Special Year in Mathematical Biology
University of Utah
Schedule and abstracts for principle lecturers.
Fall Quarter, 1995
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Steve Ellner
North Carolina State University
October 2 - October 13
LIFE HISTORY EVOLUTION IN FLUCTUATING ENVIRONMENTS
I propose to lecture on life history evolution in fluctuating
environments, from both ESS and genetic perspectives. The ESS
perspective asks how a gene should gamble against an unpredictable
future and leads to 'bet-hedging' strategies of the
eggs-in-many-baskets type (dispersal in space, disperal in time, or
phenotypic diversity). The genetic perspective involves (i) exploring
the consequences of constraints, imposed by the organism's mode of
reproduction and inheritance, on the set of possible strategies; and
(ii) asking how the mode of reproduction and inheritance themselves
should evolve. Underlieing both approaches is a toolkit based on
stochastic demography, the use of Lyapunov exponents as a measure of
fitness, and the modern theory of Markov chains on general state
spaces. The main case study will be my recent work, with Nelson
Hairston, Jr., on life history evolution in copepods in response to
fluctuations in the timing and intensity of predation late in their
growing season.
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Robert D. Holt and Richard Gomulkiewicz
University of Kansas
October 16 - October 27
THE EVOLUTION OF SPECIES NICHES: POPULATION DYNAMIC PERSPECTIVES
Many species of organisms seem surprisingly conservative over
evolutionary time in their ecological niches---the range of environmental
conditions permitting the persistence of local populations without
immigration. This phenomenon is important in basic and applied ecology
(e.g., conservation biology) and evolutionary biology. In our lectures, we
will present a survey of mathematical models involving combined population
and evolutionary dynamics that shed light on niche evolution and
conservatism. The topics to be covered, which employ a wide range of
mathematical approaches, are as follows:
(1) Natural selection and extinction in a closed environment:
deterministic approaches.
(2) Natural selection and extinction in a closed environment: stochastic
approaches.
(3) The influence of immigration on local adaptation: fresh perspectives on
an old problem.
(4) Adaptive evolution in source-sink environments.
(5) Niche evolution in temporally varying environments.
(6) Niche evolution in metapopulations.
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Odo Diekmann
October 30 - November 3
MATHEMATICAL EPIDEMIOLOGY: A BIRD'S EYE VIEW
Various ideas, concepts, questions, objectives, techniques and results
pertaining to the mathematical description of the spread of an
infectious disease in a population will be reviewed. The underlying
theme will be the analysis of the population consequences of individual
behaviour. The following key words should give an impression of the
contents: invasion (the basic reproduction ratio R0 ), contact process,
epidemic, final size, demographic time-scale, persistence, regulation,
discrete structure, age structure, spatial structure. As a case study
we focus on the spread of Phocid Distemper Virus among seals. The key
modelling issue in this context turns out to be how does the per capita
contact rate depend on colony size?
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Simon Tavare
University of Southern California
November 6 - November 17
ANCESTRAL INFERENCE IN POPULATION GENETICS
Inferring the evolutionary history of populations from molecular data is a
hot subject these days. In these lectures, I shall attempt to describe some
of the probabilistic and statistical approaches to the subject, using
mitochondrial DNA sequence data from the North American Indians as the
motivating example.
The lectures begin with the basic ingredients of the stochastic models
that are used to study the molecular variability observed in samples of
DNA sequences. The approach is a genealogical one, and makes use of
Kingman's coalescent, a stochastic process that approximates the
genealogical relationships among the individuals in the sample.
The molecular variability observed in the sample is due to the effects of
mutation. Superimposing mutation on the genealogy leads to a probability
distribution for the sample configuration. These distributions form the
basis of statistical models for the data that are used for estimation
and inference. Such sampling distributions can rarely be found explicitly
-- the Ewens sampling formula, that appeared in the early 70s as a model
for allozyme frequency data, is the lone exception -- but rather they
arise as the solution of complicated recursive linear systems. We shall
see how a Markov chain Monte Carlo approach can be used to approximate
the solution to such recursions.
There are many applications of this technique, among them Monte Carlo
maximum likelihood estimation. We use it to estimate mutation rates and
to study something about `Mitochondrial Eve'. We focus on the distribution
of the time to the most recent common ancestor of a sample, conditional on
the observed DNA sequences. Applications to processes with variable population
size and to models of recombination are also given. The time to `Y Adam' is
also be discussed.
Synopsis of lectures
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1. Overview: The biology
2. Exchangeable models and the coalescent.
3. Mutation in the coalescent and sampling formulas
4. Estimation and Monte Carlo likelihood methods
5. Variants on a theme: variable population size and recombination
6. Mitochondrial Eve and Y Adam
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Simon Levin
Princeton University
November 20 - November 22
ECOSYSTEM RESPONSES TO CLIMATE CHANGE
I will discuss the problem of understanding how forests and grasslands
respond to climate change and how these responses are manifest on
larger scales. This requires describing the spatial interactions
between individuals through shading and resource competition. Detailed
individual-based models require too much information and provide too much
detail in their predictions. Out goal, therefore, has been to develop
approaches that include enough spatial information to describe interactions
adequately without retaining huge amounts of irrelevant detail. I will
describe our application of moment-closure techniques, based on methods from
fluid dynamics, to this problem.
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Roger Nisbet
University of California at Santa Barbara
November 27 - December 8
INDIVIDUAL-BASED POPULATION MODELS (IBMs)
The lectures will cover the foundations of individual-based population
modeling, using models of the zooplankter Daphnia as a case study. The case
study will illustrate the use of a wide range of methamtical approaches,
and will help focus discussion of problems related to model testing.
Lecture 1: Overview. The need for IBMs. What ecological issues do they
address? Different formulations. The case study - why model Daphnia?
Lecture 2: Simple models of individual growth, reproduction and mortality
(Von bertalanffy, Kooijman, Kooijman-Thieme, others)
Lecture 3: Models of growth reproduction and mortality in Daphnia pulex.
Lecture 4: From individuals to populations. Equilibrium demography. Four
approaches to modeling population dynamics: big simulations, matrix models,
PDE models, delay-differential equation models.
Lecture 5: Testable models of lab populations of Daphnia.
Lecture 6: Beyond Daphnia! Conflicting demands of generality versus
testability. Testing models of field populations. Open questions.