From trees to forests. A unified theory

(1)

From Trees to Forests

A Unified Theory

(2)

(3)

From Trees to Forests

A Unified Theory

A Personal Contribution to the International Year of

(4)

(5)

From Trees to Forests

A Unified Theory

Luís Soares Barreto

Jubilee Professor of Forestry, Technical University of Lisbon, Portugal

(6)

Version revised, and improved in January, 2012

From Trees to Forests. A Unified Theory

This book is a free translation of my e-book "Árvores e Arvoredos. Geometria e Dinâmica" that I disclosed in November, 2010. In some points I abridged the Portuguese text, in other issues, I enlarged the original version.

Clip art in the cover obtained from the software 225,000 Images Focus MultimediaTM

Prof. Doutor Luís Soares Barreto

Av. do Movimento das Forças Armadas, 41 – 3D 2825-372 Costa de Caparica

Portugal

With Compliments

This e-book is freeware but neither public domain nor open source. It can be copied and disseminated only in its totality, with respect for the authorship rights. It can not be sold.

(7)

To Luísa Maria,

who kindly helped me

in the diffusion of my

works by e-mail,

with tender gratitude

(8)

(9)

Luís Soares Barreto was borne in 1935, in Chinde, a small village in the

delta of the Zambezi River, in Mozambique. In this African country, from 1962 till 1974, he did research in forestry, and he was also member of the faculty of the Universidade de Lourenço Marques (actual Universidade Eduardo Mondlane, Maputo), where he started the teaching of Forestry. While member of this university, from 1967 to 1970, he was a graduate student at Duke University, Durham, North Carolina. From this university, he received his Master of Forestry in Forest Ecology (1968), and his Ph. D. in Operations Research applied to Forestry (1970). Since 1975 till March, 2005, he taught at the Instituto Superior de Agronomia, Universidade Técnica de Lisboa, where he was is professor “catedrático” (tenure). From 1975 to 1982, simultaneously, he taught in the Department of Environmental Sciences of the Universidade Nova de Lisboa. Here, in 1977, he conceived, and created a new five years degree in environmental engineering. For the last twenty five years, he worked in the establishment of a unified theory for self-thinned forest stands, and tools for the ecologically sound management of forest resources, grounded on it.

(10)

Published recently by the author

Conceitos e Modelos da Dinâmica de uma Coorte de Árvores. Aplicação ao Pinhal. “e-book”. 2ª edição. Instituto Superior de Agronomia, Lisboa, 2004.

Pinhais Bravos. Ecologia e Gestão. “e-book”. Instituto Superior de Agronomia, Lisboa, 2004. Theoretical Ecology. A Unified Approach. “e-book”. Costa de Caparica, 2005.

The Stochastic Dynamics of Self-Thinned-Pure Stands. A Simulative Quest. Silva Lusitana, 14(2):227-238, 2006.

The Changing Geometry of Self-Thinned Mixed Stands. A Simulative Quest. Silva Lusitana, 15(1):119-132, 2007.

The Reconciliation of r-K, and C-S-R Models for Life-History Strategies. Silva Lusitana, 16(1):97-103, 2008. O Algoritmo Barcor: Classificação de Cortiça para Rolhas Recorrendo a Quatro Atributos de Qualidade. Silva Lusitana, 16(2):207-227, 2008.

Growth, Regeneration, and Survival Indices for Tree Species. Silva Lusitana, 17(1):83-95, 2009.

Caracterização da Estrutura e Dinâmica das Populações de Lince Ibérico (Lynx pardinus). Uma Digressão Exploratória. Silva Lusitana, 17(2):193-209, 2009.

Simulação do Carbono Retido no Pinhal Bravo e da sua Acreção. Silva Lusitana, 18(1):47-58, 2010.

Simulator SB-IberiQu. Simulator for self-thinned even-aged forests of Quercus robur, written in Scilab. Setember 2010.

Árvores e Arvoredos. Geomatria e Dinâmica. E-book disclosed in November, 2010.

The Blended Geometry of Self-Thinned Uneven-Aged Mixed Stands. Silva Lusitana, 18(2):225-237, 2010. Modelling and Simulating Omnivory. Silva Lusitana, 19(1):67-83, 2011.

Submitted for publication

Gause’s Competition Experiments with Paramecium sps. Revisited. Submitted to Silva Lusitana, September 2007.

Plant Growth and Kleiber's Law. Submitted to Silva Lusitana, March 2009.

A Unified Theory for Self-Thinned Pure Stands. A Synoptic Presentation. Submitted to Silva Lusitana, May 2009.

A Unified Theory for Self-Thinned Mixed Stands. A Synoptic Presentation. Submitted to Silva Lusitana, May 2009.

The Simulation of Thinning in Mixed Even-Aged Stands. Submitted to Silva Lusitana, October 2009.

An Ecological Approach to the Management of Mixed Uneven-Aged Forests. Submitted to Silva Lusitana, October 2009.

Breve Revisitação do Algoritmo BARCOR. Submitted to Silva Lusitana, November 2009.

The Total Biomass of Self-Thinned Mixed Forests. A Theoretical and Simulative Inquiry. Submitted to Silva Lusitana, February 2010.

The Global Yield and Allometry of Self-Thinned Mixed Forests. Submitted to Silva Lusitana, June 2010.

(11)

“No theory, no science"

Mario Bunge. Philosophy of Science. From Problem to Theory. Volume I, page 437.

"I know that most men, including those at ease with problems of the highest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives."

L. Tolstoy

Quotation obtained from Bohmian Mechanics,* (Section 15), Stanford Encyclopaedia of Philosophy, in the internet.

*Sheldon Goldstein. Bohmian Mechanics. The Stanford Encyclopaedia of Philosophy (Spring 2009 Edition), Edward N. Zanta (Ed.) URL=< http//plato.stanford.edu/archives/spr2009/entries-qm-bohm/>

“We can put it down as one of the principles learned from the history of science that a theory is only overthrown by a better theory, never merely by contradictory facts”.

James B. Conant, 1958. On Understanding Science. Mentor Book, New York. Page 48.

(12)

(13)

Symbols, and acronyms frequently used

CAI current annual increment

dbh diameter at breath height

E = e(-c(t-t(0))), where c is a constant characteristic of a species fc form coefficient of the tree

f(i) represents the number of trees (frequency) of age classe i, in uneven-aged stands FOV forest variable(s)

Fs forest stands

Fw Wilson's index (relative density referred to dominant heigth). Fw=100/(density1/2 x dominant height), for square spacing

GZE the Gompertz equation MAI mean annual increment

MCAI maximum current annual increment MMAI maximum mean annual increment Mg Megagramme(= 106 grammes)

Ri = yi0/yif this value is constant for a given species

RMR-21t relative mortality rate at age t. RMR-21t =-c ln (R-2)E

rmr=relative mortality rate

Rm= ratio of the rmr of two species

RVRit relative variation rate of variable yijt, at age t. RVRit=-c ln (Ri) E

SMS self-thinned mixed stands or forest

SMUS self-thinned mixed uneven-aged stand or forest SPUS self-thinned pore uneven-aged stand or forest SMES self-thinned mixed even-aged stand or forest SPES self-thinned pure even-aged stand or forest SQ site quality

t0 age when the competition among trees starts to be dominant

Uf uneven-aged forests

yijt FOV with the power of the linear dimension equal to i, with order j, at age t. See tables 2.1 and

2.2

yijf final or asymptotic value of FOV yij

yijo value of FOV yij at age t0

yt value of a generic FOV, y, at age t

(19)

1 Introduction

1.1 The Scope of this Book

In this book, I will present a unified theory for trees and forests. Pure and mixed stands, either even-aged or uneven-aged will be considered. The theory is focused on self-thinned stands, when the dominant competition is among trees, although its application to thinning will be also presented.

This text is an abridged version of Barreto (2010a), written in Portuguese, and is accompanied by it. I expurgated the original version from its more didactical, and pedagogic aspects, particularly in chapter 1. With the software available today, the curious reader can use an application for translation to explore the included Portuguese version. On the other hand, I introduced three new sections (15.8 The Mean Coefficients of Intra, and Interspecific Competition; 16.5 Evaluating the Proposed Models; 19.7 The Simultaneous Effects of the Environment, and Competition).

It is admitted that it were attempts to solve many practical problems in agriculture, fisheries, and forestry that gave origin to many of the central ideas, concepts and methods, in ecology (Kingsland, 1985: 23). I must confess that, initially, the results here presented were rooted in practical problems, as they emerge in the Portuguese forestry. The following observations led me to enlarge the initial scope of my work:

 In forestry, the structure, and dynamics of tree populations are characterized by collections of

models, loose concepts, and not by a unified theory, with an encompassing, coherent, and unifying deductive structure.

 Today, simultaneously, forests are under the pressure of increasing demands of timber, other

products, and services. Only a robust theory for tree populations can underpin a sound forest management that guarantees the sustainability of this natural resource, in its complete plenitude.

 In my opinion, individual tree models (ITM) are a testimony of the poorness of forest science.

They are the answer to the non-existence of a good predictive model for tree competition (I will present one, ahead). In my humble opinion, mixed stands are reifications of interspecific competition, and I wonder if any study of mixed forests that ignores this basic fact is a little more than Natural History, with a variable degree of descriptive sophistication.

 The quantity and longevity of data required by ITM make them prohibitive for many countries,

in the next decades, as good forest inventories are expensive and require qualified professionals. I do not mention other eventual shortcomings that may occur, generally associated to very complex regressive models. Even in a book devoted to ITM, the necessity of new models to replace ITM is recognized (Schmidt, Nagel, Skovsgaard, 2006:157).

 ITM, as yield tables, are regressive models that add very little to the theory of forestry. In my

opinion, in terms of biological realism, comparing yield tables, ITM are less sophisticated as they ignore the important biological variable age.

 ITM evince a paradox. Ecological preoccupations led to an a-ecological approach to mixed

stands. This approach completely ignores the competitive ability of the species, competitive hierarchies, competition models, and life history strategies of the species (Barreto, 2010b).

(20)

 The time series of forest data do not escape to the visibility problem. We cannot separate the

influence of density-dependence, and of interspecific interactions from the noise of the physical environment, and isolate the crucial factors. For more information on this issue see Ranta, Lundberg, and Kaitala (2006:34-38, and references therein). In a time of fast climatic change, this subject cannot be ignored. On the other hand, there is a risk that when the gathering of the forest data is finished, the environmental conditions associate to the growth of the measured forests, as climate, no longer prevails.

 As there is not any theory or science of complexity (Mitchell, 2009:14) we must remain in the

firm ground of the existing sciences.

Given this setting, my aim is to locate the study of the structure, and dynamics of tree populations in the area of population ecology, and to use a hypothetical deductive approach. At the same time, this endeavour must utilize the data already available.

There is a parallel between my theory, and statistical mechanics. In a forest, I give up on determining the individual sizes, and growths of each tree (molecules of a gas) as ITM (classical mechanics) does, and instead I predict the averages sizes, and growths of the population of trees, as statistical mechanics does for the average velocity, and position of a large number of molecules.

To avoid misinterpretations and to provide an adequate context for the results here presented, a relevant issue must be clarified: the one related to the time frame and scale (Allen and Starr, 1982). My time frame is the natural longevity of a cohort of a given tree species, and the focus of my research is the tree population of self-thinned pure stands (SAPS).

It is my understanding that the results here presented, as a conceptual framework, can be generalized to other plant populations. Forests are particularly suitable to study the dynamics of plant populations. Trees and their components are easily individualized and measurable. They also evince a conspicuous picture of self-thinning, due to their fierce competition, and longevity (Zeide, 1991:518). Briefly, this is an inquiry on the process of self-thinning, and growth of tree populations, that I started in the eighties. Trees and stands will be approached in a continuum,

without gaps between their attached concepts, and models. The necessity to bridge the gap

between tree and stand modelling is already recognized by other authors (e.g., Garcia, 2001). As this book is a text about tree population dynamics it is also of the interest of population ecologists. For these readers, I clarify some terms of the area of forestry.

1.2 The Fundamental Assumptions

The elaborations here presented assume four levels of very basic assumptions. The first level comprehends the following philosophical hypothesis (Bunge, 2005: section 5.9):

 The world external to my subjectivity has a real, concrete existence.

 The reality has a multilevel structure, and it is not a homogeneous block. Each level has its

own properties and laws. The levels are not independent of each other, but are related.

 The external world it is not lawless, but evince ontological determinism.  The external world can be known.

 Logic and mathematics are autonomous formalisms.

The second level is illustrated in figure 1.1, and clarified by the following comments:

When life appeared, in the Earth, the reality in our planet was already submitted to physical and chemical laws. There is a continuum from physics to life, through chemistry. But each one of

(21)

these levels of the organization of matter exhibits his particular set of emergent properties and regularities. The existence of life is affected by ecophysiological, chemical, and physical constraints. Organisms live in space: two dimensional space (almost all terrestrial organisms), and three dimensional space (almost all aquatic organisms).

The growth of organisms and populations, are related to the physical variables of time and space. Ecology studies the effects of environmental factors (temperature, soil fertility, etc), but pays little attention to the basic physical relation of biology with space and time. This forgotten link can be introduced, in the discourse of ecology, through the metric concepts (variables, and constants) here used (figure 1.2). This attitude facilitates the use of dimension analysis, and allometry, as detailed in the next chapter.

Figure 1.1. Second conceptual space underpinning the nomological inquiry here developed

Figure 1.2. The basic triangle of the growth of trees, and populations in the physical space

On the third level I introduce the following methodological assumption: although, in nature, self-thinned forests are stochastic systems, the analysis of a deterministic archetype can bring valuable knowledge, and make a bridge for the stochastic approach, as I already illustrated and accomplished in Barreto (2006). Figure 1.3 represents this assertion. For my conception of the growth of SPES see Barreto (1989a:241; 2002).

Linear dimension of the forest variables Linear dimension of the physical space Eco-physiological and chemical restrictions Other physical restrictions

Emergent simplicity _{structure and dynamics}Laws about the

of forests

Time and space

Power of the linear dimension of tree

variables

Power of the linear dimension of population variables

(22)

The fourth level of conjectures implies a more refined conception about the approached phenomena, and the organization of the knowledge available for the purpose of establishing the desired hypothetical deductive system, as illustrated in figure 1.4.

My inquiry to the realm of tree populations accepts the following starting hypothesis:

 Organisms and populations abide allometric or power laws. They possess scalability.  Biological growth follows the Gompertz equation.

 For any tree species, there is time-space symmetry between its even-aged and

uneven-aged stands.

 The competitive ability of a tree species is determined by its growth rate (Grime's

hypothesis), and it reflects both the growth pattern, and the life-history strategy of the species.

Figure 1.3. The assumed deterministic simplification

Figure 1.4. Basic diagram of the broad conceptual structure of my theory for thinned pure stands (SPS), and self-thinned mixed stands (SMS)

1.3 The Book

In accordance with previous formalizations of my theory (Barreto, 2003, 2004c), I did my best to elaborate a presentation that abides the following:

1. Let its hypothetical-deductive structure be easily depicted. 2. Make clear the logical sequence of the subjects approached.

Deterministic conceptual system (theory) Trees of the biosystem forest Stochastic reconciliation Deterministic forest archetype Deterministic filter Allometry The Gompertz equation Time-space symmetry between

even and uneven-aged forest stands

Theory for SPS Theory for SMS

Competitive ability of species determined by the relative

(23)

3. Thus, obtain a text that makes conspicuous the rationalizations developed, to facilitate the control of the internal coherence of the theory, by the reader.

T can symbolically represent my hypothetic-deductive theory, T=D, A, L.

Set D: it contains definitions, forest variables, and conventions, mainly the propositions that establish the object-concept relation. Basically, it states the object of the enquiry, and introduces its descriptive and operational conceptualisation.

Set A: it includes the basic assumptions. They behave as axioms but they are not postulates in the classical sense, because some of them can be proved. This set encompasses the minimum group of conjectures, or hypotheses, I must formulate about system s, in order to get more insight about it.

Set L: it embraces the corollaries and theorems or laws, deduced from D, and A. It is the nomological part of T.

I consider the elements of L (except corollaries) law statements because: a) they apply to all tree species and Fs; b) they are integrated in a hypothetic-deductive system; c) they are being satisfactorily confirmed (Mahner and Bunge, 1997, Definition 3.9). A and L can be seen as the syntactic of T.

For the benefit of clarity, and systematization, I will approach first pure forests (Pfs; Part I), and after mixed forests (Mfs; Part II). This is, as any text of ecology I deal first with isolated populations, and next with population interactions. Finally, I introduce Part III dedicated to the evaluation of the theory.

For the sake of completeness, I write: TFs=TPfs, TMfs

TPfs=Dp, Ap, Lp

TMfs=Dm, Am, Lm

I start Part I with the presentation of the basic concepts, definitions and symbolization for the forest variables (chapter 2). Chapter 3 approaches allometry. The Gompertz equation is deduced, and analyzed in chapter 4. In the next chapter, I attempt to sustain that biological growth evinces time self-similarity, and I show that the growth of self-thinned pure even-aged stands (SPES) support this conjecture. The great extension of set L for pure stands is contained in chapter 6. The time-space symmetry between SPES and SPUS is explored in chapter 7. Chapters 8 to 12 are applications and extensions of the previous results to the clarification of several problems: the pattern of growth of tree species, the life history strategy of tree species, the effects of thinning, tree growth end the Kleiber´s law. They evince the fecundity of the theory. The results of chapters 8 and 9 are necessary for a more ecological sustained approach to mixed stands. Chapter 13 is devoted to the stochasticity of SPES.

I begin Part II with a chapter (14) devoted to a conceptualization of tree competition and the establishment of the associated mathematical model BACO2. As this model underpins all subsequent elaborations, chapter 15 is dedicated to test the truthlikeness of the model. Models derived from BACO2 are deduced in chapter 16. In chapter 17, I introduce a tentative typification of the patterns of tree interaction. The changing geometry of mixed stands is analyzed in chapter 18. Self-thinned mixed uneven-aged forests are approached in chapter 19. The thinning of mixed stands occupies chapter 20.

(24)

There are professional indexers utilized by publishing companies (e. g., Haugland e Jones: 2003:335). As a good analytical index can strongly ameliorate the quality of a book, a bad one can be an unencouraging and frustrating experience. I assume that the reader will use a computer or other adequate device to read this book, with software to find the words he wants to locate. Thus I spare the reader to the experience of an index made by a no specialist. Within this assumption, I used a font easily red in the screen - calibri.

1.4 Related Works

Introductory texts to the present book are Barreto (1994, 1995 e 2004a). Detailed applications of the theory here presented to forests of Pinus pinea, and of Pinus pinaster (Ppi) can be found in Barreto (2000; 2004b).

Based on my theory, I already disclosed several simulators for pure and mixed forest, both even, and uneven-aged, of European, North-American, and North-African species (Cedrus atlantica) written in Basic, Excel, Scilab, and Visual Basic 6.

1.5 References

Allen, T. F. H., and T. B. Starr, 1982. Hierarchy. Perspectives for Ecological Complexity. The University of Chicago Press, Chicago.

Barreto, L. S., 1989. The 3/2 power law: a comment on the specific constancy of K. Ecological Modeling, 45:237-242. Barreto, L. S., 1994. Alto Fuste Regular. Instrumentos para a sua Gestão. Publicações Ciência e Vida, Lda., Lisboa.

Barreto, L. S., 1995. Povoamentos Jardinados. Instrumentos para a sua Gestão. Publicações Ciência e Vida, Lda., Lisboa

Barreto, L. S., 2000. Pinhais Mansos. Ecologia e Gestão. Estação Florestal Nacional, Lisboa.

Barreto, L. S., 2003. A Unified Theory for Self-Thinned Pure Stands. A Synoptic Presentation. Research Paper SB-03/03. Departamento de Engenharia Florestal, Instituto Superior de Agronomia, Tapada da Ajuda, Lisboa. Revised version submited to Silva Lusitana, May, 2009.

Barreto, L. S., 2004a. Conceitos e Modelos da Dinâmica de uma Coorte de Árvores. Aplicação ao Pinhal. Second edition revised, and enlarged. Departamento de Engenharia Florestal, Instituto Superior de Agronomia, Tapada da Ajuda, Lisboa.

Barreto, L. S., 2004b. Pinhais Bravos. Ecologia e Gestão.E-book. Departamento de Engenharia Florestal, Instituto Superior de Agronomia, Tapada da Ajuda, Lisboa.

A table with the descriptions of the Scilab scripts presented in Barreto, (2010a) is inserted in the Appendix. I translated the comments and the output inserted in the scripts, and fuctions to English. These scripts and functions are in folder T2FScilab. The programs are grouped by chapters (Cap=Chap.), and named by the box where they are inserted in the text. For instance, to access the program SMUSEstrut that generates the structure, and the parameters of the dynamics of a self-thinned mixed uneven-aged stand with two species, the path is T2FScilab \Cap19\Caixa192.

(25)

Barreto, L. S., 2004c. A Unified Theory for Self-Thinned Mixed Stands. A Synoptic Presentation. Research Paper SB-02/04. Departamento de Engenharia Florestal, Instituto Superior de Agronomia, Tapada da Ajuda, Lisboa. Revised version submited to Silva Lusitana, May, 2009.

Barreto, L. S., 2005. Theoretical Ecology. A Unified Approach. E-book. Costa de Caparica, Portugal.

Barreto, L. S., 2006. The Stochastic Dynamics of Self-Thinned-Pure Stands. A Simulative Quest. Silva Lusitana, 14(2):227-238.

Barreto, L. S., 2010a. Árvores e Arvoredos. Geometria e Dinâmica. E-book Costa de Caparica, Portugal. Barreto, L. S., 2010b. Book review, Hasenauer (2006b). Silva Lusitana, 18(1):115-120.

Garcia, O., 2001. On bridging the gap between tree-level and stand-level models. In: Rennolls, K. (ed.) Proceedings of IUFRO 4.11 Conference 'Forest Biometry, Modelling and Information Science', University of Greenwich, June 25-29, 2001. http://cms1.gre.ac.uk/conferences/iufro/proceedings/ .

Hasenauer, H. (Ed.), 2006b. Sustainable Forest Management. Growth Models for Europe. Springer, Berlin.

Haugland, S. e F. Jones, 2003. StarOffice 6.0. Office Suite Companion. Prentice Hall, Upper Saddle River, New Jersey. Kingsland, S. E., 1985. Modelling Nature. Episodes in the History of Population Ecology. The University of Chicago Press, Chicago.

Mahner, M., and M. Bunge, 1997. Foundations of Biophilosophy. Springer, Berlin. Mitchell, M., 2009. Complexity: A Guided Tour. Oxford University Press, New York.

Ranta, E., P. Lundberg, and V. Kaitala, 2006. Ecology of Populations. Cambridge University Press.

Schmidt, Nagel, Skovsgaard, 2006. Evaluating Individual Tree Growth Mdels. In H. Hasenauer, (Ed.). Sustainable Forest

Management. Growth Models for Europe. Springer, Berlin. Pags. 151-163.

Zeide, B., 1991. Self-tinning and stand density. Forest Science, 37(2): 517-523.

Appendix

Description of the Scilab scripts available in folder T2FScilab

Caixa (=Box) Description of the Scilab script

Caixa 4.1. Generates simulated data, and fits the 3/2 poer law

Caixa 4.2. Generates data, and fits the allometric equation y2,661=k2,666-2 y-21-1,333

Caixa 4.3. Project a SPES of Pinus pinaster

Caixa 4.4. Fits the Gompertz equation using the least squares method

Caixa 4.5. Uses the method of Khilmi to fit the GZE

Caixa 4.6. Applys algorithm SBFASTG

Caixa 5.1. Applies model KRONOKHABA

Caixa 6.2. Simulates three types of thinning

Caixa 6.3. Calculates the form coefficient

Caixa 6.4. Projects the dimensional structure of a SPES of Pinus pinaster

Caixa 6.5. Establishes the structural entropy of a SPES of Pinus pinaster

Caixa 6.6. Calculates mean annual increments

(26)

Caixa 7.1. Generates the structure of a pure self-thinned uneven-aged stand of Pinus pinaster

Caixa 7.2. Calculates the parameters of the dynamics of a pure self-thinned uneven-aged stand of Pinus pinaster

Caixa 7.4. Establishes the eigenvalues of a matrix of Leslie, the sensitivity, and eslasticity matrices of the dominant eigenvalue, the stable age distribution, and the reproductive values of the classes

Caixa 7.5. Uses the matrix of the previous program to project a SPUS of Pinus pinaster, every five years Caixa 7.6. Project a SPUS Pinus pinaster with trimester values

Caixa 8.1. Calculates the growth, regeneration, and survival indices of several species Caixa 9.1. Calculates the relative proximity of a set of species to a K archetype

Caixa 10.1. Applies method SB-BARTHIN to the simulation of a neutral thinning at age 20, and ti=20%, to a stand of Pinus

pinaster

Caixa 10.2. Applies the algorithm SB-ALODESB, that uses allometry to predict the effects of thinning Caixa 12.1. Generates a dimensional structured SPES of Pinus pinaster

Caixa 13.1 Simultes the stochastic, and cyclic dynamics of the total biomass of a SPES of Pinus pinaster Caixa 13.2. Simulates the dynamics of a SPUS, with cyclic regeneration

Caixa 13.3. Stochastic simulation of tree volume, and stand density to verify the 3/2 power law Caixa 13.4. Simulation of white, red, and blue noise

Caixa 13.5. Stochastic simulation of a SPUS of Pinus pinaster

Caixa 14.1. Simulates a self-thinned mixed even-aged stand with Quercus robur, and Pinus pinaster Caixa 15.1. Simulates the deterministic buterfly effect in a SMES with Picea sitchensis, and P. menziesii Caixa 15.2. Simulates the total biomass of a SMES with Fagus sylvatica, and Picea abies

Caixa 15.3. Simulates the total biomass of a SMES with Larix decidua, and Picea abies, and the homologous population in pure stands

Caixa 16.1. Calculate the annual values of the coefficients of competition in a SMES with Larix decidua, and Picea abies Caixa 16.2. Calculates the equilibrium values, and their stability in an application of model BACO3

Caixa 18.1. Simulates the effects of the relative size of the trees, and the proportions of the competitors upon the total yield, in order to depict overyielding

Caixa 18.2 Calculates the yield table for a SMES of Quercus robur, and Pinus pinaster Caixa 19.1 Projects a SMUS of Quercus robur, and Pinus pinaster

Caixa 19.2 Projects a SMUS with two species, and and establishes the Leslie's matrices of each species Caixa 19.3 Uses the output of the previous simulator to create a program to project their SMUS Caixa 19.4 Establishes the projection matrices of SMUS, with dimensional structures in the age classes Caixa 19.5 Stochastic simulation of a SMUS with Quercus robur, and Fraxinus excelsior

Caixa 19.6 Stochastic simulation of a SMUS with Alnus rubra, Picea sitchensis, and P. menziesii Caixa 20.3 Simulates a neutral thinning in a SMES

(27)

PART I

PURE FORESTS

Dp



Definitions, and Basic Concepts



Allometry: The Universal Mathematical Laws of Life

Ap



The Gompertz Equation: The Pattern of Biological Growth

Lp



The Time Self-Similarity of Biological Growth



The Laws that Rule the Structure, and Dynamics of SPES



Self-Thinned Pure Uneven-Aged Stands



Growth, Regeneration, and Survival Indices



Life-History Strategies of Tree Species



The Analysis of Thinning



Plant Growth and Kleber’s Law



Application

(28)

(29)

2 Definitions, and Basic Concepts

2.1 Remembering Preliminary Definitions

In this chapter I introduce a notation for forest variables (FOV), and I recall some other basic concepts that are used along the book. For a more comfortable reading, other necessary concepts will be explained in the best opportunity, ahead.

Self-thinned pure forest can be of two types: even-aged (SPES)theoreticallywhen the trees have the same ageand uneven-aged theoretically when the trees evince all possible ages (SPUS).

The better or worst favourableness of the environment of a particular location to the growth of the trees of a given species is called site quality (SQ). In this text the definition of SQ used is the mean height of the 200 trees of the stand with largest diameter at breath height (dbh), at age 40, called dominant height.

From here on, I assume that the reader has some basic knowledge of dimensional analysis.

2.2 The Phases of the Growth of SPES

Under the perspective of competition, the life of a SPES goes through three phases. In phase I, there are intraspecific competition between the small trees, and interspecific competition with the other vegetation...

In phase II (FII), when the trees became taller with age, the trees compete dominantly among them. This intraspecific competition for space and resources (mainly light, nutrients, and water) is responsible for the changes that occur in the stand: decreasing of the number of trees (self-thinning), growth of the trees that remain, standing volume and biomass increasing. My theory is particularly devoted to this phase II.

When the trees attain maturity, the growth becomes residual, and the same happens to intraspecific competition. This is phase III.

More detailed analysis of the growth of SPES can be found in forest literature, but this one properly suits my purpose (e.g., Oliver, and Larson, 1990)

Some tree species evince different growth strategies in phases I, and II as Pinus palustris and Pinus merkusii.

2.3 Forest Variables

My characterization of the FOV is grounded in the assumptions of the second level (section 1.2). Here, the dominant feature of a FOV is the power of the linear dimension (EL) associated to each one. This attitude facilitates the use of dimension analysis, and allometry.

The notation for the FOV is as follows:

(30)

i is the power of the linear dimension associated to the variable.

j is the identifier of the variable.

t is age, as usual. When dispensable, t can be omitted. The FOV here considered are described in table 2.1.

Table 2.1. The description of the variables of trees, and a population of trees A. Tree variables

i=1 i=2 i=2,6666 i=3

j Variable name j Variable name j Variable name j Variable name

1 2 2d Dbh Height Dominant height 1 2 3 4 5 6 7 Leaf biomass Biomass of live branches Biomass of dead branches Total crown biomass Leaf area Basal area Area occupied by a tree

1 Total tree biomass 1 2 3 4 5 Stem volume Biomass of the stem wood Biomass of the stem bark

Total stem biomass Total root biomass

Table 2.1. Continuation B. Population variables

As the reader can verify, the value of i of a variable referred to stand is equal to the homologous referred to the tree minus 2:

istand = itree-2, as supported by dimensional analysis.

The value of PL for root biomass probably needs a confirmation or eventual improvement.

i=-2 i=0 i=0,6666 i=1

j Variable name j Variable name j Variable name j Variable name

1 Trees ha-1 1 2 3 4 5 6 Leaf biomass ha-1 Biomass of live branches ha-1 Biomass of dead branches ha-1 Total crown biomass ha-1 Leaf area ha-1 Basal area ha-1

1 Total tree biomass ha-1 3 4 5 6 7 8 Stem volume ha-1 Biomass of the stem wood ha-1 Biomass of the stem bark ha-1 Total stem biomass ha-1

Total root biomass ha-1

(31)

At the beginning of phase II, I write yi,j, 0. The final or asymptotic value of a FOV is written as yi, j,f.

For SQ, the following equation holds

SQ=y12d40 (2.1)

Let Y be the set of FOV, encompassing two subsets: the variables of the trees (t) and stands (s). We write so Y= [YT, ys]. Each subset of these is in turn decomposed into subsets of variables with the same dimension, which are the variables of each column of table 2.1. For instance, There are two sets with i = 1: Yt1 = [y11, y12, y12d] and Ys1 = [y13, y14, y15, y16, y17, y18, y19,].

The fractional values of i (0.6666, 2.6666) allow us to assume that the trees and forest have fractal geometry. This finding sustains the conjecture that SPES evince self similarity, and structures formed by modules. The modules are defined as the smallest set of trees that abides all laws related to the dynamics of SPES. Generally, for populations of plants, I assume that the final density of a module is one plant, in our case one tree. I propose the following FOV to describe the modules:

y-22t refers to the number of trees per module area in SPES. y-23t refers to the number of trees per module area in SPUS. y07t refers to the number of modules per hectare in the SPES. y08t refers to the number of modules per hectare in SPUS. y28t refers to the area of the module in the SPES.

y29t refers to the area of the module in SPUS. Obviously the FOV are positive real numbers.

The FOV are here expressed in the customary units used in forestry. Heights and tree spacing are referred as meters. Dbh measured in cm. Tree leaf area, basal area, and occupied space referred as m2. The unit for volume is the m3. The biomasses of the tree are measured in kg. The biomasses of the stand are referred to Mg. Ages are referred to the year. The stand area is referred to the hectare.

Two questions can be asked here now:

Question 1. Does a general pattern exist for the growth of FOV in pure stands?

Question 2. How are the dynamics of the various FOV interrelated during the growth of forests? The answers to these questions are my theory to the pure forests, whose summarized presentation occupies part I of the book

2.4 Complementary Definitions

Measures of relative spacing (RS) are used to compare different stands. A generic index can be defined as follows (e.g., Cluter et al., 1983):

(32)

Generally, the FOV in the denominator has i=1 or i=3. A particular case of eq. (2.2) is the Wilson´s index. When referred to the hectare, with square spacing, being N the number of trees, is written as:

=

√ (2.3)

Let L be the linear dimension, being N adimensional. As it is verified [Fw] =L1 L-1=L0, Fw is constant during the life of a SPES.

Let us consider two SPES, with the same SQ, one with square spacing (sq) and the other triangular spacing (TR). The following relationships are verified:

Case I: the same density (y-21tr = y-21sq).

Y18tr=1.240806 y18sq (2.4)

Fwtr=1.240806 Fwsq (2.5)

b) Case II: the same tree spacing (y18tr = y18sq).

y-21tr=1.53964 y-21sq (2.6)

Fwtr=0.80591 Fwsq (2.7)

Generally, it is verified 0<Fw<1, and the smaller the density, the larger is Fw.

In eq. (2.2), when the FOV in the denominator of eq. (2.2) has i=3, we find [RS] =L-2, and the index decreases.

Now, I introduce the index of performance (s) of a pure stand(Barreto, 1995). The growth of a stand depends on the available resources, measured by SQ (y12d40), and the number of trees to take advantage of them. To estimate the occupancy of the space I will use Fw. This index is independent of both age, and SQ.

I define s as:

s = Fw y12d40 (2.8)

As in the right-hand of this equation the entities are constant, s is equal to a constant fraction of the dominant height at age 40, this is, s< y12d40.

We already saw that s is constant. At age 40 we find:

1840 40 12 40 12 1840 _y _y y y s d d   (2.9)

This is s is equal to tree spacing at age 40. The relevance of tree spacing in a forest is stressed.

(33)

y-2110=104 R-21-E40 s-2=G s-2 (2.10) where E40=exp(-c(40-t0)). As I assume t0=10 years, E40=e-30c. The density given by this equation, at age 10, is referred to the hectare. Tabulated values of G, for several species, can be found in Barreto (1995: table 1). For instance, for Quercus robur G=306404, and for Pseudotsuga menziesii G=270935. In Barreto (1995:44-46), I introduced scaling factors for the same FOV in two different stands. These scaling factors are relevant when applying yield tables to real stands.

2.5 References

Barreto, L. S., 1995.The fractal nature of the geometry of self-thinned pure stands. Silva Lusitana, 3(1):37-52.

Barreto, L. S., 2003. A Unified Theory for Self-Thinned Pure Stands. A Synoptic Presentation. Research Paper SB-03/03. Departamento de Engenharia Florestal, Instituto Superior de Agronomia, Tapada da Ajuda, Lisboa. Revised version submited to Silva Lusitana, May, 2009.

Cluter, J. L., J. C. Fortson, L. V. Pienaar, G. H. Brister, R. L. Bailey, 1983. Timber Management. A Quantitative Approach. John Wiley & Sons, New York.

(34)

(35)

3 Allometry: The Universal Mathematical Laws of Life

3.1. Introduction

Allometry is the name for the use of power laws in biology. It reflects the process of harmonization of body growth and functions, required during the life of the organism and its population. Allometric development assures the self-similarity of an organism (or population) during its life.

D’Arcy Thompson (Thompson, 1917) used the principle of geometric similarity to introduce allometry. The acceptance of this principle is justified by the general high incompressibility of organisms that have a mass per unit volume close to seawater. In this particular situation, body mass can be rescaled according to body mass, and related quantities. Other criteria of similarity can be used, such as mechanical, thermic hydrodynamic, or kinematic.

For many years, allometry was considered a numerical curiosity with poor theoretical value, and incipient explanatory power. Today, it is admitted that allometric relationships are fruitful tools to expand, and unify the knowledge, in the areas of biology, and ecology. I hope this book contributes to the illustration of this statement.

Allometric relationships (West, Brown, e Enquist, 2000:91):

 Evince systematic simple mathematical laws that emerge from the complexity of life.

 They are the few available mathematical and universal laws related to life, because they are supported by the existence of a set of principles indispensable to all forms of life.

 The clarification and interpretation of allometric equations open a new and rich field for research, and unification of knowledge in biology, and ecology.

For more insight in this subject see Brown e West (2000), Schneider (1994). Niklas (1994), approaches plant allometry, and theoretical related issues.

To close this section let me introduce a quotation from a book written by the eminent academic Gregory I. Barenblatt (Professor-in-Residence at the University of California at Berkeley, and Lawrence Berkeley National Laboratory, Emeritus G. I. Taylor Professor of Fluid Mechanics at the University of Cambridge, Adviser, Institute of Oceanology, Russian Academy of Science, Honorary Fellow, Gonville and Caius College, Cambridge):

“One may ask, why is that scaling laws are of such distinguished importance? The answer is that scaling laws never appear by accident. They always manifest a property of a phenomenon of basic importance, “self-similar” intermediate asymptotic behaviour: the phenomenon, so to speak, repeats itself on changing scales. This behaviour should be discovered if it exists, and its absence should also be recognized. The discovered of scaling laws very often allows an increase, sometimes even a drastic change, in the understanding of not only a single phenomenon but a wide branch of science. The history of science of the last two centuries knows many such examples”.

(36)

3.2. The Allometric Equation

The simplest allometric equation has the form:

yajt = β0 ybhβ1 (3.1)

β0 is a constant.

Elementary dimensional analysis shows the following relationship:

β1=a/b (3.2)

Eq. (3.1) can be logarithmized for fitting, and graphical purposes.

3.3 A Few Illustrations

The way I defined my variables, the estimation of the theoretical value of β1 for any pair of variables is trivial, using eq. (3.2)..

Let me mention some cases that are more often discussed in the literature. Two frequently approached equations are the following ones:

y-3 = β0 y2.7-1.125 (3.3)

y-2 = β0 y2.7-0.75 (3.4)

Eq. (3.4) is familiar to students of ecology. These two equations suggest that the correct method to fit the same allometric relationships is to follow the dynamics of an isolated cohort.

Some authors analyse eq. (3.4) under the following form:

y2.7= β0 y-2-1.333 (3.5)

Also, it is verified:

y5.7 = β0 y2.72.125 (3.6)

y4.7 = β0 y2.71.75 (3.7)

A particular case is the following one:

ya= β0 y1a (3.8)

(37)

y_{2 . 7} y₁₁2 . 7 _(3.9)

2 11

2 y

y  _(3.10)

Eqs. (3.9), and (3.10) illustrate a physical constriction to biology, as they can be obtained from engineering first principles, as pointed out by Niklas, and Enquist (2001:2926). These equations sustain the correctness of my definitions of the FOV.

Finally, the 3/2 power law (figure 1):

y3= β0 y-2-1.5 (3.11)

The absolute value of the power greater then 1 (3/2=1, 5>1) guarantees that the standing volume grows with age, because the growth of the trees overcompensates the losses of standing volume caused by self-thinning. If β1=1 the standing volume would be constant; β1<1 implies decreasing standing volume.

Figure 3.1. The continuous line represents the logarithmic form of the 3/2 power law. During its growth, the SPES is not always in this line, but occupies position in the space between the continuous line and the line with points (amplitude). The SPES moves from right to left, this is form larger density, and smaller trees to lower densities, and larger tree.

Allometric equations show that the growths of the tree and the stand follow patterns that are interrelated, and harmonized. We formulated an affirmative answer to Question II (section 2.3).

Sprugel (1984) in SPES of Abies balsamea observed the constancy of the basal area, and referring to the allometric equation yij= β0 y-21 β

1he found β1=-1.04 for the biomass of the leaves of the mean tree (theoretically β1=-1). This finding sustains i=2 for the tree crown. For the trunk

ln m3 tree-1

Phase III Phase II ln tree ha-1 Time

(38)

biomass he found β1=-1.43, and for all above ground biomass of the tree β1=-1.24. These values do not led to the rejection of table 2.1. Osawa e Allen (1993) in SPES of Pinus densiflora, and Nothofagus solandri verified the constancy of the biomass of the (i=0).

In Davies and Johnson (1987:57) there is empirical evidence that sustains the basal area constancy. Also, Waring and Schlesinger (1985: 55) mention empirical for the leaf area index. The close of the crowns, at the beginning of phase II, is consistent with the constancy of the stand crown biomass, and the leaf area index.

As I will illustrate ahead, interspecific competition can modify the population allometry (Barreto, 2007).

3.4 Geometric Similitude

Now, I will show that SPES evince geometric similitude. Let me retrieve eq. (2.3). Generically, it can be written:

y18= Fw y12d (3.12)

This equation can be interpreted as “tree spacing measures Fw dominant heights". It is equivalent to the statement dominant height measures 24 meters.

The new metric unit y12d changes with time – increase with age. But, concomitantly, the variable y18 grows in a way that allows the constancy of Fw. During the growth of a SPES, the tree spacing measure with metric dominant height is invariant. The geometric similitude is maintained during the stand growth.

I can write:

yaj= kajh yah (3.13)

This equation is isometric as the two FOV have the same power of the linear dimension (a), and thus β1=1. Using this notation, the isometric equation related to Fw is written as:

d d y y k 12 18 182  (3.14)

In eq. (3.1), let make ybhβ1=z. It can be written now with the form of eq. (3.13):

z

yaj 0 (3.15)

During the growth, the size of yaj is constant when measured with the variable metric z. This constancy reflects the geometric similitude, relatively to the two variables. β1 is the scale factor.

These findings can be summarized. Given the two sets of FOV, Yt and Ys (section 2.3),

consider their subsets Yi. There is a binary isometric relationship between the elements of each Yi.

(39)

them. The allometric equations inside each set (YT or Ys) guarantee the geometric similitude of the

tree, and of the stand, along their growths. The allometric binary relationships between elements of the two same sets, Yt and Ys sustain the intimate relation between the dynamics of the tree, and

of the stand. The consequences of these allometric relationships are the nomological structure of the dynamics of both trees, and stands (chapters 6 e 7). Let us elaborate a verification.

Consider two ages T1 and T2. At age T1 variables yajT1 e ybjT1 were measured. At age T2 variable ybjT2 was measured. The allometric equation let us write:

b a bjT bjT ajT ajT y y y y ) ( 1 2 1 2  _(3.16) then: 1 1 2 2 ( ) ajT b a bjT bjT ajT y y y y  _(3.17)

Assume a deterministic SPES, and let its density be ybjT1, in eq. (3.17). At age T1 we measure the density, and all variable generically named yajT1, of our interest. At any other age of the SPES we only have to count the number of trees, to evaluate the new values of the FOV previously measured, as the following equation can be used:

1 2 1 2 2 2 2 ( ) ajT a jT jT ajT y y y y     _(3.19)

The relation between the structure of a SPES at an age T>t0, and its structure at age t0, is the same that exists in engineering between a model and its prototype. The rule to scale the results of experiments with a model to a prototype is an extension of eq. (3.19) as explained in Barenblatt (2003:38-39).

SPES are geometrically similar in time (chapter 6), and SPUS are in space (chapter 7). Mixed forests do not evince self-similarity or isomorphism (Barreto, 2007). These properties of the self-thinned pure stands are very important, and can change the way we simulate, and manage them.

More information, at an introductory level, can be found in Schneider (1994:chaps. 13 and14). Scaling is a recurrent topic in papers published in the most prestigious journals in the area of ecology, and biology.

Recently, Ritchie (2010) uses a concept close to module (section 2.3), named cluster, and allometry to elaborate an integrated approach to the heterogeneity, structure, organization, and biological diversity of ecological communities.

3.5 References

Barenblatt, G. I., 2003. Scaling. Cambridge University Press, Cambridge.

Barreto, L. S., 2007. The Changing Geometry of Self-Thinned Mixed Stands. A Simulative Quest. Silva Lusitana, 15(1):119-132.

(40)

Brown, J. H., West, G. B. Edits., 2000. Scaling in Biology. Oxford University Press. Davis, L. S. and Johnson, K. N., 1987. Forest Management. McGraw-Hill, New York.

Ginzburg, L. e M. Colyvan, 2004. Ecological Orbits. How Planets Move and Populations Grow. Oxford University Press, Oxford.

Niklas, K. j., 1994. Plant Allometry. The Scaling of Form and Process. The University of Chicago Press.

Niklas K. J., B. J. Enquist, (2001) Invariant scaling relationships for interspecific plant biomass production rates and body size. Proc Nat Acad Sci U S A 98: 2922–2927.

Osawa, A. e R. B. Allen, 1993. Allometric Theory Explains Self-thinning Relationships of Montain Beech and Red Pine.

Ecology, 74(4):1020-1032.

Ritchie, M. E., 2010. Scale, Heterogeneity, and the Structure and the Diversity of Ecological Communities. Princeton University Press.

Sprugel, D. G., 1984. Density, Biomass Productivity, and Nutrient Cycling Changes During Stand Development in Wave Regenerated Balsam Fir Forests. Ecological Monographs, 54(2):165-186.

Thompson, D. A. W., 1994. Forme et Croissance. Éditions du Seuil/Éditions du CNRS. Schneider, D. C., 1994. Quantitative Ecology. Spatial and Temporal Scaling. Academic Press.

Waring, R. H. and Schlesinger, W. H., 1985. Forest Ecosystems. Concepts and Management. Academic Press, Orlando. West, G. B., J. H Brown, and B. J. Enquist, 2000. The Origin of Universal Scaling Laws in Biology. In J. H. Brown, and G. B West,. Edits., 2000. Scaling in Biology. Oxford University Press. Pages 87-112.

(41)

4 The Gompertz Equation: The Pattern of Biological Growth

4.1 Introduction

The organisms that today exist are the result of a long evolutionary process. All species are submitted to the same laws of nature, physics, and chemistry. These facts led me to the following conjectures:

1. The pattern of growth is the same for all organisms and populations. 2. This pattern is the Gompertz equation (GZE).

These conjectures revealed fecundity in the resolution of problems in the area of theoretical ecology (Barreto, 2005), and the dynamics of pure and mixed stands, as it will be illustrated ahead. My choice is justified by the following reasons:

1. Medawar (1940) deduced theoretically that biological growth should follow the GZE.

2. Several authors (e.g., Caution ad Venus, 1981; Zullinger et al., 1984;Karkach, 2006) found that the GZE is the most adequate equation in the field of biology.

In the text ahead, I will attempt to illustrate the following characteristics of the GZE: 1. The GZE is enough flexible to fit several patterns of empirical data.

2. The GZE is consistent with allometric growth.

3. The same equation generates patterns of growth that reflects the species life-history strategy.

4. The GZE allows a correct study of population interactions.

5. The GZE gives consistent results at different time scales, for the same organism or population.

6. The Royal Society, in 2002, dedicated an issue of its Philosophical Transactions to a sole subject – population growth rates. In the introduction to this issue, it can be red “The thesis put forward in this issue of Philosophical Transactions is that population growth rate is the key unifying variable linking the various facets in population ecology.” (Sibly, Hone, Clutton-Brock, 2002:1149). This unifying function implies the existence of one basic pattern for biological growth, evincing some versatility. The Gompertz equation satisfies these requirements, as it will be seen in chapters 8, and 9.

An advantage of the use of the GZE is the possibility of articulation of the tree growth and the next level, this is, the tree population dynamics, as I will show in this book. The GZE allow the establishment of a predictive and adequate model for interspecific competition. This happens because the GZE can be deduced from the perspective of the organism growth (aging conjecture), and the population growth (density-dependence hypothesis), as it will be displayed in the next section.

My first attempts to use GZE as a unique model for the FOV are Barreto (1987, 1990). Other authors also pledge for the detection of a single pattern of growth for FOV (e. g., Zeide, 1993).

4.2 The Density-dependence Perspective

Let us assume that the growth of y is proportional to the difference ln yf-ln y, being yf the final or asymptotic value of the variable. This leads to the following equation:

(42)

) ln (lny y cy dt dy f   (4.1)

This equation can be written as:

cdt y y y dy f  ) / ln( (4.2)

The integrating both members of eq. (4.2) give:

1 ) ln(ln ct C y y_f     (4.3)

Being C1 a constant of integration. Thus,

ct f e C y y   ₂ ) ln( (4.4) ct e C f e y y   2 (4.5) ct e C fe y y  2 

(4.6)

When t=0 it is found C2=ln(y0/yf). Let me write R=y0/yf., to obtain the GZE:

y= yf Rexp(-ct) (4.7)

where exp refers to the exponential function.

For a more detailed way to establish eq. (4.1), see the Appendix at the end of this chapter.

4.3 The Aging Perspective

Let assume that the resources do not constrain the growth of an organism, and it is proportional to the size of the population. This assumption leads to the differential equation of the exponential growth:

ry dt dy

 (4.8)

Now, it is postulated that the growth constant r decays with time according to the equation (c>0):

(43)

cr dt dr



 (4.9)

With initial condition (t=0) of being r=r0, the solution of eq. (4.9) is:

r=r0 exp(-ct) (4.10)

Replacing eq. (4.10) into eq. (4.8) gives:

y e r dy dy ct  ₀ (4.11)

This equation can be easily integrated writing:



yf 



 y t _ct dt e r y dy 0 0 0 (4.12) Obtaining ) . 1 ( ) ln( 0 0 ct e c r y y    (4.13) that leads to

y=y0 exp(r0/c (1-exp(-ct))) (4.14)

Coherently, the limit of eq. (4.14) when c→0, is an exponential equation:

y= y0 exp(r0t) (4.15)

Eq. (4.12), when t→∞, gives:

yf=y0 exp(r0/c) (4.16)

Replacing eq. (4.16) in eq. (4.14) it is obtained an alternative form to GZE:

y=yfexp(-r0/c exp(-ct)) (4.17)

As exp (-r0/c) =1/exp (r0/c)=y0/(y0 exp(r0/c))=y0/yf=R, I obtain eq. (4.7). The derivative of eq.(4.17) is eq. (4.1).

For small values of t, exp(-ct)≈1-ct, giving 1-exp(-ct)=ct. This implies that growth is exponential, and leads to

From trees to forests. A unified theory

From Trees to Forests

A Unified Theory

From Trees to Forests

A Unified Theory

A Personal Contribution to the International Year of

From Trees to Forests

A Unified Theory

Luís Soares Barreto

From Trees to Forests. A Unified Theory

With Compliments

To Luísa Maria,

who kindly helped me

in the diffusion of my

works by e-mail,

with tender gratitude

Published recently by the author

Contents

Symbols, and acronyms frequently used

1 Introduction

1.1 The Scope of this Book

1.2 The Fundamental Assumptions

1.3 The Book

1.4 Related Works

1.5 References

PART I

PURE FORESTS

Dp



Definitions, and Basic Concepts



Allometry: The Universal Mathematical Laws of Life

Ap



The Gompertz Equation: The Pattern of Biological Growth

Lp



The Time Self-Similarity of Biological Growth



The Laws that Rule the Structure, and Dynamics of SPES



Self-Thinned Pure Uneven-Aged Stands



Growth, Regeneration, and Survival Indices



Life-History Strategies of Tree Species



The Analysis of Thinning



Plant Growth and Kleber’s Law



Application

2 Definitions, and Basic Concepts

2.1 Remembering Preliminary Definitions

2.2 The Phases of the Growth of SPES

2.3 Forest Variables

2.4 Complementary Definitions

2.5 References

3 Allometry: The Universal Mathematical Laws of Life

3.1. Introduction

3.2. The Allometric Equation

3.3 A Few Illustrations

3.4 Geometric Similitude

3.5 References

4 The Gompertz Equation: The Pattern of Biological Growth

4.1 Introduction

4.2 The Density-dependence Perspective

4.3 The Aging Perspective



