Reaction–diffusion system

Abstract: Dynamical reaction–diffusion processes and metapopulation models are standard modelling approaches for a wide array of phenomena in which local quantities—such as density, potentials and particles—diffuse and interact according to the physical laws. Here, we study the behaviour of the basic reaction–diffusion process (given by the reaction steps B→A and B+A→2B) defined on networks with heterogeneous topology and no limit on the nodes’ occupation number. We investigate the effect of network topology on the basic properties of the system’s phase diagram and find that the network heterogeneity sustains the reaction activity even in the limit of a vanishing density of particles, eventually suppressing the critical point in density-driven phase transitions, whereas phase transition and critical points independent of the particle density are not altered by topological fluctuations. This work lays out a theoretical and computational microscopic framework for the study of a wide range of realistic metapopulation and agent-based models that include the complex features of real-world networks.

Abstract: IN 1952, Turing proposed a hypothetical molecular mechanism, called the reaction–diffusion system1, which can develop periodic patterns from an initially homogeneous state. Many theoretical models based on reaction–diffusion have been proposed to account for patterning phenomena in morphogenesis2–4, but, as yet, there is no conclusive experimental evidence for the existence of such a system in the field of biology5–8. The marine angelfish, Pomacanthus has stripe patterns which are not fixed in their skin. Unlike mammal skin patterns, which simply enlarge proportionally during their body growth, the stripes of Pomacanthus maintain the spaces between the lines by the continuous rearrangement of the patterns. Although the pattern alteration varies depending on the conformation of the stripes, a simulation program based on a Turing system can correctly predict future patterns. The striking similarity between the actual and simulated pattern rearrangement strongly suggests that a reaction–diffusion wave is a viable mechanism for the stripe pattern of Pomacanthus.

Abstract: It is pointed out that chemical reactions which show an absorbing stationary state in the master-equation approach (e.g. Schlogl's first reaction) exhibit nevertheless a second order phase transition in non-zero dimensional macroscopic systems. The relation to Reggeon field theory is given more directly than by Grassberger et al. using the functional integral formalism of statistical dynamics. As a new result the correlation length exponent ν and the order parameter exponent β are found toO(e2) in an ɛ-expansion around the upper critical dimensiondc=4.

Abstract: (1979). LP Bounds of solutions of reaction-diffusion equations. Communications in Partial Differential Equations: Vol. 4, No. 8, pp. 827-868.