Effects of anisotropic diffusion on Turing patterns in heterogeneous environment

Abstract

Diffusion plays a crucial role in the forming and evolving of Turing patterns. Generally, the diffusion processes in complex systems do not comply to the complete random walk theory, which means that the diffusion is abnormal rather than normal, such as super-diffusion, sub-diffusion and anisotropic diffusion. However, most of previous studies focused on the pattern formation mechanism under the normal diffusion. In this paper, a two-component reaction-diffusion model with anisotropic diffusion is used to study the effect of anisotropic diffusion on Turing patterns in heterogeneous environments. Three different types of anisotropic diffusions are utilized. It is shown that the system gives rise to stripe patterns when the degree of anisotropic diffusion is high. The directions of stripes are determined by the degree of the diffusion coefficient deviating from the bifurcation point. In a low degree of anisotropic diffusion, the pattern type is the same as the counterpart in a low degree of the isotropic diffusion. When the diffusion coefficient grows linearly in the space, different types of patterns compete with each other and survive in different regions under the influence of spatial heterogeneity. When the diffusion coefficient is modulated by a one-dimensional periodic function, both type and wavelength of the pattern are determined by the modulated wavelength and the intrinsic wavelength. The system can exhibit alternating two-scale mixed patterns of different types when the modulated wavelength is larger than the intrinsic wavelength. Note that each of the diffusion coefficients of some special anisotropic media is a tensor, which can be expressed as a matrix in two-dimensional cases. We also study the influence of off-diagonal diffusion coefficient <i>D</i> on Turing pattern. It is found that the Turing pattern induced by off-diagonal diffusion coefficient always selects the oblique stripe pattern. The off-diagonal diffusion coefficient <i>D</i> not only affects the pattern selection mechanism, but also expands the parameter range of Turing space. The critical diffusion coefficient <inline-formula><tex-math id="M3">\begin{document}$ {D_{\text{c}}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M3.png"/></alternatives></inline-formula> increases linearly with the diagonal diffusion coefficient <inline-formula><tex-math id="M4">\begin{document}$ {D_u} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M4.png"/></alternatives></inline-formula> increasing. The intrinsic wavelength of the oblique stripe pattern decreases as the off-diagonal diffusion coefficient <i>D</i> increases. It is interesting to note that the critical wavelength corresponding to the critical diffusion coefficient <inline-formula><tex-math id="M6">\begin{document}$ {D_{\text{c}}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M6.png"/></alternatives></inline-formula> is independent of the diagonal diffusion coefficient <inline-formula><tex-math id="M7">\begin{document}$ {D_u} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M7.png"/></alternatives></inline-formula>. These results not only provide a new insight into the formation mechanism of Turing patterns, but also increase the range and complexity of possible patterns.