Bogdanov-Takens Bifurcation of Codimension 3 in the Gierer-Meinhardt Model

Article Computer Science, Interdisciplinary Applications

Multiple bifurcations in a predator-prey system of modified Holling and Leslie type with double Allee effect and nonlinear harvesting

Zuchong Shang et al.

Summary: In this paper, a modified Leslie-type predator-prey model with simplified Holling type IV functional response is established, considering double Allee effect on prey and nonlinear prey harvesting. The analysis of the model reveals the existence of a Bogdanov-Takens singularity (focus case) and multiple nonhyperbolic and degenerate equilibria. Various bifurcations are explored, including transcritical bifurcation, saddle-node bifurcation, Bogdanov-Takens bifurcation of codimension 2, degenerate cusp type Bogdanov-Takens bifurcation of codimension 3, and degenerate focus type Bogdanov-Takens bifurcation of codimension 4. These bifurcations result in complex dynamic behaviors, such as double limit cycle, triple limit cycle, quadruple limit cycle, cuspidal loop, (multiple) homoclinic loop, saddle-node loop, and simultaneous existence of limit cycle(s) with homoclinic loop. Numerical simulations confirm the theoretical results, showing bistability, tristability, or even tetrastability in the system.

MATHEMATICS AND COMPUTERS IN SIMULATION (2023)

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Article Mathematics

A predator-prey system with generalized Holling type IV functional response and Allee effects in prey

Alessandro Arsie et al.

Summary: This paper investigates the interaction between Holling type IV functional response and both strong and weak Allee effects, revealing complex dynamics and high codimension bifurcations in the model studied. The discovery of three limit cycles in predator-prey models with multiplicative Allee effects is particularly noteworthy, along with the analysis of nilpotent cusp singularity of order 3 and degenerate Hopf bifurcation of codimension 3. This work extends existing results on predator-prey systems with Allee effects, providing biological interpretations of predator-prey interactions through bifurcation analysis and diagrams.

JOURNAL OF DIFFERENTIAL EQUATIONS (2022)

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Article Mathematics, Applied

Bifurcations in a Dynamical Model of the Innate Immune System Response to Initial Pulmonary Infection

Juan Su et al.

Summary: We revisit a dynamical model of the innate immune system response to initial pulmonary infection and provide a complete analysis on bifurcations with high codimension. The model can undergo various bifurcations as parameters vary, and numerical simulations further validate the theoretical results. Our study demonstrates the complexity of the interaction between the innate immune system and initial pulmonary bacterial infection.

QUALITATIVE THEORY OF DYNAMICAL SYSTEMS (2022)

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Article Mathematics, Interdisciplinary Applications

Hopf Bifurcation Analysis in a Gierer-Meinhardt Activator-Inhibitor Model

Rasoul Asheghi

Summary: In this paper, we study the Hopf bifurcation in a Gierer-Meinhardt model with different sources and analyze the impact of diffusion rates on system stability. The normal form of this bifurcation is computed up to the third order, and the direction of Hopf bifurcation is determined using normal form theory. Numerical simulations are also provided to illustrate the analytical results.

INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS (2022)

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Article Engineering, Multidisciplinary

Turing-Turing bifurcation and multi-stable patterns in a Gierer-Meinhardt system

Shuangrui Zhao et al.

Summary: This paper explores the coexistence of multi-stable patterns and the superposition of patterns in the classical Gierer-Meinhardt system from the perspective of Turing-Turing bifurcation. The study reveals the existence of semi-stable patterns superimposed by two different spatial resonances and the coexistence of four stable steady states with different characteristic wavelengths. Numerical simulations are consistent with the theoretical analysis. The findings suggest that experimental patterns of vascular mesenchymal cells can be interpreted as the superposition of different spatial modal patterns.

APPLIED MATHEMATICAL MODELLING (2022)

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Article Multidisciplinary Sciences

Unified framework for localized patterns in reaction-diffusion systems; the Gray-Scott and Gierer-Meinhardt cases

Fahad Al Saadi et al.

Summary: This study extends the research on canonical activator-inhibitor Schnakenberg-like models to include models with bistability of homogeneous equilibria, such as Gray-Scott. A homotopy from a Schnakenberg-like glycolysis model to the Gray-Scott model is studied, with numerical continuation used to understand the sequence of transitions in parameter regimes. Several distinct bifurcations are discovered, including cusp and quadruple zero points, under homotopy between bifurcation diagrams for different field feed scenarios.

PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES (2021)

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Article Physics, Multidisciplinary