Homogeneous Periodic Solutions Research Articles

Turing instability of periodic solutions for a general Brusselator model with cross-diffusion

This paper is devoted to a general Brusselator model with cross-diffusion under Neumann boundary conditions. We mainly consider the instability effect of cross-diffusion on stable periodic solutions bifurcating from the unique positive equilibrium point. According to Floquet theory and implicit function existence theorem, we establish some conditions on the self-diffusion and cross-diffusion coefficients under which the stable Hopf bifurcation periodic solutions can become unstable. The instability of stable spatial homogeneous periodic solutions will lead to the emergence of new irregular spatiotemporal patterns. Finally, we provide numerical simulations to support our analytical findings.

Equivariant Hopf Bifurcation in a Class of Partial Functional Differential Equations on a Circular Domain

Circular domains frequently appear in mathematical modeling in the fields of ecology, biology and chemistry. In this paper, we investigate the equivariant Hopf bifurcation of partial functional differential equations with Neumann boundary condition on a two-dimensional disk. The properties of these bifurcations at equilibriums are analyzed rigorously by studying the equivariant normal forms. Two reaction–diffusion systems with discrete time delays are selected as numerical examples to verify the theoretical results, in which spatially inhomogeneous periodic solutions including standing waves and rotating waves, and spatially homogeneous periodic solutions are found near the bifurcation points.

Bifurcations in a General Delay Sel’kov–Schnakenberg Reaction–Diffusion System

The dynamics of a delay Sel’kov–Schnakenberg reaction–diffusion system are explored. The existence and the occurrence conditions of the Turing and the Hopf bifurcations of the system are found by taking the diffusion coefficient and the time delay as the bifurcation parameters. Based on that, the existence of codimension-2 bifurcations including Turing–Turing, Hopf–Hopf and Turing–Hopf bifurcations are given. Using the center manifold theory and the normal form method, the universal unfolding of the Turing–Hopf bifurcation at the positive constant steady-state is demonstrated. According to the universal unfolding, a Turing–Hopf bifurcation diagram is shown under a set of specific parameters. Furthermore, in different parameter regions, we find the existence of the spatially inhomogeneous steady-state, the spatially homogeneous and inhomogeneous periodic solutions. Discretization of time and space visualizes these spatio-temporal solutions. In particular, near the critical point of Hopf–Hopf bifurcation, the spatially homogeneous periodic and inhomogeneous quasi-periodic solutions are found numerically.

Turing-Hopf Bifurcation Analysis in a Diffusive Ratio-Dependent Predator-Prey Model with Allee Effect and Predator Harvesting.

This paper investigates the complex dynamics of a ratio-dependent predator-prey model incorporating the Allee effect in prey and predator harvesting. To explore the joint effect of the harvesting effort and diffusion on the dynamics of the system, we perform the following analyses: (a) The stability of non-negative constant steady states; (b) The sufficient conditions for the occurrence of a Hopf bifurcation, Turing bifurcation, and Turing-Hopf bifurcation; (c) The derivation of the normal form near the Turing-Hopf singularity. Moreover, we provide numerical simulations to illustrate the theoretical results. The results demonstrate that the small change in harvesting effort and the ratio of the diffusion coefficients will destabilize the constant steady states and lead to the complex spatiotemporal behaviors, including homogeneous and inhomogeneous periodic solutions and nonconstant steady states. Moreover, the numerical simulations coincide with our theoretical results.

The effect of nonlocal interaction on chaotic dynamics, Turing patterns, and population invasion in a prey-predator model.

Pattern formation is a central process that helps to understand the individuals' organizations according to different environmental conditions. This paper investigates a nonlocal spatiotemporal behavior of a prey-predator model with the Allee effect in the prey population and hunting cooperation in the predator population. The nonlocal interaction is considered in the intra-specific prey competition, and we find the analytical conditions for Turing and Hopf bifurcations for local and nonlocal models and the spatial-Hopf bifurcation for the nonlocal model. Different comparisons have been made between the local and nonlocal models through extensive numerical investigation to study the impact of nonlocal interaction. In particular, a legitimate range of nonlocal interaction coefficients causes the occurrence of spatial-Hopf bifurcation, which is the emergence of periodic patterns in both time and space from homogeneous periodic solutions. With an increase in the range of nonlocal interaction, the whole Turing pattern suppresses after a certain threshold, and no pure Turing pattern exists for such cases. Specifically, at low diffusion rates for the predators, nonlocal interaction in the prey population leads to the extinction of predators. As the diffusion rate of predators increases, impulsive wave solutions emerge in both prey and predator populations in a one-dimensional spatial domain. This study also includes the effect of nonlocal interaction on the invasion of populations in a two-dimensional spatial domain, and the nonlocal model produces a patchy structure behind the invasion where the local model predicts only the homogeneous structure for such cases.

Spatial pattern formation driven by the cross-diffusion in a predator–prey model with Holling type functional response

In this paper, we consider a cross-diffusion predator–prey system with Holling type functional response. We study the local stability, Turing instability, spatial pattern formation, Hopf and Turing–Hopf bifurcation of the equilibrium. Numerical simulation with zero-flux boundary conditions discloses that the system under consideration experiences the occurrence of cross-diffusion-driven instability. The dynamical system in Turing space emerges spots, stripe-spot mixtures and labyrinthine patterns, which reveals that the interaction of both self- and cross-diffusions play a significant role on the pattern formation of the present system in a way to enrich the pattern. We obtain the normal form of the Turing–Hopf bifurcation and observe that the system has stably spatially homogeneous periodic solutions, stable constant and nonconstant steady-state solutions, which indicates that the intrinsic growth rate coefficient and the environmental carrying capacity coefficient are two important factors for predator–prey system, and affect the stability of predator–prey system.

Phase-synchronized oscillations in a unidirectional ring of pump-coupled lasers

The dynamics of a closed chain of lasers with optoelectronic unidirectional delayed coupling is analyzed. Assuming that the number of lasers is sufficiently large, we propose the integro-differential spatially distributed model with periodic boundary conditions. The critical value of the coupling coefficient is determined, at which the stationary state of laser generation becomes unstable. A two-dimensional complex partial differential equation of the Ginzburg–Landau type is derived as a quasi-normal form. Its simplest homogeneous periodic solutions correspond to traveling wave solutions of the distributed model and, in turn, describe phase synchronized oscillations in the ring of coupled lasers. The frequencies and amplitudes of oscillations of the radiation intensity of each laser and the phase difference for adjacent oscillators are determined.

Turing‐Turing and Turing‐Hopf bifurcations in a general diffusive Brusselator model

AbstractA general reaction‐diffusion Brusselator model subject to homogeneous Neumann boundary conditions is investigated in this paper. First, the stability of the unique positive equilibrium is studied, and we identify the existence of the Hopf bifurcation. Then, occurrence conditions of the Turing instability, the Turing‐Turing, and the Turing‐Hopf bifurcations are established. To explore the spatiotemporal solutions resulting from the bifurcation, the amplitude equations of the Turing‐Turing and the Turing‐Hopf bifurcations are established via the method of multiple time scale. It is found that the model admits the nonconstant steady state, the mixed nonconstant steady state, the spatially homogeneous periodic solution, and the spatially nonhomogeneous periodic solution. As such, the nonhomogeneous spatiotemporal solutions appear in the model. In the end, numerical simulations verify the validity of the theoretical results.

Tree-structured neural networks: Spatiotemporal dynamics and optimal control

How the network topology drives the response dynamic is a basic question that has not yet been fully answered in neural networks. Elucidating the internal relation between topological structures and dynamics is instrumental in our understanding of brain function. Recent studies have revealed that the ring structure and star structure have a great influence on the dynamical behavior of neural networks. In order to further explore the role of topological structures in the response dynamic, we construct a new tree structure that differs from the ring structure and star structure of traditional neural networks. Considering the diffusion effect, we propose a diffusion neural network model with binary tree structure and multiple delays. How to design control strategies to optimize brain function has also been an open question. Thus, we put forward a novel full-dimensional nonlinear state feedback control strategy to optimize relevant neurodynamics. Some conditions about the local stability and Hopf bifurcation are obtained, and it is proved that the Turing instability does not occur. Moreover, for the formation of the spatially homogeneous periodic solution, some diffusion conditions are also fused together. Finally, several numerical examples are carried out to illustrate the results’ correctness. Meanwhile, some comparative experiments are rendered to reveal the effectiveness of the proposed control strategy.

Bifurcations in a diffusive predator–prey system with linear harvesting

Complex spatiotemporal dynamical behaviors of a diffusive predator–prey system with Michaelis–Menten type functional response and linear harvesting are investigated. Firstly, the critical conditions for the occurrence of Turing instability, which are necessary and sufficient, are derived in a novel way. Then, the existence conditions of codimension-1 Turing bifurcation, Hopf bifurcation, and codimension-2 Turing–Turing bifurcation, Turing–Hopf bifurcation are established. Furthermore, the detailed bifurcation set is given by calculating the amplitude equation with the method of the multiple time scale near the Turing–Hopf bifurcation. We find that the system may exhibit nonconstant steady-state solutions, spatially homogeneous periodic solutions, and spatially inhomogeneous periodic solutions, which can be verified by a series of numerical simulations. These investigations not only explain the effect of diffusion and harvesting on the dynamic behavior of the system, but also reveal the mechanism of spatiotemporal complexity in the diffusive predator–prey system.

Stability and patterns of the nutrient-microorganism model with chemotaxis

Abstract In this paper, the stability and the bifurcations of the nutrient-microorganism model with chemotaxis are analyzed, subject to no-flux boundary conditions. By choosing the chemotaxis coefficient as the control parameter, it is found that the steady state bifurcation, the Hopf–Turing bifurcation, can happen in the model. The induced spatially homogeneous periodic solution, the non-constant steady state, and the spatially inhomogeneous periodic solution are exhibited. The results suggest that chemotaxis assimilated into the model could give rise to rich spatiotemporal dynamical behaviors.

Turing–Hopf Bifurcation Analysis of the Sel’kov–Schnakenberg System

In this paper, we investigate the spatiotemporal dynamics of the Sel’kov–Schnakenberg system. The stability of the positive constant steady state is studied by the linear stability theory. Hopf bifurcation and Turing–Hopf bifurcation are generated by varying two parameters in the model. The normal form near the Turing–Hopf singularity is calculated to explore the complex dynamics of the system. Finally, numerical simulations are carried out to verify the theoretical results. Our results show that the Sel’kov–Schnakenberg system exhibits complex dynamics near the Turing–Hopf singularity, including the existence of inhomogeneous steady states, homogeneous periodic solutions and inhomogeneous periodic solutions.

Spatiotemporal dynamics of a diffusive predator-prey model with delay and Allee effect in predator.

It has been shown that Allee effect can change predator-prey dynamics and impact species persistence. Allee effect in the prey population has been widely investigated. However, the study on the Allee effect in the predator population is rare. In this paper, we investigate the spatiotemporal dynamics of a diffusive predator-prey model with digestion delay and Allee effect in the predator population. The conditions of stability and instability induced by diffusion for the positive equilibrium are obtained. The effect of delay on the dynamics of system has three different cases: (a) the delay doesn't change the stability of the positive equilibrium, (b) destabilizes and stabilizes the positive equilibrium and induces stability switches, or (c) destabilizes the positive equilibrium and induces Hopf bifurcation, which is revealed (numerically) to be corresponding to high, intermediate or low level of Allee effect, respectively. To figure out the joint effect of delay and diffusion, we carry out Turing-Hopf bifurcation analysis and derive its normal form, from which we can obtain the classification of dynamics near Turing-Hopf bifurcation point. Complex spatiotemporal dynamical behaviors are found, including the coexistence of two stable spatially homogeneous or inhomogeneous periodic solutions and two stable spatially inhomogeneous quasi-periodic solutions. It deepens our understanding of the effects of Allee effect in the predator population and presents new phenomena induced be delay with spatial diffusion.

Stability switch and Hopf bifurcations for a diffusive plankton system with nonlocal competition and toxic effect

<abstract><p>Since the distribution of plankton is always uneven, the nonlocal phytoplankton competition term indicates the spatial weighted mean of phytoplankton density, which is introduced into the plankton model with toxic substances effect to study the corresponding dynamic behavior. The stability of the positive equilibrium point and the existence of Hopf bifurcations are discussed by analysing the distribution of eigenvalues. The direction and stability of bifurcation periodic solution are researched based on an extended central manifold method and normal theory. Finally, spatially inhomogeneous oscillations are observed in the vicinity of the Hopf bifurcations. Through numerical simulation, we can observe that the system without nonlocal competition term only generates homogeneous periodic solution, and inhomogeneous periodic solution will produce only when both diffusion and nonlocal competition exist simultaneously. We can also see that when the toxin-producing rate of each phytoplankton is in an appropriate range, the system with nonlocal competition generates a stability switch with inhomogeneous periodic solution, when the value of time delay is in a certain interval, then Hopf bifurcations will appear, and with the increase of time delay, the quantity of plankton will eventually become stable.</p></abstract>

Bifurcation analysis and spatiotemporal patterns in delayed Schnakenberg reaction-diffusion model

The diffusive Schnakenberg model with gene expression time delay is considered. In this paper, the stability, diffusion-driven instability, and time delay-induced Hopf bifurcation have been investigated. By linear stability analysis, we find the parameter areas where the unique positive equilibrium is stable and Turing instability can occur for a certain relationship of diffusion rates. Then we obtain a series of critical values for the time delay at which the spatially homogeneous and inhomogeneous periodic solutions may emerge. Based on the explicit formula determining the properties of the Hopf bifurcation, we employ numerical simulations for parameters both in the stable region and Turing instability region. The numerical simulations show that delay can destabilize the stability of the positive equilibrium solution and eventually induce spatially homogeneous and inhomogeneous periodic solutions. Furthermore, the spatiotemporal patterns in the two spaces dimension from the Turing instability regime provide an indication of the wealth of patterns that the delayed system can exhibit.

Pattern Formation in a Reaction-Diffusion BAM Neural Network With Time Delay: (k1, k2) Mode Hopf-Zero Bifurcation Case.

This article investigates the joint effects of connection weight and time delay on pattern formation for a delayed reaction-diffusion BAM neural network (RDBAMNN) with Neumann boundary conditions by using the (k1,k2) mode Hopf-zero bifurcation. First, the conditions for k1 mode zero bifurcation are obtained by choosing connection weight as the bifurcation parameter. It is found that the connection weight has a great impact on the properties of steady state. With connection weight increasing, the homogeneous steady state becomes inhomogeneous, which means that the connection weight can affect the spatial stability of steady state. Then, the specified conditions for the k2 mode Hopf bifurcation and the (k1,k2) mode Hopf-zero bifurcation are established. By using the center manifold, the third-order normal form of the Hopf-zero bifurcation is obtained. Through the analysis of the normal form, the bifurcation diagrams on two parameters' planes (connection weight and time delay) are obtained, which contains six areas. Some interesting spatial patterns are found in these areas: a homogeneous periodic solution, a homogeneous steady state, two inhomogeneous steady state, and two inhomogeneous periodic solutions.

Spatiotemporal dynamics on a class of [formula omitted]-dimensional reaction–diffusion neural networks with discrete delays and a conical structure

In this paper, the stability and Hopf bifurcation problems on a class of high-order reaction–diffusion neural networks are investigated, where the conical structure and signal transmission delays are introduced to make the model more accurate. By the linear stability analysis, Hopf bifurcation theory and Turing instability theorem, some criteria on the local stability and Hopf bifurcation of this system are established by selecting the time delay as bifurcation parameter. Especially, the Coates formula and the holistic element method are taken to determine the critical value, at which the purely imaginary root appears in the associated high-degree characteristic equation. On the other hand, the effects of diffusion terms on the spatial distribution of periodic solutions are explored. New sufficient conditions are derived to guarantee the occurrence of spatially homogeneous periodic solutions. Finally, some numerical examples are provided to validate the main results. It is found that the critical value can also be affected by the network size and the self-feedback rate.

Turing–Hopf bifurcation co-induced by cross-diffusion and delay in Schnakenberg system

The dynamics of Schnakenberg model subjected to gene expression time delay and cross-diffusion is investigated. It is shown that the presence of cross-diffusion allows the enlarged Turing instability region compared with self-diffusion and the variation of time delay can either lead to destabilization of the positive equilibrium or fail Turing instability through Turing–Hopf bifurcation. A bifurcation analysis of the Schnakenberg model in question has been provided to derive the stable regime, Turing instability regime and also the condition for delay-induced Turing–Hopf bifurcation. With the aid of normal form of Turing–Hopf bifurcation, the spatiotemporal dynamics near this bifurcation point can be divided into six categories where various solutions including stable spatially homogeneous or inhomogeneous periodic solutions, stable steady states and the transition from one type of spatiotemporal patterns to another would exhibit. These spatiotemporal solutions have also been illustrated through some numerical simulations, confirming the theoretical predictions.

Spatiotemporal patterns and bifurcations with degeneration in a symmetry glycolysis model

In this paper, we consider the glycolysis system with symmetry and degeneration. Based on the detailed stability of constant steady state solution and symmetry of nonlinear glycolysis system, for the Hopf bifurcations, we obtain the existence, stability and bifurcation direction not only for the homogeneous and inhomogeneous periodic solutions, but also for the degenerate bifurcating periodic solutions. The approach we adopted is the Lyapunov–Schmidt reduction method instead of the center manifold theory. On the other hand, we derive the nonexistence of the steady state solutions for small input flux of substrate. Additionally, for the steady state bifurcations, main object is to discuss degenerate bifurcations by means of Lyapunov–Schmidt procedure and singularity theory because the classical Crandall–Rabinowitz theorem cannot be applied. Some numerical simulations are achieved to illustrate the analytical results, especially the degenerate bifurcation results. It is interesting to noticed that the degenerate bifurcations can lead to richer spatiotemporal patterns, including the meeting of two Hopf bifurcation branches and the coupling of two eigenfunction models.

Normal forms of double Hopf bifurcation for a reaction-diffusion system with delay and nonlocal spatial average and applications

In this paper, we are concerned with a reaction-diffusion model incorporating delay and nonlocal effects. The normal form of double Hopf bifurcation is derived. The diffusive model of pollen tube tip growth is discussed and numerical simulations show that spatially homogeneous and inhomogeneous periodic solutions can be both stable or connected by a heteroclinic orbit under certain conditions. In addition, the diffusive Lotka-Volterra model with delay and nonlocality is considered and spatially inhomogeneous quasi-periodic solution is obtained.