Bistable Reaction-diffusion Systems Research Articles

Entire solutions of bistable reaction-diffusion systems in unbounded domain with multiple cylindrical branches

Entire solutions of bistable reaction-diffusion systems in unbounded domain with multiple cylindrical branches

Pulse Dynamics in a Bistable Reaction-Diffusion System with Chemotaxis

We consider pulse dynamics in a bistable reaction-diffusion system with chemotaxis. We derive the ordinary differential equation of interfaces by applying the multiple scales method to the reaction-diffusion system for examining the effect of the chemotaxis on pulse dynamics in one dimension. The stability of the standing pulse is considered by two different methods, and the applicability of the methods is demonstrated. The chemotaxis influences the Hopf and drift bifurcations and the collision of two traveling pulses. It also enlarges the bifurcation point and enhances the repulsive force between pulses so that the parameter region of the elastic collision becomes large. Although the ordinary differential equation of interfaces can describe the elastic collision, it cannot describe the pair annihilation of pulses caused by the collision. The conditions for the reliable calculation of pulse collision are discussed.

Exact travelling solution for a reaction-diffusion system with a piecewise constant production supported by a codimension-1 subspace

<p style='text-indent:20px;'>A generalisation of reaction diffusion systems and their travelling solutions to cases when the productive part of the reaction happens only on a surface in space or on a line on plane but the degradation and the diffusion happen in bulk are important for modelling various biological processes. These include problems of invasive species propagation along boundaries of ecozones, problems of gene spread in such situations, morphogenesis in cavities, intracellular reaction etc. Piecewise linear approximations of reaction terms in reaction-diffusion systems often result in exact solutions of propagation front problems. This article presents an exact travelling solution for a reaction-diffusion system with a piecewise constant production restricted to a codimension-1 subset. The solution is monotone, propagates with the unique constant velocity, and connects the trivial solution to a nontrivial nonhomogeneous stationary solution of the problem. The properties of the solution closely parallel the properties of monotone travelling solutions in classical bistable reaction-diffusion systems.</p>

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.

Existence of Waves for a Bistable Reaction–Diffusion System with Delay

Existence of travelling waves is studied for a delay reaction–diffusion system of equations describing the distribution of viruses and immune cells in the tissue. The proof uses the Leray-Schauder method based on the topological degree for elliptic operators in unbounded domains and on a priori estimates of solutions in weighted spaces.

Reduction approach to the dynamics of interacting front solutions in a bistable reaction–diffusion system and its application to heterogeneous media

The dynamics of pulse solutions in a bistable reaction-diffusion system are studied analytically by reducing partial differential equations (PDEs) to finite-dimensional ordinary differential equations (ODEs). For the reduction, we apply the multiple-scales method to the mixed ODE-PDE system obtained by taking a singular limit of the PDEs. The reduced equations describe the interface motion of a pulse solution formed by two interacting front solutions. This motion is in qualitatively good agreement with that observed for the original PDE system. Furthermore, it is found that the reduction not only facilitates the analytical study of the pulse solution, especially the specification of the onset of local bifurcations, but also allows us to elucidate the global bifurcation structure behind the pulse behavior. As an application, the pulse dynamics in a heterogeneous bump-type medium are explored numerically and analytically. The reduced ODEs clarify the transition mechanisms between four pulse behaviors that occur at different parameter values.

Effects of transport memory in wave fronts in a bistable reaction–diffusion system

In this paper, we analyze transport memory effects in bistable reaction−diffusion systems. Traveling wave fronts are obtained for two interesting cases: (i)The nonlinear reaction term is treated without any approximation by factorization method. We find that transport memory effects appear to play a key role as it prevents the concentration or the amplitude from taking negative values. These memory effects enter the dynamics of the reaction diffusion systems through their influence on the speed of the traveling wave fronts. (ii) The nonlinear reaction term is replaced by a piecewise linear approximate form. We obtain a system of differential equations describing the three damped harmonic oscillators, one of which has a negative mass and the others positive masses.

Existence of waves for a reaction–diffusion–dispersion system

Existence of travelling waves is studied for a bistable reaction–diffusion system of equations with linear integral terms (dispersion) and with some conditions on the nonlinearity. The proof is based on the Leray–Schauder method using the topological degree theory for Fredholm and proper operators with the zero index and a priori estimates of solutions in properly chosen weighted spaces.

Entire solutions of monotone bistable reaction–diffusion systems in $$\pmb {\mathbb {R}}^N$$ R N

This paper is concerned with entire solutions of monotone bistable reaction–diffusion systems in $$\mathbb {R}^N$$ . We first show that there is a new entire solution for bistable reaction–diffusion systems which behaves as three moving planar fronts as time goes to $$-\,\infty $$ and as a V-shaped traveling front as time goes to $$\infty $$ . Then we address the speed of the interfaces corresponding to the entire solution by appealing to the super-sub solutions technique. Finally, we apply our results to two important models in biology.

Sampling rare events in stochastic reaction-diffusion systems within trajectory looping.

In bistable reaction-diffusion systems, transitions between stable states typically occur on timescales orders of magnitude longer than the chemical equilibration time. Estimation of transition rates within explicit Brownian dynamics simulations is computationally prohibitively costly. We present a method that exploits a single trajectory, generated by a prior simulation of diffusive motions of molecules, to sample chemical kinetic processes on timescales several orders of magnitude longer than the duration of the diffusive trajectory. In this approach, we "loop" the diffusive trajectory by transferring chemical states of the molecules from the last to the first time step of the trajectory. Trajectory looping can be applied to enhance sampling of rare events in biochemical systems in which the number of reacting molecules is constant, as in cellular signal transduction pathways. As an example, we consider a bistable system of autophosphorylating kinases, for which we calculate state-to-state transition rates and traveling wave velocities. We provide an open-source implementation of the method.

State space reconstruction of spatially extended systems and of time delayed systems from the time series of a scalar variable.

The space-time representation of high-dimensional dynamical systems that have a well defined characteristic time scale has proven to be very useful to deepen the understanding of such systems and to uncover hidden features in their output signals. By using the space-time representation many analogies between one-dimensional spatially extended systems (1D SESs) and time delayed systems (TDSs) have been found, including similar pattern formation and propagation of localized structures. An open question is whether such analogies are limited to the space-time representation, or it is also possible to recover similar evolutions in a low-dimensional pseudo-space. To address this issue, we analyze a 1D SES (a bistable reaction-diffusion system), a scalar TDS (a bistable system with delayed feedback), and a non-scalar TDS (a model of two delay-coupled lasers). In these three examples, we show that we can reconstruct the dynamics in a three-dimensional phase space, where the evolution is governed by the same polynomial potential. We also discuss the limitations of the analogy between 1D SESs and TDSs.

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.

Initiation of reaction–diffusion waves of blood coagulation

Formation of blood clot in response to the vessel damage is triggered by the complex network of biochemical reactions of the coagulation cascade. The process of clot growth can be modeled as a traveling wave solution of the bistable reaction–diffusion system. The critical value of the initial condition which leads to convergence of the solution to the traveling wave corresponds to the pulse solution of the corresponding stationary problem. In the current study we prove the existence of the pulse solution for the stationary problem in the model of the main reactions of the blood coagulation cascade using the Leray–Schauder method.

Stability of planar traveling fronts in bistable reaction–diffusion systems

This paper is concerned with the multidimensional stability of planar traveling fronts in bistable reaction–diffusion systems. It is first shown that planar traveling fronts are asymptotically stable under spatially decaying initial perturbations by appealing to the comparison principle and super-subsolution method. In particular, if the perturbations belong to L1(Rn−1) in a certain sense, we obtain a convergence rate like t−n−12. Then we show that the solution of the Cauchy problem converges to the planar traveling front with rate t−n+14 for a spatially non-decaying perturbation with the help of semigroup theory. Finally, we prove that there exists a solution oscillating permanently between two planar traveling fronts, which indicates that planar traveling fronts are not always asymptotically stable in multidimensional space under general bounded perturbations.

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.

Front and Turing patterns induced by Mexican-hat–like nonlocal feedback

We consider the effects of a Mexican-hat–shaped nonlocal spatial coupling, i.e., symmetric long-range inhibition superimposed with short-range excitation, upon front propagation in a model of a bistable reaction-diffusion system. We show that the velocity of front propagation can be controlled up to a certain coupling strength beyond which spatially periodic patterns, such as Turing patterns or coexistence of spatially homogeneous solutions and Turing patterns, may be induced. This behaviour is investigated through a linear stability analysis of the spatially homogeneous steady states and numerical investigations of the full nonlinear equations in dependence upon the nonlocal coupling strength and the ratio of the excitatory and inhibitory coupling ranges.

Stationary Waves on the Sphere

In this paper, we investigate stationary waves on the sphere using the bistable reaction-diffusion system. The motivation of this study arises from the study of activation waves of B cells in immune systems. We analytically establish (i) the existence and uniqueness of stationary waves; (ii) the limiting wave profile for small diffusivity of diffusing species; and (iii) the stability of the constructed stationary waves. The stability result may suggest the critical role of stationary waves in the determination of initial data for initiating propagating waves on the sphere, which is consistent with the numerical results for the B-cell activation model.

Dynamics of reaction-diffusion patterns controlled by asymmetric nonlocal coupling as a limiting case of differential advection.

A one-component bistable reaction-diffusion system with asymmetric nonlocal coupling is derived as a limiting case of a two-component activator-inhibitor reaction-diffusion model with differential advection. The effects of asymmetric nonlocal couplings in such a bistable reaction-diffusion system are then compared to the previously studied case of a system with symmetric nonlocal coupling. We carry out a linear stability analysis of the spatially homogeneous steady states of the model and numerical simulations of the model to show how the asymmetric nonlocal coupling controls and alters the steady states and the front dynamics in the system. In a second step, a third fast reaction-diffusion equation is included which induces the formation of more complex patterns. A linear stability analysis predicts traveling waves for asymmetric nonlocal coupling, in contrast to a stationary Turing patterns for a system with symmetric nonlocal coupling. These findings are verified by direct numerical integration of the full equations with nonlocal coupling.

An Appropriate Bounded Invariant Region for a Bistable Reaction-Diffusion Model of the Caspase-3/8 Feedback Loop

The apoptotic caspase-3/8 feedback loop describes the core of the extrinsic pro-apoptotic signaling pathway, an essential part of apoptosis. Latter is a prototype of the programmed cell death, which enables organisms to remove damaged or infected cells. The reaction network of the caspase-3/8 feedback loop in a single cell is modeled by a reaction-diffusion system, which shows a bistable behavior. In this work, we present an appropriate bounded invariant region for the bistable reaction-diffusion system in order to theoretically confirm that diffusion rapidly balances the concentrations of the different caspase types. This justifies the decomposition of the dynamics into a diffusion dominated part on a very short time scale and a pure reaction driven dynamics on a large time scale.

Dynamics of two interfaces in a hybrid system with jump-type heterogeneity

We consider the dynamics of two interfaces that interact through a continuous medium with spatial heterogeneity. The dynamics of interface positions is governed by ordinary differential equations (ODEs), whereas that of the continuous field by a partial differential equation. The resulting mixed ODE-PDE system, which we call a hybrid system (HS), is derived as a singular limit of a certain bistable reaction-diffusion system (PDE), describing the dynamics of traveling pulses of front-back type. First, the traveling pulse dynamics in the heterogeneous medium is numerically studied both for the bistable reaction-diffusion system and for the hybrid system. Then, the hybrid system is analyzed to clarify the underlying mechanisms for the pulse behavior observed. In particular, we carry out a center manifold reduction for the hybrid system, which reveals not only the supercriticality of Hopf bifurcations but also the mechanism for sliding motion of an oscillating pulse observed in the heterogeneous medium.