Integration of a System Dynamics Model and 3D Tree Rendering—VISmaF Part II: Model Development, Results and Potential Agronomic Applications

Abstract

Biological–mathematical models of trees can be exploited for a wide range of agronomic applications including crop management, visualization of ecosystem changes over time, in-field phenotyping, crop load effects, testing of plant functions, biomechanics, and many others. Some models propose a 3D output of tree that, in addition to having functionality to visualize the result, offers an additional tool for the evaluation of some parameters of the model itself (interception and amount of light, temperature, obstacles, physical competition between multiple trees). The present study introduces a biological–mathematical model of tree growth with a 3D output of its structure in a realtime 3D rendering environment (Unity©). Thanks to the virtual environment created in Unity©, it was possible to obtain variable environmental parameters (amount of light, temperature) used as inputs to the mathematical simulation of growth. The model is based on ordinary differential equations (ODEs) that compute the growth of each single internode in length (primary growth) and width (secondary growth) and the accumulation of growth inhibitors regulating the seasonal cyclicity of the tree. Virtual experiments were conducted varying environmental conditions (amount of light and temperature), and the species-specific characteristics of the simulated tree (number of buds, branching angle). The results have been analyzed showing also how the model can be adapted for the creation of different tree species and discussing the potential agronomic applications of model.

1. Introduction

Biological–mathematical tree models can be used for a wide range of agronomic applications including crop management, visualization of ecosystem changes over time, field phenotyping, crop loading effects, plant function testing, biomechanics, and many others [1,2,3]. Some models propose a 3D tree output that, in addition to result visualization, offers an additional tool for the evaluation of some parameters of the model itself (light interception and quantity, temperature, obstacles, and physical competition between multiple trees) [4]. Among the various biological–mathematical models, those called Functional–Structural Plant Models (FSPMs) in the last two decades have been developed by scientists to explore and integrate plant structure and its biological/functional processes [5]. In conjunction with this, the development of 3D rendering techniques has achieved a level of complexity and realism [6] that allows the 3D structure to be used not only as a simple visual output of FSPMs but also to characterize plant phenotypes [7] or as feedback to evaluate and calculate light partitioning [8]. The 3D plants can be part of several modeling systems, being major components of many digital representations of landscapes or natural sceneries [9]. A 3D tree can be used in several scientific applications, such as synthetic forestry [10], generic digital representation of the real world [11], flow dynamics [12,13,14,15,16] and, as mentioned, in botany to determine light distribution and physiological parameters [17,18]. In addition, geometric modeling of plants allows researchers to visually validate biological processes, such as how plants interact with light and the environment. An extensive review of different modeling approaches of tree generation using FSPM that have 3D output and the state-of-the-art of current modeling approaches is provided in the companion paper, part I of this research, by the authors [19]. Among the biological–mathematical models that provide 3D output, a biological–mathematical model of a tree with a 3D output of its structure in a real-time 3D rendering (Unity©) environment is presented in this paper as expansion and further development of a proof of concept presented by authors in Part I [19]. The goal of this work is to create a bridge between biological physiological process and mathematical system dynamics model to a real-time 3D rendering engine (Unity©). To create this connection, the authors used the object-oriented programming paradigm to make the various modules of the model independent from the 3D rendering while being functionally connected to each other. The biological basis of the model is based on a system of growth units (defined as a set of nodes and internodes), photosynthetic potential, growth rate of growth units, xylematic water transport efficiency and internal accumulation of growth inhibiting metabolites. The biological model is explained in more detail in Section 2.1 and Section 2.2. The model proposed is also able to generate different architectural types of plants (globe shaped trees, columnar trees, etc.) depending on environmental variables (temperature, amount of light), species-specific genetic characteristics (number of buds, branching angle, maximum tree height, etc.) as well as biotic factors such as competition with other plants. How environmental inputs were modeled and how they affect the growth model are explained in Section 2.3. How the biological/mathematical model was linked to the 3D rendering environment is explained in Section 2.4 with references to the dumb code shown in the Appendix A. Finally, the various setups of different simulations are presented in Section 2.5, with the respective results shown in Section 3. In conclusion, in Section 4, the authors discuss the results, the planned future updates and enhancements to the model, the possible practical applications in agronomy and highlight the limitations of the study.

2. Materials and Methods

2.1. Model Overview

The proposed model is based on a hybrid approach with IBM (Individual-Based Model) elements and ODE elements interacting with each other [20] of the epigeal part of a tree species. The plant is simulated as a set of nodes and internodes whose internal dynamics of each internode are described by ODE, while the production of new internodes and interactions with the external environment are managed by the IBM part. This model (Section 2.2) is linked to Unity© [21] rendering engine in order to provide a realtime, procedural 3D output of model calculations and interactions with the environment (specifically, we only considered light and temperature as the main growth driving factors). The link can operate in both ways, i.e., by rendering a tree using parameters from ODE solutions, and/or by considering geometrical 3D data (e.g., the amount of light from the virtual scene) as input to solve the ODEs. Unity© has been chosen as 3D rendering engine because it offers a programming flexibility given by the use of scripts in C# language and a series of tools for the management of geometries and their interactions (collisions, physics, hierarchy, etc.). Unity© is also a widely used environment for many applications, mainly for the production and development of videogames. Since it was designed as a videogame development environment, it has been necessary to program from scratch some features necessary for the correct functionality of the model, such as a custom shader for the calculation of the virtual environmental light amount, and more importantly, an algorithm for the numerical resolution of the ODEs involved. In the following subsections is shown how the model has been developed from a system of ODE equations simulating growth units (Figure 1), then all transferred in Unity©, developing the modules in C# according to the Object-Oriented Programming (OOP) paradigm (Section 2.4). Being a real-time 3D rendering engine, through the development in Unity© it has been possible to implement the variation of the virtual environmental parameters during the run-time: it is not necessary to stop the simulation in order to change these parameters and then to start over but it is possible to modify them in real-time observing instantly how the model adapts to new input parameters and computing the modified tree architecture.

2.2. Branch Dynamics

The mathematical model representing the growth dynamics of the virtual plant’s branches has been developed as a set of ordinary differential equations (ODEs). Each branch is divided into several growth units representing a single node/internode couple (Figure 1). The first two equations represent primary and secondary growth, i.e., the dynamics in time of length L and width D of each growth unit (Figure 1).

In general terms, we assume that both processes follow a logistic dynamic whose growth rate is proportional to the photosynthetic potential of the plant which, for simplicity, is considered as a simple function of the two main environmental parameters temperature and light. Primary growth of each internode is limited by two factors, its position on the tree and internal metabolic factors. The first limiting factor is assumed to represent the water transport limitation as the height increases, in other terms internodes which are higher above the ground grow shorter than internodes closer to the ground. Each modelled tree is assumed to have a fixed species-specific maximum height representing the potential xylematic water transport efficiency (e.g., [22]). On the other hand, seasonal cycles of growth arrest and resumption are assumed to be regulated by the internal accumulation of growth inhibiting metabolites within the meristematic tissues [23,24]. Specifically, the primary growth arrest is caused by the internal accumulation of the inhibitors which leads to buds’ formation. We then assume a gradual decay of such molecules leading to vegetative buds endodormancy break and the possibility of growth resumption if the environmental conditions are favourable. Restarting the growth, the system will generate one or more branches depending on a species-specific parameter (number of children). Similarly, secondary growth is assumed to start after the cessation of primary growth with a rate proportional to the environmental conditions. The following set of equations represents the biological processes described above.

Specifically, the differential equation modeling the i-th internode length over time $L_i$ is defined as follows:

$${\displaystyle \frac{d{L}_i}{dt}=k_i\ast \alpha \ast L_i\ast \left(1-\frac{L_i}{L_{max}}\right)}$$

(1)

where $k_i$ is a given parameter as in Equation (5) and:

$$\alpha =c_L\ast envTemp\ast inputLight$$

(2)

with ${\displaystyle c_L}$ representing the optimal growth coefficient; ${\displaystyle envTemp}$ as temperature given by virtual environment and ${\displaystyle inputLight}$ as the amount of light calculated into virtual environment. ${\displaystyle L_{max}}$ represents the maximum length of the internode, calculated as:

$${\displaystyle L_{max}=1-\frac{\sum L_i}{H_{max}}-0.2\ast \frac{I}{I_{max}}}$$

(3)

where ${\displaystyle \sum L_i}$ is the actual tree height, ${\displaystyle H_{max}}$ is the tree potential max height as species-specific parameter, ${\displaystyle I}$ is the inhibitor concentration as calculated in Equation (4), ${\displaystyle I_{max}}$ is the set maximum value of inhibitor concentration. The differential equation modeling inhibitor concentration is defined as follows:

$${\displaystyle \frac{dI}{dt}=\sum _i^{}{c}_I\ast \alpha \ast L_i\ast \left(1-\frac{L_i}{L_{max}}\right)-k_i\ast d_I\ast I}$$

(4)

where ${\displaystyle c_I}$ is the optimal inhibitor coefficient, ${\displaystyle k_i}$ is a given parameter that changes if the tree is in seasonal stop or not. The parameter ${\displaystyle k_i}$ changes its value if the inhibitor concentration reaches the ${\displaystyle I_{max}}$ value or the ${\displaystyle I_{min}}$ value, making the tree enter and exit season stop, as follows:

$$\left\{\begin{array}{ccc}I\ge 0.95\ast I_{max}\hfill & \to k_i=1\hfill & \to seasonstop\hfill \\ I\le I_{min}\hfill & \to k_i=0\hfill & \to restartgrowth\hfill \end{array}\right.$$

(5)

The differential equation modeling secondary growth is defined as follows:

$${\displaystyle \frac{d{D}_i}{dt}=d_{actual}\ast D_i\ast k_i}$$

(6)

where ${\displaystyle d_{actual}}$ is a parameter function of virtual environment temperature calculated as:

$$d_{actual}=c_D\ast envTemp\ast inputLight$$

(7)

with ${\displaystyle c_D}$ as optimal secondary growth coefficient and ${\displaystyle k_i}$ is a given parameter as in Equation (5). An overview of all variables involved, their definitions and assigned values are shown in

Table 1. Variables involved in tree growth ODEs system.

Variable	Definition	Assigned Value
${\displaystyle L}$	Length of internode	${\displaystyle 0.1(t=0)}$
${\displaystyle I}$	Concentration of inhibitor	${\displaystyle 0.1(t=0)}$
${\displaystyle D}$	Width of internode (secondary growth)	${\displaystyle 0.1(t=0)}$
${\displaystyle t}$	Simulation time	${\displaystyle 0}$
${\displaystyle c_L}$	Optimal growth coefficient	0.5
${\displaystyle c_I}$	Optimal inhibitor coefficient	0.2
${\displaystyle c_D}$	Optimal secondary growth coefficient	0.01
${\displaystyle d_I}$	Inhibitor degradation parameter	0.1
${\displaystyle envTemp}$	Virtual environmental temperature	as set by user
${\displaystyle inputLight}$	Light amount	-
${\displaystyle H_{max}}$	Max tree height	species-specific
${\displaystyle I_{max}}$	Max inhibitor concentration	${\displaystyle 1}$
${\displaystyle I_{min}}$	Minimum inhibitor concentration	${\displaystyle 0.1}$
${\displaystyle k_i}$	Season stop parameter	${\displaystyle 0}$ or ${\displaystyle 1}$

. This set of differential equations cannot be solved directly in Unity© because there is no built-in differential equation solver. A C# script as abstract class has been written, allowing the above differential equations to be solved via the fourth-order Runge–Kutta numerical method. This script has been associated with the internode element for integration of its length and secondary growth, and with the branch element for calculation of the inhibitor concentration. For example, the internode element processes its length at each time step by numerically solving the associated differential equation (Equation (1)) at each time step thanks to the integration script created.

2.3. Environmental Inputs

2.3.1. Light Amount Calculation

A key part of the model development is the calculation the amount of light coming from the virtual environment. Several shader solutions have been tested to find the optimal one in terms of computational performance and flexibility. A shader is a program that runs in the graphics pipeline and tells the computer how to render each pixel. These programs are called shaders because they are often used to control lighting and shading effects [25]. Standard lighting shaders are unable to compute indirect light (such as diffusive light). The shaders tested have been Global Illumination (GI), Radiosity and Ray-tracing, each one showing pros and cons. The Ray-tracing is the most accurate but more computationally heavy, Radiosity and GI are computationally lighter but less accurate and not suitable for the purpose of this work because they failed to calculate diffusive and indirect light in the quality and precision desired for this particular type of application. The alternative solution, evaluating all the solutions previously shown, has been to write a custom shader specialized in calculating the amount of light. A “light meter” sphere GameObject (a fundamental object in Unity©) is placed in the scene with a camera pointed on the object (Figure 2). The position of the camera changes each frame to cover the entire surface. The camera renders a very narrow texture and records the RGB values into an array. Then, a conversion from RGB values to brightness is made thanks to Equation (8) from ITU-R BT.709 standard [26] as shown in Listing A5 as dumb code in Appendix A. The final value of brightness will be the sum of all arrays calculated at each camera positions.

$${\displaystyle Brightness=0.2126\ast R+0.7152\ast G+0.0722\ast B}$$

(8)

This way it is possible to calculate direct and indirect light, even if the object is partially in shadow, in real-time and with a little weight on performance (if the number of cameras is kept low).

As shown in Equations (1) and (2) as well as in the growing scheme in Figure 3, the light input affects internode growth with a standardized calculated value between 0 and 1. When the internode is growing in full light, with a value close to 1, its growth will be the maximum or very close to the maximum. The light meter has been placed at the top of the growing internode, and then removed whenever the internode stops growing, as it is no longer necessary to calculate the amount of light. By doing so, it has been possible to keep a relatively low number of light meters simultaneously active during growth, managing to have a certain balance between computational performance and precision. In addition, by placing the light meter at the apex of the growing internode, it has been possible to vary the growth angle ${\displaystyle {\phi }_{AAV}}$ shown in the Section 2.4, which will deviate from the ideal one as a function of the amount of light calculated.

2.3.2. Light and Temperature Implementation

The Unity© environment allows the use of virtual environmental parameters as model inputs. In the previous paragraph it has been shown how it is possible to calculate the amount of light that is a part of the environmental parameters. Unity© allows the programming and modeling of other parameters that can be classified as environmental. As shown in the model diagram in Figure 3, another environmental parameter that has been chosen as an input to the model, in addition to the amount of light, is temperature. Obviously, in a virtual environment it is not possible to record a real temperature but it is possible to model its trend and use the simulated numerical data as model input. In this work, the temperature data is an input that influences the coefficient ${\displaystyle \alpha }$ in the growth of the internode (Equations (1) and (2)). In combination with the amount of light, the temperature will affect the amount of internode growth. In Unity© is possible to set a function to simulate the non-constant temperature trend (temperature changes during seasonal and daily cycles) as well as for any parameter, value, variable in the model. In this study, the temperature has been managed by a curve that represent the variation of temperature value along a time range: the temperature changes over a period of time from 1 to 365 (as days, simulating a year) starting from the minimum value of 15 (assumed as °C) reaching the peak at 25 (assumed as °C) and then decreasing again. Once the 365th step has been passed, the cycle starts again from 1 simulating another year. The temperature value will be different at each time-step which, in the model proposed, corresponds to a day for each second of simulation (1 s of simulation = 1 day of tree growth). It is possible to create more complex curves to better simulate the real variability of temperature over time. For example, to simulate a real temperature trend over a year, it is possible to set a variation curve given by real yearly data record. In this work, the simple temperature variation curve explained above is used as generic example of temperature variation application. Moreover, this curve function can also be used to manage the variability of other environmental parameters such as the already mentioned ambient light in order to simulate the day/night cycle.

2.4. Model Implementation

The model shown in Figure 3, a concise explanation of the model developed in Simile [27] that translates in stock and flow representation a system of ordinary differential equations that resolve numerically time variable dimensions, describes as flow-chart the separation of working modules and their mutual interactions. As discussed in the work by Crimaldi et al. [19], Unity© has been the choice for a real-time 3D engine having useful features for the purpose. Unity© uses the C# language to code scripts associated with geometric elements (prefabs) in the working environment. Each element (or module) has a linked C# script that defines its behavior. The C# language complies with the objectoriented programming (OOP) paradigm: each element can inherit all common features, methods, functions and parameters from another. Unity© also allows the management of the hierarchy between elements enabling to have elements as father generating children: a situation that is well suited to model a tree where there will be child internodes descending from common father (primary branches from a common main trunk, secondary branches from primary branches, and so on). Moreover, thanks to the use of Unity© as a real-time rendering engine, it is possible to change some parameters during run-time: parameters such as the virtual environment temperature, light quantity, direction and characteristics, allowing the modeling of some typical situations of a tree growth like the day/night cycle or the seasonal cycle.

As shown in Figure 3 the tree has a number of parameters that allow the model to run. Some are species-specific, others are process parameters that change step by step in real time as the simulation runs. The species-specific parameters are set at the beginning of the simulation and do not change during the simulation. They are parameters proper of the tree species that is intended to simulate. In the model proposed in this work it has been chosen to simulate as species-specific characteristics the branching angle (defined as the angle of the new branch created in relation to the parent), the number of children (defined as the number of new branches created after the end of the seasonal stop of the tree, Figure 3), the maximum height of the tree (defined as the maximum height to which the tree will tend asymptotically, Figure 3) and the simulation time (defined as how long the simulation will run, 1 s in simulation equals to 1 day of tree life). These parameters have been coded in the C# script shown in Listing A1 in Appendix A as dumb code of Tree class.

These settings are possible in Unity© modifying the variables of the script via the inspector (a built-in Unity© editor), Figure 4, or via scripting. According to the objectoriented programming paradigm, the other modules and sub-modules will descend from the tree script, inheriting the common characteristics. Each module that inherits from tree will have the same characteristics. For example, it will inherit the maximum height of the tree as a parameter. The tree module holds also lists of all branches, internodes and buds (nodes) of tree to process the tree structure hierarchy.

In Unity© engine, the whole structure of the tree is based on a hierarchy between the elements described above. Thanks to C# scripts and the object-oriented programming paradigm, all elements are interconnected structurally but also behaviorally. Any single element can affect all other elements and vice-versa. The single elements in Unity© that have a shape and a behavior are called GameObject. The hierarchy of a tree in the very first phase of growth is simple and evolves in a much more complex form during growth. From the GameObject Tree all the other GameObject(s) descend hierarchically, starting from the first bud called seed and then moving on to subsequent branches, internodes and buds. As explained above, the GameObject “end” is the one positioned in the space where it should be the end of the length of the growing internode calculated step by step by solving the ODE (Equation (1)). By doing so, although it seems an overcomplicated procedure, it is possible to separate and make independent the calculation of the length of the internode (e.g., the position of the GameObject “end”) and the generation of the mesh or geometry of the same internode. The GameObject in charge of calculating the amount of light from the external virtual environment is the LightMeter described in detail in Section 2.3. An example of hierarchy is provided in the Appendix A, Figure A1. The tree has a final structure formed by a series of nodes (lateral and apical buds) and consequential internodes according to a hierarchy (Figure 5). The main trunk is the collection of main internodes father of all subsequent child internodes that have a lower hierarchical scale, up to the apical internodes that are the internodes having lower hierarchical scale. Branches are the entities that contain the entire sequence of nodes and internodes and that manage, thanks to their code scripts, the functions of generating new branches at certain branching angles and the concentration of inhibitor that will determine the seasonal stop of the tree. At the end of the seasonal stop, as shown in Figure 3, new branches are generated according to the number of children set in the species-specific characteristics of the tree and according to the branching angle. The actual branching angle deviates from the ideal branching angle set as a species-specific parameter based on the amount of light calculated from the virtual environment as explained in Section 2.3.1.

The Internode C# class manages the behavior of internode(s) as explained above. As shown in the dumb code in Listing A2 in Appendix A, the internode can have 3 states: awake, asleep or dead. The three states can be changed thanks to external events or by other modules of the tree (i.e., by branch module at the end of the seasonal stop). The internode only grows when it is in the awake state, is paused when it is in sleep state (it can resume growth), and can no longer grow if it is in the dead state. At tree initiation, the initial numerical conditions of the differential equations for calculating length and width (secondary growth) are set as well as the creation of a GameObject (a fundamental object in Unity©) “end” representing the end point of the internode whose position is the numerical solution of the ODE that calculates the length (see method public override voidInit(Tree t) in Listing A2). This approach has been chosen because there is no need to create a 3D shape each time and scale it according to the calculated length, but only the position of the end point is calculated, reducing the computational load according to the principle of procedural rendering as proposed by Weber and Penn [28]. More specific code explanation is provided in Appendix A.

The Branch C# class manages the behavior of branch(es) as explained before (Equation (4)). As shown in the dumb code in Listing A3 in Appendix A, the branch can have three states: awake, asleep or dead like the internode. The branch calculates the inhibitor concentration numerically resolving the ODE (Equation (4)) setting the numeric initial conditions, then calculates the numeric solution of the ODE at each time step. When the inhibitor concentration reaches its maximum, the tree (and the branches) enters in seasonal stop. When the inhibitor concentration reaches its minimum, the seasonal stop ends and the tree restarts the growth, as well as the calculation of inhibitor concentration. At the end of seasonal stop, new branches are also created as the number of children set. More specific code explanation is provided in Appendix A.

The Bud C# class manages the behavior of buds, both apical and lateral ones (Figure 5). As for internodes and branches, buds can have three states: awake, asleep and dead. In this case, as shown in Listing A4 in Appendix A, the script will check at each update if an external event affects the buds, changing their state from asleep to awake or dead and vice versa. If the bud is awakened by an external event, it will produce new branches that will be rotated at certain angle function of light amount. At the end of internode growth, when the internode reaches its maximum length, a method is called to generate a new internode child of the previous one with a rotation angle function of the amount of light (${\displaystyle {\phi }_{AAV}}$ in Figure 5). Another important function handled by the Bud class is leaf placement. In this work we decided not to model the leaves as an organ of the plant with its own ODEs but only as an object in the 3D environment with its own mesh, orientation and rotation. This is because, during preliminary tests, we found an exponential drop of computational performance as the number of leaves increases. The leaves, therefore, having their own shape and position, will contribute to the calculation of light, influencing tree own shadow and consequently its growth. The parameters of leaves angles and positions are either fixed as hard-coded value or depend on the position of the bud. A particular bud is the one named seed in the hierarchy: it is considered as the first bud from which the main trunk grows and generates the first branch as the main trunk. More specific code explanation is provided in Appendix A.

2.5. Simulations Setup

In order to test the potential of the model and its flexibility and adaptability to different conditions and parameter settings, experiments have been conducted generating growing trees in the virtual environment created in Unity© with different combinations of both species-specific and environmental parameters. Specifically, simulations have been first carried out with fixed environmental parameters: with a constant temperature (set to 25 °C) and amount of light (set to 1, corresponding to a tree growing in full light) and only changing the species-specific parameters such as number of buds and branching angle. Specifically, tests were carried out with:

• number of buds: 2–3–4;
• branching angle: 30°–40°–55°–60°.

Tests with a number of buds equal to 1 have already been done in the preliminary tests showed in the previous companion paper published by authors [19]. Tests made by changing environmental parameters have been done by keeping fixed the number of buds and varying the temperature, the quantity and the direction/origin of the light source. The temperature regimes adopted in these simulations have been as follows:

• fixed temperature at three different values: 15 °C–25 °C–35 °C;
• variable temperature over time as explained in the previous paragraph (Section 2.3.2).

Competition tests were carried out by growing 2 and 4 trees with the same speciesspecific parameters and temperature at a variable distance (1 m, 2 m). Tree competition is limited to light: no module has been modeled that simulates resources and thus competition for resources (e.g., nutrients, water). This module will be developed in forthcoming work. Finally, tests have been performed by varying species-specific parameters to simulate the growth of different species of trees; in this work, columnar-shaped trees (such as a Populus nigra) and conical-shaped trees (such as a Picea abies) have been modeled. All tests were carried out for a growth duration of 5 years (${\displaystyle 1826.25}$ s of simulation). In the following section, the results with the different combinations of parameters discussed in the previous paragraph are shown. The results of the simulations in Unity© are shown without the presence of the leaves even though the position of the leaves have been computed and they have been considered in the calculation of tree self and mutual shadow, to analyze and evaluate the architecture of the resulting trees with their species-specific characteristics.

3. Results

3.1. Species-Specific Parameter Changes—Fixed Environmental Parameters

The expected behavior of the branch dynamics, with fixed environmental parameters, is the one shown by the simulations in Figure 6. The tree branch grows by increasing internode length, until it reaches the maximum internode length calculated as in Equations (1) and (3). At the end of growth, the internode stops growing, the apical bud generates a new child internode that will grow to its maximum length. Simultaneously, the inhibitor concentration increases until it reaches its maximum. When the maximum concentration value is reached, according to Equation (4), the tree enters the seasonal stop and secondary growth starts. Inhibitor concentration begins to decrease until it reaches its minimum, the tree starts growing again, secondary growth stops, inhibitor concentration begins to increase again, as does internode growth. At the end of the seasonal stop, new branches of the same number set in the tree species-specific parameters will also be generated. The expected behavior of the simulation is, therefore, to have internodes that decrease their maximum length as they approach the maximum tree height. The result will be that the first internodes (those in the main trunk) will be longer than the apical internodes.

The result obtained in Unity© with fixed environmental parameters, 55° branching angle and two children (branches) is shown in Figure 7. The tree reached after 5 years (${\displaystyle 1826.25}$ s) of simulation the height of 2.6 m. It can be noticed that the number of internodes in the main trunk is the same as in Figure 6. It is possible to see the set branching angle and how the apical internodes are shorter than those in the main trunk.

By increasing the number of children, while leaving the environmental parameters constant, the tree appears more dense due to higher new branches generated. The apical internodes remain shorter than those in the main trunk, as can be seen in Figure 8 where the simulation generated a tree 2.55 m in height.

It is possible to notice the same effect generating a tree with 4 children, as shown in Figure 9, where it was generated a tree 2.4 m high.

By changing the branching angle instead, the results obtained are shown in Figure 10. It can be seen that by changing the branching angle from 30° (Figure 10a) to 40° (Figure 10b) and 60° (Figure 10c), the tree widens the canopy and takes up more space. The heights are also different, respectively, 2.4 m for the trees with 30° and 60° branching angles, and 2.6 m for the tree with 40° branching angle.

3.2. Virtual Environment Parameter Changes

The expected behavior, varying the virtual environmental temperature, is that the behaviors of the internode length and inhibitor concentration curves change. By lowering the temperature, internode growth is slower. The variation in inhibitor concentration also changes, slowing it down as well. At lower temperature, therefore, there will be the effect of seeing a tree grow, for the same simulation time, lower in height and less branched because it takes longer to reach the value of maximum internode length and maximum inhibitor concentration, which causes the tree to enter seasonal stop late, generating new branches at lower heights. The result of growing a tree at a low temperature is shown in Figure 11b where it also possible to see that the first branching is lower in height than the tree growing at higher temperature in Figure 11a. By contrast, increasing temperature will cause internode length and inhibitor concentration to increase faster, reaching their maximum values sooner. There will be the effect of seeing a tree, for the same simulation time, higher and more branched because it reaches the value of maximum internode length and maximum inhibitor concentration quickly, which causes the tree to enter seasonal stop soon, generating at higher heights the new branches, as shown in Figure 11a.

As described in Section 2.3.2, the temperature can also be changed dynamically over time, following a variation curve. In this study, the variation curve manages the temperature variation, starting at second 1 (day 1) from a value of 15 °C, reaching a maximum at second 182 (day 182) of 25 °C and then dropping back down to a value of 15 °C at second 365 (day 365) and then start again from day 1. The variation curve can have values and curvature at will, in this work it was set up this way to demonstrate how the growth function, and thus the resulting tree, adapts to changes in virtual environmental parameters in real time. Due to low starting temperature in the early days (seconds of simulations), the internode length growth is slower. As the temperature dynamically returns to the “standard” value of 25 °C, the growing rate approaches the ideal conditions. After reaching the changing temperature curve apex, the temperature itself starts to decrease again slowing down the growth of the internode. Inhibitor concentration follows the same trend: slow increase in the first part, ideal situation in the middle part, slowed down in the second part where the temperature has a descending trend. The resulting tree with dynamically time changing temperature is shown in Figure 12.

It is possible to notice in Figure 12 how the dynamic variation of the temperature makes the internodes grow in different lengths depending on the period in which the simulation is along the curve of variation of the temperature. In addition, the tree is shorter in height (2.0 m) than those grown under ideal conditions as growth is slowed at the periods when the temperature deviates from the ideal, increasing (first phase of the temperature change curve) and decreasing (second phase of the temperature change curve). As shown in the model diagram in Figure 3 and elaborated in the internode growth Equations (1) and (2), the amount of virtual ambient light affects the rate of internode growth. The effect, as the amount of light changes, is quite similar to that showed with changing temperature. In addition, in this case, if the amount of light is reduced, the growth rate of the internode will also be reduced, i.e., for the same simulation time the internode will grow more slowly and less. In addition to the amount of light, as explained in Section 2.4, the direction of the light source will also affect growth. Specifically, internodes will grow by changing the angle ${\displaystyle {\phi }_{AAV}}$ (apical bud angle to internode axis, Figure 5) to turn towards the light source. This effect can be clearly seen by placing a light source to the side, creating parts of the tree in full light and others in shadow, as shown in Figure 13. The shadowed parts of the tree have internodes with a low growth rate due to low light, no growth at all if the light amount is equal to 0. The brighter parts of the tree have higher internode growth rates and they are directed towards the light source. The not equally distributed light over the entire tree as in the previous results will also lead to a lower overall tree growth rate, producing a smaller tree than ideal conditions.

3.3. Competition between Neighboring Trees

The model’s ability to adapt to the amount and location of the light source allows it to simulate competing groups of nearby trees. Figure 14 shows the growth of two trees 1 m away from each other while Figure 15 shows two trees 2 m away from each other. Figure 16 shows four trees grown in two parallel rows at 1 m between trees and between rows. Lastly, Figure 17 shows the growth of four trees arranged in 2 rows at a distance of 2 m between trees and 2 m between rows.

The greatest effects of competition for light can be seen in simulations with groups of trees at shorter distances from each other (1 m) where the proximity of trees increases mutual shadow, decreasing the rate of growth of internodes closer to the nearby tree and therefore more in shade. At higher distances the amount of mutual shadow between trees is much less, causing neighboring trees to grow in near ideal conditions.

3.4. Model Adaptation to Different Tree Shapes

The tree models shown in the previous subsections all represent cluttered globeshaped trees. By changing the model parameters as described in the previous sections, it is possible to model different types of tree shapes. In the examples proposed in Figure 18, it is possible to see how by setting a branching angle of 45° and a number of children equal to 2, it has been possible to recreate a tree with a columnar shape, like a tree of the species Populus nigra (Figure 18a, with a comparison with a real photo of Populus nigra in Figure 18b). Figure 18c shows a tree with a conical shape, with a number of children equal to 5 and a branching angle of 90° for the primary branches (those directly generated by the main trunk) and 55° for the secondary and tertiary branches (those generated from the primary or subsequent branches). In the case shown in Figure 18c, it is possible to compare this type of tree with a real photo of a conifer with the same characteristics, such as a Picea abies in Figure 18d.

4. Discussion

The results presented in Section 3 show how the model is adaptable to different specifications, species-specific parameters, and environmental conditions. The flexibility given by the OOP paradigm allows a separation of modules (and their code) while remaining interconnected at the functional level. It is possible to change at any time any ODE that rules a specific module without having to modify the 3D rendering part nor affecting other parts of the code. Any change made can be immediately displayed in the generated output structure. In addition to ODEs, the results show that changes are also possible to the parameters that regulate growth, showing different outputs depending on the modifications. Modifying species-specific parameters, it is possible to model potentially any tree species with different numbers of children, different maximum heights, and different branching angles. Being a real-time rendering environment and procedurally computing the solutions of the ODEs at each frame and rendering the results, it is also possible to modify the inputs of virtual environment parameters such as temperature and light. For the temperature, the parameter is only numerical not being related to a physical component of the virtual environment and its variation modifies the amount of growth of the internode and therefore of the tree, as well as the concentration of inhibitor. As for light, in the virtual environment its value and position are linked to a physical component (light source). Changing the amount, position, and direction of the light source, the internode growth rate changes, modifying the direction of growth by simulating phototropism. Despite being a compromise between the precision of ray-tracing (but too heavy computationally) and the computational lightness of the GI (but not very accurate), the system adopted for the calculation of the amount of light turns out to be satisfactory for the aim of the work proving to be adaptable to the desired growth conditions. Although not shown in the results, the presence of the leaves contributes to the calculation of the self and mutual shadow and influences the amount of light measured. The leaves themselves were not modeled as an organ and do not have a specific module, the authors will implement this module in future work. Thanks to the calculation of the amount of light in real time, it has been also possible to model small groups of trees competing with each other for light, showing how the proximity and therefore the shadow generated by another tree, the growth of the internodes (and of the tree) is affected by modified light amount detected. Competition for resources and their implementation is another module, as the one for leaves as organs, that the authors plan to incorporate in future work. Despite some limitations given by design and modeling choices, the authors believe that the present work can be extended and refined in future research. However, at this stage, the proposed model can already be used in some practical and scientific agronomic applications as suggested in the following section.

Potential Agronomic Applications of the Model

The possible agronomic applications of the model presented in this work may be various. As shown in previous companion publication [19], the potential of FSPMs with 3D output can be numerous. In the case of the model presented in this work, the authors count on being able to use it in applications to study the effects on the architecture of planting density, to calibrate allometric models, or even for quantitative analysis of branching patterns [29,30,31]. The model proposed in this paper can be used to estimate the Tree Row Volume (TRV) index to optimize fungicide and insecticide delivery by automated and precision spray systems [32,33,34,35,36,37]. Another potential use of the presented model, being in real-time, is to implement it in an immersive visualization system to train agricultural operators in the practice of pruning or more simply to evaluate the effects of pruning on tree structure. The authors are already working on a system that allows the “slicing” in real time of the meshes that are part of the tree structure to visualize the pruning operation and its effects. In addition, the virtual pruned tree model can be used as a tool to optimize breeding form or in a scenario where automatic pruning systems are instructed to implement automatic operations [38,39,40,41,42,43]. The modularity of the proposed system can also be used to implement effects of plant pathologies and visualize their consequences on tree structure and its development in time. In the less agronomic context, the virtual trees generated by the proposed model can be used in the creation of synthetic datasets for the training of artificial intelligence in automatic species recognition [44,45,46]. More generally, the model presented can be applied in all applications that require a digital-twin [47], in this case of a tree, to perform destructive testing or operations without materially compromising a real tree.

5. Conclusions

The use of FSPM models has been well established in the modeling community for two decades [5,18]. Many models now have 3D output that allows rapid visualization of results as well as being an integral part of the model itself as a provider of input parameters, as described in the review by Crimaldi et al. [19]. The use of a real-time 3D rendering engine allowed the authors to create a link between a mathematical biological model of tree growth and the procedural creation of its architecture. This solution differs from other proposals of integration between models and 3D rendering in the capability of having as input the parameters of the virtual environment in real time (amount of light and temperature), calculating at each variation (at each simulation time step) the solutions to the ODEs that regulate the processes showing immediately the result as a modification of the structure. While showing a flexibility that is well suited to the purpose, thanks to the implementation of interconnected modules adhering to the OOP paradigm specialized in individual components of the tree, the model has some margin for improvement such as the implementation of a module for the modeling of leaves as organs, or the integration of competition for resources in neighboring trees. Further studies are underway by the authors to implement the above features, as well as direct applications of physical changes to the generated structure in the 3D environment such as a pruning operation.