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Turing Instability and Pattern Induced by Cross-Diffusion for a Nonlinear Reaction-Diffusion System of Turbulence-Shear Flow Interaction
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作者 周辉 彭亚红 《Journal of Donghua University(English Edition)》 EI CAS 2017年第5期689-693,共5页
The Turing instability and the phenomena of pattern formation for a nonlinear reaction-diffusion(RD) system of turbulence-shear flowinteraction are investigated.By the linear stability analysis,the essential condition... The Turing instability and the phenomena of pattern formation for a nonlinear reaction-diffusion(RD) system of turbulence-shear flowinteraction are investigated.By the linear stability analysis,the essential conditions for Turing instability are obtained.It indicates that the emergence of cross-diffusion terms leads to the destabilizing mechanism.Then the amplitude equations and the asymptotic solutions of the model closed to the onset of instability are derived by using the weakly nonlinear analysis. 展开更多
关键词 pattern formation amplitude equation CROSS-DIFFUSION turing instability
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Turing Instability of Diffusive Predator⁃Prey System with Gompertz Growth
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作者 LI Ying PENG Yahong 《Journal of Donghua University(English Edition)》 CAS 2021年第5期459-464,共6页
This paper mainly focus on the research of a predator⁃prey system with Gompertz growth of prey.When the system does not contain diffusion,the stability conditions of positive equilibrium and the occurring condition of... This paper mainly focus on the research of a predator⁃prey system with Gompertz growth of prey.When the system does not contain diffusion,the stability conditions of positive equilibrium and the occurring condition of the Hopf bifurcation are obtained.When the diffusion term of the system appears,the stable conditions of positive equilibrium and the Turing instability condition are also obtained.Turing instability is induced by the diffusion term through theoretical analysis.Thus,the region of parameters in which Turing instability occurs is presented.Then the amplitude equations are derived by the multiple scale method.The results will enrich the pattern dynamics in predator⁃prey systems. 展开更多
关键词 predator⁃prey system Gompertz growth stability analysis turing instability amplitude equation
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Cross-diffusion induced Turing instability of Hopf bifurcating periodic solutions in the reaction-diffusion enzyme reaction model
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作者 Haicheng Liu Wenshuo Yuan +1 位作者 Bin Ge Jihong Shen 《International Journal of Biomathematics》 SCIE 2024年第4期133-150,共18页
Aiming at the spatial pattern phenomenon in biochemical reactions,an enzyme-reaction Sporns-Seelig model with cross-diffusion is chosen as study object.Applying the central manifold theory,normal form method,local Hop... Aiming at the spatial pattern phenomenon in biochemical reactions,an enzyme-reaction Sporns-Seelig model with cross-diffusion is chosen as study object.Applying the central manifold theory,normal form method,local Hopf bifurcation theorem and perturbation theory,we study Turing instability of the spatially homogeneous Hopf bifurcation periodic solutions.At last,the theoretical results are verified by numerical simulations. 展开更多
关键词 Sporns-Seelig model diffusion Hopf bifurcation periodic solutions turing instability
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Bifurcation and Turing instability for genetic regulatory networks with diffusion
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作者 Hongyan Sun Jianzhi Cao +1 位作者 Peiguang Wang Haijun Jiang 《International Journal of Biomathematics》 SCIE 2023年第2期1-30,共30页
In this paper,a diffusive genetic regulatory network under Neumann boundary conditions is considered.First,the criteria for the local stability and diffusion-driven instability of the positive stationary solution with... In this paper,a diffusive genetic regulatory network under Neumann boundary conditions is considered.First,the criteria for the local stability and diffusion-driven instability of the positive stationary solution without and with diffusion are investigated,respectively.Moreover,Turing regions and pattern formation are obtained in the plane of diffusion coeficients.Second,the existence and multiplicity of spatially homogeneous/nonhomogeneous non-constant steady-states are studied by using the Lyapunov-Schmidt reduction.Finally,some numerical simulations are carried out to illustrate the theoretical results. 展开更多
关键词 Genetic regulatory networks DIFFUSION turing instability pattern formation BIFURCATION Lyapunov-Schmidt reduction.
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Bifurcation and Turing Pattern Formation in a Diffusion Modified Leslie-Gower Predator-Prey Model with Crowley-Martin Functional Response
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作者 Dong Wang Yani Ma 《Journal of Applied Mathematics and Physics》 2024年第6期2190-2211,共22页
In this paper, we study a modified Leslie-Gower predator-prey model with Smith growth subject to homogeneous Neumann boundary condition, in which the functional response is the Crowley-Martin functional response term.... In this paper, we study a modified Leslie-Gower predator-prey model with Smith growth subject to homogeneous Neumann boundary condition, in which the functional response is the Crowley-Martin functional response term. Firstly, for ODE model, the local stability of equilibrium point is given. And by using bifurcation theory and selecting suitable bifurcation parameters, we find many kinds of bifurcation phenomena, including Transcritical bifurcation and Hopf bifurcation. For the reaction-diffusion model, we find that Turing instability occurs. Besides, it is proved that Hopf bifurcation exists in the model. Finally, numerical simulations are presented to verify and illustrate the theoretical results. 展开更多
关键词 Modified Leslie-Gower Model Crowley-Martin Function Response Hopf Bifurcation Transcritical Bifurcation turing instability
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Turing instability in a diffusive SIS epidemiological model 被引量:1
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作者 Shaban Aly Houari B. Khenous Fatma Hussien 《International Journal of Biomathematics》 2015年第1期69-79,共11页
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... 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 "ihlring space (the space which the emergence of spatial patterns is holding) compared to the ~lhlring 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. 展开更多
关键词 SIS epidemiological model reaction-diffusion equation diffusive instability turing instability.
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Turing pattern selection for a plant-wrack model with cross-diffusion
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作者 孙颖 王进良 +2 位作者 李由 江南 夏娟迪 《Chinese Physics B》 SCIE EI CAS CSCD 2023年第9期128-136,共9页
We investigate the Turing instability and pattern formation mechanism of a plant-wrack model with both self-diffusion and cross-diffusion terms.We first study the effect of self-diffusion on the stability of equilibri... We investigate the Turing instability and pattern formation mechanism of a plant-wrack model with both self-diffusion and cross-diffusion terms.We first study the effect of self-diffusion on the stability of equilibrium.We then derive the conditions for the occurrence of the Turing patterns induced by cross-diffusion based on self-diffusion stability.Next,we analyze the pattern selection by using the amplitude equation and obtain the exact parameter ranges of different types of patterns,including stripe patterns,hexagonal patterns and mixed states.Finally,numerical simulations confirm the theoretical results. 展开更多
关键词 plant-wrack model CROSS-DIFFUSION turing instability pattern selection amplitude equation
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Superlattice Patterns in Coupled Turing Systems 被引量:1
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作者 刘富成 贺亚峰 潘宇扬 《Communications in Theoretical Physics》 SCIE CAS CSCD 2010年第5期971-976,共6页
In this paper, superlattice patterns have been investigated by using a two linearly coupled Brusselator model. It is found that superlattice patterns can only be induced in the sub-system with the short wavelength. Th... In this paper, superlattice patterns have been investigated by using a two linearly coupled Brusselator model. It is found that superlattice patterns can only be induced in the sub-system with the short wavelength. Three different coupling methods have been used in order to investigate the mode interaction between the two Turing modes. It is proved in the simulations that interaction between activators in the two sub-systems leads to spontaneous formation of black eye pattern and/or white eye patterns while interaction between inhibitors leads to spontaneous formation of super-hexagonal pattern. It is also demonstrated that the same symmetries of the two modes and suitable wavelength ratio of the two modes should also be satisfied to form superlattice patterns. 展开更多
关键词 superlattice pattern turing instability mode interaction
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Pattern selection in a predation model with self and cross diffusion
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作者 王玮明 王文娟 +1 位作者 林晔智 谭永基 《Chinese Physics B》 SCIE EI CAS CSCD 2011年第3期300-307,共8页
In this paper, we present the amplitude equations for the excited modes in a cross-diffusive predator-prey model with zero-flux boundary conditions. From these equations, the stability of patterns towards uniform and ... In this paper, we present the amplitude equations for the excited modes in a cross-diffusive predator-prey model with zero-flux boundary conditions. From these equations, the stability of patterns towards uniform and inhomogenous perturbations is determined. Furthermore, we present novel numerical evidence of six typical turing patterns, and find that the model dynamics exhibits complex pattern replications: for μ1 〈μ ≤μ2, the steady state is the only stable solution of the model; for μ2 〈 μ ≤ μ4, by increasing the control parameter μ, the sequence Hπ-hexagons→ Hπ- hexagon-stripe mixtures → stripes → H0-hexagon-stripe mixtures → H0-hexagons is observed; for μ 〉 μ4, the stripe pattern emerges. This may enrich the pattern formation in the cross-diffusive predatorprey model. 展开更多
关键词 CROSS-DIFFUSION turing instability pattern selection amplitude equations
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Pattern dynamics of network-organized system with cross-diffusion
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作者 Qianqian Zheng Zhijie Wang Jianwei Shen 《Chinese Physics B》 SCIE EI CAS CSCD 2017年第2期80-85,共6页
Cross-diffusion is a ubiquitous phenomenon in complex networks, but it is often neglected in the study of reaction–diffusion networks. In fact, network connections are often random. In this paper, we investigate patt... Cross-diffusion is a ubiquitous phenomenon in complex networks, but it is often neglected in the study of reaction–diffusion networks. In fact, network connections are often random. In this paper, we investigate pattern dynamics of random networks with cross-diffusion by using the method of network analysis and obtain a condition under which the network loses stability and Turing bifurcation occurs. In addition, we also derive the amplitude equation for the network and prove the stability of the amplitude equation which is also an effective tool to investigate pattern dynamics of the random network with cross diffusion. In the meantime, the pattern formation consistently matches the stability of the system and the amplitude equation is verified by simulations. A novel approach to the investigation of specific real systems was presented in this paper. Finally, the example and simulation used in this paper validate our theoretical results. 展开更多
关键词 cross diffusion random network turing instability amplitude equation
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Modeling the dynamics of information propagation in the temporal and spatial environment 被引量:1
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作者 Yi Zhang Linhe Zhu 《Communications in Theoretical Physics》 SCIE CAS CSCD 2023年第9期22-35,共14页
In this paper, we try to establish a non-smooth susceptible–infected–recovered(SIR) rumor propagation model based on time and space dimensions. First of all, we prove the existence and uniqueness of the solution. Se... In this paper, we try to establish a non-smooth susceptible–infected–recovered(SIR) rumor propagation model based on time and space dimensions. First of all, we prove the existence and uniqueness of the solution. Secondly, we divide the system into two parts and discuss the existence of equilibrium points for each of them. For the left part, we define R_(0) to study the relationship between R_(0) and the existence of equilibrium points. For the right part, we classify many different cases by discussing the coefficients of the equilibrium point equation. Then, on this basis, we perform a bifurcation analysis of the non-spatial system and find conditions that lead to the existence of saddle-node bifurcation. Further, we consider the effect of diffusion. We specifically analyze the stability of equilibrium points. In addition, we analyze the Turing instability and Hopf bifurcation occurring at some equilibrium points. According to the Lyapunov number, we also determine the direction of the bifurcation. When I = I_(c), we discuss conditions for the existence of discontinuous Hopf bifurcation. Finally, through numerical simulations and combined with the practical meaning of the parameters, we prove the correctness of the previous theoretical theorem. 展开更多
关键词 non-smooth system rumor propagation turing instability Hopf bifurcation
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Qualitative analysis for a Wolbachia infection model with diffusion 被引量:7
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作者 HUANG Mu Gen YU Jian She +1 位作者 HU Lin Chao ZHENG Bo 《Science China Mathematics》 SCIE CSCD 2016年第7期1249-1266,共18页
We consider a reaction-diffusion model which describes the spatial Wolbachia spread dynamics for a mixed population of infected and uninfected mosquitoes. By using linearization method, comparison principle and Leray-... We consider a reaction-diffusion model which describes the spatial Wolbachia spread dynamics for a mixed population of infected and uninfected mosquitoes. By using linearization method, comparison principle and Leray-Schauder degree theory, we investigate the influence of diffusion on the Wolbachia infection dynamics.After identifying the system parameter regions in which diffusion alters the local stability of constant steadystates, we find sufficient conditions under which the system possesses inhomogeneous steady-states. Surprisingly,our mathematical analysis, with the help of numerical simulations, indicates that diffusion is able to lower the threshold value of the infection frequency over which Wolbachia can invade the whole population. 展开更多
关键词 dengue fever Wolbachia infection dynamics cytoplasmic incompatibility mechanism attractive region turing instability non-constant steady-states
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Exploring the dynamics of a Holling-Tanner model with cannibalism in both predator and prey population 被引量:3
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作者 Aladeen Al Basheer Rana D. Parshad +2 位作者 Emmanuel Quansah Shengbin Yu Ranjit Kumar Upadhyay 《International Journal of Biomathematics》 SCIE 2018年第1期225-253,共29页
Cannibalism is an intriguing life history trait, that has been considered primarily in the predator, in predator-prey population models. Recent experimental evidence shows that prey cannibalism can have a significant ... Cannibalism is an intriguing life history trait, that has been considered primarily in the predator, in predator-prey population models. Recent experimental evidence shows that prey cannibalism can have a significant impact on predator-prey population dyna- mics in natural communities. Motivated by these experimental results, we investigate a ratio-dependent Holling-Tanner model, where cannibalism occurs simultaneously in both the predator and prey species. We show that depending on parameters, whilst prey or predator cannibalism acting alone leads to instability, their joint effect can actually stabilize the unstable interior equilibrium. Furthermore, in the spatially explicit model, we find that depending on parameters, prey and predator cannibalism acting jointly can cause spatial patterns to form, while not so acting individually. We discuss ecologicalconsequences of these findings in light of food chain dynamics, invasive species control and climate change. 展开更多
关键词 Holling-Tanner model prey cannibalism predator cannibalism stability global attraction turing instability.
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A diffusive predator-prey system with additional food and intra-specific competition among predators 被引量:1
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作者 Ruizhi Yang Ming Liu Chunrui Zhangt 《International Journal of Biomathematics》 SCIE 2018年第4期301-328,共28页
In this paper, a diffusive predator-prey system with additional food and intra-specific competition among predators subject to Neumann boundary condition is investigated. For non-delay system, global stability, Turing... In this paper, a diffusive predator-prey system with additional food and intra-specific competition among predators subject to Neumann boundary condition is investigated. For non-delay system, global stability, Turing instability and Hopf bifurcation are studied. For delay system, instability and Hopf bifurcation induced by time delay and global stability of boundary equilibrium are discussed. By the theory of normal form and center manifold method, the conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution are derived. 展开更多
关键词 PREDATOR-PREY additional food DELAY turing instability Hopf bifurcation.
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Stability and Hopf bifurcation analysis of a diffusive predator-prey model with Smith growth 被引量:4
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作者 M. Sivakumar M. Sambath K. Balachandran 《International Journal of Biomathematics》 2015年第1期163-180,共18页
In this paper, we consider a diffusive Holling-Tanner predator prey model with Smith growth subject to Neumann boundary condition. We analyze the local stability, exis- tence of a Hopf bifurcation at the co-existence ... In this paper, we consider a diffusive Holling-Tanner predator prey model with Smith growth subject to Neumann boundary condition. We analyze the local stability, exis- tence of a Hopf bifurcation at the co-existence of the equilibrium and stability of bifur- cating periodic solutions of the system in the absence of diffusion. Furthermore the Turing instability and Hopf bifurcation analysis of the system with diffusion are studied. Finally numerical simulations are given to demonstrate the effectiveness of the theoretical analysis. 展开更多
关键词 Stability analysis diffusive Holling-Tanner predator-prey model Smith growth turing instability.
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Modeling and numerical simulations for a prey-predator model with interference among predators
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作者 M.S.Surendar M.Sambath 《International Journal of Modeling, Simulation, and Scientific Computing》 EI 2021年第1期29-52,共24页
In this paper,we modeled a prey–predator system with interference among predators using the Crowley–Martin functional response.The local stability and existence of Hopf bifurcation at the coexistence equilibrium of ... In this paper,we modeled a prey–predator system with interference among predators using the Crowley–Martin functional response.The local stability and existence of Hopf bifurcation at the coexistence equilibrium of the system in the absence of diffusion are analyzed.Further,the stability of bifurcating periodic solutions is investigated.We derived the conditions for which nontrivial equilibrium is globally asymptotically stable.In addition,we study the diffusion driven instability,Hopf bifurcation of the corresponding diffusion system with zero flux boundary conditions and the Turing instability region regarding parameters are established.The stability and direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem.Numerical simulations are performed to illustrate the theoretical results. 展开更多
关键词 STABILITY Crowley-Martin model Hopf bifurcation Periodic solutions turing instability
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Pattern dynamics of a diffusive predator-prey model with delay effect
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作者 Guangping Hu Xiaoling Li Dongliang Li 《International Journal of Biomathematics》 2017年第4期353-370,共18页
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... 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. 展开更多
关键词 Hopf bifurcation turing instability spatiotemporal pattern.
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Dynamics and pattern formation in a modified Leslie-Gower model with Allee effect and Bazykin functional response
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作者 Peng Feng 《International Journal of Biomathematics》 2017年第5期301-326,共26页
In this paper, we study the dynamics of a diffusive modified Leslie-Cower model with the multiplicative Allee effect and Ba^zykin functional response. We give detailed study on the stability of equilibria. Non-existen... In this paper, we study the dynamics of a diffusive modified Leslie-Cower model with the multiplicative Allee effect and Ba^zykin functional response. We give detailed study on the stability of equilibria. Non-existence of non-constant positive steady state solutions are shown to identify the rage of parameters of spatial pattern formation. We also give the conditions of Turing instability and perform a series of numerical simulations and find that the model exhibits complex patterns. 展开更多
关键词 PREDATOR-PREY LESLIE-GOWER Bazykin functional response turing instability pattern formation global stability.
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Spatial patterns created by cross-diffusion for a three-species food chain model
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作者 Canrong Tian Zhi Ling Zhigui Lin 《International Journal of Biomathematics》 2014年第2期27-49,共23页
This paper deals with the stability analysis to a three-species food chain model with crossdiffusion, the results of which show that there is no Turing instability but crossdiffusion makes the model instability possib... This paper deals with the stability analysis to a three-species food chain model with crossdiffusion, the results of which show that there is no Turing instability but crossdiffusion makes the model instability possible. We then show that the spatial patterns are spotted patterns by using numerical simulations. In order to understand why the spatial patterns happen, the existence of the nonhomogeneous steady states is investigated. Finally, using the Leray-Schauder theory, we demonstrate that cross-diffusion creates nonhomogeneous stationary patterns. 展开更多
关键词 Spatial pattern turing instability CROSS-DIFFUSION food chain.
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