TY - JOUR
T1 - Spike density distribution for the Gierer–Meinhardt model with precursor
AU - Kolokolnikov, Theodore
AU - Xie, Shuangquan
N1 - Funding Information:
This work was supported by grants from NSERC, Canada Appendix
Publisher Copyright:
© 2019 Elsevier B.V.
PY - 2020/1/15
Y1 - 2020/1/15
N2 - We consider the Gierer–Meinhardt model in one dimension with a spatially-dependent precursor μ(x) which was proposed as a model of cell differentiation with control over pattern distribution in space. Assuming that the activator diffuses much slower than the inhibitor, such a system is well-known to admit solutions where the activator concentrates at N “spikes”. In the large-N limit, we derive the effective spike density for an arbitrary μ(x). We show that this density satisfies a first-order separable ODE. As a consequence, we derive instability thresholds for N spikes that correspond to a singularity in the ODE for the density. We recover, as a special case, the well-known stability thresholds for constant μ first derived in Iron et al. (2001), as well as cluster solutions that concentrate near the minimum of μ(x) that were recently discovered in Wei and Winter (2017). The main trick is applying Taylor expansions and geometric series to the equations of effective spike dynamics.
AB - We consider the Gierer–Meinhardt model in one dimension with a spatially-dependent precursor μ(x) which was proposed as a model of cell differentiation with control over pattern distribution in space. Assuming that the activator diffuses much slower than the inhibitor, such a system is well-known to admit solutions where the activator concentrates at N “spikes”. In the large-N limit, we derive the effective spike density for an arbitrary μ(x). We show that this density satisfies a first-order separable ODE. As a consequence, we derive instability thresholds for N spikes that correspond to a singularity in the ODE for the density. We recover, as a special case, the well-known stability thresholds for constant μ first derived in Iron et al. (2001), as well as cluster solutions that concentrate near the minimum of μ(x) that were recently discovered in Wei and Winter (2017). The main trick is applying Taylor expansions and geometric series to the equations of effective spike dynamics.
KW - Heterogeneous media
KW - Pattern formation
KW - Reaction-diffusion systems
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U2 - 10.1016/j.physd.2019.132247
DO - 10.1016/j.physd.2019.132247
M3 - Article
AN - SCOPUS:85075402640
VL - 402
JO - Physica D: Nonlinear Phenomena
JF - Physica D: Nonlinear Phenomena
SN - 0167-2789
M1 - 132247
ER -