Stationary and oscillatory corrosion patterns in a modified Barkley-Leslie-Gower model

Giancarlo Consolo; Guglielmo Inferrera; Giancarlo Consolo; Guglielmo Inferrera

doi:10.3934/mbe.2026063

Mathematical Biosciences and Engineering

2026, Volume 23, Issue 6: 1743-1767. doi: 10.3934/mbe.2026063

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Stationary and oscillatory corrosion patterns in a modified Barkley-Leslie-Gower model

Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina, Viale F. Stagno d'Alcontres 31, 98166 Messina, Italy

Received: 03 April 2026 Revised: 05 May 2026 Accepted: 14 May 2026 Published: 21 May 2026

In this work, we investigated the emergence of stationary and oscillatory patterns in an extended version of the Barkley model for corrosive and passivating species, specifically tailored to describe the initiation stage of localized corrosion at the bottom of fuel tanks. While the proposed model leaves the corrosive reaction unchanged relative to the original formulation, it modifies the kinetics of the passivating species by assuming a Leslie-Gower law with a Holling type Ⅱ functional response. A rigorous analysis of the dynamical system was first conducted to establish the boundedness of solutions within a positively invariant region and to identify the emergence of Hopf bifurcations. Subsequently, through local stability analysis performed on the full spatial model, the conditions for Turing bifurcations were determined. These analyses highlighted a wide parameter space where Turing and Hopf instabilities coexist, thereby demonstrating the model's capability to trigger stationary and oscillatory corrosion patterns. Numerical investigations in 1D and 2D domains, utilizing direct integration and continuation tools, revealed a large variety of pattern morphologies. Specifically, small computational domains exhibited stripes as well as square and hexagonal holes. In larger domains and across broader parameter ranges, labyrinthine, target, and non-Turing spiral-like patterns also emerged. Beyond validating theoretical predictions, these analyses revealed that patterned branches emerge supercritically in 1D and subcritically in 2D domains, providing a qualitative match with experimental observations. Overall, the proposed framework sheds light on the complex competition between material dissolution and protective film regeneration, serving as a potential support for predictive strategies in the safety management of fuel tanks.
- activator-inhibitor dynamics,
- turing and Hopf instabilities,
- Barkley-Leslie-Gower kinetics,
- reaction-diffusion systems,
- corrosion pattern formation
Citation: Giancarlo Consolo, Guglielmo Inferrera. Stationary and oscillatory corrosion patterns in a modified Barkley-Leslie-Gower model[J]. Mathematical Biosciences and Engineering, 2026, 23(6): 1743-1767. doi: 10.3934/mbe.2026063

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Abstract

In this work, we investigated the emergence of stationary and oscillatory patterns in an extended version of the Barkley model for corrosive and passivating species, specifically tailored to describe the initiation stage of localized corrosion at the bottom of fuel tanks. While the proposed model leaves the corrosive reaction unchanged relative to the original formulation, it modifies the kinetics of the passivating species by assuming a Leslie-Gower law with a Holling type Ⅱ functional response. A rigorous analysis of the dynamical system was first conducted to establish the boundedness of solutions within a positively invariant region and to identify the emergence of Hopf bifurcations. Subsequently, through local stability analysis performed on the full spatial model, the conditions for Turing bifurcations were determined. These analyses highlighted a wide parameter space where Turing and Hopf instabilities coexist, thereby demonstrating the model's capability to trigger stationary and oscillatory corrosion patterns. Numerical investigations in 1D and 2D domains, utilizing direct integration and continuation tools, revealed a large variety of pattern morphologies. Specifically, small computational domains exhibited stripes as well as square and hexagonal holes. In larger domains and across broader parameter ranges, labyrinthine, target, and non-Turing spiral-like patterns also emerged. Beyond validating theoretical predictions, these analyses revealed that patterned branches emerge supercritically in 1D and subcritically in 2D domains, providing a qualitative match with experimental observations. Overall, the proposed framework sheds light on the complex competition between material dissolution and protective film regeneration, serving as a potential support for predictive strategies in the safety management of fuel tanks.

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