A strongly coupled elliptic system under the homogeneous Dirichlet boundary condition denoting the steady-state system of the Lotka-Volterra two-species competitive system with cross-diffusion effects is considered. B...A strongly coupled elliptic system under the homogeneous Dirichlet boundary condition denoting the steady-state system of the Lotka-Volterra two-species competitive system with cross-diffusion effects is considered. By using the implicit function theorem and the Lyapunov- Schmidt reduction method, the existence of the positive solutions bifurcating from the trivial solution is obtained. Furthermore, the stability of the bifurcating positive solutions is also investigated by analyzing the associated characteristic equation.展开更多
This paper considers the Holling-Tanner model for predator-prey with self and cross-diffusion. From the Turing theory, it is believed that there is no Turing pattern formation for the equal self-diffusion coefficients...This paper considers the Holling-Tanner model for predator-prey with self and cross-diffusion. From the Turing theory, it is believed that there is no Turing pattern formation for the equal self-diffusion coefficients. However, combined with cross-diffusion, it shows that the system will exhibit spotted pattern by both mathematical analysis and numerical simulations. Furthermore, asynchrony of the predator and the prey in the space. The obtained results show that cross-diffusion plays an important role on the pattern formation of the predator-prey system.展开更多
A chemostat model with maintenance energy and crossdiffusion is considered,and the formation of patterns is caused by the cross-diffusion. First, through linear stability analysis, the necessary conditions for the for...A chemostat model with maintenance energy and crossdiffusion is considered,and the formation of patterns is caused by the cross-diffusion. First, through linear stability analysis, the necessary conditions for the formation of the spatial patterns are given. Then numerical simulations by changing the values of crossdiffusions in the unstable domain are performed. The results showthat the cross-diffusion coefficient plays an important role in the formation of the pattern, and the different values of the crossdiffusion coefficients may lead to different types of pattern formation.展开更多
The prey-predator system of three species with cross-diffusion pressure is known to possess a local solution with the maximal existence time T ≤ ∞.By obtaining the bounds of W21-norms of the local solution independe...The prey-predator system of three species with cross-diffusion pressure is known to possess a local solution with the maximal existence time T ≤ ∞.By obtaining the bounds of W21-norms of the local solution independent of T,it is established the global existence of the solution.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
Using finite differences and entropy inequalities, the global existence of weak solutions to a multidimensional parabolic strongly coupled prey-predator model is obtained. The nonnegativity of the solutions is also sh...Using finite differences and entropy inequalities, the global existence of weak solutions to a multidimensional parabolic strongly coupled prey-predator model is obtained. The nonnegativity of the solutions is also shown.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
In this paper,we consider the positive steady state solutions of a predator-prey model with Holling type Ⅱfunctional response and cross-diffusion,where two cross-diffusion rates represent the tendency of prey to keep...In this paper,we consider the positive steady state solutions of a predator-prey model with Holling type Ⅱfunctional response and cross-diffusion,where two cross-diffusion rates represent the tendency of prey to keep away from its predator and the tendency of the predator to chase its prey,respectively.Applying the fixed point index theory,some sufficient conditions for the existence of positive steady state solutions are established.Furthermore,the non-existence of positive steady state solutions is studied.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 derive rigorously a non-local cross-diffusion system from an interacting stochastic many-particle system in the whole space.The convergence is proved in the sense of probability by introducing an inte...In this paper,we derive rigorously a non-local cross-diffusion system from an interacting stochastic many-particle system in the whole space.The convergence is proved in the sense of probability by introducing an intermediate particle system with a mollified interaction potential,where the mollification is of algebraic scaling.The main idea of the proof is to study the time evolution of a stopped process and obtain a Gronwall type estimate by using Taylor's expansion around the limiting stochastic process.展开更多
In order to understand the transmission mechanism of West Nile virus(WNv)in birds(especially crows)and mosquitoes populations,this paper extends the traditional ordinary differential model of WNv to a reaction-diffusi...In order to understand the transmission mechanism of West Nile virus(WNv)in birds(especially crows)and mosquitoes populations,this paper extends the traditional ordinary differential model of WNv to a reaction-diffusion system with more complex cross-diffusion.We explore the relationship between the basic reproduction number and cross-diffusion coefficients involving various parameters,and investigate the effect of vertical transmission of the virus on the transmission mechanism.We use the method of upper and lower solutions to investigate the existence of the coexistence solutions.The theoretical analysis and numerical simulation show that the WNv carried by birds and mosquitoes will coexist when the low-risk threshold R0*≥1,which is a disadvantage to the prevention and control of this virus,and disappear when the high-risk threshold R*0≤1,which is an advantage to that.展开更多
We consider a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-II functional response.The main concern is the existence of positive solutions under the combined effect of cross-diffusion and Ho...We consider a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-II functional response.The main concern is the existence of positive solutions under the combined effect of cross-diffusion and Holling type-II functional response.Here,a positive solution corresponds to a coexistence state of the model.Firstly,we study the sufficient conditions to ensure the existence of positive solutions by using degree theory and analyze the coexistence region in parameter plane.In addition,we present the uniqueness of positive solutions in one dimension case.Secondly,we study the stability of the trivial and semi-trivial solutions by analyzing the principal eigenvalue of the corresponding linearized system,and then we characterize the stable/unstable regions of semi-trivial solutions in parameter plane.展开更多
This paper is concerned with the existence and stability of traveling waves with transition layers for a quasi-linear competition system with cross diffusion,which was first proposed by Shegesada,Kawasaki and Teramoto...This paper is concerned with the existence and stability of traveling waves with transition layers for a quasi-linear competition system with cross diffusion,which was first proposed by Shegesada,Kawasaki and Teramoto.When one of the random diffusion rates is small and the cross-diffusion rate is not small,by the geometric singular perturbation method,the existence of traveling waves with transition layers is obtained.Further,by the detailed spectral analysis and topological index method,the traveling waves with transition layers are proved to be locally exponentially stable with shift.展开更多
This paper is concerned with a ratio-dependent predator-prey system with diffusion and cross- diffusion in a bounded domain with no flux boundary condition. We show that under certain hypotheses, the cross-diffusion c...This paper is concerned with a ratio-dependent predator-prey system with diffusion and cross- diffusion in a bounded domain with no flux boundary condition. We show that under certain hypotheses, the cross-diffusion can create non-constant positive steady states even though the corresponding model without cross-diffusion fails.展开更多
To understand the impact of environmental heterogeneity and mutualistic interaction of species, we consider a mutualistic model with cross-diffusion in a heterogeneous environ- ment. Semi-coexistence states have been ...To understand the impact of environmental heterogeneity and mutualistic interaction of species, we consider a mutualistic model with cross-diffusion in a heterogeneous environ- ment. Semi-coexistence states have been studied by using the corresponding eigenvalue problems, and sufficient conditions for the existence and non-existence of coexistence states are given. Our results show that the model possesses at least one coexistence solution if the intrinsic populations growth rates are big or free-diffusion and cross-diffusion coefficients are weak. Otherwise, the model have no coexistence solution. The true solutions are obtained by utilizing the monotone iterative schemes. In order to illustrate our analytical results, some numerical simulations are given.展开更多
In this paper, we investigate the positive solutions of strongly coupled nonlinear parabolic systems with nonlinear boundary conditions： {ut-a（u, v）△u=g（u, v）, vt-b（u, v）△v=h（u, v）, δu/δη=d（u, v）, δu...In this paper, we investigate the positive solutions of strongly coupled nonlinear parabolic systems with nonlinear boundary conditions： {ut-a（u, v）△u=g（u, v）, vt-b（u, v）△v=h（u, v）, δu/δη=d（u, v）, δu/δη=f（u, v）.Under appropriate hypotheses on the functions a, b, g, h, d and f, we obtain that the solutions may exist globally or blow up in finite time by utilizing upper and lower solution techniques.展开更多
This paper is concerned with the global existence of solutions for a class of quasilinear cross-diffusion system describing two species competition under self and cross population pressure. By establishing and using m...This paper is concerned with the global existence of solutions for a class of quasilinear cross-diffusion system describing two species competition under self and cross population pressure. By establishing and using more detailed interpolation results between several different Banach spaces, the global existence of solutions are proved when the self and cross diffusion rates for the first species are positive and there is no self or cross-diffusion for the second species.展开更多
In this paper, a strongly coupled diffusive predator-prey system with a modified Leslie- Gower term is considered. We will show that under certain hypotheses, even though the unique positive equilibrium is asymptotica...In this paper, a strongly coupled diffusive predator-prey system with a modified Leslie- Gower term is considered. We will show that under certain hypotheses, even though the unique positive equilibrium is asymptotically stable for the dynamics with diffusion, Turing instability can produce due to the presence of the cross-diffusion. In particular, we establish the existence of non-constant positive steady states of this system. The results indicate that cross-diffusion can create stationary patterns.展开更多
In this paper,we propose a new application of a recent semi-numericalsuccessive linearization method(SLM)in solving highly coupled,nonlinear boundaryvalue problem.The method is presented in detail by solving the probl...In this paper,we propose a new application of a recent semi-numericalsuccessive linearization method(SLM)in solving highly coupled,nonlinear boundaryvalue problem.The method is presented in detail by solving the problem of steady flowof mixed convection and an incompressible viscous hydromagnetic fluid from a verticalflat plate embedded in a fluid-saturated porous medium.The governing partial differentialequations are transformed into a system of ordinary differential equations and then solvedby SLM.The effects of different physical parameters on the velocity,temperature,andconcentration profiles are determined and discussed.The skin-friction,and heat and masstransfer coefficients have been obtained and analyzed for various physical parametricvalues.The results are presented numerically through graphs and tables for both assistingand opposing flows to observe the effects of various parameters encountered in the equations.展开更多
Using the energy estimate and Gagliardo-Nirenberg-type inequalities,the existence and uniform boundedness of the global solutions to a strongly coupled reaction-diffusion system are proved. This system is a generaliza...Using the energy estimate and Gagliardo-Nirenberg-type inequalities,the existence and uniform boundedness of the global solutions to a strongly coupled reaction-diffusion system are proved. This system is a generalization of the two-species Lotka-Volterra predator-prey model with self and cross-diffusion. Suffcient condition for the global asymptotic stability of the positive equilibrium point of the model is given by constructing Lyapunov function.展开更多
基金Supported by the National Natural Science Foundation of China (10961017)"Qinglan" Talent Programof Lanzhou Jiaotong University (QL-05-20A)
文摘A strongly coupled elliptic system under the homogeneous Dirichlet boundary condition denoting the steady-state system of the Lotka-Volterra two-species competitive system with cross-diffusion effects is considered. By using the implicit function theorem and the Lyapunov- Schmidt reduction method, the existence of the positive solutions bifurcating from the trivial solution is obtained. Furthermore, the stability of the bifurcating positive solutions is also investigated by analyzing the associated characteristic equation.
基金Project supported by the National Natural Science Foundation of China (Grant No 60771026)Program for New Century Excellent Talents in University of China (Grant No NCET050271)the Special Scientific Research Foundation for the Subjects of Doctors in University of China (Grant No 20060110005)
文摘This paper considers the Holling-Tanner model for predator-prey with self and cross-diffusion. From the Turing theory, it is believed that there is no Turing pattern formation for the equal self-diffusion coefficients. However, combined with cross-diffusion, it shows that the system will exhibit spotted pattern by both mathematical analysis and numerical simulations. Furthermore, asynchrony of the predator and the prey in the space. The obtained results show that cross-diffusion plays an important role on the pattern formation of the predator-prey system.
基金National Natural Science Foundation of China(No.11571227)
文摘A chemostat model with maintenance energy and crossdiffusion is considered,and the formation of patterns is caused by the cross-diffusion. First, through linear stability analysis, the necessary conditions for the formation of the spatial patterns are given. Then numerical simulations by changing the values of crossdiffusions in the unstable domain are performed. The results showthat the cross-diffusion coefficient plays an important role in the formation of the pattern, and the different values of the crossdiffusion coefficients may lead to different types of pattern formation.
基金Supported by the Fundamental Research Funds for the Central Universities SCUT(2009ZM0014)
文摘The prey-predator system of three species with cross-diffusion pressure is known to possess a local solution with the maximal existence time T ≤ ∞.By obtaining the bounds of W21-norms of the local solution independent of T,it is established the global existence of the solution.
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supported by the National Natural Science Foundation of China (Nos. 10701024, 10601011)
文摘Using finite differences and entropy inequalities, the global existence of weak solutions to a multidimensional parabolic strongly coupled prey-predator model is obtained. The nonnegativity of the solutions is also shown.
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Supported by the National Natural Science Foundation of China(Grant No.11761063).
文摘In this paper,we consider the positive steady state solutions of a predator-prey model with Holling type Ⅱfunctional response and cross-diffusion,where two cross-diffusion rates represent the tendency of prey to keep away from its predator and the tendency of the predator to chase its prey,respectively.Applying the fixed point index theory,some sufficient conditions for the existence of positive steady state solutions are established.Furthermore,the non-existence of positive steady state solutions is studied.
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.
基金funding from the European Research Council (ERC)under the European Union's Horizon 2020 research and innovation programme,ERC Advanced Grant No.101018153support from the Austrian Science Fund (FWF) (Grants P33010,F65)supported by the NSFC (Grant No.12101305).
文摘In this paper,we derive rigorously a non-local cross-diffusion system from an interacting stochastic many-particle system in the whole space.The convergence is proved in the sense of probability by introducing an intermediate particle system with a mollified interaction potential,where the mollification is of algebraic scaling.The main idea of the proof is to study the time evolution of a stopped process and obtain a Gronwall type estimate by using Taylor's expansion around the limiting stochastic process.
基金Supported by the Anhui Province Graduate Education Quality Engineering Project(2022jyjxggyj168)Anhui Province Higher Education Provincial-Level Quality Engineering Major Project(2022jcbs020)Anhui Province Higher Education Provincial-Level Quality Engineering General Project(2022jyxm527).
文摘In order to understand the transmission mechanism of West Nile virus(WNv)in birds(especially crows)and mosquitoes populations,this paper extends the traditional ordinary differential model of WNv to a reaction-diffusion system with more complex cross-diffusion.We explore the relationship between the basic reproduction number and cross-diffusion coefficients involving various parameters,and investigate the effect of vertical transmission of the virus on the transmission mechanism.We use the method of upper and lower solutions to investigate the existence of the coexistence solutions.The theoretical analysis and numerical simulation show that the WNv carried by birds and mosquitoes will coexist when the low-risk threshold R0*≥1,which is a disadvantage to the prevention and control of this virus,and disappear when the high-risk threshold R*0≤1,which is an advantage to that.
基金supported by National Natural Science Foundation of China(Grant No.11201380)the Fundamental Research Funds for the Central Universities(Grant No.XDJK2012B007)+2 种基金Doctor Fund of Southwest University(Grant No.SWU111021)Educational Fund of Southwest University(Grant No.2010JY053)National Research Foundation of Korea Grant funded by the Korean Government(Ministry of Education,Science and Technology)(Grant No.NRF-2011-357-C00006)
文摘We consider a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-II functional response.The main concern is the existence of positive solutions under the combined effect of cross-diffusion and Holling type-II functional response.Here,a positive solution corresponds to a coexistence state of the model.Firstly,we study the sufficient conditions to ensure the existence of positive solutions by using degree theory and analyze the coexistence region in parameter plane.In addition,we present the uniqueness of positive solutions in one dimension case.Secondly,we study the stability of the trivial and semi-trivial solutions by analyzing the principal eigenvalue of the corresponding linearized system,and then we characterize the stable/unstable regions of semi-trivial solutions in parameter plane.
基金supported by National Natural Science Foundation of China (Grant No.10671131)Beijing Natural Science Foundation (Grant No.1092006)
文摘This paper is concerned with the existence and stability of traveling waves with transition layers for a quasi-linear competition system with cross diffusion,which was first proposed by Shegesada,Kawasaki and Teramoto.When one of the random diffusion rates is small and the cross-diffusion rate is not small,by the geometric singular perturbation method,the existence of traveling waves with transition layers is obtained.Further,by the detailed spectral analysis and topological index method,the traveling waves with transition layers are proved to be locally exponentially stable with shift.
基金Supported in part by the National Natural Science Foundation of China under Grant No.11601542 and 11626238
文摘This paper is concerned with a ratio-dependent predator-prey system with diffusion and cross- diffusion in a bounded domain with no flux boundary condition. We show that under certain hypotheses, the cross-diffusion can create non-constant positive steady states even though the corresponding model without cross-diffusion fails.
基金This work was partially supported by the National Natural Science Foundation of China （11771381） and Project funded by China Postdoctoral Science Foundation.
文摘To understand the impact of environmental heterogeneity and mutualistic interaction of species, we consider a mutualistic model with cross-diffusion in a heterogeneous environ- ment. Semi-coexistence states have been studied by using the corresponding eigenvalue problems, and sufficient conditions for the existence and non-existence of coexistence states are given. Our results show that the model possesses at least one coexistence solution if the intrinsic populations growth rates are big or free-diffusion and cross-diffusion coefficients are weak. Otherwise, the model have no coexistence solution. The true solutions are obtained by utilizing the monotone iterative schemes. In order to illustrate our analytical results, some numerical simulations are given.
文摘In this paper, we investigate the positive solutions of strongly coupled nonlinear parabolic systems with nonlinear boundary conditions： {ut-a（u, v）△u=g（u, v）, vt-b（u, v）△v=h（u, v）, δu/δη=d（u, v）, δu/δη=f（u, v）.Under appropriate hypotheses on the functions a, b, g, h, d and f, we obtain that the solutions may exist globally or blow up in finite time by utilizing upper and lower solution techniques.
文摘This paper is concerned with the global existence of solutions for a class of quasilinear cross-diffusion system describing two species competition under self and cross population pressure. By establishing and using more detailed interpolation results between several different Banach spaces, the global existence of solutions are proved when the self and cross diffusion rates for the first species are positive and there is no self or cross-diffusion for the second species.
文摘In this paper, a strongly coupled diffusive predator-prey system with a modified Leslie- Gower term is considered. We will show that under certain hypotheses, even though the unique positive equilibrium is asymptotically stable for the dynamics with diffusion, Turing instability can produce due to the presence of the cross-diffusion. In particular, we establish the existence of non-constant positive steady states of this system. The results indicate that cross-diffusion can create stationary patterns.
文摘In this paper,we propose a new application of a recent semi-numericalsuccessive linearization method(SLM)in solving highly coupled,nonlinear boundaryvalue problem.The method is presented in detail by solving the problem of steady flowof mixed convection and an incompressible viscous hydromagnetic fluid from a verticalflat plate embedded in a fluid-saturated porous medium.The governing partial differentialequations are transformed into a system of ordinary differential equations and then solvedby SLM.The effects of different physical parameters on the velocity,temperature,andconcentration profiles are determined and discussed.The skin-friction,and heat and masstransfer coefficients have been obtained and analyzed for various physical parametricvalues.The results are presented numerically through graphs and tables for both assistingand opposing flows to observe the effects of various parameters encountered in the equations.
基金Partially supported by the National Natural Science Foundation of China (10671158)the NSFof Gansu Province (3ZS061-A25-015)+1 种基金the Scientific Research Fund of Gansu Provincial Educatio nDepartment (0601-21)NWNU-KJCXGC-03-18, 39 Foundations
文摘Using the energy estimate and Gagliardo-Nirenberg-type inequalities,the existence and uniform boundedness of the global solutions to a strongly coupled reaction-diffusion system are proved. This system is a generalization of the two-species Lotka-Volterra predator-prey model with self and cross-diffusion. Suffcient condition for the global asymptotic stability of the positive equilibrium point of the model is given by constructing Lyapunov function.
