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Reaction-diffusion System (reaction-diffusion + system)
Selected AbstractsThe reaction-diffusion system: a mechanism for autonomous pattern formation in the animal skinGENES TO CELLS, Issue 6 2002Shigeru Kondo How do animals acquire their various skin patterns? Although this question may seem easy, in fact it is very difficult to answer. The problem is that most animals have no related structures under the skin; therefore, the skin cells must form the patterns without the support of a prepattern. Recent progress in developmental biology has identified various molecular mechanisms that function in setting the positional information needed for the correct formation of body structure. None of these can explain how skin pattern is formed, however, because all such molecular mechanisms depend on the existing structure of the embryo. Although little is known about the underlying molecular mechanism, many theoretical studies suggest that the skin patterns of animals form through a reaction-diffusion system,a putative ,wave' of chemical reactions that can generate periodic patterns in the field. This idea had remained unaccepted for a long time, but recent findings on the skin patterns of fish have proved that such waves do exist in the animal body. In this review, we explain briefly the principles of the reaction-diffusion mechanism and summarize the recent progress made in this area. [source] Roles of nodal-lefty regulatory loops in embryonic patterning of vertebratesGENES TO CELLS, Issue 11 2001Hou Juan Nodal is a signalling molecule that belongs to the transforming growth factor,, superfamily of proteins, and Lefty proteins are antagonists of Nodal signalling. The nodal and lefty genes form positive and negative regulatory loops that resemble the reaction-diffusion system. As a pair, these genes control various events of vertebrate embryonic patterning, including left-right specification and mesoderm formation. In this review, we will focus on recent studies that have addressed the roles of nodal and lefty in mouse development. [source] EFFECTS OF DOMAIN SIZE ON THE PERSISTENCE OF POPULATIONS IN A DIFFUSIVE FOOD-CHAIN MODEL WITH BEDDINGTON-DeANGELIS FUNCTIONAL RESPONSENATURAL RESOURCE MODELING, Issue 3 2001ROBERT STEPHEN CANTRELL ABSTRACT. A food chain consisting of species at three trophic levels is modeled using Beddington-DeAngelis functional responses as the links between trophic levels. The dispersal of the species is modeled by diffusion, so the resulting model is a three component reaction-diffusion system. The behavior of the system is described in terms of predictions of extinction or persistence of the species. Persistence is characterized via permanence, i.e., uniform persistence plus dissi-pativity. The way that the predictions of extinction or persistence depend on domain size is studied by examining how they vary as the size (but not the shape) of the underlying spatial domain is changed. [source] The attractor for a nonlinear reaction-diffusion system in an unbounded domainCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 6 2001Messoud A. Efendiev In this paper the quasi-linear second-order parabolic systems of reaction-diffusion type in an unbounded domain are considered. Our aim is to study the long-time behavior of parabolic systems for which the nonlinearity depends explicitly on the gradient of the unknown functions. To this end we give a systematic study of given parabolic systems and their attractors in weighted Sobolev spaces. Dependence of the Hausdorff dimension of attractors on the weight of the Sobolev spaces is considered. © 2001 John Wiley & Sons, Inc. [source] The probability approach to numerical solution of nonlinear parabolic equationsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2002G. N. Milstein Abstract A number of new layer methods for solving semilinear parabolic equations and reaction-diffusion systems is derived by using probabilistic representations of their solutions. These methods exploit the ideas of weak sense numerical integration of stochastic differential equations. In spite of the probabilistic nature these methods are nevertheless deterministic. A convergence theorem is proved. Some numerical tests are presented. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 490,522, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10020 [source] | |||||||||||||