A Stochastic Pitchfork Bifurcation in Most Probable Phase Portraits

Wang, Hui; Chen, Xiaoli; Duan, Jinqiao

doi:10.1142/s0218127418500177

Cited by 29 publications

(12 citation statements)

References 29 publications

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“…Unlike the phase portraits for deterministic dynamical systems [56,57], these sample paths ('orbit' or 'trajectories') mingled together and can not provide much information. Most probable phase portraits [48,49,52] for the stochastic gene regulation system (3) is a powerful tool to understand the stochastic system.…”

Section: Resultsmentioning

confidence: 99%

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Most probable transition pathways and maximal likely trajectories in a genetic regulatory system

Cheng

Wang

et al. 2019

Physica A: Statistical Mechanics and its Applications

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We study the most probable transition pathways and maximal likely trajectories in a genetic regulation model of the transcription factor activator's concentration evolution, with Gaussian noise and non-Gaussian stable Lévy noise in the synthesis reaction rate taking into account, respectively. We compute the most probable transition pathways by the Onsager-Machlup least action principle, and calculate the maximal likely trajectories by spatially maximizing the probability density of the system path, i.e., the solution of the associated nonlocal Fokker-Planck equation. We have observed the rare most probable transition pathways in the case of Gaussian noise, for certain noise intensity, evolution time scale and system parameters. We have especially studied the maximal likely trajectories starting at the low concentration metastable state, and examined whether they evolve to or near the high concentration metastable state (i.e., the likely transcription regime) for certain parameters, in order to gain insights into the transcription processes and the tipping time for the transcription likely to occur. This enables us: (i) to visualize the progress of concentration evolution (i.e., observe whether the system enters the transcription regime within a given time period); (ii) to predict or avoid certain transcriptions via selecting specific noise parameters in particular regions in the parameter space. Moreover, we have found some peculiar or counter-intuitive phenomena in this gene model system, including: (a) A smaller noise intensity may trigger the transcription process, while a larger noise intensity can not, under the same asymmetric Lévy noise. This phenomenon does not occur in the case of symmetric Lévy noise; (b) The symmetric Lévy motion always induces transition to high concentration, but certain asymmetric Lévy motions do not trigger the switch to transcription.These findings provide insights for further experimental research, in order to achieve or to avoid specific gene transcriptions, with possible relevance for medical advances.

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Section: Resultsmentioning

confidence: 99%

“…As in [52], the maximal likely equilibrium state is defined as a state which either attracts or repels all nearby orbits. When it attracts all nearby orbits, it is called a maximal likely stable equilibrium state, while if it repels all nearby orbits, it is called a maximal likely unstable equilibrium state.…”

Section: Maximal Likely Trajectorymentioning

confidence: 99%

Most probable transition pathways and maximal likely trajectories in a genetic regulatory system

Cheng

Wang

et al. 2019

Physica A: Statistical Mechanics and its Applications

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“…Using the method of maximal likely trajectory to explore the impact of Lévy noise on the stochastic dynamical system has achieved some results. The stochastic pitchfork bifurcation for a system under multiplicative stable Lévy noise by the maximal likely trajectory is studied in [38]. Similarly, this indicator is also used in gene regulatory systems under stable Lévy noise [39].…”

Section: The Methodsmentioning

confidence: 99%

State transitions in the Morris-Lecar model under stable Lévy noise

et al. 2020

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This paper considers the state transition of the stochastic Morris-Lecar neuronal model driven by symmetric α-stable Lévy noise. The considered system is bistable: a stable fixed point (resting state) and a stable limit cycle (oscillating state), and there is an unstable limit cycle (borderline state) between them. Small disturbances may cause a transition between the two stable states, thus a deterministic quantity, namely the maximal likely trajectory, is used to analyze the transition phenomena in non-Gaussian stochastic environment. According to the numerical experiment, we find that smaller jumps of the Lévy motion and smaller noise intensity can promote such transition from the sustained oscillating state to the resting state. It also can be seen that larger jumps of the Lévy motion and higher noise intensity are conducive for the transition from the borderline state to the sustained oscillating state. As a comparison, Brownian motion is also taken into account. The results show that whether it is the oscillating state or the borderline state, the system disturbed by Brownian motion will be transferred to the resting state under the selected noise intensity.

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“…Cheng [36] obtained the analytical results of the MPPP and showed that the MPPP can give some information about the propagation of stochastic dynamics in 1-dimensional model. Wang [37] studied the stochastic bifurcation by using the qualitative changes of the MPPP to a stochastic system driven by a stable Lévy noise. In [38], the scholars investigated the most probable trajectories of the tumor system growth with immune surveillance under correlated Gaussian noises, and derived analytical solution of the most probable steady state by using the extremum theory with the local Fokker-Planck equation in the system.…”

Section: Introductionmentioning

confidence: 99%

Most Probable Dynamics of the Single-Species with Allee Effect under Jump-diffusion Noise

Tesfay¹,

Yuan²,

Brannan³

2021

Preprint

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For a stochastic single-species model with Allee effect, we compute the most probable phase portrait (MPPP) using the non-local Fokker-Planck equation. This bistable stochastic model is driven by multiplicative Lévy noise as well as white noise. The two fixed points are stable equilibria and one is unstable state between them. In order to study biological behavior of species, we focus on the transition pathways from the extinction state to the upper fixed stable state for the transcription factor activator in a single-species model. By calculating the solution of the non-local Fokker-Planck equation corresponding to the population system of single-species model, the maximum possible path is obtained and corresponding maximum possible stable equilibrium state is determined. We also obtain the Onsager-Machlup (OM) function for the stochastic model, and solve the corresponding most probable paths. Our numerical simulation shows that: (i) The maximum of the stationary density function is located at the most probable stable equilibrium state when non-Gaussian noise is presented in the system; (ii) When the initial value increases from extinction state to the upper stable state, the most probable trajectories converge to the maximal likely equilibrium state (maximizer), in our case lies between 9 and 10; (iii) As time goes on, the most probable paths increase to stable state quickly, and remain a nearly constant level, then approach to the upper stable equilibrium state. These numerical experiment findings help researchers for further experimental study, in order to achieve good knowledge about dynamic systems in biology.

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A Stochastic Pitchfork Bifurcation in Most Probable Phase Portraits

Cited by 29 publications

References 29 publications

Most probable transition pathways and maximal likely trajectories in a genetic regulatory system

Most probable transition pathways and maximal likely trajectories in a genetic regulatory system

State transitions in the Morris-Lecar model under stable Lévy noise

Most Probable Dynamics of the Single-Species with Allee Effect under Jump-diffusion Noise

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