We address the problem of pattern formation on the surface of a sphere using Turing equations. By considering a generic reaction-diffusion model, we numerically investigate the patterns formed under different conditions on the parameter values. Our results show that a closed surface with curvature, as a sphere, imposes geometrical restrictions on the shape of the pattern. This is important in some biological systems where curvature plays an important role in guiding chemical, biochemical, and embryological processes.

In this paper we address the problem of pattern formation in confined Turing systems in two dimensions, when one assumes the enhancement of the concentration of one of the chemicals at some of the confining surfaces. This model is suitable to study biological systems, such as the skin patterns shown by some marine fish. We also study numerically the dynamical growth of the system by changing the size of the confined region while dynamical diffusion and reaction phenomena take place. This idea is tested in two different models. This allows one to estimate the robustness of stripe formation. ͓S1063-651X͑97͒06807-4͔ PACS number͑s͒: 47.54.ϩr, 82.40.Bj, 82.20.Mj In 1952, Turing established the basis to explain biological patterns using two interacting chemicals ͓1͔. The experimental observation of a ''Turing pattern'' occurred in a chemical system nearly 40 years after their prediction by Turing ͓2,3͔, but it was not until very recently that the example of a Turing pattern in a biological system was confirmed in skin patterns of the angelfish ͑Pomacanthus͒ by Kondo and Asai ͓4͔. In this work the authors propose and solve a system of two reaction-diffusion equations in a growing onedimensional domain to explain the insertion of new stripes between the older ones during the growth of Pomacanthus semicirculatus and the rearrangement of the stripe pattern of Pomacanthus imperator.Kondo and Asai's interpretation was subject to criticism by Höfer and Maini ͓5͔, who did not find enough evidence to say that reaction-diffusion systems could provide a mechanistic basis for the strip-doubling phenomenon. In particular, they claim that a two-dimensional simulation would be a more realistic representation of the fish skin than a onedimensional domain. Höfer and Maini argue that a mechanism that sets the distance between adjacent stripes and some kind of ''memory'' that conserves the location of old stripes is needed in order to explain the patterning dynamics of the Pomacanthus. Accordingly, in this work we shall show that a reaction-diffusion system is capable of describing the main features of the phenomena observed in the Pomacanthus skin. For that goal, we consider two sets of Turing equations known to form different kinds of patterns; these are solved in a two-dimensional spatial domain that simulates the fish shape, with zero flux boundary conditions. The key feature of our simulation is the enforcement of an enhanced source of the activator along some of the boundaries of the domain. This idea has close parallels with the mechanism of stripe formation in the Drosophila embryo where the pattern of the anteroposterior ͑head-tail͒ segmentation is caused by a high concentration of the Bicoid protein along the anterior ͑head͒ side ͓6͔.To see how the boundary conditions and domain shape affect the stationary patterns from an initially homogeneous state, we study a simplified version of a model for glycolisis as the specific reaction mechanism, which has been extensively studied numerically by Dillon et al. ͓7͔ in one dimension...

In this paper we study the solutions of a generalized reaction-diffusion system with a bistable reaction term, and considering directional anomalous diffusion. We use the well-known properties of fractional derivatives to model asymmetric anomalous diffusion, and obtain traveling wave solutions that propagate in a direction that depends on the metastability of the front, the fractional exponent and the asymmetry of the diffusion.

We derive the interfacial-curvature free energy for a simple fluid from density-functional theory, and find a form matching that for a two-dimensional shearless elastic media. The fourth moment of the direct correlation function and the density gradient determine the bending moduli~and~, while the spontaneous curvature co is given by asymmetry in the density. We obtain the critical indices of these quantities and corrections to the Laplace equation for curved interfaces. PACS number(s): 64.70.p, 05.70.Fh, 82.70.y The statistical-mechanical description of twodimensional shapes is a rich field, with novel and previously unforeseen features, currently undergoing rapid development [1]. Attention to this subject has been spurred, in part, by studies where the objects of interest are interfaces with small or even vanishing surface tension, such as those occurring in microemulsions, lyotropic liquid crystals, and bilayer vesicle solutions [2]. Generally, fluctuationsof an interface between coexisting phases, say a liquid and its vapor, are governed by the surface tension y, but in the case of the above-mentioned systems the interfacial Auctuations are determined solely by curvature effects [1 -3] The determination of these elf'ects has been accomplished phenornenologically via the consideration that the interface forms an incompressible, two-dimensional Quid, with shape fluctuations regulated by its elastic-curvature free energy. This free energy, known also by the name of Helfrich [4], can be written, per unit area, as fH=~(Jco) +iTK, where J=c, +c2 is the mean curvature and E =c, c2 is the Gaussian curvature of the interface and c& and c2 are the principal curvatures. The bending modulus~and the saddle-splay constant F measure, respectively, changes in fH due to deviations from the spontaneous curvature co, and due to the Gaussian curvature. Minimization of fH determines the stability of the various possible interfacial shapes or structures.In view of these developments it is of interest to consider the microscopic origins of the moduli a and~, and of the spontaneous curvature co, and to provide the necessary statistical-mechanical foundation to the Helfrich-free-energy terms, just as it has been done some time ago for the case of the surface tension [5 -7] Here we restrict ourselves to the simplest case of the liquidvapor interface of a simple one-component Auid, and consider the free energy for the density inhomogeneity p(r), i.e. , the free-energy density functional F[p(r)], descriptive of a macroscopic interface and obtain, besides the customary surface tension term, the form corresponding to fH and find the expressions for a, i7, and co in terms of the molecular distribution functions. This exercise can be done in two different ways: (i) Through the examina-fo(p(r))=kT p(r)[ in[A. p(r)] -1I +-, 'p(r) f dr'c(r', p(r)) (3a) A (p(r) ) = , ' kT f d r'r e( r-', p(r) ), (3b) (3c)In the above expressions A. is the Broglie thermal length, c(r';p(r)) is the direct correlation function for a uniform tion of the general for...

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