919 research outputs found
Pursuit-evasion predator-prey waves in two spatial dimensions
We consider a spatially distributed population dynamics model with excitable
predator-prey dynamics, where species propagate in space due to their taxis
with respect to each other's gradient in addition to, or instead of, their
diffusive spread. Earlier, we have described new phenomena in this model in one
spatial dimension, not found in analogous systems without taxis: reflecting and
self-splitting waves. Here we identify new phenomena in two spatial dimensions:
unusual patterns of meander of spirals, partial reflection of waves, swelling
wavetips, attachment of free wave ends to wave backs, and as a result, a novel
mechanism of self-supporting complicated spatio-temporal activity, unknown in
reaction-diffusion population models.Comment: 15 pages, 15 figures, submitted to Chao
Quasisolitons in self-diffusive excitable systems, or Why asymmetric diffusivity does not violate the Second Law
Solitons, defined as nonlinear waves which can reflect from boundaries or
transmit through each other, are found in conservative, fully integrable
systems. Similar phenomena, dubbed quasi-solitons, have been observed also in
dissipative, "excitable" systems, either at finely tuned parameters (near a
bifurcation) or in systems with cross-diffusion. Here we demonstrate that
quasi-solitons can be robustly observed in excitable systems with excitable
kinetics and with self-diffusion only. This includes quasi-solitons of fixed
shape (like KdV solitons) or envelope quasi-solitons (like NLS solitons). This
can happen in systems with more than two components, and can be explained by
effective cross-diffusion, which emerges via adiabatic elimination of a fast
but diffusing component. We describe here a reduction procedure can be used for
the search of complicated wave regimes in multi-component, stiff systems by
studying simplified, soft systems.Comment: 11 pages, 2 figures, as accepted to Scientific Reports on 2016/07/0
Soliton-like phenomena in one-dimensional cross-diffusion systems: a predator-prey pursuit and evasion example
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.Comment: 15 pages, 18 figures, submitted to Physica
Envelope quasisolitons in dissipative systems with cross-diffusion
Copyright © 2011 American Physical SocietyJournal ArticleWe consider two-component nonlinear dissipative spatially extended systems of reaction-cross-diffusion type. Previously, such systems were shown to support "quasisoliton" pulses, which have a fixed stable structure but can reflect from boundaries and penetrate each other. Herein we demonstrate a different type of quasisolitons, with a phenomenology resembling that of the envelope solitons in the nonlinear Schrödinger equation: spatiotemporal oscillations with a smooth envelope, with the velocity of the oscillations different from the velocity of the envelope
Classification of wave regimes in excitable systems with linear cross diffusion
Copyright © 2014 American Physical SocietyWe consider principal properties of various wave regimes in two selected excitable systems with linear cross diffusion in one spatial dimension observed at different parameter values. This includes fixed-shape propagating waves, envelope waves, multienvelope waves, and intermediate regimes appearing as waves propagating at a fixed shape most of the time but undergoing restructuring from time to time. Depending on parameters, most of these regimes can be with and without the "quasisoliton" property of reflection of boundaries and penetration through each other. We also present some examples of the behavior of envelope quasisolitons in two spatial dimensions.Russian Foundation for Basic Research (RFBR
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