Research Summary
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I use mathematical approaches to address problems that arise in biology.
Recently, my work has centered on three main areas of theoretical
ecology: competition between species (plant competition for sunlight,
interactions at multiple spatial scales, directed movement along
resource gradients, and structured resources), the spatial spread of
epidemic diseases (rabies and others), and the evolution of optimal
choice (state-based decision-making in foraging gray jays, the honeybee
nest-site selection process, and animal-mediated seed dispersal). I also
am interested in species persistence and permanence within ecological
communities; the dynamics of spatially (or otherwise) structured
populations; classical and social foraging theory; animal and plant
behavior; and formulating ecological models that make use of mechanistic
reasoning and principles.
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Modeling philosophy
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An ecological model consists of assumptions and rules that govern the behavior
of an idealized ecological system. Model formulation is the exciting process by
which these assumptions are articulated, and this requires a solid
understanding of the fundamental ecological processes that are of interest.
Sometimes, the simplest version of a model can be considered analytically, in
which case its assumptions give rise to mathematical equations and ultimately
to theorems that constitute its conclusions. Model formulation is also an
iterative process, in which successive refinements arise as both analysis and
intuition reveal departures from nature and ever better ways to capture
biological realism. As a mathematical biologist, I must combine biological
insight with appropriate analytical and computational methods to make the most
of my mathematical models.
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Competition Between Species
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Plant competition for sunlight
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I completed my doctoral dissertation on mathematical models of plant
competition for sunlight under the co-direction of Paul Roberts
(UCLA, Mathematics) and Richard Vance (UCLA, Ecology). We formulated
and analyzed a canopy partitioning model
(Vance and Nevai 2007,
Nevai and Vance 2007,
Nevai and Vance 2008)
to determine whether two plant species with clonal growth forms and
possessing distinct but overlapping vertical leaf profiles can
successfully coexist while competing for sunlight and no other
resource. The results of this base model are robust to a variety of
density-independent and density-dependent enrichments.
I am collaborating with Winfried Just (Ohio University, Mathematics) on
open questions posed in my Ph.D. dissertation. In
(Just and Nevai 2008),
we constructed an example in which two clonal plant species
that obey a canopy partitioning model
(Vance and Nevai 2007,
Nevai and Vance 2007),
and have rectangular vertical leaf profiles and distinct
photosynthesis functions, can coexist stably at multiple equilibra.
We also constructed a second example, in which the species share the
same photosynthesis function but one species has a bi-rectangular
vertical leaf profile. We recently submitted a second paper which
explores plant competition for sunlight in which at least one
species has a finitely supported vertical leaf profile
(Just
and Nevai in review).
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Species interactions at multiple scales (and spatial moment equations)
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Ben Bolker (University of Florida, Biology) and I are studying
connections between a stochastic spatial logistic equation with
diffusion and its related spatial moment equations (Bolker and
Pacala 1999, Dieckmann and Law 2000). I am also interested in
connections between the fundamental growth equation of population
ecology and its related spatial moment equations. Our results are
currently in preparation.
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Competition and directed movement along resource gradients
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Yuan Lou (Ohio State, Mathematics) and I are collaborating on a mathematical
model that describes the interactions of spatially-structured
populations. This continuous-time patch model will compare the
outcome of competition between two Lotka-Volterra type species, both
of which move randomly by diffusion but only one of which can also
move intelligently toward patches with higher resource levels. It is
expected that under some parameter combinations, coexistence will
result in which the intelligent species concentrates in patches with
high resource levels (relative to neighboring patches) and the other
species subsists in the remaining patches. This work may confirm and
extend the results of an earlier related partial differential
equation competition model with ecological diffusion and intelligent
movement along resource gradients (Lou 2005). This work will also
evolve into other related projects.
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Spatial Spread of Infectious Diseases
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Linda Allen (Texas Tech University, Mathematics and Statistics), Yuan
Lou (Ohio State, Mathematics), Ben Bolker (University of Florida,
Biology), and I are collaborating on a family of spatial epidemic disease
models (SIS) in which the movement rate of one subpopulation
(susceptible individuals) becomes very small relative to that of another
(infected individuals). The project consists of mathematical analysis of a
continuous-time discrete patch model
(Allen et al. 2007),
a continuous-time reaction-diffusion model
(Allen et al. 2008),
and a discrete-time patch model (to be submitted).
In each project, we connect spatial heterogeneity, habitat
connectivity, and different movement rates among subpopulations to
the observed spatial patterns of infectious diseases. Local
differences in disease transmission and recovery rates characterize
whether regioins are low-risk or high-risk, and these differences
collectively determine whether the spatial domain, or habitat, is
low-risk or high-risk. In low-risk habitats, the disease persists
only when the mobility of infected individuals lies below some
threshold value, but for high-risk habitats, the disease always
persists. When the disease does persist, then there exists an
endemic equilibrium (EE) which is unique and positive everywhere.
This EE tends to a spatially inhomogeneous disease-free equilibrium
(DFE) as the mobility of susceptible individuals tends to zero.
Sufficient conditions for whether high-risk regions in the limiting
DFE are empty can be given in terms of disease transmission and
recovery rates, habitat connectivity, and the infected movement
rate. For each model, we also compute the basic reproduction number
as the spectral radius of the appropriate next generation
operator.
In studying these models, we make use of comparison principles, the
theory of nonnegative and irreducible matrices, the theory of
elliptic operators, linear eigenvalue problems, the Perron-Frobenius
Theorem for the eigenvalues of a nonnegative irreducible matrix, the
Krein-Rutman Theorem for the eigenvalues of a positive linear
operator, maximum principles for discrete and continuous systems,
and super-solution and sub-solution methods for monotone dynamical
systems.
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Continuous-time disease patch model
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One of these models, which is in continuous-time and describes the movement of
individuals between discrete patches, consists of a system of differential
equations. More description to come soon...
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Reaction-diffusion disease model
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Our continuous-space version of the model consists of a system of reaction
diffusion equations. Several notable feature of this
model are that the basic reproduction number is defined as the solution
to a variational problem and the limiting DFE is defined as the solution
to a free-boundary problem.
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Discrete-time disease patch model
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Our discrete-time patch model consists of a system of difference equations.
More description to come soon...
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Evolution of Optimal Choice
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State-based decision-making in gray-jays
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Tom Waite (Ohio State, Evolution, Ecology, and Organismal Biology),
Kevin Passino (Ohio State, Electrical Engineering), and I
collaborated on a state-based individual foraging model for gray
jays (Perisoreus canadensis). This model, which is based on a
previous optimal choice model (Houston and McNamara 1999),
incorporates partial preferencing to describe a tradeoff between
maximizing reproductive value and minimizing predation risk for an
individual that hoards tens of thousands of food items over the
course of a year. This model will ultimately lead to generalized
mathematical models of optimal choice that can be applied to a wide
variety of individually foraging animal species. Two papers on this
subject have now been published (Waite et al. 2007,
Nevai et al. 2007).
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The honeybee nest-site selection process
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I am collaborating with Kevin Passino (Ohio State, Electrical
Engineering) on a mathematical model to describe the distributed
self-organizing quorum-sensing decision-making process called
nest-site selection in honeybees (Apis mellifera). We are
formulating this model in continuous-time using a series of systems
of ordinary differential equations in which new sites are discovered
and considered asynchronously. This model is based on social insect
foraging models and several previous nest-site selection models,
including a continuous-time epidemiological-based SIR model (Britton
et al. 2002), a discrete-time Leslie matrix model (Myerscough 2003)
and a recent stochastic discrete-time model (Passino and Seeley
2005). Our model will demonstrate that natural selection has tuned
certain parameters in this process so that the speed of
decision-making (measured on the order of hours) is balanced against
accuracy (so that a high quality nest-site is actually chosen). Our
results are currently in preparation.
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Animal-mediated seed dispersal
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Graduate student Vishwesha Guttal (Ohio State, Physics), Gregg
Hartvigsen (SUNY Geneso, Biology), and I are examining the role of
animal-mediated seed dispersal mechanisms (such as endozoochory and
epizoochory) and animal movement behaviors (uncorrelated random
walks, correlated random walks, and Levy flights) on the ability of
plant species to obtain leptokurtic dispersal kernels. Our results
are currently in preparation.
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Outlook
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I find it natural to describe ecological problems using the language
of mathematics because ecological communities are dynamical systems
and evolution by natural selection favors optimization. The use of
mathematics not only improves our understanding of ecology, but also
offers new and interesting challenges to mathematicians. When
mathematicians collaborate with biologists in problem solving, the
resulting work can be realistic and useful to the biological
community, it can deepen our understanding in the originating field,
it can make creative use of both pure and applied mathematics, and
it often leads to the development of new mathematical methods. I
hope that my future research into mathematical ecology will allow me
to find new connections between existing areas of mathematics and
biology, and will also inspire new mathematics.
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