7: Partial Differential Equations

MCED Chapter 07

Title

Partial Differential Equations

Authors

Michael Sieber and Horst Malchow

Abstract (from the book)

Spatially homogeneous processes of change are the subject of the preceding chapter. Partial differential equations are one method to model the interplay of these processes with spatial phenomena such as movement of individuals and/or a heterogeneous environment. Random motion of organisms might be described as diffusion, and directed motion as advection. The latter can be composed of locomotion and motion of the surrounding medium. The focus of this chapter is on classical systems of no more than two interacting and diffusing populations. The potential of such systems to exhibit spatiotemporal pattern formation is studied.

Further Reading

  • Allen LJS (2003) An introduction to stochastic processes with applications to

biology. Pearson Education, Upper Saddle River NJ

  • Holmes EE, Lewis MA, Banks JE, Veit RR (1994) Partial differential equations in ecology: Spatial interactions and population dynamics. Ecology

, 75:17–29

  • Shigesada N, Kawasaki K (1997) Biological invasions: Theory and practice.

Oxford University Press, Oxford

  • Malchow H, Petrovskii SV, Venturino E (2008) Spatiotemporal patterns in

ecology and epidemiology: Theory, models, simulations. CRC Mathematical

and Computational Biology Series, CRC Press, Boca Raton

  • Murray JD (2003) Mathematical biology. II. Spatial models and biomedical

applications, Interdisciplinary Applied Mathematics, vol 18. Springer, Berlin

  • Okubo A (1980) Diffusion and ecological problems: Mathematical models,

Biomathematics Texts, vol 10. Springer, Berlin

  • Okubo A, Levin S (2001) Diffusion and ecological problems: Modern perspectives, Interdisciplinary Applied Mathematics, vol 14. Springer, New York

Links

  • AnT 4.669 (www.ant4669.de)
    AnT is a simulation and
    Analysis Tool for dynamical systems. It allows the analysis of a wide range of dynamical systems, such as time-discrete maps, ordinary, time-delay and functional differential equations, stochastic systems, hybrid systems and external input data, which is interpreted as a time series. The possible investigation methods range from general trajectory evaluations, basic statistics, period and region analysis to spectral analysis, calculation of Lyapunov exponents and Poincaré sections. Especially useful is the scan-mode, which allows the exploration of a high-dimensional parameter space and the easy generation of detailed bifurcation diagrams.