Tian, Canrong; Ling, Zhi; Lin, Zhigui Turing pattern formation in a predator-prey-mutualist system. (English) Zbl 1231.35275 Nonlinear Anal., Real World Appl. 12, No. 6, 3224-3237 (2011). Summary: We develop a theoretical framework about spatial patterns in a three-species predator-prey-mutualist system with cross-diffusion. We concentrate on three aspects of Turing pattern formation: (1) What conditions enable the occurrence of Turing patterns? (2) What are the underlying mechanisms? (3) What are the corresponding configurations? For the first two questions, by use of the stability analysis for the positive uniform solution and the Leray-Schauder degree theory, we prove that under some conditions, the system admits at least a nonhomogeneous stationary solution. For the third question, we carry out numerical simulations for a Turing pattern, and we show that the configurations of Turing pattern are stable spotted patterns, which resemble a real ecosystem. Cited in 27 Documents MSC: 35Q92 PDEs in connection with biology, chemistry and other natural sciences 92C15 Developmental biology, pattern formation 92D25 Population dynamics (general) 35K51 Initial-boundary value problems for second-order parabolic systems Keywords:Turing pattern; cross-diffusion; nonhomogeneous stationary state PDFBibTeX XMLCite \textit{C. Tian} et al., Nonlinear Anal., Real World Appl. 12, No. 6, 3224--3237 (2011; Zbl 1231.35275) Full Text: DOI References: [1] Turing, A., The chemical basis of morphogenesis, Philos. Trans. Royal. Soc. B, 237, 37-72 (1952) · Zbl 1403.92034 [2] Maini, P. K.; Painter, Kevin J.; Phong Chau, Helene Nguyen, Spatial pattern formation in chemical and biological system, J. Chem. Soc., Faraday Trans., 93, 3601-3610 (1997) [3] Maini, P. K.; Benson, D. L.; Sherratt, J. A., Pattern formation in reaction diffusion models with spatially inhomogeneous diffusion coefficients, IMA J. Math. Appl. Med. 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