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Aslanidi, Oleg; Mornev, O.A.

doi:10.1023/a:1005119218136

Cited by 32 publications

(17 citation statements)

References 18 publications

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“…penetration of waves upon impact with each other or reflection from non-flux boundaries. This is rather uncharacteristic of the waves in (1) with the exception of narrow parametric regions on the margins of the excitability [3]. However, in systems with cross-diffusion, such "quasi-soliton' behaviour can be observed in large parametric regions [4,5].…”

Section: Introductionmentioning

confidence: 93%

Classification of wave regimes in excitable systems with linear cross diffusion

Tsyganov

Biktashev

2014

Phys. Rev. E

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We consider principal properties of various wave regimes in two selected excitable systems with linear crossdiffusion in one spatial dimension observed at different parameter values. This includes fixed-shape propagating waves, envelope waves, multi-envelope waves, and intermediate regimes appearing as waves propagating fixedshape most of the time but undergoing restructuring from time to time. Depending on parameters, most of these regimes can be with and without the "quasi-soliton" property of reflection of boundaries and penetration through each other. We also present some examples of behaviour of envelope quasi-solitons in two spatial dimensions.

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Section: Introductionmentioning

confidence: 93%

Classification of wave regimes in excitable systems with linear cross diffusion

Tsyganov

Biktashev

2014

Phys. Rev. E

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“…Soliton-like interactions of solitary waves have been observed in some reaction-diffusion systems with excitable kinetics, both in numerics [25,26,27,28,29,30] and in experiments [30,31]. Such interactions are always limited to narrow parameter ranges close to the boundaries between excitable and oscillatory (limit cycle) regimes of the reaction kinetics.…”

Section: Quasisoliton Interaction Of Pulsesmentioning

confidence: 99%

Soliton-like phenomena in one-dimensional cross-diffusion systems: a predator–prey pursuit and evasion example

Tsyganov

Brindley

Holden

et al. 2004

Physica D: Nonlinear Phenomena

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We have studied properties of nonlinear waves in a mathematical model of a predator-prey system with pursuit and evasion. We demonstrate a new type of propagating wave in this system. The mechanism of propagation of these waves essentially depends on the "taxis", represented by nonlinear "cross-diffusion" terms in the mathematical formulation. We have shown that the dependence of the velocity of wave propagation on the taxis has two distinct forms, "parabolic" and "linear". Transition from one form to the other correlates with changes in the shape of the wave profile. Dependence of the propagation velocity on diffusion in this system differs from the square-root dependence typical of reaction-diffusion waves. We demonstrate also that, for systems with negative and positive taxis, for example, pursuit and evasion, there typically exists a large region in the parameter space, where the waves demonstrate quasisoliton interaction: colliding waves can penetrate through each other, and waves can also reflect from impermeable boundaries.

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“…Solitonlike interactions of solitary waves have been observed in some reaction-diffusion systems with excitable kinetics, both in numerics [19][20][21][22][23][24] and in experiments [24,25]. Such interactions are always limited to narrow parameter ranges close to the boundaries between excitable and oscillatory (limit cycle) regimes of the reaction kinetics.…”

mentioning

confidence: 99%

Quasisoliton Interaction of Pursuit-Evasion Waves in a Predator-Prey System

et al. 2003

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We consider a system of partial differential equations describing two spatially distributed populations in a "predator-prey" interaction with each other. The spatial evolution is governed by three processes: positive taxis of predators up the gradient of prey (pursuit), negative taxis of prey down the gradient of predators (evasion), and diffusion resulting from random motion of both species. We demonstrate a new type of propagating wave in this system. The mechanism of propagation of these waves essentially depends on the taxis and is entirely different from waves in a reaction-diffusion system. Unlike typical reaction-diffusion waves, which annihilate on collision, these "taxis" waves can often penetrate through each other and reflect from impermeable boundaries.

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Cited by 32 publications

References 18 publications

Classification of wave regimes in excitable systems with linear cross diffusion

Classification of wave regimes in excitable systems with linear cross diffusion

Soliton-like phenomena in one-dimensional cross-diffusion systems: a predator–prey pursuit and evasion example

Quasisoliton Interaction of Pursuit-Evasion Waves in a Predator-Prey System

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