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Current Forestry Reports

https://doi.org/10.1007/s40725-021-00145-8

MODELLING PRODUCTIVITY AND FUNCTION (M WATT, SECTION EDITOR)

Growth Equations in Forest Research: Mathematical Basis

and Model Similarities

Christian Salas-Eljatib1,2,3 ·Lauri Meht¨

atalo4·Timothy G. Gregoire5·Daniel P. Soto6·Rodrigo Vargas-Gaete7,8

Accepted: 24 August 2021

©The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021

Abstract

Purpose of Review Growth equations have been widely used in forest research, commonly to assess ecosystem-level

behavior and forest management. Nevertheless, the large number of growth equations has obscured the growth-rate behavior

of each of these equations and several different terms for referring to common phenomena. This review presents a unified

mathematical treatment of growth-rates besides several well-known growth equations by giving their mathematical basis

and representing their behavior using tree growth data as an example.

Recent Findings We highlight the mathematical differences among several growth equations that can be better understood

by using their differential equations forms rather than their integrated forms. Moreover, the assumed-and-claimed biological

basis of these growth-rate models has been taken too seriously in forest research. The focus should be on using a plausible

equation for the organism being modelled. We point out that more attention should be drawn to parameter estimation

strategies and behavior analysis of the proposed models. Thus, it is difficult for a single model to capture all possible shapes

and rates that such a complex biological process as tree growth can depict in nature.

Summary We pointed out misleading concepts attributed to some growth equations; however, the differences come from

their mathematical properties rather than pure biological reasoning. Using the tree growth data, we depict those differences.

Thus, comparisons of some functional forms (at least simple ones) must be carried out before selecting a function for drawing

scientific findings.

Keywords Differential equations ·Growth-rates ·von Bertalanffy ·Logistic ·Trees

Introduction

Growth is a term which everybody understands but not

necessarily in the same way. Although we can track

definitions of growth as far as Aristotle [1], one of the

first obstacles in understanding tree growth is the lack of

precision in the definition of what is meant by growth.

Growth is determined by cell division, cell extension and

cell differentiation [2].

This article is part of the Topical Collection on Modelling

Productivity and Function

Christian Salas-Eljatib

cseljatib@gmail.com

Extended author information available on the last page of the article.

Ergo, growth is a phenomenon that captures the interest

of research in a suite of disciplines [3], especially in plant

sciences [4]. The growth pattern in forest trees is divided into

primary, i.e., the growth from a bud, root, tip or another

apical meristem, and secondary, i.e., growth from the

cambium [5]. Growth implies an increase in size and the

formation of new tissue; however, it growth may occur when

older organs, particularly leaves, are dropping off faster than

new ones are being formed [6]. The term net growth includes

also this component, whereas gross growth does not. In

forestry, these terms are mainly used at the stand level,

and using volume as variable of reference, as follows: net

growth refers to the difference between volume at the end

of the period and at the beginning; gross growth must also

incorporate the growth of trees which may have died in the

period [5,7]. We refer here to the growth of variables such

as height and stem diameters as the irreversible increase in

Curr Forestry Rep

leading-shoot length and stem diameter, respectively [8], or

in a more physiological context, the incorporation of carbon

into structural material [2]. Nevertheless, we follow [9],

in the sense that the level of abstraction of our modelling

framework emphasizes the physically measurable exterior

tree characteristics, which we call variables.

Let us take a closer look at the use of the terms “growth”,

“increment” and “yield” in forestry. Bruce [10] stated, using

height growth equations as an example, that there are growth

and growth-rate equations, the first one being an expression

giving the variable as a function of time, and the second as

the differential of the growth equation. He also emphasized

that growth-rate is determined at one point in time. In

forestry, growth over a time period shorter than a year

may be confusing because the growth usually occurs during

summertime. Yang et al. [11] defined the term increment

as the increase in size of an organism within a certain

time interval, i.e., as the difference in observed or predicted

growth at two points in time [10]. Current annual increment

(CAI) is the difference in the growth at the beginning and

the end of the year and periodic annual increment (PA I )

for longer time units [12]. Term yield has also been used

in the same meaning as growth [12–16], but it has mainly

been associated with volume. For example, Clutter [14]

and Curtis [15] refer to growth-rate as the derivative form

and to yield as the integral of volume. Clutter [14] defined

that if integrating the growth-rate equation gives the yield

equation, the models are compatible. Even though the

forestry literature often recognizes the works by [12]and

[14] as the first to mention the consistency between growth-

rate equations and the yield equations, many researchers

in plant ecology [17–19], animal growth [20], and also in

forestry [21–23] have been aware of this relationship long

before.

In this paper, we use the following terminology that

corresponds to the mathematical explanation of derivatives

and integrals in calculus [24]: If a state variable, y,is

measured at times t0and t1,orsizeatbothtimes,wehavea

dynamic variable, therefore we also refer to them as growth

at times t0and t1(Fig. 1).

The difference between these two measurements is the

increment of that variable, for the period of time t =t1−

t0, and is symbolized by y =y1−y0. The ratio y /t ,

is referred to as the average rate of change, between t0and

t1, which is how fast the yvariable is changing, and is the

slope of a straight line connecting f(t

0)to f(t

1)in Fig. 1.

If t becomes infinitesimally small, then we are analyzing

the derivative of y,dy/dt,inyfor any particular time

(i.e., the slope of the tangent line at points t0and t1in Fig. 1),

which we call (instantaneous) growth rate. If the increment

in the state variable from t0up to a time t1,y (or dy/dt

integrated), is added to y0,thenwegety1, the growth at time

t1.

If we were able to capture the entire lifetime of most live

organisms, their empirical growth would depict a sigmoid

curve. A sigmoid or S-type curve resembles trends in the

life cycle of many living organisms and phenomena [25]

(Fig. 2a). Virtually all exterior tree dimensions develop in

a sigmoid manner with respect to time [26]. A growth

curve is typically separated in phases (Fig. 2a). Generally

three phases have been recognized [27–29], but the pattern

remains the same. Baker [28] called them acceleration,

intermediate, and deceleration and Assmann [29] youth, full

vigor, and old age; while Zedaker et al. [30], youth, maturity,

and senescence. In phase 1, yis an exponential function of

time [27]. Trees grow relatively slowly at first, increasing

their growth-rate to the point of inflection of the growth

curve [30]. In phase 2, yis directly proportional to time [27],

and in phase 3, there is a decrease of the growth-rates [27,

30].

In addition to CAI, the forestry literature recognizes

the term mean annual increment (MAI), which is found

by dividing the growth through time tby the number of

years required to produce it (red curve in Fig. 2b), i.e.,

MAI =yt/t. Growth and increment curves have long been

used for making silvicultural decisions, using as reference

the following facts. The inflection point of the CAI curve,

which occurs at the time symbolized by tain Fig. 2,isalso

the time of maximum CAI. It has been recognized as a

good opportunity to apply thinning, since the increment of

Fig. 1 Growth definitions for the state variable yobtained from

function f. Growth curve (blue), state variable at time 0 (f(t

0))and

time 1 (f(t

0)), increment (y) between t0and t1, and growth-rate

or derivative (dy/dt)att0(f(t0))andt1(f(t1)). The derivative, or

instantaneous growth-rate, is the slope of the tangent lines (black) at

points t0and t1

Curr Forestry Rep

Fig. 2 Growth and growth-rates

curves. aThe trajectory of the

state variable (y), generally

representing an expression of

size, is the growth curve.

Meanwhile, bgrowth-rates

curves are represented by the

current (blue) and mean (red)

annual increments. tais the time

of culmination of current annual

increment and tbis the time of

maximum mean annual

increment

residual trees is going to reach its maximum. The point tais

called the culmination of current annual increment [29–31].

The inflection point of the MAI curve, which occurs at the

time symbolized by tbin Fig. 2, represents the point where

the curves of MAI and CAI cross, and MAI is maximized.

It has been recognized as a biological criterion to choose

the rotation age of even-aged stands. More specifically, it

is the rotation that maximizes the growth. It would be the

rotation that leads to maximum carbon sequestration if the

carbon of harvested biomass would never be released back

to the atmosphere [32]. Otherwise, the optimal rotation for

carbon sequestration would be longer. Point tbwould be

an economically optimal rotation if the interest rate of the

forest owner is 0%. A higher interest rate would make the

optimal rotation shorter [33].

Growth Equations

Many empirical and theoretical growth equations have been

used in forest research. Traditionally, practitioners want to

predict the value of a random variable at time t,let’ssay

yt, as a function of some variables that might affect the

value of yt. The usual approach is to build a mathematical

model that fits the pattern of the observed data. The resulting

model is called “empirical”. On the other hand, growth

equations have been developed from biological theory about

the growth process, with parameters that have (sometimes)

biological meaning. Theoretical growth equations have

become preferred in research but not always in practice.

Some of the so called theoretical growth equations have

empirical bases. Nevertheless, we prefer to call them as such

because the parameters have some biological interpretation.

Empirical models serve a different purpose than do

theoretical models. Thereupon, empirical models should not

be viewed as poorer alternatives.

Theoretical growth equations have been developed

for many biological disciplines. Several mathematical

equations are based on theoretical constructs. Since many

growth curves are non-linear in terms of their parameters,

growth models are an important family of non-linear models

[34]. There is an extensive number of growth equations in

the literature; however, it is hard to believe that a model

with three or four parameters could describe so complicated

process as growth from birth to death [35,36]. The origin

of many non-linear models in use today can be traced to

scholarly efforts to discover laws of nature, to reveal scales

of being, and to understand the forces of life [34]. There

are several growth equations with theoretical or theoretical-

empirical bases. Here, we focus on some that have been

widely used in forestry research.

Curr Forestry Rep

There are many different parameterizations of a certain

growth model. Because these models start from differential

equations, many trajectories can be obtained; as a result

numerous models have been proposed based on the

differential forms. Furthermore from different differential

equations it is possible to attain the same solution or

trajectory, a mathematical fact that is not fully understood,

as expressed by [37,38]. Several authors have reported

different parameterization of growth models for forestry

and other scientific disciplines [3,34,37,39–44]. Here, we

present differential forms and solutions of selected and most

widely use models in forest science. We want to stress that

by selecting these mathematical functions, we are aiming

to compare their mathematical differences and origins, but

not providing a detailed review of all the available growth

functions that can be found.

Hereafter, all the parameters are positive, as well as time

(t) is positive, and the state variable y. We also provide

expressions of the asymptote (i.e., the maximum level for

the state variable) and the points of inflection (i.e., where

the curve changes of curvature) of all models. We shall

use Greek letters to refer to parameters and italics font

style to refer to variables. The use of the same symbols

that represent parameters in equations through the document

do not represent the mathematical equivalence, unless that

is clearly specified. For example, βor γcan be used in

different equations, but they do not imply the same number.

Monomolecular Growth Model The monomolecular model

is the following differential equation

dy

dt=β(α −y), (1)

where yis the state variable, trepresent time, and βand

αare parameters that represents the proportional constant

and final size, respectively. For this model growth is

proportional to the remaining size of the organism [34].

Therefore, the growth-rate decreases as tincreases and the

equation cannot describe sigmoidal growth. A solution of

(1) produces the growth function

yt=α1−1−y0

αe−β(t−t0).(2)

If we further assume that t0=0,y

0=0, then (2) becomes

yt=α1−e−βt,(3)

where αis the asymptote of the state variable, but the model

does not have an inflection point (Table 1). The equation

form of (3) is known as the Mitscherlich law or Mitscherlich

equation [34].

This function was proposed for height-diameter mod-

elling by [45] as well and is therefore sometimes called

Meyer’s equation in the literature.

Tab le 1 Features of the studied growth models

Model Differential form A solution Asymptote Point of inflection

Monomolecular dy

dt=β(α −y) yt=α(1−e−βt)α None

Logistic dy

dt=βy(1−y

α)y

t=α

1+eβ0−β1tαβ0

β1;α

2

Gompertz dy

dt=βy(ln α−ln y) yt=αe−e−β(t−γ) αγ;α

e

Johnson-Schumacher dy

y=βd1

t2ln yt=α−β1

teαβ

2;eα−2

Bertalanffy-Richards dy

dt=β

γyα

yγ

−1yt=α1−e−βt1

γα−ln(γ )

β;α(1−γ)

1

γ

Wei bul l dy

dt=αβ γ t γ−1e−βt yt=α(1−e−βtγ)α (γ−1

γβ )

1

γ;α(1−e

γ−1

γ)

Schnute d2y

d2t=dy

dt[−α+(1−β)γ]yt=yβ

1+(yβ

2−yβ

1)1−e−α(t−t1)

1−e−α(t2−t1)1

βαyβ

1+yβ

2−yβ

1

1−e−α(t−t1)1

βt1+t2−1

αln(β(yβ

2eαt2−yβ

1eαt1)

yβ

2−yβ

1

);(1−β)(yβ

2eαt2−yβ

1eαt1

eαt2−eαt1)

The differential form (i.e., growth-rate equation), a solution (i.e., growth equation), the asymptote and the inflection point are provided

Curr Forestry Rep

Logistic Growth Model The logistic growth is mostly

attributed to Verhulst’s work in 1838 [46–48]. The logistic

model imposes a restriction to the exponential growth, then

representing limiting resources for population growth.

There are many parametrizations of the logistic model,

but we shall show the one most commonly cited in the

ecological literature [46,49],

dy

dt=θy −φy2=y(θ −φy), (4)

where yis the state variable, θand φare parameters.

The differential logistic equation can also be parameter-

ized as follows [48,50],basedontreatingθas βand (θ/φ)

as α,

dy

dt=βy −β

αy2=βy 1−y

α,(5)

where the constant αhas a biological interpretation as the

carrying capacity of the environment, e.g., the maximum

population size for ecological studies or the maximum tree

size, and βis a proportionality constant.

The logistic growth equation is the simplest equation

describing sigmoidal population growth in a resource-

limited environment, and it forms the basis for many models

in ecology [48]. A solution of the differential form of the

logistic is the following growth curve,

yt=α

1+[(α−y0)/y0]e−βt ,(6)

where y0is the value of the initial condition of the state

variable y.From(5) we see that when y→α,which

happen when t→∞, the growth-rate is zero, therefore the

population being modelled or the size of any organism, does

not grow without limit. The logistic equation is symmetric

around the inflection point t0, implying that the growth rate

fulfills f(t

0−t) =f(t

0+t) for any positive t. Also notice

that, when the growth-rate of the equation (Eq. 4) is plotted

as a function of the state variable y, and not time, the point

of inflection y=α/2(Table1) is when the growth-rate

reaches its maximum.

A usual parameterization found in the forestry literature

[31,51]is

yt=α

1+eβ0−β1t(7)

which is equivalent to make (α −y0)/y0of Eq. (6)

as equal to eβ0,andβ0and β1are parameters. Robertson

[52] proposed the same differential equation (as in Eq. 4)

to describe an autocatalytic monomolecular reaction in

biochemistry. Then, the logistic equation is also known

as the expression that represents the “autocatalytic law”

of physiology and chemistry. When [53] commented on

some recent studies on growth including the one by

[52], further generalized this thought to brain (and other)

growth. According to [34], Pearl and Reed promoted the

autocatalytic concept not only for individual but also for

population growth, in their work of 1924. In sociology, the

logistic model is know as the law of Verhulst established

in 1838 describing the growth of human populations with

limited resources [54].

Gompertz Growth Model This growth model was proposed

by [55], who was concerned with modelling mortality in

an arithmetical progression; however, [56] derived it as a

growth model. Gompertz’s differential model is

dlny

dt=β(ln α−ln y),

1

y

dy

dt=β(ln α−ln y),

dy

dt=βy(ln α−ln y), (8)

when the relative growth-rate declines with lnyand the

other terms are parameters.

The most common parameterization of a solution of (8)

is [3,34]

yt=αe−e−β(t−γ),(9)

where αis the asymptote and γis the time where the point

of inflection occurs (Table 1). Although βis a dimensionless

parameter, it affects the maximum growth-rate (i.e., αβ/e).

As opposed to the logistic model, the Gompertz curve is

asymmetric.

Johnson-Schumacher Growth Model This model was inde-

pendently proposed by [57]and[21], which has been also

known as the “reciprocal function”.

Schumacher [21] explained the model by saying that

the growth percent varies inversely with age, which he

expressed in a differential form as follows:

dy

y=βd1

t. (10)

A solution to this differential equation is found by first, set

u=t−1and then substitute du=−t−2dtfor d(1/t) =

−t−2dt in Eq. (10)

dy

y=βdu,

1

ydy=βdu,

ln y+γ1=βu +γ2,(11)

where γ1and γ2are the constants of integration. Second, by

assigning α=γ2−γ1, we arrive at

ln y=α+β1

t. (12)

This model has been widely used in forestry, mainly for

fitting height-age models. Equation (12) allows parameter

estimation in the context of linear modelling. However, one

should notice that model fitting for ln yleads to biased

Curr Forestry Rep

predictions of y[58,59]. Also the alternative form y=

αexp(β

t), is used in forest sciences; however, here αhas a

different interpretation than in (12).

The differential equation of [21] can be represented as the

growth-rate as a fraction of the state variable y, as follows

⎛

⎝dy

d1

t⎞

⎠/y =β. (13)

however, as mentioned by [60], this differential equation

does not represent what [21] said in words, i.e., “growth

per cent varies inversely with age”. Instead, that thought is

represented by the following differential equation

dy

dt/y =β1

t. (14)

A solution of Eq. (14) is obtained as follows,

dy

y=β1

tdt,

1

ydy=β1

tdt,

ln y+γ1=βln t+γ2,(15)

where γ1and γ2are the constants of integration, and α=

γ2−γ1, as before, we get

ln y=α+βln t, (16)

This function is not the growth equation proposed by

[21], which is being advocated to represent growth-rate

as a percentage inversely proportional to time. Using

exponential transformation, we see that the equation is a

linearized version of the extensively used power function

y=φtα,whereφ=eα.

Furnival [60] also detected the error in [21], and said “the

curve given by [21] is obtained if growth percentage is taken

as inversely proportional to the square of age”. If we proceed

as above, we arrive to

ln y=α+β−1

t. (17)

This formulation emphasizes that model (12) leads to

increasing logarithmic growth only if β<0, whereas in

(17) we need to have β>0. Therefore, we favor this

expression as a correct solution of the differential equation

dy

y=βd1

t2. (18)

Finally, we could say that the derivation of Schumacher

is correct but his formulation is wrong.

Bertanlanﬀy Growth Model This model was first published

in German [61], and later the author introduced it to the

English literature in a brief paper [62], and in a more

developed one [20].

He originally developed it for weight growth of animals.

The model is a result of a deeper developed basic idea of

P¨utter in 1920 [54]. That is, growth can be considered a

result of a counteraction of synthesis and destruction, of the

anabolism and catabolism of the building materials of the

body. The differential form of the von Bertanlanffy model is

dy

dt=ηyθ−κyφ,(19)

where ηand κare constants of anabolism and catabolism

respectively, and the exponents θand φindicate that both

are proportional to some power of the state variable y. Leary

[63]termed(19) as the Bertalanffy’s anabolic-catabolic

balance equation of biological growth. As explained by

[20], this differential equation represents “the change of

yis given by the difference between the processes of

building up and breaking down”. Nonetheless, this is a

property that is also shared by the monomolecular (1),

logistic (4)andGompertz(8) equations, in the sense that the

differential equation can be separated into different additive

terms. In the Bertalanffy model, both these processes are

described by the simple power models, but the form of these

submodels is not justified by biological theory. Ricker [47]

pointed out that von Bertalanffy did not make clear what

he meant by anabolism and catabolism, and aside from

all the widely quoted references to his model as a solid

theoretical growth model, that is not actually true. For

instance, von Bertalanffy [54] explained, when referring

to his model, that the use of the exponent θand φin

(19) has a pure empirical base, since it is well known that

the size dependence of physiological processes can well

be approximated by allometric expressions. More recently,

Seber and Wild [3] noticed that unfortunately the biological

basis of these models (referring to the logistic, Gompertz,

and von Bertalanffy) has been taken too seriously and had

led to the use of ill-fitting growth curves.

Bertalanffy [20] justified that φ=1 based on

physiological experience; hence

dy

dt=ηyθ−κy, (20)

and he also empirically found that for a wide class of

animals, the allometric power for the metabolic rate (θ)is

2/3. Thus, differential equation (19) becomes

dy

dt=ηy2/3−κy. (21)

Equation (21) is usually termed the von Bertalanffy

model [3]; nonetheless, we prefer to reserve that denomina-

tion to Eq. (20).

Bertalanﬀy-Richards Growth Model Richards [19] doubted

some theoretical considerations of the Bertalanffy model

(21), and treated some of the parameters to be related to a

Curr Forestry Rep

point of inflection [3]. The Richard’s differential equation is

dy

dt=β

γyα

yγ

−1,(22)

where α, β and γare the asymptote, a scale, and shape-

related parameters [34]. Although, this equation has been

termed as the Bertalanffy-Richards model by [64,65],

and also as a generalization of the Bertanlanffy model

[19], it is paramount to understand that it is simply a

reparametrization of (20) as shown by [38]. A solution of

this differential equation is

yt=α1−1−y0

αγe−β(t−t0)1

γ. (23)

Notice that (23) becomes the following growth equation if

y0=t0=0,

yt=α1−e−βt1

γ. (24)

Hence, (24) is a special case of (23), being the former

suitable for forestry plantations and the latter for natural

forests [66].

It is key to realize, and as aforementioned, that from

distinct differential equations, we can arrive at a same

solution. Notice that from the following linear differential

equation with a power transformation [64],

dyγ

dt=βαγ−yγ,(25)

we can obtain the same solution of (23), as explained in

detail by [38].

Chapman [67] starting from the von Bertalanffy differ-

ential equation, in the same way as [19] did, derived a new

parameterization of the model particularly suitable for fish-

eries population modelling. Because of that, sometimes this

equation has been called “Chapman-Richards” as well [40,

68]. We can argue that the denomination for the general-

ized differential equation (20) should be credited to [20],

because even though he did not state it formally in his paper,

he used it as the basis to obtain his proposed solution. Nev-

ertheless, Richards [19] made this clearer, and also gave his

famous 2/3 exponent for θin (20). Thus, the denomination

“Bertalanffy-Richards” seems a better compromise. Any

other denomination, e.g., Chapman-Richards, even though

correct in the sense that they started from the differen-

tial equation (20), is burdensome. Similar claims had been

brought by [37–39,69]. Based on the shortcomings men-

tioned above, we shall focus on the Bertalanffy-Richards

growth model (23).

Weibull Growth Model This model is a probability density

function (pdf) widely used in engineering [70]aswellas

in forestry for describing diameter distributions [71,72]

because of its versatility. Prodan [73] noticed that tree

increment and growth curves resemble a probability density

(i.e., frequency distribution) and a probability distribution

(i.e., cumulative frequencies distribution), respectively.

Later, Yeng et al. [11] exploited this relationship and used

it to fit growth curves of trees. Based on the review given

by [74], it seems that [11] were the first to notice this. The

differential form of the Weibull growth model is as follows

[40]:

dy

dt=αβ γ t γ−1e−βtγ. (26)

A solution of the Weibull growth-rate model is [11,41]

yt=α1−e−βtγ,(27)

where α,β,andγare parameters representing the

asymptote, intrinsic rate and the shape of the curve,

respectively. The last two parameters affects the point of

inflection (Table 1). Zeide [40,41] and Payandeh and

Wang [ 75] found that the Weibull growth model performed

as well as other common growth equations in modelling tree

growth of several variables.

Schnute Growth Model Schnute [76] introduced a model

first tested for fisheries research. The model is capable of

assuming a wide range of characteristic shapes that describe

asymptotic as well as non-asymptotic growth trends [65,

77]. The Schnute’s model is based upon acceleration in

growth, that is the rate of a rate, hence using second

derivatives as follows [76]:

d2y

d2t=dy

dt[−α+(1−β)1

y],(28)

where α,β,anddy/dtare parameters. A solution of the

Schnute differential equation is [76]

yt=τβ

1+(τ β

2−τβ

1)1−e−α(t−t1)

1−e−α(t2−t1)1/β

,(29)

where α= 0, β= 0, tis the time of interest, t1time

at beginning (e.g., young), t2time at end of (e.g., old),

τ1and τ2are parameters representing the state variable y

at time t1and t2, respectively. As stated by [65], φis a

constant acceleration in growth-rate, and βis incremental

acceleration. Unfortunately, the growth function (29) does

not have a single parameter with biological interpretation

(Table 1). The Schnute growth-rate model is among the very

few growth investigations based upon second derivatives,

with the exception of the lucid study of [78] when modelling

height growth.

The Use of Growth Equations in Empirical

Studies

Although the logistic model has been long used in

population dynamics in ecology, it has barely been used in

forestry to model growth [51]. Among them, Cooper [22],

Curr Forestry Rep

MacKinney and Schumacher [31], and Monserud [51]used

it for modelling growth of stand volume, stem area, and tree

height growth, respectively.

Unlike the logistic model, the Gompertz model is not

symmetric about the point of inflection.

It is a model more used in ecology than in forestry; it

performed well in population models as reported by the

comparison of [79] and was used with success by [80]. In

forestry, it was first mentioned by [81], but was not used

until the work of [82] in modelling stand volume growth.

The Bertalanffy-Richards model is used more than any

other function in studies at tree and stand level [41,43].

This equation was first used in forestry by [83], but

mostly popularized by [68], after which the model was

termed and later known as “Chapman-Richards”.

The Schumacher growth equation (17) is long-established

for modelling height growth of dominant trees, because it is

linear in its parameters. Notwithstanding, lately the Bertalanffy-

Richards model has become the preferred model.

The Weibull model was first proposed by [11]. Pro-

dan [73] earlier had mentioned the resemblance of growth

and increment curves to probability density and probability

distribution.

The Schnute model has the property of having asymptotic

as well as non-asymptotic shape depending on the parameter

values combination, therefore being versatile to a wider

ranges of situations. Bredenkamp and Gregoire [77]were

the first to introduce the model to forestry. Later on, Yuncai

et al. [65,84]and[85] used this model as well.

Comparisons among growth equations are valuable both

for practical and theoretical use; nonetheless, it seems that

researchers tend to prefer one of the models beforehand.

Researchers might conduct initial screenings, but may

not report the results in detail. Considering the large number

of growth functions [43], very few studies have empirically

compared them. In Table 2, we have summarized a

comprehensive list of studies in this regard. Overall, the

authors selected their models using traditional goodness-of-

fit statistics. The Bertalanffy-Richards, Weibull and Schnute

were the most widely selected according to these criteria.

Practical Demonstration

To illustrate the different growth-rates, we estimated the

parameters of the growth equations by using time series

height data of 107 Nothofagus dombeyi sample trees, a

native evergreen tree species of South America temperate

forests [93], growing on secondary stands in the south of the

country. The tree-level data were collected within 30 sample

plots established throughout the ecological distribution of

the species. Sample trees, at the time of sampling, ranged

from young ages to mature (between 21 and 71 years), as

well as from small trees to taller ones (between 10 and 34

m, Table 3).

Tab le 2 Empirical comparisons of growth equations in some forestry studies

Model

Source Monomolecular Logistic Gompertz Bertalanffy-Richards Weibull Schnute Other Data

[11]•• • √Tree vo lume

[86]√••Stand volume

[77]•√Tree diameter

[40]•• • • √Tree diameter

[75]•√Tree height

[87]••• √Plant disease

[65]√√

Tree diameter

[88]•√√√

Tree height

[84]√√

Tree height

[89]•• √√√

Tree height

[43]•••• √Tree height

[90]√•Tree basal a rea

[91]•√•• ••Weight

[85]√•Tree height

[92]√•√Plant disease

The symbol •and √represents that the equation was assessed and selected as the best in the study, respectively

Curr Forestry Rep

Tab le 3 Tree level variables summary for the 107 Nothofagus dombeyi

trees at the time of sampling. dis diameter at breast height, his total

height, age is total age, and bha is breast-height age

Statistic dhagebha

(cm) (m) (yr) (yr)

Minimum 5.3 9.9 21 15

Maximum 60.2 33.7 71 68

Mean 26.6 19.9 41.3 37.7

Median 26 20.8 40 37

CV(%) 19 16.6 15.8 17.3

The height growth data cover a wide variety of growing

conditions (Fig. 3). Further details on the data can be found

in [38].

The growth equations were fitted in a mixed-effects

model framework for considering the grouped structure of

thedata[66,94].

A general non-linear mixed-effects model for the kth

observation on the jth tree at the ith plot is

yij k =f(φij ,t

ij k )+εij k ,(30)

where fis a growth equation and the parameter vector φij

includes the parameters of the selected growth function.

Each parameter consists of a fixed effect (μ)andtwo

random effects, i.e., plot (bi)andtree(bij ), therefore

φij =μ+bi+bij . (31)

Fig. 3 Tree height growth series of dominant trees for Nothofagus

dombeyi in southern Chile. Gray lines join successive observations of

height on the same tree

We assume that the random effects among plots are inde-

pendent with bj∼N(0,D), as well as for the random

effects among trees within plots with bij ∼N(0,E).Also

the residual errors are independent and normally distributed

with Var(εij k )=σ2. Further details on the formulation

of non-linear mixed-effects models are presented in [59,

95–97].

The models were fitted using the nlme package [98]of

R[99].

The fixed-effects parameter estimates for each growth

model are displayed in Table 4. We also computed

the prediction statistics as in [100], i.e., the root mean

square differences (RMSD) and the aggregated difference

(AD) using the population level predicted value from the

corresponding mixed-effects fitted model. These statistics

provide some insight into model fit. Except for the Johnson-

Schmacher equation, all models have similar precision and

accuracy levels.

Using the fixed-effects estimated parameters of the non-

linear mixed-effects growth models (Table 4), we represent

the height growth or trajectory of the state variable h

(Fig. 4a), and the instantaneous growth-rates versus time

(Fig. 4b) and versus the state variable (Fig. 4c), as well.

These trajectories should be interpreted as the estimates for

a typical sample plot of the data set.

The number of parameters (i.e., coefficients) affects the

flexibility of functions. As such, the greater number of

parameters, the greater flexibility or intrinsic curvature.

We used two functions (i.e., Monomolecular and Johnson-

Schumacher) having 2 parameters, four (i.e., logistic,

Gomperts, Bertlanffy-Richards, and Weibull) having 3

parameters, and the Schnute’s function having 4 parameters.

Comparing flexibility of the functions for a given

number of parameters is of great interest when a generally

applicable function is searched for empirical growth

modelling. See [101] for such an analysis in the context

of diameter distribution models. However, we leave such

analysis for future studies.

Caveats

In physics, it is rare to analyze different models because they

are already well established, and uncertainty is not a central

feature. On the contrary, in biology-related disciplines, such

as forestry, variability and uncertainty is the rule. As such,

selecting a growth equation on which our findings are based

is not straightforward. Most practitioners might compare a

suite of equations and choose one of the merit of goodness-

of-fit indices [102,103]. On the contrary, others will start

with a preferred growth equation at the very beginning of

their studies [104–106].

In this vein, it is often not recognized the tremendous

impact that a selected growth model will have on our

Curr Forestry Rep

Tab le 4 Fixed-effects parameter estimates and prediction statis-

tics for each mixed-effects growth model. All models were fitted

by maximum-likelihood and using time series height data of 107

Nothofagus dombeyi sample trees from southern Chile. RMSD is the

root mean square differences and AD the aggregated difference

Parameters RMSD AD

Growth model α

βγ(m) % (m) %

Monomolecular 49.76 0.012 – 2.8249 26.2 0.0507 0.47

Logistic 23.41 2.242 0.10 2.8560 26.5 0.0689 0.63

Gompertz 26.28 0.057 18.59 2.8003 25.9 0.1082 1.00

Johnson-Schumacher 2.72 5.681 – 5.1607 47.9 1.6846 15.62

Bertalanffy-Richards 34.67 0.024 0.81 2.7876 25.9 0.1494 1.39

Weibull 32.70 0.012 1.18 2.7871 25.8 0.1487 1.38

τ1τ2

Schnute 0.16 -2.13 2.84 18.02 2.9419 27.3 0.2120 1.96

inferences. For instance, let us assume that the goodness-

of-fit indices are similar among the suite of growth

equations examined here as a reference, a pattern that is

not unrealistic. However, if we are interested in finding

the maximum height for the species, a critical functional

trait, the asymptote of the growth curves (Fig. 4a) varies

tremendously between an unrealistic value of 15.2 m for

the Johnson-Schumacher model (we used the [58]bias

correction for prediction) and 49.8 m for the monomolecular

one (Table 4).

Overall, there is no theoretical reason to limit either tree or

forest growth analysis by using a single equation, and claims

on different reparametrizations have been proposed [107].

As previously, most growth models are non-linear. The

only linear model (in the parameters) assessed here was the

Johnson-Schumacher (Eq. 17). There is no closed formula

to estimate non-linear models’ parameters, as we have

for the linear models. Nowadays, most statistical software

includes tools for easing the fit of non-linear equations, such

as self-starting functions that find good initial guesses for

the parameters [97]. However, a little more care and time

may be needed than with linear models, e.g., to run the

numerical estimation routines in large data sets, to find good

enough initial guesses for the parameters, and to ensure

that the estimates found are the global maximum of the

likelihood function, but not local [3].

Regardless, several attempts have been devoted to

ecological applications to offer guidance for fitting non-

linear growth models, since [108,109]to[44], highlighting

that this is still an important topic when modelling tree

and forest growth. Consequently, the possibility of working

with models having parameters with biological or physical

interpretation is an advantage for fitting these non-linear

models.

There are numerous applications for the unified mathe-

matical treatment of growth equations presented here. The

Fig. 4 Growth (a) and growth-rates (b) versus time, and growth-rates versus the state variable cfor the growth equations studied here

Curr Forestry Rep

more obvious are in forest modelling, such as the ones fully

described in [110,111]. Besides, growth equations have sev-

eral implications in decision support and scientific findings

[112]. Nonetheless, there are a handful of other applica-

tions where growth equations are used. Among them, we

can mention the studies focus on: ecophysiological aspects

of sapling growth [104–106,113]; forest-dynamics [114];

silviculture [115,116]; and restoration ecology [117–119],

where all the findings are based upon a single fitted–and–

assessed growth function.

As lucidly summarized by [3], we can use x(i.e., another

state variable) instead of tin the growth-rate equations, and

by this simple twirl, the same mathematical models shown

here, as well as their implications, can serve for representing

tree allometry [120], a major scientific endeavor in biology

[121,122]. In the forestry context, growth equations have

been used in modelling height-diameter [88,123,124], and

stem taper [125,126] allometric relationships.

Growth-rates of the tree and forest growing processes

are essential in research [23,127,128], especially to study

the partitioning of components, resources, and allocation

of them to different processes. At least a basic but correct

mathematical understanding of differential equations is

paramount for researching these complex processes. Similar

advocacy has been claimed by others in ecology [129].

It is noteworthy to realize how the forestry literature is

way more focused on growth than on the growth-rates

themselves; that is to say, forestry is a discipline where we

seem to be trained to think in an integrated form rather

than in a differential form. In contrast, we highlight the

pioneer works on differential equations in forestry led by

[64,130–133].

Although we might think that the reasons for focusing

on growth, but not on growth-rates when modelling forestry

variables are a handful; the main one could be related

to the fact that instantaneous growth rates are essentially

unobservable, whereas realized sizes can be measured and

so are amenable to exploratory data analysis. However, this

can also be applied to other biological disciplines, such

as wildlife ecology and landscape ecology. Regardless, the

use of growth-rates, or differential equations, are just a

few compared to size-time equations (i.e., a differential

equation solution). Maybe the reasons are also related to a

weak educational background in calculus of ecology-related

scientists [129,134]. Accordantly, we favor using growth-

rates functions, such as differential equations, in forest

growth modelling when justified by the research question

[23,38,128,133].

As a reflection of the above, the analyses of the

growth rates is mandatory. The growth-rates against time

for each model broadly vary (Fig. 4b). All of them

implies completely different processes. For instance, the

Monomolecular and the Johnson-Schumacher curves seem

unrealistic. Both the Bertalanffy-Richards and the Weibull

functions behave quite similar at reaching their maximum

growth-rates at much earlier stages. On the contrary, the

logistic, Gompertz and Bertalanffy-Richards seems to be

more feasible. Finally, a graphic representing the growth-

rates against time (Fig. 4b) is usually the most commonly

reported; however, plotting the growth-rates against size

(Fig. 4c) is also essential.

These two figures describe completely different pro-

cesses. For instance, the logistic growth-rate is symmetric,

and the point on which this occurs is where half of the

asymptote is reached, and the growth-rate is maximum.

Nonetheless, no biological reasons for the symmetry have

been presented. The asymmetric behavior of the other func-

tions seems more realistic. Both the Bertalanffy-Richards

and the Weibull behave quite similarly, reaching their max-

imum growth-rates at much smaller sizes. Finally, the

Monomolecular and the Johnson-Schumacher curves are

entirely unrealistic.

In general, the rules on which the growth equations are

based are rather broad and motivated more by mathematical

formulation than by a solid, established theory about the

biological processes of growth.

Concluding Remarks

We have shown that the mathematical differences among

several growth equations can be more easily explained

and understood using their differential equation than

their integrated forms. Inter alia, the assumed-and-claimed

biological basis of these growth-rate models has been

taken too seriously, and the focus should be on using a

plausible equation for the organism being modelled. More

attention should be put on parameter estimation strategies

and behavior analysis of the proposed models. It is difficult

for a single function to capture all possible shapes and rates

that such a complex biological process as tree growth can

depict in nature. Therefore, comparisons of some functional

forms (at least simple ones) must be carried out before

selecting a function for drawing scientific findings.

Funding This study was supported by the Chilean research grants

Fondecyt No. 1191816 and FONDEF No. ID19|10421, Academy of

Finland Flagship Programme (UNITE, decision number 337655), and

ANID BASAL FB210015.

Declarations

Conﬂict of Interest The authors declare that they have no conflict of

interest

.

Human and Animal Rights and Informed Consent This article does not

contain any studies with human or animal subjects performed by the

authors.

Curr Forestry Rep

Papers of particular interest, published recently, have been

highlighted as:

•Of importance

•• Of major importance

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Aﬃliations

Christian Salas-Eljatib1,2,3 ·Lauri Meht¨

atalo4·Timothy G. Gregoire5·Daniel P. Soto6·Rodrigo Vargas-Gaete7,8

1Centro de Modelaci´

on y Monitoreo de Ecosistemas, Universidad

Mayor, Santiago, Chile

2Vicerrector´

ıa de Investigaci´

on y Postgrado, Universidad de La

Frontera, Temuco, Chile

3Departamento de Silvicultura y Conservaci´

on de la Naturaleza,

Universidad de Chile, Santiago, Chile

4Natural Resources Institute Finland (Luke), Joensuu, Finland

5School of the Environment, Yale University,

New Haven, CT, USA

6Departamento de Ciencias Naturales y Tecnolog´

ıa, Universidad de

Ay s ´

en, Coyhaique, Chile

7Laboratorio de Biometr´

ıa, Departamento de Ciencias Forestales,

Universidad de La Frontera, Temuco, Chile

8Centro Nacional de Excelencia para la Industria de la Madera

(CENAMAD), Pontificia Universidad Cat´

olica de Chile,

Santiago, Chile

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