Persistent random walk of cells involving anomalous effects and random death

Fedotov, Sergei; Tan, Abby; Zubarev, A. Yu.

doi:10.1103/physreve.91.042124

Cited by 31 publications

(42 citation statements)

References 63 publications

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“…This term is responsible for the shift of the superdiffusive Lévy walk toward standard diffusion as the density ρ ± increases. The tempering effect of the repulsion and collision interactions is similar to the tempering due to the random death of walkers [42]. Figure 4 shows the results of numerical simulations corresponding to the rate (2) with f (A ± ) = 1.…”

Section: Emergence Of Lévy Walks In Systems Of Physical Review mentioning

confidence: 85%

“…where K(τ ) is the memory kernel determined by its Laplace transform [42]K(s) 1 T (1 + As μ−1 ) for 1 < μ < 2, as s → 0 (T is mean running time and A is a constant). For the nonlinear case, the expressions for i ± are not known.…”

Section: Emergence Of Lévy Walks In Systems Of Physical Review mentioning

confidence: 99%

“…To implement nonlinear effects we use the structural density approach together with a population density-dependent turning rate. This method has been used by the authors for the analysis of subdiffusive random walks [40,41] and Lévy walks [42,43].…”

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confidence: 99%

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Emergence of Lévy walks in systems of interacting individuals

Fedotov¹,

Korabel²

2017

Phys. Rev. E

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We propose a model of superdiffusive Lévy walk as an emergent nonlinear phenomenon in systems of interacting individuals. The aim is to provide a qualitative explanation of recent experiments [G. Ariel et al., Nat. Commun. 6, 8396 (2015)] revealing an intriguing behavior: swarming bacteria fundamentally change their collective motion from simple diffusion into a superdiffusive Lévy walk dynamics. We introduce microscopic mean-field kinetic equations in which we combine two key ingredients: (1) alignment interactions between individuals and (2) non-Markovian effects. Our interacting run-and-tumble model leads to the superdiffusive growth of the mean-squared displacement and the power-law distribution of run length with infinite variance. The main result is that the superdiffusive behavior emerges as a cooperative effect without using the standard assumption of the power-law distribution of run distances from the inception. At the same time, we find that the collision and repulsion interactions lead to the density-dependent exponential tempering of power-law distributions. This qualitatively explains the experimentally observed transition from superdiffusion to the diffusion of mussels as their density increases [M. [15][16][17][18]. Recently an intriguing behavior of swarming bacteria was found: they fundamentally change their collective motion from simple diffusion into a superdiffusive Lévy walk dynamics [19]. The extraordinary feature of this movement is that the emergence of a superdiffusive motility is a result of the interactions between bacteria rather than the standard mechanism of controlling the individual frequency of tumbling. However, it is still an open question how Lévy walks emerge in systems of interacting self-propelled particles. The current theory of Lévy walk [8] assumes noninteracting particles and power-law distribution of traveled distances from the inception.The collective behavior of large groups of interacting individuals such as bird flocks, fish schools, and the collective migration of cells or bacteria is another rapidly growing area of active matter research [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. There exist two main types of models used for a collective behavior: (1) Lagrangian models describing the movements of self-propelled particles in terms of nonlinear equations for the positions and velocities of all particles [20][21][22][23], and (2) kinetic models involving partial differential equations for the population densities [24][25][26][27][28][29][30]. In this Rapid Communication we are only concerned with the nonlinear kinetic and macroscopic equations. They have been used to describe interactions between individuals and investigate the formation of a large variety of spatiotemporal self-organized aggregations. Most of these models converge to the density-dependent diffusive transport and do not lead to non-Markovian Lévy walks.In this Rapid Communication we propose a kinetic nonlinear Lévy walk model for interacting individuals in which...

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Section: Emergence Of Lévy Walks In Systems Of Physical Review mentioning

confidence: 85%

Section: Emergence Of Lévy Walks In Systems Of Physical Review mentioning

confidence: 99%

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Emergence of Lévy walks in systems of interacting individuals

Fedotov¹,

Korabel²

2017

Phys. Rev. E

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“…Applying the Fourier-Laplace transform to these equations, we find expressions for i + (x,t) and i − (x,t) [21] …”

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confidence: 99%

“…To derive the governing equation for the Lévy walk, we start with the Markovian model involving structural densities with the extra running time variable τ [17][18][19][20][21]. We define the structural PDF's of walker, n + (x,t,τ ), at point x and time t that moves in the right direction, (+), with constant speed v during time τ since the last switching.…”

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confidence: 99%

Single integrodifferential wave equation for a Lévy walk

Fedotov

2016

Phys. Rev. E

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We derive the single integrodifferential wave equation for the probability density function of the position of a classical one-dimensional Lévy walk with continuous sample paths. This equation involves a classical wave operator together with memory integrals describing the spatiotemporal coupling of the Lévy walk. It is valid at all times, not only in the long time limit, and it does not involve any large-scale approximations. It generalizes the well-known telegraph or Cattaneo equation for the persistent random walk with the exponential switching time distribution. Several non-Markovian cases are considered when the particle's velocity alternates at the gamma and power-law distributed random times. In the strong anomalous case we obtain the asymptotic solution to the integrodifferential wave equation. We implement the nonlinear reaction term of Kolmogorov-PetrovskyPiskounov type into our equation and develop the theory of wave propagation in reaction-transport systems involving Lévy diffusion.

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Motility Switching and Front–Back Synchronisation in Polarised Cells

Estrada-Rodriguez

Perthame

2022

J Nonlinear Sci

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The combination of protrusions and retractions in the movement of polarized cells leads to understand the effect of possible synchronisation between the two ends of the cells. This synchronisation, in turn, could lead to different dynamics such as normal and fractional diffusion. Departing from a stochastic single cell trajectory, where a "memory effect" induces persistent movement, we derive a kinetic-renewal system at the mesoscopic scale. We investigate various scenarios with different levels of complexity, where the two ends of the cell move either independently or with partial or full synchronisation. We study the relevant macroscopic limits where we obtain diffusion, drift-diffusion or fractional diffusion, depending on the initial system. This article clarifies the form of relevant macroscopic equations that describe the possible effects of synchronised movement in cells, and sheds light on the switching between normal and fractional diffusion.

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Persistent random walk of cells involving anomalous effects and random death

Cited by 31 publications

References 63 publications

Emergence of Lévy walks in systems of interacting individuals

Emergence of Lévy walks in systems of interacting individuals

Single integrodifferential wave equation for a Lévy walk

Motility Switching and Front–Back Synchronisation in Polarised Cells

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