Comptes Rendus
Partial differential equations/Dynamical systems
Turing patterns induced by cross-diffusion in a 2D domain with strong Allee effect
Comptes Rendus. Mathématique, Volume 357 (2019) no. 11-12, pp. 863-877.

In this work, we introduce a two-dimensional domain predator-prey model with strong Allee effect and investigate the Turing instability and the phenomena of the emergence of patterns. The occurrence of the Turing instability is ensured by the conditions that are procured by using the stability analysis of local equilibrium points. The amplitude equations (for supercritical case cubic Stuart–Landau equation and for subcritical quintic Stuart–Landau equation) are derived appropriate for each case by using the method of multiple time scale and show that the system supports patterns like squares, stripes, mixed-mode patterns, spots and hexagonal patterns. We obtain the asymptotic solutions to the model close to the onset instability based on the amplitude equations. Finally, numerically simulations tell how cross-diffusion plays an important role in the emergence of patterns.

Dans cette Note, nous introduisons un modèle prédateur–proie dans un domaine de dimension deux, avec effet Allee fort. Nous étudions l'instabilité de Turing et le phénomène d'émergence de motifs. L'apparition de l'instabilité de Turing est garantie par les conditions fournies par l'analyse de la stabilité des points d'équilibre locaux. Les équations d'amplitude (une équation de Stuart–Landau cubique dans le cas supercritique et quintique dans le cas sous-critique) sont établies en utilisant la méthode des échelles de temps multiples. On montre que le système admet des motifs comme des carrés, des bandes, des motifs en mode mixte, des taches et des motifs hexagonaux. Nous obtenons les solutions asymptotiques du modèle près de l'instabilité, à partir des équations d'amplitude. Finalement, les simulations numériques montrent comment la diffusion croisée joue un rôle important dans l'apparition de motifs.

Published online:
DOI: 10.1016/j.crma.2019.10.011

Naveed Iqbal 1; Ranchao Wu 2

1 Mathematics Department, Faculty of Science, University of Ha'il, Ha'il 81451, Saudi Arabia
2 School of Mathematical Sciences, Anhui University, Hefei, Anhui, 230601, PR China
     author = {Naveed Iqbal and Ranchao Wu},
     title = {Turing patterns induced by cross-diffusion in a {2D} domain with strong {Allee} effect},
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Naveed Iqbal; Ranchao Wu. Turing patterns induced by cross-diffusion in a 2D domain with strong Allee effect. Comptes Rendus. Mathématique, Volume 357 (2019) no. 11-12, pp. 863-877. doi : 10.1016/j.crma.2019.10.011.

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