Mathematical modelling of nonlinear dynamics in activator-inhibitor systems with superdiffusion

Received: November 30, 2016
Accepted: November 30, 2016
Z. Prytula

Pidstryhach Institute for Applied Problems of Mechanics and Mathematics 
of the National Academy of Sciences of Ukraine

The nonlinear dynamics in generalized activator-inhibitor systems with space fractional derivatives is studied. As an example, the Brusselator model and the reaction–diffusion model with cubic nonlinearity, in which the classical spatial differential operators are replaced by their fractional analogues, are considered. The fractional operator reflects the nonlocal behavior of superdiffusion. The spatially homogeneous, time independent solution has been found for each system. We have also studied its linear stability and determined instability conditions of both Hopf and Turing. It was established that the anomalous diffusion (superdiffusion) leads to the qualitative change of nonlinear dynamics in mentioned systems