Conditions For Turing Instability Research Articles

Pattern Formation in a Diffusive Ratio-Dependent Holling-Tanner Predator-Prey Model with Smith Growth

The spatiotemporal dynamics of a diffusive ratio-dependent Holling-Tanner predator-prey model with Smith growth subject to zero-flux boundary condition are investigated analytically and numerically. The asymptotic stability of the positive equilibrium and the existence of Hopf bifurcation around the positive equilibrium are shown; the conditions of Turing instability are obtained. And with the help of numerical simulations, it is found that the model exhibits complex pattern replication: stripes, spots-stripes mixtures, and spots Turing patterns.

Complex Dynamical Behavior of a Predator-Prey System with Group Defense

A diffusive predator-prey system with prey refuge is studied analytically and numerically. The Turing bifurcation is analyzed in detail, which in turn provides a theoretical basis for the numerical simulation. The influence of prey refuge and group defense on the equilibrium density and patterns of species under the condition of Turing instability is explored by numerical simulations, and this shows that the prey refuge and group defense have an important effect on the equilibrium density and patterns of species. Moreover, it can be obtained that the distributions of species are more sensitive to group defense than prey refuge. These results are expected to be of significance in exploration for the spatiotemporal dynamics of ecosystems.

Complex Dynamics of a Diffusive Holling-Tanner Predator-Prey Model with the Allee Effect

We investigate the complex dynamics of a diffusive Holling-Tanner predation model with the Allee effect on prey analytically and numerically. We examine the existence of the positive equilibria and the related dynamical behaviors of the model and find that when the model is with weak Allee effect, the solutions are local and global stability for some conditions around the positive equilibrium. In contrast, when the model is with strong Allee effect, this may lead to the phenomenon of bistability; that is to say, there is 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, spots-stripes mixtures, and stripes patterns. The results show that the impact of the Allee effect essentially increases the models spatiotemporal complexity.

Turing Patterns in a Predator-Prey System with Self-Diffusion

For a predator-prey system, cross-diffusion has been confirmed to emerge Turing patterns. However, in the real world, the tendency for prey and predators moving along the direction of lower density of their own species, called self-diffusion, should be considered. For this, we investigate Turing instability for a predator-prey system with nonlinear diffusion terms including the normal diffusion, cross-diffusion, and self-diffusion. A sufficient condition of Turing instability for this system is obtained by analyzing the linear stability of spatial homogeneous equilibrium state of this model. A series of numerical simulations reveal Turing parameter regions of the interaction of diffusion parameters. According to these regions, we further demonstrate dispersion relations and spatial patterns. Our results indicate that self-diffusion plays an important role in the spatial patterns.

TURING INSTABILITY AND PATTERN FORMATION IN A SEMI-DISCRETE BRUSSELATOR MODEL

In this paper, a semi-discrete Brusselator system is considered. The Turing instability theory analysis will be given for the model, then Turing instability conditions can be deduced combining linearization method and inner product technique. A series of numerical simulations of the system are performed in the Turing instability region, various patterns such as square, labyrinthine, spotlike patterns, can be exhibited. The impact of the system parameters and diffusion coefficients on patterns can also observed visually.

Pattern formation in a predator–prey model with spatial effect

In this paper, spatial dynamics in the Beddington–DeAngelis predator–prey model with self-diffusion and cross-diffusion is investigated. We analyze the linear stability and obtain the condition of Turing instability of this model. Moreover, we deduce the amplitude equations and determine the stability of different patterns. Numerical simulations show that this system exhibits complex dynamical behaviors. In the Turing space, we find three types of typical patterns. One is the coexistence of hexagon patterns and stripe patterns. The other two are hexagon patterns of different types. The obtained results well enrich the finding in predator–prey models with Beddington–DeAngelis functional response.

Turing-Hopf instability in biochemical reaction networks arising from pairs of subnetworks

Network conditions for Turing instability in biochemical systems with two biochemical species are well known and involve autocatalysis or self-activation. On the other hand general network conditions for potential Turing instabilities in large biochemical reaction networks are not well developed. A biochemical reaction network with any number of species where only one species moves is represented by a simple digraph and is modeled by a reaction–diffusion system with non-mass action kinetics. A graph-theoretic condition for potential Turing-Hopf instability that arises when a spatially homogeneous equilibrium loses its stability via a single pair of complex eigenvalues is obtained. This novel graph-theoretic condition is closely related to the negative cycle condition for oscillations in ordinary differential equation models and its generalizations, and requires the existence of a pair of subnetworks, each containing an even number of positive cycles. The technique is illustrated with a double-cycle Goodwin type model.

Spatial dynamics in a predator-prey model with Beddington-DeAngelis functional response

In this paper spatial dynamics of the Beddington-DeAngelis predator-prey model is investigated. We analyze the linear stability and obtain the condition of Turing instability of this model. Moreover, we deduce the amplitude equations and determine the stability of different patterns. In Turing space, we found that this model has coexistence of H(0) hexagon patterns and stripe patterns, H(π) hexagon patterns, and H(0) hexagon patterns. To better describe the real ecosystem, we consider the ecosystem as an open system and take the environmental noise into account. It is found that noise can decrease the number of the patterns and make the patterns more regular. What is more, noise can induce two kinds of typical pattern transitions. One is from the H(π) hexagon patterns to the regular stripe patterns, and the other is from the coexistence of H(0) hexagon patterns and stripe patterns to the regular stripe patterns. The obtained results enrich the finding in the Beddington-DeAngelis predator-prey model well.

Complex dynamics of a reaction–diffusion epidemic model

In this paper, we investigate the complex dynamics of a reaction–diffusion S−I model incorporating demographic and epidemiological processes with zero-flux boundary conditions. By the method of Lyapunov function, the global stability of the disease free equilibrium and the epidemic equilibrium was established. In addition, the conditions of Turing instability were obtained and the Turing space in the parameters space were given. Based on these results, we present the evolutionary processes that involves organism distribution and their interaction of spatially distributed population with local diffusion, and find that the model dynamics exhibits a diffusion-controlled formation growth to “holes, holes–stripes, stripes, spots–stripes and spots” pattern replication. Furthermore, we indicate that the diseases’ spread is getting smaller with R0 increasing, and the increasing the diffusion of infectious will increase the speed of diseases spreading. Our results indicate that the diffusion has a great influence on the spread of the epidemic and extend well the finding of spatiotemporal dynamics in the epidemic model.

Turing instability for a semi-discrete Gierer–Meinhardt system

In this paper, we investigate the spatial patterns of a Gierer–Meinhardt system where the space is discrete in two dimensions with the periodic boundary condition and time is continuous, in contrast to the continuum models. The conditions of Turing instability are obtained by linear analysis and a series of numerical simulations are performed. In the instability region, we have shown that this system can produce a number of different patterns such as stripes and snowflake pattern, other than ubiquitous fix-spotted patterns. As mentioned, the results are substantiated only by means of snapshots of the apatial grid. However, we also give some analysis by using the time series at three random grids and of the average value of states, that is, the stable state patterns can be observed. On the other hand, the effects of varying parameters on pattern formation are also discussed. Moreover, simulations for fixed parameters and special initial conditions indicate that the initial conditions can alter the structure of patterns. The patterns can form as a consequence of cellular interaction. So patterns arising from a semi-discrete model can present simulations on a geometrically accurate representation in biology. As a result, our work is interesting and important in ecology.

Pattern selection in a ratio-dependent predator–prey model

In this paper, we have presented Turing pattern selection in a ratio-dependentpredator–prey model with zero-flux boundary conditions, for which we have given a generalsurvey of the linear stability analysis and determined the condition of Turing instability,and derived amplitude equations for the excited modes. From the amplitude equations, thestability of patterns towards uniform and inhomogeneous perturbations is determined.Furthermore, we have presented novel numerical evidence of typical Turing patterns, andfound that the model dynamics exhibits complex pattern replication: in the rangeμ1 < μ ≤ μ2, the steady state is the only stable solution of the model; in the rangeμ2 < μ ≤ μ4, on increasing thecontrol parameter μ, the sequence Hπ-hexagons -hexagon–stripe mixture stripes -hexagon–stripe mixture -hexagons is observed; and whenμ > μ4, anH0-hexagon–stripe mixture pattern emerges. This may enrich the pattern formation in adiffusive system.

Turing instability for a two-dimensional Logistic coupled map lattice

In this Letter, stability analysis is applied to a two-dimensional Logistic coupled map lattice with the periodic boundary conditions. The conditions of Turing instability are obtained, and various patterns can be exhibited by numerical simulations in the Turing instability region. For example, space–time periodic structures, periodic or quasiperiodic traveling wave solutions, stationary wave solutions, spiral waves, and spatiotemporal chaos, etc. have been observed. In particular, the different pattern structures have also been observed for same parameters and different initial values. That is, pattern structures also depend on the initial values. The similar patterns have also been seen in relevant references. However, the present Letter owes to pattern formation via diffusion-driven instabilities because the system is stable in the absence of diffusion.

Turing instability conditions for growing domains with divergence free mesh velocity

Questions have been asked from a theoretical point of view about the effects of domain growth on Turing [A.M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. Lond. B 237 (1952) 37–72] diffusion-driven instability analysis for reaction–diffusion systems on continuously growing domains. The increase in computational power and tools has given the scientific community an almost immediate vast amount of computational data on pattern transitions (peak splitting, peak insertion, period doubling, circular patterns etc) on growing domains but with little knowledge of the theoretical justifications of the role played by domain growth in pattern formation. This paper derives the conditions for Turing diffusion-driven instability on growing domains whose domain velocity is divergence free. In particular we establish the result that the conditions for diffusion-driven instability growing domains whose domain velocity is divergence free are equivalent to those derived on a fixed domain. For this type of growth, the advection terms have no effect on the standard diffusion-driven instability conditions. Under these conditions, travelling wave solutions are obtained. This result has implications in the selection of parameter values as these are selected such that they belong to the Turing space. Our results do not however generalise to arbitrary growth protocols such as exponential, linear or logistic growths since the divergence of the mesh velocity is time-dependent.

Linear Stability of Fractional Reaction - Diffusion Systems

Theoretical framework for linear stability of an anomalous sub-diffusive activator-inhibitor system is set. Generalized Turing instability conditions are found to depend on anomaly exponents of various species. In addition to monotonous instability, known from normal diffusion, in an anomalous system oscillatory modes emerge. For equal anomaly exponents for both species the type of unstable modes is determined by the ratio of the reactants' diffusion coefficients. When the ratio exceeds its normal critical value, the monotonous modes become stable, whereas oscillatory instability persists until the anomalous critical value is also exceeded. An exact formula for the anomalous critical value is obtained. It is shown that in the short wave limit the growth rate is a power law of the wave number. When the anomaly exponents differ, disturbance modes are governed by power laws of the distinct exponents. If the difference between the diffusion anomaly exponents is small, the splitting of the power law exponents is absent at the leading order and emerges only as a next-order effect. In the short wave limit the onset of instability is governed by the anomaly exponents, whereas the ratio of diffusion coefficients influences the complex growth rates. When the exponent of the inhibitor is greater than that of the activator, the system is always unstable due to the inhibitor enhanced diffusion relatively to the activator. If the exponent of the activator is greater, the system is always stable. Existence of oscillatory unstable modes is also possible for waves of moderate length.

Network topology and Turing instabilities in small arrays of diffusively coupled reactors

We study the effect of the network structure on the diffusion-induced instability to nonuniform steady states in arrays of diffusively coupled reactors. The kinetics is given by the Lengyel–Epstein model, and we derive the conditions for Turing instabilities in all arrays of two, three, and four reactors.

Turing bifurcation in nonlinear competition models with delay

Turing instability in reaction-diffusion and predator-prey models including diffusion and Volterra-type distributed delays in the interspecies interaction terms is considered. For general functional forms of the reaction terms/prey birth rate-predator death rate, and delays modeled by the “weak” generic kernel a exp ⁡ ( − a U ) a \exp \left ( - aU \right ) and the “strong” generic kernel a 2 U exp ⁡ ( − a U ) {a^2}U \exp \left ( - aU \right ) , the necessary and sufficient conditions for Turing instability are derived and are found to be significantly different from the classical conditions with no delay. The structure of the resulting Turing space, where steady states may be diffusionally driven unstable initiating spatial patterns, is delineated for four specific models, and compared to the corresponding regimes in the absence of delay. An alternative bifurcation-theoretic derivation of the boundary of the Turing-unstable domain is also presented.

Turing patterns in CNNs. III. Computer simulation results

For part II, see ibid., vol. 42, no. 10, pp. 612-26 (Oct. 1995). In this paper, various patterns obtained through computer simulations are presented. It is shown through simple examples that the four inequalities known as the Turing instability conditions are only necessary but not sufficient conditions for Turing patterns to exist. The influence of various parameters, initial conditions as well as sidewall forcing on the final patterns and the possibilities of eliminating defects are studied by means of computer simulations. The possibility of generating perfectly regular patterns or patterns with few defects through the application of small and short controlling signals at one boundary suggests the intriguing possibility of producing high-purity high-tech materials, such as crystals, which was not possible with current technology. In addition, novel applications in image processing, pattern recognition, and other areas can be expected. >

Turing Instability in Competition Models with Delay I: Linear Theory

Turing instability in two-component predator-prey and reaction-diffusion models including diffusion and Volterra-type distributed delays in the interspecies interaction terms is considered. For general functional forms of the prey birthrate-predator deathrate/reaction terms and delays modeled by the “weak” generic kernel $a\exp ( - aU )$ or the “strong” generic kernel $aU\exp ( - aU )$, the structure of the diffusively-unstable space is shown to be completely altered by the inclusion of delays. The necessary and sufficient conditions for Turing instability are derived using the “weak” generic kernel and are found to be significantly different from the classical conditions with no delay. The structure of the Turing space, where steady states may be diffusionally driven unstable initiating spatial pattern, is delineated for several specific models, and compared to the corresponding regimes in the absence of delay. An alternative bifurcation-theoretic derivation of the boundary of the Turing-unstable domain is also presented. Finally, the instability with delays is briefly considered for two spatial dimensions and a finite domain size.