Deterministic mechanisms of pattern formation - Noise-induced Phenomena in the Environmental Sciences - page 274

Appendix B
Deterministic mechanisms of pattern formation
B.1 Introduction
In this appendix we provide a mathematical description of the three major deter-
ministic models of self-organized pattern formation that are commonly invoked to
explain mechanisms of spatial self-organization in the biogeosciences. In these mod-
els patterns emerge from a mechanism of symmetry-breaking instability , whereby the
uniform state of the system becomes unstable, thereby leading to the emergence of
spatial patterns. Spatial interactions induce this instability, whereas the resulting pat-
terns are stabilized by suitable nonlinear terms. In Turing and kernel-based models,
symmetry breaking is the result of the interactions between short-range activation and
long-range inhibition, i.e., of positive and negative feedbacks acting at different spatial
scales. In the third class of models (i.e., differential-flow models) symmetry breaking
emerges as a result of the differential-flow rate between two (or more) species.
In Turing and differential-flow models, the nonlinearities are local (i.e., they do not
appear in the terms expressing spatial interactions), whereas in kernel-based models
the nonlinearities can be in general nonlocal, i.e., they can appear as multiplicative
functions of the term accounting for spatial interactions (e.g., Lefever and Lejeune ,
1997 ). In a particular class of kernel-based models - known as neural models (e.g.,
Murray and Maini , 1989 ) - the nonlinearities are only local and do not affect the
spatial interactions. In these models the nonlinear terms appear as additive functions
of the spatial interaction term. Here we describe the Turing and kernel-based mod-
els separately because they use different mathematical representations of the spatial
dynamics. However, Borgogno et al. ( 2009 ) showed that Turing and neural models
are based on mathematical frameworks that are closely related. Both models invoke
similar mechanisms of morphogenesis, namely symmetry-breaking instability in-
duced by spatial interactions in activation-inhibition systems and stabilization by local
nonlinearities.
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