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Relationship for dynamical properties in the vicinity of fixed points betweentwo-dimensional continuous and its positivity-preserving discretized dynamicalsystems is studied. Based on linear stability analysis, we reveal theconditions under which the dynamical structures of the original continuousdynamical systems are retained in their discretized dynamical systems, and thetypes of fixed points are identified if they change due to discretization. Wealso discuss stability of the fixed points in the discrete dynamical systems.The obtained general results are applied to Sel'kov model and Lengyel-Epsteinmodel.

Types and stability of fixed points for positivity-preserving discretized dynamical systems in two dimensions