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:
My research program typically involves four key steps:
My results help to reveal how different processes and time scales interact to govern biological dynamics and to constrain diversity.
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.
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.
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.
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.
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.
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.
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.
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.
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.