Envelope quasisolitons in dissipative systems with cross-diffusion

Solitary pulses and periodic wave trains in a bistable FitzHugh-Nagumo model with cross diffusion and cross advection

We describe analytically, and simulate numerically, traveling waves with oscillatory tails in a bistable, piecewise-linear reaction-diffusion-advection system of the FitzHugh-Nagumo type with linear cross-diffusion and cross-advection terms of opposite signs. We explore the dynamics of two wave types, namely, solitary pulses and their infinite sequences, i.e., periodic wave trains. The effects of cross diffusion and cross advection on wave profiles and speed of propagation are analyzed. For pulses, in the speed diagram splitting of a curve into several branches occurs, corresponding to different waves (wave branching). For wave trains, in the dispersion relation diagram there are oscillatory curves and the discontinuous curve of an isola with two branches. The corresponding wave trains have symmetric or asymmetric profiles. Numerical simulations show that for large values of the period there exist two wave trains, which come closer and closer together and are subject to fusion into one when the value of the period is decreasing. Other types of waves are also briefly discussed.

Nonlinear waves in a quintic FitzHugh-Nagumo model with cross diffusion: Fronts, pulses, and wave trains

We study a tristable piecewise-linear reaction-diffusion system, which approximates a quintic FitzHugh-Nagumo model, with linear cross-diffusion terms of opposite signs. Basic nonlinear waves with oscillatory tails, namely, fronts, pulses, and wave trains, are described. The analytical construction of these waves is based on the results for the bistable case [Zemskov et al., Phys. Rev. E 77, 036219 (2008) and Phys. Rev. E 95, 012203 (2017) for fronts and for pulses and wave trains, respectively]. In addition, these constructions allow us to describe novel waves that are specific to the tristable system. Most interesting is the pulse solution with a zigzag-shaped profile, the bright-dark pulse, in analogy with optical solitons of similar shapes. Numerical simulations indicate that this wave can be stable in the system with asymmetric thresholds; there are no stable bright-dark pulses when the thresholds are symmetric. In the latter case, the pulse splits up into a tristable front and a bistable one that propagate with different speeds. This phenomenon is related to a specific feature of the wave behavior in the tristable system, the multiwave regime of propagation, i.e., the coexistence of several waves with different profile shapes and propagation speeds at the same values of the model parameters.

Oscillatory multipulsons: Dissipative soliton trains in bistable reaction-diffusion systems with cross diffusion of attractive-repulsive type

One-dimensional localized sequences of bound (coupled) traveling pulses, wave trains with a finite number of pulses, are described in a piecewise-linear reaction-diffusion system of the FitzHugh-Nagumo type with linear cross-diffusion terms of opposite signs. The simplest case of two bound pulses, the paired-pulse waves (pulse pairs), is solved analytically. The solutions contain oscillatory tails in the wave profiles so that the pulse pairs consist of a double-peak core and wavy edges. Several pulse pairs with different profile shapes and propagation speeds can appear for the same parameter values of the model when the cross diffusion is dominant. The more general case of many bound pulses, multipulse waves, is studied numerically. It is shown that, dependent on the values of the cross-diffusion coefficients, the multipulse waves upon collision can pass through one another with unchanged size and shape, exhibiting soliton behavior. Moreover, multipulse collisions with the system boundaries can generate a rich variety of wave transformations: the transition from the multipulse waves to pulse-front waves and further to simple fronts or to annihilation as well the transition to solitary pulses or to multipulse waves with lower numbers of pulses. Analytical and numerical results for the pulse pairs agree well with each other.

Multifront regime of a piecewise-linear FitzHugh-Nagumo model with cross diffusion

Oscillatory reaction-diffusion fronts are described analytically in a piecewise-linear approximation of the FitzHugh-Nagumo equations with linear cross-diffusion terms, which correspond to a pursuit-evasion situation. Fundamental dynamical regimes of front propagation into a stable and into an unstable state are studied, and the shape of the waves for both regimes is explored in detail. We find that oscillations in the wave profile may either be negligible due to rapid attenuation or noticeable if the damping is slow or vanishes. In the first case, we find fronts that display a monotonic profile of the kink type, whereas in the second case the oscillations give rise to fronts with wavy tails. Further, the oscillations may be damped with exponential decay or undamped so that a saw-shaped pattern forms. Finally, we observe an unexpected feature in the behavior of both types of the oscillatory waves: the coexistence of several fronts with different profile shapes and propagation speeds for the same parameter values of the model, i.e., a multifront regime of wave propagation.

Response function framework for the dynamics of meandering or large-core spiral waves and modulated traveling waves

In many oscillatory or excitable systems, dynamical patterns emerge which are stationary or periodic in a moving frame of reference. Examples include traveling waves or spiral waves in chemical systems or cardiac tissue. We present a unified theoretical framework for the drift of such patterns under small external perturbations, in terms of overlap integrals between the perturbation and the adjoint critical eigenfunctions of the linearized operator (i.e., response functions). For spiral waves, the finite radius of the spiral tip trajectory and spiral wave meander are taken into account. Different coordinate systems can be chosen, depending on whether one wants to predict the motion of the spiral-wave tip, the time-averaged tip path, or the center of the meander flower. The framework is applied to analyze the drift of a meandering spiral wave in a constant external field in different regimes.

Oscillatory pulse-front waves in a reaction-diffusion system with cross diffusion

We explore traveling waves with oscillatory tails in a bistable piecewise linear reaction-diffusion system of the FitzHugh-Nagumo type with linear cross diffusion. These waves differ fundamentally from the standard simple fronts of the kink type. In contrast to kinks, the waves studied here have a complex shape profile with a front-back-front (a pulse-front) pattern. The characteristic feature of such pulse-front waves is a hybrid type of the speed diagram, which on the one hand reflects the typical dynamical behavior of the fronts in the FitzHugh-Nagumo model, related to the nonequilibrium Ising-Bloch bifurcation, and on the other hand exhibits also the solitary pulse scenario where several waves appear simultaneously with different speeds of propagation. We describe analytically the wave profiles and heteroclinic trajectories in the phase plane and discuss their morphology and transformation. The phenomena of wave formation and propagation are also studied by numerical simulations of the model partial differential equations. These simulations support the view that the pulse-front waves are constructed of fronts and pulses.

Oscillatory pulses and wave trains in a bistable reaction-diffusion system with cross diffusion

We study waves with exponentially decaying oscillatory tails in a reaction-diffusion system with linear cross diffusion. To be specific, we consider a piecewise linear approximation of the FitzHugh-Nagumo model, also known as the Bonhoeffer-van der Pol model. We focus on two types of traveling waves, namely solitary pulses that correspond to a homoclinic solution, and sequences of pulses or wave trains, i.e., a periodic solution. The effect of cross diffusion on wave profiles and speed of propagation is analyzed. We find the intriguing result that both pulses and wave trains occur in the bistable cross-diffusive FitzHugh-Nagumo system, whereas only fronts exist in the standard bistable system without cross diffusion.

Quasisolitons in self-diffusive excitable systems, or Why asymmetric diffusivity obeys 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.

Wavy fronts in a hyperbolic FitzHugh-Nagumo system and the effects of cross diffusion

We study a hyperbolic version of the FitzHugh-Nagumo (also known as the Bonhoeffer-van der Pol) reaction-diffusion system. To be able to obtain analytical results, we employ a piecewise linear approximation of the nonlinear kinetic term. The hyperbolic version is compared with the standard parabolic FitzHugh-Nagumo system. We completely describe the dynamics of wavefronts and discuss the properties of the speed equation. The nonequilibrium Ising-Bloch bifurcation of traveling fronts is found to occur in the hyperbolic case as well as in the parabolic system. Waves in the hyperbolic case typically propagate with lower speeds, in absolute value, than waves in the parabolic one. We find the interesting feature that the hyperbolic and parabolic front trajectories coincide in the phase plane for the FitzHugh-Nagumo model with a diagonal diffusion matrix, which is the case of self-diffusion, and differ for the system with cross diffusion.

Classification of wave regimes in excitable systems with linear cross diffusion

We 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.

Spatial dynamics in a predator-prey model with herd behavior

In this paper, a spatial predator-prey model with herd behavior in prey population and quadratic mortality in predator population is investigated. By the linear stability analysis, we obtain the condition for stationary pattern. Moreover, using standard multiple-scale analysis, we establish the amplitude equations for the excited modes, which determine the stability of amplitudes towards uniform and inhomogeneous perturbations. By numerical simulations, we find that the model exhibits complex pattern replication: spotted pattern, stripe pattern, and coexistence of the two. The results may enrich the pattern dynamics in predator-prey models and help us to better understand the dynamics of predator-prey interactions in a real environment.

Spatial dynamics in a predator-prey model with Beddington-DeAngelis functional response

In this paper spatial dynamics of the Beddington-DeAngelis predator-prey model is investigated. We analyze the linear stability and obtain the condition of Turing instability of this model. Moreover, we deduce the amplitude equations and determine the stability of different patterns. In Turing space, we found that this model has coexistence of H(0) hexagon patterns and stripe patterns, H(π) hexagon patterns, and H(0) hexagon patterns. To better describe the real ecosystem, we consider the ecosystem as an open system and take the environmental noise into account. It is found that noise can decrease the number of the patterns and make the patterns more regular. What is more, noise can induce two kinds of typical pattern transitions. One is from the H(π) hexagon patterns to the regular stripe patterns, and the other is from the coexistence of H(0) hexagon patterns and stripe patterns to the regular stripe patterns. The obtained results enrich the finding in the Beddington-DeAngelis predator-prey model well.