## Growth and Yield Models

Forests have been managed more or less intensively for the last few centuries to ensure a sustainable supply of stem-wood for the production of different goods, including lumber, pulp and paper, or fuelwood. The development of forest management plans requires estimates of forest productivity. Stand tables have been developed for many decades to provide estimates of forest productivity over the life of forest stands, mostly in terms of stem volume. They provide relatively accurate estimates of stand productivity, but only if the growing conditions of forest stands do not change appreciably. As soon as growing conditions change or silvicultural treatments are performed, stand tables become less accurate or reliable. The advent of computers has made it possible to develop sophisticated models to simulate tree and stand growth. One of the reasons that justified the development of such models was to provide forest managers with quantitative tools flexible enough to predict the outcomes of silvicultural treatments. Growth and yield models are developed using stand inventory data collected by forest agencies (e.g., government forestry departments or forest companies), usually through monitored long-term sample plot networks. Predicted state variables include tree or stand attributes, such as stemwood volume or mean diameter at breast height (dbh) at different time cycles (e.g., years) along the development pathways of forest ecosystems. Statistical methods, such as ordinary least squares or maximum likelihood, are used to estimate the parameters of the relationships. Usually, the accuracy and precision of model parameters increase with the availability of historical measurements (tree and stand data collected at different ages). In the literature, the term empirical model is also used to design a growth and yield model. A common definition of an empirical model is a model derived from empirical knowledge obtained from experience or experimentation. Empirical models do not focus on explaining the underlying mechanisms, even though they may have a theoretical foundation or predict biologically consistent patterns of tree and stand growth. Growth and yield models can be classified into two large subcategories: whole-stand models and single-tree models.

### Whole-Stand Models

Whole-stand models predict the growth of stand attributes of forest ecosystems, such as basal area, stand volume, or stand density. Basal area is the summation of the cross-sectional area at breast height (usually 1.3 m aboveground) of individual tree stems appropriately weighted to reflect a particular unit area (e.g., 1 ha).

Stand volume (e.g., m3 ha-1) is the summation of the volume of individual tree stems within a forest ecosystem and stand density is the number of trees per unit area (e.g., hectare). An example of a model form to predict basal area over age is

where AB is basal area increment rate (e.g., m2 ha-1 yr-1), T stand age (year), BT basal area (m ha- ) at age T, S site index, and e the error. Site index, a measure that estimates the potential productivity of a forest ecosystem, consists of the average tree height at a reference age. For instance, in North America the most common reference age is 50 years and average tree height is based on the average height of the hundred largest trees in dbh per hectare. In this example, basal area at age T + 1 is computed by adding AB to basal area at age T.

The derivation of whole-stand models to predict the changes in stand attributes over time has been an active field of research for the last few decades. Thus, many models with different forms ofmathematical relationships were derived, including linear as well as nonlinear relationships. For instance, the nonlinear Chapman-Richards model has been frequently used in the development of whole-stand models. Different variants were also developed, such as the integration of dbh distribution classes, the derivation of distribution functions to represent the dbh distribution, or the use of transition matrices.

A particular type of whole-stand model that has received much attention is the self-thinning model. The self-thinning model was derived for different types of plant communities, including herbaceous plants and tree species. Self-thinning can be defined as the mortality that occurs in even-aged monospecific ecosystems due to competition among plants. As stand density increases within a forest ecosystem, tree mortality is more likely to occur due to the increase in competitive stress. Indeed, as plant size increases, the intensity of competitive stress accentuates, resulting in mortality. The foundation of the self-thinning model is based on a relationship proposed by Yoda and colleagues in the early 1960s in which the logarithm of mean plant biomass is inversely related to plant population density:

where w is average plant mass, N stand density, a a species-specific parameter, and a the slope. The work by Yoda and collaborators is largely cited in the literature where the origin of the self-thinning model is discussed. However, similar relationships were also derived by different authors. For instance, Reineke derived in 1933 a stand-density index based on a relationship between stand density and average diameter. Several studies conducted for different types of plant communities, including forest ecosystems, concluded that the slope of the self-thinning model was —3/2. The self-thinning model assumes that the species-specific constant and the slope define a maximum limit of plant yield in terms of biomass or size for a given density. This limit is in fact an upper boundary of density-dependent mortality that occurs under competitive conditions in even-aged forest ecosystems. For forest stands, the self-thinning model was widely applied and average plant mass was replaced in many instances by average stem volume. Further research resulted in the development of models of stand-density management diagrams to identify different species-specific development phases related to the intensity of competition. For instance, zones were defined to indicate the initiation of competition, which could be associated with the onset of crown closure, or the lower limit of competition-induced mortality, or to better highlight the asymptotic limit of mean biomass or size as a function of density.

### Individual-Tree Models

The dbh, height, basal area, biomass, or volume growth rate of individual trees is predicted by deriving models that represent inter- or intraspecific competitive interactions that occur among a subject tree and its neighbors within a forest ecosystem. The majority ofindividual-tree models are based on the derivation of competition indices, which are generally classified into spatial and nonspatial indices. This long-used classification distinguishes competition indices that require tree growth data and information about the physical location of individual trees from those that only require tree growth data. Different terms have been used in the literature to define both types of models based on the use of competition indices. For models based on competition indices requiring spatial information, the following terms have been frequently used: single- or individual-tree distance-dependent models, single-tree spatial models, space-dependent models, or spatially explicit models. The terms single- or individual-tree distance-independent models, single-tree nonspatial models, or space-independent models have been used for models based on competition indices that do not use spatial information.

### Space-dependent models

The derivation of space-dependent models is based on the use of competition indices that integrate spatial information on stand structure to describe the intensity of competition that a subject tree experiences from neighboring trees. The level of detail in the description of competitive interactions varies among competition indices. A common classification of spatial competition indices includes four major groups:

(1) amount of overlap between the zone of influence of a subject tree and the zones of influence of competitors,

(2) absolute or relative size of competitors adjusted by distance or a distance-based factor, (3) crown interference effect, and (4) potential available area. For the first type of competition index, the zone of influence represents the area within which trees compete for site resources and is a function of the crown width of an open-grown tree of the same dbh to represent the maximum zone of influence of a subject tree and each competitor. Examples of the first three types of competition indices are presented in Table 1.

The majority of space-dependent models that were developed using spatial competition indices were derived using multiple linear regression with the following general form:

where AG,- is the growth rate in dbh, height, tree basal area, volume or biomass of the subject tree; G, the initial size of the subject tree in terms of dbh, height, tree basal area, volume, or biomass; CIic the competition index that describes the intensity of competition between a subject tree i and the neighboring competitor c, the parameters, and e the error. X represents any other stand or tree variable that may be included alone or in combination with other variables. For instance, these variables may consist of site index to represent the effect of site quality, age, or basal area or number of trees per unit area to account for the effect of stand density.

### Space-independent models

Many single-tree space-independent models (i.e., without reference to spatial information on individual trees) were developed using the multiple linear model form listed above (see eqn [4]) to predict individual-tree growth rate in terms of dbh, basal area, height, or volume. Predictor variables consist of tree and stand attributes that may include initial tree size, age, a non-spatial competition index, or other variables representing the effect of stand density or site quality. The derivation of nonlinear models has also been quite common in the last three decades, probably because advances in computer power have facilitated the use of complex algorithms for the derivation of the parameters of nonlinear models. A typical form of an individual-tree nonlinear model is

where AGi is the growth rate in dbh, height, basal area, volume, or biomass of the subject tree; CIic the nonspatial competition index that describes the intensity of competition between a subject tree , and the neighboring competitor c; r)„ the parameters; and e the error. X and Z represent tree or stand variables that influence tree growth.

The effect of competition is commonly modeled by computing nonspatial competition indices that represent the effect of the presence of competitors on the growth of individual trees. Simple competition indices based on the sum of the size of competitors in terms of diameter or basal area have been used quite extensively. Relatively complex competition indices have also been derived (Table 2). The effect of the presence of competitors on the growth of the subject tree was also modeled by computing indices or functions that considered in more detail the social status of individual trees within stands, such as the sum of the basal area, the crown surface or volume of

Table 1 Examples of three types of spatial competition indices to model competitive interactions among individual trees within a forest ecosystem

Competition index3

Type

Referenceb

H d25

Zone of influence

Size/distance of competitors

Size/distance of competitors

Crown interference

Bella (1971)

Hegyi (1974) Vettenranta (1999)

aa and b, constants; CSA,c, exposed surface of the crown of the subject tree that is not shaded by the presence of competitors; Dc, diameter at breast height (dbh) of a competitor tree; D, Dbh of a subject tree; dic, distance between a subject tree and a competitor; N,, number of competitors for a subject tree; hc, height of a competitor; h,, height of a subject tree; hd, dominant height of a forest stand; H,, distance between breast height and the base of the crown of a subject tree; T, constant that accounts for species-specific tolerance; ZA,, area of the zone of influence of a subject tree; ZO,c, area of overlap between the zone of influence of a competitor and a subject tree.

bBella IE (1971) A new competition model for individual trees. Forest Science 17: 364-372; Hegyi F (1974) A simulation model for managing jack-pine stands. In: Fries J (ed.) International Union of Forestry Research Organizations, Proceedings of Meetings in 1973, Growth Models for Tree and Stand Simulation, pp. 74-90, Royal College of Forestry, research note no. 30. Stockholm, Sweden; Vettenranta J (1999) Distance-dependent models for predicting the development of mixed coniferous forests in Finland. Silva Fennica 33: 51-72; Hatch CR, Gerrard DJ, and Tappeiner JC II (1975) Exposed crown surface area: A mathematical index of individual tree growth potential. Canadian Journal of Forest Research 5: 224-228.

Table 2 Examples of nonspatial competition indices to model competitive interactions among individual trees within a forest ecosystem

Competition index"

Reference

Edc c=1

^10000/Nd H

Steneker and Jarvis (1963)

Schroder and Gadow (1999)

Rouvinen and Kuuluvainen (1997)

aDc, diameter at breast height (dbh) of a competitor tree; D, dbh of a subject tree; H, the dominant stand height (m); N, number of competitors around a subject tree; ND, number of stems per hectare; N5, number of competitors within a 5 m distance around a subject tree; gc, basal area of a competitor tree; g, basal area of a subject tree; pc, basal area percentile of a competitor tree.

bStadt KJ, Huston C, and Lieffers VJ (2002) A comparison of non-spatial and spatial, empirical and resource-based competition indices for predicting the diameter growth of trees in maturing boreal mixedwood stands. Edmonton, AB Sustainable Forest Management Network, Project Report 2002-8; Steneker GA and Jarvis JM (1963) A preliminary study to access competition in a white spruce-trembling aspen stand. Forestry Chronicle 39: 334-336; Schroder J and Gadow K von (1999) Testing a new competition index for Maritime pine in northwestern Spain. Canadian Journal of Forest Research 29: 280-283; Rouvinen S and Kuuluvainen T (1997) Structure and asymmetry of tree crowns in relation to local competition in a natural mature Scots pine forest. Canadian Journal of Forest Research 27: 890-902.

the trees within the stand that are greater than the subject tree, percentile in dbh, or tree basal area distribution.

One of the space-independent models that has received much recognition in North America over the last few decades is the Forest Vegetation Simulator (FVS) model. It has a long history of development and calibration for different tree species, forest types, and regions in the United States and Canada. Several variants were developed for specific geographic areas. An important feature of FVS is the development of various model components to simulate different types of silvicultural treatments. FVS has its foundation in the Prognosis model, which was originally developed by Albert R. Stage at the USDA Forest Service in the early 1970s. Several modifications have been made or new components have been implemented over the last few decades to expand its capabilities, including the integration of components from other models, such as TWIGS. The model components of FVS were designed such that simulations could be conducted by using forest inventory or basic stand data to characterize the initial conditions of the forest stands for which growth predictions are desired.

For instance, the basic form of the model to predict diameter growth rate of individual trees may include the following independent variables, in addition to a competition index: current tree dbh, stand basal area, site index, or crown ratio (the ratio of crown length to stem height). For some variants, the representation of the effects of site fertility on tree growth was performed by integrating variables describing ecological characteristics, such as elevation or habitat type. Other FVS components include a mortality model to predict individual-tree death, an establishment model to predict regeneration, and other submodels to predict the impacts of disturbances on tree and stand growth, such as insect infestations, diseases, or fire.

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