Analysis of spatiotemporal patterns in a single species reaction–diffusion model with spatiotemporal delay

Employing the theories of Turing bifurcation in the partial differential equations, we investigate the dynamical behavior of a single species reaction–diffusion model with spatiotemporal delay. The linear stability and the conditions for the occurrence of Turing bifurcation in this model are obtained. Moreover, the amplitude equations which represent different spatiotemporal patterns are also obtained near the Turing bifurcation point by using multiple scale method. In Turing space, it is found that the spatiotemporal distributions of the density of this researched species have spots pattern and stripes pattern. Finally, some numerical simulations corresponding to the different spatiotemporal patterns are given to verify our theoretical analysis.     

Bifurcations and spatiotemporal patterns in a ratio‐dependent diffusive Holling‐Tanner system with time delay

In this paper, we concentrate on the spatiotemporal patterns of a delayed reaction‐diffusion Holling‐Tanner model with Neumann boundary conditions. In particular, the time delay that is incorporated in the negative feedback of the predator density is considered as one of the principal factors to affect the dynamic behavior. Firstly, a global Turing bifurcation theorem for τ = 0 and a local Turing bifurcation theorem for τ > 0 are given. Then, further considering the degenerated situation, we derive the existence of Bogdanov‐Takens bifurcation and Turing‐Hopf bifurcation. The normal form method is used to study the explicit dynamics near the Turing‐Hopf singularity. It is shown that a pair of stable nonconstant steady states (stripe patterns) and a pair of stable spatially inhomogeneous periodic solutions (spot patterns) could be bifurcated from a positive equilibrium. Moreover, the Turing‐Turing‐Hopf–type spatiotemporal patterns, that is, a subharmonic phenomenon with two spatial wave numbers and one temporal frequency, are also found and explained theoretically. Our results imply that the interaction of Turing and Hopf instabilities can be considered as the simplest mechanism for the appearance of complex spatiotemporal dynamics.     

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.     

Existence,uniqueness and stability analysis of recurrent neural networks with time delay in the leakage term under impulsive perturbations

In this paper, a class of recurrent neural networks with time delay in the leakage term under impulsive perturbations is considered. First, a sufficient condition is given to ensure the global existence and uniqueness of the solution for the addressed neural networks by using the contraction mapping theorem. Then, we present some sufficient conditions to guarantee the existence, uniqueness and global asymptotic stability of the equilibrium point by using topological degree theory, Lyapunov–Kravsovskii functionals and some analysis techniques. The proposed results, which do not require the boundedness, differentiability and monotonicity of the activation functions, can be easily checked via the linear matrix inequality (LMI) control toolbox in MATLAB. Moreover, they indicate that the stability behavior of neural networks is very sensitive to the time delay in the leakage term. In the absence of leakage delay, the results obtained are also new results. Finally, two numerical examples are given to show the effectiveness of the proposed results.     

Passivity analysis for neural networks of neutral type with Markovian jumping parameters and time delay in the leakage term

In this paper, the problem of passivity analysis is investigated for neutral type neural networks with Markovian jumping parameters and time delay in the leakage term. The delay is assumed to be time-varying and belong to a given interval, which means that the lower and upper bounds of interval time-varying delays are available. By constructing proper Lyapunov–Krasovskii functional, new delay-dependent passivity conditions are derived in terms of linear matrix inequalities (LMIs). Moreover, it is well known that the passivity behavior of neural networks is very sensitive to the time delay in the leakage term. Finally, three numerical examples are given to show the effectiveness and less conservatism of the proposed method.     

Stability and persistence of two-dimensional patterns

The canonical equations for evolution of the amplitude order parameters order parameters describing the nonlinear development and persistence of two-dimensional three-mode spatial patterns generated by Turing instability in dissipative systems are considered. The stability conditions for persistent hexagonal patterns are generalized, and the conditions under which patterns are either disrupted, exhibit bounded quasiperiodic or chaotic behavior, or decay under nonlinear evolution are derived. These conditions are applied to the specific three-mode amplitude evolution equations derived for the Schnakenberg model and a delay predator system in Chapter 3. Numerical results are presented for the persistence, disruption and decay of patterns in these systems, including fairly detailed comparisons with simulations results for the Snackenberg model.     

Turing pattern formation in a predator–prey–mutualist system

In this paper, we develop a theoretical framework about spatial patterns in a three-species predator–prey–mutualist system with cross-diffusion. We concentrate on three aspects of Turing pattern formation: (1) what conditions enable the occurrence of Turing patterns? (2) what are the underlying mechanisms? (3) what are the corresponding configurations? For the first two questions, by use of the stability analysis for the positive uniform solution and the Leray–Schauder degree theory, we prove that under some conditions, the system admits at least a nonhomogeneous stationary solution. For the third question, we carry out numerical simulations for a Turing pattern, and we show that the configurations of Turing pattern are stable spotted patterns, which resemble a real ecosystem.     

Stationary patterns of a predator-prey model with spatial effect

In this paper, spatial patterns of predator-prey model with cross diffusion are investigated. The Hopf and Turing bifurcation critical line in a spatial domain are obtained by using mathematical theory. Moreover, exact Turing space is given in two parameters space. Our results reveal that cross diffusion can induce stationary patterns, which may be useful to help us better understand the dynamics of the real ecosystems.     

Stability Switches, Hopf Bifurcations, and Spatio-temporal Patterns in a Delayed Neural Model with Bidirectional Coupling

The dynamical behavior of a delayed neural network with bi-directional coupling is investigated by taking the delay as the bifurcating parameter. Some parameter regions are given for conditional/absolute stability and Hopf bifurcations by using the theory of functional differential equations. As the propagation time delay in the coupling varies, stability switches for the trivial solution are found. Conditions ensuring the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. We also discuss the spatio-temporal patterns of bifurcating periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. In particular, we obtain that the spatio-temporal patterns of bifurcating periodic oscillations will alternate according to the change of the propagation time delay in the coupling, i.e., different ranges of delays correspond to different patterns of neural activities. Numerical simulations are given to illustrate the obtained results and show the existence of bursts in some interval of the time for large enough delay.     

Global asymptotic stability of BAM fuzzy cellular neural networks with time delay in the leakage term,discrete and unbounded distributed delays

In this article, a class of bidirectional associative memory (BAM) fuzzy cellular neural networks (FCNNs) with time delay in the leakage term, discrete and unbounded distributed delays is formulated to study the global asymptotic stability. This approach is based on the Lyapunov–Krasovskii functional with free-weighting matrices. Using linear matrix inequality (LMI), a new set of stability criteria for BAM FCNNs with time delay in the leakage term, discrete and unbounded distributed delays is obtained. Also, the stability behavior of BAM FCNNs is very sensitive to the time delay in the leakage term. In the absence of a leakage term, a new stability criteria is also derived by employing a Lyapunov–Krasovskii functional and using the LMI approach. Our results establish a new set of stability criteria for BAM FCNNs with discrete and unbounded distributed delays. Numerical examples are provided to illustrate the effectiveness of the developed techniques.     

Spatiotemporal patterns induced by delay and cross-fractional diffusion in a predator-prey model describing intraguild predation

In this article, we study a reaction-diffusion predator-prey model that describes intraguild predation. We mainly consider the effects of time delay and cross-fractional diffusion on dynamical behavior. By using delay as the bifurcation parameter, we perform a detailed Hopf bifurcation analysis and derive the algorithm for determining the direction and stability of the bifurcating periodic solutions. We also demonstrate that proper cross-fractional diffusion can induce Turing pattern, and the smaller the order of fractional diffusion is, the more easily Turing pattern is able to occur.     

Exponential synchronization of stochastic fuzzy cellular neural networks with time delay in the leakage term and reaction-diffusion

Separate studies have been published on the stability of fuzzy cellular neural networks with time delay in the leakage term and synchronization issue of coupled chaotic neural networks with stochastic perturbation and reaction-diffusion effects. However, there have not been studies that integrate the two fields. Motivated by the achievements from both fields, this paper considers the exponential synchronization problem of coupled chaotic fuzzy cellular neural networks with stochastic noise perturbation, time delay in the leakage term and reaction-diffusion effects using linear feedback control. Lyapunov stability theory combining with stochastic analysis approaches are employed to derive sufficient criteria ensuring the coupled chaotic fuzzy neural networks to be exponentially synchronized. This paper also presents an illustrative example and uses simulated results of this example to show the feasibility and effectiveness of the proposed scheme.     

Turing Instabilities at Hopf Bifurcation

Turing–Hopf instabilities for reaction-diffusion systems provide spatially inhomogeneous time-periodic patterns of chemical concentrations. In this paper we suggest a way for deriving asymptotic expansions to the limit cycle solutions due to a Hopf bifurcation in two-dimensional reaction systems and we use them to build convenient normal modes for the analysis of Turing instabilities of the limit cycle. They extend the Fourier modes for the steady state in the classical Turing approach, as they include time-periodic fluctuations induced by the limit cycle. Diffusive instabilities can be properly considered because of the non-catastrophic loss of stability that the steady state shows while the limit cycle appears. Moreover, we shall see that instabilities may appear even though the diffusion coefficients are equal. The obtained normal modes suggest that there are two possible ways, one weak and the other strong, in which the limit cycle generates oscillatory Turing instabilities near a Turing–Hopf bifurcation point. In the first case slight oscillations superpose over a dominant steady inhomogeneous pattern. In the second, the unstable modes show an intermittent switching between complementary spatial patterns, producing the effect known as twinkling patterns.     

Strategies for the existence of spatial patterns in predator–prey communities generated by cross-diffusion

Fear of predators is an important drive for predator–prey interactions, which increases survival probability but cost the overall population size of the prey. In this paper, we have extended our previous work spatiotemporal dynamics of predator–prey interactions with fear effect by introducing the cross-diffusion. The conditions for cross-diffusion-driven instability are obtained using the linear stability analysis. The standard multiple scale analysis is used to derive the amplitude equations for the excited modes near Turing bifurcation threshold by taking the cross-diffusion coefficient as a bifurcation parameter. From the stability analysis of amplitude equations, the conditions for the emergence of various ecologically realistic Turing patterns such as spot, stripe, and mixture of spots and stripes are identified. Analytical results are verified with the help of numerical simulations. Turing bifurcation diagrams are plotted taking diffusion coefficients as control parameters. The effect of the cross-diffusion coefficients on the homogeneous steady state and pattern structures of the self-diffusive model is illustrated using the simulation techniques. It is also observed that the level of fear has stabilizing effect on the cross-diffusion induced instability and spot patterns change to stripe, then a mixture of spots and stripes and finally to the labyrinthine type of patterns with an increase in the level of fear.     

Pattern formation in a nonlinear model for animal coats

Several models have been proposed for describing the formation of animal coat patterns. We consider reaction-diffusion models due to Murray, which rely on a Turing instability for the pattern selection. In this paper, we describe the early stages of the pattern formation process for large domain sizes. This includes the selection mechanism and the geometry of the patterns generated by the nonlinear system on one-, two-, and three-dimensional base domains. These results are obtained by an adaptation of results explaining the occurrence of spinodal decomposition in materials science as modeled by the Cahn-Hilliard equation. We use techniques of dynamical systems, viewing solutions of the reaction-diffusion model in terms of nonlinear semiflows. Our results are applicable to any parabolic system exhibiting a Turing instability.     

Turing patterns induced by cross-diffusion in a 2D domain with strong Allee effect

In this work, we introduce a two-dimensional domain predator-prey model with strong Allee effect and investigate the Turing instability and the phenomena of the emergence of patterns. The occurrence of the Turing instability is ensured by the conditions that are procured by using the stability analysis of local equilibrium points. The amplitude equations (for supercritical case cubic Stuart–Landau equation and for subcritical quintic Stuart–Landau equation) are derived appropriate for each case by using the method of multiple time scale and show that the system supports patterns like squares, stripes, mixed-mode patterns, spots and hexagonal patterns. We obtain the asymptotic solutions to the model close to the onset instability based on the amplitude equations. Finally, numerically simulations tell how cross-diffusion plays an important role in the emergence of patterns.     

A delay decomposition approach for robust dissipativity and passivity analysis of neutral‐type neural networks with leakage time‐varying delay

This article presents the robust dissipativity and passivity analysis of neutral‐type neural networks with leakage time‐varying delay via delay decomposition approach. Using delay decomposition technique, new delay‐dependent criteria ensuring the considered system to be ‐γ dissipative are established in terms of strict linear matrix inequalities. A new Lyapunov–Krasovskii functional is constructed by dividing the discrete and neutral delay intervals into m and l segments, respectively, and choosing different Lyapunov functionals to different segments. Further, the dissipativity behaviors of neural networks which are affected due to the sensitiveness of the time delay in the leakage term have been taken into account. Finally, numerical examples are provided to show the effectiveness of the proposed method. © 2015 Wiley Periodicals, Inc. Complexity 21: 248–264, 2016     

Pattern dynamics in a Gierer–Meinhardt model with a saturating term

Gierer–Meinhardt system is a typical mathematical model to describe chemical and biological phenomena. In this paper, the Gierer–Meinhardt model with a saturating term is considered. By the linear stability analysis, we find the parameter area where possible Turing instability can occur. Then the multiple scales method is applied to obtain the amplitude equations at the critical value of Turing bifurcation, which help us to derive parameter space more specific where certain patterns such as spot-like pattern, stripe-like pattern and the coexistence pattern will emerge. Furthermore, the numerical simulations provide an indication of the wealth of patterns that the system can exhibit and the two-dimensional Fourier transform by the image processing interface of GUI from Matlab enables us to gain intensive understanding of these patterns. Besides, the interesting patterns including labyrinthine-like patterns are also numerically observed. All the results obtained reveal the mechanism of morphogenetic processes in adult mesenchymal cells. The patterns obtained corresponding to the patterns induced by morphogens in the vascular mesenchymal cells may play a role in atherosclerotic vascular calcification.     

Spatio-temporal patterns of Hopf bifurcating periodic oscillations in a pair of identical tri-neuron network loops

Recently, the coupling time delay has been considered as the source of the occurrence of the phase-flip bifurcation in time-delay coupled system. But the analytical results of how the coupling time delay affects this phenomenon is still lacking. In this paper, we consider a pair of identical tri-neuron network coupled with time delay. By using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups, we investigate the spatio-temporal patterns of Hopf bifurcating periodic oscillations induced by the coupling time delay. The explicit intervals of delay and the regions in the plane of the coupling strength and the gain of the inherent response function for the existence of synchronized in-phase or anti-phase oscillation are obtained. Our study show that the coupling time delay does not affect the spatio-temporal patterns of the individual neural loop but it has the significant impact on the spatio-temporal patterns between the two loops. These analytic results are then verified by numerical simulations.     

Spatiotemporal dynamics in a predator-prey model with a functional response increasing in both predator and prey densities

In this paper, we studied a diffusive predator-prey model with a functional response increasing in both predator and prey densities. The Turing instability and local stability are studied by analyzing the eigenvalue spectrum. Delay induced Hopf bifurcation is investigated by using time delay as bifurcation parameter. Some conditions for determining the property of Hopf bifurcation are obtained by utilizing the normal form method and center manifold reduction for partial functional differential equation.