Studies in Living Porous Media

by

Samuel Alaii Ocko

Submitted to the Department of Physicsin partial fulfillment of the requirements for the degree of

Doctor of Philosophy

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

September 2015

Massachusetts Institute of Technology 2015. All rights reserved.

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Studies in Living Porous Media

by

Samuel Alan Ocko

Submitted to the Department of Physicson September 1, 2015, in partial fulfillment of the

requirements for the degree ofDoctor of Philosophy

Abstract

Many biological systems need to control transport of nutrients and ventilation. Un-like many nonliving porous media, they modify themselves to meet these demands;they are active. Using a combination of experiment, theory, and computation, weinvestigate several living porous media.

First we consider termite mounds, meter-sized structures built by insects nearlythree orders of magnitude smaller than the mounds themselves. It is widely acceptedthat the purpose of these mounds is to give the colony a controlled microhabitat thatbuffers the organisms from strong environmental fluctuations while allowing them toexchange energy and matter with the outside world. However, previous work towardunderstanding their functions has led to conflicting models of ventilation mechanismsand little direct evidence to distinguish them. By directly measuring air flows insidemounds of the Indian termite Odontotermes obesus, we show that they use diurnalambient temperature oscillations to drive cyclic flows inside the mound. These cyclicflows in the mound flush out CO 2 from the nest and ventilate the colony, in a novelexample of deriving useful work from thermal oscillations. We also observe the samediurnally-driven flows in mounds of the African termite Macrotermes michaelseni,evidence that this is likely a general mechanism.

We then consider the problem of honeybee swarming, wherein thousands of beescling onto each other to form a dense cluster that may be exposed to the environ-ment for several days. During this period, the cluster has the ability to maintain itscore temperature actively without a central controller. We suggest that the swarmcluster is akin to an active porous structure whose functional requirement is to ad-just to outside conditions by varying its porosity to control its core temperature.Using a continuum model that takes the form of a set of advection-diffusion equa-tions for heat transfer in a mobile porous medium, we show that the equalization ofan effective "behavioral pressure", which propagates information about the ambienttemperature through variations in density, leads to effective thermoregulation. Ourmodel extends and generalizes previous models by focusing the question of mecha-nism on the form and role of the behavioral pressure, and allows us to explain thevertical asymmetry of the cluster (as a consequence of buoyancy driven flows), the

3

ability of the cluster to overpack at low ambient temperatures without breaking upat high ambient temperatures, and the relative insensitivity to large variations in theambient temperature. Our theory also makes testable hypotheses for the responseof the cluster to external temperature inhomogeneities, and suggests strategies forbiomimetic thermoregulation.

Finally, we consider a generic model of an active porous medium where the con-ductance of the medium is modified by the flow and in turn modifies the flow, so thatthe classical linear laws relating current and resistance are modified over time as thesystem itself evolves. This feedback coupling is quantified in terms of two parametersthat characterize the way in which addition or removal of matter follows a simple lo-cal (or non-local) feedback rule corresponding to either flow-seeking or flow-avoidingbehavior. Using numerical simulations and a continuum mean field theory, we showthat flow-avoiding feedback causes an initially uniform system to become stronglyheterogeneous via a tunneling (channel-building) phase separation; flow-seeking feed-back leads to an immuring(wall-building) phase separation. Our results provide aqualitative explanation for the patterning of active conducting media in natural sys-tems, while suggesting ways to realize complex architectures using simple rules inengineered systems.

Thesis Supervisor: L. MahadevanTitle: Professor

Thesis Supervisor: Mehran KardarTitle: Professor

4

Introduction

In order to survive and reproduce, organisms must be able to cope with, take ad-

vantage of, and sometimes manipulate a complex physical environment. Often, much

of this complexity comes from fluids in the environment, which transport heat and

other resources to and from the organism through a complex geometry. Flow through

porous media is important in many problems in physics, biology, geology, and engi-

neering. Most studies are limited to transport through static media, and focus on

how geometry leads to permeability, and how permeability leads to flow. However, in

many living (and nonliving) systems, transport and flow can lead to the remodeling

of the medium itself. This leads to a coevolution of the medium with the flow through

it. Examples of active porous media abound in both living and nonliving systems.

In nonliving systems, drainage networks are formed through the interplay of ero-

sion, transport, and deposition [16, 24]. Higher flow velocities in channels can increase

erosion, a positive feedback mechanism that results in widening the channels further

[51]. Channels also grow and bifurcate based on groundwater flows that supply them

[1, 76], leading to hierarchical structures[69]. In fuse networks, commonly used to

model fracture, edges with too much current are broken. When an edge breaks, the

current has to redistribute itself through nearby edges, which can lead those edges to

break, leading to growing cracks and eventual catastrophic failure [17, 100]. There

are open questions regarding the statistics of failure and crack propagation.

In living systems, slime molds (Fig. 0-1(a)) form networks to efficiently connect

nutrient sources [28, 29]. The organism creates synchronized peristaltic flows to en-

hance transport [59, 5], and tubes with higher flow tend to grow at the expense of

those with less flow. These single celled organisms have been shown to solve maze

5

and optimization problems [57, 74, 89, 58], but the full set of rules that leads to this

behavior is not well understood.

One example of high practical importance, and possibly the most complicated, is

the remodeling of vascular networks in animals (Fig. 0-1(b)). In 1918 it was observed

that the vasculature of the chicken embryo would not develop the same if the heart

was removed [10]. Since then, many feedback mechanisms have been observed in

vascular remodeling. Vessels enlarge as a result of currents, hypoxia, or signaling,

and contract due to high transmural pressure; these factors can also control the

creation of new vessels [60, 73, 77, 45, 44, 82, 71]. One important unsolved problem is

how shunts are prevented [72]: if a low resistance vessel has high flows through it, it

can grow even larger, taking up more current without actually supplying any tissue.

Preventing shunts seems to require some level of upstream communication which is

not well understood.

Other examples of active porous media are insect-built structures. Bees, termites,

and ants (Fig. 0-1(c)) all build elaborate structures to create a microclimate for

the colony [92, 91, 13, 87, 8, 40, 42]. The structures created are remodeled due to

feedback from temperatures, gas concentrations, and airflows [34, 36] (Fig. 0-1(d)).

There are open questions regarding both the physiology of these structures as well as

the behavioral rules that give rise to them.

In other living porous media the animals themselves form the structure [7]. Swarm-

ing honeybees(Fig. 0-1(e)) and bivouacs of ants cluster together in groups of - 104

individuals, maintaining a relatively constant core temperature despite variations in

ambient conditions [11, 32, 31, 20]. In a very different context, some penguins hud-

dle together to reduce heat loss [98, 101], alternating which individuals are at the

center of the huddles. There are interesting questions of how the dynamics of these

assemblages are determined by the relatedness of the individuals that comprised them

[27].

These problems in living porous media exist at the interface between biology and

physics. How matter is transported through the medium is a physical question; how

the medium remodels itself is a biological one. What is required is a full understanding

6

Figure 0-1: Some living porous media. a) Network formation in slime molds [89]. b)Development of vascular networks in chicken embryo [44]. c) Randomly distributed

ant corpse clusters(top), are rearranged into piles aligned with a downwards imposedairflow(bottom) [36]. e) Endocast showing internal geometry of Odontotermes obesus

mound [80]. e) Thermoregulation of honeybee swarm clusters(top is contracted cluster

on a cool day, bottom is well-ventilated cluster on a warm day) [11]

of the interplay between the physics of transport and the behavior of how the system

remodels itself.

For a better understanding of these systems, we ask several questions:

* How can living porous media exchange matter and energy with their environ-

ment?

" How can a medium adjust to outside conditions to regulate itself?

" What feedback mechanisms can a medium use to regulate transport and con-

ductivity?

My thesis is composed of three parts, which are steps towards answering these

questions.

1. Diurnally driven respiration of termite mounds

2. Collective thermoregulation of honeybee swarm clusters

3. Generic model of an active porous medium

7

Diurnally driven respiration of termite mounds

A particularly impressive example of animal architecture is found in termites of the

subfamily Macrotermitinae. Individually only a few mm in body length, they are

well-known for their ability to build massive, complex structures [43, 91] without cen-

tral decision-making authority [95]. The resulting structure includes a subterranean

nest containing brood and symbiotic fungus, and a mound extending - 1 - 2m above

ground, which is primarily entered for construction and repair, but otherwise rela-

tively uninhabited. The mound contains conduits that are many times larger than a

termite [91], and widely viewed as a means to ventilate the nest [91]; the shape of the

mound results from some sort of interplay between building behavior, structure, and

transport.

Understanding the mechanism by which the mound works is important for both

understanding the solution the termites are approaching as well as the feedback pro-

cesses that gives rise to the structure. However, these mechanisms continue to be

debated due to the absence of direct in-situ measurements [93, 41, 42, 50].

In Chapter 1, by directly measuring diurnal variations in flow through the surface

conduits of the mounds of the species Odontotermes obesus, we show that a simple

combination of geometry, heterogeneous thermal mass and porosity allows the mounds

to use diurnal ambient temperature oscillations for ventilation. In particular, the

thin outer flute-like conduits heat up rapidly during the day relative to the deeper

chimneys, pushing air up the flutes and down the chimney in a closed convection

cell, with the converse situation at night. These cyclic flows flush out CO 2 from the

nest and ventilate the mound. We also directly measure and observe similar air flow

patterns inside the mounds of Macrotermes michaelseni, evidence that this is likely

to be a general mechanism among termite mounds.

8

Collective thermoregulation of honeybee swarm clus-

ters

Honeybees are masters of cooperative thermoregulation, and indeed need to be able

to do so in multiple contexts. We focus on swarming, an essential part of colony

reproduction, where a fertilized queen and about 2,000 - 20,000 bees cling onto each

other in a swarm cluster, typically hanging on a tree branch, while scouts search

for a new hive location [79]. For up to several days, the swarm cluster regulates its

temperature by forming a dense surface mantle that envelopes a more porous interior

core. Over this period, the cluster adjusts its shape and size to allow the bees to

maintain and regulate the core temperature to within a few degrees of a homeostatic

set point of 35'C over a wide range of ambient conditions.

The swarm cluster is able to perform this thermoregulatory task and modulate

its heat loss without a centralized controller to coordinate behavior, in the absence of

any long-range communication between bees in different parts of the cluster [32, 31].

Instead, the shape and temperature profile of the swarm cluster emerges from the

interplay between heat transfer throughout the cluster and the collective behavior of

thousands of bees [7] which know only their local conditions.

In Chapter 2, we present a model for swarm cluster thermoregulation that results

from the collective behavior of bees acting based on local information, yet propagates

information about ambient temperature throughout the cluster. Our model yields

good thermoregulation and is consistent with experiments at both high and low tem-

peratures, with a cluster radius, temperature profile, and density profile qualitatively

similar to observations, and makes predictions for possible experiments.

Generic model of an active porous medium

Elements across many living and nonliving active porous media are conservation of

current, feedback that changes resistance, and stochasticity. A minimal distillation of

the common elements in these different systems corresponds to a stochastically evolv-

9

ing network driven uniformly by fluxes and forces at the boundary due to pressure,

voltage, or concentration gradients.

In Chapter 3, we consider a minimal model for this feedback coupling in terms

of two parameters that characterize the way in which addition or removal of matter

follows a simple local (or non-local) feedback rule corresponding to either flow-seeking

or flow-avoiding behavior. We show that flow-avoiding feedback causes an initially

uniform system to spontaneously channelize; flow-seeking behavior causes the system

to spontaneously form a network of walls. We can understand this behavior by

constructing a continuum model, and understanding the walling and channelization

behaviors in terms of phase separations.

10

Acknowledgments

At the start of my PhD studies, I used to (badly) joke to myself that there were

two steps: 1) Move into my apartment, 2) Take classes, do research, then graduate.

By that metric, I was already halfway there! Now the second half is winding down,

and first coming to MIT feels like a odd combination of yesterday and forever ago.

Reflecting back on the last six years, I feel I've grown up as a scientist, and a lot of

that has to do with the people I've had the privilege of knowing during my time here.

First, I'd like to thank Prof. L. Mahadevan for introducing me to the world of

biophysics, as well as years of encouragement and support throughout both exciting

discoveries and dead ends. The last several years I have benefitted from his ability to

find interesting problems in everyday life, and find some level of simplicity in a messy

world. I'd also like to thank my co-advisor Prof. Mehran Kardar for his guidance over

the last several years, as well as teaching two great classes in statistical physics. I'd

also like to thank my committee members Prof. Jeff Gore and Prof. Jeremy England

for providing feedback and advice on my research.

At the Harvard School of Engineering and Applied Sciences, I have had the priv-

ilege to get to know many people the Mahadevan group and in Pierce hall: Zhiyan

Wei, Levi Dudte, Brian Lee, Laura Adams, John Kolinski, Andreas Carlson, Renaud

Bastien, Raghu Chelakkot , Jun Chung, Mattia Guzzola, Alex Isakov, Nadir Kaplan,

Ee Hou Yong, Andy McCormick, Yasmine Meroz, Orit Peleg, Ido Regev, Teresa Ruiz,

Baudoin St. Yves, Mary Salcedo, Dervis Can Vural. Special thanks to Hunter King

for teaching me about experiments, patiently disabusing me of some of my imprac-

tical theorist ideas, and his stoicism in the face of the numerous ups and downs of

fieldwork. I can only hope to one day be as level-headed a scientist as him. I want

to thank Jim MacArthur for all the electronics help, and Claudia Stearns for making

SEAS a functional place to work. I'd also like to thank my many collaborators over

the last several years: Jake Peters, Scott Turner, Rupert Soar, Sanjay Sane, David

Andreen, Amritansh Vats, Paul Bardunias, and Pallavi Sharma for helping us with

our forays into field biology. I also want to thank my roommates Ethan Dyer, Josh

11

Dillon, Cosmin Deaconu, and Ognjen Illic for their friendship, and tolerance of oc-

casional accumulating stacks of papers and misplaced cheerios, as well as countless

other friends at MIT.

Before discovering the world of Biophysics, I came to MIT as a quantum mechanic.

I would like to thank my first advisor Prof. Isaac Chuang for early guidance as a

scientist, as well as Xie Chen and Beni Yoshida for their patience and guidance to a

starry eyed first year grad student.

Also I want to thank the Human Frontiers Science Program, the Henry Kendall

physics fellowship, the Wyss Institute for Biologically Inspired Engineering, the Na-

tional Science Foundation, and the MIT Interdisciplinary Quantum Information Sci-

ence and Engineering program for funding this work.

Before I came to MIT, I had some great teachers who nurtured my love of math

and science. Special thanks to Wendy Cifelli, Bill Bernhard, Prof. Phil Allen, and

Prof. Brad Marston for helping me to get to this point in the first place.

Last of all I'd like to thank my parents Ben Ocko and Lorraine Solomon, my sister

Naomi Ocko, and my girlfriend Lin Zhu for all their love and support, and keeping

things in perspective for me.

12

Contents

1 Diurnally driven respiration of termite mounds

1.1 Odontotermes obesus

1.1.1 Mound structure . . . . . . . . . . . . . . . . . . . . . . . . .

1.1.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1.3 M odel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2 Macrotermes michaelseni .. . . . . . . . . . . . . . . . . . . . . . . . .

1.2.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2.2 Velocity measurements . . . . . . . . . . . . . . . . . . . . . .

1.2.3 Thermal measurements . . . . . . . . . . . . . . . . . . . . . .

1.3 M od el . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4 Transients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.5 CO 2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Collective thermoregulation of honeybee swarm clusters

2.1 Model assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . .

2.3 Simulations and results . . . . . . . . . . . . . . . . . . . . . . . . .

2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Generic model of an active porous medium

3.1 Transport Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2 Simulations: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3 Continuum model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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31

32

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39

40

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48

53

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59

.... .. ... . .

4 Conclusions

A Supplemental information for "Diurnally driven respiration of ter-

mite mounds" 77

A.1 Additional Measurements: Long-

A.2 Steady flow measurement . . .

term flow in a dead

. . . . . . . . . . .

A.3 Temperature measurements . . . . . . . . . . . . .

A.4 Permeability and Diffusibility . . . . . . . . . . . .

A.5 CO 2 measurements . . . . . . . . . . . . . . . . . .

A.6 Estimate of effect of temperature dependence . . .

A .7 Flow Sensor . . . . . . . . . . . . . . . . . . . . . .

A .7.1 Electronics . . . . . . . . . . . . . . . . . . .

A.7.2 Steady mode function . . . . . . . . . . . .

A.7.3 Transient mode function . . . . . . . . . . .

A.7.4 Steady flow calibration . . . . . . . . . . . .

A.7.5 Temperature and humidity dependence

A.7.6 Orientational Dependence . . . . . . . . . .

A.8 Geometry and sources of error in flow measurement

mound ...... 77

. . . . . . . . . . 78

. . . . . . . . . . 80

. . . . . . . . . . 8 2

. . . . . . . . . . 84

. . . . . . . . . . 85

. . . . . . . . . . 86

. . . . . . . . . . 86

. . . . . . . . . . 87

. . . . . . . . . . 88

. . . . . . . . . . 89

. . . . . . . . . . 9 2

. . . . . . . . . . 93

. . . . . . . . . . 95

B Supplemental Information for "Collective thermoregulation of hon-

eybee swarm clusters" 97

B.1 Units, parameter estimation, and dimensional analysis . . . . . . . . 97

B.1.1 Estimation of heat conductivity, metabolic rate, and permeability 97

B.1.2 Dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . 99

B.1.3 Units of behavioral pressure . . . . . . . . . . . . . . . . . . . 103

B.2 Behavioral pressure formalism and its antecedents . . . . . . . . . . . 104

B.3 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

B.3.1 Method of solving for equilibrium . . . . . . . . . . . . . . . . 105

B.3.2 Discretization of space . . . . . . . . . . . . . . . . . . . . . . 106

B.4 Role of temperature dependent metabolic rate . . . . . . . . . . . . . 108

B.5 Role of finite bee size on thermoregulation . . . . . . . . . . . . . . . 109

14

65

B.6 Linear stability of cluster . . . . . . . . . . . . . . . . . . . . . . . . . 111

B.6.1 Methods for calculating linear stability . . . . . . . . . . . . . 112

C Supplemental Information for "Generic model of an active porous

medium" 119

C.1 Determining if a distribution is bimodal . . . . . . . . . . . . . . . . 119

C.2 Comparison to local dynamics . . . . . . . . . . . . . . . . . . . . . . 120

C.3 Analytical Mean Field Theory . . . . . . . .. . .... . . .. .. . .. 120

C.4 Short Length Scale Continuum Model . . . . . . . . . . . . . . . . . . 122

C.5 Higher Order Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

C.5.1 Finding change in conductivity . . . . . . . . . . . . . . . . . 124

C.5.2 Change in voltage . . . . . . . . . . . . . . . . . . . . . . . . . 125

C.5.3 Change in mean flow . . . . . . . . . . . . . . . . . . . . . . . 126

C.5.4 Change in T . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

C .6 M ethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

15

THIS PAGE INTENTIONALLY LEFT BLANK

16

Chapter 1

Diurnally driven respiration of

termite mounds

Adapted from "Termite mounds harness diurnal temperature oscillations for ventila-

tion", by H. King, S. Ocko, and L. Mahadevan[38], to appear in Proceedings of the

National Academy of Sciences

Many social insects which live in dense colonies [39, 78] face the problem of keep-

ing temperature, respiratory gas, and moisture levels within tolerable ranges. They

solve this problem by using naturally available structures or building their own nests,

mounds or bivouacs [97, 201. A particularly impressive example of insect architecture

is found in termites of the subfamily Macrotermitinae, individually only a few mm

in body length, that are well-known for their ability to build massive, complex struc-

tures [43, 91] without central decision-making authority [95]. The resulting structure

includes a subterranean nest containing brood and symbiotic fungus, and a mound

extending -1-2m above ground, which is primarily entered for construction and re-

pair, but otherwise relatively uninhabited. The mound contains conduits that are

an order of magnitude larger than a termite [91], and viewed widely as a means to

ventilate the nest [94]. However, the mechanism by which it works continues to be

debated [93, 41, 42, 50].

Ventilation necessarily involves two steps: transport of gas from underground

17

metabolic sources to the mound surface, and transfer of gas across the porous exterior

walls with the environment. While diffusion can equilibrate gradients across the

mound surface [12], it does not suffice to transport gas between nest and surface'.

Thus, ventilation must rely on bulk flow inside the mound. Within the mound, a range

of indirect measurements of CO 2 concentration, local temperature, condensation, and

tracer gas pulse chase[50, 93, 41, 42, 14] show the presence of transport and mixing.

Previous studies of mound-building termites have suggested either thermal buoyancy

or external wind as possible drivers, making a further distinction between steady

(eg. metabolic driving [50], steady wind) and transient (eg. diurnal driving[41, 42],

turbulent wind[93]) sources. While a better understanding of the driving mechanism

behind these processes requires direct measurements of flow inside the mound, the

technical difficulties of direct in-situ measurements of air flow in an intact mound

and its correlation with internal and external environmental conditions has precluded

differentiating between any of these hypotheses.

Direct measurement of airflows

Direct measurement of flows inside the mound is difficult for several reasons. First,

the mound is opaque, so that any instrument must be at least partly intrusive. Sec-

ond, expected flows are small (- cm/s), outside the operating range of commercial

sensors, requiring a custom-engineered device. Third, because conduits are vertical,

devices relying on heat dissipation, or larger, high heat capacity setups can generate

their own buoyancy-driven (and geometry-dependent) flows, making measurements

ambiguous[49]. Finally, and most importantly, the mound environment is hostile and

dynamic. Termites tend to attack and deposit sticky construction material on any

foreign object, often within 10 minutes of entry. If one inserts a sensor even briefly,

termites may continue construction for hours, effectively changing the geometry and

hence the flow in the vicinity of the sensor.

To measure airflow directly, we designed and built a directional flow sensor com-

'It takes gas ~4 days to diffuse 2m.

18

posed of three linearly arranged glass bead thermistors exposed to the air (See A.7).

A brief pulse of current through the center bead creates a tiny bolus of warm air,

which diffuses outward and is measured in either neighboring bead. Directional flow

along the axis of the beads biases this diffusion, and is quantified by the ratio of the

maximum response on each bead, measured as a temperature-dependent resistance.

In a roughly conduit-sized vertical tube, this resistance-change metric depends lin-

early on flow velocity, with a slight upward bias due to thermal buoyancy. This allows

us to measure both flow speed and direction locally. The symmetry of the probe al-

lows for independent calibration and measurement in two orientations by rotating by

180' (arbitrarily labeled "upward" and "downward")(See A.7).

In addition to measuring these slowly varying flows, we used our flow sensors in a

different operating mode to also measure short-lived transient flows similar to those of

earlier reports[49, 93]. This requires a different heating protocol, wherein the center

bead is constantly heated, so that small fluctuations in flow lead to antisymmetric

responses from the outer beads. However, the steady heating leads to a trade-off, as

thermally induced buoyancy hinders our ability to interpret absolute flow rate.

Here, we use both structural and dynamic measurements to understand the mech-

anism of ventilation in two species of termites. First, we focus on the mounds of

Odontotermes obesus (Subfamily Macrotermitinae), chosen for proximity to NCBS

Bangalore and commonly found in southern Asia in a variety of habitats[52]. Sec-

ond, we focus on mounds of Macrotermes michaelseni (Subfamily Macro termitinae),

chosen for proximity to a field site in Namibia, and commonly found in southern

Africa.

1.1 Odontotermes obesus

1.1.1 Mound structure

In Fig. 1-1(a), we show the external geometry of a typical Odontotermes obesus

mound, with its characteristic buttress-like structures ("flutes") which extend radially

19

Figure 1-1: Mounds of Odontotermes obesus. Viewed from (a) the side, (b) top and by

(c) cross section. Filling the mound with gypsum, letting it set, and washing away the

original material reveals the interior volume (white regions) as a continuous network

of conduits, shown in (d). Endocast of characteristic vertical conduit in which flow

measurements were performed, near ground level, toward the end of flutes, indicated

by the arrow (e).

from the center (Fig. 1-1(b)). The internal structure of the mound can be visualized

by either making a horizontal cut (Fig. 1-1(c)) or endocasting (Fig. 1-1(d)). Both

approaches show the basic design motif of a large central chimney with many surface

conduits in the flutes; all conduits are larger than the termites, most are vertically

oriented, and the conduits are well connected 2. This macro-porous structure can

admit bulk internal flow and thus could serve as an external lung for the symbiotic

termite-fungus colony.

To understand how the mound interacts with the environment, we first note that

the walls are made of densely deposited granules of clay soil, forming a material with

high porosity (~37 - 47% air, by volume 3), and small average pore diameter (~ 5[tm,

roughly the mean clay particle size). Indeed, healthy mounds have no visible holes

to the exterior, and repairs are quickly made if the surface is breached. The high

porosity means that the mound walls provide little resistance to diffusive transport of

gases along concentration gradients. However, the small pore size makes the mound

very resistant to pressure driven bulk flow across its thickness. Thus, the mound

2A simple proof of well connectedness is that gypsum injected from a single point can fill all

interior conduits.3 Private communication from R. Kandasami, R. Borges, and T. Murthy

20

surface behaves like a breathable windbreaker. Finally, the low wind speeds observed

around the termite mounds of ~ 0-5m/s implies that they are not capable of creating

significant bulk flow across the wall, effectively ruling out wind as the primary driving

source.

1.1.2 Measurements

In live mounds, the sensor was placed in a surface conduit at the base of a flute for

< 5 minutes at a time to avoid termite attacks which damage the sensors. For a self-

check, the sensor was rotated in place, such that a given reading could be compared

on both "upward" and "downward" calibration curves. We also measured the flow

inside an abandoned ("dead"), unweathered mound that provided an opportunity for

long-term monitoring without having termites damaging the sensors. Simultaneous

complementary measurements of temperature in flutes and the center were taken. To

measure the concentrations of C0 2, a metabolic product, a tube was inserted into

the nest; in one mound it was placed in the center slightly below ground and another

in the chimney at -1.5 m above. Gas concentration measurements were made every

15 minutes by drawing a small volume of air through an optical sensor from the two

locations for most of one uninterrupted 24 hour cycle.

Steady flow measurements

Nearly all the 25 mounds that were instrumented were in a forest with little direct

sunlight. In Fig. 1-2(a), we show flow measurements in 78 individual flutes of these

mounds as a function of time of day. We see a clear trend of slight upward (positive)

flow in the flutes during the day, and significant downward (negative) flow at night.

The data saturates for many night values, as the flow speed was larger than our

range of reliable calibration (see A.2). In Fig. 1-2 (b), we show the flow rate for a

sample flute in the abandoned mound. Notably, it follows the same trend seen in live

mounds, but the flow speeds at night are not nearly as large as for the live mounds.

For both live and dead mounds, we also show the difference in temperature measured

21

between the flutes and center, AT = Tflute - Tcenter, and see that it varies in a manner

consistent with the respective flow pattern 4. This rules out metabolic heating [50]as a central mechanism, since a dead mound shows the same gradients and flows as

a live one.

Respiratory gas measurements

In Fig. 1-2(c) we show that the accumulation of respiratory gases also follows a diurnal

cycle, with two functioning states. During the day, when flows are relatively small,

CO 2 gradually builds up to nearly 6% in the nest, and drops to a fraction of 1% in the

chimney. At night, when convective flows are large, CO 2 levels remain relatively low

everywhere. While it is surprising that these termites allow for such large periodic

accumulations of C0 2 , similar tolerance has been observed in ant colonies[62].

Transients and wind

Our measurements found that transients are at most only a small fraction (- lmm/s)

of average flow speeds under normal conditions, and were not induced by applying

steady or pulsing wind from a powerful fan just outside the termite mound. Pulse

chase experiments on these mounds, in which combustible tracer gas is released in one

location in the mound and measured in another, gave an estimate of gas transport

speed (not necessarily the same as flow speed) of the same order (~ cm/s), which

indicates that our internally measured average flow is the dominant means of gas

transport and mixing with no real role for wind-induced flows. Furthermore, temper-

ature measurements in the center at different heights showed that the nest is almost

always the coolest part of the mound central axis, additional evidence against the

importance of metabolic heating[50] (See A.3).

4The thermal gradient for a live mound slightly lags behind the flow measurements. Only onelive mound's thermal gradients were measured, so this lag may not be general.

22

1.1.3 Model

Taken together, these results point strongly to the idea that diurnally driven ther-

mal gradients drive air within the mound, facilitating transport of respiratory gases.

Observations of the well connectedness of the mound and the impermeability of the

external walls imply that flow in the center of the mound, to obey continuity, must

move in the vertical direction opposite that of the flutes. When the flutes are warmer

than the interior, air flows up in the flutes, pushing down cooler air in the chimney.

The opposite occurs when the gradient is reversed at night5 (Fig. 1-3.).

This model is consistent with the quick uptake and gradual decline of CO 2 mea-

sured in the chimney; with the evening temperature inversion, convection begins to

push C0 2-rich nest air up the chimney before diffusion across the surface gradually

releases CO 2 from the increasingly mixed mound air. This forcing mechanism is inher-

ently transient; if the system ever came to equilibrium and the gradient disappeared,

ventilation would stop.

We can mathematize this model to predict flow speeds. As a model for a convective

circuit within a termite mound, we choose a pipe radius r, in the shape of a closed

vertical loop of height h, where the temperature difference between left and right side

of the loop is AT. The total driving pressure is paATgh, where p is air density and

a is the coefficient of thermal expansion, and Poiseuille's law gives

Q = paATgh - rr

Driving Pressure Poiseuille Resistance

where the factor of two comes from the resistance on both sides of the loop. Calcu-

lating flow speed:

V= paAT' (1.2)Tr 2 16/ '

and plugging in values of AT = 3C, r 3cm, y./p .16 cm2/s and a we

obtain a result of -35 cm/s. This speed is -10 times higher than those observed,

5The air flowing down the flutes is significantly warmer than ambient temperature, a situationonly possible if it was being pushed by even warmer air flowing up the chimney.

23

which is likely due to oversimplifying internal geometry; disorder, variation in con-

duit size, and connections between the flutes and the chimney favor high resistance

bottlenecks which reduce the mean flow speed. The calculation demonstrates that

observed thermal gradients and crude dimensions are sufficient to produce flow of the

order measured.

24

6

3CD)

E

0

3

3

-6

6

5

C-)

4

3

2

1

00

FlowAT -

oi-

Li n

---------

6 12

Hour of day18

6

13

-3

-6

6

3

0

- 3

-6

0

24

Figure 1-2: Diurnal temperature and flow profiles show diurnal oscillations. (Top)Scatter plot of air velocity in individual flutes of 25 different live mounds (0). Er-

ror bars represent deviation between "upward" and "downward" - 1.5 minute flow

measurements. The dashed red line is the average difference between temperaturesmeasured in 4 flutes and the center (at a similar height), AT, in a sample livemound(Representative error bar shown at left). (Middle) Corresponding flow and

AT, continuously measured in the abandoned mound. (Bottom) CO2 schedule in the

nest (*) and the chimney 1.5m above (o), measured over one cycle in a live mound.

25

-3

-6

6-- Flow

- -.---------------

Dead Mound]

0

0

- Nest* Chimney

- e04

-0

TopView

Chimney SurfaceCompliexlC C nduits

SideView

Ground Nest

Structure Day NightFigure 1-3: (Top) Thermal images of the mound in Fig. 1-1(a), during the day andnight qualitatively show an inversion of the difference between flute and nook surfacetemperature. Bases of flutes are marked with ovals to guide the eye. (Middle and

Bottom) Mechanism of convective flow illustrated by schematic of the inverting modes

of ventilation in a simplified geometry. Vertical conduits in each of the flutes are

connected at top and in the subterranean nest to the vertical chimney complex. This

connectivity allows for alternating convective flows driven by the inverting thermal

gradient between the massive, thermally damped, center and the exposed, slender

flutes, which quickly heat during the day, and cool during the night.

26

Figure 1-4: Mounds of Macrotermes michaelseni. a) Side view of a mound. b) Fillingthe mound with gypsum, letting it set, and washing away the original material revealsthe interior volume (white regions) as a continuous network of conduits, shown.

1.2 Macrotermes michaelseni

Here, we report measurements of airflows in the mounds of Macrotermes michaelseni

(Subfamily Macrotermitinae) in Namibia. Measurements were made from April 15

to May 20 in 2014 and 2015. We find underlying mechanisms which are qualita-

tively similar to those observed in obesus mounds in India, but with some additional

complexity.

1.2.1 Structure

The internal structure of the mound can be visualized by endocasting (Fig. 1-4b).

While the mound is conical and doesn't have "flutes" like in Odontotermes obesus

mounds, we still see that the mound is well connected with conduits, much larger

than the termites, connecting both the center and periphery of the mound.

27

6I

12

Hour of day18

Figure 1-5: Scatter plot of air velocity in the conduits as a function of hour of day.Error bars represent deviation between "upward" and "downward" - 1.5 minute flowmeasurements.

1.2.2 Velocity measurements

A total of 121 steady and transient measurements from - 30 mounds were made

at various times of the day. While mounds of michaelseni did not have "flutes" like

those of obesus, there are still large vertical conduits that could be accessed through

drilling with a hole saw. Flow speeds were recorded along with the time of day, the

orientation of the conduit relative to the mounds center, and the local wind speeds.

Our first observation is that flow during the daytime is generally upwards, while flow

during the nighttime is generally downwards.

1.2.3 Thermal measurements

Temperature profiles were obtained by implanting iButtons (DS1922L, Maxim) into

the mound using the hole saw and closing the openings with wet mud. Eight IButtons

were placed at a height of - 1 meter, ~ 5 - 10cm below the surface at the north,

northeast, east, etc. sides of the mound, in places where velocity measurements would

have been made. IButtons were also placed in the nest (~ .5 meters below ground),

28

100")

E

0

3

0

3 L------------ --- - -----

60O 24

-I . ------------

--.3 --- -- ---1-- - -------- -----

center(along the central axis at a height of ~1 meter) and top(along the central

axis - 15cm from the top). Like in obesus mounds, the mound periphery tracked

daily temperature changes while the center temperature was heavily damped. One

significant contrast is that michaelseni mounds are directly exposed to the sun, which

gives the thermal schedule a strong orientational dependence, unlike obesus mounds

which are in forests and shielded from direct sunlight.

- Center

40 ---- TopNest

- North

-South

25

20

1510 3 6 9 12 15 18 21 24

Hour of day

Figure 1-6: Thermal schedule of a michaelseni mound (Ordinal directions not shown).Note how the east side reaches its warmest point during the morning as the sun risesand the west side reaches its warmest point during the afternoon as the sun sets. Asthese mounds are located in the southern hemisphere, the north side receives muchmore direct sunlight than the south side, and is hotter at all times of the day.

1.3 Model

The basic model for ventilation in michaelseni is the same as that of obesus; mounds

have a daily thermal schedule, where the periphery tracks diurnal temperature vari-

ations which are more damped in the center. This gives rise to cyclic flows which

also operate on a daily schedule, driven by the gradient between the periphery and

29

center. Unlike in obesus mounds, the temperature gradient between the periphery

of the mound and the center depends both on the hour of day and orientation. To

understand this effect, the location of every velocity measurement had one of eight

orientations recorded(north, northeast, east, etc.). We estimate the thermal gradient

inside the mound with:

AT(t, #) = Tperiphery(t, #) - Tcenter (t)

Where t is the time of day and q is the orientation. This provides an estimate of the

temperature gradient driving convection, and provides a good correlation(0.76) with

the flow velocities measured (Fig. 1-7). When orientational information is not taken

into account, and the gradient is estimated from the average periphery temperature:

AT(t) = Tperiphery(t) - Tcenter(t), the correlation coefficient drops to 0.63, indication

of the relative importance between direct heating and ambient temperature variation.

U

4 -J6I

0

-- ------

~I.

Estimated temperature difference(C)AT(t, )

Figure 1-7: Measured air velocity vs. temperature gradient AT(t, #).

30

bIII

Iii

----------

I )

-1 ----I

8

1.4 Transients

Unlike in obesus mounds, we observed significant transients in michaelseni mounds.

We measure these by constant heating of the center bead; when the flow becomes

more upwards(downwards), the upper bead becomes warmer(cooler), and the lower

bead becomes cooler(warmer). This shows up as anti-correlated fluctuations in the

readings of the two beads on top of thermal drift.

15000- Upper Bead

Lower Bead

a 10000

0

4-J

-5000

1-10000

- 15000

-20000 20 40 60 80 100 120

Time(seconds)

Figure 1-8: Raw transient data. Note the anti-correlated "jumps" in the readings of

the upper and lower beads.

As a crude measure of transience, we subtract thermal drift, perform principal

component analysis and define our measure of transience to be the eigenvalue corre-

sponding to the anti-correlated principal component. Despite the existence of tran-

sients inside the mound, these transients are uncorrelated with wind speeds, suggest-

ing that these transients are not primarily driven by external wind. It is possible that

mounds are remodeled over the course of a year, or that, in addition to diffusion,

wind drives transfer of gas across the mound wall without contributing significantly

to internal flows. This would be consistent with the observations of [93] on the same

species.

Using a modified probe without a central bead for heading, we have observed

31

thermal fluctuations on the order of ~ 0.20 C(Fig. 1-10). One strong possibility

for the source of these thermal and velocity transients is a high Rayleigh number,

characterized by Ra = [agAT]D3 where k is the heat conductivity of air, C is the heat

capacity, and D is the diameter of the conduit. At high Rayleigh number, flow will

not reach a steady state; instead hot(cold) air will rise(sink) in plumes. A diameter

of 8cm and a thermal gradient of 30C give a Rayleigh number of - 3 - 105, large

enough to create this sort of turbulence [3, 2]. While we have been unable to prove

this is the case, there does not seem to be another candidate source of energy for

these fluctuations.

2.0.

I- 0

0

00

*0

0

I.,: sOSI. *

,0.0 .0 *0

0 0: 0.0,.. S

1 2 3

Wind(m/s)

Figure 1-9: The magnitude of transience is uncorrelated with wind speed.

1.5 CO 2 Measurements

For the majority of a 24-hour period, CO 2 levels were monitored inside the nest and

center (height 2m) of a mound. Unlike in obesus, there were not large gradients of

CO 2 within the mound, and levels stayed close to 5% throughout the whole period.

32

1.5

GJ)1.0

0.5

0.0 L

0

00

.0

. . 0 *

gO

040S

0

0

4 5

t

20000

100004.JCWD

0

-o.<-000

w- 20000

3000C10 20 40 60 80 100 120

Time(seconds)

Figure 1-10: Observations of thermal fluctuations, magnitude - 0.2'C.

In several other mounds, the CO 2 levels were also close to 5% just under the mound

skin, suggesting that during the months of April and May when these experiments

were performed, ventilation is limited by transfer across the surface, and the mound

interior stays well mixed. At other times of the year, lower levels and larger gradients

of CO 2 were observed [93], so it is possible that the mound walls are remodeled over

the course of the year.

1.6 Summary

It has long been thought that animal-built structures, spectacularly exemplified by

the termite mound, maintain homeostatic microhabitats that allow for exchange of

matter and energy with the external environment while buffering against strong exter-

nal fluctuations. Our study quantifies this by showing how a collectively-built termite

mound harnesses natural temperature oscillations to facilitate collective respiration.

In obesus the radiator-like architecture of the structure facilitates a large thermal

gradient between the insulated chimney and exposed flutes; mounds of michaelseni

do not have this feature, but are also exposed to much larger temperature oscillations.

33

-- Upper Bead- Lower Bead

6

04

5 -

4-

00

U

2

1-* . nest* . 2m

00 5 10 15 20

Hour of day

Figure 1-11: Daily CO 2 schedule of michaelseni mound

These two species of mounds harness this gradient by creating a closed flow circuit

which straddles it, promoting circulation, and flushing the nest of CO 2 in the case of

obesus. One notable difference is the relative saturation levels of CO2 ; depending on

the structure, mounds can exhibit either surface-limited or bulk-limited ventilation.

Lower levels, and larger gradients of CO 2 have been observed in michaelsenri [93]

during other times of the year, and it is unknown whether these mounds go between

different operating modes over a year, the relatively poor ventilation offered is suffi-

cient, or whether flows in michaelseni mounds are simply the natural consequence of

some other factor being optimized for. Our data comes from two different species on

two different continents, evidence the transport mechanism described here is generic

and is likely dominant in similarly massive mounds with no exterior holes that are

34

found around the globe in a range of climates. Diurnal temperature oscillations are

everywhere, and we suspect that this could be an important factor in driving convec-

tive flows for ventilation in other contexts. A natural question that our study raises

is that of the rules that lead to decentralized construction of a reliably functioning

mound. While the insect behavior that leads to construction of these mounds is not

well understood, it is likely that feedback cues are important. The knowledge of the

internal airflows and transport mechanisms might allow us to get a window into these

feedbacks and thus serve as a step towards understanding mound morphogenesis and

collective decision making.

The swarm-built structures described here demonstrates how work can be derived,

through architecture, from the fluctuations of an intensive environmental parameter.

This is a qualitatively different strategy than that of most human engineering, which

typically [6] extracts work from unidirectional flow of heat or matter. Perhaps this

mechanism might serve to inspire the design of similarly passive, sustainable human

architecture[90, 21].

35

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36

Chapter 2

Collective thermoregulation of

honeybee swarm clusters

Adapted from "Collective thermoregulation in bee clusters" [63]

Swarming is an essential part of honeybee colony reproduction during which a

fertilized queen leaves the colony with about 2,000-20,000 bees, which cling onto each

other in a swarm cluster, typically hanging on a tree branch, for times up to several

days, while scouts search for a new hive location [791. During this period, the swarm

cluster regulates its temperature by forming a dense surface mantle that envelopes a

more porous interior core. At low ambient temperatures, the cluster contracts and

the mantle densifies to conserve heat and maintain its internal temperature, while at

high ambient temperatures the cluster expands and the mantle becomes less dense to

prevent overheating in the core; this results in the core temperature remaining within

a few degrees of the homeostatic set point of 35'C over a wide range of ambient

conditions.

The swarm cluster is able to perform this thermoregulatory task without a central-

ized controller to coordinate behavior in the absence of any long-range communication

between bees in different parts of the cluster [32]. Instead, this behavior of a swarm

cluster emerges from the collective behavior of thousands of bees[7] who know only

their local conditions. Early work on swarm clusters, and the related problem of

37

winter clusters [86, 30], used continuum models for variations in bee density, and

temperature as determined by the diffusion of heat in a metabolically active mate-

rial. Most of these models [66, 47, 67, 18] assumed that the bees know their location

and the size of the cluster, contrary to experimental evidence. However, a new class

of models initiated by Myerscoughf54] are based on local information, as experimen-

tally observed; these models are qualitatively consistent with the presence of a core

and mantle, but are unable to explain how a high core temperature persists at low

ambient temperatures. Further refinements of these models that account for bee ther-

motaxis and also use only local information[99, 88], yield the observed mantle-core

formation and good thermoregulation properties at low ambient temperatures, while

allowing for an increased core temperature at very low ambient temperatures, ob-

served in some clusters[84, 19, 31, 32]. However, the thermotactic mechanism that

defines these models of bee behavior causes the cluster to break up at moderate to

high ambient temperatures, unlike what is observed.

Here we present a model for swarm cluster thermoregulation 1 that results from

the collective behavior of bees acting based on local information, yet propagates in-

formation about ambient temperature throughout the cluster. Our model yields good

thermoregulation and is consistent with experiments at both high and low temper-

atures, with a cluster radius, temperature profile, and density profile qualitatively

similar to observations, without leading to cluster breakup. In Section 2.1, we outline

the basic principles and assumptions behind our model. In Section 3, we formulate

our model mathematically, characterize the parameters in it. In Section 4, we solve

the governing equations using a combination of analysis and numerical simulation,

and compare our results qualitatively with observations and experiments. We con-

clude with a discussion in Section 5, where we suggest a few experimental tests of our

theory.

'While we do not consider the related problem of winter clusters, we believe a behavioral pressureformalism should also be applicable.

38

2.1 Model assumptions

Any viable mechanism for cluster thermoregulation consistent with experimental ob-

servations should have the following features: a) The behavior of a cluster must

result from the collective behavior of bees acting on local information, not through

a centralized control mechanism. b) The bee density of a cluster must form a stable

mantle-core profile. c) The cluster must expand (contract) at high (low) ambient

temperatures to maintain the maximum interior "core" temperature robustly over a

range of ambient temperatures.

This suggests that any quantitative model of the behavior of swarm clusters re-

quires knowledge of the transfer of heat through the cluster, the movement of bees

within the cluster, and how these fields couple to each other. The basic assumptions

and principles behind our model are as follows:

1. The only two independent fields are the packing fraction of bees in the cluster

p (1-porosity) and the air temperature T. These then determine the bee body

temperature and metabolism which can be written as a function of the local

air temperature; convection of air in the form of upwards air currents depends

entirely on the global bee packing fraction and temperature profiles.

2. We treat the cluster as an active porous structure, with a packing fraction-

dependent conductivity and permeability. Bees metabolically generate heat,

which then diffuses away through conduction and is also drawn upwards through

convection. The boundary of the cluster has an air temperature equal to the

ambient temperature.

3. Cold bees prefer to huddle densely, while hot bees dislike being packed densely.

In addition, bees attempt to push their way to higher temperatures. The move-

ment of bees is determined by a behavioral variable which we denote by "be-

havioral pressure" P (p, T), which we use to characterize their response to

environmental variables such as local packing fraction and temperature. In

terms of this variable, we assume that the bees move from high to low behav-

39

ioral pressure, a notion that is similar in spirit to that of "social forces" used

to model pedestrian movements [33]. Here, we must emphasize that behavioral

pressure is not a physical pressure. For packing fraction to be steady, behavioral

pressure must be constant throughout the cluster.

4. We assume that the number of bees in the cluster is fixed and that the cluster

is axisymmetric with spherical boundaries, whose radius R is not fixed (Fig.

2-1). At higher temperatures, the clusters become elongated and misshapen, so

that the assumption is no longer accurate; however, this does not change our

results for thermoregulation qualitatively. In general, to determine the shape of

the cluster, we must account for both heat and force balance, but we leave this

question aside in the current study.

A model based on these assumptions can be used to study both the equilibria

and dynamics. Since a swarm cluster is a constantly changing network of attach-

ments between bees, as bees grab onto and let go of nearby bees, and can also detach

and reattach themselves at different points on the surface of the cluster, these "mi-

croscopic" dynamical processes allow the cluster to quickly equilibrate to changes in

ambient conditions [32, 311. Thus, we will mostly focus on the resulting equilibria,

but consider the slow dynamical modes that allow it to respond to large scale weak

forcing, as they are particularly important in the context of cluster stability.

2.2 Mathematical formulation

The two independent fields in our model are bee packing fraction p(?, t) c [0, 1], and

the air temperature T(f, t); while both of these are functions of space(f) and time(t),

we will focus primarily on the equilibrium behavior of these fields. For a static cluster

of bees modeled as an active porous medium that generates heat and is permeable to

air, the heat generated metabolically must balance heat lost due to conduction and

40

convection, so that

pM(T) + - (k(p)VT) - Ci. VT =0 (2.1)

T = Ta |ieen,

where k(p) is the packing fraction-dependent conductivity (Power/[Distance x Tem-

perature]), M(T) is the metabolic heat production rate of the bees per unit volume,

and C is the volumetric heat capacity of air (Energy/[Volume x Temperature]). We

model the conductivity of the cluster as arising from a superposition of random con-

vection currents within the cluster which are suppressed at high bee packing fraction

and the bare conductivity of the bees treated as a solid, and approximate this by a

function k(p) = ko1 7. Although this form diverges as p - 0 and random convection

currents are unsuppressed, the p never vanishes in the interior of the cluster, so that

this limitation is not a problem. Likewise, by bounding p from above, we prevent k

from vanishing in the cluster. We further assume that the mean flux per unit area

u (Distance/Time) is determined by Darcy's law for the flow of an incompressible

buoyant 2 fluid through a porous medium, so that:

U = [airOaair (T - Ta) 2 - VP] s(p)/nj |ien (2.2)

V - U, = 0, P = 0 li,63m. (2.3)

Here 2.2 relates U' to the effects of thermal buoyancy and the pressure gradi-

ent3 , while 2.3 is just the incompressibility condition, with i(p) the packing fraction-

dependent permeability (Distance2), aair is the coefficient of thermal expansion (Temperature- 1 ),

7yair is the specific weight of air (Pressure/ Distance), 17 is the viscosity of air (Pressure

x Time), and P is air pressure. We assume that the permeability of the cluster may

be approximated via the Carman-Kozeny equation r'(p) = 1-O P used to describe

the permeability of randomly packed spheres [61].

2While air is compressible, this compression is small enough to be approximated by a "buoyantforce".

3We note that in Darcy's law, a buoyant "body force" may be added to the gradient of pressurein the same way that is done for the full Navier-Stokes equations.

41

C)

a)

R

Figure 2-1: A cluster at an ambient temperature of a) 11'C and b) 27'C, approxi-mately 12 cm across(Photo courtesy of [11]). Note that number of bees inside thecluster is nearly constant; change in cluster width is a result of changes in bee pack-ing fraction [79]. c) Schematic of interior (Q), and boundary (6Q), with mantle-corestructure. s, . are radial, vertical directions in polar coordinates, R is the cluster

radius.

In general, the bee metabolic activity is not a constant, and depends on a num-

ber of factors such as temperature, age, oxygen and carbon dioxide concentration

etc.[32, 85, 56]. To keep our model as simple as possible, we start by assuming that

the metabolic rate is temperature-independent, with M(T) = M0 and show that this

is sufficient to ensure robust thermoregulation, setting apart the details of the cal-

culations that show that our model can also yield robust thermoregulation with a

temperature-dependent metabolic rate (B.4).

To close our set of equations, we still must relate p(i) to T(r), which we do

by making a hypothesis that bees respond to local packing fraction and temperature

changes by changing their packing fraction to equalize an effective behavioral variable

which we call the behavioral pressure. Our formulation of this behavioral pressure

relies on two assumptions that are based on observations:

1. In their clustered state, bees have a natural packing fraction which is a function

of the local temperature. This natural packing fraction decreases with increasing

42

temperature, and increases with lower temperature, until it reaches a maximum

packing fraction pmax. Effectively, cold bees prefer to be crowded, while hot bees

dislike being crowded, consistent with a variety of experiments in the field and

in the laboratory [32, 11, 84, 30].

2. In addition to having a temperature-dependent natural packing fraction, bees

also push their way towards higher temperatures '. A result of this behavior

is that lower ambient temperatures will give rise to denser packing, even when

local temperature is unchanged; both are important for determining density.

This is consistent with observations [32].

With these constraints in mind, a minimal model for bee behavioral pressure

suggests the piecewise function 5

P(p, T) XT+ p-p.(T|| p<prax (2.4)00 P > Pmax,

where pm(T) = min {Pmax, Po - abeeT} is the natural packing fraction. The constant

po is dimensionless and represents the baseline for natural packing fraction; acbe has

units of Temperature- 1 and describes how the natural packing fraction changes with

temperature, while x also has units of Temperature- 1 and describes how bees push

their way towards higher temperatures. Behavioral pressure diverges as p - Pmax to

enforce the maximum packing density 6.

We emphasize that our model assumes that individual bees have no independent

homeostatic set points for temperature or packing fraction; the behavioral pressure

depends on a combination of p and T. Furthermore, we note that we only allow stable

packing fractions(DPb/p > 0); thus, bee packing fraction in a cluster at equilibrium

can never be lower than the natural packing fraction. In our formulation, a higher

4At some high enough temperature, they should stop pushing towards higher temperatures. Wehave ignored this effect for simplicity.

5 The linear form here is chosen for simplicity; qualitatively similar nonlinear forms gave rise toqualitatively similar behavior.

6 This discontinuity is not a problem in practice as our simulations do not work with behavioralpressure directly.

43

behavioral pressure at near-zero packing fractions represents the basic aggregation

behavior that creates the cluster. Additionally, the absolute behavioral pressure is of

no consequence; only gradients and relative values are important 7 . Further evidence

for a behavioral pressure comes from experiments [86] which observed a relation be-

tween the ambient temperature and the core density, even when the core temperature

remained approximately constant.

To complete the formulation of our problem, we need to specify boundary condi-

tions for the temperature and packing fraction fields. As we have stated earlier, the

surface bees will be at the ambient temperature; furthermore, since they can freely

expand and contract to minimize their behavioral pressure, we assume that they will

be at their natural packing fraction pm(Ta).

Comparing our model with earlier models such as the Myerscough model [54]

and the Watmough-Camazine model [991, we note that both fit into this general

behavioral pressure framework (B.2). However, our work differs from these studies

in that we synthesize and generalize the implicit relation between bee behavior and

their environmental variables in terms of the behavioral pressure, choosing a form

for behavioral pressure which combines elements from both models and is consistent

with experimental observations. In addition, we account for the role of fluid flow

and convection of heat in the cluster, which breaks the vertical symmetry, and is po-

tentially relevant in heat transport at high Rayleigh number Ra = iC (T - Ta) KR.

Dimensionless equations: To reduce the number of parameters in our model,

we make our equations dimensionless. We note that the total number of bees in the

cluster is constant; in our continuum model, this means that the total bee volume

within the cluster fff pdv is fixed. Rather than defining cluster size by number of

bees, we define it in terms of the dimensionless total bee volume V = fff pdv/VO,

where VO is the total bee volume of a typical cluster, i.e. the average volume of a bee

times the average number of bees in a cluster(B.1.1). Upon setting a characteristic

7The magnitude of these gradients and differences are also not important. This has allowed usto omit a prefactor from Ip - pm(T)|1

44

length scale to be the radius of a sphere of volume Vo, Ro - -, we write the

constraint on total bee volume as fff pdv = (47r/3) V. Scaling the temperature

so that a typical ambient temperature of 15"C -+ 0, and the goal temperature of

35"C -+ 1, we use the dimensionless variables:

T T - 150C200C

NYirtairC ( C) 2KO - be > crbee (20 0C)

rIMoRo

M4 -+ I, abee - aeee (200C)

Ta -15 0 C" 20 C

ko (200 C)

x x (20C)

and write the dimensionless form of Eqs. 2.1, 2.2, 2.3, 2.4 as:

p + V - (k(p)VT) - ' -VT -0

u = [(T - Ta) z VP]K(p), V -U= 0.

_ i-p (1 -p) 3k(p) = ko , '(p) = _ 2 3p p

T = Ta |FeG6Q, P = 0 I|G6Q

(2.5)

(2.6)

(2.7)

(2.8)

(2.9)

(2.10)

Pb (p, T) = TX + p - pm(T) P <Pa.

00 P> Pmax,

pm(T) = min {pmax, Po - abeeT},

for the packing fraction and temperature profiles of the cluster which depend on the

dimensionless parameters Pmax, Po, ebee, X, Ta, ko, K0, V 8 .

Information transfer through equalization of behavioral pressure: On the

outer boundary of the mantle, the air temperature is equal to ambient temperature,

so that minimizing the behavioral pressure requires that the packing fraction at the

mantle will be the natural packing fraction at the ambient temperature pm(Ta). At

equilibrium, (4) then implies that the behavioral pressure in the mantle and thus

8 Note this can be further reduced to five dimensionless parameters as we do in B.1.2, but thiscauses us to lose sight of what the goal temperature, typical ambient temperatures, and typicalcluster sizes are.

45

throughout the cluster will be -TX. This means that throughout the cluster, we

may write the local bee packing fraction as a function of both the local temperature

and the ambient temperature, i.e. Pb(p(T, Ta), T) -TaX which we solve to find

p(T, Ta) = min{ pm(T) + X (T - Ta), Pmax}

= min{po + T (-aee - x) - XTa, Pnax} (2.11)

= min {po - Tco - Taci, Pmax},

where we have made substitutions co = abee - x, c1 = x. Intuitively, the co term char-

acterizes the sensitivity of core temperature to ambient temperature, but the cluster

cannot fully adapt at low Ta through this term alone; adaptation at lower ambient

temperatures requires the ci term, which is eventually responsible for overheating in

the core at very low Ta. In Fig. 2-2 we graph the packing fraction p(T, Ta) obtained

by tracing contours of equal P which allows us to write the equilibrium local packing

fraction everywhere in terms of the conditions at the boundary. Bees respond to their

local conditions and move accordingly, and these variations in packing fraction prop-

agate information about ambient temperature throughout the entire cluster without

long-range communication. Although we have mapped Pb(p, T) -a p(T, Ta) for just

one choice of behavioral pressure, our approach will work for any equation of state

Pb(p, T) which uniquely defines a stable packing fraction, i.e. with DPb/ap > 0.

Having obtained 2.11 from Eqs. 2.9, 2.10, from now on we work with 2.11 directly.

The set of equations(2.5 ,2.6 ,2.7, 2.8, and 2.11) with the boundary condition that

the surface temperature of the cluster is the ambient temperature completes the

formulation of the problem to determine p(f), T(f).

2.3 Simulations and results

While the boundary of the cluster is assumed to have spherical symmetry, the tem-

perature and packing fraction fields inside do not need to have spherical symmetry

owing to the effects of convection. However, the fields still have cylindrical symmetry,

46

0.8,-

0.6 Opb(PT)p (T, T = -0.5)

_p (T, T = 0.5)

0.4 PMm(T) a

-0.5 0 T 0.5 1

Figure 2-2: Graphical representation of Eqs. 2.9, 2.10, 2.11, showing how local packing fractionis obtained through behavioral pressure (Online version in color). At low Ta, bees on the mantlewill pack at p = pmax (upper circle), and behavioral pressure is high throughout the cluster leadingto higher interior packing fraction(upper solid line) and overpacking. At high T", bees on themantle will pack more loosely(lower circle), and low behavioral pressure throughout the clusterleads to low interior packing fraction(lower solid line). The solid lines run along contour lines ofequal Pb(p, T), as behavioral pressure must be uniform through the cluster. Coefficients used are

po = 0.85, abee = 0.75, pmax = 0.8, X = 0.3.

and therefore we can represent p(f), T(f) as p(s, z), T(s, z), where s is the distance

from the central axis and z is the height. With this coordinate representation, we

solve the governing equations (2.5 ,2.6 ,2.7, 2.8, and 2.11) in a spherically bounded

domain using a simple discretization scheme with 30 values of s and 60 values of z

(B.3.2).

Our choice of the dimensionless parameter ko = 0.2 is constrained by experiments

[85], while we estimate ro = 1 though a simple calculation assuming the bees to be

randomly packed spheres, and pmax = 0.8, slightly higher than the maximum pack-

ing fraction of spheres, as bees are more flexible (B.1.1). However, the parameters

defining bee movement and behavior, namely co, ci, and po are experimentally un-

known. Guided by the general observation that physiological performance is often

improved by changing parameters while basic mechanisms remain unchanged, we op-

timize these parameters to achieve robust thermoregulation, i.e. the core temperature

remains close to 1(35 C) over a range of scaled ambient temperatures Ta E [-0.7, 0.8]

corresponding to a real ambient temperature Ta E [0, 30]'C. For the choice of param-

47

eters po = 0.85, co = 0.45, ci = 0.3, we find that within this wide range of T and a

factor of three in V, the dimensionless core temperature stays in the range 0.7 - 1.3

corresponding to a real temperature range of 29 - 410C, while the core temperature

itself increases monotonically with po in an analytically solvable way (B.1.2).

Our simulations also capture the qualitative mantle-core structure of the cluster

(Fig. 2-3) with a dense mantle surrounding a sparse core. We also find that at high

ambient temperatures, the cluster expands and the mantle thins, and at low ambient

temperatures the cluster contracts and the mantle thickens. Furthermore, we find

that the core temperature, defined as the maximum temperature of the cluster, is

higher at low ambient temperatures resulting from "overpacking", consistent with

experiments of [84, 19, 31, 32], and predicted earlier [99]. Finally, we note that the

temperature profile is vertically asymmetric due to convection, causing the point of

maximum temperature to rise above the geometric center of the cluster, as observed

in experiments [32, 31].

2.4 Discussion

We have shown that it is possible to provide a self-organized thermoregulation strategy

in bee clusters over a range of observed ambient temperatures in terms of a few

behavioral parameters. Our theory fits into a broader framework for understanding

collective behavior where the organism responds to the environment, but in doing

so, changes it, and its behavior until it reaches a steady state. Here, our model

takes the form of a two-way coupling between bee behavior and local temperature

and packing fraction, quantified in terms of an effective behavioral pressure whose

equalization suffices to regulate the core temperature of the cluster robustly. Although

our choice of the form of the behavioral pressure is likely too simplistic, it is consistent

qualitatively with experimental observations, and we think it provides the correct

framework within which we can start to quantify collective behavior. Furthermore,

our strategy might be useful in biomimetic settings. Our formulation for behavioral

pressure shows that in a cluster, bee packing fraction should depend only on local

48

Hot Ta T Cold Ta Hot Ta P Cold Taa)

1.2.0.8

.6 0.6

0.1

0 T.2

-0.7 0

V = 1.5 =1.5V -_ 1

V = 0.50-..I00,- "0\\\= 0.5

0.5-Teore 1-

0T Ta X~a_ 0.5-

-0.5

-0.5 0 0.5 -0.5 0 0 .Ta Ta

Figure 2-3: Adaptation with temperature-independent metabolism(Online version in color). a)

Comparison of temperature and packing fraction profiles at high(0.8) and low(-0.7) ambient tem-

perature, where the dimensionless total bee volume V is 1. b) Core temperature as a func-

tion of ambient temperature and total bee volume shows that as V increases the core temper-

ature increases but adaptation persists over a range of Ta(also plotted to guide the eye.) c)

Cluster radius as a function of ambient temperature and total bee volume showing how clusters

swell with temperature, consistent with experiment. For bee packing fraction, we choose coeffi-

cients of po 0.85,co = 0.45,ci = 0. 3 ,pmax = 0.8. For heat transfer, we choose coefficients of

ko = 0.2, so 1.

temperature and the temperature of bees at the boundary, which effectively control

the surface packing fraction. These dependencies might be measured by applying

different temperatures to the surface and interior of an artificial swarm cluster. This

observation provides an immediately testable prediction: a cluster may be "tricked"

into overpacking and overheating its core by warming the bees just below the surface,

while exposing the surface bees to a low temperature to increase behavioral pressure.

As pointed out earlier, experiments and observations of bee core temperature and bee

49

packing fraction [86] are consistent with these ideas, although a direct experiment of

this type does not seem to have been carried out. Preliminary analysis of winter

clusters shows that bee density can be written as a function of the local temperature

9

Our model adjusts well in response to changes in ambient temperatures, but it

doesn't have the same level of tolerance to different total bee volume that honeybees

exhibit. We have used a continuum model where the bees on the surface are exposed to

ambient temperature from all sides, but in reality the first layer of bees is hotter than

ambient temperature and supports a large temperature gradient driven by heat from

interior bees. This means that the surface bees feel an average temperature higher

than ambient due to the interior bees, and we predict this should give a large cluster

a lower behavioral pressure than a small cluster at the same ambient temperature.

This should reduce the sensitivity of core temperature to total bee volume, and we

have confirmed this in simulations (B.5).

We close with a description of some possible extensions of our study. Currently,

our model ignores changes in shape of the cluster associated with force balance via

the role of gravity, and the associated effects on thermoregulation. In reality, the

cluster is a network of connections between bees which changes shape and size due

to a combination of mechanical forces and heat balance, and a complete theory must

couple these two effects as well.

Our heat balance equation and estimates (B.1.1) suggest that while convective

terms are responsible for the asymmetry in the temperature profile, they do not play

an important global role in thermoregulation. However, our calculation assumes a

uniform packing which does not accurately represent the microscopic structure of

the cluster, and thus we may have underestimated the Rayleigh number and the im-

portance of convection. At high ambient temperatures, swarm clusters are observed

to "channelize", where channels open up to ventilate. Studying a simple "behavioral

pressure-taxis" dynamical law(B.6) , we find no linear instability that leads to chan-

nelization without an "anemotaxis" mechanism, where behavioral pressure increases

9 A. Stabentheiner, personal communication.

50

with itl; instead we see only only one kind of linear instability which comes from the

mathematical necessity of fixing cluster radius. This raises the question of whether

there are anemotaxis mechanisms. If not, how can channelization result from bee-

level dynamics and mechanics? We have also neglected active cooling, which includes

fanning, evaporative cooling(which can give up to ~ 50'C of cooling), diffusion of

heat through diffusion of bees, and effects of oxygen and CO 2 [85, 83, 96, 561. Finally,

we have also neglected any implications of bee age distribution, despite knowledge of

the fact that younger bees tend to prefer the core and produce less heat, while older

bees prefer the mantle and can produce more heat[19, 32, 11]. Accounting for these

additional effects will allow us to better characterize the ecological and possibly even

evolutionary aspects of thermoregulation.

Thermoregulation is a necessity for a wide variety of organisms. When achieved

collectively, individuals expend effort at a cost that accrues a collective benefit. The

extreme relatedness of worker bees in a cluster and near-inability to reproduce implies

that the difference between the individual and the collective is nearly nonexistent, so

that cost and benefit are equally shared. However, many other organisms are faced

with the "huddler's dilemma" [27]; expending individual metabolic effort is costly, and

benefits a group that is only partially related. Because genetic relatedness, metabolic

costs, individual temperature and spatial positions are all easily measurable[22], a

collective thermoregulatory system is an ideal context in which to study the tangible

evolution of cooperation and competition by building on our current framework both

theoretically and experimentally.

51

Table 2.1: Table of quantities and associated units

52

Quantity Symbol Units Typical ValuesBee Packing Fraction p Dimensionless - 0.2 - 0.8Bee Metabolic Rate M Power/Volume - 0.001 - 0.01W/cm 3 [31, 851Heat Conductivity k Power/(Distance x Temperature) ~ 0.0004 - 0.006W/(cm0 C)? (B.1.1)

Permeability r Distance2 (0.05cm) 2 ? (B.1.1)Darcy Velocity U Distance/Time ~ 1cm/s? (B.1.1)

Air Heat Capacity C Energy/(Volumex Temperature) 1.2 x 10- 3J/(mL 'C)

Air Specific Weight yair Pressure/Distance 1.2 x g/(cm 2 s 2)Coefficient of thermal expansion aair Temperature- 1 1/300 C

Air Viscosity 7 Pressurex Time 1.8 x 10- 4 gram/(cm s)T dependence of pm abee Temperature- 1 0.040 C-'?Behavioral Pressure Pb Model Dependent (B.1.3) Unknown (B.1.3)

Thermotactic Coefficient x Behavioral Pressure/Temperature Unknown(B.1.3)

Chapter 3

Generic model of an active porous

medium

Adapted from "Feedback-Induced Phase Transitions in Active Heterogeneous Conduc-

tors" [64]

Active porous media often involve complex interactions, such as transport of mat-

ter, gradients, and chemical signaling. However, there are universal elements across

them: some sort of conserved current, feedback that changed resistance, and stoch-

asiticity. Here, we consider a minimal distillation of these common elements, corre-

sponding to a stochastically evolving network driven uniformly by fluxes and forces

at the boundary due to pressure, voltage, or concentration gradients. The feedback

coupling is characterized by two parameters that determine way in which addition or

removal of matter follows a simple local (or non-local) feedback rule corresponding

to either flow-seeking or flow-avoiding behavior. Using numerical simulations and a

continuum mean field theory, we show that flow-avoiding feedback causes an initially

uniform system to become strongly heterogeneous via a tunneling (channel-building)

phase separation; flow-seeking feedback leads to an immuring(wall-building) phase

separation.

53

3.1 Transport Laws

Since problems involving steady state diffusion of heat, concentration gradients, or

flow through a porous material are mathematically analogous to current flow through

electrical circuits, we will use the language of circuit theory from now on. We focus

on a translationally symmetric case with periodic boundary conditions and a uniform

driving voltage in the vertical 2 direction '. The vertices are arranged in a square

network, with the current I between neighboring vertices i, j given by

Iij = I([Vi - V] + g - fii) , Iij = 0, (3.1)Qii

where Vi is the voltage at vertex i, g is the driving force, 2 is the vertical direction,

and jj is the relative position of i, j. Each vertex i either contains a particle (pi = 1)

or is empty(pi = 0), and the resistance between two vertices Qjj increases by AQ

when full; Qj =1 +AQ (p + pj) /2.

Particles are removed from their vertices at a flow-dependent rate proportional

to r(vi), where vi = 1/2j I23 is the current through the cell 2. They are then

added to an empty vertex with probability proportional to a(vj), leading to a simple

algorithm for the evolution of the medium[23]:

1. Remove a particle from filled vertex i randomly selected with probability pro-

portional to r(vj).

2. Solve for the new current through the network 3[26].

3. Add the particle to an empty vertex j randomly selected with probability pro-

portional to a(v,).

4. Solve for the new current through the network.

We emphasize that these four steps conserve particle number, but the movement

'A driving voltage concentrated at the boundary will give an identical current profile.2This particular expression for v is chosen for rotational invariance in situations with uniformcurrent.

3Currents were solved using the Eigen linear equation solver. See Supplemental Material, whichincludes Ref. [26]

54

of particles is nonlocal in that the distance between i, j may be arbitrarily large ".

Furthermore, we note that these dynamics lack detailed balance (Fig. 3-1d.)

Since the removal and addition rates can either increase or decrease with local

current in the active systems described earlier, we explore this range of possibilities

with two parameters aR, aA. For the removal process, we choose r(vi) = v,-CR; at

positive aR, particles in high current will rarely be removed(flow-seeking), and vice

versa. For the addition process, we choose a(vj) = vf'A; at positive aA, empty vertices

with high current will likely be filled(flow-seeking) and vice versa.

For simplicity, we have chosen our functional forms such that in any circuit, r(vj) Oc

9gR , a (v) OC gA. Because only the ratios of currents are important, we may set

this system of equations to be dimensionless by making the substitutions: Qo - 1 ,

AQ -+ AQ/QO, g + 1. This gives four dimensionless parameters: AQ, aR, OZA, and p,

the mean particle density. We want the difference between filled and unfilled vertices

to be large; here we choose AQ = 19 such that the resistance between filled vertices

is 20 times higher than empty vertices.

3.2 Simulations:

For each aR, aA, we start the system at a uniform density p and evolve it so that

each particle or hole moves an average of one thousand times. This procedure is

repeated twenty times for the parameters CeR, aA = (-6, -3, ... , 3, 6), p = 1/4, com-

puting the current through the entire network at every step. The results in Fig. 3-1

show that the system spontaneously forms channels (with high conductivity) at suf-

ficiently negative aR, aA; consistent with natural systems sharing these qualitative

biases. In drainage networks, erosion increases with current(CeR < 0), while deposi-

tion decreases(aA < 0), while in biology, ants have been observed to remove corpses

from high-wind areas(aR < 0) and place them in low-wind areas(aA < 0); both of

these systems form channels. Likewise, the system forms walls at sufficiently positive

4A related model involving local movement gives very similar behavior. See C.2 for comparisonwith local dynamics.

55

aR, aA, consistent with fuse networks(aA > 0). This kind of phase transition is also

similar to those seen in driven lattice gas models [48, 75].

When the system channelizes due to negative aR, we find thin channels; negative

aA gives thick channels. To understand these transitions, we consider their robustness

to perturbations. When the system has formed a set of parallel channels, occasionally

a channel gets blocked (Fig. 3-1c). When the channel is thin, current must go through

the clog blocking the channel, and therefore total current through the channel is

reduced while the clogging particle has much of this current forced through it. On

the other hand, when a channel is thick, current will go around the clog, and so is

barely impeded. Therefore, at negative aA, the thin channel has reduced current, and

this clogging will cause the thin channel to fill, while a large channel is much more

robust and will not be filled. On the other hand, for negative aR, the clog in the

wide channel has little current through it, and lingers, allowing the wide channel to

eventually be filled; the clog in the thin channel has high current forced through it and

is quickly removed. A similar argument shows that positive aA leads to thin walls,

while positive aR leads to thick walls. The system exhibits large hysteretic effects

which are especially strong at very negative aA, aR, when the system is "frozen" and

fluctuations are suppressed. Interestingly, when aR is positive and aA is negative,

both channelization and wall-building phase separations occur, as the system phase-

separates into thick clumps, which can be explained through a continuum model.

56

a)

-- 4- - -min-

c)

K2

U{, IN4fit 1

-i.t

K ~ t~i

e)

6. 'E~S~:K% :1.ooo;~ {LI ja

-r 4

g.5.-

6~

E.S .

f)

-6 -- -- - 6

'4"'.g)

Figure 3-1: a) Example system for (aA, aR) (0, 0). Filled vertices are covered bygray squares. Current is driven in the upwards direction; direction and magnitudeof current between neighboring vertices is indicated by red arrows(color online). b)Phase diagram for p = 1/4. Each individual box represents a single system that hasequilibrated for a particular (aA, aR), where aA, aR have values (-6, 3, 0, 3, 6). At

positive aR, the system forms thick walls; negative CxR gives thin channels. Positive

aA leads to a series of thin walls; negative aA gives thick channels. At positive aR,negative aA, a phase separation occurs at both orientations, giving a set of clumps.c) Robustness of thick and thin channels. d) Lack of detailed balance: aA < 0, andK2 K3, as the current through the lower right vertex has only weak dependance onthe occupation of the upper left vertex. However K1 6 K4 , as the current throughthe upper left vertex strongly depends on the occupation of the lower right vertex;therefore K 1 K 2 $ K 3K4 e) Fourier transform (p(k)2) (high amplitudes in dark, k = 0at center). f) Two-point correlation ((p(0) -- ,) (p(r) - ,)) (positive correlations indark, = 0 at center). g) Contour plot of conductivity of the medium as a function

of aR, aA-

57

I

0.3

06

k4

b) Ap - I / a )J" o N d)

c) 10 - -

103

A/1R e)102 Pi /FP

10,

0 0.2 0.4 0.6 0.8 1

P

6Walls

fclhannels

D R..c-- lumps0 Pz 1 0 Pz 1 Order Parameter

niform

1.8 -aR 2.2 continuum Model

Figure 3-2: al)(p(k)2 ) for (aR, aA) = (.1, .1). a2) Scatter plot of (p(k)2) 2

cos 2(0) for (JkJ L)/27 < 5 compared with prediction from continuum model. Notethat the majority of dependence is on direction k2/ k2 , not magnitude Jkl. b) Fokker-Planck dynamics of continuum model for wall phase separation for a slice of area A.c) Visualization of Eqs. 3.5, 3.6, for (aR, aA) = (2.6, 0) (color online). d) Comparison

of histograms of row density, aA = 0 for a 40 x 40 system. The right histogram isbimodal, so a wall phase transition is considered to have occurred. e) Comparison ofsnapshots and row density histograms for wall-building phase transition. f) Observedphase transitions though observation of order parameters on ensemble vs. predictionsof continuum model, aA, aR = (-6, 3, 0, 3, 6).

Phase Separation

To characterize the wall-building phase separation, we use the row density pz

Zx pxz/Lx as an order parameter, consistent with the fact that when the distribution

of row densities becomes bimodal(Fig. 3-2 d example, wall-building has occurred) 5 .

We characterize the channelization phase separation in terms of the column density

PX = Jz pxz/Lz; when this is bimodal, channelization has occurred (Fig. 3-1). We

characterize a clumping phase separation through local density pi = 1/A , 0 (V -

I'- f-'f)pe, where 0 is the Heaviside function 6 ; when this is bimodal and neither

column or row density are bimodal, clumping has occurred.

5See C.1 for details of determining bimodality6We are limited to small system sizes, and thus a short-ranged local density function, for com-

putational reasons.

58

al)

a2).2,

17

() o'(O) 1

3.3 Continuum model

Characterizing the distribution of filled vertices with a mean density field p, the

continuum version of Eq. 3.1 is:

J = r(p)[-VV + ge], V -J = 0, (3.2)

where /-, the conductivity, is a function of density. Similarly, in the continuum limit,

the discrete addition and removal activity are replaced with a stochastic equation for

density evolution:

p = -R(p, J, aR) + KA(p, J, aA) + , (3.3)

where R(p, J, aR) is the mean removal rate from a region of with density p, current

J, and a bias of aR. A is the mean addition rate, and J is itself a functional of p

obtained by solving Eq. 3.2. K = ff R/ ff A acts as a sort of chemical potential,

which is set to conserve total particle number. q is the stochastic noise term, whose

form will be discussed later. 7

The predictions of the continuum model depend strongly on the functions i', A, R.

To determine them, we use a hybrid approach, sampling via numerical experiment

using randomly placed particles and then varying the density to approximate the

entire functions, leaving us with no fitting parameters8 While the discrete and con-

tinuum model both lack detailed balance and thus we cannot write down a free energy

function associated with their dynamics, we can use equilibrium considerations when

certain limits or symmetries are assumed.

7A complete continuum model should involve spatial terms to account for a short wavelengthcutoff; here we do not include terms in A, R, K that involve spatial derivatives, focusing on thehomogeneous mean field limit.

8 Making analytical approximations gives similar qualitative behavior, although the agreementwith simulations is not as good. As in the discrete case, we start with a uniform density of P, Jon(p). See C.3 for details

59

Nearly-disordered limit:

In the limit where aR, aA - 0, only the linear response is important. Writing Eq.

3.3 as

p= -(p J, a) + 77,

where T (p, J, 5) = -R (p, J, aR) + /A (p, J, aA) is the time derivative functional

and a= (aR, aA). In Fourier space, we separate the effects of J and p on T:

dT(k) OTdp(k) Op

'T+iz P p

OJz(k)Op(k)

aT Jx(k)8 ap

We note that, due to symmetry, the third term is zero and may be removed. We now

must find: dJz(k)dp(k)

To do so, consider a uniform density po with a a sinusoidal perturbation Apeik.

The conductivity is, to within first order:

n = no + dp Apeik = no + Are

Giving us a mean current

J = ['o+ Aseik'] -- KOV [AVeikI -- Ko + --& Jo + AJei -,

where

AJ = eKkA[Vz - i{V[ikxi + ikz2].]

We set AV to conserve current up to first order:

V - AJ = eik'[Aikz + noAV [k' + k]] 0 -+ AV = ikz|k2 K

60

Giving us our change in current,

AJ = Ap- +dp [

2 ikiz[ikxi|k|

+ ikzZ2 = AP- Ix -- dp 11k| 2

dJz(k)dp(k)

When k oc -, density fluctuations are horizontal(channels) and cause fluctuations in

current; when k cc 2, fluctuations are vertical(walls) and current is uniform.

Plugging 3.4 into 3.3 yields:

p(k) = - ( OTnp

aJz+ a i

19P PK k 2

-- X p (k) + 17(k).(9P k 1

In general, the form of q will depend on A and R; however, in the nearly disordered

limit, 17 takes the form of white noise, where (17(k, t)l*(k, t')) = 26(t - t')D(p) 9 . The

Einstein relation then predicts the strength of fluctuations to within first order:

Oiz 1 x (+

ap opk 2

|k 2 -

We note that this is independent of the magnitude of k, and is a function of its

direction alone(Fig. 3-2 a), because of the dipole-like interactions between particles

which inhibit current upstream and downstream while increasing it laterally. Similar

interactions have been noted in fuse networks [55, 81, 4].

i translation symmetry:

To predict wall-building, we assume translation symmetry in the -i direction, so that

J(x, z) is constant throughout the system. In the discrete model the current through

any vertex is proportional to J, so that we may make the simplification R(p, J, CaR)

R(p, CeR)J- , A(p, J, aA) - A(p, aA)JIA 10.

9The magnitude of noise D comes from the two poisson processes corresponding to removal at arate of R and addition at a rate of .IA. In the nearly-disordered limitD(p) = 1/2 - [(P, s(P), aR) + nA(p, (P), a A)

1OThis is because in any region v oc J, and therefore r(V) oC V--aa DC J-- , av OC VIA OC J'A

61

dK k2dp k1 2

(3.4)

(p(k)2)~~-.1D(p) a( p Ja s

We can view a horizontal slice containing A vertices as having uniform density,

obeying the dynamics shown in Fig. 3-2, with a mean addition rate A(p, aA)J AJJ,

and a mean removal rate of 7Z(p, aR)JR. The first criteria for a phase separation

to occur is mass balance, i.e. no net particle transfer, between two horizontal slices

of densities pi, P2:

R(pi, aR)J-aR R(p 2 , aR) JR J--R

A = = (3.5)A(pi, aA)JQA A(P 2,oCA)jQA JA

where we have defined K in order to separate the dependence of J and p. Assuming

each slice is large(A -+ oc), we may find a recursion relation for the equilibrium

distribution of densities:

P(p +1/A) A(p, aA)-/

P(p) R(p, aR)

giving conditions for free energy balance:

fP2n A(p aA)M dp = 0. (3.6)

S1 (p 6', CeR))

The continuum model predicts the system to form walls when p falls between Pi, P2

which satisfy both mass and free-energy balance. Note that the wall phase separation

is independent of J.

2 translation symmetry:

To predict channelization, we assume translation symmetry in the 2 direction, so

J(x, z) = i(p(x))2. As before, each vertical slice will obey the dynamics in Fig. 3-2,

except the mean removal rate is now R(p, aR) r,(p>aR; the mean addition rate be-

comes KA(p, aA),(p)A. Following the same procedure, the criteria for mass balance

becomes:

Kr = R(p1 , aR) ,(pi)--QR _ R(p2 , CR) (,p2 )-R

A(pi, aA) k(p1)IA A(p2, aA) r{(P2)CA

62

and the condition for free energy balance becomes

jP2 InA(p' aA)AKpDQA QR dp 0.P I R(p',) aR)

We note that when CeA = -R, the criteria for wall-building and channelization

become identical; if a phase separation occurs, it will occur in both orientations,

giving rise to a clumping phase separation. This is consistent with our observations

in simulations (Fig. 3-1 (b)).

Summary

Our model for active heterogeneous porous media relies on a small number of very

simple elements: a conserved current in a medium where flow and resistance are

coupled to each other, in the presence of noise. This allows us to provide a uni-

fied coarse-grained approach that links a number of different physical and biological

systems with different underlying fine-grained mechanisms that are often considered

disparately. Despite its minimal complexity, our numerical simulations of the model

produce the same channeling and immuring phase separations through the variation

of only two parameters corresponding to biases in material addition and removal,

consistent with the biases of the systems it is inspired by; flow-avoiding drainage net-

works and ant corpse piles lead to channels, and flow-seeking fuse breaks that lead to

walls.

A complementary continuum model corroborates our numerical simulations and

also leads to the formation of walls, channels and clumps. However, the functions

characterizing conductivity and activity used in this continuum model come from nu-

merical experiments which neglect microscopic correlations, resulting in a prediction

of a continuous phase transition as opposed to the observed discontinuous one. In

addition, because there is no inherent length scale to the continuum model, it can

not explain the transition between thin and thick structures. A continuum model

considering the formation of the thinnest structures also predicts channelization and

63

walling ", but a theory combining both elements has no additional predictive power.

Additionally the model predicts scale-free dipole-like correlations observed in the dis-

crete model, which ought to exist in all nearly-disordered systems where conductivity

and current are coupled. While in these systems, addition and removal biases

complement each other, our model shows that only one type of bias is needed for a

phase separation. Opposing removal and addition biases, if they could be engineered,

offer new possibilities for patterning that explore the entire phase diagram.

"See C.4 for short length scale continuum model

64

Chapter 4

Conclusions

We have used a combination of experiment, theory, and computation to study various

aspects of transport and remodeling in biological systems. Across these systems, many

behaviors can be understood in terms of their macroscopic properties rather than the

discrete details.

In chapter 1, we have investigated how living porous media exchange matter and

energy with the environment by conducting the first direct measurements of airflows

inside termite mounds. We have shown that, in two species, Odontotermes obesus and

Macrotermes michaelseni, flows inside the mound operate on a diurnal cycle, driven

by diurnal temperature variations. During the daytime, the periphery is warmed by

the environment, driving upwards airflows in the periphery(downwards in the center);

during the nighttime the periphery is cooled, driving downwards airflows in the pe-

riphery(upwards in the center). Surprisingly, this seems to be a novel mechanism for

biological systems to extract useful work from their environment 1. We suspect this

mechanism could be generic across all termite mounds and other animal-built struc-

tures; the only requirement is a diurnal temperature variation coupled to thermal

mass capable of supporting cyclic flows. Some possibilities are tree knots, burrows

built into hillsides, and underground nests with no above ground structure. The ex-

perimental tools we have developed for measuring very small flows might help make

10ne notable similarity comes from the maple tree, which pumps sap upwards through repeatedfreezing and thawing cycles [53, 35].

65

headway into these related problems.

It is well understood that there is no "blueprint" for the mound, and the mound

does not result from any central decision making authority. Rather, the mound results

from the collective behavior of thousands of termites interacting with each other and

the partially built structure. While this is true, the actual underlying behavior and

mechanisms are not yet well understood. Feedback and olfaction rules are important

in many aspects of insect behavior, and in a closed environment like a termite mound

the usual intuition of dissipation doesn't work; transport happens through the specific

flow patterns within the mound. A better understanding of these flows will potentially

lead to a better understanding of the interplay between behavior, structure, and

transport leading to formation of these structures. For example, olfactory templates

explain the building of simple two-dimensional walls around ant colonies[9j. Could

the interaction between the diurnal flow profiles we have discovered and a colony-

produced template set the shape and size of a mound?

In chapter 2, we have investigated how living porous media adjust to outside

conditions through developing a quantitative model to explain thermoregulation and

control of heat loss in honeybee swarm clusters. Using steady-state considerations,

we posit a "behavioral pressure", which defines combinations of bee density and tem-

perature that bees prefer; the cluster reaches equilibrium when behavioral pressure

is equalized throughout the cluster, thus transmitting information about through

variations in bee density.

Our behavioral pressure model explains qualitative features of cluster behavior and

is consistent with observations. We have attempted direct experiments of behavioral

pressure on honeybee swarms by investigating the relationship between temperature

and bee density. To do so necessitated being able to both apply temperature gradients

as well as measuring bee densities across the cluster, which required forcing the swarms

into a two-dimensional geometry. Our results were inconclusive, as the bees no longer

exhibited the same clustering behavior when forced into this geometry.

While we developed the idea of behavioral pressure for swarm-clusters, there are

many self-assemblages of animals which face the problem of how to regulate heat

66

loss through huddling. A better experimental testing ground for these ideas might

be found in systems which are naturally two-dimensional, which would sidestep ex-

perimental difficulties encountered while working with swarm clusters. Data on the

related problem of winter clusters 2, where bees thermoregulate in two-dimensional

huddles in the spaces between combs, shows a relationship between temperature and

bee density. However, there is not sufficient data to determine whether a behavioral

pressure model can explain these density distributions.

While we have modeled the swarm cluster as a continuum model with uniform

local density, the effect of the bee-level structure is significant. At high ambient

temperatures, channels have observed to open up for the likely purpose of ventilation.

While channels cannot be explained through a behavioral dependence on density and

temperature alone, an anemotaxis mechanism, where bees avoid areas with faster

airfiows, will give a channelization instability. Does this anemotaxis mechanism exist,

or do these channels arise from other mechanisms? These are possible connections to

the formation of conduits in termite mounds, where the collective behavior of termites

leads to conduits which are much larger than any individual.

In chapter 3, we see how an active porous medium can regulate transport and

conductivity with generic model of active porous media that remodels itself itself in

response to flow. Our discrete model, as well as the complimentary continuum model,

allows us to provide a unified coarse-grained approach linking many systems with dif-

ferent fine-grained mechanisms. In these systems, biases corresponding to addition

and removal of material complement each other as either both flow-seeking or both

flow-avoiding. However, our model shows that only one type of bias is needed for a

phase separation; in fact, phase separations can be achieved through opposing addi-

tion biases. These opposing biases, if they could be engineered, offer new possibilities

for patterning that explore the entire phase diagram.

While only minimal complexity is required to observe the channelization and wall-

building transitions seen in these systems, real systems display much more complexity.

Some notable differences are the emergence of length scales, and the ability to form

2 A. Stabentheiner et al, private communication

67

hierarchical networks and distribute resources to an area evenly. Further study is

needed to determine how much complexity is needed for these additional features,

and at what level of complexity the similarities break down.

68

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[101] D. P. Zitterbart, B. Wienecke, J. P. Butler, B. Fabry, and M. Perc. Coordinatedmovements prevent jamming in an emperor penguin huddle. PLoS one, 6(6):e20260,2011.

76

Appendix A

Supplemental information for

"Diurnally driven respiration of

termite mounds"

A.1 Additional Measurements: Long-term flow in a

dead mound

Continuous flow in five flutes from the dead mound were measured, shown in Fig. A-1.

Four out of five flutes follow the observed trend. The heterogeneity is likely due to

the complex geometry and is consistent with the few live mound measurements which

also go against the trend.

One flute (which happened to go against the observed trend) was measured for

three days. For the first two days, the mound (at the edge of a forest) was exposed

to partial direct sunlight. It can be seen in Fig. A-2 that the measured air velocity

strictly follows a daily schedule, where fine features repeat themselves. On the third

day, a tarp was positioned above the mound, keeping the mound shaded from any

direct sunlight, but exposed to ambient temperatures. Some midday features are

missing, presumably those directly induced by solar heating, but the general trend

remains the same.

77

6

4 -

2

CD

E0 -

~2o

4-

0 3 6 9 12 15 18 21 24

Hour of day

Figure A-1: Continuous flow measurements of 5 flutes in a dead mound.

A.2 Steady flow measurement

In an area within walking distance of the campus of NCBS Bangalore, India, 25

mounds were chosen which appeared sufficiently developed (> 1m tall) and intact.

All mounds were located in at least partial shade, but received direct sunlight inter-

mittently during the day'. The scatter plot of steady flow (Fig. 1-2 (a)) represents

individual measurements of the flow in the large conduits found at the end of each

available flute.

In order to place the probe, a hole was manually cut with a hole saw fixed to a

steel rod, usually to a depth of about 1-3cm before breaking into the conduit. With

a finger, the appropriate positioning and orientation of the sensor was determined.

Occasionally, the cavity was considered too narrow (< 3 cm) diameter), or the hole

entered with an inappropriate angle, in which case it was sealed with putty and

another attempt was made.

Because internal reconstruction, involving many termites and wet mud, continues

'Direct sunlight does not appear to play a direct role in the flow schedule (see SI). Full exposure

would lead to orientational dependence of temperature and air flows.

78

Tarp ...........1.

1.~1.

01.

S0.

0.

0.

0.

0.0.

1.

0

0

00

6 12Hour of Day

18

Figure A-2: a) 3 day flute flow measurement. A tarp was hung to shade the mound

on the third day. b) The same data, with days 1, 2, 3 overlaid on top of each other.

long after any hole is made, the same flute was never measured twice. In a round

of measurements, flows in 1-3 (out of approximately seven available) flutes of each

of -8 nearby mounds were measured, a process which took several hours. Each of

the 25 mounds was visited 2-3 times to utilize unmeasured flutes, but deliberately at

different times of day to avoid possible correlations between mound location and flow

pattern.

An individual measurement of flow was taken for -2 minutes in each orientation,

such that the response curves from which the metric and then flow are calculated are

averaged over many pulses (~6 pulses per minute). The error bar of each constant

flow velocity measurement in Fig. 1-2 (a) indicates the deviation in measured flow

between orientations. Care was taken to ensure that the probe temperature remained

79

..... .. .. ... -

a)

8

04

2

0 2 18 12 18 0 12 18 0Hour of Day

IDy I8

Day 2

2.6

.2

.a 24

close to the interior flute temperature, and no long-term drift in measured velocity was

observed as the probe equilibrated. Periodically between measurements, the sensor

was tested in the same apparatus to check that it remained calibrated, especially

when thermistors were damaged or dirtied by termites and needed to be cleaned.

The dead mound referenced in the text was identified as such because no repairs

were made upon cutting holes for the sensor. As it was also intact (there were no

signs that erosion had yet exposed any of the interior cavities) and within reach of

electricity, it was possible to make continuous flow measurements which could be

compared with the brief measurements for live mounds.

A.3 Temperature measurements

Temperatures in the dead mound reported in Fig. 1-2 (b) in the main text were

obtained by implanting iButtons (DS1921G, Maxim) into the mound using the hole

saw and closing the openings with wet mud. 2 iButtons were placed in flutes at

the same location where flow data had been acquired. Another 2 iButtons were

placed 5-10 cm below the surface, in the nooks between flutes, such that they were

located roughly in the periphery of the central chimney. AT, as shown in Fig. 1-

2, was calculated as the temperature from the measured flute minus the average

temperature measured by the centrally placed iButtons. The raw data was slightly

smoothed before taking the difference, to reduce distracting jumps in data from the

iButtons, which have a thermal resolution of 0.5 C.

In the a large healthy mound, digital temperature/humidity sensors (SHT11, Sen-

siron) were implanted at different heights near the central axis. Screened windows

protected the sensors from direct contact by termites and building material, while re-

maining coupled to the interior environment. The sensors and Arduino were powered

with a high capacity 12V lead acid battery and they recorded temperature for ap-

proximately two days. iButtons were placed in the bases of flutes in four sides of the

same mound. Temperature differences reported in Fig. 1-2 (a) were calculated from

the average of flute temperatures and central axis temperature at the corresponding

80

height.

The temperature differences between interior and flutes shown in Fig. A-3 are

significantly larger at night than during the day. This and/or the vertical asymmetry

of the convective cell (in that the flutes are closer to the top of the convective cell)

might be responsible for the observed asymmetry in flow magnitudes between night

and day.

32

30

C26 - East FluteNorth Flute

- Northwest FluteSouth Flute

- Center Nest---- Center Middle

Center Top

292 3 6 9 12 15 18 21 24

Hour of day

Figure A-3: Temperatures along the center of healthy, ~.,2m tall, mound at threeheights: "nest" (~30cm below ground), "middle" (-50cm above ground) and "top"(-130cm above ground), and in bases of flutes at 4 cardinal directions. Error invalues along the center is ( 0.40C), and (+I1C) in the flutes. The independence ofbehavior on cardinal direction shows direct solar heating is not of primary importance.

81

.. .... ...... -

A.4 Permeability and Diffusibility

A hollow, conical sample of a flute was cut from a mound. The bottom was sealed

with gypsum, such that the pores in the wall material, and length of plastic tubing

are the only path in and out of the cone.

To measure the permeability of the wall material, air was pulled by a vacuum

pump from the tube through the volumetric flow meter (Alicat MC-20slpm) and

the conical sample of mound wall. In parallel was a piece of glassware inside a

water reservoir, connected to the connected to the mound sample on top (Fig. A-

4).The height differential Ah allows us to calculate the back pressure AP = gpAh.

This, combined with the measured air flow Q (measured by the mass flow controller)

gives the permeability of mound sample , = Q/(APA), where A is the area of the

sample. Fig. A-4 shows the 20cm tall conical sample(a) and relationship between back

pressure and average flow(c). Our measured values for a larger mound sample are

in rough agreement with the measurements [37] on hydraulic conductivity, as well as

the predictions of the Carman-Kozeny equation (=_,21O, w # the porosity

and D is the particle diameter [61].

From this graph, one can read the local flow induced across the mound wall due

to a pressure differential from incident wind. Wind in the area during the study was

typically in the range 0-5m/s, which could produce a maximum dynamic pressure

P = pv 2 = 0-15Pa, giving a maximum flow through the surface of 0.01mm/s. With

even the most liberal approximations, this is not enough to produce bulk flow of

the order we've measured, in agreement with the observed negligibly small transient

flows in tests with a powerful fan. If macroscopic holes penetrated the mound surface

in some locations, they would dramatically change the permeability estimate of the

mound as a whole. However, such holes were not observed in these mounds, and

the species seems to fill in even the smallest holes. This behavior contrasts that of

other species, which appear to tolerate some holes; Odontotermes obesus actively

closed narrow holes made for the C02 measurement, while we have observed that

Macrotermes michaelseni in Namibia did not.

82

a) b)Air Drawn

Through Wall

WallSample

Flo) )VacuumMeter

Ah due to back pressure

Water

c) d)E0.8-

E0.7- m=0.65mm/s per kPa -- - ,r=85min0.60

0.5 -0.4

0.4- 0.2.00.2

0 0.1-0.100 400 600 800 1000 1200 50 100 150 200

Back Pressure (Pa) Time (min)

Figure A-4: a) A hollow conical sample from a mound flute. b) Schematic of per-meability measurement apparatus. c) Flow velocity as a function of back pressuremeasured by sealing the bottom and pulling air into the sample. d) Loss of com-bustible gas by diffusion in same sample.

83

Impermeability to bulk flow of the wall does not mean non-porous or impermeable

to diffusion. Cooking gas was injected into the conical sample and measured by a

combustible gas sensor that was sealed inside the conical sample. Fig. A-4(d) shows

that it diffuses out the surface over the course of about two hours (following close to

exponential decay).

The pores are much larger than the mean-free path of air molecules ~ 100nm,

and therefore, the material will have an effective diffusivity equal to the diffusivity in

empty space times some coefficient of the geometry. This coefficient is often modeled

as Deff = , where 0 is the porosity and T is the tortuosity, the ratio between theT

total distance needed to travel through a structure and the displacement achieved

[25]. While we observe diffusion levels considerably lower than this estimate, the

microscopic geometry is not well understood. One important contrast between the

equations for diffusion and permeability is that the size of the particles that comprise

the material only effects permeability; given any geometry, a lower particle size will

lead to decreased importance of permeability, but will not affect diffusion as long as

the pore sizes remain larger than the molecule size and mean free path.

A.5 CO 2 measurements

One large (- 2m tall), apparently healthy mound was chosen (that shown in Fig. 1-1

(a-b), Fig. 1-3) for measurements. One hole was drilled from ground level diagonally

down into the nest and another into the central chimney 1.5m above ground. 1/4

inch tubing was inserted in the holes and left over night such that the termites sealed

the holes at the surface leaving the tubes snugly in place. A Cozir wide range IR

LED C0 2 sensor was fitted with a custom machined, air-tight cap with two nozzles,

such that air pulled into the cap would gradually diffuse across the sensor membrane

and the response could be recorded with an Arduino onto a laptop computer. For

most of one 24 hour cycle, every 15 minutes air was drawn from each of the tubes

in the mound through the sensor with a 50mL syringe, pulling gradually until the

response leveled out, meaning the full concentration of mound air had diffused across

84

the sensor membrane. When termites periodically sealed the end of the tube inside

the mound, a few mL of water was forced into the tube, softening and breaking the

seal so measurements could continue.

A.6 Estimate of effect of temperature dependence

6 6

Flow

E 3- AT % . - 30

- 00e

-3 - - -- - - . -3

< NoThermal-6 Correction -- 6

6 6

Cn Flw0 3 --A T . ..-..-. 3

S- 0 -

0 --3 ---- --- -- 3

With Thermal Am. se-6 Correction 6

Figure A-5: Comparison of calculated velocities where temperature-dependent sen-

sitivity is taken into account

We can estimate the effect of temperature dependence of the probe by applying a

temperature-dependent calibration, where sensitivity is assumed to vary linearly with

temperature, interpolated between the slope and offset of the 210 and 29' curves (Fig.

A.7.5). Using this with the temperature information from inside the mound, the effect

of temperature on all live mound values can be approximated, as shown in Fig. A-5.

We can see the values are only slightly modified, causing a tiny enhancement to the

trend, where daytime flows become slightly more positive.

85

A.7 Flow Sensor

Our probe consists of three 0.3mm diameter glass coated thermistor beads (Victory

engineering corporation, NJ, Ro = 20kQ) held exposed by fine leads in a line with

2.5mm spacing. The center bead is used as a heat source, either pulsed or continuous,

in steady and transient modes, respectively. As the heat diffuses outward, its bias

depends on the direction and magnitude of flow through the sensor, as depicted in

Fig. A-6(Left). The operating principle is similar to that of some sensitive pulsed

wire anemometers [65]. Flow is quantified by comparing the signals (effectively tem-

peratures) from either neighboring bead.

Figure A-6: Left: Sensor bead schematic. A brief pulse of heat diffuses towards the

nearby beads measuring temperature, with a bias in the direction of air flow. Right:

Image of the sensor head showing three beads aligned with window of cap.

The plastic housing for the sensor was drawn with Solidworks and printed on a

Object Connex500 3D printer and the individual thermistors were connected to the

electronics via shielded Cat 7 ethernet cable. The thermistor beads can be seen in

the large window in the protective cap of the sensor in Fig. A-6(Right).

A.7.1 Electronics

Data was collected and the sensor was controlled via an Arduino Uno, while connected

to a laptop computer using custom scripts written in Processing. To improve the

resolution of the tiny temperature signals, the voltages are measured with AD7793

(Analog Devices) ADCs recording at 10Hz, and a stable baseline signal was supplied

by three C batteries in series. The pulsing voltage was supplied by 5 9V batteries,

86

a)

CD

Figure A-7: a) Schematic of circuit. b) Photograph of setup. The sensor is connected

by ethernet cable to the box containing supplemental electronics and Arduino, which

is connected to a computer by USB.

connected momentarily to the middle bead by opening a SIHLZ14 (Vishay) MOSFET.

Trimmer potentiometers were used to keep the baseline temperature signal within

range of the ADCs. The basic circuit diagram can be seen in Fig. A-7(a). The

supplemental electronics, batteries, and Arduino were housed in a portable metal

box, as shown in Fig. A-7(b).

A.7.2 Steady mode function

In order to obtain a passive measurement of steady (average) flow, the center bead, in

series with 4kQ was with 47V every 10 seconds. For Odontotermes obesus in India, it

was pulsed for 30 ms to minimize buoyancy. In mounds of Macrotermes michaelseni

there are more thermal fluctuations. This rebalances the tradeoffs between thermal

buoyancy and signal-to-noise ratio, and so a longer pulse of either 140 or 700 was

used. We estimate its temperature to briefly reach 180 -300'C. This produced a small

bolus of warm air that diffused outward toward neighboring beads in either direction.

The tiny change in resistance due to the expanding heat bolus was measured in the

neighboring beads, as shown in Fig. A-8. In the absence of flow through the sensor,

there is a slight upward bias in the response magnitudes due to thermal buoyancy

87

bl

a) .000

25000

W20000

15000 h

10000 100/h2

2 4 551

Upper BeadLower Bead

lhi

5000 h 24

m 0 e nd2 4

Time(seconds)

c)

-fooo,

25000

20000

10000,

I 5000

Time(seconds)

Figure A-8: Averaged responses from two thermistors as heat bolus diffuses awayfrom center bead for a) Zero flow b) 1.4 cm/s upwards flow in calibration tube; c)Before and d) after subtraction of temperature drift in field data.

of the bolus. Upward and downward flow biases the relative responses of the two

thermistors in a reproducible way. The log of the ratio of the maxima ln(hi/h2) was

used as a metric to calculate flow velocity (see steady flow calibration section). For

experimental data sets, temperature drift must be subtracted as the probe equilibrates

with the interior flute temperature.

A.7.3 Transient mode function

While in steady mode, infrequent, brief pulsing of heat avoids troublesome induced

convective currents, it prohibits measurement of high frequency(> .1 Hz) changes

in flow. In transient mode of the same sensor, continuous voltage 47 V is applied

across the middle bead, such that transients in the flows can be measured from

the instantaneous ratio of responses in the neighboring beads. The trade-off is that

information about the baseline flow is lost, because induced currents, which depend

on several parameters, including unknown details of tunnel geometry, can dominate

the signal. Fig. A-9 shows the response of the neighboring beads in transient mode

88

Upper Bead- Lower Bead

U BAd25000 pper ead

..-- Lower Bead

2FA)00

10000

b) 30000

25000

20000

15000

CO 10,M]

Upper Bead- Lower Bead

d

150000

100000

50000

CY)

C

(1) -50000

100000

150000

20000)

Velocity(cm/s -.64)0 1 2 3 4 5 6 7 0 -1 -2 -3 -4 -5 -6 -7 0

100 200 300 400Time(seconds)

500 600 700

Figure A-9: Relative readings upper and lower beads in transient mode as flow wasvaried in increments of .64cm/s

for some prescribed transients. At ~ 0 cm/s the dependence on the difference between

the two readings is the weakest( 0,000); assuming that slope, the largest fluctuation

observed in the field corresponded to - 1mm/s.

A.7.4 Steady flow calibration

diffuser

sensor

to computer

diffuser

Iflow-direction switch

to vacuum source

Figure A-10: Schematic of tube used for calibration. Arrows show direction path of

air from inlet toward vacuum source.

89

-Upper Bead

S--Lower Bead

A 5.6 cm diameter, 2m long, vertical acrylic tube was used as a sample conduit

for calibration of the sensor (see Fig. A-10). The tube was covered with a layer

of thermal insulation to prevent external thermal gradients from influencing flow,

which was observed when the bottom of the tube was exposed to the sun through

a lab window. Air was pulled through the tube by a vacuum source and mass flow

controller (Alicat MC-20slpm) in series. To prevent heterogeneous inertial flows where

the narrow connector tube met the wider conduit, air was directed through a fine

metal mesh before entering a conical flow rectifier. The whole set-up was up-down

symmetric, such that by rotating two valves, flow direction could be reversed without

changing the setup geometry. The sensor was inserted into a hole in the middle of

the tube and oriented so that the beads were aligned with the tube.

1.5

A Upwards Probe1. Downwards Probe

1.0

-P

-0.5 'I

-1.0-3 -2 -1 0 1 2 3

Velocity (cm/s)

Figure A-11: Calibration curves for steady flow sensor measured in the plastic tube.

The upwards black triangles represent the upwards orientation of the sensor, while

the downwards blue triangles represent the downwards orientation. Error bars are

the pulse-to-pulse deviation.

As a consistency check and to eliminate possible systematic biases in the probe,

every airflow measurement is taken twice using the same probe in two orientations.

In one the (arbitrarily named) first bead is above the middle thermistor("upward");

in the other it is directly below ("downward"). To do so, we need separate calibration

90

1.5

1o A A A Upwards ProbeI V V Downwards Probe

.5 A- -

--

-. 0'-4--45

-1.5-4 -2 0 2 4

Velocity (cm/s)

Figure A-12: Shifted calibration curves for three instances of the sensor reveal apredictable relationship between metric and velocity.

tables for the "upward" and "downward" orientations.

Fig. A-11 shows the dependence of the metric on flow velocity in the calibration

tube. The typical standard deviation between pulses during calibration is small and

does not represent the dominant source of error in the field, where variation in conduit

geometry plays a larger role (see Experimental Procedure). There is an unusual,

though robust, feature at - 2cm/s, where the response briefly jumps. This is most

likely an effect due to shifting from a regime of viscously dominated laminar flow to

inertial dominated laminar flow. We note that this leads to a small range of flow

velocities where we overestimate the flow speed(but not the sign).

Fig. A-12 shows several calibration curves for the different instances of the sensor,

either a spare duplicate sensor, or the same physical sensor after thermistor beads

were replaced. Each instance requires a new calibration, as the magnitudes of bead

responses varies according to the manufacturer's 25% tolerance, but upon shifting the

vertical offset and slope, on can see the qualitative behavior of each instance is the

same.

91

A.7.5 Temperature and humidity dependence

0.7

A 21 OC0.6 -2 0

A 290C -

-1.0 -0.5 0. -. .

02 I

Velocity (cm/s)

Figure A-13: Calibration curves for two temperatures show a slight shift(~ 0.2cm/s)and slope change (~'- 40%) for this temperature difference, which is closely matchedwith the total range observed in live mounds. Horizontal and vertical lines are toguide the eye.

While the calibration curve in laboratory conditions is robust, additional tests

were performed to make sure that our metric was not terribly sensitive to other fac-

tors which can vary in the field. Two such factors which should influence the thermal

response of the sensor, and therefore its performance are background temperature

and humidity. Fig. A-13 shows the calibration curve for two different ambient tem-

peratures, which roughly represent the range of temperature in the mound. Though

the curve has a lower slope at higher temperature, which indicates that the sensor

underestimates hotter flows, this effect is small compared to the variation of the daily

mound flow schedule measured in the field, and would not account for the sign change.

The metric, continuously measured for constant flow was unaffected by a change in

ambient relative humidity from 25% to 70% (at 290C).

92

A.7.6 Orientational Dependence

0.8r

0.6

0.4

0.2

0.0

9-0.2

-0.4

2 -0.6

-0.8

-1.0

A

7

A

--- ---- ---- ----------

--------

A

-3 -2 -1 0 1 2 3Velocity (cm/s)

A A A Parallel0 0 0 Orthogonal

Figure A-14: Velocity metric as a function of flow speed, tube orientation, and relativeprobe orientation. Regardless of whether the tube is vertical (red), diagonal (450,black) or horizontal (blue), the probe is only sensitive to flow velocity when theprobe is aligned parallel with the tube (upwards triangles), and not when orthogonal(rightwards triangles)

-3 -2 -1 0Velocity (cm/s)

1 2 3

Figure A-15: Velocity metric as a function of flow speed and tube orientation for avertical probe. When the tube is vertical(90'), sensitivity is highest, diagonal(450),gives reduced sensitivity, and the probe is not sensitive to horizontal flows(90') at all.

As mentioned above, the sensor is calibrated exclusively for vertical flow. Though

it was not difficult to ensure that local conduit orientation was vertical where the

sensor was placed, and that flow was therefore predominantly vertical, it is not un-

reasonable to suspect that a horizontal flow component could be misinterpreted by

the sensor.

93

0.6

0.4

-C! 0.2

0.0

Q.-0.2

-0.4

-0.6

A Vertical TubeI Diagonal Tube0 Horizontal Tube

-- - ---- ~ - - - -- -

-1) X,

0.7

0.6

0.5

0.4

0.3

0.2 Measurement0.1 Cosine Fit

0.0.

-0.1

-0.2-90 -45 0 45 90

Angle(Degrees)

Figure A-16: Velocity metric as a function of probe rotation in a tube with upwardsflow.

Fig. A-14 shows the metric as a function of flow velocity for 3 orientations of

the calibration tube; horizontal, 450, and vertical, when the sensor is either parallel

or orthogonal (cap windows 90' from tube axis). Negligible change in metric for

all orientations in which the sensor was misaligned indicates that the sensor cap

effectively rectifies flow for all orientations; any flow component perpendicular to

the sensor would have negligible effect on the measured value. When the flow is

aligned with the sensor, for the full range of probe and tube orientations, there is

only a small shift in the curve. Additionally, Fig. A-15 shows that a vertically

placed sensor's sensitivity to flow decreases to zero as the direction of flow is changed

from vertical to horizontal. These rule out the possibility of non-trivial sensitivity to

conduit orientation.

Fig. A-16 shows the metric as a function of probe angle in a vertical tube of

constant flow velocity 3.1 cm/s, where the probe is aligned (0) or orthogonal (t90')

to the tube. The good fit to a cosine shows that flows not aligned with the probe

(horizontal flows in the field) do not significantly affect the measurement.

94

A.8 Geometry and sources of error in flow measure-

ment

In situ measurements take place in a complex geometry, and the width, shape, and

surrounding features can be highly variable. This can lead to significant variation in

local velocities, even causing some local velocities to go against the average trend; this

is a generic feature of flow through disordered, porous media [15, 46]. In addition, the

width, shape, and impedance of a channel are different than in our calibration setup,

and the position of the probe within a channel could not be exactly known. These

factors are most likely the dominant source of error for any given measurement in

the field, either over- or underestimating the flow in a particular conduit. This error,

though potentially as large as a factor of ~2, is reflective of the natural variation in

mound geometry, is not correlated to any other parameter, and cannot mistake the

direction of flow, such that the trend in average flow remains unambiguous.

95

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96

Appendix B

Supplemental Information for

"Collective thermoregulation of

honeybee swarm clusters"

B.1 Units, parameter estimation, and dimensional

analysis

B.1.1 Estimation of heat conductivity, metabolic rate, and

permeability

For the metabolic rate and heat conductivity we use estimates from the experiments

of Southwick [85], where a cluster of 4250 bees(608 grams) is put into a set of roughly

planar, parallel honeycombs, and the temperature profile and oxygen consumption

are measured. The bees are roughly uniformly distributed with a bee packing fraction

p of about 0.5, for which the metabolic rate is roughly uniform, and the temperature

is well approximated with a parabolic profile, with a temperature of 34'C at the core,

and 1 10C at the edge 9.5cm from the core, parallel to the combs. The combs insulate

well, so heat transfer occurs primarily in the two directions parallel to the combs.

97

Then, within the cluster:

T 34'C - 23 0C (X2+ )(9.5cm 2

V 2T~-4x 230C 1.00C(9.5cm 2 ) cm 2

The oxygen consumption rate is measured to be 6.5mL/min which gives a volumetric

metabolism of 0.0035W/cm 3, assuming a bee specific weight of 1 gram/cm 3, and an

oxygen to energy conversion of 3 .5mL02 - 0.0012 w . This metabolic rate agreeskgmin - gram

well with the experiments of Heinrich [321. We now have all the pieces to calculate

the conductivity using the conduction heat balance:

kV2T-Ok-1.7 x 10-3 WkV 2T + pM = 0 -* k = .. 03

At p = 0.8, the maximum packing fraction we allow, the conductivity becomes close

to the value of 2.4 x 10- 3 W/(cmC) for fur and feathers [70] as suggested by South-

wick which gives us some level of confidence in the functional form k01 " we have

p

chosen for conductivity.

To estimate KO, we use the Carman-Kozeny equation [61]. The average bee weighs

about 0.14 grams, which corresponds to a sphere of diameter - 0.65cm. In the

absence of any detailed information about the bee structure in the cluster, we treat

the cluster as a system of randomly packed spheres. It would be interesting to measure

and better understand convective gas and heat transfer within swarm clusters. Using

this diameter in the Carman-Kozeny equation, we find:

KO = -- (0.05cm)2 .180

A typical cluster has about 10,000 bees, which is about 1.4 kg, giving RO = 7cm.

Plugging these values in(B.1.2), we find dimensionless conductivity and permeability

98

to be:

ko ~-- 0. 2, KO ~

B.1.2 Dimensional analysis

Our conditions for heat balance imply that:

pM(T) +V - (k(p)VT) - Ci - VT =0

T = Ta IE6Q

U = [Lairaair (T - Ta) 7 -VP,(p)/, \FE

V -w = 0, P = 0hEls.

while our equation for behavioral pressure reads:

Pb (p, T) =XT + p -- pm(T|00

P <; Pmax

P > Pmax ,

pm(T) = min { pmax, Po - abeeT}

We set our unit of length to be R0 , so that the volume constraint becomes fff pdv =

(4-F/3) V. We make the transformation V -* V/Ro. Then our heat balance equations

read:

pM (T) + W-V - (k(p)VT) - 1C' - VT = 0 \rcQ

T = Ta |fenQ

U = [iraair (T - Ta)1R

2 Ro r~)r7t~x

V - U = 0, P = 0 IrEcQ.

On making the substitutions:

99

T - 150 C350C - 150C'

T - 150 C350C - 15C

leads to the goal temperature of 35'C yielding a dimensionless temperature of unity,

and a typical ambient temperature of 150C corresponding to a dimensionless temper-

ature that vanishes. Then our system of equations becomes:

35C C-15oCvpM(T)+ 2 10

0o- (k(p)VT)

35 0C - 150 C- Cit-VT =I |.CRo

T = Ta |eEn6

I = [(350 C - 150 C)Niraair (T - Ta) -VPRo I

V -i= 0,

r4p)/r IFeQ

P = 0 |f']6.

Pb(p, T) { -x (350C - 150C) T + p - pm(T) I

00

P < Pmax

P > Pmax,

pm(T) =min{pmax, [PO - abeel 5 C] - acbeeT[35 C - 150C]}.

We divide all terms in the heat equation by the base metabolism M 0 to yield:

pM(T)/Mo +350C - 150C

2M40RO- (k(p)VT)

35"C - 150C- M(Ci- VT) = 0 |FeEQ

(350 C - 15 C) -yairaai, (T - Ta)

T = Ta |te6Q

Ro

V -U = 0, P = 0 |rcm. .

Making the substitution

350C - 15 0C'U->U xMORo

100

and the substitution

P(350 C - 150C) yairczairRo

leads to the set of equations:

p(T)Mo+ 350C - 50C (k(p)VT)MoRo

u = [(T - Ta) S-V](35-C -

- u- VT = 0 1ieQ

T = Ta | c6Q

15 C) 2 Yai~airCA A T- GOp) rIE

v - =0, P =0 |i n.

Finally, making the substitutions for the coefficients

350 C - 150 CMoR2

(350C - 15 C) 2 'YairaairC

MoRor;

Po - Po - asbe (15'C), Cbee - abee (350C - 150C), x - x (35 0C - 15 0C).

leads to our full dimensionless set of equations for heat balance:

pM(T)+V -(k(p)VT) -- VT =0 |ieQ

T = Ta |fcG6

i= [(T Ta) 2VP](p) |FEQ

V. - = 0, p = 0 |en

101

P -

M - M /Mo,

while the dimensionless behavioral pressure reads:

Pb (p, T) = -x + p - pm(T)

pm (T) = min { pmax,

P Pmax

P> Pmax,

Po - aeeeT},

Our model has seven parameters Pmax, Po, cbee, X, Ta, ko, ro, with an additional param-

eter for the total bee volume V, which varies from cluster to cluster.

Note on further dimensional analysis

We note that, when the metabolic rate is temperature independent, the goal tem-

perature and the typical independent ambient temperature have no bearing on the

actual behavior of the model, only on whether it represents effective thermoregula-

tion. Then, we may set the temperature where the packing becomes maximally dense,

Tpacked - PO-PmPax, to be zero, and the temperature at which the packing fraction be-

comes zero, Tempty - P, to be 1. We can then make slightly different substitutions

for the coefficients:

1 k --+ k Tempty -Tpacked

MoR2

(Tempty - Tpacked) 2 'airOairC

Mo0 or

P0 Pmax, cbee --+ Pmax, X -+ X (Tempty - Tpacked) -

We may remove the parameter V by setting the length scale to be the fully packed

radius of this particular cluster rather than the fully packed radius of a typical cluster.

Doing this makes the total bee volume constraint become fff pdv = (47r/3), and

102

requires the substitutions:

k _

V 2/3 k 1/3

We now find that there are now only five free parameters Pmax, X, Ta, ko, ,o. We do

not carry out this extra analysis in the main body of the paper because this causes

us to lose sight of what the goal core temperature, typical ambient temperatures, and

typical cluster sizes are.

B.1.3 Units of behavioral pressure

Our model is based on behavioral pressure being uniform at equilibrium. The units

and typical values of behavioral pressure are unknown: any set of dynamical equations

for bee movement will result in the same equilibrium where behavioral pressure, whose

units depend on the set of dynamical equations used, remains constant. For example,

a simple taxis model is one where:

dt

would mean that behavioral pressure has units of Distance 2 /Time. A more compli-

cated evaporation /condensation model would have the form

dp~i;*) ( ff~Pb - e 2a2 1 (V 1d~dt =N] bk b(r)) U3 T(p~f) ,p~i))Jd

where o is the evaporation and condensation radius and T is some transfer coefficient

would give units of 1/Time. A yet more complicated model involving mechanical

compressibility or viscosity would have yet another set of dimensions for behavioral

pressure. All of these models would, however, yield the same static solution.

103

B.2 Behavioral pressure formalism and its antecedents

To understand how our behavioral pressure formalism fits in with previous models,

we compare them within this framework. We note that previous modes have defined

density in terms of bees/cm3 instead of packing fraction; to go between the two, bees

may be assumed to have water density, and a packing fraction of 1 corresponds to

(1 gram)/mbe bees/cm3 .

The Myerscough model assumes that the bee density depends only on local tem-

perature, and thus can be written as P(p, T) = jp-pm , where pm = 8 bees/cm3 (I _ _

The Watmough-Camazine model defines a dynamical law for the bee density via

the equations:

p =-V*-J

J = -pI(p)Vp - pX(T)VT,

where A(p) > 0 is a motility function, and X(T) is a thermotactic function. This may

be written as

i -p [ P Vp + x(T)VT = -pVPb,

where the behavioral pressure P is defined as:

Pb(p,T)= dp' + x(T')dT'.

Density Component Temperature Component

We note that as p(p) > OVp, the density component is minimized as p -* 0, and

the density at the surface must be fixed to prevent the cluster from falling apart,

unlike in our behavioral pressure framework. In contrast, the Watmough-Camazine

behavioral pressure allows the packing fraction at the mantle to become too high.

104

Additionally, because the behavioral pressure can be divided into temperature and

density components, the point of highest density will always be at the same tempera-

ture, regardless of size or ambient temperature, which is not observed experimentally.

This is in contrast with our formalism.

B.3 Numerical methods

B.3.1 Method of solving for equilibrium

To reach equilibrium, we use an iterative scheme, described by the following pseu-

docode:

T(F) <- Ta

R < Ro/ pmax

Pmax : < R

0 : r1> R,

repeat

Solve for Ui at fixed temperature and density, then solve for T at fixed M and i

Find ', P such that: U'= [(T - Ta) 2 - VP]r(p), V -ii= 0

Find Tnew(i) such that: pM(T) = -V - (k(p)VTnew) - U -VTnew

T (i) - T(f ) + (Tnew(?) - T(i ))

Pnew V) <- p(T (f), Ta)

p(f) <- pV) + cp (Pnew(i;) - P(i))

Expand or contract the bee packing fraction and temperature profiles to normal-

ize total bee volume

R V Scaling ratio

R&-7xR

p( F) +-PKIZ R)

T (F) <-T (i/ R )

105

until converged

The intermediate steps can be solved as a system of linear equations. Note that

this method does not add or remove cells to vary cluster radius and conserve the

total number of bees; it grows and shrinks a fixed number of cells. The solution is

considered to be converged when Pnew P, Tnew T, R = 1 to within 10-10, which

takes about 100 - 200 iterations, about a minute on a laptop. All simulations were

done using MATLAB. The source code is available at:

http://web.mit.edu/socko/Public/PublishedCode/PRS2013BeePaperSimulations.zip.

B.3.2 Discretization of space

To solve for the temperature and density profiles, we must first discretize the system.

While spherical symmetry is broken due to convection, the system retains rotational

symmetry about its axis. We therefore use cylindrical coordinates, where each cell is

given indices (i, j), and has coordinates which represent the distance from the central

axis si. and the vertical coordinate zij, where

Sij2 n Z2 n

n is the radius of the cluster in cells. All cells with s2 +Z < R are in the interior of

the cluster, while all cells with s2 + Z > R are at the exterior of the cluster, subject

to the boundary conditions Ti = Ta, Pi = 0, pij = 0.

The volume of each cell with coordinates (i, j) is Vi = 27rw 2sig, where w = Rn

is the width of each cell. Each cell (i, j) neighbors four other cells, (i, j + 1), (i, j -

1), (i+ I 1, j), (i - 1, j), with the exception of cells which border the axis (i = 0), which

only have three neighbors. The area shared by cell (i, j) and its outside horizontal

neighbor (i + 1, j) is 2-rw (si, + 1L). The area shared by cell (i,j) and its vertical

neighbor (i, j + 1) is 27wsi3 .

106

Figure B-1: Cell layout of a cluster 30 cells in radius. Interior cells are colored white,exterior cells are colored grey.

Heat equations

For heat balance, we discretize our (now dimensionless) heat equations as

dQi A A4(TQ = M(Ti)pjVij 'H (k(pij), k(piejp)) (Tiyjl - Ti9)dt w

Metabolism

conduction

+i ui,i,4 -0 (Ui,ij) Ti + 0 (-ui3 ,iji) T'A3~ = 0.

convection

The first term corresponds to metabolic heat generation, and the second term is heat

conduction, where Z (i's') is the sum of all (i', J') neighboring (i, j), with the heat

conductance between two neighboring cells depending on the harmonic mean of the

conductance of each cell; H (a, b) = 2 -. The third term represents convective

heat transfer, with uij,i/j, the air flow from cell (i, j) to (i', ') (Units of dimensionless

volume/time due to discretization).

We have chosen an "upwinding" scheme [68]; when air flows out of a cell, the

outwards heat flux is determined by the temperature of that cell. When air flows into

a cell from a neighboring cell, the inwards heat flux is determined by the temperature

of the neighboring cell where the air originates. This scheme uses the Heaviside step

107

function:

x<0 X<

I X > 0.

Solving for buoyancy driven flow

The air flow from cell (i, j) to neighboring cell (i', j') is given by:

U -,ej = H (ri(pij), K(piiy)) A2 gj3 P - P 1 + (zie3r - zi3) (T + Ti - Ta)w 2 )

where the air conductance between two cells again depends on the harmonic mean of

the permeability of each cell, and the pressure is set so that air flow is conserved in

every cell i.e. Z(zj') uij,i = 0 for all cells (i, i). This yields a set of linear equations

that can be solved easily.

B.4 Role of temperature dependent metabolic rate

To formulate our temperature-dependent metabolic rate, we note that bees have a

higher base metabolic rate at high temperatures than at low temperatures. At mod-

erate temperatures, bees on the mantle keep their body temperatures approximately

3YC above ambient temperature, and at below 150C, they will "shiver" to keep their

body temperature at 18'C[31]. Assuming a constant coefficient of thermal transfer

between a bee and the surrounding air, this leads to a formulation of metabolism by

shivering at air temperatures below 15 C, and gives us the full piecewise function for

metabolic rate. Our metabolic rate for high temperatures comes from experiments

involving oxygen consumption in swarm clusters [31].

1 _+ 15_C- : T < 150CM(T) Mo 30C

T+ 150 C T ;> 15 C,

where the base metabolism, M0 has units of power/volume.

The simulated cluster with temperature-dependent metabolic rate has the same

108

6-

4-M(T)

2-

01-1 -0.5 0 0.5 1

TFigure B-2: Metabolic rate as a function of temperature.

qualitative features as when we use temperature-independent metabolic rates. We set

M0 to be 0.00175W/cm 3, half as high as when M was set temperature-independent,

because now it represents the minimal metabolic rate, not the uniform one (Fig.

B-3). Using dimensional analysis, this results in the values of ko, KO being doubled,

giving ko = 0.4, {o = 2.

B.5 Role of finite bee size on thermoregulation

In the paper, we have defined the temperature at the boundary to be the ambient

temperature, and we assumed the surface bees feel the ambient temperature, which

sets the behavioral pressure accordingly. In reality, bees point their heads inwards,

and feel a temperature gradient driven by the heat produced by interior bees. There-

fore, it may be more realistic that the behavioral pressure is set by the temperature a

slight distance inwards from the surface. If we include the effects of convection, this

implies that spherical symmetry must be broken, and the temperature becomes not

just a function of distance, but also dependent on angle. Therefore, to close our set of

equations without having to delve deeper into the question of cluster shape, we must

neglect upwards convection and only consider conduction. This gives us the system

of equations:

109

Cold Ta

0 0.5Ta

T

a)

b

=

0Ta

Figure B-3: Adaptation with temperature-dependent metabolic rate (Online versionin color). Note that cluster behavior is nearly the same as for constant metabolism (cf.Fig. 2-3). a) Comparison of temperature and packing fraction profiles at high(0.8)and low(-0.7) ambient temperature, where the dimensionless total bee volume V is1. b) Adaptation is nearly the same as for temperature independent metabolic rate.Ta is also plotted to guide the eye. c) Cluster radius is also nearly the same as for

temperature independent metabolic rate. For packing fraction, we choose coefficientsof po = 0.85, co = 0.5, ci = 0.25, pmax = 0.8.

p.M 0 +V - (k(p)VT) = 0, k(p) = ko 1 -p

T = Ta |-6,

where the behavioral pressure is now set by the temperature a distance of Lbee inside

the cluster rather than by the ambient temperature, so that the equation for density

110

Hot Ta Hot Ta P/

c)

1.5-

I ? - -, I

0.5

0

Cold T,

10.80.60.40.20

V 1.5V 1V 0.5

S-1.5-10.51

0.5rcore

Ta0

-0.5

-0.5 -0 5

is:

p= min {Po - TcO - T(R - Lbee)Ci, Pmax}. (B.1)

Assuming Lbee = 1 cm, slightly shorter than the body length of a worker bee, for

an average cluster radius of 7 cm we find that the dimensionless Lbee of about 0.14.

We simulate the system for dimensionless total bee volumes of 0.5, 1, and 3, with

the same parameters of co = 0.45, c1 = 0.3, Pmax = 0.8 as were used in the paper.

In the first system, to test thermoregulation with this effect, we choose Lbee = 0.14,

and compensate for the slightly lower average behavioral pressure by increasing po

to 0.95. In the second system, we test thermoregulation without this effect, and

so we choose Lbee = 0, po = 0.85, as we did in the main body. We find that for

large clusters, the temperature gradient created by the interior bees lowers behavioral

pressure and loosens the cluster. This mitigates overheating in the core and sensitivity

to total bee volume, e.g. at Ta = -0.7, Tcore varies - 50% more with cluster size

when behavioral pressure is set by ambient temperature rather than the temperature

beneath the surface. We note that in the models of Myerscough [54] and Watmough-

Camazine[99], a similar shielding of surface bees from ambient air and reduction of

sensitivity to cluster size was achieved through use of a heat transfer coefficient, with

units of Power/(Area Temperature) between the cluster surface and ambient air.

B.6 Linear stability of cluster

Solving for linear stability using simple "behavioral pressure-taxis" dynamics (B.6.1),

we find that all clusters simulated at a temperature-independent metabolic rate are

stable. However, at low ambient temperatures, clusters with a temperature-dependent

(B.4) metabolic rate can be linearly unstable via an overheating instability, which we

believe to be a relic of fixing the boundaries of the cluster. In this instability, bees

from the core move to the mantle, increasing the mantle thickness and insulation.

This causes the core to heat up, increasing its behavioral pressure causing even more

bees to move from the core to the mantle, leading to eventual runaway. We only

see this instability when using a temperature-dependent metabolic rate, where an

111

a)_b)V=3 1.5V =I1 1 V 3

V=0.5 -

0.5 V 05 -0.5

Teore Tore0 Ta o Ta

-0.5 -0.5

-0.5 0 0.5 -0.5 0 0.5Ta Ta

Figure B-4: Comparison of core temperatures(Online version in color). a) Behavioralpressure is set by the temperature slightly below the surface of the cluster, and sen-sitivity to total bee volume is mitigated. b) Behavioral pressure is set by ambienttemperature, and sensitivity of core temperature to total bee volume is considerablyhigher. Ta is also plotted in each to guide the eye. Note that figure b) is very similarto Fig. 2-3b), as it results from the same equations except for convection.

increased core temperature results in a greater net metabolic rate, aggravating the

problem.

Dynamical behavior which allows the cluster radius to vary would suppress this

instability, as bees moving from the core to the mantle would result in an expansion

of the cluster, increasing the surface area and cooling the core. However, a dynamical

model which allows the boundaries of the cluster to change requires a better under-

standing of the bee-level structure and the mechanics within a swarm cluster, and we

leave this aside here.

B.6.1 Methods for calculating linear stability

Having solved for the equilibrium state, we want to find if this state is stable or

unstable. To do so, we must define some dynamical laws for bee movement. Choosing

a simple "behavioral pressure-taxis" behavior, where J c -PVPb, dp -V - J gives

112

Cluster Unstablea) T Profile p b) T Eigenvector P

1 r - -,- - 10.2

1 1 0.150.8

0.5 0.10.5 0.5 0.5 0.5

0.050.5 0.6

ZoZo Zo 0 Z0 0.4 -0.05

-0.5 1 -0.5 -0.5 . -0.5 .-0.52

-0.15-00..

-- 1 . .2-0.15

-1 -0.20.20.40.60.8 1 1 .2 0.20.40.60.8 1 1.2 0.20.40.60.8 1 1.2 0.20.40.60.8 1 1.2

S S S S

Figure B-5: a) Cluster profile. b)Contour plots of unstable eigenvector of linearresponse matrix(Online version in color). Circle represents boundaries of the cluster,and a temperature-dependent metabolic rate is is used. Jo = 1, V = 1.5, Ta = -0.7.po = 0.85, co = 0.5, ci = 0.25, pmax = 0.8. For heat transfer, we choose coefficients ofko = 0.4, no = 2. Note that the packing fraction very close to the boundary is fixedbecause p = Pmax-

us the complete set of dynamical equations:

pT pM(T) +V - (k(p)VT) - CzZ -VT =0 = pF[p, T], T = Ta |FEEn

U = [Kaira'air (T - Ta) 2 -VP]K(p)/T Wn, V - U 0, P = 0 |FG6Q.

dpJ = -JPVPb, j = 0 |dp6Q, - V . EQ= G[p, T]

dt

Here we vary Jo over a wide range to reflect the large variations in bee movement

and temperature time scales, and define F, G to be the functionals that determine the

dynamics of the system. We also emphasize some issues about these choices: a) Our

form assumes a substrate that the bees move on. In reality, in a swarm cluster the

bees are the substrate, and changes on one side of the cluster propagate mechanically

to other parts of the cluster at a rate faster than the taxis rate. b) Bee movement is

not necessarily a local movement down pressure gradients. Bees can disconnect from

the cluster and reattach at a different point, which doesn't fit into the local gradient

picture. c) There is a discontinuity in behavioral pressure and packing fraction at the

boundary of the cluster, so this taxis model does not explain how the boundary of

113

the cluster can change. Within these limits then, this choice of behavior gives us a

full set of differential equations. To determine if the equilibrium state of the system

is linearly stable, we must determine whether the linear response matrix:

dF dF

M dT dpdG dGdT dp /

has positive eigenvalues. We note that the equilibrium temperature and bee packing

fraction profile we solved for is symmetric with respect to rotation about the central

axis(O direction), and therefore we may partition the space of perturbations into

subspaces defined by wave number ko : AT(s, z, 0$) = eik+%6T(s, z), Ap(s, z,#) =

eikoo6p(s, z). These temperature and bee packing fraction perturbations will in turn

give a change in airflow, pressure, and behavioral pressure: AI(s, z, #) = eikkc is(S, z),

AP(s, z, #)= eik+5 P(s, z), APb(s, z, #) eike8Pb(s, z). All of these will change the

temperature and bee packing fraction time derivatives which will be proportional to

eiko4. For each wave number ko, we construct the stability matrix and study its

spectrum. To first order:

p= - - Vsz +Vsz - kVsz - k + p( ) M AT+

[MAp + VszkpVszT] Ap - Au- -V sz T

( ko) 2 PAVsz - (pVszAP) - b

dP dPAPb= Ap+ ATdp dT

where Vsz is the gradient in the s and z directions, MT - dM (T) k = dk(p)dT P dp

Regions where p = pmax are locked at pma and not allowed to vary in bee packing

fraction.

114

Solving for AU', AP

The above equations depend on AU', which is determined by:

Au= [AT2 - rV (AP)] + Ap[(T - Ta) -VP].

AU' must have the form:

Ailz- ik " AP]Au = AUsz - KVAPi =.

where AU'sz is the sz component of Al. We solve for pressure AP using the

condition V - A' = 0:

2

- (AP ik)s S )

= 0 -> APr

At ko = 0, A'U- = 0, this condition simply becomes V8 z - Usz = 0.

Therefore, at a given AT, Ap, we can solve AU', AP from this set of linear equa-

tions.

Numerical computation of linear response matrix

To solve for stability, we discretize the system in the s, z directions in the same way

that we did when solving for the equilibrium state. Because we are only solving for

linear stability and the system starts off uniform in the # direction, we don't need

to discretize in the / direction. For perturbations at a certain wavenumber ko, the

temperature derivative is, to first order:

115

(k 2

8= - V z . Usz.I

dT, _

dt pie -ik = M ([T + 6T]j) (p + p) -

Metabolism

6Tj k(pjj)

Conduction in 0 direction

1+Vil.

A ijH[k (pi + 6p2j) , k (p + 6py)] (T + 6Tj,- (T +6T) -

Conduction in s, z directions

(ui,i'j' + 6uij,i'j [-0 'j + O (Tii + 6Tij) + 0 (- [uij,'(i'j')

+ 6u 1ij~i])I (Ti'3 , + 6Tij1,)]

Air flow in s, z directions

ir flw ijirjcio

Air flow in 0 direction

Note the slight modification of the upwinding terms, where we have also included a

# component to represent the influx or outflux of air in the # direction.

The density derivative is, to first order:

(Momn) 2in + drti

Movement in q0 direction

1i z [Awji p -; [' (P + 6Pb)'j - (Pb - Pb "i

Movement in s, z directions

The airflow is solved using the set of linear equations:

-ip'~ + 6uS,,2 -+ [H (r,(pi +Tey +6 Ti, i()

(P +6P)ij - (P + 6P)1 1j + (zi'3 , - zij) + , J

6P is set such that the divergence in the 0 direction negates the divergence in the

116

e_ _ = jo

dt I

s, z directions,

z u tj,i'jl = 6Pir(pij3 )w 2 (A)

for all cells (i, j).

117

THIS PAGE INTENTIONALLY LEFT BLANK

118

Appendix C

Supplemental Information for

"Generic model of an active porous

medium"

C.1 Determining if a distribution is bimodal

Each simulation i at a particular parameter value gives us a P(N), the probability

of measuring a N particles in a row, column, or clump in simulation. Averaging

many individual simulations allows us to calculate an average P(N), as well as -(N),

the estimated standard deviation of this average measurement. We use a 3-sigma

threshold of statistical significance-we are significantly more likely to measure N

particles than N' iff

P(N) - P(N') > 3 c/-(N) 2 + a (N') 2 (C.1)

If a local maximum P(N) can reach a larger P(N") without moving through a

valley where (C.1) holds, it is considered to be a false peak. If not, it is considered to

be a true peak. A histogram with at least two true peaks is considered to be bimodal.

119

One true peak Two true peaks

Figure C-1: The distribution on the left has many false peaks, but is not considered

to be bimodal. The right distribution is.

C.2 Comparison to local dynamics

For local dynamics, we we use a slightly modified time integration step.

1. Remove a particle from filled vertex i randomly selected with probability pro-

portional to r(vi).

2. Solve for the new current through the network.

3. Add the particle to an empty vertex j randomly selected with probability pro-

portional to a(vj) e a .

4. Solve for the new current through the network.

This prohibits a removed particle from traveling non-locally.

The phase diagram generated is very similar (Fig. C-2).

C.3 Analytical Mean Field Theory

An alternate mean field theory produces some of the same qualitative behavior. In-

stead of relying on numerics to find the values of A(p, aA), (p, aR), i'(p), we rely

on a very simple model which gives analytical results.

Channels: For predicting a channelization phase transition, we assume that cur-

rent is unable to travel in the 1 direction(Fig. C-3). Therefore, the equation for

conductivity is:

120

-6 6

6* .. ~.4 ... 5. 40_

8 "Ii'~ ' *' 7.

; 511 8 *TIP T

* ~ ~ 55.5, ______6:

-6Non Local

aR

a = 16

Figure C-2: Comparison of Nonlocal and Local Dynamics

A I~

Channels Walls

Figure C-3: Schematic of approximations made to obtain analytic forms for A, R, Kfor channelization and wall-building

Kchan (P) ~

1+LAQp-

Because current cannot flow laterally, an equal current of J is pushed through the

filled and empty vertices, and so

Achan(P, OA) = (1 - P) lchan(P, OR) p-

121

6

-6 -6

- - --- ~ -ar%~

*1* -

.* -~ ~y ~.. _______ _______

* s-.. A?'!~ ~ s'~v~V ______

~ Is I . . 5

* di ~ *I-'~" _____ _____

I__ UN IIIJ~~liiiIiiIL~w

We note that Awall is a function of aA -+ aR, and has no individual dependencekwa11'wall

on aA, aR-

Walls: For predicting a wall phase transition, we assume that, between rows,

current can freely flow in the -i direction without any resistance (Fig. C-3). Therefore,

the equation for conductivity is

Kwall(P) = 1-p) +. 1 + AQ'

The total driving across a wall is , and thus the current across an empty vertex1'wall

is ,J. The total current across a filled vertex is . Therefore:

Awaii(P, aA) = (1 - ) , Rwa(P, CeR) = P CI)Kwall(P) rlwal(p) (1+ A Q)

We note that Awai" is also function of aA -+ aR, and has no individual dependence onRwall

aA, CeR-

When p .25, a channelization phase transition occurs when aA + aR , -1.55.

A wall-building phase transition occurs when aA +n aR > 2.25.

C.4 Short Length Scale Continuum Model

We can also create a continuum model on a short length scale. To do so, we select

a periodic structure with two regions labeled 1 and 2. Region 1 comprises a fraction

V, of the squares, while region 2 comprises a fraction V2 of squares.

The density will originally be uniform, s.t. Pi = P2 p, and a mean density of '

(Fig. C-4) can transfer between squares such that

122

Pi - P2=p-s/V

.6T~~~~ _iMMITM a

= 0.1 s= 0.2

Figure C-4: Example systems where p .5, s = 0, 0.1, 0.2, with a spacing of d = 2.Orientation is set to channels /pillars.

At a particular imbalance s, the probability of an particle moving from region 2

to region 1 divided by the probability of the opposite process gives us a "fugacity":

ZR 2 (P2, K(p, 5), ft) -A1 (pi, '(p, ,), OA)

R1(p, r(p, ), aR) -A 2 (P2, r, s), aA)

Therefore, the free energy of a state with an imbalance .s is

- j Ln[N(p, ', aA, aR)jds

, will be set to minimize free energy, and when the optimal - is nonzero, the continuum

model predicts the system to spontaneously "crystallize" into a form where regions 1

and 2 have different density. For thin channels, region 1 is set by 6" mod d, where d

some integer which sets the spacing between channels or pillars. This walls are the

same except that region 1 is now set by 6y mod d.

The short-length scale continuum model gives similar behavior to the uniform

continuum model, although the change in free energy is nearly always lower.

123

ras"IF'rusen5 M)nsPlllftm a W%MINLawn 11 widMw aaspIrMS00% Mirma

MMORPOr" E43 amp*own, =%a 3Of wALMALA!a 't. 40 IRMNNra an a

0 Mass IXXrWO01 0 1 Vq

L. WIPS amel an asa

off MI

11

WIL IrMPas IPa1MP21 aa

5_=0

P2 - P - -9/V2

TI 1 4 IMTMrTiTITf I i

Walls

Channels

Clumps

Uniform

6

-6Ii !jj

Uniform

-6

Figure C-5: Comparison of large and

P = .25

-- -

6

-A

-6ill-- r

small length scale continuum model where

C.5 Higher Order Terms

To find higher order dependencies of T of Ap, we go through the following steps:

1. Obtain AK from Ap up to the desired order.

2. Obtain V setting flow to be conserved up to the desired order, e.g. V -J = V - [ - rVV] = 0

3. Obtain J through J = np - VV]

4. Obtain T through p, J up to the desired order.

Here, we will carry out all steps

the number of terms increases quite

to within second order. We do not go higher as

rapidly.

C.5.1 Finding change in conductivity

We start from

po + Ap =

124

6

Po + 1 p(k)eikkk

First we must find the change in conductivity, which we split into first and second

order components: K = KO + I + K2 + O(Ap 3 )

K = KO+ AK = K+ioAP Ip + PPp2 + 0 (Ap3) =

io +i- ,s 3 Ap (k) eikok + ppE p(k')p(k")eikOO(k' + k") + O(Ap3 )k k,ki

Giving us the first order and second order change in conductivity sI, K2:

K1(k) = Kpp(k)

rl 2(k) = rppp(k')p(k")k'+k",k.

(C.2)

(C.3)

C.5.2 Change in voltage

We must now find the change in voltage, which we do by setting current to be con-

served.

V _ J=V - KVV] = K;- _V 2V - V . VV + O(A 3) = 0

We separate voltage into it's first and second order components V1, V2:

Dzki - oV2V1 + -- 22V2 - vv1 =

We solve for V by balancing all first-order terms:

V2V1 = -[ -Ik 2 Vi(k)eik*k - =Z1 - Z ikIi (k) eikkKo KO

k k

-ik~-> V1(k) = i2 1(k).

rNo IkI

(CA4)

To calculate voltage to second order, there are four contributions which must be

125

matched;

ZK 2 - -_,72V2 - vv1 =

Fortunately, there is only one factor of V2 which we can move to the left hand side:

(,V2V2 = DzK2 - V 2 V1 - Vi - VV1 4 (-o k| 2) V2(k)eik*kk

=x: ikzK2 (k)eiko*kk

- Z VI(k')ki(k") k' -

_V17 2 V V

k' - k" 1Vai .VV1]

Ik' 2 + k' - k1, 2 12(k')s1(k")

6k'+k",k

Which we then simplify using k 8 k'+k",k = (k' + k") 6k'+k",k:

[-ikzK 2(k) +ik' k' - k

KO ( Ik' 1

C.5.3 Change in mean flow

We start off from the equation for J:

J = ( -VV)

Accumulating first and second order terms

J = tog + r12 + K2Z - rOVV - I{OVV2 - I1VV 1 + O(Ap 3 )

126

(C.5)

eik*po(k' + k")

=: V2(k) = (KO I k12) -l

(rso IkI2 -Zikzrs2(k)

K 1(k') K1 (k")6k'+k",k .-

We first calculate the first-order current, J1.

J1 = nis - noVVI = Z ,l(k)eik* k2 - no Z n1(k)i (kxs- + kz ) ik z k -kko kk 1

Z 1(k)eiko*k k x , kxkz

-+ Ji(k) = i1(k) [2 - kz_ (C.6)

Calculating the second-order current J2 , we find:

J2 = J 2 (k)e ik*k = r2 - r1VV1 - rOVV2k

We expand this as:

E J2 (k)e iko kk

= I' K2z -1: V(k')/>(k")ik'eik*o(k' + k") - 1 V2(k) noikeiko*kk k' ,k" k

J2(k) = K2 - Vl(k')Kl(k")ik'6k'+k",k - iKokV2 (k)

Plugging in V1, V2 gives us:

J2(k)= K2' -

-ikzK 2 (k) +

-ik'

no Ik/ 12

ik l .k K1(k')I K1(k")k'+k",k

We simplify this as:

J2(k) = 2 (k) k2 - kx2 + '1(k')rK1(k")6 k'+k",k k' k'k 12+k (k' - k)

(C.7)

127

K1 (k"/)ik'6k'+k"l,k

-inok (( o Ik 12)-l

Here, we can confirm that

V . [eiko*kJ 2 (k)] = ieiko*k[k -J2(k)] = 0.

It will be convenient to have this answer divided into x and z components:

kxkz- 12 12 +

, -k' k 2 + kz (k'- k)k 12 k' 12

xk' Ik12 + kI (k'- k)zk1

z k 12 1k'2

(C.-8)

C.5.4 Change in T

Likewise, we may represent the change in the time derivative functional:

T(p, J,4) = pAp + T.J i + TjJ2 + 2 (7h J, + T, p2 + 2 ApJi) + O(Ap3)

We note that, due to symmetry, T is even with respect to Jx are relevant. There-

fore we may simplify the time derivative as:

T = TAO + Tp+Jlz + TjJ2 z + 2 [mT J I + Th. J1+ T AP 2 + 2Tp J1zAp] + O(Ap3 ),

where we have used the notation Jiz = Ji -2, Jix = Ji , J22 = J2.

We recover the first order component T1 as before:

71 = TpAp + TJiz Ti(k) = [TP +TJziP k2 p(k).Ik 12_

We now calculate the second order component T2:

17-=TJ 2 z -IIj2j J1z +H- 3 'TJX , +~X Tp zXP2 + 2'TJP J1ZA P]

128

[2

J2(k) = 2k) si k

rsi(k') K, (k")6k'+k",k

Which we may decompose as:

T2 (k) = TJJ2z+1[Tjz Jiz(k')Jiz(k") + T,, J1.(k')J 1 (k") + Tpp(k')p(k') + 2Tjzp Jlz(k')p(k")]k'+k",k

Substituting Jlz, J2z, J 11 , we get:

[I k , p2 ppk k)k)+k"k

]k'+k",k

1 (k")k'+k",k ,-k' Ik 2 + k, (k' - k)+o z J' k12 k'12

k' k'z k'/k"k'12 k12+Tppp(k')lk' I k"I

2

(k')

We shuffle some terms around to get

T2(k) = (p(k')p(k")6 k'+k",k) 2[Tpp + T.ppI k12

T, , -k' Ik2 + kz(k' -k)[ kz - k12 k'2 + 2

( (k')2

Ik'12(k" )2k" 12

7 JxJx+ 2

k'kz k''kTk'/ 2 k"/12

(k')2+r,1(k')p(k")6k'+k"/,kj,p |' 2

And substitute Kl(k) = p(k)rp:

T2(k) = (p(k')p(k")6 k'+k",k)

+ (p(k')p(k")6 k'+k",k) (Kp)2

+ (p(k')p(k")6k'+k",k) TJZPK

K k2l2IPP + Ti ppk1

T -k' Ik 12 + k, (k'- k)kzk 2 k'1 2

(k')2lp Ik ' |12

TJ7JZ+2

((k')2 (k")2

k'12 k"12

J 2+2

k'kz k k)z]Ik' r2 k"/12

We obtain our final answer:

T2(k) = F(k, k', k")p(k')p(k")6k'+k",k,

129

F (k')2 (k"'')2(|k' 12 1k" 2+% 3 (k) (")

+Kl1(k ) I1(k )6k'+k",k

Where our interaction function, F, is:

F(k, k', k") = + Tk+ k Z k _k'_k_2+_k_(k'-_k 2 2 o k12 k'1 2

T3j (2)2 (kx)2(k/)2 Tj (K ) 2 k' k'z k'x'k'T' (k')22 Ik' 2 k"| 2 J 2 kk' 2 k" 2) P "k' 2

We note that F(k, k', k") -# F(k', k, k"), and thus can not be written as the derivative

of some cubic free energy functional. Therefore, like the discrete model, the continuum

model at second order has no detailed balance. This gives a stochastic field equation

with no detailed balance and complicated interaction coefficients, and we choose not

to proceed further. Note that q will also change with J, p, giving spatially dependent

white noise of the form:

(7(7 t) 77(f , t')) = R(P( t), J(F t),i aR) +M(t) (p (', t), JF t),i aRl ' 6 (r -- (t - tI),

giving rise to analogous first-order and second-order terms.

C.6 Methods

The code used was written in a combination of C++, MATLAB, and Objective-C, and

can be downloaded at http://web.mit.edu/socko/Public/PublishedCode/ActivePorousMediaCode.zip,

Currents were solved using the Eigen linear equation solver [26].

130