Bistability Dynamics in Structured Ecological Models

Alternative stable states exist in many important ecosystems, and gradual change of the environment can lead to dramatic regime shift in these systems (Beisner et.al. (2003), May (1977), Klausmeier (1999), Rietkerk et.al. (2004), and Scheffer et.al. (2001)). Examples have been observed in the desertification of Sahara region, shift in Caribbean coral reefs, and the shallow lake eutrophication (Carpenter et.al. (1999), Scheffer et.al. (2003), and Scheffer et.al. (2001)). It is well-known that a social-economical system is sustainable if the life-support ecosystem is resilient (Holling (1973) and Folke et.al. (2004)). Here resilience is a measure of the magnitude of disturbances that can be absorbed before a system centered at one locally stable equilibrium flips to another. Mathematical models have been established to explain the phenomena of bistability and hysteresis, which provide qualitative and quantitative information for ecosystem managements and policy making (Carpenter et.al. (1999) and Peters et.al. (2004)). However most of these models of catastrophic shifts are non-spatial ones. A theory for spatially extensive, heterogeneous ecosystems is needed for sustainable management and recovery strategies, which requires a good understanding of the relation between system feedback and spatial scales (Folke et.al. (2004), Walker et.al. (2004) and Rietkerk et.al. (2004)). In this chapter, we survey some recent results on structured evolutionary dynamics including reaction-diffusion equations and systems, and discuss their applications to structured ecological models which display bistability and hysteresis. In Section 1, we review several classical non-spatial models with