The water demand for each individual leaf is a function of light exposure, and all the individual leaf demands are summed to determine the whole-canopy water demand. The ability of the root system to provide water is determined by root system structural biomass, the soil volume available to the tree, a user-defined soil moisture release curve, and the user-defined irrigation schedule. The ratio of canopy water demand and root water supply capability provides an index of the water stress in the tree at any given time – as the value of this ratio goes down, the impact of water stress on tree growth and photosynthesis increases. The L-PEACH model is developmental, with the buds producing new stem segments, leaves, fruit, etc.The growth of organs initiated during the previous season as well as the subsequent initiation of new organs is controlled by the amount of available carbon. If the carbon supply is insufficient for growth and/or maintenance, organs are shed by the tree. Thus the development and growth of the branching plant structure are closely coupled with the production and partitioning of carbohydrates.The formalism of L-systems automatically couples the tree structure with the topology and parameters of the carbohydrate supply network that represents the sources, sinks and conductive elements. At the heart of this coupling lies the notion of context sensitive L-systems ,drainage for plants in pots which provides a convenient means of capturing connections between elements of a growing structure at each stage of its development.

Given this information, Lsystems are used to compute the distribution of carbohydrate, its concentrations, and fluxes at each step of the simulation. Efficient implementation of this computation is the main methodological innovation of the L-PEACH model. Within L-PEACH, the plant is modeled as a growing network comprised of elements that represent individual organs such as leaves, stem segments, fruit, buds and roots. The behavior of each type of organ is given by a set of user-defined functions. For example, a mature leaf is characterized by its source strength which, in turn, depends on the amount of mobilizable carbohydrates that have been accumulated in the leaf as a result of photosynthesis. During each time-step, these accumulated carbohydrates can flow into the various sinks within the tree . Stem segments, in addition to being potential sources or sinks, act as conduits for the fluxes throughout the tree. The magnitude of these fluxes depends on the differences in carbohydrate concentrations between sources and sinks, and the resistances of the intervening paths. All elements may exhibit nonlinear behavior, meaning that the resistances may depend on concentrations. In general, the network representing a growing plant has a dynamically changing structure ; is non-stationary ; and is nonlinear . L-systems are used to ‘develop’ the plant , and to solve the set of equations defined by the network at any given point in time. These equations are solved numerically, by taking advantage of the branching topology of the network. In order to calculate the accumulation, flow and partitioning of carbohydrates between the individual components of this network, we rely on an analogy to electric circuits and employ equations developed in linear circuit theory. The underlying correspondence between biological and electrical quantities is summarized in Table 1.

The fundamental concept is to identify the amount of mobilizable carbohydrates with an electric charge. Other correspondences are a straightforward consequence of this identification. The only non-intuitive notion is the source/sink strength, the analogue of electromotive force. It can be thought of as the concentration of carbohydrates inherent in an organ , as it would be measured in the absence of flow through resistive conductive elements associated with that organ. There are two types of connection between the elements of the modeled tree: a serial connection between two consecutive elements, or a parallel connection that occurs at a branch point. Using standard rules for combining components in an electric circuit, it is then possible to reduce any two connected elements into an equivalent single element. For serial connections, the combination is done as shown in Fig. 1a, where r and eare the resistance and the source/sink strength of the network that results from the combination of the two separate elements shown on the left side of the figure. Following the same labeling conventions, the combination of two parallel elements is shown in Fig. 1b. Given these two rules, context-sensitive L-system productions are used to calculate fluxes and concentrations at the nodes , given the source/sink strengths and resistances to carbohydrate flow. The calculation is performed in two phases, which we refer to as the folding and unfolding of the network. In the folding phase, information is passed from the tips of branches to the base of the tree. The branching network is then gradually simplified by combining the elements into a sequence of ever more inclusive equivalent networks. For example, let us consider the branching axis comprised of a series of elements, shown in Fig. 2a. Working from right to left in this figure, network elements are combined, or ‘folded’, into equivalent simpler networks. In the first step, the elements shown in white are combined into a single element. The resulting simplified topology is shown in Fig. 2c, where e′ is the source/ sink strength of the network equivalent to the combined white elements from Fig. 2b, and r′ is the combined resistance. Proceeding in this way, the entire axis can be sequentially simplified to the network shown in Fig. 2d.

As there is nothing to the left of the resistance rs1 , no carbohydrate will flow through it. Therefore the carbohydrate concentration at the point of the 90° bend will be equal to the sink/source strength . Now that this concentration is known, the network can be ‘unfolded’, and the fluxes can be determined for the entire branching structure. The first step of this process is shown in Fig. 3a. Proceeding to the right, we gradually unfold the entire network into its original form, and calculate values for the concentrations and fluxes in the network . While this example deals with a simple non-bifurcating ‘branch’, structures that include bifurcations can be handled in a similar way. If the relationship between the carbohydrate flux into a sink and the carbohydrate concentration at the point where that sink is attached were linear for each of the elements in the model, the computation of the fluxes flowing to each element of the network would be completed after the unfolding phase. However,plant pot with drainage as mentioned above, elements in the model mayexhibit a nonlinear behavior, which can be thought of as the dependence of resistances and source/sink strengths on concentrations at the attachment points. Because of this, the distribution of concentrations and fluxes in the network is calculated iteratively, using an L-system implementation of the Newton–Raphson method . first, the functional characteristics of sinks and sources are linearized for the values of concentrations obtained in the previous iteration. The resulting source/sink strengths and resistances are then used to compute new values of concentrations in the next iteration of folding and unfolding. Once this process converges on a solution, fluxes out of each source and into each sink are assigned and accounted for, and the simulation advances one time-step.During a simulation, generates a dynamic visualization of the modeled tree and simultaneously quantifies and displays the output data selected by the user. These data may include global statistics, such as the overall amount of carbon assimilated and allocated to different organ types, as well as local data, characteristic of specific organs chosen by the user. The user can thus evaluate, both qualitatively and quantitatively, how different parameters of the model influence the growth and carbon partitioning in the plant. The manner in which the model handles carbon partitioning is illustrated in Fig. 6 with several still frames from a simulation run utilizing a highly simplified static ‘tree’ structure. In this example the modeled plant consists of a small number of stem segments, a root system , two leaves, and two peaches. Color coding of the stem segments during the simulation run allows for visualization of carbohydrate fluxes within the plant. Colors ranging from light blue to purple indicate flow ‘down’ the plant towards the roots, while colors from yellow through red indicate flows ‘up’ the tree in the direction of the shoot tips .

At the beginning of the simulation there is a strong flow of carbohydrates from each leaf out to the rest of the plant . Each of the fruits is consuming a considerable amount of these carbohydrates, resulting in a noticeably smaller carbohydrate flux down the plant below the fruit . Fig. 6b shows a still frame taken after the right-hand fruit was ‘pruned’ before its maturity. Without this strong sink on the right-hand branch, a significant amount of the carbon flow is ‘pulled’ to the remaining fruit on the left branch . This is indicated by the red color of the stem segment below the left-hand fruit – carbohydrates are now flowing ‘up’ this segment towards the fruit. When the fruit is fully mature it no longer acts as a sink, and all carbon coming from the leaves goes downwards through the stem system to the roots . The power of L-PEACH becomes clear when simulating the effects of management, genetic and environmental factors that can influence the plant through complex interactions between plant organs. For example, these interactions may include the influence of crop load, rate of fruit maturity, carbohydrate storage capacity, and water stress on the growth and carbohydrate partitioning within a fruit tree. The manipulation of the model consists of simple adjustments of parameters, such as the number of fruit, behavior of fruit , and storage capacity of stems. To simulate responses to canopy management, such as pruning, the model contains a pruning function that allows the user to stop a simulation; prune leaves, fruit or branches at will; and resume the simulation with the adjusted tree structure. To model responses to water stress, the user specifies the soil volume available for root exploration, an irrigation interval for replenishing soil water, and the relative sensitivities of each organ type to water stress . During the simulation, water demand is calculated based on the cumulative leaf exposure to light, and the sink strength of each organ is modified in response to the developing water shortage within the plant. Thus the differential effects of a developing water stress on root, shoot and fruit growth, as well as on carbon assimilation and partitioning, can be simulated without any empirical rules governing allometry between plant parts. As examples, we have run two different pairs of simulations. The first pair involves 2-yr-old trees with indeterminate shoot growth under two different irrigation scenarios. One tree is irrigated at regular intervals such that it is never experiences any water stress. The other tree has a limited soil volume from which to extract water, and is irrigated at long intervals so that it experiences mild water stress. We assumed that shoot growth was more sensitive than photosynthesis to water stress, thus the primary visual effect of the water stress was a reduction in shoot elongation and girth growth .The model also predicted quantitative differences in carbon partitioning . In the second pair of simulations, fruit set is altered such that crop load in one tree is twice that of the other. In response to this decrease in initial fruit set, the model produced the following results: an increase in fifinal fruit size; a decrease in the total amount of carbon partitioned to fruit growth; lower variance in fruit size within a tree; and greater partitioning of carbon to vegetative growth .L-PEACH is an L-system-based functional–structural model for simulating complex interactions within trees, including growth, carbon partitioning among organs, and responses to environmental, management and genetic factors. The use of L-systems facilitates several aspects of model construction and operation: the integration of individual organ models into a growing branching structure; the dynamic updating of the system of equations that characterizes carbon partitioning in this structure; the solution of this growing system of equations; the communication between the plant model and the model of its light environment; and the visual presentation of the simulated trees.