These results imply that the nonlinearity still plays the main role in the formation of spatial patterns. The current results for such a kind of systems mainly discovered how the temporal and spatial fractional derivatives change transient dynamical behaviors and affect structure of spatial patterns. Thereinto, spatial patterns for some fractional-derivative reaction-diffusion systems, whose first-derivative counterparts can produce some spatial patterns, were deeply discussed in 15, 16, 17, 18, 19, 20. In fact, more and more fractional-derivative differential equations have been successfully used in biological materials 12, fluid mechanics 13, quantum mechanics 14, and so on. However, many realistic processes are well described by nonlinear reaction-diffusion equations with the temporal fractional derivatives because this class of derivative can deal comfortably with memory effect in dynamical systems 10, 11. In spite of extensive applications, Turing bifurcation theory is limited in nonlinear reaction-diffusion systems with the temporal first derivative. This pioneer work of Turing not only came into being theoretical foundation for understanding diverse patterns occurring in the natural world, but also opened a new research direction-pattern dynamics that have received extensive attention and are currently still a hot topic in many scientific fields, such as molecular biology 2, 3, 4, biochemistry 5, development biology 6, epidemiology 7, 8, mechanics 9, and so on. By model analysis, he showed that if the underlying system undergoes Turing bifurcation, then a so-called Turing pattern (i.e., a spontaneously-organized spatial heterogeneous pattern away from the stable equilibriums of the system) can occur. ![]() In order to interpret the formation of the patterns observed in his experiments 1, Alan Turing proposed a reaction-diffusion model (currently popularly called as the Turing model). As a kind of organized heterogeneous macroscopic structure, spatial patterns exist extensively in the natural world ranging from chemical reaction systems to physical systems and to ecological systems.
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