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7 Reaction-Diffusion Equations – Homogeneous and Heterogeneous Environments
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Fig. 7.5. Fields in Chiapas, Mexico

A simple calculation shows that the extinction state is the intersection of a
1-dimensional stable manifold corresponding to the predator species and an
unstable one corresponding to the prey species. The other stationary state is
stable (in the linearized sense when diffusion is neglected and when ac−df > 0).
Averyinterestingphenomenonhappenswhendiffusionistakenintoaccount
in the stability analysis. Although, intuitively speaking, diffusion stabilizes par-
ticle flow, there are cases of reaction-diffusion vector systems (d> 1!) with
stationary points, which are exponentially stable in the diffusionless case AND
feature unstable modes if appropriate diffusion is taken into account. This so
called Turing instability (see [11]) is generated by sufficiently different diffu-
sivities for the different components of the vector u.Anexampleisgiven in
Chapter 4, in the context of biological pattern formation.
For analytical results on diffusive predator-prey models in heterogeneous
environments we refer to [3].
While modeling and analysis of reaction-diffusion systems in spatially un-
structuredenvironments(i.e.thenonlinearityofF andthediffusionmatrixDare