Special Year in Mathematical Biology University of Utah Schedule and abstracts for principle lecturers. Fall Quarter, 1995 ******************************************************************** 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. ******************************************************************** 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. ******************************************************************** 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? ******************************************************************** 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 ---------------------- 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 ******************************************************************** 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. ******************************************************************** 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.