Populations with self-regulated production and reproduction

• Applicant: Professor Dr. Anton Wakolbinger
• Subject Area: Mathematics
• Term: from 2003 to 2010
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 5470024

• Final Report Year: 2010

Final Report Abstract

The mathematical treatment of processes that model the evolution of populations over many generations has seen an enormous development over the past few decades, both conceptually and methodologically. In a suitable idealization, populations consist of individuals that live in a (genetic or geographic) space and have the ability to reproduce. Keeping track of the individual “lines of descent” in the scaling limit of large populations and many generations has added a new quality to population valued processes and also to the mathematics of random spatial models: key words are random genealogies, treevalued processes and the look-down representation. The individual reproduction may be regulated by a random environment or by the state of the population (e.g. through a dependence on the total number or density of the population). The project under report focussed on this aspect of “self-regulated reproduction”, and in parallel contributed to the topic of random genealogies and their evolution. Our work on alpha-stable branching and beta coalescents clarified links between the “forward time” notion of branching dynamics and the “backward time” notion of coalescents for continuum populations with heavily fluctuating offspring. Large (though not continuum) trees have also been investigated within our project. In joint work with Peter Pfaffelhuber we investigated processes derived from evoloving coalescents: the process of most recent common ancestors and the tree length process. Our research on genealogies under selection connects the two topics regulated reproduction and random genealogies. Based on structured coalescents, we derived (in joint work with Etheridge and Pfaffelhuber) an approximate sampling formula under genetic hitchhiking, which was further developed in joint work with B. Haubold. The joint work with Martin Hutzenthaler on the Ergodic behaviour of locally regulated branching populations is in the project’s core topic. By establishing a comparison to a mean field model we were able to derive a suffient criterion for local extinction for a class of spatially structured models with locally regulated reproduction. In subsequent work, M. Hutzenthaler defined and analysed the so-called Virgin Island model in which migration happens to ever new new islands and which is in close connection to the just mentioned mean-field model. Based on this, M. Hutzenthaler recently obtained a comparison theorem (and a criterion for global extinction) also for spatially structured (continuum mass) populations of finite size.

Publications

• Particle systems with locally dependent branching: long-time behaviour, genealogy and critical parameters. PhD thesis, Goethe-Universität Frankfurt, 2003
  M. Birkner
  
• A Condition for Weak Disorder for Directed Polymers in Random Environment. Electron. Comm. Probab. 9 (2004), 22-25
  M. Birkner
  
• The shape of large Galton-Watson trees with possibly infinite variance. Random Struct. Algorithms 25 (2004), 311-335
  J. Geiger and L. Kauffmann
  
• Alpha-stable branching and Beta coalescents. Electron. J. Probab. 10 (2005), 303-325
  M. Birkner, J. Blath, M. Capaldo, A. Etheridge, M. Möhle, J. Schweinsberg and A. Wakolbinger
  
• An approximate sampling formula under genetic hitchhiking. Ann. Appl. Probab. 16 (2006), 685-729
  A. Etheridge, P. Pfaffelhuber and A. Wakolbinger
  
• Approximate genealogies under genetic hitchhiking. Genetics 174 (2006), 1995-2008
  P. Pfaffelhuber, B. Haubold and A. Wakolbinger
  
• The process of most recent common ancestors in an evolving coalescent. Stoch. Processes Appl. 116 (2006), 1836-1859
  P. Pfaffelhuber and A. Wakolbinger
  
• Ergodic behaviour of locally regulated branching populations. Ann. Appl. Probab. 17 (2007), no. 2, 474 - 50
  M. Hutzenthaler and A. Wakolbinger
  
• Graphical representation of some duality relations in stochastic population models. Electron. Comm. Probab. 12 (2007), 206-220
  R. Alkemper and M. Hutzenthaler
  
• Interacting locally regulated diffusions. PhD thesis, Goethe Universität Frankfurt, 2007
  M. Hutzenthaler
  
• On Seneta’s constants for the supercritical Bellman-Harris process with E(Z+ log Z+ = ∞). Sankhya 69 (2007), 256-264
  W. Angerer
  
• How often does the ratchet click? Facts, heuristics, asymptotics. In: “Trends in Stochastic Analysis”, pp. 365-390, London Mathematical Society Lecture Note Series 353, Cambridge University Press 2009
  A. Etheridge, P. Pfaffelhuber and A. Wakolbinger
  
• Survival, extinction and ergodicity in a spatially continuous population model. Markov Process. Related Fields 15 (3) (2009), 265 - 288
  N. Berestycki, A.M. Etheridge and M. Hutzenthaler
  
• The Virgin Island Model. Electron. J. Probab. 14 (2009), paper no 39, 1117-1161
  M. Hutzenthaler
  
• Proliferation model dependence in fluctuation analysis: the neutral case. J. Math. Biol. 61(1), July 2010
  W. Angerer
  
• The tree length of an evolving coalescent. Probab. Th. Rel. Fields, published online by PTRF on June 25, 2010
  P. Pfaffelhuber, A. Wakolbinger and H. Weisshaupt