Turing Bifurcation Research Articles

Effect of delay on pattern formation of a Rosenzweig–MacArthur type reaction–diffusion model with spatiotemporal delay

In this paper, a predator–prey reaction–diffusion model with Rosenzweig–MacArthur type functional response and spatiotemporal delay is investigated through using the tool of Turing bifurcation theories. First, by taking the average time delay as a bifurcation parameter, conditions of occurrence of Turing bifurcation are obtained through employing the Routh–Hurwitz criteria. Second, as the average time delay varies the amplitude equations of Turing bifurcation patterns including spots pattern and stripes pattern are also obtained through the multiple scale perturbation method. Finally, the two kinds of spatiotemporal evolution distributions of species such as spots pattern and stripes pattern are shown to illustrate theoretical results.

Pattern forming systems coupling linear bulk diffusion to dynamically active membranes or cells.

Some analytical and numerical results are presented for pattern formation properties associated with novel types of reaction-diffusion (RD) systems that involve the coupling of bulk diffusion in the interior of a multi-dimensional spatial domain to nonlinear processes that occur either on the domain boundary or within localized compartments that are confined within the domain. The class of bulk-membrane system considered herein is derived from an asymptotic analysis in the limit of small thickness of a thin domain that surrounds the bulk medium. When the bulk domain is a two-dimensional disk, a weakly nonlinear analysis is used to characterize Turing and Hopf bifurcations that can arise from the linearization around a radially symmetric, but spatially non-uniform, steady-state of the bulk-membrane system. In a singularly perturbed limit, the existence and linear stability of localized membrane-bound spike patterns is analysed for a Gierer-Meinhardt activator-inhibitor model that includes bulk coupling. Finally, the emergence of collective intracellular oscillations is studied for a class of PDE-ODE bulk-cell model in a bounded two-dimensional domain that contains spatially localized, but dynamically active, circular cells that are coupled through a linear bulk diffusion field. Applications of such coupled bulk-membrane or bulk-cell systems to some biological systems are outlined, and some open problems in this area are discussed. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

Turing-Hopf bifurcation analysis and normal form of a diffusive Brusselator model with gene expression time delay

In this paper, by incorporating the gene expression time delay into a diffusive Brusselator model, a diffusive Brusselator model with gene expression time delay is proposed, and the Turing-Hopf bifurcation of the proposed model subject to homogeneous Neumann boundary condition is investigated. Firstly, the condition for the occurrence of Turing instability is deduced, and the existence of Turing bifurcation, Hopf bifurcation and Turing-Hopf bifurcation for the proposed model is also established. Then by a slight modification, the normal form on the center manifold near the Turing-Hopf singularity for the proposed model is derived by using the algorithm of calculating the normal form of Turing-Hopf bifurcation for the general reaction diffusion system with delay. With the aid of the obtained third-order truncated normal form of the proposed model, the spatiotemporal dynamics corresponding to six different regions in the parameter plane are analyzed. Moreover, by the corresponding relationships between the equilibrium points of the normal form and the solutions of original system, the dynamics of the original system can also be obtained. Finally, some numerical simulations are carried out to support the theoretical findings.

Self-Organization in a Plankton Community with Herd Predation and Weakly Nonlinear Diffusion

We studied a plankton community composed of phytoplankton (prey) and zooplankton (predators). The zooplankton forms small groups, cooperatively forages phytoplankton, and disperses nonlinearly to adapt to predation strategies and to avoid fierce interspecific competition. First, through linear stability analysis, we obtained the conditions of Hopf stability and the Turing bifurcation. Second, weakly nonlinear analysis helped us establish the amplitude equations that determine the type of patterns: spot patterns, stripe patterns, and mixed patterns of both kinds. Finally, the numerical simulations illustrate the stability of the system and the self-organizing behaviors of planktons. The results partly validate our analysis and help us better understand the dynamics of the algae ecosystem in the real world. Furthermore, we found that the weakly nonlinear analysis, as a tool, can be applied to the model without linearizing the weakly nonlinear diffusion, which extends the application scope of the tool.

Effect of herd-taxis on the self-organization of a plankton community

We studied a plankton community composed of phytoplankton (prey) and zooplankton (predators) with herd-taxis. The herd-taxis implies that the zooplankton would not only trend towards the higher density zone of phytoplankton but also has conformity behavior, which leads to the nonlinear cross-diffusion. The stability analysis gives out the conditions for the Hopf stability and the Turing bifurcation. In addition, the active foraging behavior of zooplankton helps to spatially stabilize the algal system and contributes to smoothing the inhomogeneous distribution of plankton, which is in line with the economic common sense: an invisible hand regulates and balances the distribution of plankton. Then we employed the weakly nonlinear analysis to establish amplitude equations to explore the effect of herd-taxis on the self-organization and found three types of patterns: spot patterns, stripe patterns, mixed patterns of the spots and stripes. The numerical simulations validated the self-organization behaviors and helped us better understand the interactions in plankton communities in the real world.

The Influence of Basic Reproduction Number on Pattern Formations in a Spatial Epidemic Model with The Susceptible Cross-diffusion

In this paper, we considered a spatial epidemic model with cross-diffusion of the susceptible. The cross-diffusion of the susceptible represents the tendency of susceptible to move away from the infected. We focused on the influence of basic reproduction numbers on the pattern formations of the model through Turing analysis. The Turing space in the parameter domain was given. From numerical simulation results, the model exhibited the following patterns, i.e., the holes, holes–stripes, stripes, spots–stripes, and spots. The holes indicated the occurrence of an outbreak in a region, whereas the spots indicated otherwise. If the moderate outbreak occurred in a region, the susceptible cross-diffusion term could change the patterns which indicated that it might prevent the disease dispersion to wider areas although the basic reproduction number was greater than its threshold. However, once the outbreak occurred spatially, the action of the susceptible to move away from the infected could not prevent the outbreak anymore. The results showed that the basic reproduction number has a big influence on the pattern formations when the values of the susceptible cross-diffusion coefficient were small. Keywords: Spatial epidemic model, cross-diffusion of the susceptible, basic reproduction number, pattern formation, Turing bifurcation. DOI: 10.7176/MTM/11-4-02 Publication date: September 30 th 2021

Quantum Turing bifurcation: Transition from quantum amplitude death to quantum oscillation death

An important transition from a homogeneous steady state to an inhomogeneous steady state via the Turing bifurcation in coupled oscillators was reported recently [Phys. Rev. Lett. 111, 024103 (2013)PRLTAO0031-900710.1103/PhysRevLett.111.024103]. However, the same in the quantum domain is yet to be observed. In this paper, we discover the quantum analog of the Turing bifurcation in coupled quantum oscillators. We show that a homogeneous steady state is transformed into an inhomogeneous steady state through this bifurcation in coupled quantum van der Pol oscillators. We demonstrate our results by a direct simulation of the quantum master equation in the Lindblad form. We further support our observations through an analytical treatment of the noisy classical model. Our study explores the paradigmatic Turing bifurcation at the quantum-classical interface and opens up the door toward its broader understanding.

Localized patterns and semi-strong interaction, a unifying framework for reaction–diffusion systems

AbstractSystems of activator–inhibitor reaction–diffusion equations posed on an infinite line are studied using a variety of analytical and numerical methods. A canonical form is considered, which contains all known models with simple cubic autocatalytic nonlinearity and arbitrary constant and linear kinetics. Restricting attention to models that have a unique homogeneous equilibrium, this class includes the classical Schnakenberg and Brusselator models, as well as other systems proposed in the literature to model morphogenesis. Such models are known to feature Turing instability, when activator diffuses more slowly than inhibitor, leading to stable spatially periodic patterns. Conversely in the limit of small feed rates, semi-strong interaction asymptotic analysis shows existence of isolated spike-like patterns. This paper describes the broad bifurcation structures that connect these two regimes. A certain universal two-parameter state diagram is revealed in which the Turing bifurcation becomes sub-critical, leading to the onset of homoclinic snaking. This regime then morphs into the spike regime, with the outer-fold being predicted by the semi-strong asymptotics. A rescaling of parameters and field concentrations shows how this state diagram can be studied independently of the diffusion rates. Temporal dynamics is found to strongly depend on the diffusion ratio though. A Hopf bifurcation occurs along the branch of stable spikes, which is subcritical for small diffusion ratio, leading to collapse to the homogeneous state. As the diffusion ratio increases, this bifurcation typically becomes supercritical and interacts with the homoclinic snaking and also with a supercritical homogeneous Hopf bifurcation, leading to complex spatio-temporal dynamics. The details are worked out for a number of different models that fit the theory using a mixture of weakly nonlinear analysis, semi-strong asymptotics and different numerical continuation algorithms.

Turing Patterns for a Nonlocal Lotka\u2013Volterra Cooperative System

This paper is devoted to investigating the pattern dynamics of Lotka–Volterra cooperative system with nonlocal effect and finding some new phenomena. Firstly, by discussing the Turing bifurcation, we build the conditions of Turing instability, which indicates the emergence of Turing patterns in this system. Then, by using multiple scale analysis, we obtain the amplitude equations about different Turing patterns. Furthermore, all possible pattern structures of the model are obtained through some transformation and stability analysis. Finally, two new patterns of the system are given by numerical simulation.

Analysis of spatially extended excitable Izhikevich neuron model near instability

The article focuses on the issue of a spatiotemporal excitable biophysical model that describes the propagation of electrical potential called spikes to model the diffusion-induced dynamics based on an analytical development of amplitude equations. Considering the Izhikevich neuron model consisting of coupled systems of ODEs, we demonstrate various results of spatiotemporal architecture (PDEs) using a suitable parameter regime. We analytically perform the saddle-node bifurcation and Hopf bifurcation analysis with bifurcating periodic solutions that show the transition phases in the system dynamics. We study different firing patterns both analytically and numerically by the formation of Riccati differential equation. To examine the characteristics of diffusive instabilities, we use Turing amplitude equations by multiscaling method. The instabilities and Turing bifurcation are established using theoretical analysis and numerical simulations. The spatial dynamics has potential effects on the deterministic system as a result of the diffusive matrices with various couplings, and the coupled oscillators with this nearest-neighbor coupling show synchronization measured by the synchronization factor analysis. Our results qualitatively reproduce different phenomena of the extended excitable system based with an efficient analytical scheme.

Patterns of a superdiffusive consumer-resource model with hunting cooperation functional response

In nature, the consumers (predator) use many strategies for hunting the resources (prey). The hunting cooperation is an effective strategy for the consumers for giving more accuracy to the hunting. This strategy works on perturbing the resources for facilitating the hunting. In this paper, we are concerned to analyze a superdiffusive resource-consumer system with hunting cooperation functional response. The presence of supperdiffusion represents the effect of the fear of the resources and the organized hunting strategy of the consumer. In fact, the effect of the superdiffusion is successfully established, where it is shown that the superdiffusion undergoes a complex patterns. Also, it noticed that the investigated system has a complex patters as Hopf bifurcation (HB), Turing bifurcation (TB), and Turing-Hopf bifurcation (THB). Furthermore, the super-diffusion can influence the stability of some equilibria. The obtained results are tested numerically.

Stationary peaks in a multivariable reaction–diffusion system: foliated snaking due to subcritical Turing instability

Abstract An activator–inhibitor–substrate model of side branching used in the context of pulmonary vascular and lung development is considered on the supposition that spatially localized concentrations of the activator trigger local side branching. The model consists of four coupled reaction–diffusion equations, and its steady localized solutions therefore obey an eight-dimensional spatial dynamical system in one spatial dimension (1D). Stationary localized structures within the model are found to be associated with a subcritical Turing instability and organized within a distinct type of foliated snaking bifurcation structure. This behavior is in turn associated with the presence of an exchange point in parameter space at which the complex leading spatial eigenvalues of the uniform concentration state are overtaken by a pair of real eigenvalues; this point plays the role of a Belyakov–Devaney point in this system. The primary foliated snaking structure consists of periodic spike or peak trains with $N$ identical equidistant peaks, $N=1,2,\dots \,$, together with cross-links consisting of nonidentical, nonequidistant peaks. The structure is complicated by a multitude of multipulse states, some of which are also computed, and spans the parameter range from the primary Turing bifurcation all the way to the fold of the $N=1$ state. These states form a complex template from which localized physical structures develop in the transverse direction in 2D.

Stationary and oscillatory localized patterns in ratio-dependent predator–prey systems

Abstract Investigations are undertaken into simple predator–prey models with rational interaction terms in one and two spatial dimensions. Focusing on a case with linear interaction and saturation, an analysis for long domains in 1D is undertaken using ideas from spatial dynamics. In the limit that prey diffuses much more slowly than predator, the Turing bifurcation is found to be subcritical, which gives rise to localized patterns within a Pomeau pinning parameter region. Parameter regions for localized patterns and isolated spots are delineated. For a realistic range of parameters, a temporal Hopf bifurcation of the balanced equilibrium state occurs within the localized-pattern region. Detailed spectral computations and numerical simulations reveal how the Hopf bifurcation is inherited by the localized structures at nearby parameter values, giving rise to both temporally periodic and chaotic localized patterns. Simulation results in 2D confirm the onset of complex spatio-temporal patterns within the corresponding parameter regions. The generality of the results is confirmed by showing qualitatively the same bifurcation structure within a similar model with quadratic interaction and saturation. The implications for ecology are briefly discussed.

Amplitude-mediated spiral chimera pattern in a nonlinear reaction-diffusion system.

Formation of diverse patterns in spatially extended reaction-diffusion systems is an important aspect of study that is pertinent to many chemical and biological processes. Of special interest is the peculiar phenomenon of chimera state having spatial coexistence of coherent and incoherent dynamics in a system of identically interacting individuals. In the present article, we report the emergence of various collective dynamical patterns while considering a system of prey-predator dynamics in the presence of a two-dimensional diffusive environment. Particularly, we explore the observance of four distinct categories of spatial arrangements among the species, namely, spiral wave, spiral chimera, completely synchronized oscillations, and oscillation death states in a broad region of the diffusion-driven parameter space. Emergence of amplitude-mediated spiral chimera states displaying drifted amplitudes and phases in the incoherent subpopulation is detected for parameter values beyond both Turing and Hopf bifurcations. Transition scenarios among all these distinguishable patterns are numerically demonstrated for a wide range of the diffusion coefficients which reveal that the chimera states arise during the transition from oscillatory to steady-state dynamics. Furthermore, we characterize the occurrence of each of the recognizable patterns by estimating the strength of incoherent subpopulations in the two-dimensional space.

The nonlinear initiation of side-branching by activator-inhibitor-substrate (Turing) morphogenesis.

An understanding of the underlying mechanism of side-branching is paramount in controlling and/or therapeutically treating mammalian organs, such as lungs, kidneys, and glands. Motivated by an activator-inhibitor-substrate approach that is conjectured to dominate the initiation of side-branching in a pulmonary vascular pattern, I demonstrate a distinct transverse front instability in which new fingers grow out of an oscillatory breakup dynamics at the front line without any typical length scale. These two features are attributed to unstable peak solutions in 1D that subcritically emanate from Turing bifurcation and that exhibit repulsive interactions. The results are based on a bifurcation analysis and numerical simulations and provide a potential strategy toward also developing a framework of side-branching for other biological systems, such as plant roots and cellular protrusions.

Effects of non-local competition on plankton-fish dynamics.

In ecology, the intra- and inter-specific competition between individuals of mobile species for shared resources is mostly non-local; i.e., competition at any spatial position will not only be dependent on population at that position, but also on population in neighboring regions. Therefore, models that assume competition to be restricted to the individuals at that position only are actually oversimplifying a crucial physical process. For the past three decades, researchers have established the necessity of considering spatial non-locality while modeling ecological systems. Despite this ecological importance, studies incorporating this non-local nature of resource competition in an aquatic ecosystem are surprisingly scarce. To this end, the celebrated Scheffer's tri-trophic minimal model has been considered here as a base model due to its efficacy in describing the pelagic ecosystem with least complexity. It is modified into an integro-reaction-diffusion system to include the effect of non-local competition by introducing a weighted spatial average with a suitable influence function. A detailed analysis shows that the non-locality may have a destabilizing effect on underlying nutrient-plankton-fish dynamics. A local system in a stable equilibrium state can lose its stability through spatial Hopf and Turing bifurcations when strength of a non-local interaction is strong enough, which eventually generates a large range of spatial patterns. The relationship between a non-local interaction and fish predation has been established, which shows that fish predation contributes in damping of plankton oscillations. Overall, results obtained here manifest the significance of non-locality in aquatic ecosystems and its possible contribution to the phenomena of "spatial patchiness."

Pattern formation induced by fractional cross-diffusion in a 3-species food chain model with harvesting

In this article, we explore the pattern formation caused by fractional cross-diffusion in a 3-species ecological symbiosis model with harvesting. Initially, all possible points of equilibrium are established and then by using Routh–Hurwitz criteria stability of an interior equilibrium point is explored. The conditions for Turing instability are obtained by local equilibrium points with stability analysis. In the neighborhood of the Turing bifurcation point weakly nonlinear analysis is used to deduce the amplitude equations. The conditions for the formation of the Turing patterns such as hexagons, rhombus, spots, squares, strips and waves patterns are identified for the amplitude equations through the dynamical analysis. Furthermore, by using the numerical simulations, the theoretical results are verified.

COMPLEX FRACTIONAL-ORDER HIV DIFFUSION MODEL BASED ON AMPLITUDE EQUATIONS WITH TURING PATTERNS AND TURING INSTABILITY

A weakly nonlinear analysis provides a system constituting amplitude equations and its related analysis is capable of predicting parameter regimes with different patterns expected to co-exist in dynamical circumstances that exhibit complex fractional-order system characteristics. The Turing mechanism of pattern formation as a result of diffusion-induced instability of the homogeneous steady state is concerned with unpredictable conditions. The Turing instability caused by fractional diffusion in a Human Immunodeficiency Virus model has been addressed in this study. It is important that the effect of the Human Immunodeficiency Virus to the immune system can be modeled by the interaction of uninfected cells, unhealthy cells, virus particles and antigen-specific. Initially, all potential equilibrium points are defined and the stability of the interior equilibrium point is then evaluated using the Routh–Hurwitz criteria. The conditions for Turing instability are obtained by local equilibrium points with stability analysis. In the neighborhood of the Turing bifurcation point, weakly nonlinear analysis is employed to deduce the amplitude equations. After applying amplitude equations, it is observed that this system has a very rich dynamical behavior. The constraints for the formation of the patterns like a hexagon, spot, mixed and stripe patterns are identified for the amplitude equations by dynamic analysis. Furthermore, by using the numerical simulations, the theoretical results are verified. Within this framework, this study through the dynamical behavior of the complex system perspective and bifurcation point based on the viral death rate can provide the basis for several researchers working on Human Immunodeficiency Virus model through various aspects. Accordingly, the Turing bifurcation point and weakly nonlinear analysis employed within the complex fractional-order dynamics addressed herein are highly relevant experimentally since the related effects can be studied and applied concerning different mathematical, physical, engineering and biological models.

Stability of Turing bifurcation in a weighted networked reaction–diffusion system

By introducing a weighted networked structure to the classical reaction–diffusion system, we investigate the Turing bifurcation which changes the trivial equilibrium to the nontrivial equilibrium. We show the existence of Turing bifurcation if the diffusion rate is large. By a weakly nonlinear analysis, we induce the amplitude equation of Turing bifurcation. By analyzing the amplitude equation, we show that the Turing bifurcation is stable.

Spatial patterns in a vegetation model with internal competition and feedback regulation

The vegetation patterns are a characteristic particularity of semiarid zones, which can be the future of modern ecology for its importance. This paper aims to study a diffusive vegetation model for the subject of studying the complex patterns generated by the presence of Turing-Hopf bifurcation. The main focus is on analyzing the effect of the locative internal rivalry between plants and feedback regulation on the pattern formations. The cross-diffusion produced by the positive feedback regulation generates surprising dynamics such as Hopf bifurcation, Turing bifurcation, Turing-Hopf bifurcation, which confirms the imbalance of the distribution of the vegetation in the semi-desert regions. For analyzing the spatiotemporal behavior near the Turing-Hopf bifurcation point, the Amplitude equation restricted at this point is used. Further, by using numerical simulations, the complex dynamics induced by the positive feedback redistribution and inner competition are explored.