Ensemble Behavior from Individual Dynamics in
Multispecies Forest Populations
Carl Boettiger
Primary Adviser: Stephen Pacala
Departmental Adviser: David Huse
April 24, 2007

Pledge
This paper represents my own work in accordance with University regulations.
Carl Boettiger
Department of Physics
Princeton University
April 2007
1

Acknowledgements
I would like than my adviser Stephen Pacala for the endless energy and inspi-
ration he has provided. I would also like the thank Jeremy Lichstein for always
being available, insightful and willing to help me throughout my research and
writing process, and Drew Purves for his help. Finally, I would like to thank
David Huse for his advice and willingness to explore new topics.
2

Abstract
ENSEMBLE BEHAVIOR FROM INDIVIDUAL DYNAMICS IN
MULTISPECIES FOREST POPULATIONS
Carl Boettiger
In this paper we present the derivation of macroscopic equations governing forest
populations and use these to explore and understand interspecies interactions.
Traditionally, complex numerical models have been used to to predict the be-
havior of forest populations from the behavior of individual trees. We explore
the possible consequences of interspecies interaction, including an analysis of
several mechanisms for coexistence of different species. Using the macroscopic
equations we also reproduce the process of ecological succession and demon-
strate how it can enable coexistence of different species whenever these species
differ by a minimum distance in trait-space.

Contents
1 Introduction 6
1.1 Forest Populations . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 From Individual to Ensemble . . . . . . . . . . . . . . . . . . . . 9
1.2.1 Trees and Forests . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.2 Atoms and Solids . . . . . . . . . . . . . . . . . . . . . . . 10
2 An Analytic Model for Forests 14
2.1 Model Setup: the Ideal Tree . . . . . . . . . . . . . . . . . . . . . 14
2.2 Perfect Plasticity Approximation . . . . . . . . . . . . . . . . . . 16
2.3 The Macroscopic Equation . . . . . . . . . . . . . . . . . . . . . . 18
2.3.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.2 Stationary States . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.3 Simplifying the Model . . . . . . . . . . . . . . . . . . . . 20
2.4 Monoculture Predictions . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.1 Invasion of an Empty Habitat . . . . . . . . . . . . . . . . 23
2.4.2 Canopy Height . . . . . . . . . . . . . . . . . . . . . . . . 24
3 Forest Competition 26
3.1 Invasion Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1.1 Terms and Notation . . . . . . . . . . . . . . . . . . . . . 26
3.1.2 Derivation of the Invasion Condition . . . . . . . . . . . . 28
3.2 Breaking down the invasion criterion . . . . . . . . . . . . . . . . 29
1

3.2.1 Identical Shading . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.2 Identical Allometry . . . . . . . . . . . . . . . . . . . . . . 32
4 Consequences of Invasion 34
4.1 Dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2 Founder Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3 Coexistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.4 Invasion Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5 Coexistence 39
5.1 Introducing a shrub . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 Survival of the tree species . . . . . . . . . . . . . . . . . . . . . . 41
5.3 Invading the resident shrub . . . . . . . . . . . . . . . . . . . . . 41
6 Succession 43
6.1 Physiological Trade-offs . . . . . . . . . . . . . . . . . . . . . . . 43
6.2 The Successional Manifold . . . . . . . . . . . . . . . . . . . . . . 44
6.3 A climax community? . . . . . . . . . . . . . . . . . . . . . . . . 46
7 Succession with Coexistence 49
7.1 Simulating Forest Cohorts . . . . . . . . . . . . . . . . . . . . . . 51
7.2 Dynamics of Closing the Canopy . . . . . . . . . . . . . . . . . . 53
7.3 Stopping Succession by Patch Disturbance . . . . . . . . . . . . . 58
7.4 Niche Structure: Understanding Successional Coexistence . . . . 64
7.5 Limiting Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . 66
7.5.1 Ecological Consequences . . . . . . . . . . . . . . . . . . . 66
7.5.2 Evolutionary Consequences . . . . . . . . . . . . . . . . . 71
8 Conclusions 74
A Tree Data 79
B Simulation Code 80
2

List of Tables
2.1 Symbols for common quantities . . . . . . . . . . . . . . . . . . . 16
2.2 Common parameters of simplified model. . . . . . . . . . . . . . 23
4.1 Possible consequences of invasion . . . . . . . . . . . . . . . . . . 34
7.1 Values of constant parameters in simulation . . . . . . . . . . . . 52
3

List of Figures
1.1 Forest growth from an open patch . . . . . . . . . . . . . . . . . 8
2.1 The ideal tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1 Possible allometries . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.1 Invasion Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6.1 Physiological trade offs . . . . . . . . . . . . . . . . . . . . . . . . 45
6.2 Diameter vs successional position µD, equation (6.2). . . . . . . . 47
7.1 Succession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7.2 Areas of individual cohorts over time . . . . . . . . . . . . . . . . 55
7.3 Canopy height and its effect on cohort height. . . . . . . . . . . . 56
7.4 Phases in total crown area during canopy formation. . . . . . . . 57
7.5 Canopy Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
7.6 Disturbances allow coexistence . . . . . . . . . . . . . . . . . . . 61
7.7 Different disturbance regimes . . . . . . . . . . . . . . . . . . . . 62
7.8 Steady state seedling reproduction . . . . . . . . . . . . . . . . . 63
7.9 Age structure niches . . . . . . . . . . . . . . . . . . . . . . . . . 65
7.10 Effects of species similarity . . . . . . . . . . . . . . . . . . . . . 66
7.11 The late successional niche vanishes . . . . . . . . . . . . . . . . 67
7.12 Limiting Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . 68
7.13 More similar face stronger competition . . . . . . . . . . . . . . . 69
4

7.14 Coexistence Diagram . . . . . . . . . . . . . . . . . . . . . . . . 70
7.15 Effects of initial conditions . . . . . . . . . . . . . . . . . . . . . . 71
7.16 Rare early-successionals invade. . . . . . . . . . . . . . . . . . . 72
7.17 Pairwise invasibility plot . . . . . . . . . . . . . . . . . . . . . . . 73
5

Figure 1.1: Forest growth from an open patch. Image courtesy of British
Columbia Ministry of Forests and Range [20], reproduced with permission.
crown area and thin in density on a finite plot. The challenge of the theory
is to derive these variables describing the forest from the variables describing
individual trees.
The Debye solid consists of quantum harmonic oscillators on a finite lattice.
The oscillators have their own set of properties: their mode or frequency of
oscillation. The solid has properties such as specific heat, temperature, and so
forth. The theory derives relations in these properties from the properties of
the oscillators.
In Debye theory the oscillators are identical. We begin with this assumption
for forests as well, which we refer to as a monoculture, indicating that all trees are
from the same species. We are interested in understanding forests consisting of
multiple different species. Clearly with higher degrees of freedom we have more
interesting behavior to deal with. We will particularly be interested in temporal
dynamics of this heterogeneous forest. Here the concept of tree competition
becomes even more central. There are a few well known facts about interspecies
8

competition in forests that we hope to recreate, as well as answering questions
about long-term forest behavior that cannot be easily observed. It will be useful
to be familiar with these general phenomena before we explore further.
The two chief phenomena of interest go by the terms succession and coexis-
tence. Succession refers to the predictable changes in an ecological community
composition, where tree species that colonize open fields quickly are eventually
replaced by more shade-tolerant species. Coexistence refers to separate species
surviving in the same environment where neither one will ever out-compete the
other. Though the two phenomena seem contradictory, they are both easily
observed in nature. Part of our objective will be to understand how both of
these can occur.
1.2 From Individual to Ensemble
1.2.1 Trees and Forests
Our first goal is to derive the behavior of the forest from the behavior of the indi-
vidual trees; that is, to find the appropriate macroscopic equations of the forest
as an ensemble. Later we will use these macroscopic equations to investigate
interspecies interactions.
We choose to begin with individual trees rather than formulating a theory
at the level of the forest directly for several reasons. Chief among these, the
individual trees are much easier to understand than the whole forest, just as
a single harmonic oscillator is much easier to understand than the entire solid.
Additionally, the properties of an individual tree are much easier to measure
than the properties of the entire forest. Lastly, existing models based on indi-
vidual trees have proven very successful in the absence of models starting at the
forest-level. Because an individual tree can be more easily measured and mod-
eled, traditional approaches have involved entering these individual properties
into a giant particle simulator, a computer program that calculates the growth,
death, reproduction and so forth of each individual tree for each instant in time.
9

Statistical mechanics proceeds by collecting the individual states together
by means of a a partition function Z, the sum of Boltzmann factors exp(−²/τ),
effectively organizing them by energy τ . Our forest model will instead organize
individual trees by common age, termed cohorts.
From the partition function we can calculate probability distributions and
hence the expectation values of the macroscopic quantities we seek (such as
energy or specific heat). What follows a quick reminder of how this derivation
proceeds in the Debye case:
The partition function for the phonons is
Z =
∞∑
s=0
exp(−s~ω/τ) (1.1)
for fundamental temperature τ = kBT . With x ≡ exp(−~ω/τ) < 1, this has
the form
∑
xs and the sum converges:
1
1− exp(−~ω/τ)
(1.2)
The probability that the system is in state s is given by:
P (s) =
exp(−s~ω/τ)
Z
(1.3)
The thermal average value of s is then
〈s〉 =
∞∑
s=0
sP (s) = Z−1
∑
s exp(−s~ω/τ) (1.4)
Defining y ≡ ~ω/τ , observe
∑
s exp(−sy) = −
d
dy
∑
exp(−sy) =
−
d
dy
(
1
1− exp(−y
)
=
exp(−y)
[1− exp(−y)]2
(1.5)
11

independent of temperature (“low” requires that τ is much smaller than the
numerator, a quantity known as the Debye temperature). Hence the energy
scales as U ∼ τ4. The specific heat CV is simply the derivative of the energy at
constant volume (V = L3), which hence scales as CV ∼ τ3, hence the T 3 law.
The road ahead
We will now do the same for forest populations. To begin, we’ll need to know
a few things about the ideal tree. We will also need an assumption about
our system to make the analytics possible, analogous to the low-temperature
limit of Debye theory. We then combine individual trees into cohorts (where
individuals share a common age rather than a common energy as they do in ZN ).
Manipulating cohorts we can determine the macroscopic equations we need.
With those equations we will be able to explore the interspecies interactions.
13

Figure 2.1: The ideal tree. The dimensions of the ideal tree can all be char-
acterized by its diameter.
need not have constant diameter along the entire length of the stem, but only a
standard height at which we will measure the diameter. Traditionally this has
been diameter at breast height, dbh (we won’t worry too much about where
exactly we’re measuring this height, merely observing that this can be a well-
defined).
In addition to height z, and diameter D, our ideal tree has a certain crown
area A. Generally, the crown of our ideal tree will be two-dimensional or flat-
topped. Some common functions of three-dimensional crown shapes behave
identically to the simpler flat-top assumption under this model. While we will
make no assumptions about the particular shape of the flat-top, (indeed it need
not even be continuous), we will generally assume its area scales with the area
of the stem by a scale constant α.
The ideal tree grows at some rate G which we measure in terms of diameter
growth. The corresponding growth in height is computed based on the tree’s al-
lometry a(D). We will not specify a particular functional form for the allometry
of the ideal tree, but two common forms are
z = H2(1− exp
(
−H1H2D
)
(2.1)
15

variable z to worry about. Essentially this allows us to successfully apply a
mean-field approximation across any plane through the forest – we assume that
the properties of the forest at any position (x, y) for a fixed height z and time
t can be effectively approximated by their mean value across that plane.
Uniform canopy height
Another key consequence of this assumption is the observation that increased
phototropism (plasticity of tree canopies) results in a decrease in the variance
of crown join heights for three-dimensional crowns. Under the limit of perfect
plasticity it is conjectured that all crowns will join at a single height in the
canopy. This results in a well-defined height dividing the sunlit canopy trees
from the shaded understory, providing a natural extension between the flat-top
and three-dimensional crown model.
Motivation: choosing between extremes
This approximation was originally introduced by Pacala and colleagues to ad-
dress one of the greatest weaknesses of the SORTIE individual-based forest
simulator [18]. The original SORTIE model assumes that trees have perfectly
rigid crowns. Real trees exhibit crown plasticity as a result of phototropism,
as well as wind, and gravity, and the modular nature of crowns. Trees move
to fill gaps both by leaning of the stem towards an opening and through indi-
vidual branches growing into gaps [17]. The PPA is the opposite extreme of
perfectly rigid crowns. Simulations built on this assumption are remarkably
accurate [12,13]. While it is certainly an oversimplification, the reality appears
closer to this extreme than to the perfectly rigid crown extreme.
We consider the area of the crowns of each tree at height z, A(z), listed in
order from tallest to shortest. We will sum these areas from the tallest tree down
until the total area of crowns equals the area of the plot of forest. The height
at which we stop determines z∗. All trees below z∗ are then completely shaded,
and z∗ effectively defines our canopy height. Forest models have suggested that
17

N(z, t) the average population density for species i at height z. This describes
the species distributions and height distributions of the forest.
2.3.1 Derivation
Consider the population inside a box at (x, y, z, t). After a time ∆t, this popu-
lation can change in the following three ways:
• Trees in the box can grow up out of the box, at rate G(z, t)
• Trees from below can grow up into the box at rate G(z −∆z, t)
• Trees in the box can die, at rate µ(z, t)
We summarize these transitions respectively as follows:
∆N(z, t) = ∆tG(z, t)N(z, t)−∆tG(z−∆z, t)N(z−∆z, t)−µ(z, t)N(z, t) (2.6)
In the limit of small ∆t and small ∆z, we can rewrite this as
∂
∂t
N(z, t) = −
∂
∂z
[G(z, t)N(z, t)]− µ(z, t)N(z, t) (2.7)
giving us a partial differential equation (pde) for N(z, t). To solve this, we need
the boundary condition that the total population of the forest is conserved. We
write this by stating that the seedling population N0 at any time t must be over
its life time produce N0 seeds:
N(z0, t) =
∫ ∞
z0
f(z, t)N(z, t)
G(z, t)
dz (2.8)
Here the function f(z) gives the individual fecundity (offspring) of trees
of height z. Given some initial condition N(s, t = 0), equations (2.7) and
(2.8) predict both transient and stationary regimes of a single-species forest.
In the stationary regime, the initial size of each cohort is determined by the
conservation of total area of the plot,
19

this resource to grow taller (and hence get more sunlight), to survive better
(decrease death-rate µ through any of a number of strategies: making denser
wood, making toxins to ward off herbivores or insects, and so forth), or invest
that carbon in seeds, increasing fecundity. The dependence on height and time
are really a consequence on the light intensity varying with z and t.
Light intensity is nearly a step function
We can take our simplification one step farther by first observing that light
intensity is essentially a step function at the canopy. Above the canopy, trees
experience direct sunlight, while below the canopy trees are in the shade. The
difference between direct sunlight and shade is far greater than the different
levels of shade that might exist under the canopy. Consequently, our functions
f(z), G(z) and µ(z) can each be written as step functions with the step occurring
at the canopy height z∗. Below the canopy the trees will then grow at a rate
GD (for growth in the dark) and at rate GL for trees above the canopy.
Additionally, we observe that a tree’s fecundity depends primarily on its
carbon uptake, which is in turn proportional to the sunlight crown area, area,
f(z, t) = F · A(z, t) ∀z > z∗ and zero otherwise. As we assume constant
diameter growth instead of constant height growth, it will be useful to change
our spatial variable to diameter rather than height. This change of variables is
governed by the allometry of the tree, D = a(z). Instead of the canopy height,
we will keep track of the diameter at canopy height D∗ = a(z∗) to determine
which trees are above or below the canopy, since for any tree of height z and
diameter D, D > D∗ whenever z > z∗.
G(z)



GD D < D
∗
GL D > D
∗
(2.13)
We make the same assumptions regarding death rate µ(z) as discontinuous
21

we add some seeds of a different species to this forest. We refer to this species
as the invader, which we usually denote as species y. For a species to invade,
it must have a non-zero net growth rate in the environment set by the resident.
Consequently a given cohort must survive to reach the canopy and more than
replace its initial population. The effects of the other species can be felt by the
trees below the resident’s canopy, which may be different than the species would
experience below its own canopy.
The parameters of the ideal tree, G, µ, F , α (for both light and dark) can
vary between different species. In general these parameters will also depend on
the environment the ideal tree finds itself in. As light is our only resource, this
refers to the light environment. The light environment in turn is set by the
characteristics of the canopy – particularly its transmittance. Transmittance
is not an explicit parameter of the ideal tree, rather, it enters in the following
way. We consider the species y. It has its own death rate in the dark, µD.
This depends on the species y to which the ideal tree belongs, but also on the
transmittance of the canopy. If the canopy consists of trees of type x, we denote
the dependence of µD on that transmittance as µD(y, x). The first variable
always lets us know which species we are considering, while the second variable
lets us know what environment we are considering. For a monoculture x we
would write µD(x, x). Though the transmittance does not get its own variable,
it is still quantified. For instance if x casts a darker shade than y, we will have
µD(y, y) < µD(y, x) and µD(x, x) < µD(x, y), that is, trees die faster under
the canopy of x than under y. All other dark-based parameters are treated
analogously.
Parameters describing the tree behavior under direct sunlight will not be
influenced by the nature of the canopy below. Consequently, we do not denote
µL as µL(y, y) but simply µL(y) indicating that it depends on species y alone.
27

Making the same order of magnitude approximation as we had in (2.23), this
simplifies:
exp
(
µD(y, x)
D˜∗(x,x)
GD(y,x)
)
<
2αFG2L
µ3L
(3.5)
Isolating D˜∗(x, x), we recover the condition,
D˜∗(x, x) <
GD(y, x)
µD(y, x)
ln
(
2FαG2L
µ3L
)
(3.6)
Which we can write as simply:
D˜∗(x, x) < D∗(y, x) (3.7)
This invasion condition requires that the invader can over top the resident’s
canopy under the conditions set by the resident. Though the condition looks
simple, it can have a wide variety of consequences, depending on how it is real-
ized. Before we explore these further, we will need one more expression. It will
be sometimes useful to rewrite the invasion condition as a comparison between
the invader’s performance under its own canopy and under the resident’s. This
is done as follows. From equation 3.5, the right-hand side can be rewritten using
equation (2.23). Then taking the logarithms, we have the invasion condition,
µD(y, x)
D˜∗(x,x)
GD(y,x)
< µD(y, y)
D∗(y,y)
GD(y,y)
(3.8)
This condition also has an intuitive interpretation: the invader y must do better
in the resident’s x setting than in its own. Since any species will simply accom-
plish steady replacement in its own equilibrium monoculture (by definition), it
must do better than this to grow from a rare population under resident x.
3.2 Breaking down the invasion criterion
The invasion criterion (3.7) can be divided into two parts: one dealing with
shading and the other with allometry. To investigate the significance of each of
29

these, we will hold one fixed independent of species and explore the consequences
of variation between species in the other.
3.2.1 Identical Shading
Our first simplification assumes that the trees cast comparable levels of shade.
In this case, µD(y, x) = µD(y, y) = µD(y) and GD(y, x) = GD(y, y) = GD(y);
all growth and death parameters depend only on the species and not the en-
vironment. Consequently the diameters depend only on the species and not
the environment set by the resident, which allows us to reduce them to func-
tions of only the first variable, i.e D˜∗(x, x) → D˜∗(x) and D∗(y, x) → D∗(y)
Equation (3.8) then simplifies:
D˜∗(x) < D∗(y) (3.9)
Invasion now depends only on parameters we can measure in each species sep-
arately, we never need actually measure trees of the invader species growing
in the environment of the resident. Instead, we need only equilibrium canopy
heights of each species and the allometries of each species. By having a thin-
ner allometry (taller trees for a given diameter), a species is more resistant to
invasion. However, such a comparison is misleading, as thinner allometries will
probably result in higher death rates, µD and µL. D∗(x) depends linearly on
inverse dark death rate µD, while most of the allometries will be less than lin-
ear, hence for larger trees it becomes harder and harder to make up in allometry
what it loses in equilibrium D∗ by thinning allometry. Consequently these trees
would do better to focus on higher D∗ than to attempt to raise actual height of
the canopy at cost to the size of D∗.
Under this assumption of identical shade cast by both species, invasion can
only be one way, (assuming allometries must be monotonically increasing). This
may not be entirely obvious, particularly if one species has the higher allometry
along a certain domain and the other has the higher allometry over another
domain of diameters, as in Figure 3.1, so let us take a moment and prove it.
30

Chapter 4
Consequences of Invasion
To investigate possible interactions between two species y and x we ask two
questions: Can y seedlings invade a resident monoculture x, and can x seedlings
invade a resident monoculture y? The answer to each question is independent
of the other, hence we have four possibilities enumerated in Table 4.1.
4.1 Dominance
Dominance is the simplest criterion to satisfy under Table 4.1. We have already
seen one example of this by assuming shading is species independent. The
invasion condition, (3.7), compares canopy diameters, each of which can be
thought to consist of two parts, seen in expression (2.23). The first consists of
the shade dependent terms: GD(y, x)/µD(y, x), while the second involves the
logarithm of the lifetime reproductive value, expression (2.20). While the shade-
dependent terms depend both on the species y and the environment (canopy
True False
True Coexistence x Dominates
False y Dominates Founder Control
Table 4.1: Possible consequences of invasion
34

height) set by the resident x, the lifetime reproductive value depends on the
species alone. With
Λ(y, x) ≡ GD(y, x)/µD(y, x) φ(y) ≡ ln
(
2FαGL(y)3
µL(y)3
)
(4.1)
we can rewrite the invasion condition (3.7) (under identical allometry) and
rearrange to obtain
φ(x)
φ(y)
<
Λ(y, x)
Λ(x, x)
(4.2)
For y to be dominant we must also insist that x cannot invade once y deter-
mines the canopy,
φ(y)
φ(x)
>
Λ(x, y)
Λ(y, y)
(4.3)
If shade tolerances are equal, dominance requires a higher lifetime reproduc-
tive value, while if lifetime reproductive values are the equal, dominance requires
better performance in the shade. Recall that φ depends only on the logarithm
of the light growth and death rates, while Λ depends linearly on this ratio, and
consequently differences in shade are more likely to be responsible in determin-
ing the dynamics. Further, we can see that it is not enough for a species to cast
deeper shade than the resident to invade, Λ(·, y) < Λ(·, x), it must also be able
to grow better in the shade, Λ(y, ·) > Λ(x, ·). Finally, recall that if the species
cast identical shade, the problem becomes one dimensional and we always have
dominance of one species (ignoring the singular case of equality). In that case
y is dominant if condition
φ(y)
φ(x)
<
Λ(x)
Λ(y)
(4.4)
and otherwise x is dominant.
4.2 Founder Control
Founder control occurs when neither species can invade the other,
35

φ(x)
φ(y)
>
Λ(y, x)
Λ(x, x)
(4.5)
φ(y)
φ(x)
>
Λ(x, y)
Λ(y, y)
(4.6)
Using the notation we have just introduced, we can write this as
φ(x)Λ(x, x)
φ(y)Λ(y, x)
> 1 >
φ(x)Λ(x, y)
φ(y)Λ(y, y)
(4.7)
To satisfy this condition the two species must cast different levels of shade,
since we have already shown that without this only dominance is possible. For
founder control (4.7) the species that casts the darker shade must also be more
shade-tolerant. That is, if Λ(·, y) < Λ(·, x) then Λ(y, y) > Λ(x, y). Similarly,
the species casting the lighter shade must also be able to take advantage of that,
Λ(x, x) > Λ(y, x). This corresponds to the shade tolerant species having a less
steep response in growth rate and mortality to differing light levels than the
shade intolerant species. The shade intolerant species is opportunistic: able to
take advantage of greater light availability but also more influenced by higher
shade. This is equivalent to requiring that each species sets a higher canopy
under its own shade (correcting for allometry; as the ∼ reminds us):
D˜∗(x, x)
D∗(y, x)
> 1 >
D∗(x, y)
D˜∗(y, y)
(4.8)
4.3 Coexistence
When each species can invade the other neither is competitively superior, re-
sulting in ecological coexistence. The condition is easily derived as before,
φ(x)Λ(x, x)
φ(y)Λ(y, x)
< 1 <
φ(x)Λ(x, y)
φ(y)Λ(y, y)
(4.9)
To be able to satisfy both inequalities, the species need the same two sources
36

Adding this matrix to its transpose gives us the coexistence matrix 4.1(b). Here
Gray has the value unity implying one species is dominant, black is zero implying
the pair exhibit founder control, while white (two) corresponds to a coexistence
pair.
The opposite shading strategy to coexistence – founder control – seems more
logical: that a tree would cast such dark shade that only its own offspring could
successfully grow [2]. Nevertheless only three of species pairs exhibit founder
control. As this interpretation of coexistence is not very satisfactory, we return
to our model to find what knobs we can turn to enable coexistence.
38

Chapter 5
Coexistence
In investigating interspecies interactions, we had two particular phenomena we
wanted to explain: succession and coexistence. Succession we will see in the
following chapter, though many of the components are already in place through
our exploration of dominance. Meanwhile, coexistence appears more challenging
to explain. We already have a simple theory of coexistence, based on (4.9).
This requires a species to favor offspring of other species over that of its own,
which seems a very unlikely evolutionary consequence and receives little support
from known values of shade tolerance and shade cast for real species [1]. In this
chapter we will modify the model directly to allow for coexistence by introducing
a species that can reproduce in the dark, which we distinguish from other species
by the appellation “shrub.” The presence of the shrub not only influences the
steady state of the tree, but can have important consequences to the forest
stability.
5.1 Introducing a shrub
We consider such a shrub species which can reproduce in the shade of a tree
canopy and form its own sub-canopy. For simplicity, the shrub will not produce
under its own (darker) shade, but only under that of the canopy tree. Our
39

overhead canopy trees that only reproduces in the light, such as we have been
describing. We adapt our notation to handle the three resulting regions, which
we denote L above the tree canopy, D below the tree canopy but above the shrub
canopy, and DD below the shrub canopy. We will also allow the parameters
(growth rates, death rates, etc) in each of these regions to be species-dependent,
which we will denote as the argument S for the shrub and T for the tree. We
will not specify the environment, explicitly, as it will always be for both species
being present. Proceeding as before, the diameter of the shrubs at the height of
their (shaded) canopy will be
D∗∗ ≈
GDD(S)
µDD(S)
log
(
2FD(S)α(S)GD(S)2
µ3D(S)
)
(5.1)
Note that this will in general be different to shrubs growing in an open field
unshaded by a tree canopy. The presence of the second canopy will reduce
the canopy height attained by the tree canopy, (assuming the second canopy is
darker and hence death-rates higher and/or growth rates less than they would
be with no shrubs present). This correction will be proportional to the height
of the shrub canopy, and can also be calculated in much the same manner:
D∗(T, S&T ) ≈ D∗(T, T )−
(
µDD(T )GD(T )
µD(T )GDD(T )
− 1
)
D∗∗ (5.2)
Where the height of the tree canopy in the presence of the shrub D∗(T, T&S)
is decreased from the height it would attain without the shrub by an amount
proportional to the shrub height. For instance, if the tree dies twice as fast and
grows twice as slowly under the shade of the shrub than under its own shade, its
canopy is reduced by twice the height of the shrub canopy. To maintain stable
coexistence in this model, a few intuitive criteria must be satisfied.
40

Chapter 6
Succession
We now turn from coexistence to succession. A fundamental concept in ecolog-
ical theory, forest succession is also taught to youth as a classic example of the
dynamic nature of ecology. Despite its ubiquity, the details of the theory are
less well understood [23]. Here we will see that succession is an immediate con-
sequence of our model when (1) the interspecies interaction is dominance based
and (2) we introduce a constraint on the parameter space of possible species. We
have already seen that dominance is the simplest consequence of our model, and
its conditions are easily satisfied by most species pairs [1]. We now introduce
the second requirement, which will turn dominance into succession.
6.1 Physiological Trade-offs
Monoculture forests can always be invaded by trees with higher canopies. This
corresponds primarily to trees that either grow faster or live longer in while in
the dark. Thus far we have considered trees drawn from a five-dimensional space
of coordinates (µD, µL, GD, GL, F ). Though α can in theory be sixth dimension,
it tends to be constant between species and enters the model in the same way
as fecundity F , hence we need not consider its role independently. Even with
a few basic constraints such as GL > GD and µD > µL, this remains a high
43

dimensional space. Real trees will not occupy this full space as a consequence
of physiological design constraints: a tree cannot arbitrarily increase its growth
rates and death rates. In fact, most species have been shown to fall on a much
lower-dimensional manifold, even when more parameters are considered [22].
This results in relationships between the parameters that represent design trade-
offs the tree faces, such as investing resources in growth or in defenses that will
increase its life expectancy.
The relationship between growth rate in the light and survival in the dark
will prove particularly interesting to us. As suggested by Figure 6.1, increased
growth rate in direct sunlight (GL) comes at the cost of a linear decrease in
survival in the shade. This can be interpreted as a difference of strategies, where
a tree bets its finite resources on either the optimistic promise of fast growth
allowing it to close the canopy first and also generate more offspring faster, or
plays it safe: pessimistically assuming it will be over topped and spending its
resources defensively to increase its survival chances, knowing it can win out in
the long run.
6.2 The Successional Manifold
Motivated by the observations in Figure 6.1, we assume that growth rate in
direct sunlight, GL can be predicted by a linear function of µD:
GL(µD) = mµD + b (6.1)
For simplicity, we will assume that the other parameters, GD and µL re-
main essentially fixed by external factors, while species differ primarily on their
position on this trade off curve (6.1). If we consider an open plot colonized by
the seeds of a variety of tree species, those with the fastest direct-light growth
rates GL will close the canopy first. However, there offspring will manage less
well than the shade-tolerant species, which will eventually over top the current
canopy. The first population will now find itself permanently in the shade, and
44

Chapter 7
Succession with Coexistence
In this chapter we introduce disturbances that destroy all of the trees living in
a certain patch of the forest. This process could represent a variety of possible
events, from forest fires or diseases to logging for timber. The patch is reseeded
from the remaining forest, in proportion to abundance of each species in the
canopy. Because the patch is once again open, the successional march of species
begins again in that area. Early-successional trees that would otherwise be
eliminated from the forest population can recover on these patches, allowing for
coexistence in a successional, dominance-regime interaction. We demonstrate
that this coexistence is indeed possible as long as two conditions are satisfied:
(1) the disturbance frequency falls within a certain window (2) the two species
have a limiting similarity: they must be separated by at least some minimum
value on the successional axis.
Role of Transients
While ecological theory has traditionally operated under the assumption that
the states of nature we observe correspond to stable equilibria of a simple model,
much as we have done here, recent work emphasizes the importance of different
timescales and transients in ecological understanding [6], including species coex-
istence problems [7,8]. Our current model has at least two important timescales:
49

Figure 7.1: Succession Top: the canopy height passes through a transient
phase, the equilibrium of the early successionals, then finally settles at the
equilibrium of the late successionals. Bottom: Without disturbances, succession
results in an early dominance of the early successional species which then vanish
as the late successionals come to dominate (lower figure). In this example,
µD(1) = 0.15, µD(2) = 0.075. All other parameter values are as stated in
Table 7.1.
50

(a) the time required for the canopy to reach an equilibrium height, and (b) the
time required for the late-successional to dominate. The dynamics that occur
within both of these timescales will play an important part in our understanding
of the interspecies interactions. Both can be seen in a traditional example of
succession, such as the simulation in Figure 7.1. To this we will also add a third
timescale: the time required for the total population densities of the forest to
reach equilibrium in a given disturbance regime. While the first two timescales
will be relevant to the dynamics of individual patches, this one pertains only
to the forest as a whole. The importance of these transients means we will no
longer be able to look only for steady state solutions, but will turn to numerical
simulations to explore the time-dependence of our macroscopic equations.
7.1 Simulating Forest Cohorts
We will introduce a simple successional regime by assuming that all species cast
identical shade and live on a manifold such as (6.1), as in Chapter 6. We can
simulate the deterministic cohort model to investigate the full dynamics of the
system. The model is parameterized by the values specified in Table 7.1. We
assume all species have the same allometry, allowing us to compare diameters
directly to determine which species tree is taller at any instant. The simulation
keeps track of the density and height of each cohort for each species, advancing
them according to our macroscopic equations. At each time step the canopy
height must be redetermined, and then growth, death, fecundity, and dispersal
can be computed. The simulation is implemented in MATLAB r©; the base code
can be found in Appendix B. As the computation of the canopy-height is the
most subtle of these steps, we take a moment to explain how this is done.
Splitting cohorts at the canopy
The simulation is essentially identical to the analytic model: population den-
sities, heights, areas, and offspring densities are all continuous variables. The
model tracks age cohorts of trees rather than individual trees: each time-step
51

worry about the boundaries between them. Patches will only interact through
seed-dispersal: each patch contributes its seeds to the total forest pool each
year, which is subsequently divided evenly between all patches to seed the next
cohorts.
We will have a total P identically sized patches, each of which may be sub-
jected to a disturbance in turn. For simplicity, we will assume this disturbance
to be perfectly regular: a different patch is cleared every M years. One can
imagine this to be the case in a forest grown for logging, though we can also
generalize this to the case in which each patch has an independent probability
P (τ) of being leveled, where τ represents the age of the patch (time since it
was last leveled). After the first M ·P years, the forest will consist of a patches
with a uniform age distribution in our case, or one reflecting P (τ) in the general
case. After an initial transient phase, the state of any patch will be uniquely
determined by its age alone. Consequently, rather than ask what a particular
patch looked like 50 years ago, we can simply find another patch in the forest
that is 50 years younger than the patch in question. In this way, the spatial
structure of the forest echoes the temporal structure of any one patch. The dif-
ferent temporal niches of early and late successional trees can now be realized
as spatial niches as well.
7.2 Dynamics of Closing the Canopy
A step back: single patch monoculture
Before we consider the patch model in detail, We are going to take a moment
to understand the time-dependence of the canopy height z∗ before it reaches
the familiar equilibrium level we have been dealing with thus far. Everything
in this section holds for all of the forests we have been describing, down to the
simplest case of monoculture on a single patch. We have only ignored it until
now because we could. However, an understanding of the patch dynamics ahead
will hinge on an understanding of this transient phase of the canopy formation.
53

So let us take a step back, and with the help of our simulation, consider what
happens when a canopy closes in the Pacala forest.
Area growth in the light
We described the highest canopy trees continually overshadowing more and more
lower-canopy cohorts. We can look at this process more precisely. For a given
cohort living entirely in the light its area A(t) grows as:
A(t) = N0(1)α[GLτ ]
2 exp(−µLτ) (7.1)
which is plotted in figure 7.2 for several different cohorts. The curve is char-
acterized by its initial increase due to the quadratic growth of the crown area,
which eventually is overtaken by the exponential death term, causing the total
area to decrease. We can easily solve for the location of this maximum, τ˜ ,
τ˜ =
2
µL
(7.2)
As this depends only on τL, each cohort growing entirely in the light will reach
its peak area at the same age, at τ˜ , regardless of its initial seedling density N0.
(In fact in the successional patch model it will be the same between species as
well.) The early cohorts may pass this maximum before the canopy closes, such
as the founding first cohort, plotted in solid black in Figure 7.2. Though they
will remain in the canopy after it first closes, the total area of these early cohorts
declines, opening new area that will be filled by the lower, younger cohorts.
The closure of the canopy is a consequence of the younger cohorts that have
yet to reach their maximum area, such as the 35th cohort in Figure 7.2. The
cohorts immediately following the founder cohort are negligibly small, as they
are seeded entirely by the founder cohort when it is of negligible area itself. The
later cohorts are larger and larger, which can be seen in comparing the 35th
and 40th cohort in Figure 7.2. The increase in seedling numbers can be seen
in Figure 7.8(a) and is discussed in more detail later. The contribution to the
54

Figure 7.2: Areas of individual cohorts over time. The areas of each
individual cohort living entirely in the direct sunlight follows equation (7.1).
total area of these cohorts younger than τ˜ more than makes up for the decrease
in area in all the cohorts older than τ˜ . When the canopy closes, the youngest
couple of these will be left in the dark, condemned to grow at a slower rate µD.
Meanwhile, the cohort at the canopy height belongs to one of those younger
than τ˜ , responsible for the still growing area. As the area of the cohorts younger
than τ˜ increases faster than space can open up from the cohorts older than τ˜ ,
those near the canopy height will soon find themselves overshadowed, and the
height of the canopy continues to move up. This effect manifests itself in several
ways. First, we can see it happen most directly in the 40th cohort in Figure 7.2,
where it jumps down from its peak. The decrease in area is due to the cohort
being split: some members were demoted to a cohort immediately below it,
beneath the canopy. Soon after, it reaches age τ˜ = 80 (using µL from Table 7.1)
and no longer increases in area.
55

Figure 7.4: Phases in total crown area during canopy formation.
with the canopy is responsible for the dramatic change in the area of of these
cohorts seen in Figure 7.2. The area of cohort 45 initially shrinks as the cohort
is partitioned successively by the finite sunlit area restriction. By the time it
reaches the light again, its area is already into its decreasing phase. Cohort
50 enters the darkness almost immediately, (the canopy closes at year 53), and
displays an arc similar to that of the light-based area growth, only smaller due
to the dark parameters. Once it breaks into the light it repeats this arc, once
again increasing area initially before declining. The inflection between these two
bumps in cohort 50 curve of Figure 7.2 corresponds exactly with its re-entrance
into the light, seen in Figure 7.3.
Dynamics of the total crown area
Now that we have an understanding of the behavior of the individual cohorts, we
can better interpret the total area of all tree crowns in the forest in Figure 7.4.
57

1) and an equal number of seeds corresponding to a later-successional type;
µD(2) = 0.075 (species 2). We track the population size of each species over
time as well as the height of the canopy (which is a property of the plot, not
a particular species), in Figure 7.1. Initially the early-successional population
increases faster: thanks to its higher growth-rate in the light, which results in
faster growing tree crowns and hence more offspring. At first the area of all the
crowns is less than the area of the total plot, and the canopy height remains at
zero. Suddenly the area of all the crowns is enough to close the canopy, and the
canopy height jumps (discontinuously) up to some small value. At this point
the youngest cohort still in the canopy may belong to the slower-growing late-
successionals. However, as the cohorts above them continue to grow taller, they
also increase their area, shutting out more and more of the cohorts below from
direct sunlight. As the highest cohorts belong to the early-successional, soon
there are no late-successionals in the canopy. Cut off from the direct sunlight,
the late successional population cease reproducing, while dying off at a constant
rate µD(2), and their population begins to shrink. Nevertheless, those remaining
continue to grow, though at the slower rate GD(2), and do not die as fast as
the new seedlings of the early-successional trees, born under the darkness of the
canopy.
Late successionals break into the canopy
The canopy of the early-successionals ceases to rise not long after it closes.
However, it fails to equilibrate: instead of finding own offspring rising to fill
the gaps that open as the older sunlit trees slowly die, it finds the remaining
late-successionals. What happens as these late successionals reach the sun-
light? They have been growing at the same rate as the early-successional’s own
seedlings, GD(1) = GD(2). However, they enjoyed an initial boost growing at
GL and dying at µL until the canopy closed (recall no late-successional cohorts
have been born since the canopy first closed). Further, they have been dying at
a slower rate than the seedlings of the early successionals. Consequently, while
59

Figure 7.5: Canopy height. The time-evolution of the canopy height z∗ for a
two-species patch is compared with that of each species alone. The equilibrium
canopy height matches with that predicted for each species by equation (2.23)
the early-successional seedlings arrive with just enough area to close the gaps
opening above, these late-successionals arrive with the potential to increase the
total area, raising the height of the canopy.
This time the canopy rises less dramatically then before, as seen in figure 7.5.
The middle curve, shows the canopy increasing from the equilibrium height
of the early successionals to that of the late successionals. This process is
more gradual as the late-successional cohorts reaching the sunlight are older
and hence farther along on the curve (7.1). Reaching the light allows them to
resume seedling production, which has been absent throughout the time they
were over-topped by the early-successionals, as seen in Figure 7.8(b). This
second generation of seedlings will eventually continue to carry the canopy up.
This gradual increase does not overshoot, settling into the steady state canopy of
the late-successionals. Meanwhile, new early-successional cohorts can no longer
reach the canopy. As the area of those early-successional cohorts above the
canopy decreases with the exponential death, so does their offspring production,
as seen in figure 7.8(a).
60

(a) (b)
Figure 7.7: Different disturbance regimes (a) As the frequency of distur-
bances increases, the early successional trees perform better. This time a differ-
ent patch is leveled every 14 years, hence the maximum age of any patch is 140
years. (b) When the disturbances are too frequent, only early successionals can
survive. A different patch is leveled every eight years, making the maximum
age eighty years. This does not give enough time for a sufficient number of
late-successionals to reach the canopy and reproduce.
Consequences of the disturbance frequency
The introduction of these disruptions halts the process of succession, allowing
the two species to reach equilibrium densities, Figure 7.6. Because the forest
experiences the disruption every M years, the equilibrium is periodic rather
than constant. The populations of both species grow immediately after a dis-
ruption and drop by 1/P every M years when a plot is leveled. The frequency
of disturbance determines the equilibrium levels each population can reach. For
large values of M · P , disturbances are just frequent enough to prevent the loss
of the early successional species, as seen in Figure 7.6. As the frequency of the
disturbances increases, the early successional population can maintain a higher
density, Figure 7.7(a). If disturbances are too rare, the early-successionals con-
tinually decrease, Figure 7.1. As densities are continuous, it is difficult to judge
if the population will equilibrate at extremely low densities at long times, though
this can be treated as effective extinction for real populations. Meanwhile, if the
62

(a) (b)
Figure 7.8: Steady state reproduction. (a) Early-successional. (b) Late-
successional. Curve represents the steady state seedling output over time for a
single patch. Consequently the dynamics of a single patch are periodic every
200 years. Asterisks plot the spatial distribution: seedlings produced by each of
the five patches against the age of each patch. At equilibrium the spatial and
temporal distributions match. Simulation included P = 5 patches, with one
leveled every M = 40 years.
patch lifetime M · P is shorter than the time required for the late successionals
to reach the canopy, they face extinction, Figure 7.7(b).
Coexistence, then, is possible even in the simple successional regime, pro-
vided patches of the forest can be disturbed. This effectively divides the forest
into niches where an early successional species can perform well and patches
where the late successional performs better. The niches are essentially tempo-
ral but have taken on a spatial reality by having patches of different ages. This
coexistence is not contingent on the particular manner in which disturbances
enter, it simply requires that patches exhibit a distribution of ages at all times.
In some sense our simple model is not the simplest, as the age structure actually
changes in between disturbances. Consequently the forest exhibits a periodic
age structure, rather than a constant age structure it would have, say, if we lev-
eled a patch every year (M = 1) and had enough patches so that the maximum
age of any patch P was high enough to permit late-successionals to over top the
early-successional canopy. Any of a variety of more complicated disturbance
63

Figure 7.10: Effects of species similarity. Decreasing the successional dis-
tance between the two species decreases the seed output of the late successionals
and increases that of the early successionals.
down the successional axis as before. The early-successional species will perform
identically well at the beginning, but will do better on older patches in the second
forest then in the first, Figure 7.10, even though its own species parameters are
identical in both forests. Meanwhile, the late-successional will perform better
in the second forest than in the first forest until the canopy closes, thanks to
its higher light-growth rate. However, it performs less well in the second phase,
due to its higher death-rate in the dark, Figure 7.10. Consequently, the size
of the niche for the late-successional shrinks for species that are more similar.
Clearly this would also be the case if we held the late successional fixed and
picked an early-successional farther down the successional axis (lower µD).
7.5 Limiting Similarity
7.5.1 Ecological Consequences
The observations of Figure 7.10 suggest that a limiting similarity exists between
two species. For a given disturbance regime, two species can only exist if each has
a niche, which requires that the species differ by a critical amount. As the species
become more similar, the size of the late-successional niche shrinks and then
66

(a) (b)
Figure 7.13: More similar species face stronger competition. When too
similar, populations cannot coexist. The closer they are, the faster the extinction
of the later-successional. (a): µD(1) = 0.15, µD(2) = 0.1278 (b): µD(1) = .1500,
µD(2) = .1499
sities sustained by lower death rates which will eventually allow it to establish
a higher canopy. Hence the late-successional niche shrinks and then vanishes,
Figure 7.11.
Limiting similarity implies that any species can coexist with another that is
either sufficiently earlier or sufficiently later on the successional axis. Exactly
what corresponds to “sufficiently” depends exactly where on the successional
axis we begin. To illustrate this, we construct a phase-diagram, Figure 7.14,
where each axis represents the successional position µD of one of the species.
As long as these species are sufficiently different, the forest exists in a region
of coexistence. Two similar species result in a position very close to the line
y = x, where coexistence becomes impossible. The size of the limiting similarity
increases as we consider higher values of µD corresponding to earlier-successional
species.
Maximum separation?
While coexistence is impossible for species that are too similar, species can be
arbitrarily different and still coexist, provided the disturbance regime allows the
69

Figure 7.17: Pairwise invasibility plot
ure 7.17, see [5, 11]. The positive region above the line y = x indicates that
the invader must have a larger value µD to invade, regardless of the trait of
the resident. Because the adaptive dynamics framework considers only nearby
invaders, the ecological coexistence at a limiting similarity does not appear.
In a population where not all patches would live to the same maximum age,
this process would result in differential rates of evolution that could eventually
achieve the limiting similarity. This would result in species different enough
to coexist, representing a mechanism of speciation not normally considered in
the present allopatric speciation models or the framework of adaptive dynamics
[5, 11].
73

presence or absence of coexistence is determined by which species first colonizes
the plot. This represents an interesting complication to the simple coexistence
and founder control possibilities previously treated.
Returning to the successional model, we have demonstrated that this too
can exhibit coexistence of different species, provided the species differ suffi-
ciently from each other in their trait-space. Because of this limiting similarity
requirement, very similar species such as might be expected to result from mu-
tational processes in evolution cannot coexist with their parental type. We
have explored why the early successional species will always out-perform the
late successional in cases of limiting similarity. Some other process, possibly
not captured by the theory, will probably be responsible for a restoring force
to generate late-successional types. The consequences of this observation merit
further investigation.
Through our exploration of this theory we have only scratched the surface.
Stephen Pacala’s lab and collaborators are currently exploring many aspects
that stem from this basic theory, ranging from a rigorous analysis of steady
states and the exclusion of limit cycles in the formation of the canopy to con-
sequences of the perfect plasticity assumption in more complicated models of
forests as a way to remove spatial dependence [3,22]. Many other aspects await
further exploration, such as the evolutionary consequences of the interactions
studied here.
This theory comes at an interesting time in theoretical ecology. Pioneering
work in simple models by theoreticians such as Robert MacArthur and Robert
May in the 1960s and 1970s gave way to skepticism and uncertainty about their
connections to available data. [18]. While numerical simulation models started
to bridge the gap, theory has emphasized the importance of spatial depen-
dence [19] in these models, as well as other fundamentally challenging elements
to treat mathematically such as stochasticity. The theory discussed represents
a breakthrough allowing the return of a simple model while still retaining close
ties to data and individual measurements that are better understood than com-
munity level measurements and models. It challenges the paradigm of spatial
75

dependence always being critical to an understanding of such complicated sys-
tems, but also shows that such dependence must nevertheless be dealt with
carefully. The perfect plasticity assumption has been much more successful
than previous mean-field approaches, representing the importance of the cor-
rect assumptions and simplifications in theory. In the end, it is these aspects
that make this more than another simple model of trees; these aspects make it
a theory.
76

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waarden. Adaptive dynamics: A geometrical study of the consequences of
77

nearly faithful reproduction. In S. J. van Strien and S. M. Verduyn Lunel,
editors, Stochastic and spatial structures of dynamical systems, pages 183–
231. KNAW Verhandelingen, Amsterdam, 1996.
[12] K. J. Mitchell. Simulation of growth of even-aged stands of white spruce.
Yale University Scholastic Forest Bulletin, 75:1–48, 1969.
[13] K. J. Mitchell. Dynamics and simulated yield of douglas fir. Forest Science
Monographs, 17:1–39, 1975.
[14] K. J. Mitchell. Distance dependent individual tree stand models. In K. M.
Brown and F. R. Clarke, editors, Forecasting forest stand dynamics: pro-
ceedings of the workshop, June 24,25, pages 100–137. Lakehead University,
Thunder Bay, Ontario, Canada, 1980.
[15] P.R. Moorcroft, G. C. Hurtt, and Stephen W. Pacala. A method for scal-
ing vegetation dynamics: The ecosystem deomgraphy model. Ecological
Monographs, 71:557–585, 2001.
[16] D. D. Munro. Forest growth models-a prognosis. In J. Fries, editor, Growth
Models for Tree and Stand Simulation, pages 7–21. Royal Coll. For. Res.,
Stockholm, 1974.
[17] C. C. Muth and F. A. Bazzaz. Tree canopy displacement at forest gap
edges. Canadian Journal of Forest Research, 32:247–254, 2002.
[18] S. W. Pacala, C. D. Canham, J. Saponara, R. K. Kobe, and E. Ribbens.
Forest models defined by field measurements: estimation, error analysis
and dynamics. Ecological Monographs, 66:1–43, 1996.
[19] Stephen W. Pacala and Douglas H. Deutschman. Details that matter: the
spatial distribution of indvidual trees mantains forest ecosystem function.
Oikos, 74:357–365, 1995.
[20] K. Polsson. British Columbia Ministry of Forests and Range.
http://www.for.gov.bc.ca/hre/gymodels/TASS/features.htm.
[21] D. W. Purves, J. P. Caspersen, P. R. Moorcroft, G. C. Hurtt, and S. W.
Pacala. Human-induced changes in US biogenic volatile organic compoind
emissions: evidence from long-term forest inventory data. Global change
biology, 10:1737–1755, 2004.
[22] D. W. Purves, J. W. Lichstein, and S. W. Pacala. Crown plasticity and
competition for canopy space: a spatially implicit model parameterized for
250 north american tree species. Unpublished manuscript, 2007.
[23] Herman Shugart. A Theory of Forest Dynamics: The Ecological Impli-
cations of Forest Succession Models. Springer-Verlag, New York, Berlin,
Heidelberg, Tokyo, 1984.
78

Appendix A
Tree Data
The following data on growth and mortality taken from the USDA Forest Inven-
tory database which contains measurements of several million individual trees.
Estimates from Drew Purves, see [21] and [1].
English name GL µL µD(x, x) GD(x, x)
Loblolly Pine 0.403 0.0123 0.098 0.072
Quaking Aspen 0.518 0.0142 0.085 0.050
Virginia Pine 0.538 0.0093 0.051 0.032
Water Oak 0.308 0.0112 0.057 0.134
Shortleaf Pine 0.365 0.0148 0.096 0.046
Southern Red Oak 0.338 0.0089 0.043 0.098
Sweetgum 0.315 0.0107 0.058 0.075
Yellow Popular 0.236 0.0192 0.134 0.118
Black Cherry 0.335 0.0079 0.058 0.062
Post Oak 0.391 0.0100 0.036 0.061
Red Maple 0.437 0.0054 0.027 0.041
Black Oak 0.576 0.0132 0.075 0.077
Paper Birch 0.283 0.0093 0.038 0.026
White Oak 0.383 0.0062 0.036 0.040
Red Oak 0.502 0.0076 0.054 0.043
White Ash 0.365 0.0104 0.058 0.080
Blackgum 0.490 0.0109 0.041 0.055
White Pine 0.572 0.0090 0.071 0.055
Chestnut Oak 0.373 0.0084 0.046 0.054
American Beech 0.433 0.0059 0.035 0.045
Sugar Maple 0.286 0.0048 0.025 0.029
Eastern Hemlock 0.420 0.0027 0.016 0.028
79

Appendix B
Simulation Code
Basic MATLAB r©code used in simulations of Chapter 7 is provided on the
following pages. To obtain this and related codes, email cboettig@princeton.edu.
80

207

4

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82

207

4

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83