Population Dynamics and Tree Growth Structure in Mathematical Ecology
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Mälardalen University Doctoral Dissertation 331
Population Dynamics and Tree
Tin Nwe Aye POPULATION DYNAMICS AND TREE GROWTH STRUCTURE IN MATHEMATICAL ECOLOGY
Growth Structure in Mathematical
Ecology
Tin Nwe Aye
Address: P.O. Box 883, SE-721 23 Västerås. Sweden
ISBN 978-91-7485-498-5
2021
Address: P.O. Box 325, SE-631 05 Eskilstuna. Sweden
E-mail: info@mdh.se Web: www.mdh.se ISSN 1651-4238
Mälardalen University Press Dissertations
No. 331
POPULATION DYNAMICS AND TREE GROWTH
STRUCTURE IN MATHEMATICAL ECOLOGY
Tin Nwe Aye
2021
School of Education, Culture and CommunicationCopyright © Tin Nwe Aye, 2021 ISBN 978-91-7485-498-5 ISSN 1651-4238 Printed by E-Print AB, Stockholm, Sweden
Mälardalen University Press Dissertations
No. 331
POPULATION DYNAMICS AND TREE GROWTH
STRUCTURE IN MATHEMATICAL ECOLOGY
Tin Nwe Aye
Akademisk avhandling
som för avläggande av filosofie doktorsexamen i matematik/tillämpad
matematik vid Akademin för utbildning, kultur och kommunikation
kommer att offentligen försvaras fredagen den 26 mars 2021, 10.00
i rum Zeta, Hus T och via Zoom, Mälardalens högskola, Västerås.
Fakultetsopponent: Professor Christian Engström, Linnéuniversitetet
Akademin för utbildning, kultur och kommunikationAbstract This thesis is based on four papers related to mathematical biology, where three papers focus on population dynamics and one paper concerns tree growth and stem structure. The first two papers are mainly devoted to studying the dynamics of physiologically structured population models by using Escalator Boxcar Train (EBT) method. The third paper concerns a class of stage-structured population systems, in both deterministic and stochastic settings. The fourth paper explores how a branch thinning model can be utilized to describe the cross-sectional area of the stem of a tree, thus generalizing the classical pipe model. In Paper I, we present a merging procedure to reduce the increasing system of ordinary differential equations generated by the EBT method. In particular, we modify the EBT method to include merging of cohorts. The accuracy of this model is explored on a colony of Daphnia Pulex. In Paper II, we study the convergence rate of the modified EBT model, allowing a general class of non- linear merging procedures. We show that this modified EBT method induces a bounded number of cohorts, independent of the number of time steps. This in turn, improves the speed of the numerical algorithm for solving the population dynamics from polynomial time to linear time, that is, the time consumption to find the solution is proportional to the number of time steps. In Paper III, a class of non-linear two-stage structured population models is studied with different growth rates for the unstructured food resource under different harvesting rates in both deterministic and stochastic settings. In the stochastic setting, we develop methods to evaluate emergent properties equivalent to the properties investigated in the deterministic case. In addition, new emergent properties, e.g. probability of extinction, are also investigated. In Paper IV, we explore the stem model which is developed by combining the pipe model and the branch thinning model. The stem model provides estimates of the heartwood, sapwood and stem cross-sectional area at any height. We corroborate the accuracy of our model with empirical data and the cross validation of our results shows a very high goodness of fit for the stem model. ISBN 978-91-7485-498-5 ISSN 1651-4238
I dedicate this work to my aunt who has been supporting me throughout my life.
Acknowledgements
This thesis becomes a reality with a kind support and help from many individuals.
First of all, I would like to express my sincere gratitude to my supervisor Associate
Professor (Docent) Linus Carlsson for the continuous support during my Ph.D
study and research, for his patience, friendship, immense knowledge and care. He
has taught me the methodology to carry out the research and the scientific tools
for my research works. It was a great privilege and honor to work and study
under his guidance. I am extending my thanks to his wife for her acceptance and
patience during the discussion I had with him during research work. My sincere
thanks also go to co-supervisor Professor Sergei Silvestrov for his encouragement,
support, kind words and suggestions to improve thesis manuscript. I am grateful to
co-supervisor Masood Aryapoor for providing suggestions on the research works
and thesis manuscript improvement.
I am extremely grateful to my parents for their love, caring and sacrifices for
me. I am very much thankful to my aunt who strongly supports me throughout my
life. I would like also to give much thanks to Professor Ohn Mar from Myanmar
who suggested and encouraged me to start this wonderful job. My appreciation
also goes to Professor Khin Myo Aye for her kind coordination of the SEAMaN
project in my home country, the project which has supported me during my studies.
My special thanks go to the International Science Programme (ISP) for financial
support. I wish to show my gratitude to Pravina, Chris and Leif at ISP for being
around me and not to let me to have any inconvenience for the whole time of my
studies. I would like to extend my sincere obligation towards all the staffs and
PhD students in Mathematics and Applied Mathematics, Mälardalen University. I
am extremely thankful to all my friends, fellow PhD students under ISP and Sida,
who encouraged and cared for me throughout my studies. I am very lucky to have
you all around and appreciate for having such a wonderful family.
7Population Dynamics and Tree Growth Structure in Mathematical Ecology
At last, my thanks and appreciation go to all my colleague and people who
have willingly helped me out with their abilities in so many ways.
Västerås, March, 2021
Tin Nwe Aye
8Populärvetenskaplig
sammanfattning
Matematisk biologi/ekologi är en snabbt växande, accepterad, intressant och mod-
ern tillämpning av matematik. Ekologi är studien av interaktioner mellan organ-
ismer, populationer och deras omgivning. En population är grupp av individer av
en art som finns inom ett visst område vid en viss tid.
Fysiologiskt strukturerade populationsmodeller (PSPMs) används för att un-
dersöka populationsdynamiken, dvs. hur populationer och miljö förändras över
tid. PSPMs är en klass av modeller som explicit sammankopplar populationsdy-
namiken med individernas livshistoria, speciellt födointag, tillväxt, reproduktion
och dödlighet. Dessa processer beror på individens tillstånd och den omgivande
miljön. Modeller av ekologiska system med PSPMs har signifikant bidragit till vår
förståelse av hur individers storlek påverkar populationsdynamiken. Individuell
tillväxt, reproduktion och dödlighet är födoberoende och varierar med popula-
tionstäthet och förändringar i den omgivande miljön. The Escalator Boxcar Train
(EBT) är en numerisk metod för att lösa storleksberoende PSPMs. Den klassiska
algoritmen för EBT genererar ett system av differentialekvationer vars antal ökar
över tid och blir ohanterliga att lösa, även i en modern dator. Vi har modifierat EBT
metoden så att systemet inte växer men bibehåller noggrannheten i lösningarna,
och därför kan mer avancerade PSPMs lösas.
En alternativ metod för att lösa PSPMs är att dela in populationsstorlekarna i
unga och vuxna individer, så kallade storleksstrukturerade populationsmodeller. Vi
studerar storleksstrukturerade, biomassabaserade konsument-resurs-modeller med
olika födotillväxtsdynamik, både deterministisk och stokastisk, för att undersöka
framträdande egenskaper i populationsdynamiken.
Inom skogsindustri och skogsekologi har den klassiska rör-modellen använts
för att uppskatta tvärsnittsarean i trädstammar. Men i originalrapporten, skriven
för mer än ett halvt sekel sedan, säger författarna explicit att deras enkla rör-
9Population Dynamics and Tree Growth Structure in Mathematical Ecology modell inte kan användas i detta syfte under trädkronan. Stammens tvärsnittsarea ökar markant med trädets storlek och ålder och är sammansatt huvudsakligen av kärnved och splintved. Sammansättningen av kärnved/splintved är viktig, både för industriella träprodukter, men även inom ekologisk skogsteori. Vi utvecklar en stammodell som förutsäger tvärsnittsarean av kärnved och splintved i stammen på träd. Vår modell överträffar den enkla rörmodellen. 10
Popular Science Summary
Mathematical biology/ecology is a fast-growing, well-recognized and useful ex-
citing modern application of mathematics. Ecology is the study of interactions
between organisms, populations and their environment. A population is a group of
interbreeding organisms of a particular species in the same geographic area over a
period of time.
Physiologically structured population models (PSPMs) investigate the popula-
tion dynamics, that is, the study of change in populations and environment over
time. PSPMs are a class of models which explicitly link population dynamics
and individual life history, in particular feeding, development, reproduction, and
mortality. These processes are dependent on the state of the individual organism
itself and the environment in which it lives. Modelling ecological systems with
PSPMs has contributed significantly to our understanding of how size-dependent
individual life history processes affect the population dynamics. Individual growth
reproduction is food-dependent and varies with population density and environ-
mental changes. The Escalator Boxcar Train (EBT) is a numerical method, used to
find solutions to size-dependent PSPMs. The classical algorithm for EBT involves
a system of differential equations where the number of equations grows with time
and eventually becomes unmanageable to solve, even on modern computers. We
have modified the EBT method in such a way that the system does not grow,
yet accurately finds the solution, which results in the possibility to solve more
advanced PSPMs. Another approach to PSPMs is to divide the population sizes
into juvenile and adult individuals, this type of model is called a size-structured
population model. We study size-structured, biomass-based, consumer-resource
models with different kind of resource growth dynamics, both deterministic and
stochastic, to investigate emergent properties of the population dynamics.
In forest industry and forest ecology, the classical pipe model has been used to
estimate the cross-sectional area of the stem in trees. However, in the original paper,
written over half a century ago, the authors explicitly state that this simple pipe
11Population Dynamics and Tree Growth Structure in Mathematical Ecology model cannot be used for this purpose below the crown of the tree. The stem cross- sectional area increases significantly with tree size and age and is composed mainly of heartwood and sapwood. The heartwood/sapwood composition is important, both in industrial products from trees as well as theoretical uses in theoretical forest ecology. We derive a stem model that predicts the heartwood and the sapwood cross-sectional areas of the stem of the tree. Our model outperforms the simple pipe model. 12
This research was financially supported by International Science Programme (ISP) in Mathematical Science (IPMS) in collaboration with South-East Asia Mathematical Network (SEAMaN) in which the University of Mandalay in Myan- mar is a partner.
Contents
1 Introduction and Background 19
1.1 Summaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.1.1 Numerical stability of the escalator boxcar train under re-
ducing system of ordinary differential equations . . . . . . 23
1.1.2 Increasing effciency in the EBT algorithm . . . . . . . . . 24
1.1.3 Method development for emergent properties in stage-
structured population models with stochastic resource growth 25
1.1.4 Pipe model theory for prediction of tree sapwood and
heartwood profiles . . . . . . . . . . . . . . . . . . . . . 27
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Paper I 37
Paper II 51
Paper III 78
Paper IV 110
15List of Papers
This thesis is based on the following papers:
Paper I. Tin Nwe Aye, Linus Carlsson (2017). Numerical stability of the escalator box-
car train under reducing system of ordinary differential equations. 17th ASMDA
Conference, London UK, 2017.
Paper II. Tin Nwe Aye, Linus Carlsson (2020). Increasing Efficiency in the EBT Algorithm.
In: Skiadas C., Skiadas C. (eds) Demography of Population Health, Aging and
Health Expenditures. The Springer Series on Demographic Methods and Population
Analysis, vol 50. Springer, Cham.
Paper III. Tin Nwe Aye, Linus Carlsson (2019). Method development for emergent properties
in stage-structured population models with stochastic resource growth. Accepted
for publication in: Stochastic Processes/ Modern statistical methods in theory and
practice. SPAS 2019. Springer Proceedings in Mathematics and Statistics 2020.
Paper IV. Tin Nwe Aye, Åke Brännström, Linus Carlsson (2020). Pipe Model Theory for
Prediction Tree Sapwood and Heartwood Profiles. Submitted for publication in:
Tree Physiology, 2020.
Reprints were made with permission from the respective publishers. Parts of this thesis
have been presented in communications given at the following international conferences:
1: ASMDA2017 - 17th Applied Stochastic Models and Data Analysis International
Conference with the 6th Demographics Workshop, 6-9 June 2017, London, UK.
2: ASMDA2019 - 18th Applied Stochastic Models and Data Analysis International
Conference with the Demographics 2019 Workshop, 11-14 June 2019, Florence,
Italy.
3: SPAS2019 - International Conference on Stochastic Processes and Algebraic Struc-
tures from Theory towards Applications, Västerås, Sweden, 30th September – 2nd
October 2019.
17Chapter 1
Introduction and Background
This thesis focuses on the formulation, analysis and application of the parts of mathematical
ecology that describe the dynamics of biological populations but also the part which
concerns tree growth and tree structure. In this scope, the relations between organisms
and their environment are studied by using many different methods. In a geographical
area, the number of individuals of a species changes over time. The variation of these
species depends on the mechanisms of reproduction, the physiology of individuals, the
resources supplied by the environment and the interactions between individuals of the
same or different species. Ecology studies the relations between living organisms and
their environment. The phenomena of life and the interaction between selected species
and the environment turn biological into an advanced field. To handle these complicated
interactions, mathematics is one of the fundamental tools for the development of life
phenomena.
Traditionally, ecological and evolutionary theory is based on unstructured population
models, that is, models that ignore the presence of population structure. Unstructured
population models are based on the assumption that all individuals within a population
are identical. In a simple population model for continuous time, an ordinary differential
equation (ODE) is often used to represent the population dynamics of ecological and
biological systems, as follows
dN
= f(N)N,
dt
where N = N(t) is the population size at time t and f(N) is the net individual growth
rate function. The growth rate f(N) is often given by
f(N) = β(N) − µ(N),
where β(N) is the birth rate and µ(N) is the mortality rate which depends on the population
size.
19Population Dynamics and Tree Growth Structure in Mathematical Ecology
The Malthusian growth model and the Verhulst–Pearl equation are well-known state-
ments of f(N) [7, 80, 87]. The Malthusian growth assumes f(N) = r, where r is a
constant growth rate. The population for this model will grow exponentially with time.
This model can be used for bacteria populations over a finite time period, otherwise it is
not reasonable due to the limitation of the resources and environment. Using the principle
of the Malthusian growth and adding a limit to the population capacity (called the carry-
ing capacity), the Verhulst–Pearl equation proposes f(N) = r(K − N), where K is the
carrying capacity. The Verhulst model can be used in a limited environment on the growth
of populations, for example, bacteria, yeast etc.
Population dynamics deals with the population growth over time [25, 67]. The changes
over time in the number of individuals in the population are determined by reproduction,
death, growth rates and food supply under the environmental changes. Two important
characteristics of a population for its future developments are population size and the
population structure. These two characteristics define the population state. The changes
in the population are activated due to the events that happen with individual organisms,
for example birth and death of individual organisms.
When modelling population, we want to distinguish individual organisms from each
other on the basis of a number of physiological characteristics, such as age, body size or
gender. One of the reasons why we need to distinguish the individuals is to keep track
of the age or size distribution of a population [30, 64], for example, reproduction starts
at a certain age or size of the individual. The collection of physiological traits is the
individual states, that are used to characterize individual organisms within a population.
In addition, the environmental properties that a population is exposed to are also important
for population dynamics. For a specific individual organism, the environment is not only
made up by the ambient temperature or food abundance but also by the number and type
of fellow members within the population.
Many animal populations restrict their reproductive activities to specific times of
the year when food is abundant and survival and reproductive success are high [69]. It
means that the changes in population depend not only on time which may be assumed to
continuous or discrete but also on the environment. Discrete time models only determine
the state of the population at specific points in time and do not mention what happens
in-between.
Under environmental condition, many phenomena appear to be stochastic. Envi-
ronmental stochasticity is unpredictable fluctuations in environmental conditions. The
environment is typically defined as any set of abiotic (e.g. temperature and nutrient avail-
ability) and biotic (e.g. predator, competitor and food) conditions to which organisms are
subject. Environmental stochasticity influences how population abundance fluctuates and
effects the fate (e.g. persistence and extinction) of populations. For the stochastic process,
Brownian motion is one of the most important formulations in mathematical biology.
Brownian motion was first introduced to describe the random movement exhibited by tiny
20particles that are suspended in a medium such as a gas or a liquid. The name of Brownian
motion is derived from the Scottish botanist Robert Brown, who noticed pollen grains
moving erratically in water.
In contrast to unstructured population models. a more accurate description of pop-
ulation dynamics is given by physiologically structured population models. In order to
specify a physiologically structured population model, these rates are defined to be of the
forms the death rates, µ(x, Et ), growth rates, g(x, Et ), and birth rates, β(x, Et ) where x
is the size of an individual and Et is the environment that individuals experience at time t.
With these assumptions, the density u(x, t) of individuals of state x at time t is given by
the first order, nonlinear, nonlocal hyperbolic partial differential equations with nonlocal
boundary condition
∂ ∂
u(x, t) + g(x, Et )u(x, t) = −µ(x, Et )u(x, t), (1.1a)
∂t ∂x Z ∞
g(xb , Et )u(xb , t) = β(ξ, Et )u(ξ, t)dξ, (1.1b)
xb
u(x, 0) = u0 (x), (1.1c)
in which we assume that xb is the birth size for all new individuals, xb ≤ x < ∞ and
t ≥ 0.
In general, the above partial differential equation, cannot be solved analytically, instead
one commonly use numerical methods to find approximate solutions. To solve physio-
logically structured population models, several numerical methods have been proposed.
These methods include the fixed-mesh upwind (FMU) method, the moving-mesh upwind
(MMU) method, the characteristic method (CM) and the Escalator Boxcar train (EBT)
method [65, 103]. The upwind methods was first applied to population dynamics by Deb-
orah Sulsky [97]. The EBT method is schemed by André M. de Roos [24], which follows
the evolution of the population. The EBT method is similar to the CM which follows the
trajectories of characteristic. The method of characteristic is a classical method for solving
non-linear first-order partial differential equations.
To develop any kind of structured population model, we need to choose one or more
variables in terms of which the population structure is described. Most organisms on
earth undergo major changes in size and resource use over their life period. Changes in
size over ontogeny mean that an individual uses different resource types. Many studies
have been devoted to examining the age and stage-structured population model [20, 19,
17, 55, 59] investigating the effects of age and stage variation on population dynamics and
communities. Structured population models are less detailed PSPMs and link individual
life history and population dynamics in different physiological states. These models
capture relevant properties such as population density, extinction, yield, resilience, and
recovery potential, the basis for these models are different processes, in particular; feeding,
21Population Dynamics and Tree Growth Structure in Mathematical Ecology
development, reproduction and mortality. These processes are dependent on the state of
the individual organism itself and the environment in which it lives. In this thesis, we study
the PSPMs by using EBT method and stage-structured population model with deterministic
and stochastic resource growth rate as well.
The second part of this thesis is related to forest ecology. Studies of tree form are
necessary in order to estimate the biomass and to understand clearly the structural features
of forest communities. Several new studies [18, 53, 62, 63, 71, 77] on the tree form analysis
have appeared since Shinozaki et al. [91, 92] proposed the pipe model theory of tree form.
The influential pipe model theory of tree form states that each unit of leaf area is
supplied by a fixed number of pipes, implying that the sapwood cross-sectional area of
the stem is proportional to the total leaf area above the cut. This simple, quantitative
relationship has been extensively used in the literature, see, e.g., [53, 85]. Importantly, the
pipe model theory also predicts that the heartwood cross-sectional area which is interpreted
as a collection of disused pipes. This is proportional to the total leaf area lost above the
cut due to branch thinning. As it is much more difficult to measure extinct leaf area than
extant leaf area, this part of the pipe model theory has not attracted the same interest.
Recently, Hellström et al. [46] developed a theory of branch thinning describing
the ontogenetic development of trees and, in particular, the leaf area that is lost as trees
grow. This branch thinning model is utilized to estimate the total number of leaves above
height h. We then propose the stem model to estimate the heartwood, sapwood and stem
cross-sectional area by synthesizing the pipe model with a recent framework of tree growth
and branch thinning model.
In Papers I and II of this thesis we use the Escalator Boxcar Train (EBT) which is
one of the most popular numerical methods used to study the dynamics of physiologically
structured population models. The reason for its widespread use in theoretical biology
is that the components of numerical scheme can be given a biological interpretation.
Rather than approximating the solution directly, it approximates the measure induced by
the solution. The EBT method requires that the initial population is divided into a finite
number of cohorts, each cohort is given an index i, where 0 ≤ i ≤ M. The number of
individuals, Ni (t), and the mean size, Xi (t) are tracked by the EBT model. The EBT
model accumulates an increasing system of ODEs to solve for each time step.
In Paper I, we present a merging procedure to reduce the increasing system of or-
dinary differential equations which does not affect the convergence of the solution. In
particular, we apply the reproduction model of Daphnia to present a mathematical proof of
convergence of our merging procedure combined with the EBT method. Furthermore, we
compare the results from simulations of the Daphnia model, with and without merging.
In Paper II, we introduce the general class of reproduction functions which cover
for example the Daphnia model. For this class of reproduction functions, we prove that
the convergence rate is preserved for the EBT model in which we modify the original
EBT formulation, allowing merging of cohorts. We show that this modified EBT method
22Summaries
induces a bounded number of cohorts, independent of the number of time steps. This in
turn, improves the numerical algorithm from polynomial to linear time with respect to the
number of time steps. We also illustrate the results of an EBT simulation of the Daphnia
model.
In Paper III, we use an aquatic ecological system containing one fish species and an
underlying resource to study a class of nonlinear stage-structured population models, both
in the deterministic and stochastic settings. The reason why we introduce randomness in the
models is to include stochastic growth rate for the resource since many phenomena appear
to be stochastic in the real world. New properties emerge when introducing randomness
in the model that cannot be studied in the deterministic cases, such as the probability of
extinction. Furthermore, emergent properties usually studied in the deterministic setting,
can now be explained by its expected value and its dispersion. We have also developed
methods to understand emergent properties, studied in deterministic models, when these
models are extended to stochastic models.
In Paper IV, we propose a stem model which is an extension of the pipe model by
synthesizing it with the recent developed framework of the branch thinning model. The
resulting theory of our stem model allows for species-dependent branching structures and
stem area profiles, as well as sapwood and heartwood area profiles with corroboration of
empirical data. To cross-validate our model, we calibrate the age and height of the tree of
same species on a similar growing environments to estimate the heartwood and sapwood
area as well as the stem area.
1.1 Summaries
Paper I up to Paper IV correspond respectively to [4], [5], [6], and [3] whose contents we
summarize below. In this chapter, we give a brief introduction to the topics in this thesis
and a summary of four papers.
1.1.1 Numerical stability of the escalator boxcar train under
reducing system of ordinary differential equations
In this paper (Paper I), we propose a merging procedure to overcome computational
disadvantageous of the EBT method. The merging is done as an automatic feature. We
present a way of how to merge cohorts in order to stabilize the number of ODEs to solve in
each time step. We consider two cohorts (Xa (t), Na (t)) and (Xb (t), Nb (t)) at time t, where
N denotes the number of individuals in a cohort and X the mean size of the individuals
in the cohort. When merging the two cohorts (Xa (t), Na (t)) and (Xb (t), Nb (t)) into one
merged cohort (Xm (t), Nm (t)) at time t, the number of individuals for the merged cohort
23Population Dynamics and Tree Growth Structure in Mathematical Ecology
is
Nm (t) = Na (t) + Nb (t)
and the merged cohort size is
s
Na (t)Xa (t)2 + Nb (t)Xb (t)2
Xm = .
Na (t) + Nb (t)
Under these assumptions, we prove that the number of newborn individuals for merging
cohorts, bm converges to the number of newborn individuals for the sum of two non-
merging cohorts, bw , which is
bm (∆t) = bw (∆t) + O(∆x0 · ∆t),
when the difference between the size of individuals for two cohorts, ∆x0 , and the time
step, ∆t, is small. Here O is the big O notation. We also show by simulations that
the relationship between the number of time steps and execution time decreases from
polynomial to linear when an automatic feature of merging cohorts is introduced. In
particular we apply the model including merging to a colony of Daphnia Pulex. The
results from simulations of the Daphnia model, with and without merging, are compared
and we show that the merging procedure does not affect the order of the error.
1.1.2 Increasing effciency in the EBT algorithm
The main goal of this paper (which is Paper II below) is to overcome computational disad-
vantageous of the EBT method, by including our proposed non-linear merging procedure.
The class of models we consider have the reproduction rate per individual of the form
(
c(F ) X n , if X > xj ,
b(X, F ) = (1.2)
0, otherwise,
where xj is the length at maturity, b(X, F ) denotes the birth rate of adult individuals,
n ≥ 1 is a constant and c(F ) > 0 is a bounded function depending on the food density F .
We consider merging when the number of individuals in an internal cohort falls below
a certain threshold. This is checked for all cohorts in each time step. When this is the
case, merging is conducted of this cohort with a cohort of similar size. We show that
for the general class of reproduction functions given by equation (1.2), the convergence
rate of the EBT model is not affected when merging the two cohorts (Xa (t), Na (t)) and
(Xb (t), Nb (t)) into one cohort (Xm (t), Nm (t)) at time t, using a specified weighted mean
of merged cohort size, given by
1
Na (t)Xan (t) + Nb (t)Xbn (t) n
Xm = .
Na (t) + Nb (t)
24Summaries
For the number of individuals of a merged cohort, we naturally add in both cohorts, i.e.
Nm (t) = Na (t) + Nb (t).
We also show that the number of cohorts is bounded, when using this non-linear merging
procedure in the EBT method, independent of the number of time steps. In addition,
we prove that the computational time of this modified EBT model is proportional to the
number of time steps. The computational time in the classical EBT method is polynomial
with respect to the number of time steps. An EBT simulation of the Daphnia model is
used as an illustration of these findings.
The main result we prove is that the convergence of the number of newborn individ-
uals, when merging internal cohorts has a convergence rate of O(∆t) for arbitrary long
simulation times, that is
bm (s) = bw (s) + O(∆t),
for each s > t. Here bm is the resulting number of newborn individuals when merging
was done at time t and bw is the sum of the number of newborn individuals from the two
original cohorts.
1.1.3 Method development for emergent properties in stage-
structured population models with stochastic resource
growth
In this paper (Paper III below), we have used a simple aquatic ecological system as the
base model which consists of one fish species and an underlying food resource. The model
we use is a stage-structured model, in which we divide the fish population in two stages,
juveniles and adults. In population dynamics, one usually decides on a specific growth
rate model for the underlying resource. Different authors use different models, where the
two dominating models are the semi-chemostat growth rate and the logistic growth rate.
Our paper includes both these models, where we show and argue that the logistic growth
rate model is unstable and unrealistic. For this reason, we propose a new stable model
which captures, in a realistic way, the biological features of the logistic model, i.e.,
dRcomb dRsc dRlg
=p + (1 − p) ,
dt dt dt
where p is the proportion of the semi-chemostat growth. The semi-logistic growth rate is
denoted by Rcomb . The semi-chemostat growth rate and logistic growth rate are denoted
by Rsc and Rlg respectively.
The stage structured population model is investigated with three types of resource
dynamics which depend on the different harvesting rates for the stochastic case, and
includes the deterministic model as a base case. The main purpose of this paper is
25Population Dynamics and Tree Growth Structure in Mathematical Ecology
the study of stochastic growth rate models. For the stochastic setting of growth rate,
we add a white noise in terms of a Brownian motion to the deterministic growth rate
model. There are properties that emerge when introducing randomness in the model that
cannot be studied in the deterministic cases, such as the probability of extinction. We
study the impact on, and emergent properties of, a staged-structured population model
using different growth rates for the unstructured resource which are extended to stochastic
models. Monte Carlo methods are used to evaluate the juvenile/adult/resource biomass,
yield, impact on biomass, impact on size-structure, resilience, recovery potential and the
probability of extinction.
For the deterministic stage model, the recovery potential is introduced in Meng et al.
[64] by assuming an equilibrium in the governing differential equations. The measure of
recovery potential is closely related to the basic reproduction ratio in a virgin environment,
i.e., when the resource biomass is close to its maximum value. Since the stochastic model
never reaches equilibrium, we derive an equivalent formulation for the recovery potential
as
J 0 +(A0 +(M+F )A)
, wJ (R) = M + F,
A(M+F )
R(F ) =
J 0 −(wJ∗ (R)−M−F )J+(A0 +(M+F )A) A0 +(M+F )A
, otherwise,
A(M+F ) 0 ∗
A +(M+F )A−(wJ (R)−M−F )J
where J and A are juvenile and adult biomasses, respectively. Here, M is the natural
mortality rate and F is the harvesting rate of juveniles and adults. Then wJ (R) is the net
biomass production rate for juveniles and wJ∗ (R) is the unique solution of the net biomass
production rate for juveniles.
In stochastic population models, the population can go extinct within a finite period
of time under environmental fluctuation. One way of finding the probability of extinction
is through the minimum viable population (MVP) which is expressed by
A + PA J < MVP
where PA is the probability that juveniles will become adults. The minimum viable
population is a lower bound on the population of a species, that can survive in the wild.
In addition, we introduce a new formulation of the probability of extinction by
P (R(F ) < 1) = CDF(1, µR (F ), σR (F )).
where CDF is the normal cumulative distribution function. Normal approximation is
appropriate, since the recovery potential is evaluated by means of many trajectories, i.e.
one can invoke the central limit theorem. The CDF is evaluated at 1, since this is the
critical value for the recovery potential. We show by simulations that our new formulation
of the probability of extinction gives the same results as the MVP formulation.
26Summaries
1.1.4 Pipe model theory for prediction of tree sapwood and
heartwood profiles
In this paper (Paper IV below), we introduce the extended pipe-model theory by synthe-
sizing the pipe model of plant form [91, 92] with a developed framework of tree growth
and branch thinning [46]. As an application of the extended pipe model theory, we calcu-
late the area of sapwood and heartwood by using the branch-thinning theory [46] which
depends on the present leaf area of the tree and the leaf area which has at some points in
the past been supplied by pipes.
In the branch thinning model [46], a branch that was formed n growth cycles ago has
the expected number of tips
b(n) = min (µn , β(n + 1)d ), (1.3)
where β is a species and location specific constant. The average number of tips formed at
a growth model is denoted by µ. Then d is an exponent in the branch carrying capacity.
The simple pipe model of plant form with the analytical framework of branch thinning
is synthesized. This allows us to explore the amount of living leaves on the tree by
summing all the values of growth modules, g(k, n), for the whole life span of leaves. The
expected number of growth modules, g(k, n), of k growth cycles from the proximal end
of a branch, n growth cycle old, is given by
b(n)
g(k, n) = , (1.4)
b(n − k)
for 1 ≤ k < n.
In this paper, we introduce the stem model to present a merging of the simple pipe
model of plant form and the branch thinning model which provides estimates for both the
heartwood and the sapwood at any height above breast height. We find the total sapwood
area at height h by
Xn
SA (h, n) = cS g(l, n),
l=max {h,n−lg }
where cS is the area of sapwood per pipe and lg is the number of growth cycles that a leaf
bud stays active, thus the leaf bud is producing new leafs during this number of growth
cycles. The total heartwood area at height h is also estimated by
Xn m−1
X
HA (h, n) = cH B(h, m) + g(l, m − 1) − g(l, m) ,
m=h+1 l=max{h,m−1−lg }
where B(h, m) = g(m − 1 − lg , m) if m > 1 + lg + h and B(h, m) = 0 otherwise. Here,
cH is the area of heartwood per pipe.
27Population Dynamics and Tree Growth Structure in Mathematical Ecology
To investigate the sapwood and heartwood stem area, the expression for the sapwood
stem area is derived by
Sarea (h, n) = κlog2 g(h,n) SA (h, n), (1.5)
and the heartwood stem becomes
Harea (h, n) = κlog2 g(h,n) HA (h, n), (1.6)
where, κ is the proportion of cross sectional area that remains on stem after a branching.
We use these expressions when corroborating our model to empirical data. Our stem model
corroborates with empirical data which have a good fit with R2 (84–99%), depending on
species and location. At last, we also investigate the cross validation of our results with
R2 (62–98 %).
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