Turing Space Research Articles

Heterogeneity induces spatiotemporal oscillations in reaction-diffusion systems.

We report on an instability arising in activator-inhibitor reaction-diffusion (RD) systems with a simple spatial heterogeneity. This instability gives rise to periodic creation, translation, and destruction of spike solutions that are commonly formed due to Turing instabilities. While this behavior is oscillatory in nature, it occurs purely within the Turing space such that no region of the domain would give rise to a Hopf bifurcation for the homogeneous equilibrium. We use the shadow limit of the Gierer-Meinhardt system to show that the speed of spike movement can be predicted from well-known asymptotic theory, but that this theory is unable to explain the emergence of these spatiotemporal oscillations. Instead, we numerically explore this system and show that the oscillatory behavior is caused by the destabilization of a steady spike pattern due to the creation of a new spike arising from endogeneous activator production. We demonstrate that on the edge of this instability, the period of the oscillations goes to infinity, although it does not fit the profile of any well-known bifurcation of a limit cycle. We show that nearby stationary states are either Turing unstable or undergo saddle-node bifurcations near the onset of the oscillatory instability, suggesting that the periodic motion does not emerge from a local equilibrium. We demonstrate the robustness of this spatiotemporal oscillation by exploring small localized heterogeneity and showing that this behavior also occurs in the Schnakenberg RD model. Our results suggest that this phenomenon is ubiquitous in spatially heterogeneous RD systems, but that current tools, such as stability of spike solutions and shadow-limit asymptotics, do not elucidate understanding. This opens several avenues for further mathematical analysis and highlights difficulties in explaining how robust patterning emerges from Turing's mechanism in the presence of even small spatial heterogeneity.

Emergent structures in reaction-advection-diffusion systems on a sphere.

We demonstrate unusual effects due to the addition of advection into a two-species reaction-diffusion system on the sphere. We find that advection introduces emergent behavior due to an interplay of the traditional Turing patterning mechanisms with the compact geometry of the sphere. Unidirectional advection within the Turing space of the reaction-diffusion system causes patterns to be generated at one point of the sphere, and transported to the antipodal point where they are destroyed. We illustrate these effects numerically and deduce conditions for Turing instabilities on local projections to understand the mechanisms behind these behaviors. We compare this behavior to planar advection which is shown to only transport patterns across the domain. Analogous transport results seem to hold for the sphere under azimuthal transport or away from the antipodal points in unidirectional flow regimes.

Domain-growth-induced patterning for reaction-diffusion systems with linear cross-diffusion

In this article we present, for the first time, domain-growth induced pattern formation for reaction-diffusion systems with linear cross-diffusion on evolving domains and surfaces. Our major contribution is that by selecting parameter values from spaces induced by domain and surface evolution, patterns emerge only when domain growth is present. Such patterns do not exist in the absence of domain and surface evolution. In order to compute these domain-induced parameter spaces, linear stability theory is employed to establish the necessary conditions for domain-growth induced cross-diffusion-driven instability for reaction-diffusion systems with linear cross-diffusion. Model reaction-kinetic parameter values are then identified from parameter spaces induced by domain-growth only; these exist outside the classical standard Turing space on stationary domains and surfaces. To exhibit these patterns we employ the finite element method for solving reaction-diffusion systems with cross-diffusion on continuously evolving domains and surfaces.

Pattern dynamics of a diffusive predator–prey model with delay effect

We study the spatiotemporal dynamics in a diffusive predator–prey system with time delay. By investigating the dynamical behavior of the system in the presence of Turing–Hopf bifurcations, we present a classification of the pattern dynamics based on the dispersion relation for the two unstable modes. More specifically, we researched the existence of the Turing pattern when control parameters lie in the Turing space. Particularly, when parameter values are taken in Turing–Hopf domain, we numerically investigate the formation of all the possible patterns, including time-dependent wave pattern, persistent short-term competing dynamics and stationary Turing pattern. Furthermore, the effect of time delay on the formation of spatial pattern has also been analyzed from the aspects of theory and numerical simulation. We speculate that the interaction of spatial and temporal instabilities in the reaction–diffusion system might bring some insight to the finding of patterns in spatial predator–prey models.

Emergent impacts of quadratic mortality on pattern formation in a predator–prey system

In this paper, we have investigated the effects of spatial diffusion in the pattern formation of a predator–prey system with quadratic mortality rate of the predator in the presence of habitat complexity. Using linear stability analysis, the regions of parameter space are plotted, and several kinds of instability regions are identified. Special attention has been given to investigate the selection of spatio-temporal patterns in the neighborhood of a critical parameter using the amplitude equation formalism. Choosing control parameter from the Turing space, the existence conditions for stable patterns are derived using the amplitude equations. Results obtained from theoretical analysis of amplitude equations agree very well with the numerical simulation results near the critical parameter value. Numerical simulations are done to show the existence of different spatio-temporal patterns. Spiral patterns are obtained in the suitable parameter region. It is shown that varying only the death rate of the predator, the spatio-temporal chaos can be controlled. Our investigations show that patterns are sensitive to the variation of habitat complexity and death rate of predator and patterns can be controlled by tuning the death rate of the predator.

Existence of spatial patterns in reaction–diffusion systems incorporating a prey refuge

Abstract. In real-world ecosystem, studies on the mechanisms of spatiotemporal pattern formation in a system of interacting populations deserve special attention for its own importance in contemporary theoretical ecology. The present investigation deals with the spatial dynamical system of a two-dimensional continuous diffusive predator–prey model involving the influence of intra-species competition among predators with the incorporation of a constant proportion of prey refuge. The linear stability analysis has been carried out and the appropriate condition of Turing instability around the unique positive interior equilibrium point of the present model system has been determined. Furthermore, the existence of the various spatial patterns through diffusion-driven instability and the Turing space in the spatial domain have been explored thoroughly. The results of numerical simulations reveal the dynamics of population density variation in the formation of isolated groups, following spotted or stripe-like patterns or coexistence of both the patterns. The results of the present investigation also point out that the prey refuge does have significant influence on the pattern formation of the interacting populations of the model under consideration.

EEG slow-wave mediates the fragmentation and coupling of cortical networks in propofol-induced general anesthesia

Electroencephalogram (EEG) recorded from propofol-induced general anesthesia is characterized by large amplitude slow-waves (0.1--1.5 Hz). Clinically, these lowest frequency components of the EEG signal become dominant over other higher frequency components during and after loss of consciousness [1]. However, it remains unclear how these slow oscillations are produced and to what extent they reflect changes in cortical network connectivity. Modeling anesthesia as a moderate reduction in interneuronal gap-junction coupling, a recent theoretical work by Steyn-Ross et al [2] predicts emergence of anesthetic slow-waves with chaotic dynamics. In the modeled anesthesia state, the weakened gap-junction coupling supports a codimension-2 bifurcation point where competing Turing (space) and Hopf (time) dynamics coexist, signifying spontaneous symmetry-breaking instabilities in the firing behavior of cortical neurons. Further, these chaotic slow-waves are found to perturb the neuronal coupling across the cortex, leading to a dramatic drop in global phase-coherence compared to its high level during consciousness. In this study, we analyze clinically-recorded EEG data to examine the model prediction for changes in phase-coherence between pairs of EEG channels in the sub-delta band during propofol anesthetic induction. Our study finds a coherence decrease in the frontal and occipital regions (see left panel of Figure ​Figure1),1), as well in the connection between them. Concomitantly, more strongly coupled neuronal activities are disclosed in the temporal-frontal, temporal-occipital and left-right temporal regions (right panel). Our clinical observation of reduced EEG coherence is consonant with cortical model predictions. However, our EEG study indicates that the coherence alternation is regional in nature, while the cortical model describes a spatially-uniform trend. Moreover, we did not find any theoretical prediction for the left- and right-temporal increased-coherence patterns. As the cortical model by Steyn-Ross et al. is spatially homogenous, i.e., there are no explicit front-to-back or right-to-left directionality, it is unable to produce regional coherence changes. It appears that the Steyn-Ross cortical model best represents the cortical dynamics in the frontal region. Figure 1 Correlation representations showing electrode pairs with significantly reduced (left, blue) or increased (right, pink) phase-coherence for sub-delta band (0.05-1.5 Hz) EEG induced by propofol anesthesia. Only electrode pairs (from 128 scalp electrodes) ...

Pattern formation and spatiotemporal chaos in a reaction–diffusion predator–prey system

In this paper, we consider a diffusive predator–prey system with modified Holling–Tanner functional response under homogeneous Neumann boundary condition. The qualitative analysis and Hopf bifurcation of the original ODE system are discussed, the conditions of the Turing instability for the reaction–diffusion system are derived, and the Turing space in the parameters space is achieved. We present the results of numerical simulations in order to validate the obtained analytical findings. We found some interesting spatiotemporal patterns when parameter values are taken in Turing–Hopf domain, in which the dynamics shows spatiotemporal behavior that is influenced by temporal oscillations as well as by Turing instabilities. With the help of numerical simulations, we identified the different types of spatial patterns in this diffusive predator–prey system, including stationary spatial pattern, periodic competing dynamics, and spatiotemporal chaos.

Turing instability in a diffusive SIS epidemiological model

Modeling and simulation of infectious diseases help to predict the likely outcome of an epidemic. In this paper, a spatial susceptible-infective-susceptible (SIS) type of epidemiological disease model with self- and cross-diffusion are investigated. We study the effect of diffusion on the stability of the endemic equilibrium with disease-induced mortality and nonlinear incidence rate. In the absence of diffusion the stationary solution stays stable but becomes unstable with respect to diffusion and that Turing instability takes place. We show that a standard (self-diffusion) system may be either stable or unstable, cross-diffusion response can stabilize an unstable standard system or decrease a Turing space (the space which the emergence of spatial patterns is holding) compared to the Turing space with self-diffusion, i.e. the cross-diffusion response is an important factor that should not be ignored when pattern emerges. Numerical simulations are provided to illustrate and extend the theoretical results.

Effect of prey refuge on spatiotemporal dynamics of the reaction–diffusion system

The present investigation deals with the problem of a two-dimensional diffusive predator–prey model incorporating prey refuge. In this investigation, we determine the Turing space in the spatial domain and perform extensive numerical simulation from both the mathematical and the biological points of view in order to study the effect of diffusion coefficients and other momentous parameters of the system in Turing instability. Furthermore, the stability behaviours of the proposed model in the absence of diffusion about the feasible equilibrium points are explored. In particular, the exact Turing domain is delineated in the parameter space. Various spatial patterns in spatial domain through diffusion-driven instability of the present model are depicted and analysed. The results indicate that the effect of prey refuge plays a significant role on the control of pattern formation of the populations.

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.

Feedback, receptor clustering, and receptor restriction to single cells yield large Turing spaces for ligand-receptor-based Turing models

Turing mechanisms can yield a large variety of patterns from noisy, homogenous initial conditions and have been proposed as patterning mechanism for many developmental processes. However, the molecular components that give rise to Turing patterns have remained elusive, and the small size of the parameter space that permits Turing patterns to emerge makes it difficult to explain how Turing patterns could evolve. We have recently shown that Turing patterns can be obtained with a single ligand if the ligand-receptor interaction is taken into account. Here we show that the general properties of ligand-receptor systems result in very large Turing spaces. Thus, the restriction of receptors to single cells, negative feedbacks, regulatory interactions among different ligand-receptor systems, and the clustering of receptors on the cell surface all greatly enlarge the Turing space. We further show that the feedbacks that occur in the FGF10-SHH network that controls lung branching morphogenesis are sufficient to result in large Turing spaces. We conclude that the cellular restriction of receptors provides a mechanism to sufficiently increase the size of the Turing space to make the evolution of Turing patterns likely. Additional feedbacks may then have further enlarged the Turing space. Given their robustness and flexibility, we propose that receptor-ligand-based Turing mechanisms present a general mechanism for patterning in biology.

Spatial pattern in a diffusive predator–prey model with sigmoid ratio-dependent functional response

In this paper, spatial patterns of a diffusive predator–prey model with sigmoid (Holling type III) ratio-dependent functional response which concerns the influence of logistic population growth in prey and intra-species competition among predators are investigated. The (local and global) asymptotic stability behavior of the corresponding non-spatial model around the unique positive interior equilibrium point in homogeneous steady state is obtained. In addition, we derive the conditions for Turing instability and the consequent parametric Turing space in spatial domain. The results of spatial pattern analysis through numerical simulations are depicted and analyzed. Furthermore, we perform a series of numerical simulations and find that the proposed model dynamics exhibits complex pattern replication. The feasible results obtained in this paper indicate that the effect of diffusion in Turing instability plays an important role to understand better the pattern formation in ecosystem.

Effect of time delay on pattern dynamics in a spatial epidemic model

Time delay, accounting for constant incubation period or sojourn times in an infective state, widely exists in most biological systems like epidemiological models. However, the effect of time delay on spatial epidemic models is not well understood. In this paper, spatial pattern of an epidemic model with both nonlinear incidence rate and time delay is investigated. In particular, we mainly focus on the effect of time delay on the formation of spatial pattern. Through mathematical analysis, we gain the conditions for Hopf bifurcation and Turing bifurcation, and find exact Turing space in parameter space. Furthermore, numerical results show that time delay has a significant effect on pattern formation. The simulation results may enrich the finding of patterns and may well capture some key features in the epidemic models.

Turing patterns and long-time behavior in a three-species food-chain model

We consider a spatially explicit three-species food chain model, describing generalist top predator-specialist middle predator–prey dynamics. We investigate the long-time dynamics of the model and show the existence of a finite dimensional global attractor in the product space, L2(Ω). We perform linear stability analysis and show that the model exhibits the phenomenon of Turing instability, as well as diffusion induced chaos. Various Turing patterns such as stripe patterns, mesh patterns, spot patterns, labyrinth patterns and weaving patterns are obtained, via numerical simulations in 1d as well as in 2d. The Turing and non-Turing space, in terms of model parameters, is also explored. Finally, we use methods from nonlinear time series analysis to reconstruct a low dimensional chaotic attractor of the model, and estimate its fractal dimension. This provides a lower bound, for the fractal dimension of the attractor, of the spatially explicit model.

Spatio-temporal dynamics of a reaction-diffusion system for a predator–prey model with hyperbolic mortality

We investigate the effects of diffusion on the spatial dynamics of a predator–prey model with hyperbolic mortality in predator population. More precisely, we aim to study the formation of some elementary two-dimensional patterns such as hexagonal spots and stripe patterns. Based on the linear stability analysis, we first identify the region of parameters in which Turing instability occurs. When control parameter is in the Turing space, we analyse the existence of stable patterns for the excited model by the amplitude equations. Then, for control parameter away from the Turing space, we numerically investigate the initial value-controlled patterns. Our results will enrich the pattern dynamics in predator–prey models and provide a deep insight into the dynamics of predator–prey interactions.

Turing pattern formation on periodic geometrical figures with continuous growing: numerical experiments

Several numerical examples of reaction–diffusion equations with increasing dominance are presented. Schnakenberg reaction model is solved with parameters in the Turing space. Therefore, numerical tests are performed in spherical and cylindrical surfaces. For the solution of the reaction–diffusion equations we provide a method of solution on surfaces in three dimensions using the finite element method with the total Lagrangian formulation. The results show that the formation of Turing patterns depends on the growth rate of the surface, the type of wave number predicted in the theory of square domains and the stabilization time of the same. These results may explain some phenomena of change of pattern on the surface of the skin of animals that exhibit characteristic spots and stripes.

Existence of spatial patterns in a predator–prey model with self- and cross-diffusion

In this work, we investigate the spatiotemporal dynamics of reaction–diffusion equations subject to cross-diffusion in the frame of a two-dimensional ratio-dependent predator–prey model. The conditions for diffusion-driven instability are obtained and the Turing space in the parameters space is achieved. Moreover, the criteria for local and global asymptotic stability of the unique positive homogeneous steady state without diffusion are discussed. Numerical simulations are carried out in order to validate the feasibility of the obtained analytical findings. Different types of spatial patterns through diffusion-driven instability of the proposed model are portrayed and analysed. Lastly, the paper finishes with an external discussion of biological relevance of the analysis regarding cross-diffusion and pattern issues.

Complex dynamics of a diffusive epidemic model with strong Allee effect

In this paper, we investigate the dynamics of a diffusive epidemic model with strong Allee effect in the susceptible population. We show some properties of solutions of the model, the asymptotic stability of the equilibria. Especially, we show that there exists a separatrix curve that separates the behavior of trajectories of the system, implying that the model is highly sensitive to the initial conditions. Furthermore, we give the conditions of Turing instability and determine the Turing space in the parameters space. Based on these results, we perform a series of numerical simulations and find that the model exhibits complex pattern replication: spots, spot–stripe mixtures and stripes patterns.

Pattern Dynamics in a Spatial Predator-Prey System with Allee Effect

We investigate the spatial dynamics of a predator-prey system with Allee effect. By using bifurcation analysis, the exact Turing domain is found in the parameters space. Furthermore, we obtain the amplitude equations and determine the stability of different patterns. In Turing space, it is found that predator-prey systems with Allee effect have rich dynamics. Our results indicate that predator mortality plays an important role in the pattern formation of populations. More specifically, as predator mortality rate increases, coexistence of spotted and stripe patterns, stripe patterns, spotted patterns, and spiral wave emerge successively. The results enrich the finding in the spatial predator-prey systems well.