Abstract
It is well known that temporal first-derivative reaction-diffusion systems can produce various fascinating Turing patterns. However, it has been found that many physical, chemical and biological systems are well described by temporal fractional-derivative reaction-diffusion equations. Naturally arises an issue whether and how spatial patterns form for such a kind of systems. To address this issue clearly, we consider a classical prey-predator diffusive model with the Holling II functional response, where temporal fractional derivatives are introduced according to the memory character of prey’s and predator’s behaviors. In this paper, we show that this fractional-derivative system can form steadily spatial patterns even though its first-derivative counterpart can’t exhibit any steady pattern. This result implies that the temporal fractional derivatives can induce spatial patterns, which enriches the current mechanisms of pattern formation.
Highlights
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
Immediately arises a question whether or not there is a certain causal relationship between the fractional derivative and Turing instability, that is, for a temporal fractional-derivative reaction-diffusion system whose temporal first-derivative counterpart cannot form any spatial pattern, can the temporal fractional derivative induce the Turing instability and produce spatial patterns? To answer this question, we consider a classical diffusive prey-predator system with the Holling II functional response, where the temporal fractional derivative is introduced into this system because of the memory for the prey’s and predator’s behaviors, as follows:
This implies that the steady-state spatial patterns of the prey and predator form for the fractional-derivative system (1), which is respectively shown in Fig. 5C and D
Summary
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. 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) By model analysis, he showed that if the underlying system undergoes Turing bifurcation, a so-called Turing pattern (i.e., a spontaneously-organized spatial heterogeneous pattern away from the stable equilibriums of the system) can occur. 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. These results imply that the nonlinearity still plays the main role in the formation of spatial patterns. A brief discussion and conclusive remark are given
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