A modelling framework to simulate foliar fungal epidemics using functional–structural plant models

Background and Aims

Sustainable agriculture requires the identification of new, environmentally responsible strategies of crop protection. Modelling of pathosystems can allow a better understanding of the major interactions inside these dynamic systems and may lead to innovative protection strategies. In particular, functional–structural plant models (FSPMs) have been identified as a means to optimize the use of architecture-related traits. A current limitation lies in the inherent complexity of this type of modelling, and thus the purpose of this paper is to provide a framework to both extend and simplify the modelling of pathosystems using FSPMs.

Methods

Different entities and interactions occurring in pathosystems were formalized in a conceptual model. A framework based on these concepts was then implemented within the open-source OpenAlea modelling platform, using the platform''s general strategy of modelling plant–environment interactions and extending it to handle plant interactions with pathogens. New developments include a generic data structure for representing lesions and dispersal units, and a series of generic protocols to communicate with objects representing the canopy and its microenvironment in the OpenAlea platform. Another development is the addition of a library of elementary models involved in pathosystem modelling. Several plant and physical models are already available in OpenAlea and can be combined in models of pathosystems using this framework approach.

Key Results

Two contrasting pathosystems are implemented using the framework and illustrate its generic utility. Simulations demonstrate the framework''s ability to simulate multiscaled interactions within pathosystems, and also show that models are modular components within the framework and can be extended. This is illustrated by testing the impact of canopy architectural traits on fungal dispersal.

Conclusions

This study provides a framework for modelling a large number of pathosystems using FSPMs. This structure can accommodate both previously developed models for individual aspects of pathosystems and new ones. Complex models are deconstructed into separate ‘knowledge sources’ originating from different specialist areas of expertise and these can be shared and reassembled into multidisciplinary models. The framework thus provides a beneficial tool for a potential diverse and dynamic research community.     

Towards aspect-oriented functional--structural plant modelling

Background and Aims

Functional–structural plant models (FSPMs) are used to integrate knowledge and test hypotheses of plant behaviour, and to aid in the development of decision support systems. A significant amount of effort is being put into providing a sound methodology for building them. Standard techniques, such as procedural or object-oriented programming, are not suited for clearly separating aspects of plant function that criss-cross between different components of plant structure, which makes it difficult to reuse and share their implementations. The aim of this paper is to present an aspect-oriented programming approach that helps to overcome this difficulty.

Methods

The L-system-based plant modelling language L+C was used to develop an aspect-oriented approach to plant modelling based on multi-modules. Each element of the plant structure was represented by a sequence of L-system modules (rather than a single module), with each module representing an aspect of the element''s function. Separate sets of productions were used for modelling each aspect, with context-sensitive rules facilitated by local lists of modules to consider/ignore. Aspect weaving or communication between aspects was made possible through the use of pseudo-L-systems, where the strict-predecessor of a production rule was specified as a multi-module.

Key Results

The new approach was used to integrate previously modelled aspects of carbon dynamics, apical dominance and biomechanics with a model of a developing kiwifruit shoot. These aspects were specified independently and their implementation was based on source code provided by the original authors without major changes.

Conclusions

This new aspect-oriented approach to plant modelling is well suited for studying complex phenomena in plant science, because it can be used to integrate separate models of individual aspects of plant development and function, both previously constructed and new, into clearly organized, comprehensive FSPMs. In a future work, this approach could be further extended into an aspect-oriented programming language for FSPMs.     

Assessment of uncertainty in functional-structural plant models

Background and Aims

Constructing functional–structural plant models (FSPMs) is a valuable method for examining how physiology and morphology interact in determining plant processes. However, such models always have uncertainty concerned with whether model components have been selected and represented effectively, with the number of model outputs simulated and with the quality of data used in assessment. We provide a procedure for defining uncertainty of an FSPM and how this uncertainty can be reduced.

Methods

An important characteristic of FSPMs is that typically they calculate many variables. These can be variables that the model is designed to predict and also variables that give indications of how the model functions. Together these variables are used as criteria in a method of multi-criteria assessment. Expected ranges are defined and an evolutionary computation algorithm searches for model parameters that achieve criteria within these ranges. Typically, different combinations of model parameter values provide solutions achieving different combinations of variables within their specified ranges. We show how these solutions define a Pareto Frontier that can inform about the functioning of the model.

Key Results

The method of multi-criteria assessment is applied to development of BRANCHPRO, an FSPM for foliage reiteration on old-growth branches of Pseudotsuga menziesii. A geometric model utilizing probabilities for bud growth is developed into a causal explanation for the pattern of reiteration found on these branches and how this pattern may contribute to the longevity of this species.

Conclusions

FSPMs should be assessed by their ability to simulate multiple criteria simultaneously. When different combinations of parameter values achieve different groups of assessment criteria effectively a Pareto Frontier can be calculated and used to define the sources of model uncertainty.     

Modelling seagrass growth and development to evaluate transplanting strategies for restoration

Background and Aims

Seagrasses are important marine plants that are under threat globally. Restoration by transplanting vegetative fragments or seedlings into areas where seagrasses have been lost is possible, but long-term trial data are limited. The goal of this study is to use available short-term data to predict long-term outcomes of transplanting seagrass.

Methods

A functional–structural plant model of seagrass growth that integrates data collected from short-term trials and experiments is presented. The model was parameterized for the species Posidonia australis, a limited validation of the model against independent data and a sensitivity analysis were conducted and the model was used to conduct a preliminary evaluation of different transplanting strategies.

Key Results

The limited validation was successful, and reasonable long-term outcomes could be predicted, based only on short-term data.

Conclusions

This approach for modelling seagrass growth and development enables long-term predictions of the outcomes to be made from different strategies for transplanting seagrass, even when empirical long-term data are difficult or impossible to collect. More validation is required to improve confidence in the model''s predictions, and inclusion of more mechanism will extend the model''s usefulness. Marine restoration represents a novel application of functional–structural plant modelling.     

Identifying ontogenetic, environmental and individual components of forest tree growth

Background and Aims

This study aimed to identify and characterize the ontogenetic, environmental and individual components of forest tree growth. In the proposed approach, the tree growth data typically correspond to the retrospective measurement of annual shoot characteristics (e.g. length) along the trunk.

Methods

Dedicated statistical models (semi-Markov switching linear mixed models) were applied to data sets of Corsican pine and sessile oak. In the semi-Markov switching linear mixed models estimated from these data sets, the underlying semi-Markov chain represents both the succession of growth phases and their lengths, while the linear mixed models represent both the influence of climatic factors and the inter-individual heterogeneity within each growth phase.

Key Results

On the basis of these integrative statistical models, it is shown that growth phases are not only defined by average growth level but also by growth fluctuation amplitudes in response to climatic factors and inter-individual heterogeneity and that the individual tree status within the population may change between phases. Species plasticity affected the response to climatic factors while tree origin, sampling strategy and silvicultural interventions impacted inter-individual heterogeneity.

Conclusions

The transposition of the proposed integrative statistical modelling approach to cambial growth in relation to climatic factors and the study of the relationship between apical growth and cambial growth constitute the next steps in this research.     

Modelling fruit-temperature dynamics within apple tree crowns using virtual plants

Background and Aims

Fruit temperature results from a complex system involving the climate, the tree architecture, the fruit location within the tree crown and the fruit thermal properties. Despite much theoretical and experimental evidence for large differences (up to 10 °C in sunny conditions) between fruit temperature and air temperature, fruit temperature is never used in horticultural studies. A way of modelling fruit-temperature dynamics from climate data is addressed in this work.

Methods

The model is based upon three-dimensional virtual representation of apple trees and links three-dimensional virtual trees with a physical-based fruit-temperature dynamical model. The overall model was assessed by comparing model outputs to field measures of fruit-temperature dynamics.

Key Results

The model was able to simulate both the temperature dynamics at fruit scale, i.e. fruit-temperature gradients and departure from air temperature, and at the tree scale, i.e. the within-tree-crown variability in fruit temperature (average root mean square error value over fruits was 1·43 °C).

Conclusions

This study shows that linking virtual plants with the modelling of the physical plant environment offers a relevant framework to address the modelling of fruit-temperature dynamics within a tree canopy. The proposed model offers opportunities for modelling effects of the within-crown architecture on fruit thermal responses in horticultural studies.     

Mathematical model of fructan biosynthesis and polymer length distribution in plants

Background and Aims

There are many unresolved issues concerning the biochemistry of fructan biosynthesis. The aim of this paper is to address some of these by means of modelling mathematically the biochemical processes.

Methods

A model has been constructed for the step-by-step synthesis of fructan polymers. This is run until a steady state is achieved for which a polymer distribution is predicted. It is shown how qualitatively different distributions can be obtained.

Key Results

It is demonstrated how a set of experimental results on polymer distribution can by simulated by a simple parameter adjustments.

Conclusions

Mathematical modelling of fructan biosynthesis can provide a useful tool for helping elucidate the details of the biosynthetic processes.     

Dry matter partitioning models for the simulation of individual fruit growth in greenhouse cucumber canopies

Background and Aims

Growth imbalances between individual fruits are common in indeterminate plants such as cucumber (Cucumis sativus). In this species, these imbalances can be related to differences in two growth characteristics, fruit growth duration until reaching a given size and fruit abortion. Both are related to distribution, and environmental factors as well as canopy architecture play a key role in their differentiation. Furthermore, events leading to a fruit reaching its harvestable size before or simultaneously with a prior fruit can be observed. Functional–structural plant models (FSPMs) allow for interactions between environmental factors, canopy architecture and physiological processes. Here, we tested hypotheses which account for these interactions by introducing dominance and abortion thresholds for the partitioning of assimilates between growing fruits.

Methods

Using the L-System formalism, an FSPM was developed which combined a model for architectural development, a biochemical model of photosynthesis and a model for assimilate partitioning, the last including a fruit growth model based on a size-related potential growth rate (R_P). Starting from a distribution proportional to R_P, the model was extended by including abortion and dominance. Abortion was related to source strength and dominance to sink strength. Both thresholds were varied to test their influence on fruit growth characteristics. Simulations were conducted for a dense row and a sparse isometric canopy.

Key Results

The simple partitioning models failed to simulate individual fruit growth realistically. The introduction of abortion and dominance thresholds gave the best results. Simulations of fruit growth durations and abortion rates were in line with measurements, and events in which a fruit was harvestable earlier than an older fruit were reproduced.

Conclusions

Dominance and abortion events need to be considered when simulating typical fruit growth traits. By integrating environmental factors, the FSPM can be a valuable tool to analyse and improve existing knowledge about the dynamics of assimilates partitioning.     

Assessing the effects of management on forest growth across France: insights from a new functional–structural model

Background and Aims

The structure of a forest stand, i.e. the distribution of tree size features, has strong effects on its functioning. The management of the structure is therefore an important tool in mitigating the impact of predicted changes in climate on forests, especially with respect to drought. Here, a new functional–structural model is presented and is used to assess the effects of management on forest functioning at a national scale.

Methods

The stand process-based model (PBM) Castanea was coupled to a stand structure module (SSM) based on empirical tree-to-tree competition rules. The calibration of the SSM was based on a thorough analysis of intersite and interannual variability of competition asymmetry. The coupled Castanea–SSM model was evaluated across France using forest inventory data, and used to compare the effect of contrasted silvicultural practices on simulated stand carbon fluxes and growth.

Key Results

The asymmetry of competition varied consistently with stand productivity at both spatial and temporal scales. The modelling of the competition rules enabled efficient prediction of changes in stand structure within the Castanea PBM. The coupled model predicted an increase in net primary productivity (NPP) with management intensity, resulting in higher growth. This positive effect of management was found to vary at a national scale across France: the highest increases in NPP were attained in forests facing moderate to high water stress; however, the absolute effect of management on simulated stand growth remained moderate to low because stand thinning involved changes in carbon allocation at the tree scale.

Conclusions

This modelling approach helps to identify the areas where management efforts should be concentrated in order to mitigate near-future drought impact on national forest productivity. Around a quarter of the French temperate oak and beech forests are currently in zones of high vulnerability, where management could thus mitigate the influence of climate change on forest yield     

Modelling temperature,photoperiod and vernalization responses of Brunonia australis (Goodeniaceae) and Calandrinia sp. (Portulacaceae) to predict flowering time

Background and Aims

Crop models for herbaceous ornamental species typically include functions for temperature and photoperiod responses, but very few incorporate vernalization, which is a requirement of many traditional crops. This study investigated the development of floriculture crop models, which describe temperature responses, plus photoperiod or vernalization requirements, using Australian native ephemerals Brunonia australis and Calandrinia sp.

Methods

A novel approach involved the use of a field crop modelling tool, DEVEL2. This optimization program estimates the parameters of selected functions within the development rate models using an iterative process that minimizes sum of squares residual between estimated and observed days for the phenological event. Parameter profiling and jack-knifing are included in DEVEL2 to remove bias from parameter estimates and introduce rigour into the parameter selection process.

Key Results

Development rate of B. australis from planting to first visible floral bud (VFB) was predicted using a multiplicative approach with a curvilinear function to describe temperature responses and a broken linear function to explain photoperiod responses. A similar model was used to describe the development rate of Calandrinia sp., except the photoperiod function was replaced with an exponential vernalization function, which explained a facultative cold requirement and included a coefficient for determining the vernalization ceiling temperature. Temperature was the main environmental factor influencing development rate for VFB to anthesis of both species and was predicted using a linear model.

Conclusions

The phenology models for B. australis and Calandrinia sp. described development rate from planting to VFB and from VFB to anthesis in response to temperature and photoperiod or vernalization and may assist modelling efforts of other herbaceous ornamental plants. In addition to crop management, the vernalization function could be used to identify plant communities most at risk from predicted increases in temperature due to global warming.     

Linking water stress effects on carbon partitioning by introducing a xylem circuit into L-PEACH

Background and Aims

Many physiological processes such as photosynthesis, respiration and transpiration can be strongly influenced by the diurnal patterns of within-tree water potential. Despite numerous experiments showing the effect of water potential on fruit-tree development and growth, there are very few models combining carbohydrate allocation with water transport. The aim of this work was to include a xylem circuit into the functional–structural L-PEACH model.

Methods

The xylem modelling was based on an electrical circuit analogy and the Hagen–Poisseuille law for hydraulic conductance. Sub-models for leaf transpiration, soil water potential and the soil–plant interface were also incorporated to provide the driving force and pathway for water flow. The model was assessed by comparing model outputs to field measurements and published knowledge.

Key Results

The model was able to simulate both the water uptake over a season and the effect of different irrigation treatments on tree development, growth and fruit yield.

Conclusions

This work opens the way to a new field of modelling where complex interactions between water transport, carbohydrate allocation and physiological functions can be simulated at the organ level and describe functioning and behaviour at the tree scale.     

Combining FISH and model-based predictions to understand chromosome evolution in Typhonium (Araceae)

Background and Aims

Since the advent of molecular phylogenetics, numerous attempts have been made to infer the evolutionary trajectories of chromosome numbers on DNA phylogenies. Ideally, such inferences should be evaluated against cytogenetic data. Towards this goal, we carried out phylogenetic modelling of chromosome number change and fluorescence in situ hybridization (FISH) in a medium sized genus of Araceae to elucidate if data from chromosomal markers would support maximum likelihood-inferred changes in chromosome numbers among close relatives. Typhonium, the focal genus, includes species with 2n = 65 and 2n = 8, the lowest known count in the family.

Methods

A phylogeny from nuclear and plastid sequences (96 taxa, 4252 nucleotides) and counts for all included species (15 of them first reported here) were used to model chromosome number evolution, assuming discrete events, such as polyploidization and descending or ascending dysploidy, occurring at different rates. FISH with three probes (5S rDNA, 45S rDNA and Arabidopsis-like telomeres) was performed on ten species with 2n = 8 to 2n = 24.

Key Results

The best-fitting models assume numerous past chromosome number reductions. Of the species analysed with FISH, the two with the lowest chromosome numbers contained interstitial telomeric signals (Its), which together with the phylogeny and modelling indicates decreasing dysploidy as an explanation for the low numbers. A model-inferred polyploidization in another species is matched by an increase in rDNA sites.

Conclusions

The combination of a densely sampled phylogeny, ancestral state modelling and FISH revealed that the species with n = 4 is highly derived, with the FISH data pointing to a Robertsonian fusion-like chromosome rearrangement in the ancestor of this species.     

Quantitative characterization of clumping in Scots pine crowns

Background and Aims

Proper characterization of the clumped structure of forests is needed for calculation of the absorbed radiation and photosynthetic production by a canopy. This study examined the dependency of crown-level clumping on tree size and growth conditions in Scots pine (Pinus sylvestris), and determined the ability of statistical canopy radiation models to quantify the degree of self-shading within crowns as a result of the clumping effect.

Methods

Twelve 3-D Scots pine trees were generated using an application of the LIGNUM model, and the crown-level clumping as quantified by the crown silhouette to total needle area ratio (STAR_crown) was calculated. The results were compared with those produced by the stochastic approach of modelling tree crowns as geometric shapes filled with a random medium.

Key Results

Crown clumping was independent of tree height, needle area and growth conditions. The results supported the capability of the stochastic approach in characterizing clumping in crowns given that the outer shell of the tree crown is well represented.

Conclusions

Variation in the whole-stand clumping index is induced by differences in the spatial pattern of trees as a function of, for example, stand age rather than by changes in the degree of self-shading within individual crowns as they grow bigger.     

Insights into secondary growth in perennial plants: its unequal spatial and temporal dynamics in the apple (Malus domestica) is driven by architectural position and fruit load

Background and Aims

Secondary growth is a main physiological sink. However, the hierarchy between the processes which compete with secondary growth is still a matter of debate, especially on fruit trees where fruit weight dramatically increases with time. It was hypothesized that tree architecture, here mediated by branch age, is likely to have a major effect on the dynamics of secondary growth within a growing season.

Methods

Three variables were monitored on 6-year-old ‘Golden Delicious’ apple trees from flowering time to harvest: primary shoot growth, fruit volume, and cross-section area of branch portions of consecutive ages. Analyses were done through an ANOVA-type analysis in a linear mixed model framework.

Key Results

Secondary growth exhibited three consecutive phases characterized by unequal relative area increment over the season. The age of the branch had the strongest effect, with the highest and lowest relative area increment for the current-year shoots and the trunk, respectively. The growth phase had a lower effect, with a shift of secondary growth through the season from leafy shoots towards older branch portions. Eventually, fruit load had an effect on secondary growth mainly after primary growth had ceased.

Conclusions

The results support the idea that relationships between production of photosynthates and allocation depend on both primary growth and branch architectural position. Fruit load mainly interacted with secondary growth later in the season, especially on old branch portions.     

An assembly and alignment-free method of phylogeny reconstruction from next-generation sequencing data

Background

Next-generation sequencing technologies are rapidly generating whole-genome datasets for an increasing number of organisms. However, phylogenetic reconstruction of genomic data remains difficult because de novo assembly for non-model genomes and multi-genome alignment are challenging.

Results

To greatly simplify the analysis, we present an Assembly and Alignment-Free (AAF) method (https://sourceforge.net/projects/aaf-phylogeny) that constructs phylogenies directly from unassembled genome sequence data, bypassing both genome assembly and alignment. Using mathematical calculations, models of sequence evolution, and simulated sequencing of published genomes, we address both evolutionary and sampling issues caused by direct reconstruction, including homoplasy, sequencing errors, and incomplete sequencing coverage. From these results, we calculate the statistical properties of the pairwise distances between genomes, allowing us to optimize parameter selection and perform bootstrapping. As a test case with real data, we successfully reconstructed the phylogeny of 12 mammals using raw sequencing reads. We also applied AAF to 21 tropical tree genome datasets with low coverage to demonstrate its effectiveness on non-model organisms.

Conclusion

Our AAF method opens up phylogenomics for species without an appropriate reference genome or high sequence coverage, and rapidly creates a phylogenetic framework for further analysis of genome structure and diversity among non-model organisms.

Electronic supplementary material

The online version of this article (doi:10.1186/s12864-015-1647-5) contains supplementary material, which is available to authorized users.     

Present,past and future of the European rock fern Asplenium fontanum: combining distribution modelling and population genetics to study the effect of climate change on geographic range and genetic diversity

Background and Aims

Climate change is expected to alter the geographic range of many plant species dramatically. Predicting this response will be critical to managing the conservation of plant resources and the effects of invasive species. The aim of this study was to predict the response of temperate homosporous ferns to climate change.

Methods

Genetic diversity and changes in distribution range were inferred for the diploid rock fern Asplenium fontanum along a South–North transect, extending from its putative last glacial maximum (LGM) refugia in southern France towards southern Germany and eastern-central France. This study reconciles observations from distribution models and phylogeographic analyses derived from plastid and nuclear diversity.

Key Results

Genetic diversity distribution and niche modelling propose that genetic diversity accumulates in the LGM climate refugium in southern France with the formation of a diversity gradient reflecting a slow, post-LGM range expansion towards the current distribution range. Evidence supports the fern''s preference for outcrossing, contradicting the expectation that homosporous ferns would populate new sites by single-spore colonization. Prediction of climate and distribution range change suggests that a dramatic loss of range and genetic diversity in this fern is possible. The observed migration is best described by the phalanx expansion model.

Conclusions

The results suggest that homosporous ferns reproducing preferentially by outcrossing accumulate genetic diversity primarily in LGM climate refugia and may be threatened if these areas disappear due to global climate change.     

New views of tapetum ultrastructure and pollen exine development in Arabidopsis thaliana

Background and Aims

The Arabidopsis thaliana pollen cell wall is a complex structure consisting of an outer sporopollenin framework and lipid-rich coat, as well as an inner cellulosic wall. Although mutant analysis has been a useful tool to study pollen cell walls, the ultrastructure of the arabidopsis anther has proved to be challenging to preserve for electron microscopy.

Methods

In this work, high-pressure freezing/freeze substitution and transmission electron microscopy were used to examine the sequence of developmental events in the anther that lead to sporopollenin deposition to form the exine and the dramatic differentiation and death of the tapetum, which produces the pollen coat.

Key Results

Cryo-fixation revealed a new view of the interplay between sporophytic anther tissues and gametophytic microspores over the course of pollen development, especially with respect to the intact microspore/pollen wall and the continuous tapetum epithelium. These data reveal the ultrastructure of tapetosomes and elaioplasts, highly specialized tapetum organelles that accumulate pollen coat components. The tapetum and middle layer of the anther also remain intact into the tricellular pollen and late uninucleate microspore stages, respectively.

Conclusions

This high-quality structural information, interpreted in the context of recent functional studies, provides the groundwork for future mutant studies where tapetum and microspore ultrastructure is assessed.     

Linking ecophysiological modelling with quantitative genetics to support marker-assisted crop design for improved yields of rice (Oryza sativa) under drought stress

Background and Aims

Genetic markers can be used in combination with ecophysiological crop models to predict the performance of genotypes. Crop models can estimate the contribution of individual markers to crop performance in given environments. The objectives of this study were to explore the use of crop models to design markers and virtual ideotypes for improving yields of rice (Oryza sativa) under drought stress.

Methods

Using the model GECROS, crop yield was dissected into seven easily measured parameters. Loci for these parameters were identified for a rice population of 94 introgression lines (ILs) derived from two parents differing in drought tolerance. Marker-based values of ILs for each of these parameters were estimated from additive allele effects of the loci, and were fed to the model in order to simulate yields of the ILs grown under well-watered and drought conditions and in order to design virtual ideotypes for those conditions.

Key Results

To account for genotypic yield differences, it was necessary to parameterize the model for differences in an additional trait ‘total crop nitrogen uptake’ (N_max) among the ILs. Genetic variation in N_max had the most significant effect on yield; five other parameters also significantly influenced yield, but seed weight and leaf photosynthesis did not. Using the marker-based parameter values, GECROS also simulated yield variation among 251 recombinant inbred lines of the same parents. The model-based dissection approach detected more markers than the analysis using only yield per se. Model-based sensitivity analysis ranked all markers for their importance in determining yield differences among the ILs. Virtual ideotypes based on markers identified by modelling had 10–36 % more yield than those based on markers for yield per se.

Conclusions

This study outlines a genotype-to-phenotype approach that exploits the potential value of marker-based crop modelling in developing new plant types with high yields. The approach can provide more markers for selection programmes for specific environments whilst also allowing for prioritization. Crop modelling is thus a powerful tool for marker design for improved rice yields and for ideotyping under contrasting conditions.     

Root–shoot growth responses during interspecific competition quantified using allometric modelling

Background and Aims

Plant competition studies are restricted by the difficulty of quantifying root systems of competitors. Analyses are usually limited to above-ground traits. Here, a new approach to address this issue is reported.

Methods

Root system weights of competing plants can be estimated from: shoot weights of competitors; combined root weights of competitors; and slopes (scaling exponents, α) and intercepts (allometric coefficients, β) of ln-regressions of root weight on shoot weight of isolated plants. If competition induces no change in root : shoot growth, α and β values of competing and isolated plants will be equal. Measured combined root weight of competitors will equal that estimated allometrically from measured shoot weights of each competing plant. Combined root weights can be partitioned directly among competitors. If, as will be more usual, competition changes relative root and shoot growth, the competitors'' combined root weight will not equal that estimated allometrically and cannot be partitioned directly. However, if the isolated-plant α and β values are adjusted until the estimated combined root weight of competitors matches the measured combined root weight, the latter can be partitioned among competitors using their new α and β values. The approach is illustrated using two herbaceous species, Dactylis glomerata and Plantago lanceolata.

Key Results

Allometric modelling revealed a large and continuous increase in the root : shoot ratio by Dactylis, but not Plantago, during competition. This was associated with a superior whole-plant dry weight increase in Dactylis, which was ultimately 2·5-fold greater than that of Plantago. Whole-plant growth dominance of Dactylis over Plantago, as deduced from allometric modelling, occurred 14–24 d earlier than suggested by shoot data alone.

Conclusion

Given reasonable assumptions, allometric modelling can analyse competitive interactions in any species mixture, and overcomes a long-standing problem in studies of competition.