Research Statement

Overview
Mathematical approaches are necessary to understand the causes and consequences of the extraordinary diversity
in form and function that exists in biological systems. Much of my research focuses on the construction of
mathematical models to discover how organismal physiology influences biological structure and dynamics. As
explained in the sections below, I am studying the intrinsic organismal and extrinsic environmental factors that
influence: 1. the diversity and structure of plant and animal vascular networks, 2. sleep times, 3. tumor growth, 4.
ecosystem structure and dynamics, and 5. speciation-extinction dynamics. I study these problems using statistics,
dynamical systems methods, ordinary, partial, and stochastic differential equations, asymptotic methods,
optimization methods, numerical methods, fluid mechanics, and models for branching architecture. These studies
are necessary both to firmly establish a basic understanding of biological processes and to address practical
matters like the effects of climate change on biological systems and the medical treatment of tumor growth.
                    My research program typically involves four key steps. First, I analyze large empirical data sets to
discover recurring, statistically significant patterns that hold across multiple scales in space and time. Examples
of patterns I have found are allometric scaling curves for population growth and abundance as well as simple
functions for time scales of speciation-extinction dynamics. Second, I construct mathematical models based on
hypotheses for the underlying causes of the patterns observed in step 1, and I then derive further predictions from
these models. Third, I test these new predictions using extensive empirical data that I compile from the literature
or that are collected by my collaborators. Fourth, I revise my model based on deviations of the empirical data
from the original model’s predictions. My results help to reveal how different processes and time scales interact
to govern biological dynamics and to constrain diversity.

PAST WORK

Origins of Biological Scaling
In my previous work, I was instrumental in constructing mathematical models to explain a suite of allometric
scaling relationships that were observed yet unexplained for nearly a century. These relationships show that there
are powerful constraints on the diversity of biological systems. Knowing the origins of these constraints is
critical for understanding and managing the forces that shape physiological structures (e.g., plant and animal
vascular systems) and ecological systems (e.g., community structure and biodiversity). The theory focuses on
biochemical reaction kinetics and the structure and fluid dynamics of vascular networks in animals and plants. To
test these theories I compiled the largest existing comparative data set for metabolic rate and body size across
626 species of mammals and performed a series of analyses that demonstrate metabolic rate scales as body size
to the 3/4 power (CV Ref. 15). Similarly, I helped compile large comparative datasets that show metabolic rate
varies with body temperature as an exponential Boltzmann-Arrhenius factor (CV Ref. 22). Surprisingly, most of
the variation in metabolic rate across a broad range of organisms, including unicellular organisms, plants, and
animals, can be explained by body mass and temperature, collapsing 15 orders of magnitude variation to just one.

Applications of Biological Scaling
I have also developed theories that explicitly and mechanistically link this biological scaling theory to, for
example, population growth dynamics and cell size. My work is among the first to take these directions. At the
population level, I constructed a theory based on classic population growth equations, life history theory, and
allometric scaling theory to predict population growth rates and abundances (CV Ref. 13). I compiled data from
the literature for a broad assortment of species at temperatures that vary over a range of 30C and include field
mortality rates. These data strongly support my predictions. For cell size, empirical and theoretical allometric
relationships reveal that cellular metabolic rate and cellular mass cannot both remain constant as body size and
body temperature vary, indicating a fundamental constraint. Within mammals I have shown that neurons and fat
cells change cell size but maintain a constant metabolic rate across species, while liver cells and most other cell
types change cellular metabolic rate but maintain a constant cell size (CV Ref. 2).
            Much of this work can be used as a point of departure for my current and future work on the forces that
constrain and organize diversity as well as possible ways to manage and control this diversity. I now discuss my
specific goals in the sections below.

CURRENT AND FUTURE WORK

What Controls the Diversity and Evolution of Plant and Animal Vascular Networks?
The vascular networks of plants and animals exhibit remarkably consistent patterns in their branching
architecture, as compared to the huge diversity that is possible. Models for the architecture and evolution of
vascular networks are necessary to understand how plants adapt to their landscapes and environments. Over the
past decade, several models have focused on the structure of vascular networks to explain the allometric scaling
of metabolic rate with body size. Among these models, the West, Brown, Enquist (WBE) model is the first and
most prominent. Despite considerable success, the WBE description of vascular networks uses many simplifying
accumptions, and the selective and developmental forces that mold these networks are almost completely
unexplored. I have been working to develop more realistic models for the vascular networks of plants and
animals and to discover the biological forces that shape these networks.
            Intense debate has surrounded WBE and the accuracy of its predictions. In particular, critics have pointed
to the rich diversity in vascular networks and scaling relationships. Some of this richness could be explained with
a more biologically realistic and mathematically sophisticated model of vascular networks. To evaluate this, I
first performed a comprehensive analysis of the WBE model by working with Deeds and Fontana (CV Ref. 24).
We used analytical, asymptotic, and numerical techniques to show that all previous studies have derived wrong
predictions for the WBE model because they effectively assume that all organisms are massive in size. Without
making this approximation, our analyses show that the allometric slope is actually predicted to be 0.81, larger
than 3/4=0.75 and excluded by the 95% C.I. determined from empirical data for mammals (CV Ref. 15). I also
make new, more specific predictions about the direction and amount of curvature that should be observed in the
data, and those predictions are again at odds with empirical data for mammals. To develop a more biologically
realistic model of vascular networks than WBE, I have begun to perform calculations of fluid resistance and flow
in vascular systems that rely on detailed treatments of Navier-Stokes equations coupled to Navier equations for
the elastic vessel walls. Perhaps more importantly, I am working to devise analytical and numerical methods for
networks composed of asymmetric branching—when one vessel branches into two or more daughter vessels that
have different radii, lengths, and flow rate. My formalism allows for any degree of asymmetry in vessel radius,
length, or both. As detailed in a recently awarded NSF grant, I plan to test this model across a wide range of
vascular networks, using data for Ponderosa pines (Brian Enquist (University of Arizona)), for plants in which
apical dominance is common (e.g., vines and conifers), and for the highly asymmetric coronary artery (Mair
Zamir (University of Western Ontario)).
                            My new, more mathematically sophisticated models are essential to investigate the selective and
developmental forces that mold vascular networks in plants and animals, and thus, the complicated feedbacks
between vegetation and landscape as mediated, for example, by light and water availability. Price, Enquist, and I
analyzed empirical data that include many desert plants in order to determine which features of networks are
under strong selection and how these features affect the fitness of the plants. We found that vascular structures
are more attuned to energy minimization and water transportation (primarily constraining scaling ratios of radii)
than to space filling and light gathering (primarily constraining scaling ratios of branch lengths) (CV Ref. 6).
These more sophisticated models for vascular networks should aid understanding of how plants adapt to their
landscapes and environments, and that will also inform how plants shape the landscapes in which they grow.


How Does Tumor Vasculature Differ from and Interact with Healthy Host Vasculature?
Many researchers recognize that there is a serious need for testable mathematical models that describe and
predict the vast amount of experimental data generated by cancer labs, particularly if these results are to reach
clinical significance. To model tumor growth, I advised a student, Alex Herman, and we extended the scaling
theory for the cardiovascular system to model the abnormal vascular development in tumors and how that
interfaces with the host vasculature. We then combined this extended model with growth equations to model the
growth dynamics of tumors. From this theory we predict the scaling of tumor metabolic rate with tumor and host
size and predict tumor growth trajectories. Our predictions are consistent with available data. The theory we have
developed provides a general framework for understanding detailed vascular properties of tumors, including
predictions for growth trajectories, degree of necrosis, rates of nutrient supply, and dependence on host size and
cellular properties. Moreover, our model enables quantitative comparisons of tumor growth across species and
connects whole tumor phenomena from cellular to organismal levels. As a result, our work predicts stages when
tumor growth is most rapid, helping to resolve Peto's paradox, namely, the failure of whales and humans to be
more cancer prone than mice. These predictions potentially provide important insights for drug discovery and
intervention in slowing tumor growth.

What Sets the Time Scales for Sleep and What Is the Purpose of Sleep?
Sleep, like eating and breathing, is one of the most ubiquitous phenomena in all of biology, yet its function
remains unknown. Many hypotheses exist for the function of sleep, but these are mostly based on qualitative
models and possibly tests of correlation using empirical data. I developed the first mathematical models and truly
quantitative, comparative tests for sleep function (CV Ref 1). Different models correspond to different
predictions for the scaling exponent for how sleep times vary with body and brain size. Using empirical data for
mammals (97 species and 79 genera), I showed that sleep is driven by rates and processes in the brain, not the
body, and that hypotheses for sleep related to neuronal repair and reorganization are the ones that are most
consistent with empirical data. This work was heralded by Steven Strogatz as a cause for optimism in the new
year (CV Press). I plan to continue this work by analyzing developmental sleep data in humans. As humans grow
from birth to adulthood, their sleep times, brain size, brain metabolic rate, and neural reorganization all change,
providing a further test of sleep theories and a potential means for quantifying the relative importance of neural
reorganization and repair. Proceeding in this manner, I hope to disentangle the evolutionary need for sleep and
the necessary functions it currently serves in mammals and other organisms.

How Do Species Interactions and Invasions Respond to Temperature Change?
Invasive species can quickly alter biodiversity levels and community composition, so management strategies for
biodiversity critically depend on the anticipation of invasions. One of my goals is to understand how global
warming will affect the control of population abundance and the ability of species to invade. More generally, I
want to use models of predator-prey systems to understand when invasions can occur and why we see the
predator-prey links that are observed in nature.
                    The Metabolic Theory of Ecology (MTE) (CV Ref. 16) has done remarkably well at explaining the
effects of temperature and body mass on ecological and some evolutionary patterns. This achievement is
somewhat paradoxical given that this body of theory mainly focuses on inherent properties of organisms (e.g.,
vascular networks, body mass, and body temperature) and essentially ignores species interactions. Since most
definitions of ecology include how organisms interact with each other and their environment, this paradox is all
the more troubling. Certainly, many ecological and evolutionary processes crucially depend on how species
interact with each other and the environment, and how these interactions play out across different spatial and
geographic scales. Therefore, it is crucial to understand how these processes are influenced by body size and
temperature and when those variables fail to adequately explain observed patterns or to predict changes.
                            I have begun to develop theory to combine MTE with these types of species and environmental
interactions. To accomplish this, I am constructing novel theory and analyzing extensive empirical data—tens of
thousands of data points—to determine the effects of body temperature and size on predator-prey interactions.
The theory uses a random movement model for interaction rate that depends on body size and temperature and
accounts for differences in behavior (e.g., predator and/or prey strategies such as active capture or sit-and-wait)
and environmental influence on organismal physiology (e.g., ectotherm-ectotherm versus endotherm-ectotherm
interactions). Moreover, I can make predictions for the effects of body temperature and size on encounter rate,
relative speed of a predator-prey pair, mortality rate, growth rate, and more. These predictions can then be used
as input to Lotka-Volterra-type equations that now naturally and explicitly include the scaling of body size and
body temperature. These equations are then used to predict the temperature and body size dependence of
consumption rate, equilibrium population size, optimal predator-prey relationships, and conditions for species
coexistence. To test these assumptions and predictions, my collaborator, Tony Dell (James Cook University), and
I surveyed the literature and compiled empirical data that represent a huge diversity of predator-prey functional
types, species, and environments. We are still analyzing these data but preliminary results are encouraging. By
combining this framework with food-web data, we also hope to predict the prevalence of certain predator or prey
strategies as well as the amount and organization of diversity in empirical food webs at different temperatures.



How Does Ecosystem Structure Respond to Climate Change?
I am helping to develop a framework for studying real ecosystems in which a changing environment affects a
suite of interconnected traits. My fundamental goal with this work is to understand how trait functional groups,
species, and ecosystems can respond to rapid environmental change. Specifically, I want to understand how
climate change affects biodiversity, how much time is required for significant shifts in community structure, and
to identify suitable habitats across the globe for a variety of species and ecosystems. Norberg, Webb, and I have
developed a trait-based approach to explore the effects of a changing environment on biomass distributions and
community structure for traits. Using this framework requires carefully constructed fitness/growth functions,
often involving stochastic differential equations that couple traits to the environment, so that trait distributions
can be calculated numerically and followed through time. To work on this, I have received funding and am
organizing a meeting (ARC-NZ WG36) in Sydney, Australia this February that will bring together experts in the
field. I have already devised a framework that can be used to study multiple, correlated environmental variables
(e.g., temperature and precipitation) and associated multiple, correlated traits (optimal temperature for growth
and water use efficiency) to discover non-intuitive effects that arise from correlations (CV Ref. 3). This
framework also extends the theory to include frequency dependence and functional complementarity. Apart from
numerical simulations, insight into the dynamics of these systems is enabled by a moment expansion and
appropriate closure method for the equations. In NSF funded research, Norberg, Webb, and I are beginning to
apply this framework to real ecosystem data for plants (Konza Prairie LTER site). Combining this work with
predictions for changes in global environmental conditions (e.g., temperature and carbon dioxide), we hope to
predict how the diversity of trait functional groups and ecosystems respond to rapid environmental change. The
following schematic summarizes the overarching themes of my ecological research.

What Sets Time Scales for Speciation-Extinction Dynamics?
Correctly predicting the time scales for speciation and extinction is a litmus test for any theory of evolution, a
lynchpin in arguments that life as we know it could have evolved during the history of the earth, and necessary
for understanding contemporary and future biodiversity. Although the neutral theory of biodiversity (NBT)
successfully predicts several characteristics of biodiversity and links ecology to evolution, it fails miserably at
predicting the time scales of speciation and extinction, being off by many orders of magnitude (~800 Myr).
Moreover, its assumption that mutational and phenotypic changes are neutral has created much controversy,
echoing earlier debates from the evolutionary neutral theory that inspired NBT. Using stochastic partial
differential equations and Green’s function methods, I have helped to formulate a new theory that incorporates
aspects of NBT but is more consistent with biological knowledge (CV Ref. 4). Specifically, this theory allows
new species to arise with more than one individual, as opposed to NBT and the point mutation model. This is
much truer to real biological systems in which new species often arise with a very large number of individuals
(~106 individuals for germinate shrimp). Moreover, Allen and I allow time scales to be affected by environmental
stochasticity, not just demographic stochasticity. Consequently, we allow for niche differences and obtain
predictions for long times during which selective pressures may average out. We have both contemporary and
fossil data for Foraminifera that give empirical estimates of contemporary levels of biodiversity and average time
scales of speciation and extinction events over the past 30 Myr. Our new model yields predictions for time scales
and species-abundance distributions that are consistent with this extensive and unique data set for Foraminifera.
We now plan to employ more sophisticated models of environmental variation and correlation by using different
stochastic processes. We also intend to more fully integrate these evolutionary and ecological time scales by
modeling stochastic fluctuations in population abundance that partially determine incipient species abundances.
These studies are important for predicting the effects that climate change will have on speciation-extinction
dynamics, on contemporary levels of biodiversity, and on extant species.

Summary
I want to use mathematical models to understand how diversity is organized, constrained, and controlled in
biological systems. My models are constructed using fractal and asymmetric branching networks along with
ordinary, partial, and stochastic differential equations. I analyze these models using asymptotic methods,
optimization theory, dynamical systems methods, and numerical methods. I test the predictions of these models by
performing statistical analyses on large empirical data sets obtained by my collaborators or compiled from the
literature. I would contribute to the department by using and teaching a variety of mathematical methods and by
bringing to the department my diverse experience constructing quantitative models from conceptual ideas, a suite
methods for analyzing large empirical and comparative data sets, and a fresh perspective on many scientific and
other applied problems.