Bifurcation analysis and spatiotemporal patterns in delayed Schnakenberg reaction-diffusion model

References

• Schnakenberg J. Simple chemical reaction systems with limit cycle behaviour. J Theor Biol. 1979;81(3):389–400.
   PubMed Web of Science ®Google Scholar
• Garzon-Alvarado DA, Garcia-Aznar JM, Doblare M. A reaction-diffusion model for long bones growth. Biomech Model Mechanobiol. 2009;8(5):381–395.
   PubMed Web of Science ®Google Scholar
• Maini PK, Woolley TE, Baker RE, et al. Turing's model for biological pattern formation and the robustness problem. Interface Focus. 2012;2(4):487–496.
   PubMed Web of Science ®Google Scholar
• Liu P, Shi J, Wang Y, et al. Bifurcation analysis of reaction-diffusion Schnakenberg model. J Math Chem. 2013;51(8):2001–2019.
   Web of Science ®Google Scholar
• Chaplain M, Ganesh M, Graham IG. Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumour growth. J Math Biol. 2001;42(5):387–423.
   PubMed Web of Science ®Google Scholar
• Ricard MR, Mischler S. Turing instabilities at Hopf bifurcation. J Nonlinear Sci. 2009;19(5):467–496.
   Web of Science ®Google Scholar
• Fragnelli G, Mugnai D. Turing patterns for a coupled two-cell generalized Schnakenberg model. Complex Var Elliptic Equ. 2020;65(8):1343–1359.
   Web of Science ®Google Scholar
• Khudhair HK, Zhang YZ, Fukawa N. Pattern selection in the Schnakenberg equations: from normal to anomalous diffusion. Numer Methods Partial Differ Equ. DOI:10.1002/num.22842
   Web of Science ®Google Scholar
• Liu GQ, Wang YW. Pattern formation of coupled two-cell Schnakenberg model. Discrete Contin Dyn Syst Ser B. 2017;10:1051–1062.
   Web of Science ®Google Scholar
• Wong T, Ward MJ. Spot patterns in the 2-D Schnakenberg model with localized heterogeneities. Stud Appl Math. 2021;146(4):779–833.
   Web of Science ®Google Scholar
• Ishii Y. Stability of multi-peak symmetric stationary solutions for the Schnakenberg model with periodic heterogeneity. Comm Pure Appl Math. 2020;19:2965–3031.
   Google Scholar
• Doelman A, Kaper TJ. Semistrong pulse interactions in a class of coupled reaction-diffusion equations. SIAM J Appl Dyn Syst. 2003;2:53–96.
   Web of Science ®Google Scholar
• Iron D, Wei J, Winter M. Stability analysis of Turing patterns generated by the Schnakenberg model. J Math Biol. 2004;49:358–390.
   PubMed Web of Science ®Google Scholar
• Gomez D, Mei LF, Wei JC. Stable and unstable periodic spiky solutions for the Gray-Scott system and the Schnakenberg system. J Dyn Differ Equ. 2020;32:441–481.
   Web of Science ®Google Scholar
• Ishii Y, Kurata K. Existence of multi-peak solutions to the Schnakenberg model with heterogeneity on metric graphs. Comm Pure Appl Math. 2021;20:1633–1379.
   Google Scholar
• Ishii Y. The effect of heterogeneity on one-peak stationary solutions to the Schnakenberg model. J Differential Equations. 2021;285:321–382.
   Web of Science ®Google Scholar
• Ishii Y, Kurata K. Existence and stability of one-peak symmetric stationary solutions for the Schnakenberg model with heterogeneity. Discrete Contin Dyn Syst. 2019;39:2807–2875.
   Web of Science ®Google Scholar
• Al Noufaey KS. Semi-analytical solutions of the Schnakenberg model of a reaction-diffusion cell with feedback. Results Phys. 2018;9:609–614.
   Web of Science ®Google Scholar
• Madzvamuse A. Time-stepping schemes for moving grid finite elements applied to reaction-diffusion systems on fixed and growing domains. J Comput Phys. 2006;214:239–263.
   Web of Science ®Google Scholar
• Madzvamuse A, Ndakwo HS, Barreira R. Cross-diffusion-driven instability for reaction-diffusion systems: analysis and simulations. J Math Biol. 2015;70:709–743.
   PubMed Web of Science ®Google Scholar
• Kaper HG, Wang S, Yari M. Dynamical transitions of Turing patterns. Nonlinearity. 2009;22:601–626.
   Web of Science ®Google Scholar
• Xu C, Wei J. Hopf bifurcation analysis in a one-dimensional Schnakenberg reaction-diffusion model. Nonlinear Anal Real World Appl. 2012;13:1961–1977.
   Web of Science ®Google Scholar
• Rodriguez-Ricard M, Mondaini RP. Multiscale analysis for diffusion-driven neutrally stable states. Math Comput Model. 2009;50:1167–1176.
   Google Scholar
• Li Y. Steady-state solution for a general Schnakenberg model. Nonlinear Anal Real World Appl. 2011;12:1985–1990.
   Web of Science ®Google Scholar
• Al-Zarka A, Alagha A, Timoshin S. Transient behaviour in RDA systems of the Schnakenberg type. J Math Chem. 2015;53:111–127.
   Web of Science ®Google Scholar
• Yadav A, Milu SM, Horsthemke W. Turing instability in reaction-subdiffusion systems. Phys Rev E. 2008;78:026116.
   Google Scholar
• Garzon-Alvarado DA, Galeano CH, Mantilla JM. Turing pattern formation for reaction-convection-diffusion systems in fixed domains submitted to toroidal velocity fields. Appl Math Model. 2011;35:4913–4925.
   Web of Science ®Google Scholar
• Lee SS, Gaffney EA, Baker RE. The dynamics of Turing patterns for Morphogen-regulated growing domains with cellular response delays. Bull Math Biol. 2011;73:2527–2551.
   PubMed Web of Science ®Google Scholar
• Hu G, Li W. Hopf bifurcation analysis for a delayed predator-prey system with diffusion effects. Nonlinear Anal Real World Appl. 2010;11:819–826.
   Web of Science ®Google Scholar
• Xu R, Gan Q, Ma Z. Stability and bifurcation analysis on a ratio-dependent predator-prey model with time delay. J Comput Appl Math. 2009;230:187–203.
   Web of Science ®Google Scholar
• Zuo W, Wei J. Stability and Hopf bifurcation in a diffusive predator-prey system with delay effect. Nonlinear Anal Real World Appl. 2011;12:1998–2011.
   Web of Science ®Google Scholar
• Song Y, Peng Y, Zou X. Persistence, stability and Hopf bifurcation in a diffusive ratio-dependent predator-prey model with delay. Int J Bifurcat Chaos. 2014;24:1450093.
   Google Scholar
• Chen S, Shi J. Global attractivity of equilibrium in Gierer-Meinhardt system with activator production saturation and gene expression time delays. Nonlinear Anal Real World Appl. 2013;14:1871–1886.
   Web of Science ®Google Scholar
• Yi F, Gaffney EA, Seirin-Lee S. The bifurcation analysis of Turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete Contin Dyn Syst B. 2017;22:647–668.
   Web of Science ®Google Scholar
• Jiang W, Wang H, Cao X. Turing instability and Turing-Hopf bifurcation in diffusive Schnakenberg systems with gene expression time delay. J Dyn Differ Equ. 2019;31:2223–2247.
   Web of Science ®Google Scholar
• Yang R. Turing-Hopf bifurcation co-induced by cross-diffusion and delay in Schnakenberg system. Chaos Solit Fractals. 2022;164:112659.
   Web of Science ®Google Scholar
• Wu J. Theory and applications of partial functional differential equations. New York: Springer; 1996.
   Google Scholar
• Faria T. Normal forms and Hopf bifurcation for partial functional differential equations with delays. Trans Amer Math Soc. 2000;352:2217–2238.
   Web of Science ®Google Scholar
• Chow SN, Hale JK. Methods of bifurcation theory. New York: Springer; 1998.
   Google Scholar
• Garvie MR. Finite-difference schemes for reaction-diffusion equations modeling predator-prey interactions in Matlab. Bull Math Biol. 2007;69:931–956.
   PubMed Web of Science ®Google Scholar