Abid, Walid; Yafia, R.; Aziz-Alaoui, M. A.; Aghriche, Ahmed Turing instability and Hopf bifurcation in a modified Leslie-Gower predator-prey model with cross-diffusion. (English) Zbl 1392.35316 Int. J. Bifurcation Chaos Appl. Sci. Eng. 28, No. 7, Article ID 1850089, 17 p. (2018). Summary: This paper is concerned with some mathematical analysis and numerical aspects of a reaction-diffusion system with cross-diffusion. This system models a modified version of Leslie-Gower functional response as well as that of the Holling-type II. Our aim is to investigate theoretically and numerically the asymptotic behavior of the interior equilibrium of the model. The conditions of boundedness, existence of a positively invariant set are proved. Criteria for local stability/instability and global stability are obtained. By using the bifurcation theory, the conditions of Hopf and Turing bifurcation critical lines in a spatial domain are proved. Finally, we carry out some numerical simulations in order to support our theoretical results and to interpret how biological processes affect spatiotemporal pattern formation which show that it is useful to use the predator-prey model to detect the spatial dynamics in the real life. Cited in 11 Documents MSC: 35Q92 PDEs in connection with biology, chemistry and other natural sciences 35B36 Pattern formations in context of PDEs 35B32 Bifurcations in context of PDEs 35K57 Reaction-diffusion equations 35B10 Periodic solutions to PDEs 35B35 Stability in context of PDEs 92D25 Population dynamics (general) Keywords:cross-diffusion; Hopf bifurcation; pattern formation; Turing instability PDFBibTeX XMLCite \textit{W. Abid} et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 28, No. 7, Article ID 1850089, 17 p. (2018; Zbl 1392.35316) Full Text: DOI References: [1] Abid, W.; Yafia, R.; Aziz-Alaoui, M. 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