Spatial dynamics in Biology

Fifth edition of the scientific course offered by the Doctoral Program in Information Technology of the Dipartimento di Elettronica, Informazione e Bioingegneria (DEIB), 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:


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). Where needed, other mathematical tools will be briefly introduced during the course.


The course will be held at the Dipartimento di Elettronica, Informazione and Bioingegneria. The Aula Seminari Alessandra Alario is in building 21 at the fourth floor.

Lecturers and program

Monday 3th June 2013, 09.00-13.00
Marino Gatto
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.


  • 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 5th June 2013, 09.00-13.00
Marino Gatto
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. Diffusive instability.


  • 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.
Friday 7th June 2013, 09.00-13.00
Renato Casagrandi, Lorenzo Mari & Paco Melià
Models for movement ecology: multi-layered networks and coupled physical-biological models — What movement ecology is and why it is important. Eulerian vs Lagrangian approaches to study the movement of organisms. Two paradigmatic and important cases: the spread of an invasive species (the zebra mussel Dreissena polymorpha) along river networks and the long journey of eels (Anguilla anguilla) from the Sargasso sea to Europe.


  • Nathan et al (2008) A movement ecology paradigm for unifying organismal movement research PNAS 105: 19050
  • Holyoak et al (2008) Trends and missing parts in the study of movement ecology PNAS 105: 19052
  • Casagrandi et al (2007) Modelling the local dynamics of the zebra mussel (Dreissena polymorpha) Freshwater Biology 52: 1223
  • Mari et al (2011) Hydrologic controls and anthropogenic drivers of the zebra mussel invasion of the Mississippi-Missouri river systemWater Resources Research47, DOI:10.1029/2010WR009920
  • Melià et al (2013) Melià, P, Schiavina M, Gatto M, Masina S, Bonaventura L, Casagrandi R. In Press. Integrating field data into individual-based models for the migration of European eel larvae Marine Ecology Progress Series, in press
Monday 10th June 2013, 09.00-13.00
Renato Casagrandi & Lorenzo Mari
From mean-field to spatially realistic epidemiological models — The classical SIR-like approaches and their use in waterborne diseases (e.g. cholera). Modelling the spread along river systems via PDEs or via realistic networks. Dealing with real epidemics by mathematical models: the Haitian and South African epidemics.


  • Codeco (2001) Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir BMC Infectious Diseases 1:1
  • Righetto et al. (2010) Modelling human movement in cholera spreading along fluvial systems Ecohydrology 4:49
  • Bertuzzo et al (2009) On spatially explicit models of cholera epidemics Interface 43:321
  • Mari et al. (2011) Modelling cholera epidemics: the role of waterways, human mobility and sanitation Interface 67:376
  • Gatto et al. (2012) Generalized reproduction numbers and the prediction of patterns in waterborne disease PNAS 109:19703
Wednesday 12th June 2013, 09.00-13.00
Renato Casagrandi & Marino Gatto
Population extinction in spatially explicit landscapes — Site and bond lattice models. Definition of percolation as a spatial stochastic process. Percolation thresholds. Interacting particle systems (IPS) 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.


  • Grimmett GR (1999) Percolation, Springer-Verlag, New York
  • 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.

Application for participation

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


Slides and papers

After having logged in with username and password, it is possible to download many of the references and the slides in PDF format.


Page mantained by Renato Casagrandi (last update 15 May 13)