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Stationary multiple spots for reaction-diffusion systems. (English) Zbl 1141.92007

Summary: We review analytical methods for a rigorous study of the existence and stability of stationary, multiple spots for reaction-diffusion systems. We consider two classes of reaction-diffusion systems: activator-inhibitor systems, such as the Gierer-Meinhardt system, and activator-substrate systems, such as the P. Gray and S. K. Scott system [Chem. Eng. Sci. 38, 29–43 (1983); ibid. 39, 1087–1097 (1984)], or the J. Schnakenberg model [J. Theor. Biol. 81, 389–400 (1979)]. The main ideas are presented in the context of the Schnakenberg model, and these results are new to the literature. We consider the systems in a two-dimensional, bounded and smooth domain for small diffusion constants of the activator. Existence of multi-spots is proved using tools from nonlinear functional analysis such as Lyapunov-Schmidt reduction and fixed-point theorems. The amplitudes and positions of spots follow from this analysis.

Stability is shown in two parts, for eigenvalues of order one and eigenvalues converging to zero, respectively. Eigenvalues of order one are studied by deriving their leading-order asymptotic behavior and reducing the eigenvalue problem to a nonlocal eigenvalue problem (NLEP). A study of the NLEP reveals a condition for the maximal number of stable spots. Eigenvalues converging to zero are investigated using a projection similar to the Lyapunov-Schmidt reduction and conditions on the positions for stable spots are derived. The Green’s function of the Laplacian plays a central role in the analysis. The results are interpreted in biological, chemical and ecological contexts. They are confirmed by numerical simulations.

MSC:
92C15Developmental biology, pattern formation
35K57Reaction-diffusion equations
46N60Applications of functional analysis in biology and other sciences
35K45Systems of second-order parabolic equations, initial value problems
35J55Systems of elliptic equations, boundary value problems (MSC2000)
References:
[1]Benson D.L., Maini P.K. and Sherratt J.A. (1998). Unravelling the Turing bifurcation using spatially varying diffusion coefficients. J. Math. Biol. 37: 381–417 · Zbl 0919.92006 · doi:10.1007/s002850050135
[2]Castets V., Dulos E., Boissonade J. and De Kepper P. (1990). Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern. Phys. Rev. Lett. 64: 2953–2956 · doi:10.1103/PhysRevLett.64.2953
[3]Crampin E.J., Gaffney E.A. and Maini P.K. (1999). Reaction and diffusion on growing domains: scenarios for robust pattern formation. Bull. Math. Biol. 61: 1093–1120 · doi:10.1006/bulm.1999.0131
[4]Crampin, E.J., Gaffney, E.A., Maini, P.K.: Mode doubling and tripling in reaction–diffusion patterns on growing domains: a piece-wise linear model. J. Math. Biol. 44, 107–128, 1093–1120 (1999)
[5]Dancer E.N. (2001). On stability and Hopf bifurcations for chemotaxis systems. Methods Appl. Anal. 8: 245–256
[6]De Kepper P., Castets V., Dulos E. and Boissonade J. (1991). Turing-type chemical pattern in the chlorite-iodide-malonic acid reaction. Phys. D 49: 161–169 · doi:10.1016/0167-2789(91)90204-M
[7]Doelman A., Gardner R.A. and Kaper T.J. (2001). Large stable pulse solutions in reaction–diffusion equations. Indiana Univ. Math. J. 50: 443–507 · Zbl 0994.35058 · doi:10.1512/iumj.2001.50.1873
[8]Doelman A., Gardner A. and Kaper T.J. (1998). Stability analysis of singular patterns in the 1-D Gray–Scott model: a matched asymptotic approach. Phys. D 122: 1–36 · Zbl 0943.34039 · doi:10.1016/S0167-2789(98)00180-8
[9]Doelman, A., Gardner, A., Kaper, T.J.: A stability index analysis of 1-D patterns of the Gray–Scott model. Mem. Am. Math. Soc. 155(737), xii+64 (2002)
[10]Doelman A., Kaper T. and Zegeling P.A. (1997). Pattern formation in the one-dimensional Gray–Scott model. Nonlinearity 10: 523–563 · Zbl 0905.35044 · doi:10.1088/0951-7715/10/2/013
[11]Dufiet V. and Boissonade J. (1992). Conventional and unconventional Turing patterns. J. Chem. Phys. 96: 664–673 · doi:10.1063/1.462450
[12]Ei S. (2002). The motion of weakly interacting pulses in reaction–diffusion systems. J. Dyn. Diff. Equ. 14: 85–87 · Zbl 1007.35039 · doi:10.1023/A:1012980128575
[13]Ei S., Nishiura Y. and Ueda K. (2001). 2 n splitting or edge splitting: a manner of splitting in dissipative systems. Jpn J. Ind. Appl. Math. 18: 181–205 · Zbl 0983.35061 · doi:10.1007/BF03168570
[14]Fife, P.C.: Stationary patterns for reaction–diffusion systems. In: Nonlinear Diffusion. Research Notes in Math., vol. 14, pp. 81–121. Pitman, London (1977)
[15]Fife P.C. (1979). Large time behaviour of solutions of bistable nonlinear diffusion equations. Arch. Rat. Mech. Anal. 70: 31–46 · Zbl 0435.35045 · doi:10.1007/BF00276380
[16]Gidas B., Ni W.M. and Nirenberg L. (1981). Symmetry of positive solutions of nonlinear elliptic equations in R N . Adv. Math. Suppl. Stud. 7A: 369–402
[17]Gierer A. and Meinhardt H. (1972). A theory of biological pattern formation. Kybernetik (Berlin) 12: 30–39 · doi:10.1007/BF00289234
[18]Gray P. and Scott S.K. (1983). Autocatalytic reactions in the isothermal, continuous stirred tank reactor: isolas and other forms of multistability. Chem. Eng. Sci. 38: 29–43 · doi:10.1016/0009-2509(83)80132-8
[19]Gray P. and Scott S.K. (1984). Autocatalytic reactions in the isothermal. continuous stirred tank reactor: oscillations and instabilites to the system A + 2B 3B, B C. Chem. Eng. Sci. 39: 1087–1097 · doi:10.1016/0009-2509(84)87017-7
[20]Hale J.K., Peletier L.A. and Troy W.C. (2000). Exact homoclinic and heteroclinic solutions of the Gray–Scott model for autocatalysis. SIAM J. Appl. Math. 61: 102–130 · Zbl 0965.34037 · doi:10.1137/S0036139998334913
[21]Hale J.K., Peletier L.A. and Troy W.C. (1999). Stability and instability of the Gray–Scott model: the case of equal diffusion constants. Appl. Math. Lett. 12: 59–65 · Zbl 0936.92034 · doi:10.1016/S0893-9659(99)00035-X
[22]Iron D., Wei J. and Winter M. (2004). Stability analysis of Turing patterns generated by the Schnakenberg model. J. Math. Biol. 49: 358–390 · Zbl 1057.92011 · doi:10.1007/s00285-003-0258-y
[23]Kolokolnikov T. and Ward M.J. (2003). Reduced wave Green’s functions and their effect on the dynamics of a spike for the Gierer–Meinhardt model. Eur. J. Appl. Math. 14: 513–545 · Zbl 1063.35024 · doi:10.1017/S0956792503005254
[24]Kolokolnikov T. and Ward M.J. (2004). Bifurcation of spike equilibria in the near-shadow Gierer–Meinhardt model. Discret. Contin. Dyn. Syst. Ser. B 4: 1033–1064 · Zbl 1063.35084 · doi:10.3934/dcdsb.2004.4.1033
[25]Kolokolnikov T., Ward M.J. and Wei J. (2005). The existence and stability of spike equilibria in the one- dimensional Gray–Scott model: the low-feed regime. Stud. Appl. Math. 115: 21–71 · Zbl 1145.65328 · doi:10.1111/j.1467-9590.2005.01554
[26]Kolokolnikov T., Ward M.J. and Wei J. (2005). The existence and stability of spike equilibria in the one- dimensional Gray–Scott model: the pulse-splitting regime. Phys. D 202: 258–293 · Zbl 1136.35003 · doi:10.1016/j.physd.2005.02.009
[27]Kondo S. and Asai R. (1995). A viable reaction–diffusion wave on the skin of Pomacanthus, a marine Angelfish. Nature 376: 765–768 · doi:10.1038/376765a0
[28]Koch A.J. and Meinhardt H. (1994). Biological pattern formation from basic mechanisms to complex structures. Rev. Mod. Phys. 66: 1481–1507 · doi:10.1103/RevModPhys.66.1481
[29]Kwong M.K. and Zhang L. (1991). Uniqueness of positive solutions of Δu + f(u) = 0 in an annulus. Diff. Integral Equ. 4: 583–599
[30]Lengyel I. and Epstein I.R. (1991). Modeling of Turing structures in the chlorite-iodide-malonic acid-starch reaction system. Science 251: 650–652 · doi:10.1126/science.251.4994.650
[31]Lee K.J., McCormick W.D., Pearson J.E. and Swinney H.L. (1994). Experimental observation of self- replicating spots in a reaction–diffusion system. Nature 369: 215–218 · doi:10.1038/369215a0
[32]Lee K.J., McCormick W.D., Ouyang Q. and Swinney H.L. (1993). Pattern formation by interacting chemical fronts. Science 261: 192–194 · doi:10.1126/science.261.5118.192
[33]Levin S.A. (1992). The problem of pattern and scale in ecology. Ecology 73: 1943–1967 · doi:10.2307/1941447
[34]Madzvamuse A., Maini P.K. and Wathen A.J. (2005). A moving grid finite element method for the simulation of pattern generation by Turing models on growing domains. J. Sci. Comput. 24: 247–262 · Zbl 1080.65091 · doi:10.1007/s10915-004-4617-7
[35]Madzvamuse A., Wathen A.J. and Maini P.K. (2003). A moving grid finite element method applied to a model biological pattern generator. J. Comput. Phys. 190: 478–500 · Zbl 1029.65113 · doi:10.1016/S0021-9991(03)00294-8
[36]Maini P.K., Baker R.E. and Chuong C.M. (2006). The Turing model comes of molecular age. Science 314: 1397–1398 · doi:10.1126/science.1136396
[37]Maini P.K., Painter K.J. and Chau H. (1997). Spatial pattern formation in chemical and biological systems. J. Chem. Soc. Faraday Trans. 93: 3601–3610 · doi:10.1039/a702602a
[38]Meinhardt H. (1982). Model of Biological Pattern Formation. Academic, London
[39]Meinhardt H. (1995). The Algorithmic Beauty of Sea Shells. Springer, Berlin
[40]Mimura, M.: Reaction–diffusion systems arising in biological and chemical systems: applications of singular limit procedures. In: Mathematical Aspects of Evolving Interfaces (Funchal, 2000). Lecture Notes in Mathematics, vol. 1812. Springer, Berlin (2003)
[41]Muratov C.B. and Osipov V.V. (2000). Static spike autosolitons in the Gray–Scott model. J. Phys. A Math. Gen. 33: 8893–8916 · Zbl 01572437 · doi:10.1088/0305-4470/33/48/321
[42]Muratov C.B. and Osipov V.V. (2002). Stability of the static spike autosolitons in the Gray–Scott model. SIAM J. Appl. Math. 62: 1463–1487 · Zbl 1012.35042 · doi:10.1137/S0036139901384285
[43]Murray J.D. (2003). Mathematical Biology II: Spatial Models and Biomedical Applications, Interdisciplinary Applied Mathematics, vol. 18. Springer, Heidelberg
[44]Ni W.-M. (1998). Diffusion, cross-diffusion and their spike-layer steady-states. Not. Am. Math. Soc. 45: 9–18
[45]Nishiura, Y.: Far-From-Equilibrium-Dynamics, Translations of Mathematical Monographs, vol. 209. AMS publications, Providence, Rhode Island (2002)
[46]Nishiura Y. (1982). Global structure of bifurcating solutions of some reaction–diffusion systems. SIAM J. Math. Anal. 13: 555–593 · Zbl 0501.35010 · doi:10.1137/0513037
[47]Nishiura Y. and Fujii H. (1987). Stability of singularly perturbed solutions to systems of reaction–diffusion equations. SIAM J. Math. Anal. 18: 1726–1770 · Zbl 0638.35010 · doi:10.1137/0518124
[48]Nishiura Y., Teramoto T. and Ueda K. (2003). Scattering and separators in dissipative systems. Phys. Rev. E 67(5): 56210 · doi:10.1103/PhysRevE.67.056210
[49]Nishiura Y. and Ueyama D. (1999). A skeleton structure of self-replicating dynamics. Phys. D 130: 73–104 · Zbl 0936.35090 · doi:10.1016/S0167-2789(99)00010-X
[50]Nishiura Y. and Ueyama D. (2001). Spatio-temporal chaos for the Gray–Scott model. Phys. D 150: 137–162 · doi:10.1016/S0167-2789(00)00214-1
[51]Ouyang Q. and Swinney H.L. (1991). Transition from a uniform state to hexagonal and striped Turing patterns. Nature 352: 610–612 · doi:10.1038/352610a0
[52]Ouyang Q. and Swinney H.L. (1991). Transition to chemical turbulence. Chaos 1: 411–420 · Zbl 0900.92171 · doi:10.1063/1.165851
[53]Painter K.J., Maini P.K. and Othmer H.G. (1999). Stripe formation in juvenile pomacanthus explained by a generalized Turing mechanism with chemotaxis. Proc. Nat. Acad. Sci. USA Dev. Biol. 96: 5549–5554 · doi:10.1073/pnas.96.10.5549
[54]Pearson J.E. (1993). Complex patterns in a simple system. Science 261: 189–192 · doi:10.1126/science.261.5118.189
[55]Pearson J.E. and Horsthemke W. (1989). Turing instabilities with nearly equal diffusion constants. J. Chem. Phys. 90: 1588–1599 · doi:10.1063/1.456051
[56]Reynolds J., Pearson J. and Ponce-Dawson S. (1994). Dynamics of self-replicating patterns in reaction diffusion systems. Phys. Rev. Lett. 72: 2797–2800 · doi:10.1103/PhysRevLett.72.2797
[57]Reynolds J., Pearson J. and Ponce-Dawson S. (1997). Dynamics of self-replicating spots in reaction–diffusion systems. Phys. Rev. E 56: 185–198 · doi:10.1103/PhysRevE.56.185
[58]Sandstede B. and Scheel A. (2005). Absolute inequalities of standing pulses. Nonlinearity 18: 331–378 · Zbl 1109.37053 · doi:10.1088/0951-7715/18/1/017
[59]Schnakenberg J. (1979). Simple chemical reaction systems with limit cycle behaviour. J. Theor. Biol. 81: 389–400 · doi:10.1016/0022-5193(79)90042-0
[60]Segel, L.A., Levin, S.A.: Appliations of nonlinear stability theory to the study of the effects of dispersion on predator–prey interactions. In: Piccirelli, R. (ed.) Selected Topics in Statistical Mechanics and Biophysics. Conference Proceedings no. 27. American Inst. Physics, New York (1976)
[61]Sick S., Reinker S., Timmer J. and Schlake T. (2006). WNT and DKK determine hair follicle spacing through a reaction–diffusion mechanism. Science 314: 1447–1450 · doi:10.1126/science.1130088
[62]Sun W., Ward M.J. and Russell R. (2005). The slow dynamics of two-spike solutions for the Gray–Scott and Gierer–Meinhardt systems: competition and oscillatory instabilities. SIAM J. Appl. Dyn. Sys. 4: 904–953 · Zbl 1145.35404 · doi:10.1137/040620990
[63]Takagi I. (1986). Point-condensation for a reaction–diffusion system. J. Diff. Equ. 61: 208–249 · Zbl 0627.35049 · doi:10.1016/0022-0396(86)90119-1
[64]Turing A.M. (1952). The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B 237: 37–72 · doi:10.1098/rstb.1952.0012
[65]Vastano J.A., Pearson J.E., Horsthemke W. and Swinney H.L. (1987). Chemical pattern formation with equal diffusion coefficients. Phys. Lett. A 124: 320–324 · doi:10.1016/0375-9601(87)90019-3
[66]Vastano J.A., Pearson J.E., Horsthemke W. and Swinney H.L. (1988). Turing patterns in an open reactor. J. Chem. Phys. 88: 6175–6181 · doi:10.1063/1.454456
[67]Ward M.J. (2006). Asymptotic methods for reaction–diffusion systems: past and present. Bull. Math. Biol. 68: 1151–1167 · doi:10.1007/s11538-006-9091-y
[68]Wei J. (1999). On single interior spike solutions of the Gierer–Meinhardt system: uniqueness and spectrum estimates. Eur. J. Appl. Math. 10: 353–378 · Zbl 1014.35005 · doi:10.1017/S0956792599003770
[69]Wei J. (1999). Existence, stability and metastability of point condensation patterns generated by Gray–Scott system. Nonlinearity 12: 593–616 · Zbl 0984.35160 · doi:10.1088/0951-7715/12/3/011
[70]Wei J. (2001). Pattern formations in two-dimensional Gray–Scott model: existence of single-spot solutions and their stability. Phys. D 148: 20–48 · doi:10.1016/S0167-2789(00)00183-4
[71]Ward M.J. and Wei J. (2002). The existence and stability of asymmetric spike patterns for the Schnakenberg model. Stud. Appl. Math. 109: 229–264 · Zbl 1152.35397 · doi:10.1111/1467-9590.00223
[72]Ward M.J. and Wei J. (2003). Hopf bifurcations and oscillatory instabilities of solutions for the one-dimensional Gierer–Meinhardt model. J. Nonlinear Sci. 13: 209–264 · Zbl 1030.35011 · doi:10.1007/s00332-002-0531-z
[73]Wei J. and Winter M. (1999). On the two-dimensional Gierer–Meinhardt system with strong coupling. SIAM J. Math. Anal. 30: 1241–1263 · Zbl 0955.35006 · doi:10.1137/S0036141098347237
[74]Wei J. and Winter M. (2000). Spikes for the two-dimensional Gierer–Meinhardt system: the strong coupling case. J. Diff. Equ. 178: 478–518 · Zbl 1042.35005 · doi:10.1006/jdeq.2001.4019
[75]Wei J. and Winter M. (2001). Spikes for the two-dimensional Gierer–Meinhardt system: the weak coupling case. J. Nonlinear Sci. 11: 415–458 · Zbl 1141.35345 · doi:10.1007/s00332-001-0380-1
[76]Wei J. and Winter M. (2003). Existence and stability of multiple-spot solutions for the Gray–Scott model in 2 Phys D 176: 147–180 · Zbl 1014.37036 · doi:10.1016/S0167-2789(02)00743-1
[77]Wei J. and Winter M. (2003). Asymmetric spotty patterns for the Gray–Scott model in R 2. Stud. Appl. Math. 110: 63–102 · Zbl 1141.35401 · doi:10.1111/1467-9590.00231