## 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

### Keywords:

Gray-Scott model; pulse-like pattern

HomCont; AUTO
Full Text:

### References:

 [1] Pearson, J.E., Complex patterns in a simple system, Science, 216, 189-192, (1993) [2] Reynolds, W.N.; Pearson, J.E.; Ponce-Dawson, S., Dynamics of self-replicating patterns in reaction diffusion systems, Phys. rev. lett., 72, 17, 1120-1123, (1994) [3] Rasmussen, K.E.; Mazin, W.; Mosekilde, E.; Dewel, G.; Borckmans, P., Wave-splitting in the bistable GS model, Int. J. bifurcation chaos, 6, 6, 1077-1092, (1996) · Zbl 0881.92038 [4] Petrov, V.; Scott, S.K.; Showalter, K., Excitability, wave reflection and wave splitting in a cubic autocatalysis reaction-diffusion system, Phil. trans. roy. soc. lond. A, 347, 631-642, (1994) · Zbl 0867.35047 [5] De Kepper, P.; Perraud, J.J.; Rudovics, B.; Dulos, E., Experimental study of stationary Turing patterns and their interaction with traveling waves in a chemical system, Int. J. bifurcation chaos, 4, 5, 1215-1231, (1994) · Zbl 0877.92032 [6] Lee, K.J.; McCormick, W.D.; Pearson, J.E.; Swinney, H.L., Experimental observation of self-replicating spots in a reaction-diffusion system, Nature, 369, 215-218, (1994) [7] Lee, K.J.; Swinney, H.L., Lamellar structures and self-replicating spots in a reaction-diffusion system, Phys. rev. E, 51, 1899-1915, (1995) [8] Reynolds, W.N.; Ponce-Dawson, S.; Pearson, J.E., Self-replicating spots in reaction-diffusion systems, Phys. rev. E, 56, 1, 185-198, (1997) [9] Doelman, A.; Kaper, T.J.; Zegeling, P.A., Pattern formation in the one-dimensional GS model, Nonlinearity, 10, 523-563, (1997) · Zbl 0905.35044 [10] A. Doelman, R.A. Gardner, T.J. Kaper, Stability analysis of singular patterns in the 1D GS model: a matched asymptotic approach, preprint. · Zbl 0943.34039 [11] Nishiura, Y.; Ueyama, D., A hidden bifurcational structure for self-replicating dynamics, ACH-models in chemistry, 135, 3, 343-360, (1998) · Zbl 0907.35013 [12] Gray, P.; Scott, S.K., Autocatalytic reactions in the isothermal continuous stirred tank reactor: oscillations and instabilities in the system A+2B→3B, B→C, Chem. eng. sci., 39, 1087-1097, (1984) [13] E.J. Doedel, A.R. Champneys, T.F. Fairgrieve, Y.A. Kuznetsov, B. Sandstede, X. Wang, AUTO97: continuation and bifurcation software for ordinary differential equations (with HomCont), ftp://ftp.cs.concordia.ca/pub/doedel/auto, (1997). [14] D. Ueyama, Dynamics of self-replicating dynamics in the one-dimensional GS model, PhD thesis. · Zbl 0987.34031 [15] Fujii, H.; Mimura, M.; Nishiura, Y., A picture of the global bifurcation diagram in ecological interacting and diffusing systems, Physica D, 5, 1-42, (1982)
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