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Analysis of bifurcations in reaction–diffusion systems with no-flux boundary conditions: the Sel'kov model

  • J. E. Furter (a1) and J. C. Eilbeck (a2)

Abstract

A plot of the bifurcation diagram for a two-component reaction-diffusion equation with no-flux boundary conditions reveals an intricate web of competing stable and unstable states. By studying the one-dimensional Sel'kov model, we show how a mixture of local, global and numerical analysis can make sense of several aspects of this complex picture. The local bifurcation analysis, via the power of singularity theory, gives us a framework to work in. We can then fill in the details with numerical calculations, with the global analytic results fixing the outline of the solution set. Throughout, we discuss to what extent our results can be applied to other models.

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1Armbruster, D. and Dangelmayr, G.. Corank-two bifurcations for the Brusselator with non-flux boundary conditions. Dynamics Stability Systems 1 (1986), 187200.
2Armbruster, D. and Dangelmayr, G.. Coupled stationary bifurcations in non-flux boundary conditions. Math. Proc. Cambridge Philos. Soc. 101 (1987), 167191.
3Ashkenazi, M. and Othmer, H. G.. Special patterns in coupled biochemical oscillators. J. Math. Biol. 5(1978), 305350.
4Aston, P.. Scaling laws and bifurcation. In Singularity Theory and its Applications. Warwick 1989, Part II, vol. 1463, eds Stewart, I. N. and Roberts, R. M., pp. 121 (Berlin: Springer, 1991).
5Brown, K. J. and Eilbeck, J. C.. Bifurcation, stability diagrams, and varying diffusion coefficients in reaction-diffusion equations. Bull. Math. Biol. 44 (1982), 87102.
6Canuto, C., Hussaini, M. Y., Quarteroni, A. and Zang, T. A.. Spectal Methods in Fluid Mechanics (Berlin: Springer, 1987).
7Catalano, G., Eilbeck, J. C., A. Monroy and E. Parisi. A mathematical model for pattern formation in biological systems. Phys. D 3 (1981), 439456.
8Crawford, J., Golubitsky, M., Gomes, M. G. M., Knobloch, E. and Stewart, I. N.. Boundary conditions as symmetry constraints. In Singularity Theory and its Applictions. Warwick 1989, Part II, vol. 1463, eds Stewart, I. N. and Roberts, R. M., pp. 6379 (Berlin: Springer, 1991).
9Duncan, K. and Eilbeck, J. C.. Numerical studies of symmetry-breaking bifurcations in reaction-diffusion systems. In Biomathematics and Related Computational Problems, ed. Ricciardi, L. M., pp. 439448 (Dordrecht: Kluwer, 1988).
10Eilbeck, J. C.. The pseudo-spectral method and path following in reaction-diffusion bifurcation studies. SIAM J. Sci. Statist. Comput. 7 (1986), 599610.
11Eilbeck, J. C.. Numerical studies of bifurcation in reaction-diffusion models using pseudo-spectral and path-following methods. In Bifurcation: Analysis, Algorithms, Applications, eds Kiipper, T., Seydel, R. and Troger, H., pp. 4760 (Basel: Birkhauser, 1987).
12Eilbeck, J. C.. Pattern formation and pattern selection in reaction-diffusion systems. In Theoretical Biology: Epigenetic and Evolutionary Order (Waddington Memorial Conference papers), eds Goodwin, B. and Saunders, P. T., pp. 3141 (Edinburgh: Edinburgh University Press, 1989).
13Eilbeck, J. C. and Furter, J. E.. Understanding steady-state bifurcation diagrams for a model reaction-diffusion system. In Continuation and Bifurcation: Numerical Techniques and Applications, eds Spence, A., Roose, D. and Dier, B. de, pp. 2543 (Dordrecht: Kluwer, 1990).
14Fujii, H., Mimura, M. and Nishiura, Y.. A picture of the global bifurcation diagram in ecological interacting and diffusing system. Phys. D 5 (1982), 142.
15Fujii, H., Nishiura, Y. and Hosono, Y.. On the structure of multiple existence of stable stationary solutions in systems of RD equations. Stud. Math. Appl. 18 (1986), 157219.
16Furter, J. E.. Remarks on scaling laws and bifurcation problems equivariant under a monoid of injective linear maps (Preprint, University of Warwick, 1990).
17Furter, J. E.. On the bifurcation of subharmonics in reversible systems. In Singularity Theory and its Applications. Warwick 1989, Part II, vol. 1463, eds Stewart, I. N. and Roberts, R. M., pp. 167192 (Berlin: Springer, 1991).
18Golubitsky, M. and Schaeffer, D.. Singularities and Groups in Bifurcation Theory, Appl. Math. Sci. 51 (Berlin: Springer, 1985).
19Golubitsky, M., Schaeffer, D. and Stewart, I. N.. Singularities and Groups in Bifurcation Theory II, Appl. Math. Sci. 69 (Berlin: Springer, 1988).
20Gomes, G.. Bifurcations on generalized rectangles: the right context (Ph.D Thesis, University of Warwick, 1992).
21Hunding, A.. Dissipative structures in reaction-diffusion systems: numerical determination of bifurcations in the sphere. J Chem. Phys. 72 (1980), 52415248.
22Hunding, A. and Sorensen, P.. Size adaptation in Turing prepatterns. J. Math. Biol. 26 (1988), 2739.
23Lopez-Gómez, J., Duncan, K. N., Eilbeck, J. C. and Molina, M.. Structure of solution manifolds in a strongly coupled elliptic system. IMA J. Numer. Anal. 12 (1992), 405428.
24Merkin, J. H., Needham, D. J. and Scott, S. K.. Oscillatory chemical reactions in closed vessels. Proc. Roy. Soc. London Ser. A 406 (1986), 299323.
25Murray, J. D.. Nonlinear Differential Equation Models in Biology (Oxford: Clarendon Press, 1977).
26Murray, J. D.. Mathematical Biology, Biomaths. Texts 19 (Berlin: Springer, 1989).
27Nishiura, Y. Global structure of bifurcating solutions of some reaction-diffusion systems. SIAM J. Math. Anal. 13 (1982), 555593.
28Prigogene, I. and Lefever, R.. Symmetry breaking instabilities in dissipative systems. II. J. Chem. Phys. 48(1968), 1665–700.
29Schaff, R.. A class of hamiltonian systems with increasing period. J. Reine Angew. Math. 363 (1985), 96109.
30Schnakenberg, J.. Simple chemical reaction systems with limit cycle behaviour. J. Theoret. Biol. 81 (1979), 389400.
31Scott, S. and Gray, P.. Chemical reactions in isothermal systems: oscillations and instabilities. In Non Linear Phenomena and Chaos, ed. Sarker, S., pp. 7096 (Bristol: Adam Hilger, 1986).
32Scovel, K. C., Kevrekidis, I. G. and Nicolaenko, B.. Scaling laws and the prediction of bifurcations in systems modelling pattern formation. Phys. Lett. A. 130 (1988), 7380.
33Sel'kov, E. E.. Self-oscillations in glycolysis. European J. Biochem. 4 (1968), 7986.
34Sevryuk, M. B.. Reversible Systems, Lecture Notes in Mathematics 1211 (Berlin: Springer, 1986).
35Seydel, R.. From Equilibrium to Chaos–Practical Bifurcation and Stability Analysis (London; Elsevier, 1988).
36Takagi, I.. Point condensation for a reaction–diffusion system. Differential Equations 61 (1986), 208249.
37Vanderbauwhede, A.. Branching of periodic solutions in time-reversible systems (Preprint, Rijksuniversiteit Gent, 1990).
38Wake, G. C., Graham-Eagle, J. G. and Gray, B. F.. Oscillating Chemical Reactions: The Well-Stirred and Spatially Distributed Cases (Providence, RI: American Mathematical Society, 1988).
39Wolfram, S.. Mathematica, 2nd edn (New York: Addison-Wesley, 1991).

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