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A skeleton structure of self-replicating dynamics. (English) Zbl 0936.35090

The aim of this paper is to present a key mechanism for the self-replicating dynamics for a class of model systems including the Gray-Scott model. As a model which shows self-replicating patterns of propagating type it is considered the system: \[ \partial u/\partial t=D_u\nabla^2u+ u(u-v^2-\alpha),\;\partial v/\partial t=D_v\nabla^2 v+mu-v, \] where \(u\) and \(v\) are concentrations of the chemical materials \(U\) and \(V\), respectively; \(D_u,D_v\) are diffusion coefficients, \(\alpha\) and \(m\) are non-negative parameters. This system is considered on a finite interval subject to Neumann (zero flux) boundary condition. The length of interval and \(\alpha\) are control parameters.
The paper contains results concerning the behaviour of solutions. There are three different phases of dynamics of self-replicating patterns: 1. Steady or traveling (an isolated pulse-like pattern stays or travels almost without changing its shape). 2. Splitting phase (some of the pulses split into two parts). 3. Convergence to a final state.

MSC:

35K57 Reaction-diffusion equations
35B32 Bifurcations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations

Software:

HomCont; AUTO
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References:

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