Xie, Zhifu Turing instability in a coupled predator-prey model with different Holling type functional responses. (English) Zbl 1214.92072 Discrete Contin. Dyn. Syst., Ser. S 4, No. 6, 1621-1628 (2011). Summary: In a reaction-diffusion system, diffusion can induce the instability of a positive equilibrium which is stable with respect to a constant perturbation; therefore the diffusion may create new patterns when the corresponding system without diffusion fails, as shown by Turing in 1950s. In this paper we study a coupled predator-prey model with different Holling type functional responses, where cross-diffusions are included in such a way that the prey runs away from predator and the predator chase preys. We conduct the Turing instability analysis for each Holling functional response. We prove that if a positive equilibrium solution is linearly stable with respect to the ODE system of the predator-prey model, then it is also linearly stable with respect to the model. So diffusion and cross-diffusion in the predator-prey model with Holling type functional responses given in this paper can not drive Turing instability. However, diffusion and cross-diffusion can still create non-constant positive solutions for the model. Cited in 14 Documents MSC: 92D40 Ecology 92C15 Developmental biology, pattern formation 35K57 Reaction-diffusion equations 37N25 Dynamical systems in biology 35Q92 PDEs in connection with biology, chemistry and other natural sciences Keywords:reaction-diffusion system; cross-diffusion; stationary pattern; Holling type functional response PDFBibTeX XMLCite \textit{Z. Xie}, Discrete Contin. Dyn. Syst., Ser. S 4, No. 6, 1621--1628 (2011; Zbl 1214.92072) Full Text: DOI