given below. Predicted values obtained from a regression equation based on a subset of dependent variables are generally biased. A 'total squared error' (CP) was used as a criterion to judge the adequacy of an equation. CP measures the sum of the squared biases plus the squared random errors in Y at all data points. Hence, it is a simple function of the residual sum of squares from each fitting equation (Daniel and Wood, 1971). The best subset among possible multiple regressions was selected according to the regression which gave the lowest CP value. 9 9 The I statistic corresponds to a coefficient of determination (R ), but it applies to curvilinear regressions. In the reciprocal models for yield-density relationships, where the dependent variable (y) was transformed, the adequacy of the models in the original scale of y was judged by the I statistic (Ezekiel and Fox, 1959). The regression which gave o highest I was considered the best. Fig. 3.1 Residual vs. predicted plot of orchardgrass shoot dry weight per plant (untransformed y) Where more than 9 observations occur at any location on the plot, additional numbers of observations are indicated by letters A, B, C...T (equivalent to 10, 11, 12,...29) ORCHARDGRASS RESIDUAL SHOOT DRY WEIGHT PER PLANT g pd H3 c+ 4 P CQ H) O 4 3 CD O a u Q ixi t» CO CO hd pd U i—i o Pd td co O o td pd I—I Q 1-3 o b i i u s OB O 1 1 i • • • • * • 1 1 1 1 | 1 1 1 U 0> • (0 — o • O -4 Jk -» M a -v CD fa) U u m -t > 01 -* - 01 I > n oi A. CO OB o cn ci M - > I -n Ol 01 U -» — * U CJ •A * -4 CD -4 £. -» M 01 A U Ol & M M U -» M U -» -A M M M 0> M U —t U M U M & -» M _* u MM-. U A -» — M IO IO - U M -t — -» -* M M — M o • «• I I. * I I I I + I 00 49 Fig. 3.2 Residual vs. predicted plot of orchardgrass shoot dry weight per plant (log e y) 1 1.8 • 1 1 1 1 2 1 - 1 2 1 1 •90 • 11 1 11 1 1 21 11 11 1 1 12 13 111 11211 1 11 1 1 3 1 11 3 1221 123111 11 1 211 11 1 122111113214613 3 1111 11 1 1 121 141211343325213113 2242 1 21 1 13 11 2234124545252913112231 1 1114 431435553 56444321 5 231211 0.0 • 1 2 43621412264622A2625217113 11 - 2 121 3 21431218613333151412211 12 1 2 4111325 335475221 1 4 1122 2 221 12 2 342 31441 21 11 - 11 11212 1 231 413 3 14 1 2 1 11 11212 11 2 11 2 1111 1 2 3 1 1 1 1 1 1 1 1 11 11 1 - .90 • 1 1 11 111 1 1 2 11 1 1 1 1 1 11 1 1 1 2 1 1 2 1 1 11 -1.8 + 1 1 1 1 • • • . -2 .7 - .90 -3 .6 -1 .8 0.0 ORCHARDGRASS PREDICTED SHOOT DRY WEIGHT PER PLANT g (log y) 51 3.4.2 B i o l o g i c a l C o n s i d e r a t i o n s A mathematical model, which describes the relationship between a dependent variate and one or more independent variates, should meet the usual statistical requirements. It also is useful for a model to have a sound biological foundation so that a model relates plant growth to meaningful model parameters. In plant growth analysis, the curve fitting procedure should redescribe the growth trends realistically. Statistical hypothesis testing plays an important role in scientific research. However, significance tests have a limited role in biological experiments because significance refers merely to plausibility, not biological importance; significant treatment differences may exist but may be biologically trivial (Perry, 1986). The use of multiple comparison procedures in biological sciences has been discouraged through several reviews (Chew, 1976; Maindonald and Cox 1984; Perry 1986; Jones and Matloff 1986). Inappropriate statistical analysis procedure and indiscriminate use of statistical hypothesis testing may lead to misinterpretation of the experimental results. Accordingly, in the present investigations, I attempted to use appropriate analytical procedures and significance tests. Subsequently, the effectiveness of those procedures were evaluated on biological grounds. 52 4. RESULTS 4.1 Analysis of Variance 4.1.1 Homogeneity of Variance Test Homogeneity of variance or homoscedasticity is a precondition for several statistical tests. Partitioned Layard's test for heteroscedasticity was applied to all primary variates and ratio indices to assess this assumption. The results are presented (Table 4.1) as per cent of dependent variates having nonsignificant heteroscedasticity (P > 0.05) as influenced by population density and mixture proportions. In the main, most variates showed a homogeneous variance for both treatment factors. However, the population density by species proportions in mixture interaction showed higher heteroscedasticity than did the main effects. These results are based on considerable data (up to 840 observations per dependent variate) and it was reasonable to assume that data usually met the assumption of homogeneity of variance. Moreover, moderate heterogeneity of variance may not be too serious a problem for the overall test of significance in the analysis of variance (ANOVA), and data transformations were not considered essential at this stage of the analysis. 4.1.2 Univariate Analysis of Variance ANOVA was performed on all primary variates and ratio indices for each species at each harvest time. The primary objectives of this ANOVA were to test the statistical significance of treatment effects, and to see whether the significance changed over time. The degrees of freedom for treatment sources of variation were further partitioned into orthogonal polynomials up to the cubic polynomial. This further partitioning was done as a preliminary to regression analysis, as regression analysis provides a more complete Table 4.1 Summary of homogeneity of variance test: Percentage of variates non-significant (P > 0.05) as influenced by treatment factors 53 Source of v a r i a t i o n Species Orchardgrass Timothy Populat ion densi ty (D) 72 59 Mixture proport ions (M) 67 57 D X M 55 50 Overal l mean 65 53 54 definition of the nature of the response to quantitative factors (Chew, 1976; Maindonald and Cox, 1984). The results are presented below as significant variance ratios. 4.1.2.1 The Overall ANOVA Results Univariate analysis of variance was performed on all primary variates and ratio indices, (up to 9 variates for orchardgrass and 12 variates for timothy, respectively), for each species and each harvest. The complete ANOVA results are summarized in Tables 4.2 to 4.10 and in Appendices 8.2 and 8.4. The ANOVA results clearly show that most variates responded very strongly to total population density and species proportions in mixture, and significant interactions occurred in addition to the main effects. Before considering the specific results for each variate, I will review the occurrence of significant responses to treatment factors, taking all variates together for each species. This was done by pooling the ANOVA results for different variates from individual harvests, and expressing the frequency (%) of significant variance ratios (P < 0.05). The overall ANOVA results (Table 4.2) for orchardgrass revealed significant responses by most variates to population density and mixture treatments at most harvests. However, the mixture effects were more common than population density effects. The effects of density were more prominent early during growth than at later harvests. Linear and quadratic polynomials were common significant components of density effects. Cubic polynomials also were important at harvests from 63 to 90 days after planting. The variance ratios for mixture responses increased with age until 77 days, but declined at the final harvest. Linear and quadratic components were the main components of mixture effects. Significant interactions occurred most frequently at harvests from 48 to 77 days. In many ways, the overall ANOVA results for timothy were similar to the orchardgrass results, except that the frequency of population density effects often exceeded 55 Table 4.2 Summary of the overall ANOVA results: Percent frequency of significant variance ratios (P < 0.05) for primary variates and ratio indices in orchardgrass Time a f t e r p lant ing (days) va r ia t Ion d . f . 37 48 63 77 90 Blocks 3 71 44 56 22 22 Density (5) 86 78 78 67 33 (b) D L inear 1 86 89 67 56 67 D Quadrat ic 1 43 78 44 33 44 D Cubic 1 - - 33 33 11 D Deviat ions 2 - - - - 11 Mixture (5) 57 89 100 100 67 (M) M Linear 1 57 89 100 100 67 M Quadrat ic 1 29 56 56 56 11 M Cubic 1 - - - - -M Deviat ions 2 - - - - -D * M (25) 43 67 11 33 -DL * ML 1 71 67 44 33 11 DL * MQ 1 - 22 - 11 33 DL * MC 1 - - - - -DL * MD 2 14 33 - - -DQ * ML 1 57 44 11 22 -DQ * MQ 1 - 22 - 22 -DQ * MC 1 - - 11 22 -DQ * MD 2 - 22 - - -DC * ML 1 - 11 22 - -DC * MQ 1 - 44 11 - -DC * MC 1 - - - - -DC * MD 2 - - - - -DD * ML 2 - 56 - 11 -DD * MQ 2 - - 11 22 -DD * MC 2 - - - 33 -DD * MD 4 - - - - -No. of v a r i a t e s 7 9 9 9 9 Table 4.3 Summary of the overall ANOVA results: Percent frequency of significant variance ratios (P < 0.05) for primary variates and ratio indices in timothy Source of v a r i a t i o n Time a f t e r p lant ing (days) d . f . 37 48 63 77 90 B locks 3 71 78 83 67 67 Density (5) 86 78 67 83 83 (D) D Linear 1 86 78 67 83 85 D Quadrat ic 1 43 44 50 42 42 D Cubic 1 - 22 33 8 8 D Dev iat ions 2 - - 17 - -Mixture (5) 57 67 58 67 42 (M) M L inear 1 57 67 50 58 50 M Quadrat ic 1 - 33 25 25 -M Cubic 1 - - - 25 -M Deviat ions 2 - 56 8 - -D * M (25) 14 - 25 8 -DL * ML 1 57 22 50 17 17 DL * MQ 1 - - 8 17 -DL * MC 1 - - 8 17 -DL * MD 2 - - - 8 -DQ * ML 1 - - 17 - 25 DQ * MQ 1 - - - 17 8 DQ * MC 1 - - - 8 -DQ * MD 2 - - - 8 -DC * ML 1 - 11 17 - -DC * MQ 1 - - 17 17 8 DC * MC 1 - - - - -DC * MD 2 - - - - -DD * ML 2 - - 17 8 8 DD * MQ 2 - - - - -DD * MC 2 - 11 - - -DD * MD 4 14 - - S -No. o f va r ia tes 7 9 12 12 12 57 Table 4.4 Summary of ANOVA results: Variance ratios for the effects of total population density and mixture proportions on orchardgrass shoot dry weight per plant Time a f t e r p lant ing (days) Source of v a r i a t i o n d . f . 37 48 63 77 90 Blocks 3 3 .4* 3 .5* - - -Density (5) 4.8*** 46.4*** 55.2*** 56.2*** 46.9*** (D) D L inear 1 22.2*** 216.5*** 219.3*** 218.6*** 196.8*** D Quadrat ic 1 - 14.0*** 50.1*** 48.4*** 34.8*** D Cubic 1 - - 6.4* 11.2* -D Deviat ions 2 - - - - -Mixture (5) 12.8*** 7.4*** 8.1*** _ (M) M Linear 1 - 55.6*** 36.3*** 35.4*** 9.1** M Quadrat ic 1 - - 4 . 0 * -M Cubic 1 - - - - -M Dev iat ions 2 - - - - -D * M (25) - 2.7*** - 1.9* -DL * ML 1 4 .6* 7.6** 17.6*** 13.0*** DL * MQ 1 - - - 4.3* 4 .3* DL * MC 1 - - - - -DL * MD 2 5.6* 8.0** - - -DQ * ML 1 - 5.3* 6 .7* 5 .1* -DQ * MQ 1 - - - 6.5* -DQ * MC 1 - - - - -DQ * MD 2 - 6.3* - - -DC * ML 1 - - - - -DC * MQ 1 - 8.8* - - -DC * MC 1 - - - - -DC * MD 2 - - - - -DD * ML 2 - 5.6* - - -DD * MQ 2 - - - - -DD * MC 2 - - - - -DD * MD 4 - - - - -* f * * t *** S i g n i f i c a n t at 5*, 1% and 0.1% l e v e l , r e s p e c t i v e l y - Not s i g n i f i c a n t 58 Table 4.5 Summary of ANOVA results: Variance ratios for the effects of total population density and mixture proportions on orchardgrass biomass density Time a f t e r p lant ing (days) Source of v a r i a t ion d . f . 37 48 63 77 90 Blocks 3 7.7*** 3.3* - -Density (5) 7.6*** 6.7*** (D) D L inear 1 28.8*** 21.0*** - -D Quadrat ic 1 7.8** 10.4** - -D Cubic 1 - - - -D Deviat ions 2 - - - -Mixture (5) 28.9*** 142.9*** 118.8*** 104.0*** 64.4*** (M) M Linear 1 141.7*** 683.7*** 576.1*** 513.4*** 318.4*** M Quadrat ic 1 - 27.0*** 15.4*** -M Cubic 1 - - - -M Dev iat ions 2 - - - -D * M (25) 1.8* 2.4** - -DL * ML 1 16.9*** 21.0*** _ _ _ DL * MQ 1 - - - -DL * MC 1 - - - -DL * MD 2 - - - -DQ * ML 1 5.4* - - -DQ * MQ 1 - - - -DQ * MC 1 - - 4.7* -DQ * MD 2 - - - -DC * ML 1 - - - -DC * MQ 1 - 7.0** - -DC * MC 1 - - - -DC * MD 2 - - - -DD * ML 2 - 9.2** - -DD * MQ 2 - - - 5.1* DD * MC 2 - - - -DD * MD 4 - - - -* t * * f *** S i g n i f i c a n t at 5%, 1% and 0 . H l e v e l , r e s p e c t i v e l y Not s i g n i f i c a n t 59 Table 4.6 Summary of ANOVA results: Variance ratios for the effects of total population density and mixture proportions on timothy shoot dry weight per plant Time a f te r p lant ing (days) Source of v a r i a t i o n d . f . 37 48 63 77 90 Blocks 3 9.7*** 9.9*** 5.9*** - -Density (5) 11.2*** 62.6*** 154.0*** 36.7*** 25.6*** (D) D L inear 1 52.9*** 263.6*** 558.7*** 159.1*** 101.1*** D Quadrat ic 1 - 41.8*** 163.5*** 21.6*** 24.0*** D Cubic 1 - 5.9* 38.0*** - -D Deviat ions 2 - - 9.1** - -Mixture (5) 7.9*** 19.5*** 5.0*** 4.0** (M) M Linear 1 8.2** 34.0*** 86.5*** 24.5*** 17.4*** M Quadrat ic 1 - - 9.1** - -M Cubic 1 - - - - -M Dev iat ions 2 - - - - -D * M (25) - - 3.5*** - -DL * ML 1 _ 34.9*** 6 .5* 9.6** DL * MQ 1 - - 5.1* - -DL * MC 1 - - - - -DL * MD 2 - - - - -DQ * ML 1 - - 12.6*** - 4 . 6 * DQ * MQ 1 - - - - -DQ * MC 1 - - - - -DQ * MD 2 - - - - -DC * ML 1 - - 6.2* - -DC * MQ 1 - - 5.4* - -DC * MC 1 - - - - -DC * MD 2 - - - - -DD * ML 2 - - 4.5* - -DD * MQ 2 - - - - -DD * MC 2 - - - - -DD * MD 4 - - - - -* , * * , *** S i g n i f i c a n t at 5%, 1% and 0.1% l e v e l , r e s p e c t i v e l y - Not s i g n i f i c a n t 6 0 Table 4.7 Summary of ANOVA results: Variance ratios for the effects of total population density and mixture proportions on timothy biomass density Time a f te r p lant ing (days) Source of v a r i a t i o n d . f . 37 48 63 77 90 Blocks 3 13.5*** 12.2*** 5.9*** 3 .6* 7.8*** Density /n \ (5) 19.6*** 6.0*** - - -D L inear 1 85.9*** 27.5*** _ _ 4.2* D Quadrat ic 1 8 .3* - - 9.8** -D Cubic 1 - - - - 4 . 9 * D Deviat ions 2 - - - - -Mixture (5) 41.9*** 68.9*** 64.7*** 45.1*** 41.4** (M) M Linear 1 205.5*** 325.9*** 313.5*** 211.7*** 201.8** M Quadrat ic 1 - 7.3** 7.0** 11.7*** -M Cubic 1 - - - - -M Deviat ions 2 - 7.9** - - -D * M (25) - - - - -DL * ML 1 6 .6* _ DL * MQ 1 - - - - -DL * MC 1 - - - - -DL * MD 2 - - - - -DQ * ML 1 - - - - -DQ * MQ 1 - - - - -DQ * MC 1 - - - - -DQ * MD 2 - - - - -DC * ML 1 - - • - - -DC * MQ 1 - - - - -DC * MC 1 - - - - -DC * MD 2 - - - - -DD * ML 2 - - - - -DD * MQ 2 - - - - -DD * MC 2 - - - - -DD * MD 4 - - - - -* t * * t *** S i g n i f i c a n t at 5%, 1% and 0.1% l e v e l , r e s p e c t i v e l y - Not s i g n i f i c a n t 61 Table 4.8 Summary of ANOVA results: Percent sum of squares for the effect of total population density and mixture proportions on total mixture biomass density Source of variation Time after planting (days) d.f. 37 48 63 77 90 Blocks 3 17*** 13*** 3* 4* 4* Density (D) 5 35*** 19*** ins 5* 5* Mixture (M) 6 10*** 15*** 38*** 30*** 31*** D * M 30 5ns 11ns 10"s 11ns 1ins Error 123 33 42 48 50 49 Total 167 100 100 100 100 100 * ** #** Significant at 5%, 1% and 0.1% probability level, respectively ns Not significant Table 4.9 Summary of ANOVA results: Variance ratios for the effects of total population density and mixture proportions on other orchardgrass variates Source of variation Variate 37 Time after planting (days) 48 63 77 90 Density (D) Mixture (M) D • M W/TL TLN/A TLN/PL LA/PL LA/TL W/TL TLN/A TLN/PL LA/PL LA/TL W/TL TLN/A TLN/PL LA/PL LA/TL N 15.2*** 8.5*** 6.2*** N 54.5*** 4.6*** N 2.4** 7.9*** 53.1*** 22.8*** 2.6* 8.1*** 141.9*** 13.7*** 12.4*** 7.3*** 2.6** 3.5*** 2.9*** 9.4*** 38.1*** 23.5*** 2.3* 4.3** 139.2*** 7.5*** 4.5*** 2.9* 2.0** 4.7*** 59.8*** 35.9*** 2.6* 8.0*** 122.7*** 5.5*** 4.3** 6.9*** 1.8* 2.0*** 3.4** 34.4*** 7.5*** 82.7*** 2.3* *, **, *** Significant at 5%, 1, and 0.1% level, respectively - Not significant N Data not recorded Table 4.10 Summary of ANOVA results: Variance ratios for the effects of total population density and Mixture proportions on other tinothy variates Source of Time after planting (days) Variate variation 37 48 63 77 90 Density ID) W/TL N - 4.0** 5.3*** 13.7*** WPAN/PL N N 43.7*** 39.4*** 15.5*** WPAN/TL N N 3.1* 9.7*** 7.2*** TLN/A 29.4*** 15.5*** 6.0*** 10.6*** 9.6*** TLN/PL 9.3*** 91.4*** 74.8*** 49.0*** 18.8*** LA/PL 5.6*** 35.9*** 29.1*** 40.3*** 34.3*** LA/TL - 3.1* 5.2*** 10.0*** 19.0*** Mixture (M) M W/TL N - - - 3.6»* WPAN/PL N N 4.6*** 2.8* WPAN/TL N N TLN/A 49.0*** 86.2*** 35.7*** 64.3*** 59.8*** TLN/PL 2.9* 8.1*** 12.2*** 4.3** LA/PL - - 3.3** -LA/TL - 2.9* - 2.6* W/TL N WPAN/PL N N WPAN/TL N N TLN/A 2.1** - - 1.9* TLN/PL - - 2.2** LA/PL . . . . LA/TL -* f ** f *•* Significant at 5%, 1, and 0.1% level, respectively - Not significant N Data not recorded 64 the mixture effects (Table 4.3). As with orchardgrass, significant linear and quadratic components were common for timothy, and higher polynomial terms were sometimes significant for both treatment factors. Details of the treatment responses varied with harvest time. In addition to the main effects, there were several significant interactions. Significant interactions were most frequent during the 63 to 90 day period. 4.1.2.2 Orchardgrass Yield Responses 4.1.2.2.1 Orchardgrass Shoot Dry Weight per Plant The ANOVA for orchardgrass shoot dry weight per plant (Table 4.4) indicated that density effects, were strongly significant at all harvests. In addition to the linear component, quadratic and sometimes cubic components of the density effect also were significant after the first harvest. Mixture effects also were significant after the first harvest, and at 77 days there was a significant quadratic component of this effect. There also were a number of significant interactions, but the types of those interactions varied during growth. Some of these relationships can be visualized in Fig. 4.1. Shoot dry matter accumulation of orchardgrass continued to increase with age. Higher population density resulted in severe reduction in shoot dry matter accumulation in all the mixtures, whereas shoot dry matter accumulation was increased by increased orchardgrass proportions in the mixtures. 4.1.2.2.2 Orchardgrass Shoot Biomass Density Orchardgrass shoot biomass density (i.e. shoot dry weight per unit land area) was also significantly influenced by both population density and mixture proportions (Table 4.5). On the basis of the size of the variance ratios, the mixture effects were more powerful and remained strong as the crop grew. Quadratic components of the mixture effects were significant at 48 and 63 days. Population density effects were significant only 65 Fig. 4.1 Time course of orchardgrass shoot d r y weight per p l a n t One m i x t u r e per sub-plot. Proportions of Orchardgrass: T i m o t h y as indicated on sub-plots Symbol Orchardgrass population density (plants m"2) a Q 668 o o 1 3 3 6 A A 2004 X X 2671 0 0 3339 V V 4007 A 1 : 5 B 2 : 4 66 TIME AFTER PLANTING d 67 at the 37 and 48 day harvests when quadratic components were also significant. Again, there were a few significant interactions which had no obvious pattern among harvests. Fig. 4.2 shows the corresponding time course of orchardgrass shoot biomass density. Shoot biomass density continued to increase until the last harvest. Population density effects were less apparent than on weight per shoot (Fig. 4.1). There was a substantial increase in orchardgrass yield with its increasing proportion in the mixtures. 4.1.2.3 Timothy Yield Responses 4.1.2.3.1 Timothy Shoot Dry Weight per Plant The ANOVA results for timothy shoot dry weight per plant (Table 4.6) were similar to orchardgrass results. However, an additional higher polynomial component was also significant for population density at the 63 day harvest. For the mixture treatments, the linear component source of variation was significant at all harvests, with an additional significant quadratic component at 63 days. Significant interactions were most common during the 63 to 90 day period. Higher densities reduced timothy shoot dry weight per plant in most mixtures (Fig. 4.3). As indicated by the size of the variance ratios, density effects became more pronounced as the crop aged. Shoot dry weight per plant also seemed to undergo a gradual increase in response to increasing timothy proportions the mixtures. 4.1.2.3.2 Timothy Shoot Biomass Density For the other yield variates considered so far, block effects declined with age (Tables 4.4 to 4.6). The ANOVA for timothy shoot biomass density, however, indicated that significant block effects persisted throughout growth (Table 4.7). The main effects for density and mixtures were usually highly significant, except for density effects from 63 days onwards. Linear, quadratic and sometimes cubic components of density responses were again important, except at 63 days. The trends for mixtures were mostly linear, 68 F i g . 4.2 Time course of orchardgrass shoot biomass density One m i x t u r e per sub-plot. Proportions of Orchardgrass: T i m o t h y as indicated on sub-plots Symbol Orchardgrass population density (plants m"2) a a o o A A X X * • v V 668 1336 2004 2671 3339 4007 900.0-750.0-600.0-450.0-300.0-150.0 H C J I A 1:5 0.0 E 900.0 n 750 .0H >-t : CO 600.0-z Ul Q 450.0-to CO 300.0-1 C 3 : 3 B 2 : 4 6 9 F 6 : 0 35 - i r— 50 65 80 95 TIME AFTER PLANTING d F i g . 4.3 Time course of timothy shoot d r y weight per p l a n t 70 One m i x t u r e per sub-plot. Proportions of Orchardgrass: Timothy as indicated on sub-plots Symbol T i m o t h y population density (plants m"2) a a 653 O o 1307 A A I960 X X 2613 • 0 3267 v v 3 9 2 0 TIME AFTER PLANTING * 72 although quadratic and higher polynomials were also significant during the 48 to 77 day period. Interactions were mostly non-significant. The time course of timothy shoot biomass density (Fig. 4.4) was similar to the corresponding orchardgrass results (Fig. 4.2). Timothy, however, was the more productive species in both the mixtures and monocultures. Timothy yields in mixture were generally lower than in monoculture. In accord with the earlier observations on yield per plant, timothy shoot biomass density was increased as its proportion increased in the mixtures. 4.1.2.4 Total Mixture Shoot Biomass Density In order to assess overall yield performance, an ANOVA was also performed on the data for total shoot biomass density. This total combined both component species in the mixtures or included each species alone for the monocultures (Table 4.8). The effects of blocks, population density and mixtures were usually highly significant throughout growth. Block and population density effects declined with age. Mixture effects peaked at 63 days and interaction effects were not significant. Fig. 4.5 plots the data related to this ANOVA. Mixture total biomass density increased with increasing population density during early growth. This response, however, was reversed at later growth stages. Mixture effects were more pronounced during late growth and higher yield was favored by timothy proportions. The timothy monocultures were the most productive cases and orchardgrass monocultures proved to be the least productive. Mixture yields fell between the monoculture extremes. The overall yields increased linearly until 63 days, followed by a slower additional increase during later growth. 4.1.2.5 Orchardgrass: Other Variates Table 4.9 summarizes ANOVA results for the main effects of treatment factors on other measured and derived quantities for orchardgrass. The variates assessed include 73 F i g . 4.4 Time course of timothy shoot biomass density One m i x t u r e per sub-plot. Proportions of Orchardgrass: T i m o t h y as indicated on sub-plots Symbol T i m o t h y population density (plants m'2) a a o o A A X X 2613 0 $ 3267 v - - - - v 3 9 2 0 653 1307 1960 B 1 : 5 D 3 : 3 F 5 :1 35 50 TIME AFTER PLANTING d 75 F ig . 4.5 Time course of total shoot biomass density in the mixtures One mixture per sub-plot. Proportions of Orchardgrass: Timothy as indicated on sub-plots Symbol Range of total ^ population densities (plants m"2) o O 655 to 668 O o 1312 to 1331 A A 1967 to 1997 X X 2623 to 2662 A $ 3279 to 3327 V " V 3935 to 3989 Values for mixtures taken from Table 3.1. Where orchardgrass was present in higher proportions the total population density was higher than when timothy was the more prevalent. E 1800.0-1500.0->-\— co 1200.0-z 1 • 1 L U Q 900.0-CO CO < 600.0-O 300.0-CD _ J < o.o-1— o 1800.0-1500.0-1200.0-900.0-600.0-300.0-0.0-B 2 : 4 1800.0-1500.0-1200.0-900.0-600.0-300.0-0.0 A 1 : 5 C 3 : 3 TIME AFTER PLANTING 77 tiller dry weight (W/TL), tiller number per area (TLN/A), tiller number per plant (TLN/PL), leaf area per plant (LA/PL) and leaf area per tiller (LA/TL). In addition, the complete ANOVA results are provided in Appendices 8.2.1 to 8.2.5. All of these variates responded significantly to both population density and/or mixture treatments at most harvests. Population density treatments had no significant effects on dry weight per tiller, and did not affect tiller number per plant at 90 days or leaf area per tiller at the 37 and 90 day harvests. Except for tiller number per area, the significant main effects of mixtures showed a different pattern among harvests than the density effects. For example, dry weight per tiller significantly responded to mixture treatments at every harvest tillers were present for observation. Significant interaction responses were less common than main effects. Significant interactions did not occur for tiller dry weight per tiller and leaf area per tiller. A few significant interactions, however, were found for tiller number and leaf area per plant. Interactions were significant for tillers per area, however, at the first four harvests. Figs. 4.6 to 4.10 provide a graphical representation of these responses. Dry weight per tiller continued to accumulate linearly until 77 days in most mixtures and densities, after which the direction of the trends varied among different cases (Fig. 4.6). The higher population densities often produced a lower tiller dry weight; however, such differences were relatively small. The response to mixture, although significant according to the ANOVA, also appears to be small. Tiller numbers, both on an area (Fig. 4.7) and a per plant (Fig. 4.8) basis were increased with increasing crop age. There was little indication of tiller mortality on an area basis during the 48 to 90 day growth period. Higher population density usually increased tillers per area, whereas tillers per plant showed a strong negative response. Mixture effects on tillers per plant were not obvious although this response is significant at intermediate harvests according to the ANOVA. Tiller number per area increased substantially with increasing orchardgrass proportion in mixtures. F i g . 4 .6 Time course of orchardgrass d r y weight per t i l l e r 78 One m i x t u r e per sub-plot. Proportions of Orchardgrass: T i m o t h y as indicated on sub-plots Symbol Orchardgrass population density (plants m"2) • O 668 O o 1336 A A 2004 X X 2671 6 $ 3339 V - - - - v 4007 7 9 TIME AFTER PLANTING d F i g . 4.7 Time course of orchardgrass t i l l e r number per unit 80 land area One mixture per sub-plot. Proportions of Orchardgrass: T i m o t h y as indicated on sub-plots Symbol Orchardgrass population density (plants m"2) o • 668 O O 1336 A A 2004 X X 2671 6 0 3339 V - - - - - - - - v 4007 TIME AFTER PLANTING d 82 F i g . 4.8 Time course of orchardgrass t i l l e r number per p l a n t One m i x t u r e per sub-plot. Proportions of Orchardgrass: T i m o t h y as indicated on sub-plots Symbol Orchardgrass population density (plants m"2) Q =_ • 6 6 G o o 1 3 3 6 A A 2004 X X 2671 $ ••• 0 3339 v v 4 0 Q 7 TIME AFTER PLANTING d 84 F i g . 4.9 Time course of orchardgrass l e a f a r e a per p l a n t One m i x t u r e per sub-plot. Proportions of Orchardgrass: T i m o t h y as indicated on sub-plots Symbol Or c h a r d g r a s s population density (plants m'2) • Q 668 O O 1336 A A 2004 X X 2671 6 0 3339 V V 4007 0.025-0.020-0.015-0.010-0.005H A 1 : 5 O.OOO-i r 0.025 -1 ~ E 0.020-1 < 0.015-^ C 3 : 3 0.000 0.025 -1 0.020-0.015-E 5 : 1 B 2 : 4 D 4 : 2 F 6 : 0 TIME AFTER PLANTING d Fig. 4.10 Time course of orchardgrass l e a f area per t i l l e r One m i x t u r e per sub-plot. Proportions of Orchardgrass: T i m o t h y as indicated on sub-plots Symbol Orchardgrass population density (plants m"2) TIME AFTER PLANTING d 88 Leaf area expansion per plant usually continued throughout the study and declined rapidly with increasing population density in all mixtures (Fig. 4.9). Mixture effects were less obvious; at low population densities leaf areas in mixtures were higher than in monocultures. Leaf area per tiller (Figure 4.10) increased strongly with time, but the effects of population density and mixture proportions are less obvious. 4.1.2.6 Timothy: Other Variates ANOVA results for the main effects of treatment factors on other measured and derived quantities for timothy are summarized in Table 4.10. The variates include dry weight per tiller (W/TL), panicle dry weight per plant (WPAN/PL), panicle dry weight per tiller (WPAN/TL), tiller number per unit area (TLN/A), tiller number per plant (TLN/PL), leaf area per plant (LA/PL) and leaf area per tiller (LA/TL). Complete ANOVA results for these variates are given in Appendices 8.2.6 to 8.2.12. All variates responded significantly to both treatment factors. However, significant mixture effects were not as common-as population density effects. Population density effects on the shoot dry weight components (W/TL, WPAN/TL, WPAN/TL) were significant only at harvests from 63 days onward. For the remaining variates, population density effects were significant throughout the experimental period, except for leaf area per tiller at initial harvest. Tiller numbers both on an area and per plant basis, however, responded significantly to mixture treatments at all harvests, except tiller number per plant at 90 days. Except for panicle dry weight per tiller, which did not respond significantly to treatments, other variates exhibited a significant mixture response at some time during the study. There was no obvious pattern, however, in the time of occurrence of significant mixture effects among those variates. Interaction effects were usually not significant. Figs. 4.11 to 4.17 plot the actual data related to this analysis. Component dry weights usually continued to accumulate with age, at least until the 77 day harvest Fig. 4.11 Time course of timothy dry weight per tiller One mixture per sub-plot. Proportions of Orchardgrass: Timothy as indicated on sub-plots Symbol Timothy population density (plants m"2) TIME AFTER PLANTING d 91 Fig. 4.12 Time course of t i m o t h y panicle d r y weight per p l a n t One mixture per sub-plot. Proportions of Orchardgrass: T i m o t h y as indicated on sub-plots Symbol T i m o t h y population density (plants m"2) a a 6 5 3 1307 1960 o o A A X X 2613 $ 0 3267 V v 3920 93 Fig. 4.13 Time course of timothy panicle dry weight per tiller One mixture per sub-plot. Proportions of Orchardgrass: Timothy as indicated on sub-plots Symbol Timothy population density (plants m"2) N Q 653 O o 1307 A A I960 X X 2613 A 0 3267 V ' " V 3920 X o U J >-O CJ < Q_ 0.10-1 0.08 0.06-0.04-0.02-A 0:6 B 1 : 5 D 3 : 3 F 5 : 1 35 50 TIME AFTER PLANTING Fig. 4.14 Time course of timothy tiller number per unit land area One mixture per sub-plot. Proportions of Orchardgrass: Timothy as indicated on sub-plots Symbol Timothy population density (plants m"2) a a 653 O O 1307 A A i960 X X 2613 0 $ 3267 V - - - - - - - - - - - - - V 3920 TIME AFTER PLANTING d 97 Fig. 4.15 Time course of timothy t i l l e r number per plant One m i x t u r e per sub-plot. Proportions of Orchardgrass: T i m o t h y as indicated on sub-plots Symbol T i m o t h y population density (plants m"2) a a 653 O O 1307 A A i960 X X 2613 v 0 3 2 6 7 V - " V 3920 TIME AFTER PLANTING d Fig. 4.16 Time course of timothy leaf area per plant One mixture per sub-plot. Proportions of Orchardgrass: Timothy as indicated on sub-plots Symbol Timothy population density (plants m*2) A 0 : 6 0.000 0.025-1 CM 0.020-E < 1 . 1 0.015-< 0.010-< U J _ J 0.005-C 2 : 4 100 B 1:5 D 3 : 3 F 5 :1 35 50 65 80 95 TIME AFTER PLANTING d Fig. 4.17 Time course of timothy leaf area per tiller One mixture per sub-plot. Proportions of Orchardgrass: Timothy as indicated on sub-plots Symbol Timothy population density (plants m'2) TIME AFTER PLANTING d 103 (Figs. 4.11 to 4.13). Dry weights tended to decline with increasing population density, and this response tended to become stronger with time. Mixture effects on component dry weights were relatively small. Fig. 4.14 and 4.15 plot the time course of timothy tiller number per unit land area and per plant, respectively. Tiller number per area responded positively to population density, whereas tiller number per plant followed the opposite trend. There was an initial decline in tiller numbers per area at higher population densities. This was followed by a period of relative stability from 48 to 77 DAP. In some cases, tiller numbers per area increased at final harvest (90 DAP) which corresponded to a resumption of more active growth which was visible in the field. As the proportion of timothy in the mixtures increased, tiller number per area also increased. On the other hand, tiller number per plant was relatively stable among mixtures. Leaf area per plant and leaf area per tiller often reached a peak at intermediate harvests, and the later decline was associated with some visible leaf senescence (Figs. 4.16 and 4.17). High population density often decreased leaf area, particularly on a per plant basis. Mixture effects on leaf area were relatively weak. It is evident, both from the ANOVA results and the corresponding graphs, that both species responded strongly to the experimental factors with respect to most primary variates (W/PL, W/TL, BD, TLN/PL). These variates usually increased in response to time after planting, while they declined with higher population density and decreasing species proportion in the mixtures. However, species responses to experimental factors differed for tiller numbers per land area, leaf area per plant and leaf area per tiller. Orchardgrass tiller number per land area tended to increase with time, whereas timothy followed an opposite trend. Leaf area per plant and per tiller in orchardgrass usually increased linearly with age. In the case of timothy, these variates peaked in the intermediate harvest and declined at later harvests. Variates per plant in both species 104 also showed an increased response in mixtures. Timothy was the more productive species both in monocultures and mixtures. The ANOVA results for each species have indicated that the responses of most variates to experimental treatments were statistically significant. The ANOVA results and associated graphical representations have shown how different variates differed in their responses to treatments. The primary role of the ANOVA is to partition and test the significance of sources of variation, and the relative importance of treatment factors can be inferred only to a limited extent from this procedure alone. 4.2 Best Subset Multiple Regression Analysis The purpose of this regression procedure was to construct models, for each dependent variate, which express the best subset of potential independent variates, including time after planting, population density of each species, and interactions among those factors (Eq. 3.1). Results of the procedure are expressed as standard partial regression coefficients for the best subset of independent variates according to Mallow's CP criterion. This simplifies interpretation because the absolute size of the standard partial regression coefficients denotes the relative importance of the independent variates causing yield responses (Steel and Torrie 1980). (The partial regression coefficients themselves cannot be compared directly as their values are dependent on the measurement scales of the independent variates. On the other hand, the standard partial regression coefficients are expressed in units of the standard deviation of independent variates.) 4.2.1 Orchardgrass Best Subset Models Significant best subset regressions were obtained for all nine dependent variates analyzed for orchardgrass (Tables 4.11a and 4.11b). In all cases, a considerable portion Table 4.11a Parameters and regression statistics for the best subset models in orchardgrass 1 0 5 P o t e n t i a l Dependent v a r i a t e s i n d e p e n d e n t v a r i a t e s BD W/PL W/TL TLN/A TLN/PL S t d . p a r t i a l r e g r e s s i o n c o e f f i c i e n t s I n t e r c e p t - 1 . 7 - 7 . 0 - 5 . 7 0 . 6 1 . 4 t* 2 . 6 3 . 0 2 . 7 0 . 4 0 . 8 0 1 . 6 - 0 . 2 1 . 9 -T - - - 0 . 6 0 . 5 to - 0 . 8 - 0 . 7 - - - 1 . 0 t T - 2 . 0 - 2 . 1 - - 2 . 7 - 3 . 7 0T 1 .3 0 . 9 - - -tOT - - 0 . 3 0 . 6 1.1 t | - 1 . 7 - 2 . 1 - 1 . 9 - -t 2 0 - - - - 0 . 8 - 0 . 8 t 2 T 0 . 7 0 . 7 - 0 . 4 1 . 0 1.1 t z 0 T - - - - -0 2 - 0 . 9 - - 0 . 2 - 1 . 4 - 0 . 5 0 2 t 0 . 4 0 . 2 - 0 . 7 1 .3 0 2 T - 0 . 8 - 0 . 5 - 0 . 3 - 0 . 2 0 2 t T 0 . 2 0 . 2 - 0 . 2 _ T2 _ - -T 2 t 0 . 7 0 . 6 - 0 . 7 1.1 T 2 0 - 0 . 5 - 0 . 3 _ - 0 . 3 _ T 2 t 0 - - - - 0 . 3 O ther r e g r e s s i o n s t a t i s t i c s CP 1 1 . 5 1 8 . 3 5 RMS 0 . 3 0 0 . 2 6 R 2 0 . 8 0 0 . 7 6 d . f . 706 708 1 .48 1 0 . 6 0 1 0 . 4 7 0 . 4 5 . 0 8 0 . 4 1 0 . 7 0 0 . 7 9 0 . 5 6 568 707 707 * t d e n o t e s t i m e a f t e r p l a n t i n g ( d a y s ) 0 d e n o t e s o r c h a r d g r a s s p o p u l a t i o n d e n s i t y ( p l a n t s m" 2 ) T d e n o t e s t i m o t h y p o p u l a t i o n d e n s i t y ( p l a n t s m " 2 ) Table 4.11b Parameters and regression statistics for the best subset models in orchardgrass P o t e n t i a l Dependent v a r i a t e s i n d e p e n d e n t v a r i a t e s LA/PL LA/TL LAI LAR S t d . p a r t i a l r e g r e s s i o n c o e f f i c i e n t s I n t e r c e p t - 1 . 5 - 1 . 2 - 3 . 9 - 1 2 . 6 t* 0 . 9 1 . 0 1 . 9 7 . 6 0 1.1 0 . 1 2 . 3 -T 1 . 3 - 0 . 7 -to - 1 . 9 - - 1 . 5 -t T - 3 . 5 - 0 . 5 - 3 . 7 -OT - 0 . 7 - - -tOT 2 . 2 1 .0 2 . 3 -t 2 0 . 8 - - 1 . 0 - 7 . 1 t 2 0 - 1 . 0 - 0 . 2 _ - 0 . 1 _ - 1 . 8 _ t 2 0 T - 0 . 9 _ 0 2 - 1 . 0 - 1 . 5 -0 2 t 1 .8 0 . 9 _ 0 2 T _ _ - 0 . 7 _ 0 2 t T - 0 . 5 - 0 . 5 - _ T 2 0 . 6 0 . 2 _ _ T2t 1 . 7 0 . 7 T 2 0 _ • _ T 2 t 0 - 0 . 6 - 0 . 4 - 0 . 6 -Other r e g r e s s i o n s t a t i s t i c s CP 1 6 . 7 7 RMS 0 . R 2 0 . 6 4 d . f - 704 3 . 5 3 1 2 . 1 6 - 7 . 1 0 . 0 . 4 6 0 . 0 . 7 1 0 . 7 1 0 . 6 8 711 705 572 * t d e n o t e s t i m e a f t e r p l a n t i n g ( d a y s ) 0 d e n o t e s o r c h a r d g r a s s p o p u l a t i o n d e n s i t y ( p l a n t s m" 2 ) T d e n o t e s t i m o t h y p o p u l a t i o n d e n s i t y ( p l a n t s m " 2 ) of the variability was explained by the regressions, with coefficients of determination (R ) ranging from 0.56 to 0.8. Thirteen of the nineteen potential independent variables were included in the best subset model for shoot biomass density (Table 4.11a). As indicated above, the standard partial regression coefficients can be used to compare the relative importance of significant independent variates. Time after planting (t), time by timothy population density interaction (tT), time quadratic (t ), orchardgrass population density (O), and orchardgrass by timothy population density interaction (OT) were the most important factors (yielding standard partial regression coefficients greater than 1.0). The signs of the standard partial regression coefficients indicate the direction of response. Among the most important factors, time, orchardgrass population density and the orchardgrass by timothy population density interaction had positive relationships with orchardgrass shoot biomass density, while the other factors were related to significant yield reduction. In the case of shoot dry weight per plant (W/PL, Table 4.11a), linear and quadratic responses to time after planting, and time by timothy density interaction, stand out as important factors. Of these, only time after planting caused a positive response. Similar results were found for orchardgrass tiller dry weight (W/TL, Table 4.11a), except in this case the time by timothy density interaction was not significant. As would be expected, orchardgrass tiller number per area (TLN/A, Table 4.11a) i was significantly increased in relation to orchardgrass population density, but negative relationships with the time by timothy population density interaction and the quadratic orchardgrass population density term were also strong. For tiller number per plant (TLN/PL, Table 4.11a), however, the most important factors were different. Tiller number per plant had a strong negative relationship with the time by timothy population density interaction, and to a lesser extent with the time by orchardgrass population density o o 9 interaction. A set of quadratic and interaction terms (tO , tOT, t T, and tT ) had moderately strong positive relationships with tiller number per plant. 108 Relatively strong positive relationships existed between leaf area per plant (LA/PL, Table 4.11b) and a large set of independent variates (tOT, 0 2 T , T 2 t , T and O), and there 9 9 were also several negative relationships (tT, tO, t O, O ). Standard partial regression coefficients for the model for leaf area per tiller (LA/TL, Table 4.11b) were relatively small. The pattern of significant independent variates for the orchardgrass leaf area index (LAI, Table 4.1 lb) model was largely similar to that for leaf area per plant, with two 2 2 independent variates (t and t T) being notable exceptions to this generalization. Time after planting was the only important independent variate in the leaf area ratio (LAR, Table 4.1 lb) model; leaf area ratio was positively related to time but negatively to the time quadratic term. 4.2.2 Timothy Best Subset Models Twelve dependent variates were analysed for timothy by the best subset multiple regression procedure. Parameters and regression statistics for the regressions are summarized in Tables 4.12a and 4.12b. All the regressions were statistically significant, 9 with coefficients of determination (R ) being higher than 0.4 except in the case of harvest 9 index where R was only 0.05. In the regression for timothy biomass density (BD, Table 4.12a), time after planting, timothy population density and the time quadratic by timothy population density interaction were the most important positive independent variates. Strong negative relationships existed for time by population density interactions (tO, tT) of each species 9 and time quadratic (t ). Shoot dry weight per plant (W/PL, Table 4.12a) had strong positive relationships with time and the time quadratic by timothy population density interaction (t T). The most important negative factors were time quadratic, time by timothy population density interaction and the time by orchardgrass population density interaction. Table 4.12a Parameters and regression statistics for the best subset models in timothy 1 0 9 P o t e n t i a l 1ndependent Dependent v a r i a t e s v a r i a t e s BD W/PL W/TL WPAN/PL WPAN/TL TLN/A S t d . p a r t i a l r e g r e s s i o n c o e f f i c i e n t s I n t e r c e p t - 3 . 2 - 8 . 2 - 6 . 7 1 . 3 - 1 7 . 1 2 . 6 t* 4 . 6 4 . 4 3 . 0 - 5 . 3 -0 - 0 . 4 - - - - 0 . 7 T 2 . 3 0 . 7 0 . 6 - - 3 . 5 to - 1 . 5 - 1 . 5 - - 1 . 1 - 0 . 4 - 0 . 4 tT - 3 . 1 - 3 . 0 - - 1 . 8 - - 5 . 7 0T 0 . 9 - - - 0 . 3 1 . 3 tOT - 0 . 1 - 0 . 5 - -t f - 3 . 5 - 3 . 4 - 1 . 8 0 . 5 - 4 . 7 _ t 2 0 0 . 7 0 . 7 - 0 . 2 - - -t 2 T 1 .4 1 . 4 - 1 . 0 _ - 0 . 3 3 . 5 t 2 0 T _ _ _ . 02 0 . 5 0 . 1 0 . 1 - - 0 . 4 Oh _ _ _ 0 . 4 _ 0 . 3 0 2 T - 0 . 6 _ - 0 . 6 0 2 t T 0 . 2 - - - - -T 2 - 0 . 8 - - 0 . 8 - - - 0 . 5 T 2 t 0 . 5 0 . 5 0 . 9 1.1 _ _ T 2 0 - 0 . 4 - - - - - 0 . 6 T 2 t 0 - - - - - 0 . 3 -Other r e g r e s s i o n s t a t i s t i c s CP RMS R2 d . f . 1 1 . 3 6 0 . 1 4 0 . 8 6 705 7 . 6 5 0 . 1 2 0 . 8 9 708 3 . 6 4 0 . 0 1 0 . 7 9 567 - 1 . 1 6 0 . 0 1 0 . 4 7 425 0 . 0 9 0 . 0 . 4 1 425 9 . 5 8 4 8 . 9 3 0 . 7 4 708 * t d e n o t e s t i m e a f t e r p l a n t i n g ( d a y s ) 0 d e n o t e s o r c h a r d g r a s s p o p u l a t i o n d e n s i t y ( p l a n t s m"2) T d e n o t e s t i m o t h y p o p u l a t i o n d e n s i t y ( p l a n t s m"2) 1 1 0 Table 4.12b Parameters and regression statistics for the best subset models in timothy P o t e n t i a l Dependent v a r i a t e s Independent v a r i a t e s TLN/PL LA/PL LA/TL LAI LAR H S t d . p a r t i a l r e g r e s s i o n c o e f f i c i e n t s I n t e r c e p t 5 . 5 - 1 0 . 9 - 6 . 4 - 1 . 7 0 . 7 4 . 3 t* - 3 . 0 5 . 8 2 . 5 - 2 . 2 -0 - - - - - - 0 . 2 T - 0 . 8 1 .4 3 . 0 - -to - 0 . 9 - 0 . 8 - 0 . 5 - 1 . 3 - -t T - 4 . 0 - 3 . 6 - 1 . 9 - 3 . 2 - 0 . 4 OT - 0 . 2 - - - -tOT 0 . 4 - - 1 .3 - -t 2 - - 2 . 8 - 5 . 4 - 2 . 2 1 . 4 _ t 2 0 - - - - - 0 . 1 -t 2 T 2 . 5 1 . 6 _ 0 . 8 - 0 . 1 _ t 2 0 T _ 0 . 1 _ _ _ O 2 - 0 . 2 - 0 . 2 - 0 . 3 - -0 2 t 0 . 4 0 . 2 0 . 7 _ _ 0 2 T _ _ _ _ _ 0 2 t T - - - - 0 . 5 T 2 1 . 0 - - 1 . 0 - 1 . 4 0 . 1 -T 2 t 0 . 6 1.1 1 . 0 _ - 0 . 5 T 2 0 _ • T 2 t 0 - - - - 0 . 6 - -Other r e g r e s s i o n s t a t i s t i c s CP RMS R2 d . f . 3 . 9 4 0 . 0 7 0 . 7 9 712 6 . 8 4 0 . 2 2 0 . 6 1 710 5 . 3 4 0 . 0 . 5 1 710 11 .51 0 . 1 8 0 . 6 0 707 1 .27 0 . 0 9 0 . 8 9 570 0 . 5 6 0 . 0 . 0 5 4 . 2 8 * t d e n o t e s t i m e a f t e r p l a n t i n g ( d a y s ) 0 d e n o t e s o r c h a r d g r a s s p o p u l a t i o n d e n s i t y ( p l a n t s m" 2 ) T d e n o t e s t i m o t h y p o p u l a t i o n d e n s i t y ( p l a n t s m " 2 ) I l l A strong positive relationship with time was evident for dry weight per tiller (W/TL, Table 4.12a). Prominent negative factors were the time quadratic alone and its interaction with timothy population density (t T). The model was quite different, however, for panicle dry weight per plant (WPAN/PL, Table 4.12a). Here, there was a positive relationship of panicle dry weight per plant with the interaction between timothy population density and time (tT*), whereas negative relationships existed with the time by species population density interactions (tO, tT). For panicle dry weight per tiller (WPAN/TL, Table 4.12a), however, time (t, t2) was the only important factor. Timothy tiller number per unit land area (TLN/A, Table 4.12a) had a strong positive relationship with timothy population density, including a few interactions (t2T, OT). There was a very strong negative relationship with the time by timothy population density interaction. A similar relationship also existed for tiller per plant (TLN/PL, Table 4.12b). In that case, however, timothy population density and the orchardgrass by timothy population density interaction were not among the most important factors. In addition, the quadratic time term was positively related to tiller number per plant. The pattern of standard partial regression coefficients for leaf area per plant (LA/PL, Table 4.12b) and leaf area per tiller (LA/TL, Table 4.12b) was very similar. Both variates had a strong positive relationship with respect to time and negative relationships with time quadratic and the time by timothy population density interaction. Leaf area per plant also had a strong positive relationship with the time quadratic by timothy population density interaction. For leaf area per tiller, however, additional important positive factors were timothy population density and the timothy population density by time interaction 9 (tT ), while a strong negative relationship also existed for the quadratic timothy 9 population density term (T ). A very complex relationship was evident for leaf area index (LAI, Table 4.12b). Here, the major positive factors were time, timothy population density, a three factor interaction (tOT), and the quadratic timothy population density by time interaction (tT2). 112 The strong negative factors were the time and timothy population density quadratic terms O 9 (t , T*), and two interactions (tO, and tT). The model for leaf area ratio (LAR, Table 9 4.12b) was relatively simple and time was the key factor (t, t ). For harvest index (H, Table 4.12b), the standard partial regression coefficients were relatively small, and a large portion of the overall variation in harvest index was 9 unexplained by the model (R = 0.05). A comparison of the best subset models for orchardgrass and timothy indicates that models for variates expressed on per unit land area basis (e.g. BD, TLN/A, LAI) were more complex than models for variates expressed on a per plant basis. Generally, the . 9 main sources of yield variation were the crop age (t, t ), and its interactions with species population density (tO, tT). 4.2.3 Total Mixture Best Subset Model A best subset multiple regression analysis was also performed on the total mixture shoot biomass density (MBD, Table 4.13) data. Total mixture biomass density was positively related to the linear component of the main experimental factors (t, O, T) and 9 9 the time quadratic by species population density interactions (t T, t O). Strong negative factors included time by species population density interactions as well as the time quadratic. The goodness of fit for the model was very satisfactory (R 2 = 0.88) with a negligible residual variation (0.09). The model for total mixture shoot biomass density was similar to the corresponding models for the component species. The standard partial regression coefficient for the timothy population density term was higher than that of orchardgrass indicating their relative importance in determining combined mixture performance. This also implies that timothy was a higher yielding species in mixtures. Table 4.13 Parameters and regression statistics for the best subset model for total mixture shoot biomass density P o t e n t i a l S t a n d a r d p a r t i a l r e g r e s s i o n i n d e p e n d e n t c o e f f i c i e n t s v a r i a t e s I n t e r c e p t - 3 . 2 t * 5 . 1 0 1 . 9 T 2 . 5 to - 3 . 0 t i ' - 3 . 2 OT - 0 . 4 tOT 0 . 4 t 2 - 3 . 9 t 2 0 1.1 t 2 T 1 .4 t 2 0 T _ 0 2 - 0 . 6 0 2 t 0 . 6 0 2 T 0 2 t T _ T 2 - 0 . 8 T 2 t 0 . 4 T 2 0 _ T 2 t 0 - 0 . 1 o t h e r r e g r e s s i o n s t a t i s t i c s CP 1 4 . 7 4 RMS 0 . 0 9 R2 0 . 8 8 d . f . 8 2 4 . * t d e n o t e s t i m e a f t e r p l a n t i n g (days ) 0 d e n o t e s o r c h a r d g r a s s p o p u l a t i o n d e n s i t y ( p l a n t s m"2) T d e n o t e s t i m o t h y p o p u l a t i o n d e n s i t y ( p l a n t s m~2) 114 4.3 Yield-density Relationships' Reciprocal equations were used to define relationships between shoot dry weight per plant and population density (Part 3.4.2.2). Through their paramater values, those equations will provide some insight into interference. 4.3.1 Monoculture Yield-density Relationships Eq. 2.20, was fitted to the shoot dry weight data of each species in monoculture at individual harvests as described in Part 3.4.2.2. Before discussing the results obtained, it may be helpful to consider how the form of the yield-density relationship is related to different values of the exponent. 4.3.1.1 Role of the Exponent in Describing Monoculture Yield-density Relationships Hyperbolic and non-hyperbolic yield-density responses, which are common in biological systems, can be accounted for by varying the exponent in the monoculture model (Eq. 2.20). The effect of varying the exponent on the form of monoculture yield-density relationships, where yield is expressed either as shoot dry weight per plant or per area, is shown in Fig. 4.18. The curves were generated by varying the exponent (-Q) while other parameters (a^  and by) were held constant and equal. The curves were also normalized to a maximum value of 1.0 (within the range of Xj values used) on the ordinate axis to facilitate plotting. Yield per plant declines proportionally with increasing population density, but the decline is proportionately more rapid for less negative values of the exponent (Fig. 4.18A). When the exponent equals -1.0 a hyperbolic relationship exists between yield per area and population density (Fig.4.18B). 'Parabolic' responses (i.e. responses where yield per land area rises to a maximum and then declines) exist when Fig. 4.18 Effect of different exponent values on the form of monoculture yield-density relationships Values of the exponent (-Q) ranging from -0.3 to -2.0 are indicated on the plots. A. Yield per plant, and B. Yield per land area. Relative population density was varied from 0 to 10 (arbitrary units), with aj and bjj set at 1.0, to predict yield according to Eq. 2.20. For any of the arrays so produced, maximum yield in the array was set at 1.0 and the rest of the array was adjusted correspondingly. Without such standardization, relative yields would be highest for less negative values of the exponent. R E L A T I V E P O P U L A T I O N D E N S I T Y 117 the exponent is greater than -1.0 (e.g. -0.8). When the exponent is less than -1.0, yield per land area rises without limit. If the other model parameters, and by, are varied, the form of the trends in Fig. 4.18 is unaffected. As noted earlier (Part 2.3.1), biological meaning has been attributed to the three model parameters. 4.3.1.2 Orchardgrass Monocultures Parameter values and statistics for the reciprocal relationships (Eq. 2.20) developed from the orchardgrass monoculture data for each harvest are summarized in Table 4.14. The relationships were statistically significant in all cases, with I 2 values reaching as high as 0.85. The residual sum of squares for the models were also low. The o 9 I values varied among harvests, however, with a low Y" at the first harvest and a maximum at 63 days. The size of exponent, which is related to the acquisition and utilization of resources within the space available per plant (Watkinson, 1980, 1984; Jolliffe, 1986), was close to zero (-0.01) during early growth and at the final harvest (37, 48 and 90 days), indicating poor acquisition and utilization of resources in the space available per plant (1/Xj). The exponent reached as low as -0.94 at 77 days, which suggests a higher efficiency of resource acquisition and utilization. As scaled by the exponent, the intercept (ap represents the yield (shoot dry weight) of an isolated plant. Hence, the size of this parameter is a measure of the yield potential of a plant in the absence of intraspecific interference. Parameter b^ measures how per plant yield declines with increasing population density of the same species. Hence, this parameter expresses plant responsiveness to intraspecific interference. Because the relationship of both aj and bjj to yield is scaled by Q, comparisons between harvests and species in Table 4.14 are hampered. Where values of Q were similar (at 37, 48 and 90 118 Table 4.14 Parameters and regression statistics for orchardgrass monoculture models (Eq. 2.20, n=24) Age Q P a r a m e t e r s ± 95% C . L . I 2 RSS ( d a y s ) V a l u e 37 0 . 0 1 a 1 .0258 ± 0 . 0 0 4 4 0 . 2 0 * 0 . 0 1 b 0 . 0 0 2 3 ± 0 . 0 0 1 7 48 0 . 0 1 a 1 .0124 ± 0 . 0 0 2 4 0 . 6 8 * * * 0 . 0 4 b 0 . 0 0 3 5 ± 0 . 0 0 0 9 63 0 . 8 0 a 0 . 6 5 9 8 ± 0 . 6 3 0 9 0 . 8 5 * * * 0 . 1 3 b 1 .2195 ± 0 . 2 4 2 6 77 0 . 9 4 a 0 . 2 9 8 1 ± 0 . 8 6 1 8 0 . 6 2 * * * 0 . 7 4 b 1 . 4 4 1 9 ± 0 . 3 3 1 4 90 0 . 0 1 a 1 .0017 ± 0 . 0 0 3 1 0 . 5 3 * * * 0 . 7 7 b 0 . 0 0 4 2 ± 0 . 0 0 1 2 * , * * , * * * S i g n i f i c a n t a t 5%, 1% and 0 . 1 % l e v e l , r e s p e c t i v e l y f o r o v e r a l l r e g r e s s i o n model r e l a t e d t o y _ Q . N o t e , f o r t h e s e models t h e d i m e n s i o n o f X i s t h o u s a n d s o f p l a n t s per s q u a r e m e t r e , and y Is shoot d ry w e i g h t (g) per p l a n t . 119 DAP), there was no significant difference among the monocultures in values for b ;^ differences in values for aj, however, were significant but small in magnitude. Graphs showing the effects of monoculture population density on orchardgrass shoot dry weight per plant (Fig. 4.19A) and shoot biomass density (Fig. 4.20A) were developed using the model parameters (Table 4.14). Shoot dry weight per plant showed a declining response to higher population density, whereas the shoot biomass density reached an asymptote at about 63 DAP. The effect of intraspecific interference was low early during growth and became more intense with time. 4.3.1.3 Timothy Monocultures Model parameters and related regression statistics for timothy monocultures are given in Table 4.15. Again, all the regressions were statistically significant (P < 0.01). A large portion of the variation in yield was accounted for by the models as indicated by the o high V" values, reaching as high as 0.98 at 63 DAP. Residual variation, i.e. residual sum of squares for the models, was very low. The exponent value was closest to zero (-0.55) at the first harvest, attained a value of -1.20 at 48 days, and was intermediate in value at later harvests. As with orchardgrass, these variations suggest the occurrence of changing patterns of resource acquisition and utilization during plant growth. Generally, the exponent values for timothy were more negative than for orchardgrass. Figs. 4.19B and 4.20B illustrate the fitted responses, drawn from the model parameters (Table 4.15), for timothy monoculture yields both on a per plant and a per land area basis. The yield-density relationships in timothy monocultures were similar to orchardgrass. In the case of timothy, however, shoot biomass density approached the asymptote at about 77 DAP. The intraspecific interference in timothy was higher than orchardgrass as indicated by the steeper slopes in Fig. 4.19B and by the values of b-^ in Table 4.15. 120 Fig. 4.19 Effects of monoculture population density on shoot dry weight per p l a n t i n orchardgrass and timothy A. Orchardgrass B. T i m o t h y Symbol T i m e after p l a n t i n g (d) a a 37 O o 48 A A 63 X * 77 0 0 90 POPULATION DENSITY Plants m 122 F i g . 4.20 Effects of monoculture population density on shoot biomass density in orchardgrass and timothy A. Orchardgrass B. Timothy Symbol Time after planting (d) a • 3 7 ° ° 48 A A 6 3 A 77." TT 777.7 77.".. A 7 7 1 2 4 Table 4-. 15 Parameters and regression statistics for timothy monoculture models (Eq. 2.20, n=24) Age Q P a r a m e t e r s ±. 95% C L . I 2 RSS ( d a y s ) V a l u e 37 0 . 5 5 a 2 . 5 6 3 0 ± 0 . 7 0 2 0 0 . 6 1 * * * 0 . 0 2 b 0 . 7 2 8 7 ± 0 . 2 7 5 9 48 1 . 2 0 a - 0 . 7 1 5 3 + 1 . 2 8 6 9 0 . 9 4 * * * 0 . 0 3 b 4 . 2 6 6 2 ± 0 . 5 0 5 8 63 0 . 9 5 a - 0 . 0 7 2 4 ± 0 . 2 7 3 7 0 . 9 8 * * * 0 . 1 0 b 1 .2646 t 0 . 1 0 7 6 77 0 . 9 0 a 0 . 2 5 3 4 + 0 . 3 8 8 5 0 . 9 5 * * * 0 . 1 7 b 0 . 8 1 5 3 ± 0 . 1 5 2 7 90 0 . 8 2 a 0 . 0 8 7 2 ± 0 . 3 0 6 9 0 . 7 7 * * * 2 . 5 6 b 0 . 7 3 7 1 ± 0 . 1 2 0 6 * , * * , * * * S i g n i f i c a n t a t 5%, 1% and 0 . 1 % l e v e l , r e s p e c t i v e l y f o r o v e r a l l r e g r e s s i o n model r e l a t e d t o y " " . N o t e , f o r t h e s e models t h e d i m e n s i o n o f X i s t h o u s a n d s o f p l a n t s per s q u a r e m e t r e , and y i s d r y w e i g h t (g ) per p l a n t . 125 4.3.2 Mixture Yield-density Relationships The additional interference which may occur due to the presence of additional species in a plant population can be accounted for by extending the monoculture model. Two types of model, having the same overall structure but differing in method of development (Eq. 2.20 and 2.23), were fitted to the mixture yield-density data as described in Part 3.4.2.2.2. Before dealing with the results of those procedures, the contribution of model parameters to different types of yield response will be considered briefly. 4.3.2.1 Variation in Model Parameters and Mixture Yield-density Relationships Eq. 2.23 was rearranged to describe yield per land area of each species in a binary mixture: Y x i = y x i x i = <ai + b i i x i + W 1 / Q  Y xj = y x j Xj = (aj +bijxj + bjiXi)-i/Q The effects of varying the model parameters (a, b, and Q) on the form of the mixture yield-density response in a replacement diagram were then evaluated (Fig. 4.21). Each parameter was varied in turn, while keeping the other parameters at nominal values. Variation in aj and Q affected the magnitude of Y x (results not shown), but did not change the patterns produced. That is, all the results resembled Fig. 4.2 IC when either aj or Q were varied and the coefficients (b) were held at 1.0. Variation in the relative values of the coefficients, however, altered the form of the replacement diagrams. Fig. 4.21 shows the results of changing the ratio b /^by while holding b:: = b- = 1.0. In this case, concave trends were found for each species when Fig. 4.21 Effects of variation in the ratio of model coefficients (bjj/by) on the form of mixture yield-density relationships in replacement diagrams Curves were generated and standardized the same as Fig. 4.18. The other model coefficients, bj: and b::, were held at 1.0. 1 2 7 A. b RATIO 0.10 < < - J cr Ld o_ Q D. b RATIO 1.50 o.o-h-—i—-i r vs-i c. b RATIO 1.00 ^ 1.5 i 1.2 H 0.9-0.6-0.3-0.0 6 •» 1 1 1 1 c 1 2 3 4 5 6 SPECIES A 5 4 3 2 1 0 SPECIES B •f 1 r -| E. b RATIO 2.00 -t 1 1 r -| F. b RATIO 4.00 T 1 2 SPECIES A 6 5 4 3 2 1 SPECIES B RELATIVE SPECIES ABUNDANCE 128 bjj/bjj < 1.0, and convex trends when b^/hy > 1.0. As a result of the similar direction of response of each species, total mixture yield also varied. 4.3.2.2 Comparison between Regression Approaches Describing Mixture Yield-density Relationships As described in Part 3.4.2.2.2, the mixture yield-density regressions were either fitted in one step ('conventional multiple regression', Eq. 2.23) or in two steps ('regression in stages', Eq. 2.23). 9 Statistics (I ) for the conventional multiple regressions are provided for each species, total population density and harvest in Appendix 8.3. Those regressions were statistically significant in 37 out of 60 cases. The highest I value obtained for any of these regressions was 0.7. Compared with the regressions developed for the monoculture yield-density relationships discussed above (Tables 4.14 and 4.15), however, the I 2 values were relatively low. The inclusion of an interaction term in this model (bXjXj) resulted in o little improvement in I (results not shown). Parameters and statistics for the regressions in stages are provided in Tables 4.16 9 and 4.17. In 53 out of 60 cases, significant regressions were obtained. I values were comparable in range with those obtained in the monoculture models (Table 4.14 and 4.15). Negative values of I are an oddity which can arise when the differences between the observed values of y and the mean value for y approach zero. These results indicate that the conventional multiple regression procedure is less satisfactory, on statistical grounds, than the regression in stages. In an independent study, Penney (1986) reached a similar conclusion. The regression in stages has the additional advantage of automatically collapsing to the monoculture yield-density model when the density of the competing species is zero. The regression in stages, therefore, was used to define the models utilized in the following sections. 129 Table 4 . 1 6 Parameters and regression statistics for orchardgrass mixture models based on regression in stages (Eq. 2.23, n=5) Age R e l a t i v e Q p o p u l a t i o n v a l u e aj bn ± 95% bi* ± 95% I 2 T e s t ( d a y s ) d e n s i t y C . L . C . L . t F 37 1/6 0 . 0 1 1 .0258 0 . 0 0 2 3 ± 0 . 0 0 1 7 - 0 . 0 0 4 7 ± 0 . 0 1 8 6 0 . 1 6 2/6 0 . 0 0 2 5 t 0 . 0 0 2 3 - 0 . 0 1 -3/6 0 . 0 0 2 3 i 0 . 0 0 4 3 - 0 . 0 3 - -V 6 0 . 0 0 5 3 ± 0 . 0 0 3 4 - 0 . 3 0 - * 5/6 0 . 0 0 3 4 ± 0 . 0 0 1 2 0 . 3 4 - ** 6/6 0 . 0 0 3 6 ± 0 . 0 0 1 6 0 . 0 7 - ** 48 1/6 0 . 0 1 1 .0124 0 . 0 0 3 5 ± 0 . 0 0 0 9 0 . 0 1 4 9 ± 0 . 0 1 4 7 0 . 2 6 * 2/6 0 . 0 1 0 6 ± 0 . 0 0 3 1 0 . 5 5 * ** 3/6 0 . 0 0 6 5 ± 0 . 0 0 3 1 - 6 . 0 6 - ** 4/6 0 . 0 0 6 8 ± 0 . 0 0 1 4 0 . 6 7 * ** 5/6 0 . 0 0 7 8 + 0 . 0 0 1 1 0 . 8 7 * *** 6/6 0 . 0 0 6 6 ± 0 . 0 0 0 8 0 . 9 0 * **# 63 1/6 0 . 8 0 0 . 6 5 9 8 1 . 2 1 9 5 ± 0 . 2 4 2 6 3 . 2 4 3 7 ± 0 . 9 6 2 4 0 . 6 4 * *** 2/6 2 . 2 8 9 5 ± 0 . 3 2 3 4 0 . 5 9 * **# 3/6 2 . 9 1 8 1 ± 0 . 6 3 2 9 0 . 2 4 * ** 4/6 2 . 3 7 2 4 + 0 . 5 7 5 2 0 . 3 2 ** 5/6 1 . 9 6 4 6 ± 0 . 6 9 3 7 0 . 2 5 - ** 6/6 2 . 2 4 5 0 + 0 . 5 0 4 1 0 . 6 6 * ** 77 1/6 0 . 9 4 0 . 2 9 8 1 1 . 4 4 1 9 ± 0 . 3 3 1 4 3 .7721 ± 2 . 9 2 6 7 - 0 . 0 4 _ * 2/6 2 . 6 5 0 5 + 0 . 6 5 6 2 0 . 1 1 * ** 3/6 2 . 9 6 6 8 + 0 . 9 8 9 4 0 . 4 3 * #* 4/6 2 . 2 5 3 1 ± 0 . 8 4 2 0 0 . 3 5 - ** 5/6 4 . 0 8 4 9 ± 2 . 2 2 0 0 0 . 3 2 * ** 6/6 3 . 5 7 6 1 ± 2 . 2 4 7 2 0 . 1 9 - * 90 1/6 0 . 0 1 1 . 0 0 1 7 0 . 0 0 4 2 ± 0 . 0 0 1 2 0 . 0 0 1 7 + 0 . 0 0 8 3 - 0 . 6 4 _ 2/6 0 .0081 t 0 . 0 0 3 3 0 . 4 4 - ** 3/6 0 . 0 0 6 6 ± 0 . 0 0 1 4 0 . 0 6 - ** 4/6 0 .0071 t 0 . 0 0 1 4 0 . 3 4 * ** 5/6 0 . 0 0 6 4 ± 0 . 0 0 0 6 0 . 6 7 * ** 6/6 0 . 0 0 5 5 t 0 . 0 0 1 1 0 . 2 8 - ** « > * * f * * * S i g n i f i c a n t at 5%, 1%, and 0 . 1 % l e v e l , r e s p e c t i v e l y - Not s i g n i f i c a n t a t P < 0 . 0 5 Q, aj and bfj a r e f i x e d at v a l u e s d e r i v e d f rom m o n o c u l t u r e model ( T a b l e 4 . 1 4 ) •F t e s t ' r e f e r s t o t h e s t a t i s t i c a l s i g n i f i c a n c e f o r o v e r a l l r e g r e s s i o n model r e l a t e d t o y " ^ , w h e r e a s , ' t t e s t 1 was a p p l i e d f o r c o m p a r i n g d i f f e r e n c e between r e g r e s s i o n c o e f f i c i e n t s i . e . bjj vs bjj . In t h e r e g r e s s i o n , X j and X^ were e x p r e s s e d i n t h o u s a n d s o f p l a n t s pe r s q u a r e m e t r e , and y x was shoot d ry we ight (g ) per p l a n t . Table 4.17 Parameters and regression s t a t i s t i c s for timothy mixture models based on regression i n stages (Eq. 2.23, n=5) Age Relat ive Q population value (days)density JJ 95% C L . j i ± 95% C L . I 2 Test 37 V 6 0.55 2.5630 0.7287 t 0.2759 0.4136 t 0.5222-0.49 - -2/6 0.3274 ± 0.3067-0.52 - * 3/6 0.1226 t 0.1233 0.89 * -4/6 0.3096 ± 0.1857 0.520 - ** 5/6 0.1296 ± 0.2701 0.624 * -6/6 0.2638 ± 0.2024 0.657 - * 48 1/6 1.20 -0.7153 4.2662 ± 0.5058 3.8504 ± 1.4027 0.10 ** 2/6 2.2690 t 1.4718 0.71 * * 3/6 1.6318 ± 0.7548 0.85 * ** 4/6 1.5289 ± 0.3853 0.94 * *+ 5/6 1.6251 ± 0.5707 0.97 * ** 6/6 1.5098 ± 0.8397 0.68 • ** 63 1/6 0.95 -0.0724 1.2646 ± 0.10766 0.6551 t 0.2987 0.76 * ** 2/6 2.2895 ± 0.3234 0.66 * ** 3/6 0.5529 ± 0.2462 0.77 * ** 4/6 0.5163 ± 0.2004 0.83 * ** 5/6 0.5728 t 0.1188 0.88 * ** 6/6 0.4407 ± 0.0661 0.97 * ** 77 1/6 0.90 0.2534 0.8153 ± 0.1527 0.1249 t 0.2107 0.19 * _ 2/6 0.1041 ± 0.0880 0.92 * * 3/6 0.2989 ± 0.2543 0.42 * 4/6 0.1980 ± 0.0339 1.00 * ** 5/6 0.3137 ± 0.0911 0.90 • ** 6/6 0.4297 t 0.2671 0.69 - * 90 1/6 0.82 0.0872 0.7371 ± 0.1206 0.3609 t 0.3489 0.64 _ * 2/6 0.2885 ± 0.1954 0.79 * * 3/6 0.3546 t 0.0993 0.46 * ** 4/6 0.3046 ± 0.1480 0.89 # #* 5/6 0.3996 * 0.0802 0.83 * ** 6/6 0.2628 ± 0.1310 0.90 • * *, **, *** Signi f ican t at 5%, 1%, and 0.1% l e v e l , respect ively - Not s ign i f i can t at P < 0.05 Q, a4 and b** are fixed at values derived from monoculture model (Table 4.157 'F t e s t ' refers to the s t a t i s t i c a l s ignif icance for overa l l regression model related to y~Q , whereas, ' t t e s t 1 was applied for comparing difference between regression coeff ic ients i . e . bj-t vs b j j . In the regression, Xj and Xj were expressed in thousands of plants per square meter, and y x was shoot dry weight (g) per p lant . 131 4.3.2.3 Mixture Yield-density Relationships: Orchardgrass The model parameters for orchardgrass yield in mixture are provided in Table 4.16. Except for the values of the second coefficient, the parameters are identical to the monoculture models; thus, the second coefficient expresses the strength of the additional interference which arises from the presence of the second species in the mixture after intraspecific interference has been taken into account. The regressions for each total population density and at each harvest, were usually significant. However, the lower population densities yielded non-significant regressions for 3 cases (twice at 37 DAP and once at 90 DAP). Jolliffe (1986) has suggested that the 3'ield-density model (Eq. 2.23) should not necessarily be expected to be successful unless competition for resources was determining yield variation. In that sense, the relative lack of success of the model early in growth and at low population densities (where competition for resources may not be strong) may be used as evidence for the validity of the model. o The I values varied widely in response to population density and harvest time, reaching o as high as 0.90. In a few instances the I values were negative, particularly for lower total population densities, indicating that the model gave a poor expression of the yield-density relationship. Significant differences between the regression coefficients (b^ and by), as indicated by paired't tests', existed at most total population densities and harvests (Table 4.16). Where those differences occurred, by exceeded b—, indicating that interference of orchardgrass by timothy was more powerful than intraspecific interference in orchardgrass. At each harvest, the 95% confidence limits for values of by overlap. Hence, interference of orchardgrass by timothy was not significantly affected by total population densities within harvests. As the scale of the coefficients was influenced by different exponent values, direct comparisions of the coefficients between harvests can not be made. The fitted trends for orchardgrass yield per area in mixture, as shown in replacement diagrams (Figs. 4.22 to 4.26), were computed from the model parameters 132 (Table 4.16) for each population density and harvest. The trends for orchardgrass yield tended to become more concave with increasing age and total population density. In the conventional interpretation of replacement diagrams, this response suggests that timothy is competing strongly with orchardgrass for the same 'space' (de Wit 1960). This will be considered in more detail in Part 4.3.2.5 below. Orchardgrass yields per area were increased by higher population density at the earlier harvests but this response became less obvious as the crop aged. These results complement the ANOVA results (Part 4.1.2.2). 4.3.2.4 Mixture Yield-density Relationships: Timothy Parameters and related regression statistics for timothy mixture yield-density models (Eq. 2.23) are provided in Table 4.17. The pattern of significant regressions among harvests and population densities was generally similar to the results for orchardgrass. In this case, however, the I values were usually higher, The model parameters also varied with total population density and harvest time. Again, significant differences between by and bjj were common. In contrast with the results for orchardgrass, the regressions indicate that timothy was more strongly affected by intraspecific interference than interspecific interference (b^  > bjj). Also in contrast to the orchardgrass results, some of the differences among values of bjj were significant at certain harvests (48, 63, and 77 DAP), according to the 95% confidence limits. Fitted trends for timothy yield per land area in mixture, as shown in replacement diagrams (Figs. 4.22 to 4.26), were computed from the model parameters in Table 4.17. The trends for timothy yield tended to become more convex with increasing age and total population density. In the conventional interpretation of replacement diagrams, this response suggests that timothy is able to utilize different 'space' than orchardgrass (de Wit 1960). This will be considered in more detail in Part 4.3.2.5 below. Like orchardgrass, timothy yields per area were increased by higher population density at the earlier Fig. 4.22 Replacement diagrams for shoot biomass density of each species and for the combined mixtures at 37 days after planting Approximate total population densities given on each sub-plot. Lines on plots are interpolated regressions from Eq. 2.26 and 2.27 (summed to provide total mixture curves), using the parameter values in Tables 4.16 and 4.17 Symbols: Orchardgrass: Open triangles, broken lines Timothy: Crosses, chaindot lines Total mixture: Open squares, solid lines 1 3 4 280 i E O >-on o o o 00 D. DENSITY = 2640 210 A 140-1 70 0 280-210-1 140 H C. DENSITY = 1980 70H 0 1 2 3 4 5 6 ORCHARDGRASS 6 5 4 3 2 1 0 TIMOTHY E. DENSITY = 3300 F. DENSITY = 3960 1 r ~ r T * • ORCHARDGRASS 6 5 4 3 2 1 0 TIMOTHY R E L A T I V E S P E C I E S A B U N D A N C E Fig. 4.23 Replacement diagrams for shoot biomass density of each species and for the combined mixtures at 48 days after planting Approximate total population densities given on each sub-plot. Lines on plots are interpolated regressions from Eq. 2.26 and 2.27 (summed to provide total mixture curves), using the parameter values in Tables 4.16 and 4.17 Symbols: Orchardgrass: Open triangles, broken lines Timothy: Crosses, chaindot lines Total mixture: Open squares, solid lines 136 CN E o UJ >-Q O O CO 500n 375 4 250 H A . DENSITY = 660 B. DENSITY = 1320 375 H 250 H 375-! 250 H 125 A C. DENSITY = 1980 \ 0 1 2 3 4 5 6 ORCHARDGRASS 6 5 4 3 2 1 0 TIMOTHY D. DENSITY = 2640 -o a-T r E. DENSITY = 3300 F. DENSITY = 3960 1 2 3 4 5 6 ORCHARDGRASS 5 4 3 2 1 0 TIMOTHY R E L A T I V E S P E C I E S A B U N D A N C E Fig. 4.24 Replacement diagrams for shoot biomass density of each species and for the combined mixtures at 63 days after planting Approximate total population densities given on each sub-plot. Lines on plots are interpolated regressions from Eq. 2.26 and 2.27 (summed to provide total mixture curves), using the parameter values in Tables 4.16 and 4.17 Symbols: Orchardgrass: Open triangles, broken lines Timothy: Crosses, chaindot lines Total mixture: Open squares, solid lines 138 RELATIVE SPECIES ABUNDANCE Fig. 4.25 Replacement diagrams for shoot biomass density of each species and for the combined mixtures at 77 days after planting Approximate total population densities given on each sub-plot. Lines on plots are interpolated regressions from Eq. 2.26 and 2.27 (summed to provide total mixture curves), using the parameter values in Tables 4.16 and 4.17 Symbols: Orchardgrass: Open triangles, broken lines Timothy: Crosses, chaindot lines Total mixture: Open squares, solid lines 1 4 0 CN I E O UJ >-Q O O CO 1600-1 1200H 8 0 C H A. DENSITY = 660 B. DENSITY = 1320 1200H 800H 4004 + a 0 1600-1 1200H 800H 400 H T ~ 1 r T r C. DENSITY = 1980 ORCHARDGRASS 6 5 4 3 2 1 TIMOTHY D. DENSITY = 2640 E. DENSITY = 3300 F. DENSITY = 3960 0 1 2 3 4 5 ORCHARDGRASS 6 5 4 3 2 1 TIMOTHY t 6 RELATIVE SPECIES ABUNDANCE Fig. 4.26 Replacement diagrams for shoot biomass density of each species and for the combined mixtures at 90 days after planting Approximate total population densities given on each sub-plot. Lines on plots are interpolated regressions from Eq. 2 . 2 6 and 2 . 2 7 (summed to provide total mixture curves), using the parameter values in Tables 4 . 1 6 and 4 . 1 7 Symbols: Orchardgrass: Open triangles, broken lines Timothy: Crosses, chaindot lines Total mixture: Open squares, solid lines 1 4 2 1800-1 1350 9Q0H A. DENSITY = 660 450H CM 1800-.'E cn 1350-' r— o 900-U J 450->-Q 0^ r — 1800-o o ~T~ 1350-CO B. DENSITY = 1320 o 9 0 0 H 450M "5 2 3 * ORCHARDGRASS 6 5 4 3 2 1 0 TIMOTHY D. DENSITY = 2640 • • • a - i r E. DENSITY = 3300 +*—i 1 r F. DENSITY = 3960 0 1 2 3 4 5 6 ORCHARDGRASS 5 4 3 2 TIMOTHY i RELATIVE SPECIES ABUNDANCE 143 harvests, and this response became less obvious as the crop aged. Timothy proved to be the more productive species in the mixtures. These results compliment the ANOVA results (Part 4.1.2.3) 4.3.2.5 Species Relative Competitive Abilities Spitters (1983a) has proposed that the ratio of regression coefficients (b-j/bjj or by/bjj) provides a meaningful index for the relative response of intraspecific interference to interspecific interference. If the ratio is less than 1.0 interspecific interference exceeds intraspecific interference and vice versa. Such ratios were calculated for orchardgrass and timothy, using the data in Tables 4.16 and 4.17 respectively, for each total population density and harvest (Table 4.18). The results for orchardgrass shows that interspecific interference was predominant. The orchardgrass results, however, did not show an obvious pattern among total population densities or harvest times. In the case of timothy, the ratio of regression coefficients exceeded 1.0. This implies that intraspecific interference in timothy was consistently greater than interspecific interference. At the 48 and 63 day harvests, there appears to be an increase in the ratio with increasing population densities. Relatively intense intraspecific interference was evident at the 77 day harvest, producing the highest ratio of regression coefficients (7.8). This occurred at a time when a resumption of rapid growth in timothy was visible in the field. Thus, the regression coefficients (Eq. 2.23) can provide a convenient way to partition species interference into intraspecific and interspecific components. Similar inferences about interference are usually drawn from inspection of replacement diagrams. Fitted responses for shoot biomass density were presented in replacement diagrams (Figs. 4.22 to 4.26). Curves were usually concave for orchardgrass and the convex for timothy. The responses tended to intensify with crop age and with increasing population density. Curves for both species were linear at lower population densities early during Table 4.18 Relative sensitivity of orchardgrass and timothy to intraspecific and interspecific intererence R e l a t i v e T ime a f t e r p l a n t i n g (days ) p o p u l a t i o n d e n s i t y 37 48 63 77 90 R a t i o o f r e g r e s s i o n c o e f f i c i e n t s * O r c h a r d g r a s s 1/6** ns*** . 2 3 5 . 3 7 6 . 3 8 2 n s 2/6 . 9 2 2 .331 . 5 3 3 . 5 4 4 . 5 2 5 3/6 ns . 5 3 9 . 4 1 8 . 4 8 6 . 6 3 5 4/6 . 4 3 5 . 5 1 1 . 5 1 4 . 6 4 0 .591 5/6 . 6 7 3 . 4 3 9 .621 . 3 5 3 .661 6/6 . 6 4 0 . 5 3 1 . 5 4 3 . 4 0 3 . 7 6 3 T i m o t h y 1/6 ns 1.11 1 . 9 3 ns 2 . 0 4 2/6 2 . 2 3 1 .88 1 . 7 9 7 . 8 4 2 . 5 5 3/6 ns 2 . 6 2 2 . 2 9 2 . 7 8 2 . 0 8 4/6 2 . 3 5 2 . 7 9 2 . 4 5 4 . 1 2 2 . 4 2 5/6 ns 2 . 6 3 2 . 2 1 2 . 6 0 1 .84 6/6 2 . 7 6 2 . 8 3 2 . 8 7 1 . 9 0 2 . 8 0 * i . e . bji/bij d e r i v e d f rom t h e r e g r e s s i o n m o d e l . I f t h e r a t i o i s l e s s than 1 i n t e r s p e c i f i c I n t e r f e r e n c e e x c e e d s i n t r a s p e c i f i c i n t e r f e r e n c e ; I f t h e r a t i o i s more t h a n 1 i n t r a -s p e c i f i c i n t e r f e r e n c e e x c e e d s i n t e r s p e c i f i c i n t e r f e r e n c e . * * Maximum p o p u l a t i o n d e n s i t y (6/6 = 4007 p l a n t s m"2 f o r o r c h a r d g r and 3920 p l a n t s m f o r t i m o t h y . * * * no s i g n i f i c a n t e f f e c t ( n s ) o f t h e c o m p e t i n g s p e c i e s 145 growth (37 DAP) indicating little interference. These results also show that timothy caused greater interference, both for itself and for orchardgrass. 4.3.2.6 Differential Yield Responses The net differential yield responses (i.e. underyielding and overyielding) for individual species and the combined mixtures are shown in Figs. 4.27C to 4.32C. These fitted curves were generated using Eqs. 2.27 and 2.28 and the model parameters given in Tables 4.16 and 4.17. Overyielding commonly occurred for the.combined mixtures (Figs. 4.28C to 4.32C) because overyielding in timothy exceeded underyielding in orchardgrass. Net overyielding was most pronounced in mixtures having about a 2:1 proportion of the two species (Orchardgrass: Timothy). The net yield advantage of mixtures reached over 300 g m"^, which is equivalent to 3 metric tonnes per hectare (Fig 4.32C). At the two lowest total population densities, net overyielding did not become pronounced until the third harvest (63 DAP). Otherwise, net overyielding usually tended to increase with increased crop age. The effect of total population density was less obvious, although overyielding reached its greatest value at the highest total population density. For the individual species, there was usually underyielding in orchardgrass (Figs.4.27A to 4.32A), except at the initial and final harvests at low total population densities. Underyielding in orchardgrass was most intense at the 48, 63 and 77 day harvests. Timothy always contributed to net yield advantage except at low population density on the first harvest (Figs. 4.27B to 4.32B). Overyielding in timothy increased rapidly until the 63 day harvest, and subsequent changes were less pronounced. Fig. 4.27 Differential yield responses for component species and combined mixtures at 1/6 relative total population density (approximately 660 plants m"^ ) A. Orchardgrass B. Timothy C. Mixture total. Yield advantage was computed from the difference between Eq. 2.27 and 2.28, using parameter values from Tables 4.16 and 4.17. 147 Fig. 4.28 Differential yield responses for component species and combined mixtures at 2/6 relative total population density (approximately 1320 plants m"^ ) A. Orchardgrass B. Timothy C. Mixture total. Yield advantage was computed from the difference between Eq. 2.27 and 2.28, using parameter values from Tables 4.16 and 4.17. 149 Fig. 4.29 Differential yield responses for component species and combined mixtures at 3/6 relative total population density (approximately 1980 plants m"^ ) A. Orchardgrass B. Timothy C. Mixture total. Yield advantage was computed from the difference between Eq. 2.27 and 2.28, using parameter values from Tables 4.16 and 4.17. Fig. 4.30 Differential yield responses for component species and combined mixtures at 4/6 relative total population density (approximately 2640 plants m"^ ) A. Orchardgrass B. Timothy C. Mixture total. Yield advantage was computed from the difference between Eq. 2.27 and 2.28, using parameter values from Tables 4.16 and 4.17. Fig. 4.31 Differential yield responses for component species and combined mixtures at 5/6 relative total population density (approximately 3300 plants m"^ ) A. Orchardgrass B. Timothy C. Mixture total. Yield advantage was computed from the difference between Eq. 2.27 and 2.28, using parameter values from Tables 4.16 and 4.17 «it «vt i»» »«• - n Fig. 4.32 Differential yield responses for component species and combined mixtures at 6/6 relative total population density (approximately 3960 plants m"^ ) A. Orchardgrass B. Timothy C. Mixture total. Yield advantage was computed from the difference between Eq. 2.27 and 2.28, using parameter values from Tables 4.16 and 4.17. 157 158 4.4 Plant Growth Analysis Modern techniques of plant growth analysis were used to follow the dynamics of plant response to experimental treatments, using growth indices which have physiological significance. Warren Wilson (1981) organized growth indices relating to crop growth rate: CGR = (XjXAGR) = (BD)(R) = (BD) (LAR) (ULR) = (LAI)(ULR) (4.1) One interesting aspect of this set of relationships is that species population density is included directly as a component of crop growth rate. Eq. 4.1 is true according to the definitions of the various growth indices (Table 3.5). Because the same primary variates (e.g. W, LA) appear at several places in the indices contained in Eq. 4.1, Hardwick (1984) warned about the possibility of tautology (i.e. the useless restatement of the same information). Each growth index, however, is useful by itself as an expression of some particular aspect of growth. In the following presentation, the indices will be considered separately, although the conceptual framework offered by Eq. 4.1 can be kept in mind. Results for each species will be considered in turn. The ratio indices (LAI, LAR and H), which were analyzed by ANOVA as well as graphical inspection, will be considered first, followed by indices involving first derivatives (AGR, R, CGR and ULR). 4.4.1 Orchardgrass: Growth Indices 4.4.1.1 Orchardgrass: Leaf Area Index Leaf area index of orchardgrass showed a strong response to treatment factors (Appendix 8.4.1). On the basis of larger F-values, mixture effects were more powerful than effects of population density. Population density effects were significant up to 63 DAP, and most of the variation was accounted for by linear and quadratic components. The response of LAI to mixtures was mainly linear throughout the study, although a 159 quadratic response remained significant until 63 DAP. Few significant interactions existed. In accord with the trend in leaf area per plant noted earlier, there was a general ingrease in LAI in orchardgrass up to the last harvest (Fig. 4.33). LAI of orchardgrass in monocultures was usually higher than in mixtures, and this was presumably related directly to relative species proportions. The effects of total population density were small compared with mixture effects. A maximum LAI value of about 14 was obtained in orchardgrass monocultures at the highest population densities. 4.4.1.2 Orchardgrass: Leaf Area Ratio ANOVA results (Appendix 8.4.2.2), indicated that both population density and mixture effects on LAR were significant at later harvests; however, the variance ratios were relatively small. Across all mixtures and population densities, orchardgrass LAR tended to decline with age until 77 DAP (Fig. 4.34). At the final harvest, however, the LAR curves began to rise in most cases. Higher population density seemed to depress LAR, athough the effects were not strong. Similarly, the differences among mixtures were small. 4.4.1.3 Orchardgrass: Absolute Growth Rate Fig. 4.35 illustrates the time course of orchardgrass absolute growth rate per shoot. As with the other indices involving first derivatives, the statistical significance of treatment effects on AGR could not be tested directly by ANOVA. The statistical significance of the responses, however, should be inferred from the ANOVA results on the primary variates (e.g. shoot dry weight, Table 4.4) from which the fitted growth curves were derived. AGR exhibited wide variations in response to crop age, population densities and mixtures (Fig. 4.35). Generally, AGR remained relatively low and stable at high 160 Fig. 4.33 Time course of leaf* area index in orchardgrass One mixture per sub-plot. Proportions of Orchardgrass: Timothy as indicated on sub-plots Symbol Orchardgrass population density (plants m" )^ ° ° 1336 A A 2004 X X 2671 $ A 3339 v V 4007 TIME AFTER PLANTING d Fig. 4.34 Time course of leaf area ratio in orchardgrass One mixture per sub-plot. Proportions of Orchardgrass: Timothy as indicated on sub-plots Symbol Orchardgrass population density (plants m"^ ) 0.030-0.025-0.020-1 0.015 0.010-0.005-- 0.000 A 1 : 5 cr. 0.030-CM E 0.025-o r— 0.020-< < 0.015-UJ < 0.010-u. < UJ 0.005-_J 0.000-0.030-0.025-0.020-0.015-0.010-0.005-C 3 : 3 E 5 : 1 Q 0.000 35 —r~ 50 —r— 65 B 2:4 -I — l 1 D 4:2 1 r i 50 80 95 35 TIME AFTER PLANTING d 65 80 164 Fig. 4.35 Time course of absolute growth rate in orchardgrass One mixture per sub-plot. Proportions of Orchardgrass: Timothy as indicated on sub-plots Symbol Orchardgrass population density (plants m'^ ) 668 1336 2004 2671 3339 4007 A 1:5 i T 3 U l h -< o o U l 0.010-0.005-0.000-o to CD < -0.005 -0.005 B 2:4 i r 1 1 1 — — l 35 50 65 80 95 F 6:0 — i 1 1 i 35 50 65 80 95 TIME AFTER PLANTING d 166 population densities throughout growth. The wide variations in AGR under lower population densities are less easy to summarize. AGR reached its highest values, up to 0.024 g per daj' at about 55 days, at the lowest population densities. The occurrence of high AGR values is another indication of the lower intensity of interference at low population densities. Negative AGR values at the early harvest (Fig. 4.35A) suggest some over-fitting of the spline regressions, although this problem does not seem to be too serious. Differences in AGR due to mixtures were less obvious, although the plots may suggest that AGR tended to rise with increasing orchardgrass proportion in the mixtures. 4.4.1.4 Orchardgrass: Relative Growth Rate The trends for R of orchardgrass shoots were highly variable among the different treatments. In many cases R was higher during initial growth, and then declined (Fig. 4.36). This pattern was less apparent for the 1:5 mixture where R was often higher late in growth (77 to 90 days). Early in growth, R was relatively high at the lowest population density. Otherwise the effects of population density and mixture treatments were inconsistent. Relative growth rate was originally introduced as an 'efficiency index' of dry matter accumulation for (average) individual plants. It is a widely used measure of the combined growth performances of the plant, or in this case the shoot. As with the other dynamic growth indices (those including first derivatives) it appears that population density and mixture treatments strongly disturb the chronology of growth in a very complex way. 4.4.1.5 Orchardgrass Crop Growth Rate Crop growth rate (CGR) expresses the rate of shoot biomass density accumulation. The progress curves for CGR (Fig. 4.37) do not show any obvious patterns in response to population density. Mixtures having lower orchardgrass proportion coupled with lower population density had relatively stable values of CGR over the entire course of growth. 167 F ig . 4.36 Time course of relative growth rate in orchardgrass One mixture per sub-plot. Proportions of Orchardgrass: Timothy as indicated on sub-plots Symbol Orchardgrass population density (plants m"^) 6 6 8 1 3 3 6 2 0 0 4 2 6 7 1 3 3 3 9 4 0 0 7 0.20-i 0.15-0.10-0.05-0.00-cn < A 1:5 -0.05 0.20 0.15 H o O U J > < U J -0.05 0.20-1 0.15 H C 3:3 -0.05 E 5 :1 1 1 1 1 35 50 65 80 95 B 2:4 «• - T •«• « / \ - - ^ i — i — i — i D 4 : 2 F 6:0 1 1 1 1 35 50 65 80 95 TIME AFTER PLANTING d 169 F ig . 4.37 Time course of crop growth rate in orchardgrass One mixture per sub-plot. Proportions of Orchardgrass: Timothy as indicated on sub-plots Symbol Orchardgrass population density (plants m"^) 6 6 8 1 3 3 6 2 0 0 4 2 6 7 1 3 3 3 9 4 0 0 7 2 4 . 0 - 1 16.0H 8 . 0 H o.oH CVJ I A 1 : 5 E - 8 . 0 2 4 . 0 - | L d < 16.0H i r O on o Q_ O Ol O C 3 : 3 8 . 0 H o.oH - - ^ * ^ — - 8 . 0 2 4 . 0 - 1 1 6 . 0 H 8 . 0 H o.oH i r E 5 : 1 - 8 . 0 3 5 —r— 5 0 —J— 6 5 I 8 0 I 9 5 1 7 0 B 2:4 1 1 D 4:2 i r F 6 : 0 3 5 5 0 6 5 8 0 9 5 TIME AFTER PLANTING d 171 Rapid fluctuations in CGR were especially evident in monocultures and in mixtures where orchardgrass proportions were high (4:2 and 5:1). 4.4.1.6 Orchardgrass : Unit Leaf Rate Unit leaf rate (ULR), sometimes called net assimilation rate, is strongly dependent on the major processes of resource acquisition, especially net photosynthesis (Hunt 1982). In the majority of treatments, ULR of orchardgrass decreased regularly during the course of plant growth (Fig. 4.38). In most cases, the lowest population density resulted in the highest ULR values, and such differences were more pronounced in early stages of growth. There was no clear pattern in variations in ULR among different mixtures. 4.4.2 Timothy: Growth Indices 4.4.2.1 Timothy: Leaf Area Index ANOVA results for LAI of timothy (Appendix 8.4.3) show that, as with orchardgrass, mixture effects were relatively prominent compared to population density -effects. Population density effects were significant, however, at early harvests and at the final harvest (37, 48 and 90 DAP). Significant components of the population density effects were mostly linear and quadratic. For mixtures, the linear component was significant throughout the experiment. An additional quadratic component for mixtures was significant at the 77 day harvest. A few interactions were significant but they had no obvious pattern. Fig. 4.39 shows that the time course of changes in LAI for timothy were quite different from orchardgrass. LAI usually rose during early growth, reached the peak at about 48 DAP and then declined during later stages. The declines in LAI at later harvests appear to be related to decreasing leaf area per plant (Fig. 4.16) and per tiller (Fig. 4.17) because tillering in timothy was not reduced during that period (Fig. 4.14). Higher LAI values were obtained with higher population densities, and with mixtures which contained 172 F ig . 4.38 Time course of unit leaf rate in orchardgrass One mixture per sub-plot. Proportions of Orchardgrass: Timothy as indicated on sub-plots Symbol Orchardgrass population density (plants m"^) 668 1336 2004 2671 3339 4007 TIME AFTER PLANTING d Fig; 4.39- T i m e course of l e a f a r e a index i n t i m o t h y One mixture per sub-plot. Proportions of Orchardgrass: Timothy as indicated on sub-plots Symbol Timothy population density (plants m'^) TIME AFTER PLANTING d 176 high timothy proportions. A maximum LAI of 12 was achieved at about 50 DAP in the monoculture having approximately 3250 plants m . The graphs (Fig. 4.39) support the conclusion drawn from the ANOVA that LAI responses to mixture proportions were more prominent than those due to population density. 4.4.2.2 Timothy: Leaf Area Ratio The ANOVA indicated that LAR in timothy did not respond strongly to either of the treatment factors (Appendix 8.4.4). However, block effects were significant at all harvests. The time course of LAR is shown in Fig 4.40. LAR was higher initially and then declined rapidly over time. In plant growth analysis, LAR is used to evaluate the extent of the major assimilatory organs (the leaves) within plants. The relative insensitivity of LAR in timothy to population density and mixture treatments, compared to orchardgrass (Fig. 4.34), may contribute to the differences between the two species in their responsiveness to interference. 4.4.2.3 Timothy: Harvest Index Harvest index (H) expresses the the ratio of some aspect of economic yield to some measure of overall yield. In this thesis, H is arbitrarily used as the ratio of panicle dry weight to total shoot dry weight per plant. H was evaluated only for timothy because orchardgrass did not initiate panicles during the experiment. The ANOVA (Appendix 8.4.5) indicated that H did not respond significantly to either of the treatment factors, except for one case: the results at 77 DAP, where population density effects were significant. Again, block effects were significant at all harvests. Fig. 4.41 shows that H often tended to peak at about 77 days and declined thereafter. On average, dry matter allocation into panicles amounted to about 11 per cent of total shoot dry weight. F i g . 4.40 Tune course of leaf area ratio in timothy One mixture per sub-plot. Proportions of Orchardgrass: Timothy as indicated on sub-plots Symbol Timothy population density (plants m*2) 1 7 8 B i r i 1 D 3 : 3 1 1 F 5 : 1 0.000 35 50 -T- r-65 80 •—1 95 TIME AFTER PLANTING d 179 Fig. 4.41 Time course of harvest index in timothy One mixture per sub-plot. Proportions of Orchardgrass: Timothy as indicated on sub-plots Symbol Timothy population density (plants m"^ ) a a 6 5 3 a O 1 3 0 7 As — A 1 9 6 0 jt '. * 2 6 1 3 A , $ 3 2 6 7 V — T T - 3 9 2 0 TIME AFTER PLANTING d 181 4.4.2.4 Timothy: Absolute Growth Rate Fig. 4.42 shows the progress curves for timothy absolute growth rate. In some treatments, AGR was initially close to zero, peaked at about 63 DAP and declined during later growth. In other cases, especially at higher population densities, AGR in timothy remained relatively constant throughout growth. As with orchardgrass, AGR was relatively high at lower population densities. The results for AGR in timothy are opposite to those for orchardgrass, however, in that AGR tended to increase as the proportion of timothy in mixtures decreased. This difference in response is another indication that interspecific and intraspecific interference by timothy are both more powerful than by orchardgrass. 4.4.2.5 Timothy: Relative Growth Rate Results for R for timothy indicated high values in the early stages of growth and subsequent declines (Fig. 4.43). Higher population densities resulted in reduced R up to about the middle of the study; thereafter this response was often reversed. This reversal may be due to the offset in values for W as suggested by Jolliffe and Ehret (1985). Compared to monocultures, mixtures tended to give higher R values. R values in timothy and orchardgrass (Fig. 4.36) were about the same during early growth at low population densities. Increasing population density, however, reduced R more in orchardgrass than timothy. Again, this coincides with other assessments this research has made of the relative responsiveness of the two species to interference. 4.4.2.6 Timothy: Crop Growth Rate Timothy CGR usually underwent large fluctuations during the course of growth (Fig.4.44). Although the responses varied among mixtures, CGR was often low initially, reached a peak value at about 63 days and then declined. In some monocultures and mixtures, mainly those having higher timothy proportions, CGR increased after 63 days. 182 Fig. 4.42 Time course of absolute growth rate in timothy One mixture per sub-plot. Proportions of Orchardgrass: Timothy as indicated on sub-plots Symbol Timothy population density (plants m"^ ) 653 1307 1960 2613 3267 3920 0.16-0.12-0.08-0.04-o.oo H I T3 ro -0.04 0.16-1 183 A 0:6 1 1 1 1 UJ I— 2 0.12-1 0.00 H o to m < C 2 : 4 - 1 95 B 1 : 5 1 r D 3 : 3 -i r F 5 : 1 35 50 I 65 80 -l 95 TIME AFTER PLANTING d 184 F i g . 4.43 Time course of relative growth rate in timothy One mixture per sub-plot. Proportions of Orchardgrass: Timothy as indicated on sub-plots Symbol Timothy population density (plants m"^ ) 653 1307 1960 2613 3267 3920 0.20-1 0.15-0.10 0.05 J° 0.00-j cn c n .0.05 0.20 -i A 0:6 i r C 2:4 U J -0.05 0.20-1 0.15 H 0.10 0.05H 0.00 1 1 1 1 E 4:2 -0.05-1 1 1 T 1 35 50 65 80 95 1 8 5 B 1 : 5 — i r D 3 : 3 F 5 : 1 35 50 —r" 65 i 80 95 TIME AFTER PLANTING d 186 Fig. 4.44 Time course of crop growth growth rate in timothy One mixture per sub-plot. Proportions of Orchardgrass: Timothy as indicated on sub-plots Symbol Timothy population density (plants m"^) 653 1307 1960 2613 3267 3920 187 TIME AFTER PLANTING d 188 This surge in growth was visible in the field. Low population density gave the widest range of CGR values. Since population density is a component of CGR, this indicates the contribution of strong variations in AGR to the CGR results. 4.4.2.7 Timothy: Unit Leaf Rate The results for ULR in timothy were highly variable (Fig. 4.45). In a moderate number of cases ULR decreased early in growth and then increased. Relatively wide fluctuations in ULR were found at the lowest population density. There was no obvious pattern in ULR responses to mixtures. Although it is not as clear as with orchardgrass, the results for ULR and the other dynamic indices of growth (those indices containing first derivatives) suggest that population density and mixture treatments can cause large adjustments in the time course of plant growth. It is difficult to interpret such shifts, however, because the patterns of the growth responses were highly variable. 4.5 Allometric Plant Relationships The results so far, including results from ANOVA and plant growth analysis, indicate that different aspects of plant growth were affected differently by the experimental treatments. Those analyses were performed separately on different aspects of growth. That type of approach also could be applied to yield-density responses, where different regressions could be developed for different yield variates. For example, the reciprocal model (Eq. 2.20) was used to model yield-density data on leaf area per plant and tiller number per plant in orchardgrass and timothy monocultures. It was found, however, that the model failed to describe those relationships effectively (I < 0.2, results not shown). Another way to consider the responses of different yield variates is through the use of the allometric model (Eq. 2.29). Experimental factors may influence some measure of 189 Fig. 4.45 Time course of unit leaf rate in timothy One mixture per sub-plot. Proportions of Orchardgrass: Timothy as indicated on sub-plots Symbol Timothy population density (plants m"^ ) 653 1307 1960 2613 3267 3920 TIME AFTER PLANTING d 191 yield per plant directly, or through allometric relationships with other aspects of growth. Allometric relationships underlying treatment responses were evaluated for monocultures and mixtures by using Eq. 8.8 and 8.9 and the best subset multiple regression procedure. 4.5.1 Allometric Relationships: Monocultures Parameters and regression statistics for the best subset models for monocultures are summarized in Table 4.19 (for orchardgrass) and Table 4.20 (for timothy). Eq. 8.8 proved to be an effective model for describing the relationship between shoot dry weight per plant and the plant parts (leaf area per plant and tiller number per plant). The coefficient of determination (R ) for the monoculture models was >^  0.90, and all those models had a normally distributed residual sum of squares. The allometric relationship between shoot dry weight and leaf area per plant in orchardgrass was strongly dependent on the allometric coefficient (a') and some partly non-allometric factors including time after planting (t), population density (Xj) and their interaction (tXj, Table 4.19a). Population density had a significant negative effect, whereas time after planting and the interaction between time and population density influenced the allometry positively. The only significant term involving a component of the allometric exponent was ^°Se(ymp[) which was positive and had the largest value for the standard partial regression coefficient after a'. A different model resulted for the allometric relationship between shoot dry weight per plant and tiller number per plant in orchardgrass (Table 4.19b). The effect of treatment factors on allometric coefficient (a') was similar to that of leaf area per plant, except that the interaction between time after planting and population density (tXj) was not a significant term. The l °g e (y m pi ) term was positive and was the only significant term involving a component of the allometric exponent. For timothy, the allometric relationship of shoot dry weight per plant and leaf area per plant also contained loge(t) and loge(Xj) as significant factors in addition Table 4-. 19a Parameters and regression statistics for the best subset allometric model for orchardgrass in mono-culture (leaf area per plant) P o t e n t i a l Independent v a r i a t e s @ P a r a m e t e r v a l u e s S t d . p a r t i a l r e g r e s s i o n c o e f f i c i e n t s I n t e r c e p t l o g e ( t ) l o g e ( X j ) l o g e ( t X j ) ]oge(ympi) t l o g e ( y m p i ) x i l o 9 e ( y m p i \ t X 5 l o g e ( y m p i ) c 1 c 2 bo b3 •8.437 3 . 0 8 1 - 1 . 3 0 5 0 . 2 5 2 0 . 0 1 1 - 9 . 3 1 * 1 . 0 9 * * - 0 . 8 8 * 0 . 9 7 * 1 . 1 3 * * * Other r e g r e s s i o n s t a t i s t i c s CP RMS R2 d . f . 4 . 3 9 0 . 0 6 0 . 9 3 115 @ t = T ime a f t e r p l a n t i n g i n d a y s , X = P o p u l a t i o n d e n s i t y , y m j = Shoot d ry we ight per p l a n t , y m p i = L e a f a r e a per p l a n t * f *#> * * * s i g n i f i c a n t a t 5%, 1% and 0 . 1 % l e v e l , r e s p e c t i v e l y Table 4.19b Parameters and regression statistics for the best subset allometric model for orchardgrass in mono-culture ( t i l ler number per plant) P o t e n t i a l Independent v a r i a t e s § P a r a m e t e r v a l u e s S t d . p a r t i a l r e g r e s s i o n c o e f f i c i e n t s I n t e r c e p t a " - 9 . 5 6 9 - 1 0 . 5 6 * * * l o g e ( t ) c\ 2 . 1 3 5 0 . 7 6 * * * ] o g e ( X t ) c 2 - 0 . 1 7 0 - 0 . 1 1 * l o g e ( t X i _ ) c 3 " 1(>ge(ympi) *<) ° - 8 8 5 0 . 4 8 * * * t l 0 9 e ( y m p i > b 1 x i l o g e ( y m p i ) b 2 -t X i l o g e ( y m p I ) b 3 -Other r e g r e s s i o n s t a t i s t i c s CP RMS R2 d . f . 2 . 7 6 0 . 0 8 0 . 9 0 116 @ t = T ime a f t e r p l a n t i n g i n d a y s , X = P o p u l a t i o n d e n s i t y , y m j = Shoot d ry we ight per p l a n t , y m p j = L e a f a r e a per p l a n t * t * * f * * * S i g n i f i c a n t a t 5%, 1% and 0 . 1 % l e v e l , r e s p e c t i v e l y Table 4 .20a Parameters and regression s t a t i s t i c s for the best subset allometric model for timothy i n monoculture (leaf area per plant) Potential Parameter Std. partial Independent values regression variates @ coefficients Intercept a ' -12.419 -12.80*** log e(t) c\ 4.375 1.45*** loge(X,) c 2 -0.633 -0.40*** loge(tXj) C3 -l og e(ympi) DQ -t ] ° g e ( y m p i ) B 1 x i l°ge (ympi ) b 2 tXilogefyinpi) b3 0.26 X 10"5 0.74*** Other regression statistics CP 2.99 RMS 0.04 R2 0.96 d.f. 116 @ t = Time after planting in days, X = Population density, ymj = Shoot dry weight per plant, ympj = Leaf area per plant *, **, *** Significant at 5%, 1% and 0.1% level, respectively Table 4.20b Parameters and regression statistics for the best subset allometric model for timothy in monoculture ( t i l ler number per plant) Potential Parameter Std. partial Independent values regression variates @ coefficients Intercept a ' -17.951 -18.50*** log e(t) cj 5.780 1.92** loggfXj) C2 0.957 0.60 ns loge(tX|) C3 l oge ( vmpi>- bO t ] °ge ( ympi> b1 x i l o g e ( ym P i ) b 2 -tXiloge(ympI> b 3 Other regression statistics CP RMS R2 d.f. 4.52 0.07 0.92 116 @ t = Time after planting in days, X = Population density ymj = Shoot dry weight per plant, ympj = Leaf area per plant *, **, *** Significant at 5%, 1% and 0.1% level, respectively ns = not signfleant at 5% level 196 to a' (Table 4.20a). Again the population density term was negative while time was positive. A complex interaction, 6gtXjloge(ympj), was positive and was the only significant component of the allometric exponent. The allometric relationship between timothy shoot dry weight per plant and tiller number per plant is given in Table 4.20b. This was the simplest model of all. Loge(t) was positive and was the only significant factor having an effect on the allometric coefficient. 4.5.2 Allometric Relationships: Mixtures The analysis of allometric responses to experimental treatments in mixtures followed a similar approach. The number of potential independent variables was increased through the inclusion of the population density of the second species (Xj) as an additional factor (Eq. 8.9). The monoculture data were not included in this analysis. The mixture models (Tables 4.21 and 4.22) were all relatively complex compared to the monoculture models. Again, the models were effective in accounting for variation in the dependent variable, with R 2 values >^  0.88. Parameters and regression statistics for the allometric relationship between mixture shoot dry weight per plant and leaf area per plant in orchardgrass are given in Table 4.21a. Time after planting (loge(t)) made a strong positive contribution to the allometric coefficient, whereas the species population density interaction (loge(XjXj)) resulted in a negative effect. Significant factors contributing to the allometric exponent included log e(y x p i), tloge(yxpj) and Xjlog e(yx p i), all of which were positive. It should be noted that several terms which had been significant for this allometry in the monoculture model (loge(t), loge(tXj) and loge(Xj), Table 4.19a) were not significant in the mixture (4.21a). Also the loge(yxpj) term was significant in the mixture while loge(ympi) was not significant in the monoculture. This suggests that the presence of a second species may cause important developmental adjustments. Table 4.21a Parameters and regression statistics for the best subset allometric model for orchardgrass in mixtures (leaf area per plant) 197 Potential Parameter Std. partial independent values regression variates @ coefficients Intercept « ' -5.813 -5.61*** log e(t) c-| 2.380 0.74*** log e ( x i ) c2 ~ loge(X4) c 3 loge(t.X t) -loge(tX*) C5 -] 0g e(X }X,) c6 -0.154 -0.18*** loge(tXjX^) 07 -log e (y x p l 1 b 0 0.411 0.41** t l o q „ ( y v n i ) bl 0.004 0.38*** : l °ge*yxp i ) ( « l o a < . ( v v n i x i l o 9e ( yxDi? b 2 Xj ggCyxJi) l»3 0.14 X 10"* 0.08*** t Xjlog.?y xpi) tX 4 i og e Ty x p l ) XI 4 l o g e ( y x p i ) t X l ^ l o g e ( y x p i ) b 5 b6 -0.60 X 10"8 -0.04 ns Other regression statistics CP 3.32 RMS 0.11 R2 0.90 d.f. 593 i> t = Time after planting in days, Xj = Population density of species i , X4 = Population density of species j , y x j = Shoot dry weight per plant, y x p i = Leaf area per plant *, **, *** Significant at 5%, 1% and 0.1% level, respectively ns = not significant at 5% level Table 4.21b Parameters and regression statistics for the best subset allometric model for orchardgrass ln mixture ( t i l ler number per plant) Potent ia l Parameter Std . p a r t i a l Independent values regression var iates @ c o e f f i c i e n t s Intercept a ' -9.692 -9.36*** l o g e ( t ) 2.063 0.64*** log e (X j ) C2 log e(X*) c 3 -0 .147 -0.12*** l o g e ( t X t ) log e ( tX*) i o g e ^ A * ; l o g e(XjX4) loggftXjXj) l o g e ( y X p i ' t l o g e ( y x p i ) X i i o g e ^ x p i ) x J l 0 9e<yxpi^ t XjlogeCyxpj) tX j log e ( y X p| ) X f X J l o g e ( y x p i > t X I x j l o g e ( y x p j ) ' c 1 c 2 -c 3 - .  C4 -c 5 -C6 -C7 -bo 0.9090 bl -b2 -b 3 0.31 X b4 -b5 - 0 . 4 0 X b6 -0 .44 X b7 0.61 X 0.58*** 0.29** -0 .24* -0.40*** 0.36*** Other regression s t a t i s t i c s CP RMS R2 d . f . 5.19 0.13 0.88 592 @ t = Time after plant ing in days, Xj = Population density of species 1, X.* = Population density of species j , y x j = Shoot dry weight per p lant , y x p * = Leaf area per plant * f ** f *** S i g n i f i c a n t at 5%, 1% and 0.1% l e v e l , respect ive ly Table 4.22a Parameters and regression statistics for the best subset allometric model for timothy in mixture (leaf area per plant) P o t e n t i a l P a r a m e t e r S t d . p a r t i a l i n d e p e n d e n t v a l u e s r e g r e s s i o n v a r i a t e s @> c o e f f i c i e n t s I n t e r c e p t l o g e ( t ) l o g e ( X j ) 1 o g e ( X « ) l o g e ( t X i ) l o g e ( t X * ) l o g e ( X j X j ) l o g e ( t X j X * ) t l o g e ( y x p l ) x i ] ° g e ( v x p i > X 4 ] o g e ( y x p l ) t X t l o g e t y x p f ) t X 4 l o g e ( y x p l ) X l x j 1 ° g e ( y x p i ) t X l X l l o 9 e ( y x p i ) c 1 c 2 c 3 c/+ c 5 C6 c 7 b0 t»1 l>2 b 3 b* b5 b6 b7 - 8 . 4 5 0 3 . 1 2 1 - 0 . 2 2 2 - 0 . 0 4 9 0 . 5 2 8 - 0 . 2 2 X 1 0 " * 0 . 5 6 X 1 0 - 6 0 . 1 9 X 1 0 - 6 - 0 . 1 1 X 1 0 " 9 - 8 . 0 2 * * * 0 . 9 5 * * * - 0 . 1 8 * * * - 0 . 0 6 ns 0 . 3 8 * * * - 0 . 1 1 * 0 . 2 1 * * * 0 . 0 7 ns - 0 . 0 6 ns O ther r e g r e s s i o n s t a t i s t i c s CP RMS R2 d . f . 8 . 4 9 0 . 0 9 0 . 9 2 591 @ t = T ime a f t e r p l a n t i n g i n d a y s , Xj = P o p u l a t i o n d e n s i t y o f s p e c i e s i , X-* = P o p u l a t i o n d e n s i t y o f s p e c i e s j , y x j = Shoot d ry we igh t pe r p l a n t , y x p j = L e a f a r e a per p l a n t * , * * , * * * S i g n i f i c a n t a t 5%, 1% and 0 . 1 % l e v e l , r e s p e c t i v e l y n s = not s i g n i f i c a n t a t 5% l e v e l Table 4.22b Parameters and regression statistics for the best subset allometric model for timothy in mixture ( t i l ler number per plant) Potent ia l Parameter Std . p a r t i a l Independent values regression var ia tes @ c o e f f i c i e n t s Intercept l o g e ( t ) Iog e (X t ) log e (X t ) l o g e ( t x i ) log e ( tX*) log e(XjX*) log e ( tX jX j I o 9 e ( y x p l 7 t l o g e ( y x p l ) X f ^ g e ^ x p i ^ X 4 l o g e ( y x p i ) t Xilog^Cyxpi) t X j l o g e ( y x p i ) x ^ X t l o g e ^ x p i ) t X i X j l o g e ( y x p l ) a' -10.333 -9 .80*** C 1 2.642 0.80*** c 2 -0 .125 -0.10*** c 3 - -- -^5 - -c 6 -0 .08 -0.10*** - -^ 0.285 0.15* b? 0.009 0.32*** 0.25 X 1 0 " 3 0.13* h -bj -0 .81 X IO" 5 -0 .26*** b 5 - -b6 - -by - -Other regression s t a t i s t i c s CP RMS R2 d . f . 5.10 0.14 0.88 592 @ t = Time af ter p lant ing in days, Xj = Population density of species I, Xj = Population density of species J , y x j = Shoot dry weight per p lant , y x p i = Leaf area per plant • f ** f **# s i g n i f i c a n t at 5%, 1% and 0.1% l e v e l , respect ive ly 201 The model for the allometric relationship between shoot dry weight per plant and tiller number per plant for orchardgrass in mixtures is given in Table 4.21b. Again, the model differred from the corresponding monoculture model (Table 4.19b), in that the loge(Xp term was not significant in mixture. In mixtures, the density of the timothy (loge(Xj)) made a significant negative contribution to the allometric coefficient. Several terms contributed significantly to the allometric exponent. Loge(yXpj), Xjloge(yXpj) and tX iXjlog e(y x p i) had positive effects while tXjloge(yxpj) and XjXjloge(yxpj) were significant negative terms. The model for the allometric relationship between shoot dry weight per plant and leaf area per plant for timothy in mixtures is summarized in Table 4.22a. Here, the mixture model is essentially an extension of the monoculture model (Table 4.20a). Additional significant negative components of the allometric exponent were X;log„(y„T.:) and tXjXjloge(yXpj), while tX|loge(yXpp was a significant positive component (as well as loge(yx p i)). The model for the allometric relationship between shoot dry weight per plant and tiller number per plant for timothy in mixtures is summarized in Table 4.22b. Here again, the mixture model may be considered an extension of the monoculture model (Table 4.20b), although the loge(Xj) term differs in sign. An additional significant negative component of the allometric coefficient existed in mixtures, log„(X-X:). An additional significant negative component of the allometric exponent was tXjloge(yXpj), while tloge(yxpp and Xjloge(yXpj) were significant positive components (as well as loge(yx_j)). 202 5. DISCUSSION In view of the extensive literature which exists on plant responses to interference, it was not surprising to find that crop yield was powerfully affected by time of harvest, population density and species proportions in mixtures. It was evident from the ANOVA results (Tables 4.4 to 4.7) that significant yield responses had already begun to develop by the first harvest and were commonly expressed at later harvests as well. An overview of the ANOVA results indicates that the other primary variates and derived quantities of orchardgrass and timothy also responded strongly to time of harvest, population density and mixture proportions (Appendices 8.2.1 to 8.2.12 and 8.4.1 to 8.4.5, respectively). The pattern of significant responses among treatments and variates, however, was not uniform. Hence, the ANOVA results provide an initial indication of the variability in sensitivity of different features of plant growth to the experimental treatments. Over all the primary variates, the mixture effects were often more significant than the population density effects for orchardgrass (Table 4.2), and the reverse was true for timothy (Table 4.3). It is possible to infer from this that the relative effects of intraspecific and interspecific interference on the two species may be different. That is, orchardgrass may be more sensitive to interspecific interference than intraspecific interference, while the reverse may be true for timothy. This inference is supported by the results from the regression analyses, as will be discussed later. In the ANOVAs, the partitioning of treatment degrees of freedom into orthogonal polynomials of linear, quadratic and higher components provided an initial indication of the form of different responses to treatment factors. It was clear that the responses were often quadratic, as well as linear, and cubic or higher order relationships were sometimes significant. These findings were later borne out in many of the regression analyses. For example, non-linear relationships were effective in describing yield-density responses in 203 monocultures and mixtures. Also, quadratic terms were often significant in the best subset regression models of treatment factor effects. Compared with the main effects of treatment factors, individual interaction effects were less commonly significant (Tables 4.2 and 4.3). Also, interactions appeared to be less common for timothy than orchardgrass. The relative importance of different treatment interactions was established by the best subset multiple regression approach, as will be discussed later. ANOVA also demonstrated the occurrence of significant treatment effects on total yield (combined shoot biomass density of the entire population in the mixtures or monocultures, Table 4.8). The strong mixture effects on total yield may be related to differential yield responses, but this cannot be established from the ANOVA alone because the monoculture yields of the two species tended to differ. This study involved a complete factorial experiment, and its factorial nature offered statistical advantages in testing for the main effects of each treatment factor as well as for their interactions. Because population densities and species proportions were varied widely, the study permitted a broadly based assessment of plant response to interference, albeit within one growing season. As Mead (1979) has pointed out, a complete factorial experiment is statistically superior to certain other experimental approaches. Replacement or additive series experiments, for example, are incomplete factorial experiments and such experiments have limited utility for ANOVA. It is clear from the present study that the ANOVA results established the justification for subsequent analyses, because significant treatment responses were identified. It is also clear that the ANOVA results foreshadowed many of the findings of later analyses. This was a large experiment, including 21 measured and derived quantities and up to 840 observations per quantity. Most variates were found to have homogenous variance in relation to both population density and mixture proportions (Table 4.1). This provides some reassurance that the inferences drawn about the treatment effects from the ANOVA results tests are valid on statistical grounds. Variances due to interactions were less 204 homogenous, and so the ANOVA results for interaction effects must be interpreted with some caution. The A N O V A results alone, however, do not allow inferences to be drawn about the actual magnitude or direction of treatment effects. Statistical significance is not the same as the biological significance, and it is difficult to visualize the biological importance of treatment effects from the ANOVA results. Because of this, the graphical presentation of the results was a helpful adjunct to the ANOVA (Figs. 4.1 to 4.17). The main responses evident from those figures are the expected tendency of different measured and derived quantities to grow with time, and the depression in many of those quantities by increased population densities. Mixture effects were less consistent, although they had often been highly significant in the ANOVA. When variates were expressed on a per plant basis, their values often increased as the proportion of that species increased in the mixtures. For both orchardgrass and timothy, yield per land area was the product of mean dry weight per shoot and species population density. Shoot dry weight per plant behaved like most primary variates in that it declined rapidly with increasing population density (Figs. 4.4 and 4.6). This may reflect a decrease in available environmental resources per plant with higher population density, but other components of interference could also contribute to the decline. In common with most previous research on interference, it has proved easier to observe the effects of interference than to understand the specific causes. The intensity of population density effects on shoot dry weight per plant increased with crop age, as indicated by the divergence of the lines on the plots. This was not the case, however, for many of the other primary variates. Orchardgrass shoot dry weight per plant was altered in different mixtures of timothy and vice versa, as can be seen by comparing the different sub-plots in Figs. 4.1 and 4.3. Orchardgrass shoot dry weight per plant was less in mixtures with timothy than in its monoculture (Fig. 4.1). Hence, that graph indicates that interspecific interference of timothy on orchardgrass was relatively strong compared with intraspecific interference in orchardgrass. In contrast, timothy 205 shoot dry weight per plant was higher in mixtures with orchardgrass than in its monoculture, and the relative strengths of interspecific and intraspecific interference were reversed. Again, this preliminary analysis of the relative strengths of different components of interference was consistent with subsequent findings. While shoot dry weight per plant declined with higher population density, shoot biomass density of each species increased because of the compensating effect of population density (Figs 4.2 and 4.4). Population density effects on shoot biomass density were relatively small in both species. Similar results have been found in a number of crop species (Holliday 1960a, 1960b, 1960c). Mixture effects on shoot biomass density were similar to their effects shoot dry weight per plant. Total mixture shoot biomass density (i.e. both species combined) was increased with higher population density early during growth but this response was reversed at later harvests (Fig. 4.5). Timothy was the most productive species in monocultures (Fig. 4.4A) and orchardgrass proved to be least productive (Fig. 4.2F). Total mixture shoot biomass densities fell between the monoculture extremes. It is interesting that timothy appeared to cause stronger interference and was also the higher yielding species. More research is required to establish whether there is a common relationship between relative competitive abilities and j'ield potential of species. An assessment of overall mixture yield performance is very important, because a main objective of research in this subject is to find crop mixtures that overyield the corresponding monocultures. There are sometimes difficulties in considering the combined yields from mixed species, however, because the nature of agricultural yield may differ among the species (e.g. grain yield vs. oil yield). One way to overcome this problem is to express yield on some standard basis (e.g. monetary value). Research on this aspect is limited and as yet there is no standard method how to express combined mixture yield. My research involved two forage species, so total shoot biomass density can be considered to be a suitable measure of overall mixture yield, at least in an agricultural context. In 206 the interpretation of the origins of yield responses, however, shoot dry weight per plant was considered to be a more revealing variate. The interpretation of the effects of experimental treatments was greatly extended using the best subset multiple regression procedure. The procedure summarized the effects of treatments in a simple but comprehensive way; the relative effects of different treatment factors were judged using the standard partial regression coefficients produced by the regressions. The procedure extended the earlier findings of the ANOVA by describing how the main effects and interactions of treatments influenced different yield variates. Except for harvest index in timothy, the best subset multiple regression models were effective in describing treatment effects on different yield variates. The regressions revealed that the important independent variates differed widely among models for different yield variates (Tables 4.11 and 4.12). Also, the importance of different treatment factors varied even for related yield variates (e.g. components of shoot dry weight). This suggests that plant responses to treatment factors were highly complex. For example, total mixture shoot biomass density (Table 4.13) was positively related to the linear component of the main experimental factors (t, O, T) and the time quadratic by species population density interactions (t T, t O). Strongly negative factors included the time by species population density interactions as well as the time quadratic. Hence, total mixture shoot biomass density depended in a complex way on the duration of growth, the population densities of component species and the interactions among those factors. The summary offered by the best subset multiple regression procedure, however, is helpful in that it details and quantifies such complexity. The present research may be the first application in which the best subset regression approach has been used to evaluate treatment effects on different yield variates. A primary objective of this research was to assess the relative contributions of intraspecific and interspecific interference to yield responses. This issue was addressed mainly through the interpretation of yield-density responses. Simple and multiple 207 regression models were developed to define yield-density relationships of monocultures and mixtures, respectively. As Willey and Heath (1969) and Thornley (1983) have emphasized, the relationships between crop yield (per land area) and population density are of considerable agronomic importance. In addition to the evaluation of interference per se, the yield-density models were also relevant to several other objectives of the study. They provided a further quantitative definition of treatment responses. They provided additional insight into biological relationships in the study through the model parameters. Finally, they were used to account for the contributions of each species to differential yield responses. Reciprocal models were used for describing yield-density relationships because such models seem to have a good biological foundation and because they have proved to be effective in practice (Willey and Heath 1969). Bleasdale's (1966b, 1967b) simple model (Eq. 2.20) described the monoculture yield-density relationships well for both orchardgrass and timothy. The model was able to account for up to 85% of the overall variation in shoot dry weight per plant in orchardgrass and up to 98% in timothy (Tables 4.14 and 4.15; Figs. 4.19 and 4.20). To evaluate the regressions, the index of multiple determination (I2) 9 was used as a statistical criterion rather than the coefficient of determination (R ). This was because I evaluated the models on the original scale of yield (y), whereas R operated in the transformed scale in yield (y'Q). The form and magnitude of the yield-density models is controlled by the size of the model parameters (aj, by, by and Q). The various parameters, however, play different roles in defining the yield-density responses. As pointed out earlier, there exist two basic biological relationships, namely asymptotic and parabolic (Holliday 1960a; Willey and Heath 1969) and the size of exponent (Q) in Eq. 2.20, expresses those responses (Fig. 4.18). The form of the yield responses in Fig. 4.18 is unaffected if other model parameters are varied, but the magnitude of yield is changed. 208 The model parameters are not simply empirical entities but they contribute information on several aspects of interference. For example, the size of exponent has been related to the acquisition and utilization of resources within the space available per plant (Watkinson 1980; Jolliffe 1986) and the rate of decay of the competitive effect (Vandermeer 1984). In the present study, the exponent value for the orchardgrass monoculture models (Table 4.14) was close to zero (0.01) early during growth and at the final harvest (37, 48 and 90 DAP), indicating poor acquisition and utilization of the resources in the space available per plant. A value of exponent close to -1.0 has commonly been been found suitable in many crop species (see review by Willey and Heath 1969), and accordingly Eq. 2.17 has been considered appropriate for use in many yield-density studies. However, the majority of previous studies are limited to yield-density data from a single harvest, usually late in growth. The choice of an exponent of -1.0 is arbitrary, however, and there is no reason a priori to consider an exponent value of -1.0 suitable for all growth stages. The requirements for resources in a growing population usually varies with the stage of growth and in that light the exponent in Eq. 2.20 should vary. Early during growth and after the main growth period, the value of the exponent for orchardgrass monoculture was close to zero. This may have been because the plant's demand for growth resources was low at those times. The other parameters are related to plant size in the absence of interference (ap, and the responsiveness of yield to population density (bjj), as scaled by the exponent. Hence, the size of aj"*^ expresses yield potential per plant, and this parameter has in fact been used as a measure of the 'genetic potential' of a plant (Bleasdale and Thompson 1966; Nichols 1974). This index may vary with growing conditions, however, and as with the other parameters it is relevant to the particular time of observation. The occurrence of negative values of aj for timothy monoculture (Table 4.15) must be considered to be a flaw in the model, although the confidence limits about the estimates of a^  were large. Because of the scaling effect of Q, the regression coefficient (bjj) is small when the value of 209 exponent is also small (Tables 4.14 and 4.15). The value of b^ can be used as an index of intraspecific interference. The method used to evaluate interference in mixtures also exploited reciprocal models, in accord with developments during the past several years (Wright 1981; Spitters 1983a; Vandermeer 1984; Firbank and Watkinson 1985, 1986; Jolliffe 1986). It is clear, however, that all those mixture models are extended versions of the monoculture model (Eq. 2.20), including the present study (Eq. 2.23). Two types of regression models, having a similar structure but developed from the data in different ways, were used to model yield-density responses of the mixtures. The conventional multiple regression approach in one step yielded model parameters for the mixtures (Appendix 8.3) which were quite different from the corresponding values for 9 monocultures (Tables 3.14 and 3.15). In addition, the goodness of fit (I ) for the conventional multiple regressions was relatively low (a maximum value of 0.7 in mixtures compared to 0.98 in monocultures), and the regressions were significant for only 37 out of 60 cases (Appendix 8.3). Hence, in the absence of interspecific interference the mixture models produced by the conventional multiple regressions do not default to the monoculture model. The two stage regression approach gave I values comparable to those achieved by the monoculture models (Tables 4.16 and 4.17). In 53 out of 60 cases the regressions were statistically significant. The fitted curves developed from the regressions in stages agreed very well with the observed mixture yield values (Figs. 4.23 to 4.27). The regression in stages approach automatically defaults to the monoculture yield-density model when interspecific interference is zero. For these reasons, the conventional multiple regression approach seems less satisfactory than the two stage regression approach. Also, the regression in stages has an additional advantage in that it can be used for the assessment of differential yield responses, as discussed later. Recently, in a study of the effects of ozone air pollution on plant interference, Penney (1986) compared three different interference analysis procedures, namely: 210 multiple regression analysis (Spitters 1983a), regression in stages (Jolliffe 1986) and the analysis of Thomas (1970). She also found that the regression in stages approach was superior and concluded that the regression in stages approach has more biological meaning than the conventional multiple regression approach. The role of the yield-density model parameters (Eq. 2.23) in describing mixture yield-density responses was similar to monocultures for the parameters a^ , b^ and Q, as discussed above. Variation in Q and aj (while keeping the other parameters at nominal values) affected the magnitude of mixture yield per land area (Yx), but did not change the form of the replacement diagrams (Fig. 4.21). The form of the replacement diagrams, however, was determined by the relative values of regression coefficients (b /^by). Clearly, convex or concave forms in the replacement diagrams, which have both been reported in previous research, were controlled by the relative strengths of different interferences in the mixed species. It follows that the trends for total mixture yield responses were also controlled by those interferences according to the sum of the responses of each species. As reviewed earlier in this thesis, several indices have been proposed to assess the effects of interference on plant growth and mixture productivity. Many previously used indices of interference, however, are based on replacement series experiments, usually consisting of 50:50 mixtures and monocultures of each species. Those indices are expressed as either simple ratios or simple differences between mixture yield and monoculture yield (e.g. aggressivity, relative yield, land equivalent ratio, competitive ratio). In such indices a large body of information is concentrated in a single value. The construction of such indices involves a large loss in information about the details of interference in a mixture, and there can be difficulties in computing suitable statistics for such indices. The regression coefficients from the reciprocal models (Eq. 2.20 and 2.23) also represent a concentration of information. However, they are derived from regression approaches which attempt to describe yield-density relationships which are central to the analysis of interference. Also, the coefficients have a distinct biological meaning, and the 211 present study shows that they provide a powerful tool for expressing interference. In these studies, the ratio of the regression coefficients (bjj/by) provided a direct measure of the relative strengths of intraspecific to interspecific interference for each species. In orchardgrass the ratio was always less than 1.0, indicating that interspecific interference exceeded intraspecific interference (Table 4.18). In timothy the ratio was always greater than 1.0, indicating that intraspecific interference was greater than interspecific interference. These relative competitive abilities of the two species were maintained throughout growth at all population densities. Thus, interference in a mixture can be meaningfully partitioned into the component sources. This approach can be extended to multispecific mixtures. The pattern of the replacement diagrams (Figs. 4.22 to 4.26) was 'compensation' (Trenbath 1974; Willey 1979a). Generally the curves were concave for orchardgrass and convex for timothy. The utility of Eq. 2.23 can be seen again in the fitted curves shown in replacement diagrams. Compensation is the most common pattern in crop mixtures, where yield of one species exceeds the 'expected' yield while the other species falls below its 'expected' yield. Where compensation is equal between the species RYT = LER = 1.0 (Trenbath 1974; Willey 1979a). Such a result has been taken to imply that species are competing for the 'same resources' and/or 'same space' (de Wit 1960). In the present experiment, the compensations of the two species were usually unequal, with overyielding occurring in most cases. This is evidence for niche differentiation between the two species. The evaluation of differential yield responses in mixtures is central to intercropping research. The replacement diagrams provide some information about differential yield responses. A more detailed analysis of the differential yield responses of the individual species and combined mixtures was provided by taking the difference between Eq. 2.27 and 2.28. Net overyielding of the total mixture was common and it was most pronounced in mixtures having about 2:1 proportions of orchardgrass: timothy (Figs. 4.28C to 4.32C). Overyielding in the mixture as a whole occurred because overyielding in timothy exceeded 212 underyielding of orchardgrass. Such differential yield responses are directly related to relative responses of species to interference, because differential yields are determined by non-zero values of (by - by)Xj. If, for the overall mixture, intraspecific interference is greater than interspecific interference (i.e. by > by) overyielding will occur in the mixture. This is the first application of this approach for the assessment of differential yield responses. It has great potential for future application because it is the only method yet developed for the quantitative analysis of differential yields on a species by species basis in a multispecific mixture. It permits such an analysis to be undertaken over a range of total population densities and mixture proportions (not just a 50:50 mixture). Much previous interpretation of replacement diagrams has been simply by visual inspection. This new approach has a better statistical foundation because it is based on the yield-density regression models. An assumption made in the development of the yield-density models was that species population densities remained constant throughout the study. Plant mortality can be promoted by high population densities (e.g. Firbank and Watkinson 1985). Early in my study it became clear that it was not feasible to follow species population densities in the different plots throughout the experiment. The two species were very similar in appearance, and the plants quickly mingled closely together, especially at high population densities. I have therefore assumed that the population densities established at the time of crop emergence applied for the whole experiment. This assumption may be reasonable considering the relatively short duration of the study. The possibility that mortality was not a serious source of error is suggested by the data on tiller number per land area (Figs. 4.7 and 4.14), where there is little indication of tiller losses late in the season and at high population densities where interference was intense. In row crops, the spatial arrangement of plants may contribute to population density effects on crop yield (Berry 1967; Rogers 1972, 1976; Bleasdale 1967a). In the present study the crop mixtures were established by random distribution, and changes in 213 spatial arrangement among different treatments (aside from the changes innately related to species frequency) may be a less serious complication. The indices used to evaluate the progress of growth in plant growth analysis are developed from a few primary variates (e.g. W, LA). Since the underlying primary variates responded significantly to experimental treatments, as discussed earlier, it is not surprising that the growth indices often exhibited strong responses. In support of the ANOVA results it is clear from the growth curves that many of the growth indices had responded to density and mixture treatments by the time of the first harvest. The shoot biomass density results were discussed earlier. Among the other ratio indices only LAI responded strongly to treatments, especially in the case of timothy LAI in populations containing high timothy proportions (Fig. 4.38). A prominent feature of the growth analysis was the large variability of the dynamic indices of growth (AGR, R, CGR and ULR) in response to crop age, population densities and mixture proportions (Figs. 4.35 to 4.38 and Figs. 4.42 to 4.45). The results indicate that those variations were driven by changes in AGR (Figs. 4.35 and 4.42). The results also indicate that interference among members of the different populations affected growth at different times and to different degrees in a way which seemed to be specific to each population. Relatively large variations in ULR occurred (Figs. 4.38 and 4.45), indicating variations in the efficiency of resource assimilation per unit leaf area. As with all of the dynamic growth indices, however, the patterns of response were complex and difficult to interpret. The growth indices also provided some additional information concerning competitive abilities of the two species. For example, the difference between the effects of mixture proportions on AGR of the two species matched earlier observations about the relative sensitivity of the species to intraspecific and interspecific interference. Some previous studies have used methods of growth analysis to interpret the effects of interference on growth. The application of plant growth analysis to crop-weed associations has recently been reviewed by Radosevich and Holt (1984). The comparision 214 of growth indices of various crops and weeds may provide a basis for the interpretation of the competitive nature of weeds; however, research on this is relatively limited. Radosevich and Holt (1984) further pointed out the difficulty of assessing whether resource limitation causes competition or results from it. Since the ultimate effect of resource availability is to influence plant growth, it is likely that indices of growth can be used to interpret and/or predict the effects of competitive interference between crop and weed species. Total dry matter production and leaf area expansion are important integrative aspects of vegetative growth. The growth indices which might be expected to be influenced by plant interference are relative growth rate, unit leaf rate, leaf area ratio. As reviewed in Radosevich and Holt (1984) the maximum relative growth rate of individual plants has been suggested as an indicator of the potential competitive ability of species. Radosevich and Holt (1984) also stressed the applicability of plant growth analysis for comparative studies between crops and weeds in an agricultural context. Plant growth analysis was recently used to interpret results from a replacement series experiment which was done to study growth and competitiveness of four annual weed species (Roush and Radosevich 1985). The species did not vary for relative growth rates; however, plant dry weights, plant heights, LAR and ULR were significantly different. Hence, growth measures other than relative growth rate were possibty related to the relative competitive abilities of the species studied. Techniques of plant growth analysis also may provide insight into differential yield responses in intercropping situations. Plant growth analysis was used to study bean: maize intercroppings (Gardiner and Craker 1981; Clark and Francis 1985). Growth and yield of bean plants was reduced in mixtures with corn, and the yield reductions were associated with reductions in LAI, R and ULR. Higher total mixture yields were correlated with higher mixture LAI values, due to a staggered leaf development between the two species (Clark and Francis 1985). The present study involved a relatively comprehensive application of modern methods plant growth analysis, including the use of cubic spline regressions. The 215 inferences drawn from this part of the work have supported and extended my other findings. Compared with the other analytical approaches I used, however, plant growth analysis was relative^ complex to apply, and the growth indices were relatively complex to interpret. The analysis of plant allometry also indicated that different aspects of plant growth responded differently to treatment factors. It extended previous analyses, however, by establishing quantitative models between different features of growth as they were influenced by treatment factors. Allometric models for the monocultures were relatively simple. This might be due to the relative simplicity of the monocultures as environments for plant growth, but it could simply be due to the fact that there were fewer potential independent variables available for the monoculture allometry models. In the models, significant relationships were always established with terms contributing through the allometric coefficient (e.g. loge(t)). Significant relationships also were established with terms contributing through the allometric exponent (e.g. l°ge(ympi)) except for the allometry between shoot dry weight per plant and tiller number per plant (Fig. 4.20b). In the more complex mixture models, there were generally a larger number of significant terms associated with components of the allometric exponent than the allometric coefficient. Since the allometric exponent is the ratio of the relative growth rates of y and y p ) it is of considerable physiological significance. The best subset multiple regression procedure allowed the allometric exponent to be partitioned into significant terms which can be quantitatively compared via the standard partial regression coefficients. It is difficult to tell at this time whether this will prove to be a valuable approach for exploring sources of variation in relative growth rate. Nevertheless, the output from the allometric analyses do provide a further assessment of the quantitative effects of treatment factors on growth. As noted in the Literature Review, this new procedure may have a better foundation than other approaches used to combine allometric and yield-density models (e.g. Bleasdale 1967b, Watkinson and Firbank 1985, Willey 1981). It should also be noted that 216 the new approach can be applied to analyze the effects of other experimental factors (Appendix 8.1). This study involved a single growing season and only one pair of species, so the general relevance of the results cannot be determined. The experimental approach was intended to detail species responses to interference and not the mechanisms of interference. It is uncertain whether interference was more intense in the soil or aerial environments, and whether the mechanisms of interference were similar for the two species and consistent throughout the experimental period. Belowground measures of growth were not performed because it was not feasible to separate the roots of the two species from the soil. In considering the present studies of growth, productivity and interference, it should be borne in mind that an important potential contributor to variation in productivity, the root system, was not assessed. The investigation of how plants perform when exposed to interference is a complex task. The present study has been limited to the study of a relatively simple set of situations involving monocultures and binary mixtures during one growing season. Obviously, much more complex situations exist when additional genotypes are included, the growing period is extended and environmental conditions are changed. Some of the new techniques used in the present research could readily be applied to more complex situations. For example, the analysis of differential yield responses can be extended to multi-specific mixtures (Jolliffe 1986). It is also evident that the different approaches used in this study led to similar conclusions when they addressed the same questions. For example, similar inferences on the relative competitive abilities of orchardgrass and timothy arose from the following analyses: ANOVA; best subset multiple regressions of yield against treatment factors; yield-density models; replacement diagrams; plant growth analysis; and the analysis of plant allometry. Such reinforcement is satisfying, and if it proves to be the case in future work then only a few, suitable methods might be required in any single study. 6. C O N C L U S I O N S 1. Using the results from a field study on the growth of orchardgrass and timothy in monocultures and mixtures, the relative strengths of interspecific and intraspecific interference on shoot dry weight per unit land area were compared. Several techniques of data analysis, including best subset multiple regression analysis of treatment effects, the interpretation of replacement diagrams, the construction of yield-density regression models, and plant growth analysis, all indicated that timothy caused stronger interference within and between species. Of the various techniques used, the yield-density models provided the most direct assessment of the relative strengths of interferences. 2. Different measures of growth and yield exhibited large differences, both within and between species, in their responses to interference. This diversity of responsiveness was indicated by ANOVA, best subset multiple regression analysis of treatment effects, plant growth analysis and the analysis of allometry. One implication of this diversity of responsiveness is that, at least in part, different results may be obtained in different studies of plant interference when different measures of growth and yield are used. 3. The relative effects of different treatment factors (time after planting, total population density, species proportions in the populations) also were measured using several approaches: ANOVA, best subset multiple regression analysis of treatment effects, yield-density models, replacement diagrams, plant growth analysis and the analysis of allometry. The main effects of treatment factors often were significant. Interaction effects also occurred but varied widely among different measures of growth. The best subset multiple regression analysis of treatment effects provided the most direct and comprehensive quantitative assessment of the effects of different treatment factors. 218 4. Bleasdale's simple model (Eq. 2.20) was found to be effective in describing yield-density relationships in monocultures. An extension of Bleasdale's simple model (regression in stages, Eq. 2.23) was effective in describing yield-density relationships in mixtures. The yield-density models were used to account for the contributions of orchardgrass and timothy to differential yield responses. Net overyielding occurred in most mixtures because overyielding in timothy more than compensated for underyielding in orchardgrass. Mixtures containing orchardgrass and timothy in proportions of about 2:1 had the highest overyielding. Differential yield responses were directly related to the size of the difference between intraspecific and interspecific interference, as measured by the coefficients developed by the multiple regression in stages. 5. Significant interference had developed in both monocultures and mixtures by the time of the first harvest (37 days after planting). Responses to interference continued throughout the rest of the study (until 90 days after planting), but the type and extent of the responses depended on which measure of growth was considered. Plant growth analysis indicated that the time course of growth was altered in a very complex way through population-dependent shifts in absolute growth rate. 6. The allometric analysis indicated that treatment factors affected the allometric exponent relating two measures of plant growth. That is, one way treatment factors altered growth was by changing the ratio of relative growth rates of different growth characteristics. 219 7. The growth and yield of plants growing under the influence of interference are difficult to study because of the complexity of possible interactions which may occur. This study applied several new techniques for the analysis of plant responses to interference. The best subset multiple regression procedure was used for the first time to measure the relative effects of time and species population densities on growth. A yield-density regression model, developed in two stages, was found to be superior to previous multiple regression approaches. That model allowed, for the first time, a direct accounting to be made of differential yield responses for each species in mixtures. In addition, the approach used to analyze allometric responses was new. These techniques, together with more conventional approaches (e.g. ANOVA, plant growth analysis), were all valuable in different ways, and they have considerable promise for future applications in research on interference. 220 7. 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According to Eq. 8.1, experimental factors will influence loge(ymj) and its relationship with loge(3rmpj) through loge(a), 6, and loge(e). Those effects can be taken into account by expanding the model. For example, consider the effects of two treatment factors, time after planting (t) and monoculture population density (Xj), on k>ge(ymj): loge(a) = loge(a0) + d 1log e(a 1t) + d2loge(a2Xj) + d^og^a^tX^) = loge(o0) + d1loge(a1) + d2loge(a2) + d 3 l o S e ( a 3) + d \ i o S e ( t ) + d2loge(Xj) + d3loge(tXj) (8.2) b = 6 0 + 6 xt + 6 2Xj + b3tX{ (8.3) loge(e) = loge(e0) + ejlogglejt) + e 2log e(c 2 xi) + e 3 l o g e ^3 t X i ) = loge(e0) + ^1loge(e1) + #2loge(e2) + ^3 1 °g e ( e 3 ) + S^oge(t) + g2\oge{Xi) + g3\oge(tX{) (8.4) In Eq. 8.2, 8.3 and 8.4 the coefficients (dj...dg, b^...b%, and g-^.-.g^) control the deviations from the nominal states (a n, 6n, and en) which exist if treatment effects are not significant. Terms from Eq. 8.2, 8.3 and 8.4 can now be grouped. First, a set of terms which will be inseparable in the best subset multiple regression analysis are grouped: loge(a') = loge(a0) + d1loge(a1) + d2loge(o2) + d3loge(a3) (8.5) loge(e') = loge(e0) + (?1loge(e1) + £ 2log e(e 2) + £ 3log e(e 3) (8.6) Eq. 8.5 and 8.6 can then be incorporated into the overall model from Eq. 8.2, 8.3 and 8.4: log e(ym i) = loge(a') + &ologe(ympi) + 6l t l ogefrmpi) + 62Xi l oge^mpi^ + 6 3tX ilog e(ym p i) + (d1 + ^1)loge(t) + (d2 + g2)loge(Xi) + (d 3 + ^^logg^Xj) + loge(e') (8.7) Because the terms are not separable, let c^  = (dj + g^) ... c 3 = (d 3 + g 3). Then: l o g e k W = l 0 g e ( o , ) + 60 loge(ympi) + 6l t l oe e(ympi) + 62Xi1°ge(ympi) + o ^ l o g ^ y ^ ) + C lloge(t) + c2loge(Xi) + Cglog^tXj) + loge(e') (8.8) Eq. 8.8 is now in a form in which the parameters loge(o'), 6Q...&3, c^...c3 can be tested for possible significant contributions to variation in loge(ymj) using the best subset multiple regression technique. 229 It should be noted that the above derivation has only considered the linear relationships and interactions due to the two treatment factors t and X-. It should be apparent from the derivation how additional factors, or higher order relationships (e.g. t , 9 t Xj, etc.) could readily be incorporated into the approach (e.g. Eq. 8.9 below). Also, the choice of factors was arbitrary; the same derivation could be used for factors other than t and Xj (i.e. this approach to studying allometric responses is not restricted to the analysis of the effects of interference or time). In studies of allometry, the main interest lies in the allometric exponent, 6. The allometric coefficient, a, is strongly dependent on the arbitrary choice of units of measurement. The allometric exponent, however, is dimensionless, and because it is equivalent to the ratio of relative growth rates of y m j and ympj (Whitehead and Myerscough 1962) it has considerable biological meaning. Jolliffe and Courtney (1984) showed how 6 was central to additive relationships in plant growth analysis. Eq. 8.8 provides an effective way of assessing how allometry is affected by treatment factors through significant contributions by the terms containing 6^...6g. The terms containing the coefficients c^ ...Cg are composite in origin. They contain some potential contributions from a (through dj...dg) and e (through ej.-.eg). Therefore, they can represent both allometric and non-allometric sources of yield response. 8.1.2 Allometry in Mixtures The effects of interspecific interference in binary mixtures can be dealt with by considering the density of the competing species (Xj) to be a third treatment factor (in addition to t and Xj). Then, expanding the above derivation, the model for mixtures would be: 230 loge(yxi) = loge(o') + 6 0 log e (y x p i ) + &itlog e(y x p i) + b2Ki\oge(yxpi) + b3X^Se(yxpi) + 6 4 t Xi l oge(y Xpi) + & 5 t X j l 0 My X pi) + b 6 ^ S e ( Y K p i ) + ftytXiXjlogeCy^i) + c^og^t) + c2loge(X i) + c3loge(Xj) + c4loge(tXi) + c5loge(tXj) + c6log e(X iX j) + cyloggaXjXj) + loge(e') (8.9) As with the monoculture model above, this model can also be applied to experimental data using the best subset multiple regression procedure. This was the procedure used for the analysis of mixture allometry presented in the thesis. Because the Xj and Xj arrays were inversely related to each other, matrix singularities arose in the regression analysis. For that reason the terms: loge(tXj), loge(tXj), and loge(tXjXj) were not included in the regression analyses. Also, for the same reason, loge(Xj) was not included in the analyses of orchardgrass mixture allometry and loge(Xj) was not included in the analyses of timothy mixture allometry. Such an approach to the analysis of mixture allometry might be considered to be inconsistent with the approach developed for fitting regressions to the yield-density data. In that case, it was found most suitable to develop the model for mixture yield-densit3' relationships in two stages: the first stage defined the parameters which accounted for intraspecific responses in monocultures, and the second stage took into account the additional interspecific responses in mixtures. The goal in the allometric analysis, however, was to determine the joint effects of species population densities and time, and for that purpose Eq. 8.9 is appropriate. Potentially, the regression in stages approach could also be applied to the allometric analysis. In that case the effects of interspecific interactions on allometry, after intraspecific effects were removed, could be assessed according to the departure from the monoculture response. Let y x i equal y^ - y m i and let y x p i equal y x p i - y m p j . Then an allometric relationship between the mixture responses of y: and y can be expressed: loge( V X P = I o S e ( a ) + 6 1 ° £ e ( y X pi ) + l o g e ( e ) (8.10) The effects of the competing species through its population density, Xj, can then be represented by: loge( y x i) = log^a1) + 60loge( y x p i ) + o ^ l o g ^ y x p i ) + c^og^Xj) + loge(e') (8.11) Eq. 8.11 is the outcome of a derivation like that in Part 8.1.1 involving only one treatment factor. Again, the best subset multiple regression approach could be used to define the model parameters. This method, however, was not used for the data analysis presented in the thesis. There was an insufficient number (30) of directly paired mean values of y m j and y^ (or y m p j and y x pj) to make the analysis effective. On the other hand, 600 mixture observations were available for analysis using Eq. 8.9 Appendix 8.2.1 Summary of ANOVA r e s u l t s : Variance r a t i o s f o r the e f f e c t s of t o t a l population density and mixture proportions on orchardgrass t i l l e r dry weight per t i l l e r Time af te r p lant ing (days) Source of — v a r i a t i o n d . f 48 63 77 90 Blocks 3 - 6.0*** 5.2*** Density (5) - - - -(D) D L inear 1 4.2* - - 6 .1* D Quadratic 1 4 .8* -D Cubic 1 - 4 .9* D Deviat ions 2 Mixture (5) 8.1*** 4.3** 8.0*** 7.5*** (M) M Linear 1 23.5*** 19.2*** 26.7*** 36.1*** M Quadratic 1 12.0*** - 8.8** M Cubic 1 M Deviat ions 2 D » M (25) -DL * ML 1 DL * MQ 1 DL * MC 1 DL * MD 2 DQ * ML 1 DQ * MQ 1 - - 5.3* DQ * MC 1 - - 4 .1* DQ * MD 2 - - -DC * ML 1 DC * MQ 1 DC * MC 1 DC * MD 2 DD * ML 2 DD * MQ 2 DD * MC .2 DD * MD 4 * , **, *** S i g n i f i c a n t at 5%, 1% and 0.1% l e v e l , respec t i ve l y Not s i g n i f i c a n t 2 3 3 Summary of ANOVA results: Variance ratios for the effects of total population density and mixture proportions on orchardgrass per unit land area Appendix 8.2.2 Time a f te r p lant ing (days) Source of v a r i a t i o n d . f 37 48 63 77 90 Blocks 3 3.0* . . . 4 .5** Density (5) 15.2*** 7.9*** 9.4*** 4.7*** 3.4** (D) D Linear 1 53.9*** 20.2*** 46.3*** 19.9*** 12.8*** D Quadratic 1 18.6*** 18.5*** -D Cubic 1 D Deviat ions 2 Mixture (5) 54.5*** 141.9*** 139.2*** 122.7** 82.7*** (M) M Linear 1 266.3*** 684.1*** 671.2*** 611.0*** 411.0*** M Quadrat ic 1 4 .5* 24.5*** 24.4*** M Cubic 1 M Deviat ions 2 D * M (25) 2.4** 2.6*** 2.0** 1.8* DL * ML 1 30.1*** 17.7*** 24.7*** 6.8* 6 .7* DL * MQ 1 DL * MC 1 DL * MD, 2 DQ * ML 1 5.9* 6 .7* DQ * MQ 1 DQ * MC 1 DQ * MD 2 DC * ML 1 DC * MQ 1 DC * MC 1 DC * MD 2 DD * ML 2 - 4 .0* DD * MQ 2 - - 5.0* 4 .3* DD * MC 2 - - - 4 .2* DD * MD 4 * } * * f *** S i g n i f i c a n t at 5%, 1% and 0.1% l e v e l , r e s p e c t i v e l y - Not s i g n i f i c a n t 234 Appendix 8.2.3 Summary of ANOVA results: Variance ratios for the effects of total population density and mixture proportions on orchardgrass t i l l e r number per plant Time a f te r p lant ing (days) va r ia t ion d . f 37 48 63 77 90 Blocks 3 - - - -Densi ty (5) 8.5*** 53.1*** 38.1*** 59.8*** _ (D) D Linear 1 41.7*** 254.1*** 151.9*** 240.7*** 50.0*** D Quadratic 1 - 11.3*** 32.9*** 49.4*** 202.7*** D Cubic 1 - - • 5.7* 7.1** 40.3*** D Deviat ions 2 - - - - 4 . 9 * Mixture (5) _ 13.7*** 7.5*** 5.5*** _ (M) M Linear 1 - 63.4*** 33.9*** 24.5*** -M Quadrat ic 1 - - - - -M Cubic 1 - - - - -M Deviat ions 2 - - - - -D * M (25) - 3.5*** - 2.0*** -DL * ML 1 4 .6* 9.9** 5.4* 8.1** DL * MQ 1 - 4.2* - - -DL * MC 1 - - - - -DL * MD 2 - 11.2** - - -DQ * ML 1 - 4.4* - 8.2** -DQ * MQ 1 - 9.4** - -DQ * MC 1 - - - - -DQ * MD 2 - 5.0* - - -DC * ML 1 - 4.7* - - -DC * MQ 1 - 13.2** - - -DC * MC 1 - - - - -DC * MD 2 - - - - -DD * ML 2 - - - - -DD * MQ 2 - - - - -DD * MC 2 - - - - -DD * MD 4 - - - - -*, **, *** S i g n i f i c a n t at 5%, ^% and 0 . U l e v e l , respec t i ve l y Not s i g n i f i c a n t 235 Appendix 8.2.4 Summary of ANOVA results: Variance ratios for the effects of total population density and mixture proportions on orchardgrass leaf area per plant Time after planting (days) Source of variation d.f 37 48 63 77 90 Blocks 3 - 3.9* Density (5) 6.2*** 22.8*** 23.5*** 35.9*** 34.4*** (D) D Linear 1 30.8*** 1111.7*** 101.9*** 147.3*** 148.5*** D Quadratic 1 - - 14.0*** 25.1*** 22.0*** D Cubic 1 4.6* D Deviations 2 - -Mixture (5) - 12.4*** 4.5*** 4.3** (M) M Linear 1 - 55.0*** 21.0*** 13.9*** M Quadratic 1 - - - 7.1** -M Cubic 1 - . . . M Deviations 2 D » M (25) - 2.9*** -DL * ML 1 - 8.9** 4.5* DL * MQ 1 - 4.2* - - 6.0* DL * MC 1 DL * MD 2 - 5.2* -DQ * ML 1 5.4* -DQ * MQ 1 - 7.8** -DQ * MC 1 DQ * MD 2 DC * ML 1 - - - - -DC * MQ 1 - 9.4** -DC * MC 1 D C * M D 2 DD * ML 2 DD * MQ 2 DD * MC 2 DD * MD 4 - - - -*, ** , *** Significant at. 5%, 1% and 0.1% level , respectively - Not significant 2 3 6 Appendix 8.2.5 Summary of ANOVA r e s u l t s : Var iance r a t i o s for the e f f e c t s of t o t a l populat ion density and mixture proport ions on orchardgrass lea f area per t i l l e r Time a f te r p lant ing (days) Source of ; v a r i a t i o n d . f 37 48 63 77 90 Blocks 3 4.5** 3 .1* 2.9* 7.6*** Density (5) - 2 .6* 2.3* 2.6* (D) D Linear 1 - 7.1** 4.2* 5.8* 6.5* D Quadratic 1 - 4.4* -D Cubic 1. D Deviat ions 2 Mixture (5) 4.6*** 7.3*** 2.9* 6.9*** 2.3* (M) M Linear 1 19.6*** 18.2*** 6 .1* 10.0** M Quadrat ic 1 - 13.7*** 7.9** 21.1*** 7.8** M Cubic 1 -M Deviat ions 2 D * M (25) -DL * ML 1 - - - -DL * MQ 1 - - - - 4 .2* DL * MC 1 DL * MD 2 DQ * ML 1 DQ * MQ 1 - - - - 4 .2* DQ * MC 1 DQ * MD 2 DC * ML 1 DC * MQ 1 DC * MC 1 D C * M D 2 DD * ML 2 - 4 .6* DD * MQ 2 DD * MC 2 DD * MD 4 * , **, *** S i g n i f i c a n t at 5%, 1% and 0 . 1 * l e v e l , r e s p e c t i v e l y - Not s i g n i f i c a n t 2 3 7 Appendix 8.2.6 Summary of ANOVA results: Variance ratios for the effects of total population density and mixture proportions on timothy dry weight per t i11er Time a f te r p lant ing (days) Source of — — v a r i a t i o n d . f 48 63 77 90 Blocks 3 7.7*** 7.0*** 3.0* -Density (5) _ 4.0** 5.3*** 13.7*** (D) D Linear 1 - 18.2*** 20.1*** 63.5*** D Quadratic 1 4 .8* - - -D Cubic 1 - - - -D Deviat ions 2 - - - -Mixture (5) 3.6** (M) M Linear 1 - - - 12.8*** M Quadratic 1 - - -M Cubic 1 - - - -M Deviat ions 2 - 9.1** - -D * M (25) - - - -DL * ML 1 9.2** DL * MQ 1 - - - -DL * MC 1 - - 7.6** -DL * MD 2 - - - -DQ * ML 1 - - - 4.3* DQ * MQ 1 - - - -DQ * MC 1 - - 4.8* -DQ * MD 2 - - - -DC * ML 1 - - - -DC * MQ 1 - - - -DC * MC 1 - - - -DC * MD 2 - - - -DD * ML 2 - - - 4.5* DD * MQ 2 - - - -DD * MC 2 - - - -DD * MD 4 - -* f ** > *** S i g n i f i c a n t at 5%, 1% and 0.1% l e v e l , r e s p e c t i v e l y Not s i g n i f i c a n t 238 Appendix 8.2.7 Summary of ANOVA results: Variance ratios for the effects of total population density and mixture proportions on timothy panicle dry •eight per plant Source of v a r i a t i o n Time af te r p lant ing (days) d . f 63 77 90 Blocks 3 8.1*** - -Density (5) 43.7*** 39.4*** 15.5*** (D) D Linear 1 169.5*** 165.2*** 64.8*** D Quadrat ic 1 39.3*** 26.8*** 11.0** D Cubic 1 6.9** 4.2* -D Deviat ions 2 - - -Mixture (5) 4.6*** 2.8* — (M) M Linear 1 19.5*** 9.1** -M Quadratic 1 - - -M Cubic 1 - - -M Deviat ions 2 - - -D * M (25) - - -DL * ML 1 _ _ _ DL * MQ 1 - 4.2* -DL * MC 1 - - -DL * MD 2 - - -DQ * ML 1 - - -DQ * MQ 1 - 4.9* -DQ * MC 1 - - -DQ * MD 2 - - -DC * ML 1 - - -DC * MQ 1 - 4.1* -DC * MC 1 - - -DC * MD 2 - - _ DD * ML 2 - - -DD * MQ 2 - - -DD * MC 2 - - -DD * MD 4 - - -* , **, *** S i g n i f i c a n t at 5%, 1% and 0.1% l e v e l , r e s p e c t i v e l y - Not s i g n i f i c a n t Appendix 8.2.8 Summary of ANOVA results: Variance ratios for the effects of total population density and mixture proportions on timothy panicle dry weight per t i l ler Source of var i at ion Time a f te r p lant ing (days) d . f 63 77 90 Blocks 3 16.2*** - 5.6** Density (5) 3 .1* 9.7*** 7.2*** (D) D Linear 1 13.3*** 42.0*** 35.1*** D Quadratic 1 - - -D Cubic 1 - - -D Deviat ions 2 - - -Mixture (5) _ (M) - - -M Linear 1 - - _ M Quadratic 1 - - -M Cubic 1 - - -M Deviat ions 2 6.0* - -D * M (25) - - -DL * ML 1 5.2* DL * MQ 1 - - -DL * MC 1 - -DL * MD 2 - - -DQ * ML 1 - - 4.1* DQ * MQ 1 - - -DQ * MC 1 - - -DQ * MD 2 - - -DC * ML 1 - -DC * MQ 1 - - -DC * MC 1 - - -DC * MD 2 - - -DD * ML 2 - - -DD * MQ 2 - - . -DD * MC 2 - - -DD * MD 4 - 4.5* -* , **, *** S i g n i f i c a n t at 5%, 1% and 0.1% l e v e l , r e s p e c t i v e l y Not s i g n i f i c a n t 240 Appendix 8.2.9 Summary of ANOVA results: Variance ratios for the effects of total population density and mixture proportions on timothy t i l ler number per unit land area Time a f te r p lant ing (days) v a r i a t i o n d . f 37 48 63 77 90 Blocks 3 - - - 3.1* 2 .9* Density (5) 29.4*** 15.5*** 6.0*** 10.6*** 9.6*** (D) D Linear 1 132.1*** 76.0*** 26.5*** 49.6*** 44.9*** D Quadrat ic 1 11.7*** - - - -D Cubic 1 - - - - -D Deviat ions 2 - - - - -Mixture (5) 49.0*** 86.2*** 35.7*** 64.3*** 59.8*** (M) M Linear 1 242.3*** 411.4*** 166.2*** 304.5*** 295.8*** M Quadrat ic 1 - 10.2** 10.5** 13.1*** -M Cubic 1 - - - 4.0* -M Deviat ions 2 - 8.6** - - -D * M (25) 2.1** - - 1.9* -DL * ML 1 28.0*** 4 .8* 7.5* 8.7** 7.0** DL * MQ 1 - - - - -DL * MC 1 - - - - -DL * MD 2 - - - - -DQ * ML 1 - - - - -DQ * MQ 1 - - - - -DQ * MC 1 - - - - -DQ * MD 2 - - - 5.3* -DC * ML 1 - - - - -DC * MQ 1 - - - - 4.3* DC * MC 1 - - - - -DC * MD 2 - - - - -DD * ML 2 - - - - -DD * MQ 2 - - - - -DD * MC 2 - - - - -DD * MD 4 - - - - -* ? * * } *** S i g n i f i c a n t at 5%, 1% and 0.1% l e v e l , r e s p e c t i v e l y Not s i g n i f i c a n t 2 4 1 A p p e n d i x 8.2.10 Summary o f ANOVA r e s u l t s : V a r i a n c e r a t i o s f o r t h e e f f e c t s o f t o t a l p o p u l a t i o n d e n s i t y and m i x t u r e p r o p o r t i o n s on t i m o t h y t i l l e r numbers p e r p i a n t Time a f te r p lant ing (days) Source of var ia t ion d . f 37 48 63 77 90 Blocks 3 - 3.4* - - 3.0* Density (5) 9.3*** 91.4*** 74.8*** 49.0*** 18.8*** (D) D Linear 1 46.0*** 370.1*** 265.3*** 210.7*** 78.6*** D Quadrat ic 1 - 78.1*** 80.5*** 31.1*** 13.1*** D Cubic 1 - 8.0** 20.0*** - -D Deviat ions 2 - - 7.8** - -Mixture (5) 2 .9* 8.1*** 12.2*** 4.3** _ (M) M Linear 1 9 .9* 34.9*** 57.9*** 16.6*** -M Quadrat ic 1 - - - - -M Cubic 1 - - - 4 . 1 * -M Deviat ions 2 - 5.4* - - -D * M (25) - - 2.2** - -DL * ML 1 _ 23.6* DL * MQ 1 - - - - -DL * MC 1 - - - - -DL * MD 2 - - - - -DQ * ML 1 - - 6.6* - -DQ * MQ 1 - • - - - -DQ * MC 1 - - - - -DQ * MD 2 - - - - -DC * ML 1 - - 5.0* - -DC * MQ 1 - - - 4.8* -DC * MC 1 - - - - -DC * MD 2 - - - - -DD * ML 2 - - - 4.6* -DD * MQ 2 - - - - -DD * MC 2 - - - - -DD * MD 4 - - - - -* f * * } *•* S i g n i f i c a n t at 5%, 1% and 0.1% l e v e l , r e s p e c t i v e l y Not s i g n i f i c a n t 242 Appendix 8.2.11 Summary of ANOVA results: Variance ratios for the effects of total population density and mixture proportions on timothy leaf area per plant Time a f t e r p lant ing (days) Source of v a r i a t i o n d . f 37 48 63 77 90 Blocks 3 4.5** 4.8** 3.0* 5.3** 4.4** Density (5) 5.6*** 35.9*** 29.1*** 40.3*** 34.3*** ~7u) D Linear 1 26.9*** 152.8*** 100.4*** 169.0*** 136.9*** D Quadratic 1 - 25.3*** 35.0*** 28.3*** 30.3*** D Cubic 1 - - 7.6** D Deviat ions 2 Mixture (5) - - 3 .3** (M) M Linear 1 - 4 .8* 15.3*** 4.4* M Quadrat ic 1 M Cubic 1 M Deviat ions 2 - 5.0* . . . D * M (25) -DL * ML 1 - - 8.7** DL * MQ 1 - - - 5.3* DL * MC 1 DL * MD 2 DQ * ML 1 DQ * MQ 1 - - - 4 . 1 * DQ * MC 1 DQ * MD 2 DC * ML 1 DC * MQ 1 DC * MC 1 D C * M D 2 - - - . -DD * ML 2 DD * MQ 2 D D * M C 2 DD * MD 4 * f ** f *** S i g n i f i c a n t at 5%, 1% and 0.1% l e v e l , r e s p e c t i v e l y Not s i g n i f i c a n t 243 Summary of ANOVA results: Variance ratios for the effects of total population density and mixture proportions on timothy leaf area per t i l le r Appendix 8.2.12 Time a f te r p lant ing (days) Source of v a r i a t i o n d . f 37 48 63 77 90 Blocks 3 9.3*** 7.3** 21.7*** 7.4*** 4.5** Density (5) - 3 .1* 5.2*** 10.0*** 19.0*** <D) D Linear 1 - 11.6*** 18.9*** 43.7*** 85.8*** D Quadratic 1 - - 6.0* - 5.9* D Cubic 1 D Deviat ions 2 -Mixture (5) - 2 .9* - 2 .6* (M) M Linear 1 - 13.0*** -M Quadrat ic 1 - - - - -M Cubic 1 4 .5* M Deviat ions 2 D * M (25) -DL * ML 1 5.4* -DL * MQ 1 DL * MC 1 -DL * MD 2 - - - 6.9** -DQ * ML 1 DQ * MQ 1 DQ * MC 1 DQ * MD 2 - 5.0* . . . DC * ML 1 DC * MQ 1 - - 4 .2* DC * MC 1 D C * M D 2 DD * ML 2 DD * MQ 2 - - -D D * M C 2 DD * MD 4 * ? ** f *** S i g n i f i c a n t at. 5%, 1% and 0.1% l e v e l , r e s p e c t i v e l y - Not. s i g n i f i c a n t 244 Appendix 8.3 Goodness of f i t and s t a t i s t i c a l significance for mixture models based on multiple regression in one stage (Eq. 2.23, n=24) Relative Time after planting (days) Species Population Density 37 48 63 77 90 Orchardgrass 1/6 -0.10 0.33*** 0.35** 0.23*** 0.02 2/6 -0.15 0.64*** 0.18** 0.25* 0.34** 3/6 -0.17 0.00 0.34* 0.12 -0.01 4/6 0.39*** 0.22* 0.17* 0.43** 0.05 5/6 0.07 0.67*** 0.04 0.31*** 0.18 6/6 -0.03 0.70*** 0.32* 0.29*** 0.10 Timothy 1/6 - -0.12 0.02 0.49** 0.07 0.13 2/6 -0.06 0.19 0.32* 0.36** 0.41** 3/6 0.12 0.34* 0.64*** 0.40* 0.35** 4/6 0.23 0.59*** 0.53*** 0.60*** 0.70*** 5/6 0.22 0.36** 0.51** 0.30** 0.18 6/6 0.34* 0.50*** 0.55*** 0.42** 0.51*** * f ** t *** Significant at 5%, 1% and 0.1% level , respectively for overall regression model related to y ~ ^ . 245 Appendix 8.4.1 Summary of ANOVA results: Variance ratios for the effects of total population density and mixture proportions on Leaf Area Index in orchardgrass Time after planting (days) Source of . variation d.f 37 48 63 77 90 Blocks 3 3.6* - 5.2** Density (5) 8.2*** 5.6*** 2.5* (D) D Linear 1 22.3*** 12.9*** 7.3** D Quadratic 1 14.6*** 14.6*** -D Cubic 1 D Deviations 2 Mixture (5) 35.3*** 69.5*** 62.8*** 50.9*** 41.5*** (M) M Linear 1 164.7*** 336.7*** 307.3*** 252.1*** 205.4*** M Quadratic 1 9.2** 9.6** 6.3* M Cubic 1 - -M Deviations 2 D * M (25) 2.3** 1.8* -DL * ML 1 17.1*** 9.1** DL * MQ 1 DL * MC 1 DL * MD 2 DQ * ML 1 8.0** 6.8* DQ * MQ 1 DQ * MC 1 DQ * MD 2 DC * ML 1 - - 5.1* DC * MQ 1 - - 6.4* DC * MC 1 DC * MD 2 DD * ML 2 - 5.5* DD * MQ 2 DD * MC 2 - - - 5.5* DD * MD 4 * ? * * f *** Significant at 5%, 1% and 0.1% level , respectively Not significant 246 A p p e n d i x 8 . 4 . 2 Summary o f ANOVA r e s u l t s : V a r i a n c e r a t i o s f o r t h e e f f e c t s o f t o t a l p o p u l a t i o n d e n s i t y and m i x t u r e p r o p o r t i o n s on L e a f A r e a R a t i o i n o r c h a r d g r a s s Time a f te r p lant ing (days) Source of v a r i a t i o n d . f 48 63 77 90 Blocks 3 2.2* 32.3*** - 4 .9** Density (5) - 2 .8* 2.5* (D) D Linear 1 D Quadratic 1 - 5.0* - 4 .5* D Cubic 1 D Deviat ions 2 Mixture (5) - 3.6** 3.5** 6.0*** (M) M Linear 1 - 7.5** 7.5** 27.4*** M Quadrat ic 1 - 8.6** 4 .8* M Cubic 1 M Deviat ions 2 D * M (25) - - -DL * ML 1 DL * MQ 1 DL * MC 1 DL * MD 2 - - - -DQ * ML 1 DQ * MQ 1 DQ * MC 1 - - 4 .4* DQ * MD 2 DC * ML 1 - 5 .9* DC * MQ 1 DC * MC 1 DC * MD 2 DD * ML 2 - - 5.6* DD * MQ 2 DD * MC 2 - - 4 .9* DD * MD 4 * ? ** ? *** S i g n i f i c a n t at 5%, 1% and 0 . 1 % l e v e l , r e s p e c t i v e l y - Not s i g n i f i c a n t 247 Appendix 8.A-.3 Summary of ANOVA results: Variance ratios for the effect of total population density and mixture proportions on Leaf Area Index in timothy Source of v a r i a t i o n Time a f te r p lant ing (days) d . f 37 48 63 77 90 Blocks 3 5.3** - 10.7*** 7.7*** -Density (5) 15.1*** 9 # 2*** _ 4.3** D Linear 1 67.5*** 44.8*** - -D Quadrat ic 1 7.2** - 5.4* -D Cubic 1 - - - -D Dev iat ions 2 - - - -Mixture (5) 26.4*** 42.3*** 24.3*** 32.1*** 22.3*** (M) M Linear 1 130.3*** 204.4*** 117.1*** 153.1*** 111.0*** M Quadrat ic 1 - - 4.7* -M Cubic 1 - - - -M Deviat ions 2 - - - -D * M (25) - - - -DL * ML 1 10.3** _ DL * MQ 1 - - - -DL * MC 1 - - - -DL * MD 2 - - - -DQ * ML 1 - - - -DQ * MQ 1 - - - 4 . 7 * DQ * MC 1 - - - -DQ * MD 2 - - - -DC * ML 1 - - - -DC * MQ 1 - - - -DC * MC 1 - - - -DC * MD 2 - - - -DD * ML 2 - - 4.3* -DD * MQ 2 - - - -DD * MC 2 - 4.8* - -DD * MD 4 4 .6* - - -* * f *** S i g n i f i c a n t at 5%, 1% and 0.1% l e v e l , r e s p e c t i v e l y Not s i g n i f i c a n t Appendix 8.4.4 Summary of ANOVA results: Variance ratios for the effects of total population density and mixture proportions on Leaf Area Ratio in timothy Source of v a r i a t i o n d . f Time a f te r p lant ing (days) 48 63 77 90 Blocks 3 4 .0** 51.9*** 6.1*** 6.1*** Density (5) 2.6* 4.2** (D) D Linear 1 - 10.9** 16.8*** D Quadrat ic 1 - - - -D Cubic 1 - -D Dev iat ions 2 - -Mixture (5) _ 3.9** (M) M Linear 1 - 11.7*** M Quadrat ic 1 4 . 1 * -M Cubic 1 - -M Deviat ions 2 -D * M (25) - 1.5* DL * ML 1 _ 11.2** DL * MQ 1 - -DL * MC 1 - 10.7** DL * MD 2 -DQ * ML 1 - - - _ DQ * MQ 1 - _ DQ * MC 1 - -DQ * MD 2 - -DC * ML 1 - -DC * MQ 1 - -DC * MC 1 - -DC * MD 2 - - - -DD * ML 2 - -DD * MQ 2 - -DD * MC 2 - -DD * MD 4 - _ #*f *** S i g n i f i c a n t at. 5%, 1* and 0.1% l e v e l , r e s p e c t i v e l y Not s i g n i f i c a n t 249 Appendix 8.4.5 Summary of ANOVA results: Variance ratios for the effects of total population density and mixture proportions on Harvest Index in timothy Time af te r p lant ing (days) Source of v a r i a t i o n d . f 63 77 90 Blocks 3 12.4*** 3.7* 5.3** Density (5) 3.1* - ( D ) — D Li near 1 - 13.8*** -D Quadrat ic 1 - - -D Cubic 1 - - -D Deviat ions 2 - - -Mixture (5) _ (M) M Linear 1 - - 4 .6* M Quadrat ic 1 - - -M Cubic 1 - - -M Deviat ions 2 - - -D * M (25) - - -DL * ML 1 DL * MQ 1 - - -DL * MC 1 6 .3* - -DL * MD 2 - - -DQ * ML 1 - - -DQ * MQ 1 - - -DQ * MC 1 - - -DQ * MD 2 - - -DC * ML 1 - - -DC * MQ 1 - - -DC * MC 1 - - -DC * MD 2 - - -DD * ML 2 - - -DD * MQ 2 - - -DD * MC 2 - - -DD * MD 4 - - -* ? * * j *** S i g n i f i c a n t at 5%, 1% and 0.1% l e v e l , r e s p e c t i v e l y Not. s i g n i f i c a n t PUBLICATIONS Potdar, M.V. and P.A. Jolliffe. 1986. Plant growth and yield in mixtures of timothy and orchardgrass. Abstract only. CSPP National Meeting, University of Saskatchewan, Sasktoon, June 16-18. Potdar, M.V., K.R. Pawar and L. Sreenivas, 1986. Yield response and transpiration behaviour of sunflower (Helanthus annuus) to leaf stripping and foliar application of antitranspirants-phenyl mercuric acetate (PMA) and kaolin. Agriculture and Agro-industries Journal, Bombay. (In Press). Potdar, M.V. and P.A. Jolliffe. 1983. Growth and yield responses of sunflower (Helianthus annuus) to water deficits. Abstract only. CSPP Silver Anniversary Meeting, University of Waterloo, Waterloo, Ontario, June 19-22. Potdar, M.V. and D.H. Pawar. 1981. Effect of irrigation and kaolin on yield and yield attributing characters in wheat variety H.D.(M) 1593. Journal of Maharashtra Agricultural Universities. _5:169. Potdar, M.V. and K.R. Pawar. 1980. Effect of leaf stripping on the grain yield of sunflower (Helianthus annuus). Journal of Maharashtra Agricultural Universities. _5:164. Potdar, M.V., V.D. Sondge, K.R. Pawar, and A.N. Gite. 1980. Estimation of leaf area constants for green gram (Phaseolus sp.) varieties. Journal of Maharashtra Agricultural Universities. _5:169 Potdar, M.V. and U.C. Upadhyay. 1978. Weed control studies related to yield contributing characters in sugarcane. All India Weed Science Conference, Tamil Nadu Agricultural University, Coimbatore, Tamil Nadu, India. February 3-4. Potdar, M.V., K.R. Pawar and L. Sreenivas. 1977. Simple correlation and regression studies between grain yield and yield attributing characters in sunflower (Helianthus annuus). Indian Journal of Agronomy. 22:115. 

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