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activator-inhibitor model; cubic polynomial source; Turing pattern; global stability; weakly linear coupling
The dynamics of an activator-inhibitor model with general cubic polynomial source is investigated. Without diffusion, we consider the existence, stability and bifurcations of equilibria by both eigenvalue analysis and numerical methods. For the reaction-diffusion system, a Lyapunov functional is proposed to declare the global stability of constant steady states, moreover, the condition related to the activator source leading to Turing instability is obtained in the paper. In addition, taking the production rate of the activator as the bifurcation parameter, we show the decisive effect of each part in the source term on the patterns and the evolutionary process among stripes, spots and mazes. Finally, it is illustrated that weakly linear coupling in the activator-inhibitor model can cause synchronous and anti-phase patterns.
[1] Abid, W., Yafia, R., Alaoui, M. A. Aziz, Bouhafa, H., Abichou, A.: Instability and pattern formation in three-species food chain model via Holling type II functional response on a circular domain. Int. J. Bifurcation Chaos Appl. Sci. Eng. 25 (2015), Article ID 1550092, 25 pages. DOI 10.1142/S0218127415500923 | MR 3357413 | Zbl 1317.92059
[2] Antwi-Fordjour, K., Nkashama, M.: Global existence of solutions of the Gierer-Meinhardt system with mixed boundary conditions. Appl. Math., Irvine 8 (2017), 857-867. DOI 10.4236/am.2017.86067 | MR 3286973
[3] Bendahmane, M., Ruiz-Baier, R., Tian, C.: Turing pattern dynamics and adaptive discretization for a super-diffusive Lotka-Volterra model. J. Math. Biol. 72 (2016), 1441-1465. DOI 10.1007/s00285-015-0917-9 | MR 3483181 | Zbl 1338.35041
[4] Bray, D., Thomas, C.: Unpolymerized actin in fibroblasts and brain. J. Mol. Biol. 105 (1976), 527-544. DOI 10.1016/0022-2836(76)90233-3
[5] Brayton, R. K., Moser, J. K.: Theory of nonlinear networks. I. Q. Appl. Math. 22 (1964), 1-33. DOI 10.1090/qam/169746 | MR 0169746 | Zbl 0242.94021
[6] Feng, P.: Dynamics and pattern formation in a modified Leslie-Gower model with Allee effect and Bazykin functional response. Int. J. Biomath. 10 (2017), Article ID 1750073, 26 pages. DOI 10.1142/S1793524517500735 | MR 3648133 | Zbl 1369.35098
[7] Freitas, P., Rocha, C.: Lyapunov functionals and stability for FitzHugh-Nagumo systems. J. Differ. Equations 169 (2001), 208-227. DOI 10.1006/jdeq.2000.3901 | MR 1808465 | Zbl 0974.35051
[8] Gierrer, A., Meinhardt, H.: A theory of biological pattern formation. Biol. Cybern. 12 (1972), 30-39. DOI 10.1007/bf00289234
[9] Henine, S., Youkana, A.: Large-time behaviour and blow up of solutions for Gierer-Meinhardt systems. Math. Methods Appl. Sci. 39 (2016), 570-582. DOI 10.1002/mma.3502 | MR 3454195 | Zbl 1333.35108
[10] Justh, E. W., Krishnaprasad, P. S.: Pattern-forming systems for control of large arrays of actuators. J. Nonlinear Sci. 11 (2001), 239-277. DOI 10.1007/s00332-001-0392-x | MR 1867072 | Zbl 1007.93060
[11] Justh, E. W., Krishnaprasad, P. S.: A Lyapunov functional for the cubic nonlinearity activator-inhibitor model equation. Proceedings of the 37th IEEE Conference on Decision and Control, 1998 IEEE Control Systems Society, Piscataway (2002), 1404-1409. DOI 10.1109/cdc.1998.758483
[12] Koch, A. J., Meinhardt, H.: Biological pattern formation: from basic mechanisms to complex structures. Rev. Mod. Phys. 66 (1994), 1481-1507. DOI 10.1103/revmodphys.66.1481
[13] Schnakenberg, J.: Simple chemical reaction systems with limit cycle behaviour. J. Theoret. Biol. 81 (1979), 389-400. DOI 10.1016/0022-5193(79)90042-0 | MR 0558661
[14] Sun, G.-Q., Wang, C.-H., Wu, Z.-Y.: Pattern dynamics of a Gierer-Meinhardt model with spatial effects. Nonlinear Dyn. 88 (2017), 1385-1396. DOI 10.1007/s11071-016-3317-9
[15] Turing, A. M.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond, Ser. B, Biol. Sci. 237 (1952), 37-72. DOI 10.1098/rstb.1952.0012 | MR 3363444 | Zbl 06853054
[16] Wang, J., Hou, X., Jing, Z.: Stripe and spot patterns in a Gierer-Meinhardt activator-inhibitor model with different sources. Int. J. Bifurcation Chaos Appl. Sci. Eng. 25 (2015), Article ID 1550108, 16 pages. DOI 10.1142/S02181274155010841550108 | MR 3376214 | Zbl 1321.35006
[17] Wei, J., Winter, M.: Existence of spikes for the Gierer-Meinhardt system in one dimension. Mathematical Aspects of Pattern Formation in Biological Systems Applied Mathematical Sciences 189, Springer, London (2014), 13-39. DOI 10.1007/978-1-4471-5526-3_2
[18] Wei, J., Winter, M., Yang, W.: Stable spike clusters for the precursor Gierer-Meinhardt system in $\mathbb R^2$. Calc. Var. Partial Differ. Equ. 56 (2017), 40 pages. DOI 10.1007/s00526-017-1233-6 | MR 3704777 | Zbl 1386.35166
[19] Wu, R., Shao, Y., Zhou, Y., Chen, L.: Turing and Hopf bifurcation of Gierer-Meinhardt activator-substrate model. Electron. J. Differ. Equ. 2017 (2017), Paper No. 173, 19 pages. MR 3690200 | Zbl 1370.35050
[20] Yang, R., Song, Y.: Spatial resonance and Turing-Hopf bifurcations in the Gierer-Meinhardt model. Nonlinear Anal., Real World Appl. 31 (2016), 356-387. DOI 10.1016/j.nonrwa.2016.02.006 | MR 3490848 | Zbl 1344.37061
[21] Yi, F., Gaffney, E. A., Seirin-Lee, S.: The bifurcation analysis of Turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete Contin. Dyn. Syst., Ser. B 22 (2017), 647-668. DOI 10.3934/dcdsb.2017031 | MR 3639134 | Zbl 1360.35016
[22] Zhou, J.: Bifurcation analysis of a diffusive predator-prey model with ratio-dependent Holling type III functional response. Nonlinear Dyn. 81 (2015), 1535-1552. DOI 10.1007/s11071-015-2088-z | MR 3367172 | Zbl 1348.92139
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