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

Hexagonal spike clusters for some PDE's in 2D

Theodore Kolokolnikov 1,, and  Juncheng Wei 2,

1. 

Dalhousie University, Halifax, Canada
2. 

University of British Columbia, Vancouver, Canada

* Corresponding author: Juncheng Wei

Received  August 2019 Published  October 2020 Early access  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.

Keywords: Reaction-diffusion systems, spike solutions, hexagonal clusters, aggregation model, pattern formation.
Mathematics Subject Classification: Primary: 35K57, 35B36; Secondary: 35Jxx.
Citation: Theodore Kolokolnikov, Juncheng Wei. Hexagonal spike clusters for some PDE's in 2D. Discrete and 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.

[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.

[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.

[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.

[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.

[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.

[7]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.  doi: 10.1007/BF00289234.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[7]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.  doi: 10.1007/BF00289234.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

Figure 1.  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)
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Figure 4.  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
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Figure 2.  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$
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Figure 3.  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)
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Figure 5.  u and f
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