Spatial dynamics in Biology

Fourth edition of the scientific course offered by the Doctoral Program in Information Technology of the Dipartimento di Elettronica e Informazione (DEI), Politecnico di Milano

Aim of the course

The course is intended for graduate students and researchers working in all disciplines and interested in the study of spatial dynamics characterizing many biological systems. The presented topics are complex, but the course aims at illustrating them in an introductory, yet rigourous, way. The main modelling approaches to the spatiotemporal dynamics in biology we present are:

Prerequisites

Basic differential and integral calculus, fundamentals of probability (mean, variance, probability densities and distributions, conditional probability, Normal and binomial distributions), basics of dynamical system theory (equilibria, linearization criterion, stability, nodes and saddles). Basic notions on Fourier series and Fourier transform will be briefly introduced during the course.

Lecturers and program

Tuesday 11th October 2011, 14.00-18.00
Marino Gatto — Sala Seminari (DEI)
Introduction and basics of stochastic processes — Why space matters in biology. The importance of networks. Discrete organisms and interacting particle systems (IPS). Markov chains in discrete and continuous time. Random walks and birth-death processes. Extinction risk in Malthusian and density-dependent populations.

References

  • Pielou EC (1977) Mathematical Ecology, Wiley & Sons, New York.
    [available in the Politecnico library, book id EH-0546]
  • D. Ludwig D (1974) Stochastic Population Theories, Lecture Notes in Biomathematics, No. 3, Springer-Verlag, Berlin.
  • Schinazi R (1999) Classical and Spatial Stochastic Processes, Birkhauser, Boston.
Wednesday 12th October 2011, 14.00-18.00
Marino Gatto — Aula Seminari (DEI)
Reaction-diffusion equations and travelling waves — Random walks and diffusion. Advection-diffusion equation. Diffusion and Malthusian growth in unbounded and bounded domains. Critical domain size. Diffusion-reaction equation in nonlinear density-dependent populations. Wave fronts. Front celerity. Telegraph equation.

References

  • Okubo A (1980) Diffusion and ecological problems: Mathematical models, Springer Verlag, Berlin.
    [available in the libraries of Dept. of Mathematics and DEI]
  • Hoppensteadt FC (1982) Mathematical Methods of Population Biology Cambridge University Press, Cambridge, UK.
  • Nisbet RM and Gurney WSC (1982) Modelling Fluctuating Populations J. Wiley & Sons, Chichester.
  • Holmes EE (1993) Are diffusion-models too simple? A comparison with telegraph models of invasion American Naturalist 142:779-795.
Thursday 13th October 2011, 14.00-16.00
Carlo Piccardi — Aula Alpha (Via Golgi 42)
Complex networks — Definitions, properties, indicators. Different topological models: random, small-world and scale-free networks.

References

  • Strogatz SH (2001) Exploring complex networks Nature 410: 268-276.
  • Albert R and Barabási AL (2002) Statistical mechanics of complex networks Reviews of modern physics 74: 47-97.
  • Newman MEJ (2003) The Structure and Function of Complex Networks SIAM Review 45: 167-256.
  • Boccaletti S, et al. (2006) Complex networks: Structure and dynamics Physics Reports 424: 175-308.
Thursday 13th October 2011, 16.00-18.00
Lorenzo Mari — Aula Alpha (Via Golgi 42)
Networks and dynamical systems — Dynamics and propagation of epidemics. Models of water-borne diseases.

References

  • Mari L, et al. (2011) Modelling cholera epidemics: the role of waterways, human mobility and sanitation Interface, in the press.
  • Bertuzzo E, et al. (2011) Prediction of the spatial evolution and effects of control measures for the unfolding Haiti cholera outbreak Geophysical Research Letters, 38: L06403.
  • Piccardi C and Casagrandi R (2008) Inefficient epidemic spreading in scale-free networks Physical Review E, 77: 026113.
  • Pastor-Satorras R and Vespignani A (2001) Epidemic dynamics and endemic states in complex networks Physical Review E, 63: 066117.
Wednsday 19th October 2011, 14.00-16.00
Marino Gatto — Aula 3B (DEI)
Percolation theory — Site and bond lattice models. Definition of percolation as a spatial stochastic process. Percolation thresholds.

References

  • Grimmett GR (1999) Percolation, Springer-Verlag, New York
  • Murray JD (1989) Mathematical Biology, Springer-Verlag, Berlin.
Wednsday 19th October 2011, 16.00-18.00
Fabio Della Rossa — Aula 3B (DEI)
Pattern formation — Diffusion and reaction in systems with two state variables. Turing instability. Formation of spatial patterns. Examples.

References

  • Peletier LA et al. (1999) Dynamical Systems and Nonequilibrium Pattern Formation e-book at Leiden University.
  • A software Ilya to study pattern fromation.
  • Cross MC and Hohenberg (1993) Pattern formation outside of equilibrium Reviews of Modern Physics 65:851-1123.
Friday 21st October 2011, 14.00-18.00
Marino Gatto & Lorenzo Righetto — Aula 3B (DEI)
Contact processes and Metapopulations — Interacting particle systems and contact processes. Local vs global survival and relationships with percolation. Fragmented populations: introduction and problems. Metapopulations as IPS's. The contact process and the mean-field approximation. Extinction risk in metapopulations. Various approaches: Markov chains, moment closure, stochastic cellular automata. The influence of landscape structure. Reserve design.

References

  • Hanski I and Gilpin ME editors (1997) Metapopulation Biology: Ecology, Genetics and Evolution, Academic Press, San Diego, USA.
  • Hanski I (1998) Metapopulation dynamics Nature 396:41-49.
  • Casagrandi R and Gatto M (1999) A mesoscale approach to extinction risk in fragmented habitats Nature 400:560-562.
  • Casagrandi R and Gatto M (2006) The intermediate dispersal principle in spatially explicit metapopulations Journal of Theoretical Biology 239:22-32.

Venue

The course will be held at the Dipartimento di Elettronica e Informazione. The Sala Seminari is at the ground floor, while the room 3B is at the third floor.

Application for participation

The course is free and open primarily to Politecnico di Milano PhDs, but also to other researchers interested to the presented topics. Please fill in the following participation form if you want to participate:

 

Slides and papers

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Page mantained by Renato Casagrandi (last update 08 Oct 11)