Pattern selection in the 2D FitzHugh–Nagumo model

We construct square and target patterns solutions of the FitzHugh–Nagumo reaction–diffusion system on planar bounded domains. We study the existence and stability of stationary square and super-square patterns by performing a close to equilibrium asymptotic weakly nonlinear expansion: the emergence of these patterns is shown to occur when the bifurcation takes place through a multiplicity-two eigenvalue without resonance. The system is also shown to support the formation of axisymmetric target patterns whose amplitude equation is derived close to the bifurcation threshold. We present several numerical simulations validating the theoretical results.

     

Bifurcation for a reaction–diffusion system with unilateral and Neumann boundary conditions

We consider a reaction–diffusion system of activator–inhibitor or substrate-depletion type which is subject to diffusion-driven instability if supplemented by pure Neumann boundary conditions. We show by a degree-theoretic approach that an obstacle (e.g. a unilateral membrane) modeled in terms of inequalities, introduces new bifurcation of spatial patterns in a parameter domain where the trivial solution of the problem without the obstacle is stable. Moreover, this parameter domain is rather different from the known case when also Dirichlet conditions are assumed. In particular, bifurcation arises for fast diffusion of activator and slow diffusion of inhibitor which is the difference from all situations which we know.     

Rise and Fall of Periodic Patterns for a Generalized Klausmeier–Gray–Scott Model

In this paper we introduce a conceptual model for vegetation patterns that generalizes the Klausmeier model for semi-arid ecosystems on a sloped terrain (Klausmeier in Science 284:1826–1828, 1999). Our model not only incorporates downhill flow, but also linear or nonlinear diffusion for the water component. To relate the model to observations and simulations in ecology, we first consider the onset of pattern formation through a Turing or a Turing–Hopf bifurcation. We perform a Ginzburg–Landau analysis to study the weakly nonlinear evolution of small amplitude patterns and we show that the Turing/Turing–Hopf bifurcation is supercritical under realistic circumstances. In the second part we numerically construct Busse balloons to further follow the family of stable spatially periodic (vegetation) patterns. We find that destabilization (and thus desertification) can be caused by three different mechanisms: fold, Hopf and sideband instability, and show that the Hopf instability can no longer occur when the gradient of the domain is above a certain threshold. We encounter a number of intriguing phenomena, such as a ‘Hopf dance’ and a fine structure of sideband instabilities. Finally, we conclude that there exists no decisive qualitative difference between the Busse balloons for the model with standard diffusion and the Busse balloons for the model with nonlinear diffusion.     

Analysis of pattern formation in reaction diffusion models with spatially inhomogenous diffusion coefficients

We consider a reaction diffusion system in one spatial dimension in which the diffusion coefficients are spatially varying. We present a non-standard linear analysis for a certain class of spatially varying diffusion coefficients and show that it accurately predicts the behaviour of the full nonlinear system near bifurcation. We show that the steady state solutions exhibit qualitatively different behaviour to that observed in the usual case with constant diffusion coefficients. Specifically, the modified system can generate patterns with spatially varying amplitude and wavelength. Application to chondrogenesis in the limb is discussed.     

Space–time chaos in a system of reaction–diffusion equations

We find conditions for the bifurcation of periodic spatially homogeneous and spatially inhomogeneous solutions of a three-dimensional system of nonlinear partial differential equations describing a soil aggregate model. We show that the transition to diffusion chaos in this model occurs via a subharmonic cascade of bifurcations of stable limit cycles in accordance with the universal Feigenbaum–Sharkovskii–Magnitskii bifurcation theory.     

Bifurcation to locked fronts in two component reaction–diffusion systems

We study invasion fronts and spreading speeds in two component reaction–diffusion systems. Using a variation of Lin's method, we construct traveling front solutions and show the existence of a bifurcation to locked fronts where both components invade at the same speed. Expansions of the wave speed as a function of the diffusion constant of one species are obtained. The bifurcation can be sub or super-critical depending on whether the locked fronts exist for parameter values above or below the bifurcation value. Interestingly, in the sub-critical case numerical simulations reveal that the spreading speed of the PDE system does not depend continuously on the coefficient of diffusion.     

Spatio-temporal dynamics of a reaction–diffusion–advection food-limited system with nonlocal delayed competition and Dirichlet boundary condition

Spatio-temporal dynamics of a reaction–diffusion–advection food-limited population model with nonlocal delayed competition and Dirichlet boundary condition are considered. Existence and stability of the positive spatially nonhomogeneous steady state solution are shown. Existence and direction of the spatially nonhomogeneous steady-state-Hopf bifurcation are proved. Stable spatio-temporal patterns near the steady-state-Hopf bifurcation point are numerically obtained. We also investigate the joint influences of some important parameters including advection rate, food-limited parameter and nonlocal delayed competition on the dynamics. It is found that the effect of advection on Hopf bifurcation is opposite with the corresponding no-flux system. The theoretical results provide some interesting highlights in ecological protection in streams or rivers.     

Multiple bifurcations in a delayed predator–prey diffusion system with a functional response

The present paper is concerned with a delayed predator–prey diffusion system with a Beddington–DeAngelis functional response and homogeneous Neumann boundary conditions. If the positive constant steady state of the corresponding system without delay is stable, by choosing the delay as the bifurcation parameter, we can show that the increase of the delay can not only cause spatially homogeneous Hopf bifurcation at the positive constant steady state but also give rise to spatially heterogeneous ones. In particular, under appropriate conditions, we find that the system has a Bogdanov–Takens singularity at the positive constant steady state, whereas this singularity does not occur for the corresponding system without diffusion. In addition, by applying the normal form theory and center manifold theorem for partial functional differential equations, we give normal forms of Hopf bifurcation and Bogdanov–Takens bifurcation and the explicit formula for determining the properties of spatial Hopf bifurcations.     

Pattern formation of a predator–prey model

In this paper, we analyze the spatial pattern of a predator–prey system. We get the critical line of Hopf and Turing bifurcation in a spatial domain. In particular, the exact Turing domain is given. Also we perform a series of numerical simulations. The obtained results reveal that this system has rich dynamics, such as spotted, stripe and labyrinth patterns, which shows that it is useful to use the reaction–diffusion model to reveal the spatial dynamics in the real world.     

Turing patterns and spatiotemporal patterns in a tritrophic food chain model with diffusion

In this paper, the temporal, spatial, and spatiotemporal patterns of a tritrophic food chain reaction–diffusion model with Holling type II functional response are studied. Firstly, for the model with or without diffusion, we perform a detailed stability and Hopf bifurcation analysis and derive criteria for determining the direction and stability of the bifurcation by the center manifold and normal form theory. Moreover, diffusion-driven Turing instability occurs, which induces spatial inhomogeneous patterns for the reaction–diffusion model. Then, the existence of positive non-constant steady-states of the reaction–diffusion model is established by the Leray–Schauder degree theory and some a priori estimates. Finally, numerical simulations are presented to visualize the complex dynamic behavior.     

Turing patterns induced by cross-diffusion in a predator-prey system in presence of habitat complexity

In this paper, we have investigated the phenomena of Turing pattern formation in a predator-prey model with habitat complexity in presence of cross diffusion. Using the linear stability analysis, the conditions for the existence of stationary pattern and the existence of Hopf bifurcation are obtained. It is shown analytically that the presence of cross diffusion in the system supports the formation of Turing pattern. Two parameter bifurcation analysis are done analytically and corresponding bifurcation diagrams are presented numerically. A series of simulation results are plotted for different biologically meaningful parameter values. Effects of variation of habitat complexity and the predator mortality rate and birth rate of prey on pattern formation are also reported. It is shown that cross-diffusion can lead to a wide variety of spatial and spatiotemporal pattern formation. It is found that the model exhibits spot and stripe pattern, and coexistence of both spot and strip patterns under the zero flux boundary condition. It is observed that cross-diffusion, habitat complexity, birth rate of prey and predator’s mortality rate play a significant role in the pattern formation of a distributed population system of predator-prey type.     

Turing Instability and Pattern Formation for the Lengyel–Epstein System with Nonlinear Diffusion

In this work we study the effect of density dependent nonlinear diffusion on pattern formation in the Lengyel–Epstein system. Via the linear stability analysis we determine both the Turing and the Hopf instability boundaries and we show how nonlinear diffusion intensifies the tendency to pattern formation; in particular, unlike the case of classical linear diffusion, the Turing instability can occur even when diffusion of the inhibitor is significantly slower than activator’s one. In the Turing pattern region we perform the WNL multiple scales analysis to derive the equations for the amplitude of the stationary pattern, both in the supercritical and in the subcritical case. Moreover, we compute the complex Ginzburg–Landau equation in the vicinity of the Hopf bifurcation point as it gives a slow spatio-temporal modulation of the phase and amplitude of the homogeneous oscillatory solution.     

Diffusion-driven instability and bifurcation in the Lengyel–Epstein system

Lengyel–Epstein reaction–diffusion system of the CIMA reaction is considered. We derive the precise conditions on the parameters so that the spatial homogenous equilibrium solution and the spatial homogenous periodic solution become Turing unstable or diffusively unstable. We also perform a detailed Hopf bifurcation analysis to both the ODE and PDE models, and derive conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution.     

Influence of discrete delay on pattern formation in a ratio-dependent prey–predator model

In this paper we explore how the two mechanisms, Turing instability and Hopf bifurcation, interact to determine the formation of spatial patterns in a ratio-dependent prey–predator model with discrete time delay. We conduct both rigorous analysis and extensive numerical simulations. Results show that four types of patterns, cold spot, labyrinthine, chaotic as well as mixture of spots and labyrinthine can be observed with and without time delay. However, in the absence of time delay, the two aforementioned mechanisms have a significant impact on the emergence of spatial patterns, whereas only Hopf bifurcation threshold is derived by considering the discrete time delay as the bifurcation parameter. Moreover, time delay promotes the emergence of spatial patterns via spatio-temporal Hopf bifurcation compared to the non-delayed counterpart, implying the destabilizing role of time delay. In addition, the destabilizing role is prominent when the magnitude of time delay and the ratio of diffusivity are comparatively large.     

一类反应扩散系统的分歧与斑图

本文研究了一类发生在密闭容器中的不可激活的高次自催化反应扩散系统．在适当的条件下,用渐进近似的方法讨论了系统平衡态的稳定范围;用多重尺度的方法证明了当扩散系数λ充分小时,系统出现两种类型的斑图,一类是由Hopf分歧引出的驻波斑图;另一类是由 Pitchfork分歧引出的定波斑图．进一步还讨论了,在分歧点附近,对于大于空间或等于空间波数的小扰动,斑图是局部稳定的,而小于自身空间波数的小扰动,斑图是不稳定的．     

Nonconstant Positive Steady States and Pattern Formation of 1D Prey-Taxis Systems

Prey-taxis is the process that predators move preferentially toward patches with highest density of prey. It is well known to have an important role in biological control and the maintenance of biodiversity. To model the coexistence and spatial distributions of predator and prey species, this paper concerns nonconstant positive steady states of a wide class of prey-taxis systems with general functional responses over 1D domain. Linearized stability of the positive equilibrium is analyzed to show that prey-taxis destabilizes prey–predator homogeneity when prey repulsion (e.g., due to volume-filling effect in predator species or group defense in prey species) is present, and prey-taxis stabilizes the homogeneity otherwise. Then, we investigate the existence and stability of nonconstant positive steady states to the system through rigorous bifurcation analysis. Moreover, we provide detailed and thorough calculations to determine properties such as pitchfork and turning direction of the local branches. Our stability results also provide a stable wave mode selection mechanism for thee reaction–advection–diffusion systems including prey-taxis models considered in this paper. Finally, we provide numerical studies of prey-taxis systems with Holling–Tanner kinetics to illustrate and support our theoretical findings. Our numerical simulations demonstrate that the \(2\times 2\) prey-taxis system is able to model the formation and evolution of various striking patterns, such as spikes, periodic oscillations, and coarsening even when the domain is one-dimensional. These dynamics can model the coexistence and spatial distributions of interacting prey and predator species. We also give some insights on how system parameters influence pattern formation in these models.     

A finite difference method for modeling the formation of animal coat patterns

We consider a nonlinear reaction–diffusion system for the formation of animal coat patterns proposed by J.D. Murray. We approximate the nonlinear partial differential equations by an extension of the Peaceman–Rachford finite difference method for linear equations. We then run experiments where we execute our method until steady-state is reached to obtain a variety of animal coat patterns.     

Pattern formation in the Gray–Scott model

We investigate an elliptic system that arises in cubic autocatalytic reactions known as the Gray–Scott model. Complicated patterns were reported by Pearson in a numerical study of this system. We produce the bifurcation analysis to support the existing numerical evidence for patterns. Specifically bifurcation results and C² bounds for nonuniform steady states are derived.     

Period-three route to chaos induced by a cyclic-fold bifurcation in passive dynamic walking of a compass-gait biped robot

This paper presents a study of the passive dynamic walking of a compass-gait biped robot as it goes down an inclined plane. This biped robot is a two-degrees-of-freedom mechanical system modeled by an impulsive hybrid nonlinear dynamics with unilateral constraints. It is well-known to possess periodic as well as chaotic gaits and to possess only one stable gait for a given set of parameters. The main contribution of this paper is the finding of a window in the parameters space of the compass-gait model where there is multistability. Using constraints of a grazing bifurcation on the basis of a shooting method and the Davidchack–Lai scheme, we show that, depending on initial conditions, new passive walking patterns can be observed besides those already known. Through bifurcation diagrams and Floquet multipliers, we show that a pair of stable and unstable period-three gait patterns is generated through a cyclic-fold bifurcation. We show also that the stable period-three orbit generates a route to chaos.     

Phase differences in reaction-diffusion-advection systems and applications to morphogenesis

The authors study the effect of advection on reaction-diffusionpatterns. It is shown that the addition of advection to a two-variablereaction–diffusion system with periodic boundary conditionsresults in the appearance of a phase difference between thepatterns of the two variables which depends on the differencebetween the advection coefficients. The spatial patterns movelike a travelling wave with a fixed velocity which depends onthe sum of the advection coefficients. By a suitable choiceof advection coefficients, the solution can be made stationaryin time. In the presence of advection a continuous change inthe diffusion coefficients can result in two Turing-type bifurcationsas the diffusion ratio is varied, and such a bifurcation canoccur even when the inhibitor species does not diffuse. It isalso shown that the initial mode of bifurcation for a givendomain size depends on both the advection and diffusion coefficients.These phenomena are demonstrated in the numerical solution ofa particular reaction–diffusion system, and finally apossible application of the results to pattern formation inDrosophila larvae is discussed.