Numerical simulations show that a simple model for the formation of Liesegang precipitation patterns, which takes into account the dependence of nucleation and particle growth kinetics on supersaturation, can explain not only simple patterns like parallel bands in a test tube or concentric rings in a petri dish, but also more complex structural features, such as dislocations, helices, "Saturn rings," or patterns formed in the case of equal initial concentrations of the source substances. The limits of application of the model are discussed. (c) 1994 American Institute of Physics.

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In certain experimental conditions, bacteria form complex spatial-temporal patterns. A striking example of such kind was reported by Budrene and Berg (1991), who observed a wide variety of different colony structures ranging from arrays of spots to radially oriented stripes or arrangements of more complex elongated spots, formed by Escherichia coli. We discuss the relevant mechanisms of intercellular regulation in bacterial colony which may cause pattern formation, and formulate the corresponding mathematical model. In numerical experiments a variety of patterns, observed in real systems, is reproduced. The dynamics of their formation is investigated.

In this paper we investigate the properties and linear stability of traveling premixed combustion waves and the formation of pulsating combustion waves in a model with two-step chain-branching reaction mechanism. These calculations are undertaken in the adiabatic limit, in one spatial dimension and for the case of arbitrary Lewis numbers for fuel and radicals. It is shown that the Lewis number for fuel has a significant effect on the properties and stability of premixed flames, whereas varying the Lewis number for the radicals has only qualitative (but not qualitative) effect on the combustion waves. We demonstrate that when the Lewis number for fuel is less than unity, the flame speed is unique and is a monotonically decreasing function of the dimensionless activation energy. Moreover, in this case, the combustion wave is stable and exhibits extinction for finite values of activation energy as the flame speed decreases to zero. However, for the fuel Lewis number greater than unity, the flame speed is a C-shaped and double valued function. The linear stability of the traveling wave solution was determined using the Evans function method. The slow solution branch is shown to be unstable whereas the fast solution branch is stable or exhibits the onset of pulsating instabilities via a Hopf bifurcation. The critical parameter values for the Hopf bifurcation and extinction are found and the detailed map for the onset of pulsating instabilities is determined. We show that a Bogdanov—Takens bifurcation is responsible for both the change in the behavior of the traveling wave solution near the point of extinction from unique to double valued type as well as for the onset of pulsating instabilities. We investigate the properties of the Hopf bifurcation and the emerging pulsating combustion wave solutions. It is demonstrated that the Hopf bifurcation observed in our present study is of supercritical type. We show that the pulsating combustion wave propagates with the average speed smaller than the speed of the traveling combustion wave and at certain parameter values the pulsating wave exhibits a period doubling bifurcation.

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