Analysis of bifurcation, chaos and pattern formation in a discrete time and space Gierer Meinhardt system

Wang, Jinliang; Li, You; Zhong, Shihong; Hou, Xiaojie

doi:10.1016/j.chaos.2018.11.013

Cited by 27 publications

(12 citation statements)

References 38 publications

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“…However, even with the same bifurcations, discrete models can still generate more complex patterns, due to the flexible of both time and space steps. Although continuous PDEs are still the main type of model that are used in pattern formation, many research studies on discrete dynamic systems have shown that spatiotemporal discrete models are not only more related to realistic processes but also can generate more complex dynamics [38,39,43,44]. And more work could be done to explore the bifurcations induced by time step variation.…”

Section: Discussionmentioning

confidence: 99%

“…Wang et al studying the discrete forms of the Gierer-Meinhardt system found that discrete systems lead to more complex nonlinear behaviors and new dynamic phenomena than continuous forms. In addition, for the discrete time and space G-M system, they found a new instability mechanism, pronunciation flip-Turing instability [44]. In addition, when we apply Turing analysis on discrete models, there are some differences.…”

Section: Introductionmentioning

confidence: 99%

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Spatiotemporal Patterns Formed by a Discrete Nutrient-Phytoplankton Model with Time Delay

et al. 2020

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In this research, a continuous nutrient-phytoplankton model with time delay and Michaelis–Menten functional response is discretized to a spatiotemporal discrete model. Around the homogeneous steady state of the discrete model, Neimark–Sacker bifurcation and Turing bifurcation analysis are investigated. Based on the bifurcation analysis, numerical simulations are carried out on the formation of spatiotemporal patterns. Simulation results show that the diffusion of phytoplankton and nutrients can induce the formation of Turing-like patterns, while time delay can also induce the formation of cloud-like pattern by Neimark–Sacker bifurcation. Compared with the results generated by the continuous model, more types of patterns are obtained and are compared with real observed patterns.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Spatiotemporal Patterns Formed by a Discrete Nutrient-Phytoplankton Model with Time Delay

et al. 2020

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“…Since μ < 0 for τ < τ c , we have proved condition (C2) for τ c . Finally, substituting μ = 0, φ = φ c and τ = τ c into (12) and 14, we have…”

Section: Lemma 1 Necessary and Sufficient Conditions For Existence Ofmentioning

confidence: 99%

“…In recent years, time-delay differential equation and difference equation models have been studied by many authors (see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]) as they are useful tools for modeling a wide variety of systems in areas including traditional areas such as physics and engineering and newer areas such as disease transmission, medical research, optimal drug treatment, bioeconomics, agriculture, finance, insurance, and environmental protection. In many of these time-delay models, bifurcations occur as values of parameters are changed.…”

Section: Introductionmentioning

confidence: 99%

Andronov–Hopf and Neimark–Sacker bifurcations in time-delay differential equations and difference equations with applications to models for diseases and animal populations

2020

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In many areas, researchers might think that a differential equation model is required, but one might be forced to use an approximate difference equation model if data is only available at discrete points in time. In this paper, a detailed comparison is given of the behavior of continuous and discrete models for two representative time-delay models, namely a model for HIV and an extended logistic growth model. For each model, there are seven different time-delay versions because there are seven different positions to include time delays. For the seven different time-delay versions of each model, proofs are given of necessary and sufficient conditions for the existence and stability of equilibrium points and for the existence of Andronov-Hopf bifurcations in the differential equations and Neimark-Sacker bifurcations in the difference equations. We show that only five of the seven time-delay versions have bifurcations and that all bifurcation versions have supercritical limit cycles with one having a repelling cycle and four having attracting cycles. Numerical simulations are used to illustrate the analytical results and to show that critical times for Neimark-Sacker bifurcations are less than critical times for Andronov-Hopf bifurcations but converge to them as the time step of the discretization tends to zero.

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“…Many authors have argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations, for example, see Freedman [8], Murry [30], Punithan et al [33], Huang et al [15] and Wang et al [45]. By using the different discrete methods, from equation (2) we can obtain the discrete logistic equations:…”

Section: Introductionmentioning

confidence: 99%

The diffusion-driven instability and complexity for a single-handed discrete Fisher equation

Zhang

Yan

2020

Applied Mathematics and Computation

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For a reaction diffusion system, it is well known that the diffusion coefficient of the inhibitor must be bigger than that of the activator when the Turing instability is considered. However, the diffusion-driven instability/Turing instability for a single-handed discrete Fisher equation with the Neumann boundary conditions may occur and a series of 2-periodic patterns have been observed. Motivated by these pattern formations, the existence of 2-periodic solutions is established. Naturally, the periodic double and the chaos phenomenon should be considered. To this end, a simplest two elements system will be further discussed, the flip bifurcation theorem will be obtained by computing the center manifold, and the bifurcation diagrams will be simulated by using the shooting method. It proves that the Turing instability and the complexity of dynamical behaviors can be completely driven by the diffusion term. Additionally, those effective methods of numerical simulations are valid for experiments of other patterns, thus, are also beneficial for some application scientists.

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Analysis of bifurcation, chaos and pattern formation in a discrete time and space Gierer Meinhardt system

Cited by 27 publications

References 38 publications

Spatiotemporal Patterns Formed by a Discrete Nutrient-Phytoplankton Model with Time Delay

Spatiotemporal Patterns Formed by a Discrete Nutrient-Phytoplankton Model with Time Delay

Andronov–Hopf and Neimark–Sacker bifurcations in time-delay differential equations and difference equations with applications to models for diseases and animal populations

The diffusion-driven instability and complexity for a single-handed discrete Fisher equation

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