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October  2020, 25(10): 4057-4070. doi: 10.3934/dcdsb.2020039

## Hexagonal spike clusters for some PDE's in 2D

 1 Dalhousie University, Halifax, Canada 2 University of British Columbia, Vancouver, Canada

* Corresponding author: Juncheng Wei

Received  August 2019 Published  February 2020

Fund Project: The authors are supported by NSERC discovery grants

We study hexagonal spike cluster patterns for Gierer-Meinhardt reaction-diffusion system with a precursor on all of $\mathbb R^2$. These clusters consist of $N$ spikes which form a nearly hexagonal lattice of a finite size. The lattice density is locally nearly constant, but globally non-uniform. We also characterize a similar hexagonal spike cluster steady state for a simple elliptic PDE $0 = \Delta u - u +u^2 + \varepsilon |x|^2$ with a small "confinement well" $\varepsilon |x|^2$. The key idea is to explicitly exploit the local hexagonality structure to asymptotically approximate the solution using certain lattice sums. In the limit of many spikes, we derive the effective spike density as well as the cluster radius. This effective density is a solution to a certain separable first-order ODE coupled to an integral boundary condition.

Citation: Theodore Kolokolnikov, Juncheng Wei. Hexagonal spike clusters for some PDE's in 2D. Discrete & Continuous Dynamical Systems - B, 2020, 25 (10) : 4057-4070. doi: 10.3934/dcdsb.2020039
##### References:
 [1] A. J. Bernoff and C. M. Topaz, Nonlocal aggregation models: A primer of swarm equilibria, SIAM Rev., 55 (2013), 709-747.  doi: 10.1137/130925669.  Google Scholar [2] Y. Chen and T. Kolokolnikov, A minimal model of predator-swarm interactions, J. Royal Soc. Interface, 11 (2014). doi: 10.1098/rsif.2013.1208.  Google Scholar [3] M. R. D'Orsogna, Y.-L. Chuang, A. L. Bertozzi and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Phys. Rev. Lett., 96 (2006). doi: 10.1103/PhysRevLett.96.104302.  Google Scholar [4] R. C. Fetecau, Y. Huang and T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity, 24 (2011), 2681-2716.  doi: 10.1088/0951-7715/24/10/002.  Google Scholar [5] G. Flierl, D. Grünbaum, S. Levins and D. Olson, From individuals to aggregations: The interplay between behavior and physics, J. Theoret. Biol., 196 (1999), 397-454.  doi: 10.1006/jtbi.1998.0842.  Google Scholar [6] B. Gidas, Symmetry of positive solutions of nonlinear elliptic equations in Rn, in Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981,369-402.  Google Scholar [7] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.  doi: 10.1007/BF00289234.  Google Scholar [8] C. Gui and J. Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differential Equations, 158 (1999), 1-27.  doi: 10.1016/S0022-0396(99)80016-3.  Google Scholar [9] D. Iron, M. J. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Phys. D, 150 (2001), 25-62.  doi: 10.1016/S0167-2789(00)00206-2.  Google Scholar [10] T. Kolokolnikov, P. Kevrekidis and R. Carretero-González, A tale of two distributions: From few to many vortices in quasi-two-dimensional Bose-Einstein condensates, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 18pp. doi: 10.1098/rspa.2014.0048.  Google Scholar [11] T. Kolokolnikov, H. Sun, D. Uminsky and A. L. Bertozzi, Stability of ring patterns arising from two-dimensional particle interactions, Phys. Rev. E, 84 (2011). doi: 10.1103/PhysRevE.84.015203.  Google Scholar [12] T. Kolokolnikov and M. J. Ward, Reduced wave Green's functions and their effect on the dynamics of a spike for the Gierer-Meinhardt model, European J. Appl. Math., 14 (2003), 513-545.  doi: 10.1017/S0956792503005254.  Google Scholar [13] T. Kolokolnikov, M. J. Ward and J. Wei, Spot self-replication and dynamics for the Schnakenburg model in a two-dimensional domain, J. Nonlinear Sci., 19 (2009), 1-56.  doi: 10.1007/s00332-008-9024-z.  Google Scholar [14] T. Kolokolnikov and S. Xie, Spike density distribution for the Gierer-Meinhardt model with precursor, Phys. D, in press. doi: 10.1016/j.physd.2019.132247.  Google Scholar [15] M. K. Kwong and L. Zhang, Uniqueness of the positive solution of $\delta u+f(u) = 0$ in an annulus, Differential Integral Equations, 4 (1991), 583-599.   Google Scholar [16] W.-M. Ni and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math., 48 (1995), 731-768.  doi: 10.1002/cpa.3160480704.  Google Scholar [17] M. J. Ward, D. McInerney, P. Houston, D. Gavaghan and P. Maini, The dynamics and pinning of a spike for a reaction-diffusion system, SIAM J. Appl. Math., 62 (2002), 1297-1328.  doi: 10.1137/S0036139900375112.  Google Scholar [18] J. Wei and M. Winter, Stable spike clusters for the one-dimensional Gierer-Meinhardt system, European J. Appl. Math., 28 (2017), 576-635.  doi: 10.1017/S0956792516000450.  Google Scholar [19] J. Wei and M. Winter, Stationary multiple spots for reaction-diffusion systems, J. Math. Biol., 57 (2008), 53-89.  doi: 10.1007/s00285-007-0146-y.  Google Scholar [20] J. Wei and M. Winter, Spikes for the Gierer-Meinhardt system in two dimensions: The strong coupling case, J. Differential Equations, 178 (2002), 478-518.  doi: 10.1006/jdeq.2001.4019.  Google Scholar [21] J. Wei and M. Winter, Existence and stability analysis of asymmetric patterns for the Gierer-Meinhardt system, J. Math. Pures Appl., 83 (2004), 433-476.  doi: 10.1016/j.matpur.2003.09.006.  Google Scholar [22] J. Wei, M. Winter and W. Yang, Stable spike clusters for the precursor Gierer-Meinhardt system in $R^{2}$, Calc. Var. Partial Differential Equations, 56 (2017), 40pp. doi: 10.1007/s00526-017-1233-6.  Google Scholar [23] S. Xie, P. G. Kevrekidis and T. Kolokolnikov, Multi-vortex crystal lattices in Bose-Einstein condensates with a rotating trap, Proc. A, 474 (2018), 21pp. doi: 10.1098/rspa.2017.0553.  Google Scholar

show all references

##### References:
 [1] A. J. Bernoff and C. M. Topaz, Nonlocal aggregation models: A primer of swarm equilibria, SIAM Rev., 55 (2013), 709-747.  doi: 10.1137/130925669.  Google Scholar [2] Y. Chen and T. Kolokolnikov, A minimal model of predator-swarm interactions, J. Royal Soc. Interface, 11 (2014). doi: 10.1098/rsif.2013.1208.  Google Scholar [3] M. R. D'Orsogna, Y.-L. Chuang, A. L. Bertozzi and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Phys. Rev. Lett., 96 (2006). doi: 10.1103/PhysRevLett.96.104302.  Google Scholar [4] R. C. Fetecau, Y. Huang and T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity, 24 (2011), 2681-2716.  doi: 10.1088/0951-7715/24/10/002.  Google Scholar [5] G. Flierl, D. Grünbaum, S. Levins and D. Olson, From individuals to aggregations: The interplay between behavior and physics, J. Theoret. Biol., 196 (1999), 397-454.  doi: 10.1006/jtbi.1998.0842.  Google Scholar [6] B. Gidas, Symmetry of positive solutions of nonlinear elliptic equations in Rn, in Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981,369-402.  Google Scholar [7] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.  doi: 10.1007/BF00289234.  Google Scholar [8] C. Gui and J. Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differential Equations, 158 (1999), 1-27.  doi: 10.1016/S0022-0396(99)80016-3.  Google Scholar [9] D. Iron, M. J. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Phys. D, 150 (2001), 25-62.  doi: 10.1016/S0167-2789(00)00206-2.  Google Scholar [10] T. Kolokolnikov, P. Kevrekidis and R. Carretero-González, A tale of two distributions: From few to many vortices in quasi-two-dimensional Bose-Einstein condensates, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 18pp. doi: 10.1098/rspa.2014.0048.  Google Scholar [11] T. Kolokolnikov, H. Sun, D. Uminsky and A. L. Bertozzi, Stability of ring patterns arising from two-dimensional particle interactions, Phys. Rev. E, 84 (2011). doi: 10.1103/PhysRevE.84.015203.  Google Scholar [12] T. Kolokolnikov and M. J. Ward, Reduced wave Green's functions and their effect on the dynamics of a spike for the Gierer-Meinhardt model, European J. Appl. Math., 14 (2003), 513-545.  doi: 10.1017/S0956792503005254.  Google Scholar [13] T. Kolokolnikov, M. J. Ward and J. Wei, Spot self-replication and dynamics for the Schnakenburg model in a two-dimensional domain, J. Nonlinear Sci., 19 (2009), 1-56.  doi: 10.1007/s00332-008-9024-z.  Google Scholar [14] T. Kolokolnikov and S. Xie, Spike density distribution for the Gierer-Meinhardt model with precursor, Phys. D, in press. doi: 10.1016/j.physd.2019.132247.  Google Scholar [15] M. K. Kwong and L. Zhang, Uniqueness of the positive solution of $\delta u+f(u) = 0$ in an annulus, Differential Integral Equations, 4 (1991), 583-599.   Google Scholar [16] W.-M. Ni and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math., 48 (1995), 731-768.  doi: 10.1002/cpa.3160480704.  Google Scholar [17] M. J. Ward, D. McInerney, P. Houston, D. Gavaghan and P. Maini, The dynamics and pinning of a spike for a reaction-diffusion system, SIAM J. Appl. Math., 62 (2002), 1297-1328.  doi: 10.1137/S0036139900375112.  Google Scholar [18] J. Wei and M. Winter, Stable spike clusters for the one-dimensional Gierer-Meinhardt system, European J. Appl. Math., 28 (2017), 576-635.  doi: 10.1017/S0956792516000450.  Google Scholar [19] J. Wei and M. Winter, Stationary multiple spots for reaction-diffusion systems, J. Math. Biol., 57 (2008), 53-89.  doi: 10.1007/s00285-007-0146-y.  Google Scholar [20] J. Wei and M. Winter, Spikes for the Gierer-Meinhardt system in two dimensions: The strong coupling case, J. Differential Equations, 178 (2002), 478-518.  doi: 10.1006/jdeq.2001.4019.  Google Scholar [21] J. Wei and M. Winter, Existence and stability analysis of asymmetric patterns for the Gierer-Meinhardt system, J. Math. Pures Appl., 83 (2004), 433-476.  doi: 10.1016/j.matpur.2003.09.006.  Google Scholar [22] J. Wei, M. Winter and W. Yang, Stable spike clusters for the precursor Gierer-Meinhardt system in $R^{2}$, Calc. Var. Partial Differential Equations, 56 (2017), 40pp. doi: 10.1007/s00526-017-1233-6.  Google Scholar [23] S. Xie, P. G. Kevrekidis and T. Kolokolnikov, Multi-vortex crystal lattices in Bose-Einstein condensates with a rotating trap, Proc. A, 474 (2018), 21pp. doi: 10.1098/rspa.2017.0553.  Google Scholar
Cluster steady-state solution to (1) consisting of 20 spikes. Contour plot of $a$ and $h$ are shown in (a) and (b) respectively. Parameter values are $\varepsilon = 0.15$ and $\mu(x) = 1+0.02\left\vert x\right\vert ^{2}.$ Computational domain was taken to be $x\in(-15,15)^{2}$; increasing the computational domain did not change spike locations. (c): Centers of spikes from the PDE simulation compared with centers generated by the reduced system (15). Dashed line denotes spike boundary computed asymptotically from (17). (d): Spike height $h(x_{j})$ versus $\left\vert x_{j}\right\vert .$ Comparison between full numerical simulation, the reduced system (15) and theoretical prediction (17)
LEFT: Steady state for (15) with $N = 500,$ $\mu(x) = 1+0.025x^{2}$ and $\varepsilon = 0.08.\$Dots represent the steady state $x_{j};$ their size and colour are proportional to $H_{j}.$ Dashed line represents the theoretical boundary of the steady state in the continuum limit $N\gg1.$ MIDDLE: scatter plot of the average distance $u(x_{j})$ from a point to any of its neighbours, as a function of $\left\vert x_{j}\right\vert .$ Solid curve is the analytical prediction of the continuum limit as given by (17). RIGHT: Scatter plot of the $H_{j}$ as a function of $\left\vert x_{j}\right\vert$ and comparison to theory
Left: steady state solution to the one-dimensional equation (5). Right: inter-spike spacing, comparison between asymptotics (10) and the steady state of (5) computed numerically. Parameters are $a = 0.1$ and $N = 50$
LEFT: steady state for (13) with $N = 500$ and $a = 0.1.$ Dots represent the steady state $x_{j}.$ Dashed line represents the theoretical boundary of the steady state in the continuum limit $N\gg1.$ RIGHT: scatter plot of the average distance $u(x_{j})$ from a point to any of its neighbours, as a function of $\left\vert x_{j}\right\vert .$ Solid curve is the analytical prediction of the continuum limit as given by equations (14)
u and f
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