Dissecting external effects on logistic-based growth: equations, analytical solutions and applications.

Characteristic growth patterns of individual organs are, to a large extent, determined by genetic factors, but can also be affected by intra-plant competition for resources and by environmental conditions. The current study proposes a dynamical system for modelling this dual control for logistic-based growth. The state of the system is defined by two state variables: size (s), and developmental age (α). The intrinsic properties of the system are represented by the potential relative growth rate as a function of α. This formulation allows dissection of the external effects on the system dynamics into two components: one that changes the duration of growth without affecting the final size and one that affects the final size without much effect on the duration. The former component determines the relationship between α and time, while the latter determines the effect on the system trajectory, s(α). The presented dynamical system is simpler and has a wider range of potential applications than the system proposed by Thornley and France (2005) for modelling logistic growth under resource limitation. The current approach can be also useful in ecology and in comparative studies of different genotypes and their responses to environmental conditions.

[1]  C. Granier,et al.  Plasticity to soil water deficit in Arabidopsis thaliana: dissection of leaf development into underlying growth dynamic and cellular variables reveals invisible phenotypes. , 2006, Plant, cell & environment.

[2]  F. Tardieu,et al.  Is thermal time adequate for expressing the effects of temperature on sunflower leaf development , 1998 .

[3]  C. Granier,et al.  Correlation between leaf growth variables suggest intrinsic and early controls of leaf size in Arabidopsis thaliana , 2005 .

[4]  F. J. Richards.  A Flexible Growth Function for Empirical Use , 1959 .

[5]  H. Schultz.  An empirical model for the simulation of leaf appearance and leaf area development of primary shoots of several grapevine (Vitis vinifera L.) canopy-systems , 1992 .

[6]  G. Farquhar,et al.  Foliar stage in wheat correlates better to photothermal time than to thermal time , 1989 .

[7]  Granier,et al.  Water deficit and spatial pattern of leaf development. Variability In responses can Be simulated using a simple model of leaf development , 1999, Plant physiology.

[8]  K. Chenu,et al.  Integrated responses of rosette organogenesis, morphogenesis and architecture to reduced incident light in Arabidopsis thaliana results in higher efficiency of light interception. , 2005, Functional plant biology : FPB.

[9]  François Tardieu,et al.  Modelling leaf expansion in a fluctuating environment: are changes in specific leaf area a consequence of changes in expansion rate? , 1999 .

[10]  J. Thornley,et al.  An open-ended logistic-based growth function , 2005 .

[11]  Roderick Hunt,et al.  Plant Growth Analysis: The Rationale Behind the Use of the Fitted Mathematical Function , 1979 .

[12]  Boris Zeide,et al.  Analysis of Growth Equations , 1993 .

[13]  C. Granier,et al.  A dynamic analysis of the shade-induced plasticity in Arabidopsis thaliana rosette leaf development reveals new components of the shade-adaptative response. , 2006, Annals of botany.

[14]  Alla N Seleznyova,et al.  Independent control of organogenesis and shoot tip abortion are key factors to developmental plasticity in kiwifruit (Actinidia). , 2007, Annals of botany.

[15]  J. Shepherd,et al.  An open-ended logistic-based growth function: Analytical solutions and the power-law logistic model , 2007 .

[16]  P. Prusinkiewicz,et al.  Using L-systems for modeling source-sink interactions, architecture and physiology of growing trees: the L-PEACH model. , 2005, The New phytologist.

[17]  Evelyne Costes,et al.  Quantitative analysis of shoot development and branching patterns in Actinidia. , 2002, Annals of botany.

[18]  Katrin Kahlen,et al.  Towards functional-structural modelling of greenhouse cucumber , 2007 .

[19]  Alla N. Seleznyova,et al.  MODELLING EFFECT OF TEMPERATURE ON AREA EXPANSION AT THE LEAF, THE SHOOT AND THE WHOLE PLANT LEVEL , 2006 .

[20]  Alla N. Seleznyova,et al.  Effects of Temperature and Leaf Position on Leaf Area Expansion of Kiwifruit (Actinidia deliciosa) Shoots: Development of a Modelling Framework , 2001 .

[21]  Prof. Dr. Francis Hallé,et al.  Tropical Trees and Forests , 1978, Springer Berlin Heidelberg.

[22]  P. Medawar.  The ‘Laws’ of Biological Growth , 1941, Nature.

[23]  Karine Chenu,et al.  Individual leaf development in Arabidopsis thaliana: a stable thermal-time-based programme. , 2002, Annals of botany.

[24]  M. Gilpin,et al.  Global models of growth and competition. , 1973, Proceedings of the National Academy of Sciences of the United States of America.

[25]  Y. Mizukami,et al.  A matter of size: developmental control of organ size in plants. , 2001, Current opinion in plant biology.

[26]  F. Steward.  Analysis of growth : behavior of plants and their organs , 1969 .

[27]  Alla N. Seleznyova,et al.  From controlled environments to field simulations: leaf area dynamics and photosynthesis of kiwifruit vines (Actinidia deliciosa). , 2004, Functional plant biology : FPB.

[28]  F. J. Richards.  The Quantitative Analysis of Growth , 1969 .

[29]  P. Prusinkiewicz,et al.  Modeling the architecture of expanding Fraxinus pennsylvanica shoots using L-systems , 1994 .

[30]  Przemyslaw Prusinkiewicz,et al.  Numerical methods for transport-resistance sink-source allocation models , 2007 .

[31]  D. R. Causton,et al.  The Biometry of Plant Growth , 1982 .

[32]  P. de Reffye,et al.  A dynamic, architectural plant model simulating resource-dependent growth. , 2004, Annals of botany.

[33]  K. Chenu,et al.  Simulations of virtual plants reveal a role for SERRATE in the response of leaf development to light in Arabidopsis thaliana. , 2007, The New phytologist.