Kinematic phase-wave propagation with weak diffusion sensitivity under an imposed phase gradient in a bounded IP<sub>3</sub>-mediated calcium model

Minsoo Kim; Minsoo Kim

doi:10.3934/math.20261105

AIMS Mathematics

2026, Volume 11, Issue 9: 27633-27664. doi: 10.3934/math.20261105

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Research article

Kinematic phase-wave propagation with weak diffusion sensitivity under an imposed phase gradient in a bounded IP₃-mediated calcium model

Minsoo Kim ^,

Major of Data Science, Korea Maritime & Ocean University, Busan 49112, Republic of Korea

Received: 27 April 2026 Revised: 17 August 2026 Accepted: 21 August 2026 Published: 01 September 2026
MSC : 35K57, 92C40

We study a one-dimensional IP₃-mediated calcium reaction–diffusion model in a bounded domain, separating localized no-flux simulations from an imposed periodic phase-gradient diagnostic used to measure the phase-wave speed $ v_\phi $. We establish global well-posedness, positivity, invariant bounds, and homogeneous-state properties. Under an explicit smooth-branch assumption, a small-wavenumber phase-wave calculation yields $ v_\phi(k) = \omega_0/k+\omega_2 k+O(k^3) $ with $ \omega_2 = \delta_C D_{\mathrm{eff}}{}_{, C}+\delta_P D_{\mathrm{eff}}{}_{, P} $, so diffusion enters through a finite-$ k $ correction along a coherent phase-wave branch. Extended simulations use pre-specified Fourier-mode coherence and phase-linearity screening. In the calcium-diffusion sweep, coherent stationary $ m = 1 $ fits persist through $ \delta_C = 0.05 $, while no speed is assigned for $ \delta_C\ge0.1 $ because the prescribed mode loses coherence. In the IP₃-diffusion sweep, all runs over $ \delta_P\in[0.005, 1.0] $ remain coherent with no systematic speed dependence resolved. For the accepted coherent states, reaction-parameter tests show that $ v_\phi $ tracks the kinematic reference $ v_{{\rm{kin}}} = L_\phi/T_{\rm osc} $. The reported propagation is therefore phase transport at a prescribed wavenumber rather than a Fisher-KPP-type invasion speed. These conclusions are specific to the present model, tested parameter ranges, and imposed phase-gradient protocol.
- reaction–diffusion system,
- calcium signaling,
- IP₃ dynamics,
- finite-domain propagation,
- phase waves,
- receptor dynamics,
- diffusion effects
Citation: Minsoo Kim. Kinematic phase-wave propagation with weak diffusion sensitivity under an imposed phase gradient in a bounded IP₃-mediated calcium model[J]. AIMS Mathematics, 2026, 11(9): 27633-27664. doi: 10.3934/math.20261105

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Abstract

We study a one-dimensional IP₃-mediated calcium reaction–diffusion model in a bounded domain, separating localized no-flux simulations from an imposed periodic phase-gradient diagnostic used to measure the phase-wave speed $ v_\phi $. We establish global well-posedness, positivity, invariant bounds, and homogeneous-state properties. Under an explicit smooth-branch assumption, a small-wavenumber phase-wave calculation yields $ v_\phi(k) = \omega_0/k+\omega_2 k+O(k^3) $ with $ \omega_2 = \delta_C D_{\mathrm{eff}}{}_{, C}+\delta_P D_{\mathrm{eff}}{}_{, P} $, so diffusion enters through a finite-$ k $ correction along a coherent phase-wave branch. Extended simulations use pre-specified Fourier-mode coherence and phase-linearity screening. In the calcium-diffusion sweep, coherent stationary $ m = 1 $ fits persist through $ \delta_C = 0.05 $, while no speed is assigned for $ \delta_C\ge0.1 $ because the prescribed mode loses coherence. In the IP₃-diffusion sweep, all runs over $ \delta_P\in[0.005, 1.0] $ remain coherent with no systematic speed dependence resolved. For the accepted coherent states, reaction-parameter tests show that $ v_\phi $ tracks the kinematic reference $ v_{{\rm{kin}}} = L_\phi/T_{\rm osc} $. The reported propagation is therefore phase transport at a prescribed wavenumber rather than a Fisher-KPP-type invasion speed. These conclusions are specific to the present model, tested parameter ranges, and imposed phase-gradient protocol.

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