Math 747: Topics in mathematical biology
Winter/spring 2012; Ben Bolker, bolker@mcmaster.ca

This is a first cut at a list of topics of interest in stochastic modeling of biological dynamics. The emphases are on (1) a broad (but hopefully not too shallow) survey of the ways in which stochasticity can affect the dynamics of biological systems, and (2) developing a toolbox of methods that are useful for tackling stochastic models, with emphasis on practical approximate and/or computational methods rather than rigorous proof.

Bestiary of models; numerical algorithms for stochastic simulation
Basics: continuous vs. discrete time, space, state (e.g. http://www.slideshare.net/bbolker/mbi-intro-to-spatial-models); discrete and continuous Markov chains, random walks, etc. Numerical algorithms for simulation: PRNGs, choice of random deviates (Press et al., 1994), Gillespie algorithm and generalizations (Gillespie, 2007), rejection samplers for spatial models, etc.
Static sampling: averaging effects
Effects of stochasticity without dynamical feedbacks. Jensen’s inequality (Ruel and Ayres, 1999), portfolio effects (Doak et al., 1998), bet-hedging.
State-based approaches
Master equations, forward/backward Kolmogorov equations, van Kampen expansion, etc. (Alonso et al., 2007)
Persistence and extinction
Effects of zeros in population models, stochastic extinction (especially spatial/epidemiological): (Bobashev et al., 2007; Boerlijst et al., 1993; Boerlijst and van Ballegooijen, 2010; King et al., 2009)
Correlation and diffusion equations for non-spatial systems
Master equation; Kolmogorov forward/backward approaches; diffusion approximations; moment equations (Lloyd, 2004) …
Spatial dynamics: correlation approaches
Neighborhood dynamics and the effects of demographic stochasticity: spatial correlation (pair/moment) equations (Durrett and Levin, 1994; Levin, 1994; Bolker and Pacala, 1999)
Spatial dynamics: diffusion equations
Spatial dynamics: individual movement
Patlak’s equation, diffusion/telegraph equations (Moorcroft & Lewis)
Other/miscellaneous/uncategorized

References

   Alonso, D., A. J. McKane, and M. Pascual (2007, June). Stochastic amplification in epidemics. Journal of The Royal Society Interface 4(14), 575 –582.

   Bobashev, G. V., M. D. Goedecke, F. Yu, and J. M. Epstein (2007, December). A hybrid epidemic model: Combining the advantages of Agent-Based and Equation-Based approaches. In Simulation Conference, 2007 Winter, pp. 1532–1537. IEEE.

   Boerlijst, M. C., M. E. Lamers, and P. Hogeweg (1993, July). Evolutionary consequences of spiral waves in a Host–Parasitoid system. Proceedings: Biological Sciences 253(1336), 15–18.

   Boerlijst, M. C. and W. M. van Ballegooijen (2010, December). Spatial pattern switching enables cyclic evolution in spatial epidemics. PLoS Computational Biology 6, e1001030.

   Bolker, B. M. and S. W. Pacala (1999). Spatial moment equations for plant competition: understanding spatial strategies and the advantages of short dispersal. American Naturalist 153, 575–602.

   Brown, S. P., S. J. Cornell, M. Sheppard, A. J. Grant, D. J. Maskell, B. T. Grenfell, and P. Mastroeni (2006, October). Intracellular demography and the dynamics of salmonella enterica infections. PLoS Biol 4(11), e349.

   Denny, M. and S. Gaines (2002, September). Chance in Biology: Using Probability to Explore Nature. Princeton University Press.

   Doak, D. F., D. Bigger, E. K. Harding, M. A. Marvier, R. E. O’Malley, and D. Thomson (1998, March). The statistical inevitability of StabilityDiversity relationships in community ecology. The American Naturalist 151(3), 264–276.

   Durrett, R. and S. Levin (1994). The importance of being discrete (and spatial). Theoretical Population Biology 46(3), 363–394.

   Gaines, S. D. and M. W. Denny (1993). The largest, smallest, highest, lowest, longest, and shortest: extremes in ecology. Ecology 74, 1677–1692.

   Gillespie, D. T. (2007). Stochastic simulation of chemical kinetics. Annual Review of Physical Chemistry 58, 35–55. PMID: 17037977.

   King, A. A., S. Shrestha, E. T. Harvill, and O. N. Bjrnstad (2009, April). Evolution of acute infections and the InvasionPersistence TradeOff. The American Naturalist 173(4), 446–455.

   Levin, R. D. S. A. (1994). Stochastic spatial models: A user’s guide to ecological applications. Philosophical Transactions: Biological Sciences 343, No. 1305, 329–350.

   Lloyd, A. L. (2004, February). Estimating variability in models for recurrent epidemics: assessing the use of moment closure techniques. Theoretical Population Biology 65(1), 49–65.

   Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery (1994). Numerical Recipes in C: the Art of Scientific Computing. Cambridge University Press.

   Rand, D. A. and H. B. Wilson (1991, nov 22). Chaotic stochasticity: a ubiquitous source of unpredictability in epidemics. Proceedings of the Royal Society B 246(1316), 6.

   Ruel, J. J. and M. P. Ayres (1999, September). Jensen’s inequality predicts effects of environmental variation. Trends in Ecology & Evolution 14(9), 361–366.