Journal articles: 'Turing instability'

1

Anma, Atsushi, Kunimochi Sakamoto, and Tohru Yoneda. "Unstable subsystems cause Turing instability." Kodai Mathematical Journal 35, no. 2 (June 2012): 215–47. http://dx.doi.org/10.2996/kmj/1341401049.

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Guiu-Souto, Jacobo, Lisa Michaels, Alexandra von Kameke, Jorge Carballido-Landeira, and Alberto P. Muñuzuri. "Turing instability under centrifugal forces." Soft Matter 9, no. 17 (2013): 4509. http://dx.doi.org/10.1039/c3sm27624d.

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Chen, Mengxin, Ranchao Wu, and Liping Chen. "Pattern Dynamics in a Diffusive Gierer–Meinhardt Model." International Journal of Bifurcation and Chaos 30, no. 12 (September 30, 2020): 2030035. http://dx.doi.org/10.1142/s0218127420300359.

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The purpose of the present paper is to investigate the pattern formation and secondary instabilities, including Eckhaus instability and zigzag instability, of an activator–inhibitor system, known as the Gierer–Meinhardt model. Conditions on the Hopf bifurcation and the Turing instability are obtained through linear stability analysis at the unique positive equilibrium. Then, the method of weakly nonlinear analysis is used to derive the amplitude equations. Especially, by adding a small disturbance to the Turing instability critical wave number, the spatiotemporal Newell–Whitehead–Segel equation of the stripe pattern is established. It is found that Eckhaus instability and zigzag instability may occur under certain conditions. Finally, Turing and non-Turing patterns are obtained via numerical simulations, including spotted patterns, mixed patterns, Eckhaus patterns, spatiotemporal chaos, nonconstant steady state solutions, spatially homogeneous periodic solutions and spatially inhomogeneous solutions in two-dimensional or one-dimensional space. Theoretical analysis and numerical results are in good agreement for this diffusive Gierer–Meinhardt model.

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XU, L., L. J. ZHAO, Z. X. CHANG, J. T. FENG, and G. ZHANG. "TURING INSTABILITY AND PATTERN FORMATION IN A SEMI-DISCRETE BRUSSELATOR MODEL." Modern Physics Letters B 27, no. 01 (November 26, 2012): 1350006. http://dx.doi.org/10.1142/s0217984913500061.

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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.

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Cai, Yongli, Shuling Yan, Hailing Wang, Xinze Lian, and Weiming Wang. "Spatiotemporal Dynamics in a Reaction–Diffusion Epidemic Model with a Time-Delay in Transmission." International Journal of Bifurcation and Chaos 25, no. 08 (July 2015): 1550099. http://dx.doi.org/10.1142/s0218127415500996.

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In this paper, we investigate the effects of time-delay and diffusion on the disease dynamics in an epidemic model analytically and numerically. We give the conditions of Hopf and Turing bifurcations in a spatial domain. From the results of mathematical analysis and numerical simulations, we find that for unequal diffusive coefficients, time-delay and diffusion may induce that Turing instability results in stationary Turing patterns, Hopf instability results in spiral wave patterns, and Hopf–Turing instability results in chaotic wave patterns. Our results well extend the findings of spatiotemporal dynamics in the delayed reaction–diffusion epidemic model, and show that time-delay has a strong impact on the pattern formation of the reaction–diffusion epidemic model.

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Szili, L., and J. Tóth. "Necessary condition of the Turing instability." Physical Review E 48, no. 1 (July 1, 1993): 183–86. http://dx.doi.org/10.1103/physreve.48.183.

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Guo, Yan, and Hyung Ju Hwang. "Pattern formation (II): The Turing Instability." Proceedings of the American Mathematical Society 135, no. 09 (May 14, 2007): 2855–67. http://dx.doi.org/10.1090/s0002-9939-07-08850-8.

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8

Hoang, Tung, and Hyung Ju Hwang. "Turing instability in a general system." Nonlinear Analysis: Theory, Methods & Applications 91 (November 2013): 93–113. http://dx.doi.org/10.1016/j.na.2013.06.010.

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9

Meng, Lili, Yutao Han, Zhiyi Lu, and Guang Zhang. "Bifurcation, Chaos, and Pattern Formation for the Discrete Predator-Prey Reaction-Diffusion Model." Discrete Dynamics in Nature and Society 2019 (April 1, 2019): 1–9. http://dx.doi.org/10.1155/2019/9592878.

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In this paper, a discrete predator-prey system with the periodic boundary conditions will be considered. First, we get the conditions for producing Turing instability of the discrete predator-prey system according to the linear stability analysis. Then, we show that the discrete model has the flip bifurcation and Turing bifurcation under the critical parameter values. Finally, a series of numerical simulations are carried out in the Turing instability region of the discrete predator-prey model; some new Turing patterns such as striped, bar, and horizontal bar are observed.

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LI, AN-WEI, ZHEN JIN, LI LI, and JIAN-ZHONG WANG. "EMERGENCE OF OSCILLATORY TURING PATTERNS INDUCED BY CROSS DIFFUSION IN A PREDATOR–PREY SYSTEM." International Journal of Modern Physics B 26, no. 31 (December 4, 2012): 1250193. http://dx.doi.org/10.1142/s0217979212501937.

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In this paper, we presented a predator–prey model with self diffusion as well as cross diffusion. By using theory on linear stability, we obtain the conditions on Turing instability. The results of numerical simulations reveal that oscillating Turing patterns with hexagons arise in the system. And the values of the parameters we choose for simulations are outside of the Turing domain of the no cross diffusion system. Moreover, we show that cross diffusion has an effect on the persistence of the population, i.e., it causes the population to run a risk of extinction. Particularly, our results show that, without interaction with either a Hopf or a wave instability, the Turing instability together with cross diffusion in a predator–prey model can give rise to spatiotemporally oscillating solutions, which well enrich the finding of pattern formation in ecology.

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11

Hu, Guangping, and Zhaosheng Feng. "Turing Instability and Pattern Formation in a Strongly Coupled Diffusive Predator–Prey System." International Journal of Bifurcation and Chaos 30, no. 08 (June 30, 2020): 2030020. http://dx.doi.org/10.1142/s0218127420300207.

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We are concerned with the Turing instability and pattern caused by cross-diffusion in a strongly coupled spatial predator–prey system. We explore how cross-diffusion destabilizes the spatially uniform steady state which is stable in reaction–diffusion systems, and explicitly describe the Turing space under certain conditions. Particularly, when the parameter values are taken in the Turing–Hopf domain, in which the spatiotemporal dynamical behavior is influenced by both Hopf and Turing instabilities, we investigate the formation of all possible patterns, including non-Turing structures such as wave pattern, competing dynamics as well as stationary Turing pattern. Furthermore, numerical simulations are illustrated to verify our theoretical findings.

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12

Ji, Yansu, and Jianwei Shen. "Turing Instability of Brusselator in the Reaction-Diffusion Network." Complexity 2020 (October 5, 2020): 1–12. http://dx.doi.org/10.1155/2020/1572743.

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Turing instability constitutes a universal paradigm for the spontaneous generation of spatially organized patterns, especially in a chemical reaction. In this paper, we investigated the pattern dynamics of Brusselator from the view of complex networks and considered the interaction between diffusion and reaction in the random network. After a detailed theoretical analysis, we obtained the approximate instability region about the diffusion coefficient and the connection probability of the random network. In the meantime, we also obtained the critical condition of Turing instability in the network-organized system and found that how the network connection probability and diffusion coefficient affect the reaction-diffusion system of the Brusselator model. In the end, the reason for arising of Turing instability in the Brusselator with the random network was explained. Numerical simulation verified the theoretical results.

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13

Al Basheer, Aladeen, Jingjing Lyu, Adom Giffin, and Rana D. Parshad. "The “Destabilizing” Effect of Cannibalism in a Spatially Explicit Three-Species Age Structured Predator-Prey Model." Complexity 2017 (2017): 1–15. http://dx.doi.org/10.1155/2017/3896412.

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Cannibalism, the act of killing and consumption of conspecifics, is generally considered to be a stabilising process in ODE models of predator-prey systems. On the other hand, Sun et al. were the first to show that cannibalism can cause Turing instability, in the classical Rosenzweig-McArthur two-species PDE model, which is an impossibility without cannibalism. Magnússon’s classic work is the first to show that cannibalism in a structured three-species predator-prey ODE model can actually be destabilising. In the current manuscript we consider the PDE form of the three-species model proposed in Magnússon’s classic work. We prove that, in the absence of cannibalism, Turing instability is an impossibility in this model, forany range of parameters. However, theinclusionof cannibalism can cause Turing instability. Thus, to the best of our knowledge, we report thefirstcannibalism induced Turing instability result, in spatially explicit three-species age structured predator-prey systems. We also show that, in the classical ODE model proposed by Magnússon, cannibalism can act as a life boat mechanism,for the prey.

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14

Zhang, Huayong, Xuebing Cong, Tousheng Huang, Shengnan Ma, and Ge Pan. "Neimark-Sacker-Turing Instability and Pattern Formation in a Spatiotemporal Discrete Predator-Prey System with Allee Effect." Discrete Dynamics in Nature and Society 2018 (June 26, 2018): 1–17. http://dx.doi.org/10.1155/2018/8713651.

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A spatiotemporal discrete predator-prey system with Allee effect is investigated to learn its Neimark-Sacker-Turing instability and pattern formation. Based on the occurrence of stable homogeneous stationary states, conditions for Neimark-Sacker bifurcation and Turing instability are determined. Numerical simulations reveal that Neimark-Sacker bifurcation triggers a route to chaos, with the emergence of invariant closed curves, periodic orbits, and chaotic attractors. The occurrence of Turing instability on these three typical dynamical behaviors leads to the formation of heterogeneous patterns. Under the effects of Neimark-Sacker-Turing instability, pattern evolution process is sensitive to tiny changes of initial conditions, suggesting the occurrence of spatiotemporal chaos. With application of deterministic initial conditions, transient symmetrical patterns are observed, demonstrating that ordered structures can exist in chaotic processes. Moreover, when local kinetics of the system goes further on the route to chaos, the speed of symmetry breaking becomes faster, leading to more fragmented and more disordered patterns at the same evolution time. The rich spatiotemporal complexity provides new comprehension on predator-prey coexistence in the ways of spatiotemporal chaos.

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15

Yang, Wenjie, Qianqian Zheng, Jianwei Shen, and Qing Hu. "Pattern Dynamics in a Predator-Prey Model with Diffusion Network." Complexity 2022 (July 31, 2022): 1–8. http://dx.doi.org/10.1155/2022/9055480.

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Diffusion plays an essential role in the distribution of predator and prey. We mainly research the diffusion network’s effect on the predator-prey model through bifurcation. First, it is found that the link probability and diffusion parameter can cause Turing instability in the network-organized predator-prey model. Then, the Turing stability region is obtained according to the sufficient condition of Turing instability and the eigenvalues’ distribution. Finally, the biological mechanism is explained through our theoretical results, which are also illustrated by numerical simulation.

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16

Shi, Hong-Bo, Shigui Ruan, Ying Su, and Jia-Fang Zhang. "Spatiotemporal Dynamics of a Diffusive Leslie–Gower Predator–Prey Model with Ratio-Dependent Functional Response." International Journal of Bifurcation and Chaos 25, no. 05 (May 2015): 1530014. http://dx.doi.org/10.1142/s0218127415300141.

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This paper is devoted to the study of spatiotemporal dynamics of a diffusive Leslie–Gower predator–prey system with ratio-dependent Holling type III functional response under homogeneous Neumann boundary conditions. It is shown that the model exhibits spatial patterns via Turing (diffusion-driven) instability and temporal patterns via Hopf bifurcation. Moreover, the existence of spatiotemporal patterns is established via Turing–Hopf bifurcation at the degenerate points where the Turing instability curve and the Hopf bifurcation curve intersect. Various numerical simulations are also presented to illustrate the theoretical results.

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17

Song, Danxia, Yongli Song, and Chao Li. "Stability and Turing Patterns in a Predator–Prey Model with Hunting Cooperation and Allee Effect in Prey Population." International Journal of Bifurcation and Chaos 30, no. 09 (July 2020): 2050137. http://dx.doi.org/10.1142/s0218127420501370.

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In this paper, we are concerned with a diffusive predator–prey model where the functional response follows the predator cooperation in hunting and the growth of the prey obeys the Allee effect. Firstly, the existence and stability of the positive equilibrium are explicitly determined by the local system parameters. It is shown that the ability of the hunting cooperation can affect the existence of the positive equilibrium and stability, and the intrinsic growth rate of the predator population does not affect the existence of the positive equilibrium, but affects the stability. Then the diffusion-driven Turing instability is investigated and the Turing bifurcation value is obtained, and we conclude that when the ability of the cooperation in hunting is weaker than some critical value, there is no Turing instability. The standard weakly nonlinear analysis method is employed to derive the amplitude equations of the Turing bifurcation, which is used to predict the types of the spatial patterns. And it is found that in the Turing instability region, with the parameter changing from approaching Turing bifurcation value to approaching Hopf bifurcation value, spatial patterns emerge from spot, spot-stripe to stripe. Finally, the numerical simulations are used to support the analytical results.

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18

Miao, Liangying, and Zhiqian He. "Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation." Open Mathematics 20, no. 1 (January 1, 2022): 986–97. http://dx.doi.org/10.1515/math-2022-0474.

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Abstract In this article, we study Hopf bifurcation and Turing instability of a diffusive predator-prey model with hunting cooperation. For the local model, we analyze the stability of the equilibrium and derive conditions for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solution by the center manifold and the normal form theory. For the reaction-diffusion model, first it is shown that Turing instability occurs, then the direction and stability of the Hopf bifurcation is reached. Our results show that hunting cooperation plays a crucial role in the dynamics of the model, that is, it can be beneficial to the predator population and induce the rise of Turing instability. Finally, numerical simulations are performed to visualize the complex dynamic behavior.

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19

Van Gorder, Robert A. "Influence of temperature on Turing pattern formation." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2240 (August 2020): 20200356. http://dx.doi.org/10.1098/rspa.2020.0356.

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The Turing instability is one of the most commonly studied mechanisms leading to pattern formation in reaction–diffusion systems, yet there are still many open questions on the applicability of the Turing mechanism. Although experiments on pattern formation using chemical systems have shown that temperature differences play a role in pattern formation, there is far less theoretical work concerning the interplay between temperature and spatial instabilities. We consider a thermodynamically extended reaction–diffusion system, consisting of a pair of reaction–diffusion equations coupled to an energy equation for temperature, and use this to obtain a natural extension of the Turing instability accounting for temperature. We show that thermal contributions can restrict or enlarge the set of unstable modes possible under the instability, and in some cases may be used to completely shift the set of unstable modes, strongly modifying emergent Turing patterns. Spatial heterogeneity plays a role under several experimentally feasible configurations, and we give particular consideration to scenarios involving thermal gradients, thermodynamics of chemicals transported within a flow, and thermodiffusion. Control of Turing patterns is also an area of active interest, and we also demonstrate how patterns can be modified using time-dependent control of the boundary temperature.

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20

Setayeshgar, S., and M. C. Cross. "Turing instability in a boundary-fed system." Physical Review E 58, no. 4 (October 1, 1998): 4485–500. http://dx.doi.org/10.1103/physreve.58.4485.

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21

Temmyo, Jiro, Richard Nötzel, and Toshiaki Tamamura. "Semiconductor nanostructures formed by the Turing instability." Applied Physics Letters 71, no. 8 (August 25, 1997): 1086–88. http://dx.doi.org/10.1063/1.119735.

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22

Buchanan, J. Robert. "Turing instability in pioneer/climax species interactions." Mathematical Biosciences 194, no. 2 (April 2005): 199–216. http://dx.doi.org/10.1016/j.mbs.2004.10.010.

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23

Huang, Y. "Interspecific Influence on Mobility and Turing Instability." Bulletin of Mathematical Biology 65, no. 1 (January 2003): 143–56. http://dx.doi.org/10.1006/bulm.2002.0328.

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Moussa, Ayman, Benoît Perthame, and Delphine Salort. "Backward Parabolicity, Cross-Diffusion and Turing Instability." Journal of Nonlinear Science 29, no. 1 (June 26, 2018): 139–62. http://dx.doi.org/10.1007/s00332-018-9480-z.

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25

Balinsky (Khazan), Y., and L. M. Pismen. "Differential flow induced chemical instability and Turing instability for Couette flow." Physical Review E 58, no. 4 (October 1, 1998): 4524–31. http://dx.doi.org/10.1103/physreve.58.4524.

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26

Weide Rodrigues, Vagner, Diomar Cristina Mistro, and Luiz Alberto Díaz Rodrigues. "Pattern Formation and Bistability in a Generalist Predator-Prey Model." Mathematics 8, no. 1 (December 20, 2019): 20. http://dx.doi.org/10.3390/math8010020.

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Generalist predators have several food sources and do not depend on one prey species to survive. There has been considerable attention paid by modellers to generalist predator-prey interactions in recent years. Erbach and collaborators in 2013 found a complex dynamics with bistability, limit-cycles and bifurcations in a generalist predator-prey system. In this paper we explore the spatio-temporal dynamics of a reaction-diffusion PDE model for the generalist predator-prey dynamics analyzed by Erbach and colleagues. In particular, we study the Turing and Turing-Hopf pattern formation with special attention to the regime of bistability exhibited by the local model. We derive the conditions for Turing instability and find the region of parameters for which Turing and/or Turing-Hopf instability are possible. By means of numerical simulations, we present the main types of patterns observed for parameters in the Turing domain. In the Turing-Hopf range of the parameters, we observed either stable patterns or homogeneous periodic distributions. Our findings reveal that movement can break the effect of hysteresis observed in the local dynamics, what can have important implication in pest management and species conservation.

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Yin, Hongwei, Xiaoyong Xiao, and Xiaoqing Wen. "Turing Patterns in a Predator-Prey System with Self-Diffusion." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/891738.

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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.

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Bhattacharyya, Saraswat, and Julia M. Yeomans. "Coupling Turing stripes to active flows." Soft Matter 17, no. 47 (2021): 10716–22. http://dx.doi.org/10.1039/d1sm01218e.

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We numerically solve the active nematohydrodynamic equations of motion, coupled to a Turing reaction–diffusion model, to study the effect of active nematic flow on the stripe patterns resulting from a Turing instability.

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29

Aly, Shaban, Houari B. Khenous, and Fatma Hussien. "Turing instability in a diffusive SIS epidemiological model." International Journal of Biomathematics 08, no. 01 (January 2015): 1550006. http://dx.doi.org/10.1142/s1793524515500060.

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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.

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Baek, Hunki, Dong Ick Jung, and Zhi-wen Wang. "Pattern Formation in a Semi-Ratio-Dependent Predator-Prey System with Diffusion." Discrete Dynamics in Nature and Society 2013 (2013): 1–14. http://dx.doi.org/10.1155/2013/657286.

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We investigate spatiotemporal dynamics of a semi-ratio-dependent predator-prey system with reaction-diffusion and zero-flux boundary. We obtain the conditions for Hopf, Turing, and wave bifurcations of the system in a spatial domain by making use of the linear stability analysis and the bifurcation analysis. In addition, for an initial condition which is a small amplitude random perturbation around the steady state, we classify spatial pattern formations of the system by using numerical simulations. The results of numerical simulations unveil that there are various spatiotemporal patterns including typical Turing patterns such as spotted, spot-stripelike mixtures and stripelike patterns thanks to the Turing instability, that an oscillatory wave pattern can be emerged due to the Hopf and wave instability, and that cooperations of Turing and Hopf instabilities can cause occurrence of spiral patterns instead of typical Turing patterns. Finally, we discuss spatiotemporal dynamics of the system for several different asymmetric initial conditions via numerical simulations.

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SUN, G., S. SARWARDI, P. J. PAL, and Md S. RAHMAN. "THE SPATIAL PATTERNS THROUGH DIFFUSION-DRIVEN INSTABILITY IN MODIFIED LESLIE-GOWER AND HOLLING-TYPE II PREDATOR-PREY MODEL." Journal of Biological Systems 18, no. 03 (September 2010): 593–603. http://dx.doi.org/10.1142/s021833901000338x.

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Formation of spatial patterns in prey-predator system is a central issue in ecology. In this paper Turing structure through diffusion driven instability in a modified Leslie-Gower and Holling-type II predator-prey model has been investigated. The parametric space for which Turing spatial structure takes place has been found out. Extensive numerical experiments have been performed to show the role of diffusion coefficients and other important parameters of the system in Turing instability that produces some elegant patterns that have not been observed in the earlier findings. Finally it is concluded that the diffusion can lead the prey population to become isolated in the two-dimensional spatial domain.

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Klika, Václav, and Eamonn A. Gaffney. "History dependence and the continuum approximation breakdown: the impact of domain growth on Turing’s instability." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2199 (March 2017): 20160744. http://dx.doi.org/10.1098/rspa.2016.0744.

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A diffusively driven instability has been hypothesized as a mechanism to drive spatial self-organization in biological systems since the seminal work of Turing. Such systems are often considered on a growing domain, but traditional theoretical studies have only treated the domain size as a bifurcation parameter, neglecting the system non-autonomy. More recently, the conditions for a diffusively driven instability on a growing domain have been determined under stringent conditions, including slow growth, a restriction on the temporal interval over which the prospect of an instability can be considered and a neglect of the impact that time evolution has on the stability properties of the homogeneous reference state from which heterogeneity emerges. Here, we firstly relax this latter assumption and observe that the conditions for the Turing instability are much more complex and depend on the history of the system in general. We proceed to relax all the above constraints, making analytical progress by focusing on specific examples. With faster growth, instabilities can grow transiently and decay, making the prediction of a prospective Turing instability much more difficult. In addition, arbitrarily high spatial frequencies can destabilize, in which case the continuum approximation is predicted to break down.

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Li, You, Jinliang Wang, and Xiaojie Hou. "Stripe and Spot Patterns for General Gierer–Meinhardt Model with Common Sources." International Journal of Bifurcation and Chaos 27, no. 02 (February 2017): 1750018. http://dx.doi.org/10.1142/s0218127417500183.

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This paper focuses on the Turing patterns in the general Gierer–Meinhardt model of morphogenesis. The stability analysis of the equilibrium for the associated ODE system is carried out and the stability conditions are obtained. Furthermore, we perform a detailed Hopf bifurcation analysis for this system. The results show that the equilibrium undergoes a supercritical Hopf bifurcation in certain parameter range and the bifurcated limit cycle is stable. With added diffusions, we then show that both the stable equilibrium and the Hopf periodic solution experience Turing instability with unequal spatial diffusions and obtain the instability conditions. Numerical simulations are given to illustrate the theoretical analysis, which show that the Turing patterns are of either spot or stripe type.

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LI, YAN, LINYAN ZHANG, DAGEN LI, and HONG-BO SHI. "SPATIOTEMPORAL DYNAMICS OF A DIFFUSIVE LESLIE-TYPE PREDATOR–PREY MODEL WITH BEDDINGTON–DEANGELIS FUNCTIONAL RESPONSE." Journal of Biological Systems 28, no. 03 (August 28, 2020): 785–809. http://dx.doi.org/10.1142/s0218339020500175.

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In this paper, we study the spatiotemporal dynamics of a diffusive Leslie-type predator–prey system with Beddington–DeAngelis functional response under homogeneous Neumann boundary conditions. Preliminary analysis on the local asymptotic stability and Hopf bifurcation of the spatially homogeneous model based on ordinary differential equations is presented. For the diffusive model, firstly, it is shown that Turing (diffusion-driven) instability occurs which induces spatial inhomogeneous patterns. Next, it is proved that the diffusive model exhibits Hopf bifurcation which produces temporal inhomogeneous patterns. Furthermore, at the points where the Turing instability curve and Hopf bifurcation curve intersect, it is demonstrated that the diffusive model undergoes Turing–Hopf bifurcation and exhibits spatiotemporal patterns. Numerical simulations are also presented to verify the theoretical results.

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Zhong, Shihong, Jinliang Wang, Junhua Bao, You Li, and Nan Jiang. "Spatiotemporal Complexity Analysis for a Space-Time Discrete Generalized Toxic-Phytoplankton-Zooplankton Model with Self-Diffusion and Cross-Diffusion." International Journal of Bifurcation and Chaos 31, no. 01 (January 2021): 2150006. http://dx.doi.org/10.1142/s0218127421500061.

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In this paper, a couple map lattice (CML) model is used to study the spatiotemporal dynamics and Turing patterns for a space-time discrete generalized toxic-phytoplankton-zooplankton system with self-diffusion and cross-diffusion. First, the existence and stability conditions for fixed points are obtained by using linear stability analysis. Second, the conditions for the occurrence of flip bifurcation, Neimark–Sacker bifurcation and Turing bifurcation are obtained by using the center manifold reduction theorem and bifurcation theory. The results show that there exist two nonlinear mechanisms, flip-Turing instability and Neimark–Sacker–Turing instability. Moreover, some numerical simulations are used to illustrate the theoretical results. Interestingly, rich dynamical behaviors, such as periodic points, periodic or quasi-periodic orbits, chaos and interesting patterns (plaques, curls, spirals, circles and other intermediate patterns) are found. The results obtained in the CML model contribute to comprehending the complex pattern formation of spatially extended discrete generalized toxic-phytoplankton-zooplankton system.

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Revina, S. V., and S. A. Lysenko. "Sufficient Turing instability conditions for the Schnakenberg system." Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki 31, no. 3 (September 2021): 424–42. http://dx.doi.org/10.35634/vm210306.

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A classical reaction-diffusion system, the Schnakenberg system, is under consideration in a bounded domain $\Omega\subset\mathbb{R}^m$ with Neumann boundary conditions. We study diffusion-driven instability of a stationary spatially homogeneous solution of this system, also called the Turing instability, which arises when the diffusion coefficient $d$ changes. An analytical description of the region of necessary and sufficient conditions for the Turing instability in the parameter plane is obtained by analyzing the linearized system in diffusionless and diffusion approximations. It is shown that one of the boundaries of the region of necessary conditions is an envelope of the family of curves that bound the region of sufficient conditions. Moreover, the intersection points of two consecutive curves of this family lie on a straight line whose slope depends on the eigenvalues of the Laplace operator and does not depend on the diffusion coefficient. We find an analytical expression for the critical diffusion coefficient at which the stability of the equilibrium position of the system is lost. We derive conditions under which the set of wavenumbers corresponding to neutral stability modes is countable, finite, or empty. It is shown that the semiaxis $d>1$ can be represented as a countable union of half-intervals with split points expressed in terms of the eigenvalues of the Laplace operator; each half-interval is characterized by the minimum wavenumber of loss of stability.

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37

Goras, Liviu, Paul Ungureanu, and Leon O. Chua. "On Turing Instability in Nonhomogeneous Reaction-Diffusion CNN’s." IEEE Transactions on Circuits and Systems I: Regular Papers 64, no. 10 (October 2017): 2748–60. http://dx.doi.org/10.1109/tcsi.2017.2701199.

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Nec, Y., and A. A. Nepomnyashchy. "Turing instability in sub-diffusive reaction–diffusion systems." Journal of Physics A: Mathematical and Theoretical 40, no. 49 (November 21, 2007): 14687–702. http://dx.doi.org/10.1088/1751-8113/40/49/005.

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Tanaka, Dan. "Turing Instability Leads Oscillatory Systems to Spatiotemporal Chaos." Progress of Theoretical Physics Supplement 161 (2006): 119–26. http://dx.doi.org/10.1143/ptps.161.119.

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Kocarev, L. M., and P. A. Janjic. "On Turing instability in two diffusely coupled systems." IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 42, no. 10 (1995): 779–84. http://dx.doi.org/10.1109/81.473587.

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NEC, Y., and A. A. NEPOMNYASHCHY. "Turing instability of anomalous reaction–anomalous diffusion systems." European Journal of Applied Mathematics 19, no. 3 (June 2008): 329–49. http://dx.doi.org/10.1017/s0956792508007389.

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Linear stability theory is developed for an activator–inhibitor model where fractional derivative operators of generally different exponents act both on diffusion and reaction terms. It is shown that in the short wave limit the growth rate is a power law of the wave number with decoupled time scales for distinct anomaly exponents of the different species. With equal anomaly exponents an exact formula for the anomalous critical value of reactants diffusion coefficients' ratio is obtained.

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42

Temmyo, Jiro, and Toshiaki Tamamura. "Self-Organizing Semiconductor Epitaxial Films by Turing Instability." Japanese Journal of Applied Physics 37, Part 1, No. 3B (March 30, 1998): 1487–92. http://dx.doi.org/10.1143/jjap.37.1487.

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Li, Xin-Zheng, Zhan-Guo Bai, Yan Li, and Kun Zhao. "Numerical investigation on oscillatory Turing patterns in a two-layer coupled reaction–diffusion system." Modern Physics Letters B 30, no. 07 (March 20, 2016): 1650085. http://dx.doi.org/10.1142/s0217984916500858.

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In this paper, various kinds of spontaneous dynamic patterns are investigated based on a two-layer nonlinearly coupled Brusselator model. It is found that, when the Hopf mode or supercritical Turing mode respectively plays major role in the short or long wavelength mode layer, the dynamic patterns appear under the action of nonlinearly coupling interactions in the reaction–diffusion system. The stripe pattern can change its symmetrical structure and form other graphics when influenced by small perturbations sourced from other modes. If two supercritical Turing modes are nonlinearly coupled together, the transition from Turing instability to Hopf instability may appear in the short wavelength mode layer, and the twinkling-eye square pattern, traveling and rotating pattern will be obtained in the two subsystems. If Turing mode and subharmonic Turing mode satisfy the three-mode resonance relation, twinkling-eye patterns are generated, and oscillating spots are arranged as square lattice in the two-dimensional space. When the subharmonic Turing mode satisfies the spatio-temporal phase matching condition, the traveling patterns, including the rhombus, hexagon and square patterns are obtained, which presents different moving velocities. It is found that the wave intensity plays an important role in pattern formation and pattern selection.

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Mi, Shao-Yue, Bang-Sheng Han, and Yu-Tong Zhao. "Turing Patterns for a Nonlocal Lotka–Volterra Cooperative System." Journal of Nonlinear Mathematical Physics 28, no. 4 (August 13, 2021): 363–89. http://dx.doi.org/10.1007/s44198-021-00002-z.

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AbstractThis 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.

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Carletti, Timoteo, and Riccardo Muolo. "Finite propagation enhances Turing patterns in reaction–diffusion networked systems." Journal of Physics: Complexity 2, no. 4 (October 19, 2021): 045004. http://dx.doi.org/10.1088/2632-072x/ac2cdb.

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Abstract We hereby develop the theory of Turing instability for reaction–diffusion systems defined on complex networks assuming finite propagation. Extending to networked systems the framework introduced by Cattaneo in the 40s, we remove the unphysical assumption of infinite propagation velocity holding for reaction–diffusion systems, thus allowing to propose a novel view on the fine tuning issue and on existing experiments. We analytically prove that Turing instability, stationary or wave-like, emerges for a much broader set of conditions, e.g., once the activator diffuses faster than the inhibitor or even in the case of inhibitor–inhibitor systems, overcoming thus the classical Turing framework. Analytical results are compared to direct simulations made on the FitzHugh–Nagumo model, extended to the relativistic reaction–diffusion framework with a complex network as substrate for the dynamics.

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Iqbal, Naveed, and Ranchao Wu. "Pattern formation by fractional cross-diffusion in a predator–prey model with Beddington–DeAngelis type functional response." International Journal of Modern Physics B 33, no. 25 (October 10, 2019): 1950296. http://dx.doi.org/10.1142/s0217979219502965.

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In this paper, we explore the emergence of patterns in a fractional cross-diffusion model with Beddington–DeAngelis type functional response. First, we explore the stability of the equilibrium points with or without fractional cross-diffusion. Instability of equilibria can be induced by cross-diffusion. We perform the linear stability analysis to obtain the constraints for the Turing instability. It is found by theoretical analysis that cross-diffusion is an important mechanism for the appearance of Turing patterns. For the dynamics of pattern, the weakly nonlinear multi-scaling analysis has been performed to obtain the amplitude equations. Finally, we ensure the existence of Turing patterns such as squares, spots and stripes by using the stability analysis of the amplitude equations. Moreover, with the assistance of numerical simulations, we verify the theoretical results.

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Liu, Biao, Ranchao Wu, Naveed Iqbal, and Liping Chen. "Turing Patterns in the Lengyel–Epstein System with Superdiffusion." International Journal of Bifurcation and Chaos 27, no. 08 (July 2017): 1730026. http://dx.doi.org/10.1142/s0218127417300269.

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Turing instability and pattern formation in the Lengyel–Epstein (L–E) model with superdiffusion are investigated in this paper. The effects of superdiffusion on the stability of the homogeneous steady state are studied in detail. In the presence of superdiffusion, instability will occur in the stable homogeneous steady state and more complex dynamics will exist. As a result of Turing instability, some patterns are formed. Through stability analysis of the system at the equilibrium point, conditions ensuring Turing and Hopf bifurcations are obtained. To further explore pattern selection, the weakly nonlinear analysis and multiple scale analysis are employed to derive amplitude equations of the stationary patterns. Then complex dynamics of amplitude equations, such as the existence of homogeneous solutions, stripe and hexagon patterns, mixed structure patterns, their stability, interaction and transition between them, are analyzed. Then different patterns occur immediately. Finally, the numerical simulations are presented to show the effectiveness of theoretical analysis and patterns are identified numerically. Whereas in the existing results of such model with normal diffusion, no amplitude equations are derived and patterns are only identified through numerical simulations.

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Song, Qiannan, Ruizhi Yang, Chunrui Zhang, and Leiyu Tang. "Bifurcation Analysis of a Diffusive Predator–Prey Model with Monod–Haldane Functional Response." International Journal of Bifurcation and Chaos 29, no. 11 (October 2019): 1950152. http://dx.doi.org/10.1142/s0218127419501529.

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In this paper, we consider a diffusive predator–prey model with Monod–Haldane functional response. We study the Turing instability and Hopf bifurcation of the coexisting equilibriums. We investigate the Turing–Hopf bifurcation through some key bifurcation parameters. In addition, we obtain a normal form for the Turing–Hopf bifurcation. Finally, we show numerical simulations to illustrate the theoretical results. For parameters around the critical value of the Turing–Hopf bifurcation, we demonstrate that the predator–prey model exhibits complex spatiotemporal dynamics, including spatially homogeneous periodic solutions, spatially inhomogeneous periodic solutions, and spatially inhomogeneous steady-state solutions.

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Zhong, Shihong, Jinliang Wang, You Li, and Nan Jiang. "Bifurcation, Chaos and Turing Instability Analysis for a Space-Time Discrete Toxic Phytoplankton-Zooplankton Model with Self-Diffusion." International Journal of Bifurcation and Chaos 29, no. 13 (December 10, 2019): 1950184. http://dx.doi.org/10.1142/s0218127419501840.

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The spatiotemporal dynamics of a space-time discrete toxic phytoplankton-zooplankton model is studied in this paper. The stable conditions for steady states are obtained through the linear stability analysis. According to the center manifold theorem and bifurcation theory, the critical parameter values for flip bifurcation, Neimark–Sacker bifurcation and Turing bifurcation are determined, respectively. Besides, the numerical simulations are provided to illustrate theoretical results. In order to distinguish chaos from regular behaviors, the maximum Lyapunov exponents are shown. The simulations show new and complex dynamics behaviors, such as period-doubling cascade, invariant circles, periodic windows, chaotic region and pattern formations. Numerical simulations of Turing patterns induced by flip-Turing instability, Neimark–Sacker Turing instability and chaos reveal a variety of spatiotemporal patterns, including plaque, curl, spiral, circle, and many other regular and irregular patterns. In comparison with former results in literature, the space-time discrete version considered in this paper captures more complicated and richer nonlinear dynamics behaviors and contributes a new comprehension on the complex pattern formation of spatially extended discrete phytoplankton-zooplankton system.

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Guin, Lakshmi Narayan, and Prashanta Kumar Mandal. "Spatial pattern in a diffusive predator–prey model with sigmoid ratio-dependent functional response." International Journal of Biomathematics 07, no. 05 (August 20, 2014): 1450047. http://dx.doi.org/10.1142/s1793524514500478.

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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.

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Journal articles on the topic 'Turing instability'

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