Numerical experiments on variation of species
diversity in size-dependent evolutionary system

Takanori Sasaki
Department of Earth and Planetary Science,
The University of Tokyo
2005

Contents

1 General Introduction

52 Introduction of Numerical Ecology

10

2.1 Implications of Fossil Records . . . . . . . . . . . . . . . . . . 10
2.2 Previous Studies . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 My Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Methods

20

3.1 Outline of the System. . . . . . . . . . . . . . . . . . . . . . 20

3.2 Size-Dependent Evolutionary System . . . . . . . . . . . . . . 24
4 Results

26

4.1 Regulations of Numerical Simulations . . . . . . . . . . . . . . 26
4.2 Diversity Change in The Food Web System . . . . . . . . . . . 29
4.3 Effects of Alteration of Evolving System . . . . . . . . . . . . 34
4.3.1

Effects of Alteration of Evolution Timescale . . . . . . 35

4.3.2

Effects of Alteration of Feeding Parameters . . . . . . . 37

4.4 Comparing of Size-Dependent/Independent Evolution . . . . . 38

1

5 Discussions

43

5.1 Analysis of Maintained State . . . . . . . . . . . . . . . . . . . 43
5.1.1

Is Size-dependent Evolution Always Necessary? . . . . 43

5.1.2

Alteration of Size-Dependent/Independent Evolution . 45

5.1.3

Implication for Observed Food Web System . . . . . . 45

5.2 Analysis of Interspecies Interactions . . . . . . . . . . . . . . . 46
5.2.1

Apparent Discrepant Results . . . . . . . . . . . . . . . 46

5.2.2

Simplification of Feeding System

5.2.3

Map of Interspecies Interactions in Food Web System . 47

5.2.4

Implication for Alteration of Parameter Dimension . . 51

. . . . . . . . . . . . 46

5.3 Punctuated Equilibria . . . . . . . . . . . . . . . . . . . . . . 52
6 Conclusions

54

Acknowledgments

56

Bibliography

57

2

List of Figures
2.1 Ecological evolutionary units and evolutionary faunas. . . . . . 11
2.2 Correlation between body size and metabolic rate. . . . . . . . 19
4.1 Overflow on very small species world. . . . . . . . . . . . . . . 28
4.2 Examples of evolutionary patterns (1) . . . . . . . . . . . . . . 30
4.3 Examples of evolutionary patterns (2) . . . . . . . . . . . . . . 31
4.4 Partition of evolutionary pattern of species diversity . . . . . . 32
4.5 Examples of evolutionary patterns (3) . . . . . . . . . . . . . . 36
4.6 Comparing of size-dependent/independent evolution . . . . . . 40
4.7 Size distributions in food web system . . . . . . . . . . . . . . 41
5.1 Comparing of size-dependent/independent system stability . . 44
5.2 Examples of evolutionary patterns (4) . . . . . . . . . . . . . . 48
5.3 Maps of interspecies interactions (1) . . . . . . . . . . . . . . . 49
5.4 Maps of interspecies interactions (2) . . . . . . . . . . . . . . . 50

3

List of Tables
4.1 Initial Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 Effects of altering evolving systems . . . . . . . . . . . . . . . 33

4

Chapter 1
General Introduction
Over 130 extrasolar giant planets are known to orbit nearby Sun-like stars,
including several in multiple-planet systems 1 . These planetary systems are
stepping stones for the search for Earth-like planets; the technology development, observational strategies, and science results can all be applied to
Earth-like planets. The search for extra-solar Earth-like planets, and possible signatures of biological lifeforms, is one of the most exciting prospects
for planetary science in the future; this disciplinary is called ”Astrobiology”.
Astrobiology seeks to understand the origin of the building blocks of life, howthese biogenic compounds combine to create life, how life affects - and is affected by the environment from which it arose, and finally, whether and how
life expands beyond its planet of origin. None of these questions is by anymeans new, however for the first time since they were posed, these questions
may now be answerable.
By now, I have done research on the origin and evolution of planets using avariety of numerical and analytic methods or some spectroscopic observations
1

California & Carnegie Planet Search, http://exoplanets.org/

5

CHAPTER 1. GENERAL INTRODUCTION

6

with Subaru telescope to examine how the planets and the material of planets,
which naturally include the building blocks of life on a terrestrial planet,
have been produced and evolved (Sasaki et al. 2004, 2005; Sasaki and Abe
submitted). Then in the next step, I would like to discuss the ecologicalsystem as a kind of astrobiology to examine the general behavior of life. Life
developed on Earth, but what about other planets throughout the universe?

Previously, not only an ecologist but other many field scientists haveworked on their own studies to understand the same thesis ”How life havebeen developed?”.
Paleontologists, evolutionary biologists and perhaps even archaeologists
will be called upon to help understand the record of previous life on Earth ina planetary context (e.g. Raup and Sepkoski 1982). Some scientists considered that changes in planetary environments lead or follow periods of change
in life (Cloud 1968, 1972; Hoffman et al. 1998). Hoffman et al. (1998) paid
attention to negative carbon isotope anomalies in carbonate rocks bracketing
Neoproterozoic glacial deposits in Namibia, which suggested that biologicalproductivity in the surface ocean collapsed for millions of years. They explained this collapse by a global glaciation, that is ”snowball Earth”, andthis period ended abruptly when subaerial volcanic outgassing that rose awarming of the snowball Earth to extreme greenhouse conditions. And otherscientists investigated the temporal distribution of the major extinctions overthe past 250 million years statistically using various forms of time series anal-

CHAPTER 1. GENERAL INTRODUCTION

7

ysis (Raup and Sepkoski 1984). They suggested that 12 extinction events
showed a statistically significant periodicity with a mean interval between
events of 26 million years. Does species complexity arise at a constant rateor does it happen in spurts?
Astronomers, planetary geologists, and paleontologists will be called upon
to assess the effect that large impacts have upon life on Earth (e.g. Kaiho
et al. 1999). A clear record of bombardment in the early history of the solar
system has been found across many planets and moons (Cohen et al. 2000).
The lack of impact melt older than 3.92 billion years ago supports the concept of a short, intense period of bombardment in the Earth-moon system
at about 3.92 billion years ago. And, even to this day, several years ago,
we watched a comet hit Jupiter with many times the force of our planet’s
collective nuclear arsenal. On Earth it is clear that large ecosystem-busting
impacts have occurred with some regularity. Alvares et al. (1980) frameda hypothesis that accounted for the mass extinctions and concentrations of
iridium in deep-sea sediments. They considered that impact of a large earthcrossing asteroid, which contained excess of iridium, would inject about 60
times the object’s mass into the atmosphere as pulverized rock and a fractionof this dust would stay in the stratosphere for several years and be distributed
worldwide. They suggested the resulting darkness would suppress photosynthesis, and the expected biological consequences matched quite closely theextinctions observed in the paleontological record. Are planetary impacts
actual component of life’s extinction on a planet?

CHAPTER 1. GENERAL INTRODUCTION

8

As described above, many phenomenological researches have done to discuss the development of life and gave us many explanations of some biologicalevents. Meanwhile, we still do not understand the mechanisms of diversification or extinction of species under several circumstances. Numericalsimulations are absolutely necessary to discuss and answer these questions.
Analysis of evolution, diversification, stabilization, and extinction of species
should hold the key to understand not only the Earth’s ecosystem but also
general ecosystems. In field of astrobiology research, biochemists is assigned
to the origin of life from the building blocks of life, planetary scientists is
assigned to the formation of planet that hold and nurture life, and ecologists
is assigned to the behaviour of life on general terrestrial planets.

However, unfortunately no numerical model for meaningful discussion ofgeneral ecology is available at the present stage; almost all models have someproblem even to describe the terrestrial phenomena of ecosystem, and few
studies have investigated the dynamics of numerical models (as will hereinafter be described in detail in Chapter 2). So we first should establish theappropriate ecosystem model and discuss the feature of it as a preliminary
step toward general ecology on universe.
Dynamics of an evolving interaction web and food web will be the firstsubjects of the research. In this master’s thesis, I will focus on the evolutionary patterns of hypothetical communities in computer simulations anddiscuss what will play the important role in diversification and stabilization

CHAPTER 1. GENERAL INTRODUCTION

9of the ecosystem.
Because evolution would be essentially a non-directional phenomenon,
the evolution of extra-terrestrial life will most likely be drastically different
from the evolution of life on Earth. The different conditions present on
different planets will lead extraterrestrial life down different evolutionary
pathways. However while life elsewhere in the universe is probably drasticallyand even unrecognizably different from life on Earth, the same concepts andmechanisms that direct evolution, such as natural selection, will apply. On
Earth and beyond, it would be a matter of the survival of the fittest.

Chapter 2
Introduction of Numerical
Ecology
2.1

Implications of Fossil Records

The evolution of biodiversity can be seen in the fossil record. Many paleontologists have studied the fossil record and have elucidated the evolutionary
patterns of taxonomic diversity throughout the Phanerozoic period in manytaxonomic groups (e.g. Larwood 1988; Sepkoski 1984). Sepkoski (1984) examined standing diversity of marine families through the Phanerozoic, andrecognized three successive evolutionary faunas (Cambrian, Paleozoic, andModern); family diversity rose to three progressively higher plateaus during
the successive evolutionary faunas (Fig.2.1). Available fossil records haveshown that the biosphere on the earth has experienced remarkable fluctuations in the taxonomic diversity since organisms first appeared more than 3.8
billion years ago. The Cambrian Period witnessed one of the most significant
events in the history of life, which was an exponential increase in animal diversity and complexity, commonly referred to as the ”Cambrian explosion”.

10CHAPTER 2. INTRODUCTION OF NUMERICAL ECOLOGY

11

Figure 2.1: Revised ecological evolutionary units and the patterns of
Phanerozoic diversity of families recognized by Sepkoski (1984). Centered
figure shows the number of marine families at each era estimated from fossil records. Leftward time scale bar is scaled by millions of years ago, andrightward bars show range of available fossil records for some representative species. This figure was adopted from The Milwaukee Public Museum,
http://www.mpm.edu/index.htm, which is alteration from Sepkoski (1984)
originally.

CHAPTER 2. INTRODUCTION OF NUMERICAL ECOLOGY

12After this explosion, the species diversity increased from one evolutionary
fauna to the next, which can be accounted for by the invasion of previously
unoccupied environments (Sepkoski 1984).
On the other hand, the fossil record also showed at least five times mass
extinction events occurred ever (Sepkoski 1984; Erwin 1993; Hallam andWignall 1997): which occurred about 450 million years ago, 350 million
years ago, 250 million years ago (P-T boundary), 200 million years ago, and65 million years ago (K-T boundary). These mass extinction events are also
showed in Fig.2.1.
Terrestrial ecological system has experienced some biological major incidents (mass explosions and mass extinctions) as described above. While weknow such a fact from fossil records, it remains to be seen what triggered
these events and how these events occurred. To reveal the mechanism of theevents, theoretical biologists have kept up establishing ecological modelingand trying to reproduce real world’s events.

2.2

Previous Studies

The relationship between the complexity and stability of an ecosystem has
been one of the most fascinating topics in theoretical biology for decades.
In the 1950s and 1960s, the proposition that highly complex communities
are more stable than simple ones was widely supported (MacArthur 1955).
May (1972) challenged this proposition. He considered a large dimensional
ecological equation with an n-dimensional random interaction matrix, andCHAPTER 2. INTRODUCTION OF NUMERICAL ECOLOGY

13

concluded that an ecological system cannot be stable if it is complex. However, there is the discrepancy between the observed complexity of ecosystem
in nature and the results of these mathematical studies. While it is foundthat the stability of a randomly linked food web model tends to decrease withthe proportion of links and the number of species, real food webs display ahigh degree of stability, in spite of being very complex.
The search for what controls the dynamics of taxonomic diversity in ataxonomic group is an important biological and paleontological issue. There
are three points of view in what controlled the diversity change: (1) Effects
of environmental changes, (2) Effects of stochastic processes, and (3) Effectsof biological interactions.

(1) Effects of environmental changes
Many researchers have found that environmental changes may cause changes
in biodiversity. It is generally accepted that the major taxonomic diversity
changes recorded in the fossil record, especially mass extinctions occurred
contemporaneously with catastrophic environmental changes such as meteorite impact (Alvarez et al. 1980; Hut et al. 1987; Sigurdsson et al. 1991),
sea level change (Schopf 1974; Simberloff 1974), global climate change (Stanley 1984), large scale volcanic activity (Hallam 1987; Officer et al. 1987),
and so on. Several authors also pointed out that the timing of the second
and third-order diversity changes are correlated with that of some kinds ofenvironmental change (Kennedy 1977, Hirano et al. 1999).

CHAPTER 2. INTRODUCTION OF NUMERICAL ECOLOGY

14(2) Effects of stochastic processes
Raup et al. (1973) carried computer simulations to evaluate the effect ofstochastic processes on the dynamics of taxonomic diversity. In their system,
appearance, extinction, and survival of species were determined by a rollinga dice. Though they revealed that an exceedingly simple stochastic model
could produce branching and diversity patterns very like those describedin the real world, detailed mechanism of diversity change was not clear,because there was no inter-species interaction and its evolution. Evolutionof the system in the study was just changes in the elements of the matrix
randomly, however real food webs have features that are ignored in randomly
linked models. Their model was criticized by some researchers for failingto reproduce temporal diversity patterns (e.g. catastrophic mass extinctions(Sepkoski 1984; Erwin 1993; Hallam and Wignall 1997)). When one considers
long-term biological phenomena as recorded in the fossil record, one must not
avoid the evolution of species and the evolution of their interactions.

(3) Effects of biological interactions
The effects of biological interactions on the dynamics of taxonomic diversity have been discussed in empirical analyses of the fossil record, however,there is insufficient information about the actual cause of biological interactions. Empirical and statistical analysis of the long-term taxonomic diversity
changes based on the fossil record has a limitation to consider the mechanismof macroevolutionary dynamics. The study of temporal change of biologicalCHAPTER 2. INTRODUCTION OF NUMERICAL ECOLOGY

15

communities by experiments or observations of real communities is very difficult, because real communities are often too large and complex for long-term
experiments and observations, and temporal changes of interactions among
species constituting communities have not been preserved in the fossil record.
A computer simulation is a good alternative approach that can be used. If
we are able to construct a model biological community on a computer, we
can easily investigate changes in interspecific interactions and their effects ondiversity changes on an evolutionary timescale. Of course, a community on
a computer is not the same as an ecosystem in the real world, however, theanalysis of a model community can improve our understanding of the processof temporal changes in taxonomic diversity in the real world.
Following above concepts, Gilpin (1994) carried computer simulations
by mixing two hypothetical communities, a random set (so-called random
interaction matrix in which species in the system interact randomly) and an
assembled set (created by decreasing the number of species from 10 to 5 in arandom interaction web). Based on the result, he suggested that communities
having experienced extinction have cohesiveness that battle against othercommunities as a team. And Happel & Stadler (1998) represented evolution
with mutations of interaction coefficient. They revealed that strong symbiotic
interactions developed through repeats of mutations and that an immigrant
easily forces the extinction of other species. These classical ecological theories
and hypotheses assume fixed species traits. Species, however, have evolved
different phenotypes or strategies in response to interspecific interactions or

CHAPTER 2. INTRODUCTION OF NUMERICAL ECOLOGY

16

abiotic conditions depending on the assembly and evolutionary history of thecommunity.
Tokita & Yasutomi (1999, 2003) have simulated the ecological evolution
from a random matrix point of view. In the computer created models, interspecific interactions are assigned randomly. Evolution of the species constituting the interaction web system was modeled by assuming that a randomly
chosen species in the system was given a chance of speciation. While they
have given us much knowledge for the diversity-stability debate, their model
would have an unrealistic assumption such as random matrix ecological system. Although there are many constraints on interspecific interactions in
the real world, there is no such constraint in random matrix systems: a land
animal can not feed on animals living in the ocean, or it is impossible for any
species to increase its biomass infinitely by mutualism without food.
Yoshida (2002, 2003) modeled another interaction web in which each animal species fed on other species according to feeding preference. The foodweb system evolved via the appearance and extinction of species. Interspecies
interactions are decided by the properties of species and change with theevolution of those properties. Their simulations showed the number of preyspecies decreased and that of predatory species increased with time, and thespecies became extinct as a result. This result suggested that species extinction often seen in the fossil record were not always connected to an external
event; they can be caused by slight changes of interspecies interactions.

CHAPTER 2. INTRODUCTION OF NUMERICAL ECOLOGY

2.3

17

My Study

While Yoshida (2003) seemed to be successful to describe the behavior of real
ecosystems, and give us important implications for the evolution of species
at last, I will point out two unrealistic assumptions in his model: ”one-way
evolution” and ”randomly chosen evolution”.
First, in his model, with evolution, descendant species of predator canfeed on prey species more effectively than the ancestor species, and descendant species of prey can have less intensity of predation by predator species
than the ancestor species. However, the evolution would not necessarily raise
an improvement of the species. It would not be caused by the urge of individual organisms toward something better. So I will assume the random-way
evolution without direction in my model.
Second, in his model, speciation occurs for randomly chosen species. However, speciation would depend on the each species property, specifically bodysize, because smaller species should have faster timescales of the alterationof generations; the relationship between body mass (M ) and metabolic rate
has an exponent of 3/4 so that characteristic timescale can scale with bodymass as M 1/4 (Peters 1983, Schmidt-Nielsen 1984, Fig. 2.2). Evolutionary
intervals of each species would also scale as the same relationship, so I willassume that speciation occurs at intervals scaled as M 1/4 . While it is known
that relative body size of the component species has often been identified asa major determinant of food web structure (Warren & Lawton 1987, CohenCHAPTER 2. INTRODUCTION OF NUMERICAL ECOLOGY18

et al. 1993, Memmott et al. 2000), there is no study to consider the evolutionary timescale for each species. Size-dependent evolution interval would
be a quite new idea for ecological modeling.
On the basis of above assumptions, I reconsider an evolving food web
model and discuss the mechanism of diversification, stabilization, and extinction of species.

CHAPTER 2. INTRODUCTION OF NUMERICAL ECOLOGY

19

Figure 2.2: The baffling correlation between body size and total metabolicrate. This correlation may stem from nutrient distribution. This figure wasadopted from Schmidt-Nielsen (1984).

Chapter 3
Methods
In the following, I defined diversity as the total number of species in thesystem. I reconsider an evolving food web system model based on mainly
Yoshida (2003).

3.1

Outline of the System

I modeled a food web in a closed area (e.g. a separate continent) whereinterspecific interaction, speciation and extinction occur. Assuming that immigration from a distant continent is very rare, the appearance of new cladesis neglected. The main routine in this computer simulation of a hypotheticalinteraction web system is the calculation of the biomass of each species inthe system by using the following multidimensional Lotka-Volterra equations(Lotka 1925; Volterra 1926):


n
X



dMi
= Mi ri +aij Mj 
dt
j=1

(3.1)

where Mi is the total biomass of species i; ri is the intrinsic growth rate ofspecies i; n is the number of species in the system and aij is the effect of20

21

CHAPTER 3. METHODS

species j on species i.
With this set of equations, I calculated the dynamics of the total biomassof each species using a computer. In the computer, the interaction web
system is represented in a matrix form; this matrix aij is then called the”interaction matrix.” The calculation was done using the following equations:

Mi (t + ∆t) = Mi (t) + Mi (t) ri +

n
X

j=1



aij Mj (t) · ∆t

(3.2)

where Mi (t) is the total species biomass of species i at a given time step t;and ∆t is equal to a thousandth of one time step. The biomass of species
Mi (t) represents the proportion of each species in carrying capacity (thisvalue is 5000). And time step ∆t represents the alternation of generations
for the minimal species. Using this equation, the fluctuation of biomass ofeach species is calculated at each time step. When the total biomass of a
species becomes lower than its individual bodyweight (wi ) as a result of thecalculation of the biomass, the species becomes extinct. When the total
biomass in the system exceed 5000 after the calculation of biomass at each
step, the biomass of each species was normalized, so that the total biomass ofthe system was equal to 5000. This normalization implied that the carrying
capacity of the system was 5000.

The system consists of both animal and plant species. Plant species areable to grow without feeding on other species. The value of r in plant species
is set to 100. On the other hand, that in animal species is given as a mass-

CHAPTER 3. METHODS

22

dependent number described below.
For simplicity, carnivorous and parasitic plants are not considered in thepresent study. My assumption is that a predatory animal species can feed onother prey animal species only when the bodyweight of the predatory species
is larger than that of the prey species, although there can be exceptions tothis in the real world. This assumption is based on the observation that most
predators are larger than their prey (Warren & Lawton 1987, Cohen et al.
1993, Pahl-Wostl 1997, Neubert et al. 2000, Jennings et al. 2001). Animal
species are free of bodyweight constraints when they feed on plant species.
Each species has the following characteristics: defense property (D), offense property (A), and feeding range (P ). These parameters are used toconstruct interspecific interactions. Both A and D are arrays consisting of
10 elements. The values of each element of A and D, and the value of P
are given by random numbers uniformly distributed in [0, 100] and [0, 10]
ranges, respectively.
The construction of a food web is explained in detail in the followingcase example where species i feeds on species j. If the defense properties of
species j (Dj ) meet the feeding preferences of species i, which is determinedby Ai and Pi , species i can feed on species j. In my model, this judgement
is made by counting the number of elements of D that satisfy the following
condition:
Ai [k] − Pi ≤ Dj [k] ≤ Ai [k] + Pi

(3.3)

23

CHAPTER 3. METHODS

where Ai [k] is the kth element of A of species i. If the number of elements
n is greater, the interaction with i and j will increase. In this case, also
considering the ingestion efficiency of energy i.e. mass ratio between preyand predator, the aij value is given by the following equations:aij = 0.01 · n ·

wj
wi

(3.4)

and aji is set to −aij .
In the real world, neighboring plant species often hamper each other’s
growth in competition for light, soil resources and germination sites. I assumed that plant species tend to hinder each others growth when they caninteract each other. The judgement of how plant species i reduces the growthrate of another plant species j is made by the same method as the case ofanimal species. If species i reduces the growth rate of species j, the growthrate of aji is set byaji = −0.001 · n(3.5)

otherwise, aij is set to 0.
The diagonal elements of an interaction matrix have negative values that
represent intraspecific competition. While intrinsic growth rate ri represents
energy gain by photosynthesis for plant or energy loss by metabolism foranimal, the diagonal elements of an interaction matrix aii represents energy
loss by sharing sunlight or food in a same species community. When species i
is a plant species, the diagonal element aii is set to −0.001, and when species
i is an animal species, aii is set to −0.01. The values (0.01 or 0.001) of

24

CHAPTER 3. METHODS

aij depend on energy conversion efficiency of each species in fact, however,to simplify I set them constant value in this model. And energy conversion
efficiency of animal species is assumed to be larger than that of plant species.

3.2

Size-Dependent Evolutionary System

In general, the larger the species, the longer the life. This relationshipholds true with remarkable precision: Life timescale tends to lengthen, andmetabolism slows down, in proportion to the animal’s body weight. Many
characteristics of organisms vary with body size, as described by allometricequations of the form
Y = Y0 · M b(3.6)

where Y is the dependent variable, M is body mass, b is a power exponent
and Y0 is a normalization constant that varies with the nature of Y and withthe kind of organism. Studies of animals suggest that many variables scalewith quarter-powers of mass, for example b = 43 for metabolic rate, and 14 forlifespan (Peters 1983, Schmidt-Nielsen 1984).
I give each species the additional parameter τi , characteristic time, which
is used to determine the interval of speciation:
τi =



wi
0.03

1
4

(3.7)

And the value of ri in animal species, the metabolic rate of the species, isgiven bywi
ri = −2 ·
0.03


3
4

(3.8)

CHAPTER 3. METHODS25

In this system, a new species appears only via speciation of mature
species. Speciation of species occurs at intervals of 5000 time steps for plant
species, and 30000 × τi time steps for animal species. A subpopulation of thespecies is separated from the main population and becomes a new species.
Characteristics of a new species are set by adding slight changes to thoseof its ancestor species. These changes are given by random numbers drawn
from Gaussian distributions (G1 ) with a mean of 1.0 and a standard deviation of 0.1. When species d is born from species i, the values of wd , Pd , andeach element of Ad and Dd are set to wi + wi · G, Pi + 1.0 · G, Ai + 10.0 · G,
Di + 10.0 · G, and the values of rd of animal species and τd are determinedby the same equations above. The total biomass of species d is set to 5% ofthat of species i, and interaction coefficients of species d are decided by using
the same method as mentioned above.

1

I apply the Gaussian random number function in Numerical Recipes in C, Cambridge
University Press.

Chapter 4
Results
4.1

Regulations of Numerical Simulations

At the beginning of a run, a food web comprising of 50 animal species and
50 plant species was constructed. Using the random function (R 1 ), initialvalues of each parameters are given by the formula shown in Table 4.1. Runs
with 10000000 steps were iterated 10 times. Only the data of animal species
were analyzed in the present study. Due to the constraint of computational
capacity, the simulation was aborted if the number of species reached beyond
1000, and this case is called ”overflow” of species.
In my numerical system, the smaller species is, the faster it will evolve.
Then once very small species is generated, the world will be filled with thesmall species soon (Fig. 4.1). This result would be consistent with real living
world. Very small species (like bacteria) have enormous amount of variety
in fact. However, In this case, because the number of species go beyond
the numerical limit (=1000 species) in a minute, the analysis of diversity

1

I apply the random number function in Numerical Recipes in C, Cambridge University
Press.

26

27

CHAPTER 4. RESULTS

Table 4.1: Initial values of each species parameters in the system
Biomass (M )Bodyweight (w)Intrinsic growth rate (r)Defense property (D)Offense property (A)Feeding range (P )Characteristic time (τ )animal species plant species
10
10
0.03R
0.03R
3
−2(w/0.03) 4
100
100R
100R
100R
100R
10R
10R
1
4
(w/0.03)0

Parameters of animal species and those of plant species are shown respectively. R is therandaom function.

28

Number of species

CHAPTER 4. RESULTS

1000
900
800
700
600
500
400
300
200
100
0
0

50

100

150

200

250

Data/1000 steps

(a)

0.035
0.03
Bodywight

0.025
0.02
0.015
0.01
0.005
0
0

100

200

300

400

500

Order of species(b)Figure 4.1: (a) Temporal change of species diversity in the food web systemplotted every 1000 steps, and (b) bodyweight of each species in ascendingorder at 200000 step.

CHAPTER 4. RESULTS

29change of species cannot be made by the simulation, so that lower limits ofbodyweight of species was fitted to 0.003 in following simulations.

4.2

Diversity Change in The Food Web System

The results of the present study had two kinds of diversity pattern, and eachresult was not greatly affected when initial values of most species characters
were altered, as the values of characters changed through the evolution of
species. Within the first 100 steps of each simulation, the diversity in thesystem decreased to fewer than 20 species, then after this stage, two diversity
patterns were observed. The first pattern was characterized by the overflowof species, I call this phenomenon a ”overflow pattern” (Figs. 4.2(a) and4.2(b)). And the other pattern was characterized by the following sequence:the diversity drastically increased to several hundreds of species, then drastically decreased, and after the decline, the diversity maintained several dozenof species with a small degree of fluctuations, I call this phenomenon a ”maintain pattern” (Figs. 4.3(a) and 4.3(b)). In the 10 simulations performed, theoverflow and maintain pattern appeared 4 and 6 times, respectively (Table
4.2). The maintain pattern was partitioned into 4 stages roughly in Fig. 4.4.
Stages II, III, and IV will be called ”increasing stage”, ”decreasing stage”,
and ”stable stage” in this paper.
The results of maintain pattern are consistent with the fossil record; we
can see that many clades diversified rapidly at the beginning of a new geologic

30

1000
900
800
700
600
500
400
300
200
100
0

Number of species

Number of species

CHAPTER 4. RESULTS

0

1000
2000
3000
Data/1000 steps

1000
900
800
700
600
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Figure 4.2: Examples of evolutionary patterns for the overflow pattern: (a)and (b) species diversity, and (c) and (d) average rate of interaction between
each species.

31

CHAPTER 4. RESULTS

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(c)

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Figure 4.3: Examples of evolutionary patterns for the maintain pattern: (a)and (b) species diversity, and (c) and (d) average rate of interaction between
each species.

32

Number of species

CHAPTER 4. RESULTS

1000
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0

stage II stage III
stage I

0

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stage IV

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Data/1000 steps

10000

Figure 4.4: Partition of evolutionary pattern of species diversity for the maintain pattern: stage II, increasing stage; stage III, decreasing stage; stage IV,
stable stage.

33

CHAPTER 4. RESULTS

Table 4.2: Effects of altering evolving systems

Current version
・・・ Interval of speciation is halved
・・・ Interval of speciation is doubled
Dimensions of A and D are decreased to 2
・・・ Feeding range P is fixed to 2
・・・ Feeding range P is fixed to 8

Maintain Diversity Limit C
6/10
53.4
0.31
1/10
25
0.19
9/10
45.7
0.27
4/10
50.2
0.25
3/10
73.3
0.23
4/10
182.5
0.45

Results of 10 runs of each case are shown. Maintain, number of times that maintainpattern occurred; Diversity, means of ten runs of total species diversity in the system atthe end of run; Limit C, means of ten runs of connectance maximum values that gave rise
to a decline of diversity of species.

CHAPTER 4. RESULTS

34

era like after the Cambrian explosion and at the end of the Cretaceous mass
extinction (Gould et al. 1977). And they also are consistent with a pictureof ”punctuated equilibria”; the history of evolution is a story of homeostatic
equilibria, disturbed only rarely by rapid and episodic events of speciation(Eldredge and Gould 1972).
As well as diversity, I investigated the average connectance rate of interaction between each species, which was defined as the ratio of the number of
non-zero elements to the total number of elements in the interaction matrix
except the diagonal elements that represent intraspecific competition at eachtime step. In the case of the overflow pattern, the connectance was kept
to low value about 0.1 (Figs. 4.2(c) and 4.2(d)). On the other hand, inthe case of the maintain pattern, the high connectance (0.2-0.25) gave rise
to a rapidly decline of diversity of species (Figs. 4.3(c) and 4.3(d)). These
results are consistent with May (1972), which found that the stability of arandomly linked food web model tends to decrease with the proportion oflinks of species. My results, which showed the same connectance (0.2-0.25)
raise decline of diversity independent of initial values of species characters,also indicate that the limiting value of connectance would exist.

4.3

Effects of Alteration of Evolving System

To consider what affected the diversity of species or evolutionary patterns of
the system, I altered interval of speciation (see Eq. 3.7) and some parametersof species. To be more precise, I halved and doubled interval of speciation,

CHAPTER 4. RESULTS

35

decreased dimensions of A and D to 2, and fixed feeding range P to 2 and 8(Table 4.2).

4.3.1

Effects of Alteration of Evolution Timescale

The evolution timescale had a large effect on the determination of evolutionary patterns. Qualitatively, slower evolution results in occurrence probabilityof maintain pattern (Table 4.2). However, the limiting connectance was notaffected by the evolution timescale (in the system that interval of speciationis halved, however, only one case showed the maintain pattern. Thus, thiscase is not statistically significant). The diversity of species declined when
the connectance exceeds about 0.2, and the connectance also declined subsequently in any case. Assuming that these declination represent natural
selection, the following explanation could be derived: these results indicatethat number of species increased too far before the system was stabilized in
the faster evolution system, that is, a timescale of stabilization of the system
was longer than a that of natural selection in the food web system. It means
too fast evolution cannot stabilized the system. Comparing the maintainevolution pattern between in the slow and fast evolution, the stage II and III
is shorter in the latter case (Figs. 4.5(a) and 4.5(b)). In the fast evolution
system, the period of increasing stage and the decreasing stage was shorter
than that of slow evolution system.

36

250

300

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CHAPTER 4. RESULTS

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0

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6000
Data/1000 steps

(c)

8000

10000

0

2000

4000
6000
Data/1000 steps(d)Figure 4.5: Examples of evolutionary patterns of species diversity for cases:(a) interval of speciation is halved, (b) interval of speciation is doubled, (c)dimensions of A and D are decreased to 2 and feeding range P is fixed to 2,
and (d) dimensions of A and D are decreased to 2 and feeding range P is
fixed to 8.

CHAPTER 4. RESULTS

4.3.2

37

Effects of Alteration of Feeding Parameters

The range of the feeding zone P value and the dimensions of A and D had
a large effect on the behaviour of the system (Table 4.2 and Figs. 4.5(c) and
4.5(d)): species diversity increased gradually not so rapidly in the beginning,then stabilized at high diversity with passing little or no decreasing stage.
This result shows that these parameters related directly to the constructionof interspecific interactions.
The species diversity in the system related positively to the P value (Table 4.2). This is seemed to be inconsistent with May (1972), because it
suggests high connectance system hold high species diversity. I will provide
an explanation for that in section 5.2 later.
On the other hand, the dimensions of A and D affected the increasingstage. As compared Figs. 4.5(a) and 4.5(b) to Figs. 4.5(c) and 4.5(d), while
in default version’s system that has 10 dimensions of A and D, the number of
species increased to several hundreds first, then decreased to several dozens ora few hundreds, there were little or no decreasing stage in the 2 dimensional
system. It would due to the availability of parameter space, that is, freespaces exist everywhere in the large dimensional system. So many species
could spread large spaces and continue to increase the diversity over the longterm in the increasing stage. However, maintained diversity at the end of
the simulation was larger in the smaller dimension than the larger one. I will
discuss the issue in the Discussions chapter later

CHAPTER 4. RESULTS

4.4

38

Comparing of Size-Dependent/Independent
Evolution

Effects of size-dependent speciation frequency is a most important idea forthis study, because no study considered the evolutionary timescale for eachspecies to modeled complicated food web system. To examine the effects of
that, simulations of size-independent evolutionary system just like previous
studies were carried out and compared to my study’s results.
First, the interval of speciation was transformed to size-independent form;
any species evolve the same interval of 5000 time steps. The species diversity fluctuated randomly. Then, in many cases, the species became extinct
in time (Figs. 4.6(c) and 4.6(d)). Size distribution of species in the foodweb system was also greatly affected by the change of speciation frequency.
While size distribution of size-dependent system was followed a power law
distribution, that of size-independent system was plotted approximately asa linear function (Figs. 4.7(a), (b) and 4.7(c), (d)).
Next, additionally, metabolism of species (see Eq. 3.8) was transformed
to size-independent form; any species burn up its energy at the same rate asthat the intrinsic growth rate r = −2, that is in this model, larger species canlive with on the same amount of food as smaller species. The species diversity
was contained to very low (around 10 species) and fluctuated randomly, thenthe species became extinct soon (Figs. 4.6(e) and 4.6(f)). Size distribution
was plotted as a linear function like the model that has size-independent

CHAPTER 4. RESULTS

39

speciation frequency (Figs. 4.7(e) and 4.7(f)). The difference of species
diversity between these two models would due to the presence of large species.
Because of the same metabolic rate for any species, larger species was easier
to survive and eat out more small species than size-dependent metabolism
model, so species diversity of the total food web system was kept to lower.
Moreover, I also carried out simulations that evolution time scale wasdoubled to check the speciation frequency could not affect the species diversity just like the results of Section 4.1. Temporary change of species
diversity and the size distribution of that model were showed in Figs. 4.6(g),(h) and 4.7(g), (h). As anticipated from the previous results, the species
diversity was very low at the value approximately the same as the former
model (size-independent speciation frequency and metabolic rate), and thesize distribution was also similar to that. However, the fluctuation of species
diversity became a little larger because evolutionary rate was set to fastervalue and more species appeared at a shorter timescale.

40

CHAPTER 4. RESULTS

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Data/1000 steps

(g)

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Data/1000 steps(h)Figure 4.6: Figure caption is in the following page.

4000

41

0.006

0.006

0.005

0.005

Bodyweight

Bodyweight

CHAPTER 4. RESULTS

0.004

0.003

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0

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(c)

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0.008

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(g)

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20
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Order of species(h)Figure 4.7: Figure caption is in the following page.

40

CHAPTER 4. RESULTS

42

Figure 4.6: Examples of evolutionary patterns of species diversity forcases: (a) and (b) size-dependent evolution, (c) and (d) size-independent evolution interval, (e) and (f) size-independent evolution interval and metabolic
rate, (g) and (h) evolution timescale is doubled for the model (e) and (f).

Figure 4.7: Size distributions of each food web system. Bodyweight ofeach species was plotted in ascending order. (a), (b), ..., (h) represent thesame system as Fig. 4.6.

Chapter 5
Discussions
5.1

Analysis of Maintained State

5.1.1

Is Size-dependent Evolution Always Necessary?

The results of my numerical experiments indicates that size-dependent evolutionary system gave us solutions that have the similar behavior of evolutionof species such as (increasing stage - decrease stage - stable stage) patternirrespective of altering the evolution timescale or some parameters of species.
However, size-independent evolutionary system never evolved like that, andnever maintained high diversity. Therefore, at least, size-dependent evolutionary system is considered to be necessary to go through the evolutionary
pattern (increasing stage - decreasing stage - stable stage) for a food web system. Then, the next question is, whether size-dependent evolutionary system
is also necessary to maintain high species diversity in the system after entering a period of stability? To examine that, evolutionary system changed
from size-dependent to size-independent after entering a maintain stage (at10000000 time step), and compared the result to size-dependent system (Fig.
5.1).
43

44

CHAPTER 5. DISCUSSIONS

400
Number of species

350
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(a)

400
Number of species

350
300
250
200
150
100
50
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9000

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Data/1000 steps(b)

Figure 5.1: Examples of evolutionary patterns of species diversity forcases: (a) size-dependent evolutionary system and (b) the system that sizedependent system was replaced with size-independent system after enteringa maintain stage.

CHAPTER 5. DISCUSSIONS

5.1.2

45

Alteration of Size-Dependent/Independent Evolution

Species diversity gradually decreased and then the species became extinct
soon for the case that evolutionary system transformed from size-dependent
system into size-independent system at 10000000 time step (Fig. 5.1(b)).
To be specific, while size-dependent evolutionary system maintained about
50 species with low standard deviation after at 10000000 time step, sizeindependent evolutionary system reduced the diversity constantly and relegated the species to extinction for about 4500000 time steps (10000000 time
step to 14500000 time step). It shows that size-independent evolutionary system causes the collapse of species diversity in the food web system even after
the system has been stabilized, that is, size-dependent evolutionary system
is also necessary at all times to maintain the food web system. Therefore,saying from another viewpoint, the state that smaller species evolve andincrease its diversity faster than larger species is required to maintain thespecies diversity.

5.1.3

Implication for Observed Food Web System

This results imply that size-dependent evolutionary system should be observed in the real world if our food web system is stabilized and maintain thespecies diversity for a long term. In other words, power law size distributionof species as shown in Figs. 4.5(a) and (b) (not linear size distribution asshown in Fig. 4.5(c), ..., (h)) should be observed in the stabilized system. I

CHAPTER 5. DISCUSSIONS

46

have not come across detailed study of size distribution of species for global
food web system, however, present study would provide a new point of viewon ecology. I believe analysis of size distribution of present species and extinct species, which is investigated from fossil records, will be important toexamine the evolution of species diversity and extinction events in the animal
history.

5.2

Analysis of Interspecies Interactions

5.2.1

Apparent Discrepant Results

As described in section 4.1, The species diversity in the system correlated
positively to the P value (Table 4.2), and maintained diversity at the end of
the simulation was larger in the smaller dimension system than the larger one(Table 4.2), those are seemed to be inconsistent with May (1972) that showed
high connectance system cannot hold high species diversity. What are theirapparent discrepant results indicate for the food web system? I analyze thestate of interspecies interactions in the system, and examine these problems
in this section.

5.2.2

Simplification of Feeding System

For the sake of simplicity, It is assumed that the predator species can feed
the prey species when all elements of D of prey species j satisfy the following
condition:
Ai [k] − Pi ≤ Dj [k] ≤ Ai [k] + Pi

(5.1)

CHAPTER 5. DISCUSSIONS

47

where Ai [k] is the kth element of A of predator species i. In other words, if
prey species lives in predator species’ feeding zone (A ± P ) in all aspects, theprey species are preyed on. It’s just a black-and-white rule of the food web
system. In order to give a graphic representation of interspecies interactions,
2 dimensions A and D system are dealt with here. I have confirmed thatspecies diversity in the system behaved in much the same way as the system
dealt in the previous sections (Fig. 5.2), that is, the behaviour of evolution
is independent of dimensions of A and D.

5.2.3

Map of Interspecies Interactions in Food Web
SystemFigures 5.3 and 5.4 show ”maps” of interspecies interactions in the food web
system; Fig. 5.3 represents the case that feeding range P is set to 2, andFig. 5.4 represents the case P is set to 8, respectively. Each figure shows theparameter place ((x, y) = (D[1], D[2]) and (x, y) = (A[1], A[2]); for details
to figure caption of Fig. 5.3) where animal species (blue star) and plant
species (green star) live, and also shows a feeding zone of each animal species(red square). Bigger animal species can feed smaller animal species and allplant species, if those prey species place in the predator’s feeding zone. And,you can follow temporal changes of interspecies interactions, and see themaintained states for the food web systems in Figs. 5.3 and 5.4.
Both Figs.5.3(c) and 5.4(c) show that there are much more green stars(plant species living place) are inside red squares (feeding zone of each animal

48

CHAPTER 5. DISCUSSIONS

Number of species

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6000

Data/1000 steps(b)

Figure 5.2: Examples of evolutionary patterns of species diversity for cases:(a) dimensions of A and D are decreased to 2 and feeding range P is fixed
to 2 and (b) dimensions of A and D are decreased to 2 and feeding range P
is fixed to 8.

49

CHAPTER 5. DISCUSSIONS

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(c)

Figure 5.3: Maps of interspecies interactions in the food web system forthe case of P = 2. Blue star: animal species living parameter place((x, y) = (D[1], D[2])); Green star: plant species living parameter place((x, y) = (D[1], D[2])); Red square: feeding zone of each animal species((x, y) = (A[1] ± 2, A[2] ± 2)). (a) Initial state, (b) midterm state, and (c)final state of the simulation.

50

CHAPTER 5. DISCUSSIONS

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Figure 5.4: Maps of interspecies interactions in the food web system for thecase of P = 8.

CHAPTER 5. DISCUSSIONS

51

species) than blue stars (animal species living place). This seems to indicate
animal species feed on a plant species more than animal species. On thecontrary, it would mean that animal species are easy to grow at places where
no animal species is feeding. And both cases have some concentration of
feeding zone, and free places that are not in use in the parameter place.
Especially in the case of smaller feeding zone (Fig. 5.3), more free parameter
spaces exist. In the case of larger feeding zone, parameter space was used up
across the board, relatively. Thus more species, those had a large variety ofparameters A and D, could exist in the latter case than in the former case.
Because of that, the species diversity at the stable stage in the system related
positively to the P value.

5.2.4

Implication for Alteration of Parameter Dimension

The result, that maintained diversity at the end of the simulation was larger
in the smaller dimension system than the larger one (Table 4.2), could be
explained from the same viewpoint. In larger dimension system, many freeparameter space would exist. It means that species cannot expand to broad
parameter places and thus cannot increase its diversity in larger dimension
system. As compared to smaller dimension system, distance between eachfeeding zone is large, relatively. Then, it is difficult for animal species tochange feeding zone from successive feeding zone to other one by slight changeof properties through evolution. Therefore, the species diversity in the larger

CHAPTER 5. DISCUSSIONS

52

dimension system was smaller than that of the smaller dimension system inpresent study. On the contrary, considering invasion or new-production of
species that use free parameter spaces, the species diversity could increase
more.

5.3

Punctuated Equilibria

Paleontology’s view of speciation has been dominated by the picture of”phyletic gradualism” (Moore et al. 1952). It holds that new species arise
from the slow and steady transformation of entire populations. Under its
influence, they sought unbroken fossil series linking two forms by insensible
graduation as the only complete mirror of Darwinian processes; they ascribed
all breaks to imperfections in the record.
On the other hand, the theory of allopatric (or geographic) speciationsuggests a different interpretation of paleontological data (Raup & Stanley1971). If new species arise very rapidly in small, peripherally isolated local
populations, then the great expectation of insensibly graded fossil sequences
is a chimera. A new species does not evolve in the area of its ancestors; itdoes not arise from the slow transformation of all its forbears. Many breaks
in the fossil record are real.
Then, considering of the fossil records objectively, Eldredge & Gould(1972) concluded that the history of life is more adequately represented by apicture of ”punctuated equilibria” than by the notion of phyletic gradualism.
The history of evolution is not one of stately unfolding, but a story of home-

CHAPTER 5. DISCUSSIONS

53

ostatic equilibria, disturbed only rarely by rapid and episodic events of speciation such as mass extinction and mass explosion. My present study also
showed extinction just after the beginning of the simulation (mass extinction), rapidly increasing (mass explosion) and following rapidly decreasing(natural selection) of the species diversity, and then stabilization (long-term
equilibrium state) finally. I have not consider environmental variations in thepresent evolutionary system, however, environmental variations could affectthe behaviour of evolution. Especially, very large environmental variations
like the mass extinction events could collapse the food web system once, andthen the system could re-evolve (go through the same evolutionary route, increasing stage - decreasing stage - stable stage) again. Thus, although present
simulation showed only one transient stage (increasing stage - decreasingstage) respectively, multiple transient stages could be reproduced if environmental variations are considered in the numerical system (and could reproduce multiple evolutionary fauna like Cambrian-Paleozoic-Modern fauna(Fig. 2.1)). Considering environmental variations built into the evolutionary
system is an important issue in the future study of numerical ecology.

Chapter 6
Conclusions
By adopting the size-dependent evolutionary system for numerical experiments on variation of species diversity in the food web system, I derived
a particular evolutionary pattern: (1) rapidly decrease of species diversity
like mass extinction, (2) rapidly increase like mass explosion, (3) rapidly
decrease like natural selection, which was observed especially in larger parameter dimension model, then (4) equilibrated state that maintained overthe long term. And I confirmed that size-dependent evolutionary system isstable whereas size-independent evolutionary system raises the fluctuationand extinction of species diversity. The results implied that considering thesize-dependent system would be essential in establishing food web numericalmodel.
For meaningful discussions of general ecology, it is necessary to improve
the numerical model of food web system in future. Considering environmental variations, which represent fluctuations of a planet’s surface or external
disturbing, would be important to discuss long-term life history. And also,
we should consider precise energy balance between prey and predator in the54

CHAPTER 6. CONCLUSIONS

55

system, precise meaning of species’ parameters, precise way of evolution (interval of speciation or range of fluctuation of parameters), and so on. I
wish present study serve as a starting point toward making proper numericalmodels and understanding of general ecological system in the universe.

Acknowledgments
I would like to express my sincere appreciation to my supervisor Dr. Y. Abe.
His advice and comments that have guided this study properly. And many
discussions with him improve not only this thesis but also my philosophy of
research.
I would like to thank Dr. H. Kayane, Dr. E. Tajika, and other members in Earth and Planetary System Science Seminar for fruitful suggestions
and comments. I would also like to thank Prof. S. Sasaki, R. Machida, K.
Hamano, and the other members in L Seminar for their constructive comments on this study.
I would like to thank H. Genda for many advice on research in all aspects.
Finally, I would like to thank S. Watanabe, T. Ishii, T. Konishi, T. Yuuki,
Y. Uehara, and other members of the Department of Earth and Planetary
Science for making my everyday life a great time.

56

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[2] California & Carnegie Planet Search, http://exoplanets.org/.
[3] Cloud, P. 1968 Atmospheric and hydorospheric evolution on the primitive Earth. Science, 160, 729.
[4] Cloud, P. 1972 A working model of the primitive Earth. Amer. J. Scie.,
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