The Fisher equation invasion biology and reaction diffusion equations

As with life history invariants, the creation of dimensionless combinations of variables is very useful. A good place to start getting more details about these methods is Lin and Segel (1988 (1974)). One could study the Fisher equation for different kinds of genetic models, such as heterozygote superiority, in which case we replace sw(1 — w) in the Fisher equation by a function f (w) with the properties that f (0) = f (a) = f (1) = 0 and with the requirement that w = a be a stable steady state, or heterozygote inferiority, in which w = a is an unstable steady state. The literature on reaction-diffusion equations is enormous. And once one begins with systems that involve two variables and two spatial dimensions, the variety of interesting patterns and solutions is nearly endless. The books by Grindrod (1996), Kot (2001), and Murray (2002) are a good place to start learning about these; the paper by Levin and Segel (1985) is a classic. An interesting alternative approach for logistic growth in space and time is offered in the paper by Law et a/. (2003); Medvinsky et a/. (2002) and McLeod et a/. (2002) use such models to understand plankton blooms; by Klein et a/. (2003) to understand patterns of pollen dispersal, and by Kot et a/. (1996) to understand invasions.

Chapter 3

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