In this lecture series I intend to illustrate the role of diffusion and cross-diffusion in nonlinear systems from the point of view of pattern formation. Using examples including Turing's "diffusion-driven instability" and the classical Lotka-Volterra competition model, I shall discuss the emergence of spatially inhomogeneous patterns as the diffusion rates and/or the cross-diffusion rates vary. Qualitative properties of these patterns will be included, and, the stability and dynamics will be emphasized.

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