Forest Stands as Dynamical Systems : An Introduction

Forest management planning relies heavily on mathematical models that involve time. Concerns about climate change and ecosystem services have highlighted the limitations of traditional growth and yield prediction tools. Modern dynamical system theory provides a framework for a flexible representation of varying environments, as well as of responses to intensive silviculture and natural disturbances. Emphasis changes from trying to directly model functions of time to modelling rates of change. The fundamental concepts are introduced here in a non-technical manner. The theory is illustrated with a recent whole-stand growth model for even-aged stands, but it is noted that it applies to any system that evolves over time. It is shown also how a modular approach can improve balance and efficiency in the development of such models.


Introduction
Due to the long planning horizons involved, experience and experimentation are not as useful in forestry as in other disciplines.Therefore, mathematical modelling plays an important role, and has been used in one form or another for centuries (Lowood, 1990).In particular, growth forecasting through yield tables and their modern successors is an important prerequisite for rational forest management.
Simple yield tables or traditional growth and yield models are often all that is needed.However, by essentially specifying fixed functions of time, they fail to address satisfactorily issues that are becoming increasingly important, including a changing environment, growth and carbon cycling in immature stands, and response to disturbances (García, 2011).Dynamical systems theory provides an alternative point of view that can fulfil those needs.The ideas are not new, and are routinely applied in other areas.But perhaps due to unfamiliarity and entrenched traditions, their adoption in forest modelling has been slow (Weiskittel et al., 2011, Burkhart & Tomé, 2012).A non-technical explanation of the basic concepts is presented here.Although illustrated through growth modelling examples, the ideas are general and apply to any system that evolves in time.

From Trajectories to Rates of Change, and Back
Yield tables and many growth and yield models define functions of time, typically including stand height, number of trees, basal area, and volumes, for various ages and site qualities.Consider the height and basal area columns from a yield table, for a given site quality.The trajectory followed by a stand can be represented as in Figure 1, with points on the curve corresponding to ages.
But what about thinning, as in the lower curves of Figure 1? Frequent light thinnings, as commonly practised in Europe, are often approximated by smooth curves, or a small number of standard regimes may be represented in managed yield tables.Evaluation of a full range of management alternatives is impractical, in particular the timing and intensity of a few heavy thinnings typical of plantation forestry in many countries.Other disturbances, and the updating of projected yields with inventory data, cause similar discontinuities.

Inputs, Outputs, Dimensionality
The rate of change equations may depend on other variables, "inputs", such as a site quality or productivity index q.Note that q does not need to be constant, it can be time-dependent possibly due to climate or nutrient level changes.All that happens then is that the length and/or direction of the arrows vary over time.We simply use whatever arrow happens to be at the current point when we get there.
In addition to the state variables H and B, one may be interested in things like volume, or size distribution parameters ("outputs").These may be estimated from the current values of the state variables.For instance, total volume may be obtained through stand volume functions such as  García et al. (2011).Right: interior spruce plot measurements (García, 2011).A dynamic growth model predicts rates of change for the three variables at any point So far, we have assumed that the behaviour of the two state variables is determined by their current values.For some purposes, a two-dimensional state may be a good enough approximation.Stand density management diagrams are a good example (Farnden, 1996).However, two stands with the same H and B, but different number of trees per hectare, may differ in their basal area growth.Also, merchantable volumes are affected by average tree size in addition to H and B. The model can be improved by adding the number of trees per hectare N (or the average spacing, or mean diameter) as a third state variable.The principles are the same, but now there is a 3-dimensional state space and 3 rate equations (Figure 3).
More state variables can be used.A stand immediately after thinning may not fully occupy the site, growing less than another stand with the same H, B and N but not recently thinned.Therefore, a fourth variable might improve accuracy, especially with heavy thinning and pruning (e.g., García et al., 2011).Iindividual-tree based models can be described in the same way, but they may contain hundreds of state variables, at least one diameter for each tree in the stand being modelled.
In the model, predicted rates and outputs are determined by the current state description.In other words, the state summarizes the relevant information about the system past.This should be seen more as a definition than as an assumption.In principle, it is always possible to add variables until they constitute a proper state, up to any desired degree of approximation.The appropriate dimensionality is a practical compromise between accuracy, parsimony, available data, and other considerations.For management purposes, it is also important to choose a scale at which reliable estimates of the initial state are possible, and at which the dynamics can be predicted (García, 2010).

Dynamical Systems and System Dynamics
The basic idea of modelling through rates of change probably originated with Isaac Newton in the 17th Century, and is standard in physics and engineering.In the 1960's, System Theory abstracted the general principles from the physical details, paving the way for wider applications (Zadeh & Polak, 1969;Kalman et al., 1969;Patten, 1971).There were contributions also from Cybernetics and Optimal Control Theory.Today the subject is part of Dynamical Systems Theory, although interest has shifted to chaotic behaviour and other aspects not directly relevant here (Wikipedia, 2012a).
The general idea is to describe a system through a suitable number of state variables.Then, a set of difference equations (Note 1) specify the change of these variables over a given time interval, depending on the current state and possibly also on one or more input variables (Figure 4, left).Continuous time can also be used, with infinitesimal time intervals, and in that case the rate of change is given by differential equations.Outputs are represented as functions of the current state.
The System Dynamics graphical notation of Forrester (1961) can facilitate communication, especially with less mathematically inclined researchers, practitioners, and students (Wikipedia, 2012b).There is also software that performs simulations based on the diagrams, requiring little or no mathematical or programming knowledge.The diagram represents each state variable as the stock of some material contained inside a box or compartment (Figure 4, right).Material moves in or out through pipes, with valves controlling the flow and therefore the rate of change in stock.Arrows indicate the dependence of a flow on stocks and on auxiliary variables (inputs or parameters).Arrows also define outputs as functions of stocks.The stock/flow analogy might be stretched too far when dealing with variables such as height, and then a more general level/rate terminology may be more appropriate.Essentially, the stocks or levels correspond to state variables, and the flows or rates to difference or differential equations.
A convenient mathematical shorthand substitutes a single symbol for a list of numbers, a vector, usually distinguished by boldface or underlining.Notation can thus be simplified, especially in theoretical developments that are valid for any number of variables (Figure 5).

An Example
How does this work in forest stand modelling?We illustrate with an even-aged whole-stand model from García (2011), García et al. (2011).With extensive data, as on the left-hand side of Figure 3, observed behaviour can be summarized by flexible purely empirical equations on three or four state variables, free from the influence of preconceived ideas.With sparse data (Figure 3, right), it is desirable to constrain the options under the guidance of eco-physiological theory and previous experience, through models that are parsimonious and consistent with biological knowledge.This is the case shown here.Although explained for this specific example, it should be where p is the proportion of pine basal area in the stand.These are differential equations (continuous time t) rather than difference equations (discrete time).Although at first sight difference equations seem simpler, their use is cumbersome when measurement and projection intervals are not uniform and multiples of each other.Rates relative to height increment were independent of site quality (an extension of an old forestry hypothesis known as Eichhorn's law); they can be expressed as conventional time rates multiplying by the first equation.

Modules
The model can be extended by interfacing to other environmental or management components through suitable input and output variables.For example, the productivity index q could be driven by a climate model, and altered by genetic improvement.Similarly, nutrient cycling and fertilizing may control foliage and fine roots formation.Carbon cycling and the fate of dead biomass can be modelled further in a separate module.It is common for modellers to focus on certain elements depending on their background and interests, at the expense of oversimplifying other components of the system.The result is typically monolithic models with detailed environmental components and simplistic stand dynamics, or vice-versa.A modular approach would facilitate an independent development of sub-models by specialists, and their later "mix and match" into more balanced wholes according to requirements.
Modules can also be interfaced to describe more complex forests.For instance, spruce and aspen models like the one previously described can be coupled through the occupancy variable to build a model for two-storied spruce-aspen mixtures.Individual-based models can be interpreted as a large number of interacting tree dynamic models.

Conclusions
Long-term forest planning requires mathematical models, and the principles of Dynamical System Theory provide a solid foundation for these.The state-space approach makes it possible to accommodate disturbances and a varying environment.The concepts were demonstrated with a biologically consistent, parsimonious and robust class of semi-empirical models.Modular strategies can improve balance and efficiency.
the opportunity of explaining this material for a non-specialist audience.I am grateful to an anonymous reviewer for suggestions that contributed to improve the text.

Figure 3 .
Figure 3. Three-dimensional trajectories observed in permanent sample plots.Left: thinned and unthinned loblolly pine data from García et al. (2011).Right: interior spruce plot measurements(García, 2011).A dynamic growth model predicts rates of change for the three variables at any point

Figure 4 .
Figure 4. Left: system equations (rate equations, output function).Right: representation in a Forrester System Dynamics diagram

Figure 5 .
Figure 5. Writing lists of numbers as vectors (boldface) simplifies and generalizes the notation

Figure 7
Figure 7 depicts such a modular structure.The central block is the stand growth model from Figure 6.The other components are shown as simplified examples for illustration only.Additional feedback or feed-forward links, e. g., between soil organic matter and nutrition, are possible but not shown.

Figure 7 .
Figure 7.The example stand model linked to blocks representing simplified nutrition, climate, and dead biomass components.Reproduced with permission from García et al. (2011)