Turing Instability with Time-Varying Diffusivities

1 Turing Instability with Time-Varying Diffusivities 

Ecological systems can display temporal oscillations in parameter values due to seasonal variations. The effects of time-varying diffusivities on the Turing instability were first considered by Timm and Okubo [436] in a predator-prey model describing the interaction between zooplankton and phytoplankton. Temporal variations in the horizontal diffusion coefficients arise from the interaction of vertical current shear with vertical mixing processes. 

We impose the first inequality in (11.3), since a Turing instability can occur in a two-variable reaction-diffusion system with constant parameters only if 1. The second inequality ensures the positivity of D (t). We assume that the system (11.1) possesses a uniform steady state, (PuU)-PvU)) = (Pu.Pv). with Ej(Pu,Pv) = E2G0U, Pv) = which fulfills the stability conditions (10.23), and U is an activator and V an inhibitor. 

To assess the stability of the uniform steady state, we carry out a linear stability analysis. We set 

If the corresponding system with constant parameters, i.e e = 0, is at the Turing 

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