Dynamical climatic model for time to flowering in Vigna radiata

Background Phenology data collected recently for about 300 accessions of Vigna radiata (mungbean) is an invaluable resource for investigation of impacts of climatic factors on plant development. Results We developed a new mathematical model that describes the dynamic control of time to flowering by daily values of maximal and minimal temperature, precipitation, day length and solar radiation. We obtained model parameters by adaptation to the available experimental data. The models were validated by cross-validation and used to demonstrate that the phenology of adaptive traits, like flowering time, is strongly predicted not only by local environmental factors but also by plant geographic origin and genotype. Conclusions Of local environmental factors maximal temperature appeared to be the most critical factor determining how faithfully the model describes the data. The models were applied to forecast time to flowering of accessions grown in Taiwan in future years 2020-2030.


Grammatical evolution method
The method was developed recently to recover the analytic form of a function from known values [O'Neill andRyan, 2001, Noorian et al., 2016]. Formal definition of a context-free grammar (CFG): A formal grammar where every production rule, formalized by the pair (n, V ), is in form of n → V . The CFG is defined by the 4-tuple (T , N , R, S), where -T is the finite set of terminal symbols, -N is the finite set of non-terminal symbols, -R is the production rule set, -S ∈ N is the start symbol.
A production rule n → V is realized by replacing the non-terminal symbol n ∈ N with the symbol v ∈ V , where V ∈ (T ∪ N ) * is a sequence of terminal and/or non-terminal symbols [Aho et al., 2006].

S4 Differential Evolution Entirely Parallel method
An effective stochastic method for function minimization termed Differential Evolution (DE), proposed in [Storn and Price, 1995], operates on a set (population) parameter vectors (individuals). The initial population is generated randomly, a size of population N P is fixed. DEEP one can be applied to solve both unconstrained and constrained optimization problems. Constraints may be imposed in the form of inequalities or equalities for a subset of parameters or their combinations. DEEP method [Kozlov and Samsonov, 2011] incorporates the "trigonometric mutation" rule proposed in [Fan and Lampinen, 2003] and used to take into account a value of the objective function for each individual at the recombination step, and the adaptive scheme for selection of internal parameters based on the control of the population diversity developed in [Zaharie, 2002].
In DEEP the age of individual is defined as a number of iterations, during which the individual survived without changes. The number of oldest individuals is substituted with the same number of the best ones after the predefined number of iterations to avoid local minima. Calculations are terminated when the objective function variation becomes less than a predefined value during the several consecutive steps or the maximal number of iterations is exceeded.
DE operates on floating point parameters, while two algorithms for parameter conversion from real to integer are implemented in DEEP [Kozlov et al., 2013]. The first method rounds off a real value to the nearest integer number. Another two step procedure firstly sorts parameters in ascending order and then uses the indices as integer parameters.
DEEP employs the pool of worker threads with asynchronous queue of tasks to evaluate the individual solutions in parallel. The calculation of objective function for each trial vector is pushed to the asynchronous queue and starts as soon as there is an available thread in the pool [Kozlov et al., 2016].
DEEP is implemented in C programming language as console application and using interfaces from GLIB project https://developer.gnome.org/glib/, e.g. Thread Pool API. DEEP method is an open source and free software distributed under the terms of GPL licence version 3. The sources are available at https://gitlab.com/mackoel/deepmethod.