A new type of localized oscillatory pattern is presented in a two-layer coupled reaction-diffusion system under conditions in which no Hopf instability can be discerned in either layer.The transitions from stationary ...A new type of localized oscillatory pattern is presented in a two-layer coupled reaction-diffusion system under conditions in which no Hopf instability can be discerned in either layer.The transitions from stationary patterns to asynchronous and synchronous oscillatory patterns are obtained.A novel method based on decomposing coupled systems into two associated subsystems has been proposed to elucidate the mechanism of formation of oscillating patterns.Linear stability analysis of the associated subsystems reveals that the Turing pattern in one layer induces the other layer locally,undergoes a supercritical Hopf bifurcation and gives rise to localized oscillations.It is found that the sizes and positions of oscillations are determined by the spatial distribution of the Turing patterns.When the size is large,localized traveling waves such as spirals and targets emerge.These results may be useful for deeper understanding of pattern formation in complex systems,particularly multilayered systems.展开更多
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
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
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
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
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
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
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
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
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
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
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
In this paper,a delayed diffusive predator-prey model with fear effect under Neumann boundary conditions is considered.For the system without diffusion and delay,the conditions for the existence and local stability of...In this paper,a delayed diffusive predator-prey model with fear effect under Neumann boundary conditions is considered.For the system without diffusion and delay,the conditions for the existence and local stability of equilibria are obtained by analyzing the eigenvalues.Then,the instability induced by diffusion and delay-diffusion of the positive constant stationary solutions are discussed,respectively.Moreover,the regions of instability and pattern formation can be achieved with respect to diffusion and delay coefficients.Furthermore,the existence and direction of Hopf bifurcation and the properties of the homogeneous/nonhomogeneous bifurcated periodic solutions are driven by using the center manifold theorem and the normal form theory.Finally,some numerical simulations are carried out to verify the theoretical results.展开更多
Traditionally,the spatiotemporal dynamics of neural networks have been analyzed in the locally continuous domain.While this modeling approach is straightforward and practical,it fails to encapsulate real-world network...Traditionally,the spatiotemporal dynamics of neural networks have been analyzed in the locally continuous domain.While this modeling approach is straightforward and practical,it fails to encapsulate real-world networks’intricate topological structures and evolution.Currently,the mechanisms of stability switches induced by time delays and diffusion effects in networkorganized systems are still vague.Besides,effective dynamic optimization strategies for network-organized models remain to be devised.In this study,we develop a delayed reaction-diffusion neural network model on complex networks.This system incorporates the effects of inter-nodal diffusion coupling and exhibits profound architectural complexity.We then pioneer the proportional-integral-derivative(PID)feedback control into the network-organized model to modulate dynamic behaviors.The linear stability analysis is conducted firstly,demonstrating that Turing patterns and Hopf bifurcation can be induced in the controlled neural network by varying diffusion coefficients and time delays.Subsequently,the bifurcation direction is deduced via the center manifold theorem.Finally,a series of simulations are performed to validate the theoretical analysis and substantiate the efficacy of the PID control strategy.The results exhibit that the PID feedback controller can flexibly regulate the dynamics of the network-organized systems and possesses excellent disturbance rejection capabilities.展开更多
This paper is concerned with a vegetation-water model with cross-diffusion and intra-plant competitive feedback under Neumann boundary conditions.First,we found that the equilibrium with small vegetation density is al...This paper is concerned with a vegetation-water model with cross-diffusion and intra-plant competitive feedback under Neumann boundary conditions.First,we found that the equilibrium with small vegetation density is always unstable,and if the cross-diffusion coefficient is suitably large,the equilibrium with relatively large vegetation density loses its stability,and Turing instability occurs.A priori estimates of positive steady-state solutions are also established by the maximum principle of elliptic equations.Moreover,some qualitative analyses on the steady-state bifurcations for both simple and double eigenvalues are conducted in detail.Space decomposition and the implicit function theorem are used for double eigenvalues.In particular,the global continuation is obtained,and the result shows that there is at least one non-constant positive steady-state solution when cross-diffusion is large.Finally,numerical simulations are provided to prove and supplement theoretic research results,and some vegetation patterns with the increase of the soil water diffusion feedback intensity are formed,where the transition appears:gap→stripe→spot.展开更多
In this paper,we consider a Leslie-Gower type reaction-diffusion predator-prey system with an increasing functional response.We mainly study the effect of three different types of diffusion on the stability of this sy...In this paper,we consider a Leslie-Gower type reaction-diffusion predator-prey system with an increasing functional response.We mainly study the effect of three different types of diffusion on the stability of this system.The main results are as follows:(1)in the absence of prey diffusion,diffusion-driven instability can occur;(2)in the absence of predator diffusion,diffusion-driven instability does not occur and the non-constant stationary solution exists and is unstable;(3)in the presence of both prey diffusion and predator diffusion,the system can occur diffusion-driven instability and Turing patterns.At the same time,we also get the existence conditions of the Hopf bifurcation and the Turing-Hopf bifurcation,along with the normal form for the Turing-Hopf bifurcation.In addition,we conduct numerical simulations for all three cases to support the results of our theoretical analysis.展开更多
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
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
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
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
基金Project supported by the National Natural Science Foundation of China(Grant Nos.12275065,12275064,12475203)the Natural Science Foundation of Hebei Province(Grant Nos.A2021201010 and A2024201020)+3 种基金Interdisciplinary Research Program of Natural Science of Hebei University(Grant No.DXK202108)Hebei Provincial Central Government Guiding Local Science and Technology Development Funds(Grant No.236Z1501G)Scientific Research and Innovation Team Foundation of Hebei University(Grant No.IT2023B03)the Excellent Youth Research Innovation Team of Hebei University(Grant No.QNTD202402)。
文摘A new type of localized oscillatory pattern is presented in a two-layer coupled reaction-diffusion system under conditions in which no Hopf instability can be discerned in either layer.The transitions from stationary patterns to asynchronous and synchronous oscillatory patterns are obtained.A novel method based on decomposing coupled systems into two associated subsystems has been proposed to elucidate the mechanism of formation of oscillating patterns.Linear stability analysis of the associated subsystems reveals that the Turing pattern in one layer induces the other layer locally,undergoes a supercritical Hopf bifurcation and gives rise to localized oscillations.It is found that the sizes and positions of oscillations are determined by the spatial distribution of the Turing patterns.When the size is large,localized traveling waves such as spirals and targets emerge.These results may be useful for deeper understanding of pattern formation in complex systems,particularly multilayered systems.
基金National Natural Science Foundation of China(No.11371087)
文摘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.
基金National Natural Science Foundation of China(No.11971143)。
文摘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.
基金supported by Scientific Research and Innovation Fund for PhD Student:Research on the bifurcation problems of diffusive oncolytic virotherapy system(No.3072022CFJ2401).
文摘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.
文摘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.
基金the National Natural Science Foundation of China(No.12171135)Natural Science Foundation of Hebei Province(Nos.A2019201106 and A2020201021)Post-Graduate's Innovation Fund Project of Hebei University(No.HBU2022bs022).
文摘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.
文摘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.
基金Supported by the National Natural Science Foundation of China under Grant Nos. 10975043, 10947166 and 10775037the Foundation of Bureau of Education, Hebei Province, China under Grant No. 2009108the Natural Science Foundation of Hebei Province, China under Grant No. A2008000564)
文摘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.
基金the National Natural Science Foundation of China(Grant Nos.10971009,11771033,and12201046)Fundamental Research Funds for the Central Universities(Grant No.BLX201925)China Postdoctoral Science Foundation(Grant No.2020M670175)。
文摘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.
基金partly supported by the National Natural Science Foundation of China (Grant No. 12002135)China Postdoctoral Science Foundation (Grand No. 2023M731382)the Young Science and Technology Talents Lifting Project of Jiangsu Association for Science and Technology。
文摘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.
基金supported by the Natural Science Foundation of Zhejiang Province,China (Grant No. Y7080041)the Shanghai Postdoctoral Scientific Program,China (Grant No. 09R21410700)
文摘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.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11272277,11572278,and 11572084)the Innovation Scientists and Technicians Troop Construction Projects of Henan Province,China(Grant No.2017JR0013)
文摘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.
基金supported by the National Natural Science Foundation of China(Nos.12171135 and 11771115)the Natural Science Foundation of Hebei Province(Nos.A2020201021 and A2019201396)+1 种基金the Post Doctor Start-up Foundation of Zhejiang Normal University(No.ZC304021906)the Research Funding for High-Level Innovative Talents of Hebei University(No.801260201242).
文摘In this paper,a delayed diffusive predator-prey model with fear effect under Neumann boundary conditions is considered.For the system without diffusion and delay,the conditions for the existence and local stability of equilibria are obtained by analyzing the eigenvalues.Then,the instability induced by diffusion and delay-diffusion of the positive constant stationary solutions are discussed,respectively.Moreover,the regions of instability and pattern formation can be achieved with respect to diffusion and delay coefficients.Furthermore,the existence and direction of Hopf bifurcation and the properties of the homogeneous/nonhomogeneous bifurcated periodic solutions are driven by using the center manifold theorem and the normal form theory.Finally,some numerical simulations are carried out to verify the theoretical results.
基金supported by the National Natural Science Foundation of China(Grant Nos.62073172,62233004,62073076)the Natural Science Foundation of Jiangsu Province of China(Grant No.BK20221329)Jiangsu Provincial Scientific Research Center of Applied Mathematics(Grant No.BK20233002).
文摘Traditionally,the spatiotemporal dynamics of neural networks have been analyzed in the locally continuous domain.While this modeling approach is straightforward and practical,it fails to encapsulate real-world networks’intricate topological structures and evolution.Currently,the mechanisms of stability switches induced by time delays and diffusion effects in networkorganized systems are still vague.Besides,effective dynamic optimization strategies for network-organized models remain to be devised.In this study,we develop a delayed reaction-diffusion neural network model on complex networks.This system incorporates the effects of inter-nodal diffusion coupling and exhibits profound architectural complexity.We then pioneer the proportional-integral-derivative(PID)feedback control into the network-organized model to modulate dynamic behaviors.The linear stability analysis is conducted firstly,demonstrating that Turing patterns and Hopf bifurcation can be induced in the controlled neural network by varying diffusion coefficients and time delays.Subsequently,the bifurcation direction is deduced via the center manifold theorem.Finally,a series of simulations are performed to validate the theoretical analysis and substantiate the efficacy of the PID control strategy.The results exhibit that the PID feedback controller can flexibly regulate the dynamics of the network-organized systems and possesses excellent disturbance rejection capabilities.
基金supported by the National Natural Science Foundation of China(Nos.12101075 and 61872227).
文摘This paper is concerned with a vegetation-water model with cross-diffusion and intra-plant competitive feedback under Neumann boundary conditions.First,we found that the equilibrium with small vegetation density is always unstable,and if the cross-diffusion coefficient is suitably large,the equilibrium with relatively large vegetation density loses its stability,and Turing instability occurs.A priori estimates of positive steady-state solutions are also established by the maximum principle of elliptic equations.Moreover,some qualitative analyses on the steady-state bifurcations for both simple and double eigenvalues are conducted in detail.Space decomposition and the implicit function theorem are used for double eigenvalues.In particular,the global continuation is obtained,and the result shows that there is at least one non-constant positive steady-state solution when cross-diffusion is large.Finally,numerical simulations are provided to prove and supplement theoretic research results,and some vegetation patterns with the increase of the soil water diffusion feedback intensity are formed,where the transition appears:gap→stripe→spot.
文摘In this paper,we consider a Leslie-Gower type reaction-diffusion predator-prey system with an increasing functional response.We mainly study the effect of three different types of diffusion on the stability of this system.The main results are as follows:(1)in the absence of prey diffusion,diffusion-driven instability can occur;(2)in the absence of predator diffusion,diffusion-driven instability does not occur and the non-constant stationary solution exists and is unstable;(3)in the presence of both prey diffusion and predator diffusion,the system can occur diffusion-driven instability and Turing patterns.At the same time,we also get the existence conditions of the Hopf bifurcation and the Turing-Hopf bifurcation,along with the normal form for the Turing-Hopf bifurcation.In addition,we conduct numerical simulations for all three cases to support the results of our theoretical analysis.
基金supported by National Natural Science Foundation of China (Grant Nos. 11471085, 91230104 and 11301103)Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT1226)+1 种基金Program for Yangcheng Scholars in Guangzhou (Grant No. 12A003S)Guangdong Innovative Research Team Program (Grant No. 2011S009)
文摘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.
文摘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.
基金The authors wish to express their gratitude to the editors and the reviewers for the helpful comments. This research is supported by the National Nature Science Foundation of China （No. 11601070） and Heilongjiang Provincial Natural Science Foundation （No. A2015016）.
文摘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.
文摘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.
