In the 1960s, Brouwer and others developed the concept of functional equilibrium [6,7] to attempt to
describe mathematically the observed dependence of root/shoot ratios on the availability of substrates.
This model can be depicted mathematically as follows [7]:
WrSr∝WsSs (2)where Wrroot mass (g)
Srspecific absorption rate of the root for a particular nutrient (g g^1 day^1 )
Wsshoot mass (g)
Ssspecific photosythesis rate of the shoot (g g^1 day^1 )
The functional equilibrium concept is based upon the idea that substrates required for growth are
composed of elements obtained either above or below ground and that growth of any part of the plant is
dependent upon the availability of all the required growth substrates. Furthermore, it assumes that each
substrate is first available to the plant parts in the region in which the substrate is first acquired. For ex-
ample, N, acquired from the soil by uptake processes occurring in the roots, would first be available for
root shoot growth. Likewise, C, obtained through photosynthesis in the shoots, would first be available
for shoot growth. Thus, for a given substrate, limitation of that substrate would reduce the growth of re-
gions away from the acquisition site. For instance, if N is limited, root growth would continue and the
shoot growth would decrease because N is first available to the root through uptake from the soil. Con-
versely, if C was limiting, shoot growth would continue and the root growth would decrease because C is
first available to the shoot through photosynthesis.
This concept was further refined by Thornley and colleagues [8–10], who added the assumption that
dry matter distribution between root and shoot was only indirectly regulated by the uptake activities of
the shoot and root, as indicated in Eq. (1) but rather was controlled by the availabilityof those substrates
in the form of labile storage pools. Pool sizes in turn would depend not only on supply (i.e., WsSsandWrSr)
but also on utilization and, more important, transport of the substrates between shoot and root. Thus,
Thornley’s model takes a much more mechanistic approach than simple utilization of the functional equi-
librium equation. The Thornley model has since been extended to the point of being an ecosystem model
for grassland crops and was extensively characterized in a recent volume [11]. In this latest rendition, the
detail for the plant submodel alone points to the complexity of the models that must be invoked to simu-
late plant growth effectively [11].
III. COMPUTER SIMULATION OF PLANT GROWTH
The ability to express plant growth responses as mathematical functions as predicted by these models
makes it highly attractive to input these equations into computer programs to use as predictors of plant
growth under various environmental conditions. These computer simulations have been used by biolo-
gists to predict plant growth for about 30 years [2,12], but until recently their use has been limited to a
small group of researchers. With the advent of more powerful and inexpensive computational equipment,
the past few decades have seen a huge increase in the number of simulations published for horticultural
and agronomic purposes [13,14]. As personal computers continue to become more accessible and more
powerful, and coupled with vast improvements in equipment for field data collection, simulations for
plant growth are becoming more practical and have a much wider range of applications than when these
models were first developed. Users of a simulation now simply supply input to a computer program,
which may include information defining the growing environment to which the plants are expected to re-
spond, such as weather, soil, and water, to name a few. Input also generally includes a set of initial con-
ditions, the starting point for growth to be simulated, and information regarding what is being simulated.
The computer program then simulates the growing process of the plants using a combination of models
within the simulation that model the process of growth by way of mathematical equations and relation-
ships. The output of the program is then the growth data based on the input provided by the user. The
growth data can be in many forms depending on what is required by the user.
A. Methods Used to Model Allocation
So how do simulations model partitioning? We can first consider some methods used to model partition-
ing in an optimum situation. Scientists have been able to make generalized observations of plant growth
910 DUBAY AND MADORE