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Bifurcation of Positive Solutions for a Cross-Diffusion Prey-Predator Model in Spatially Heterogeneous Environment 认领 引用
1
作者 Tian Qiaoyu Ma Zhanping 《Journal of Mathematical Research with Applications》 CSCD 2026年第4期509-526,共18页
This work analyzes a quasilinear elliptic system describing a cross-diffusion preypredator model with Dirichlet boundary conditions and spatially heterogeneous environment.Treating the predator's intrinsic mortali... This work analyzes a quasilinear elliptic system describing a cross-diffusion preypredator model with Dirichlet boundary conditions and spatially heterogeneous environment.Treating the predator's intrinsic mortality rate k as a bifurcation parameter,we establish the global bifurcation of positive solutions from the semi-trivial solution branch using bifurcation theory and the method of upper and lower solutions.We prove that when the prey birth rate r is relatively high(i.e.,r>λ1[-D1(0)∆;m(x)])and k lies within a moderate interval(i.e.,H(r)<k<λ1[-D2(0)∆]),coexistence of prey and predator occurs,that is,the system admits positive solutions,where λ1[-D1(0)∆;m(x)]represents the principal eigenvalue of the boundary value problem(2.4)and H(r)is defined in(4.1).Furthermore,we derive precise conditions determining the direction of this bifurcation.Our results highlight the critical influence of the spatial distribution of prey resources m(x)on the existence of positive solutions for the system. 展开更多
关键词 prey-predator model cross-diffusion spatial heterogeneity environments positive solutions Bifurcation direction
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Pattern formation induced by cross-diffusion in a predator-prey system 认领 引用 被引量：7
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作者 孙桂全 靳祯 +1 位作者 刘权兴 李莉 《Chinese Physics B》 SCIE EI CAS 2008年第11期3936-3941,共6页
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
关键词 Holling-Tanner cross-diffusion pattern structures
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Positive Solutions Bifurcating from Zero Solution in a Lotka-Volterra Competitive System with Cross-Diffusion Effects 认领 引用 被引量：8
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作者 ZHANG Cun-hua YAN Xiang-ping 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2011年第3期342-352,共11页
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... 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
关键词 Lotka-Volterra competitive system Cross-Diffusion Positive solution Steady state bifurcation Stability.
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Global existence of weak solutions to a prey-predator model with strong cross-diffusion 认领 引用
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作者 李慧玲 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2009年第6期727-740,共14页
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
关键词 prey-predator model strong cross-diffusion entropy functional existenceof weak solutions Orlicz space
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Convergence Towards the Population Cross-Diffusion System from Stochastic Many-Particle System 认领 引用 被引量：1
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作者 Yue Li Li Chen Zhipeng Zhang 《Communications in Mathematical Research》 CSCD 2024年第1期43-63,共21页
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
关键词 Stochastic particle systems cross-diffusion system mean-field limit population dynamics
Spatial Pattern Induced by Cross-Diffusion in a Chemostat Model with Maintenance Energy 认领 引用 被引量：1
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作者 LIU Qingsheng PENG Yahong 《Journal of Donghua University(English Edition)》 CAS 2018年第6期469-472,共4页
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
关键词 pattern formation chemostat model cross-diffusion Turing instability stability
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Turing pattern selection for a plant-wrack model with cross-diffusion 认领 引用
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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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Turing Instability and Pattern Induced by Cross-Diffusion for a Nonlinear Reaction-Diffusion System of Turbulence-Shear Flow Interaction 认领 引用
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作者 周辉 彭亚红 《Journal of Donghua University(English Edition)》 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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The Existence and Non-Existence of Positive Steady State Solutions for a Cross-Diffusion Predator-Prey Model with Holling Type Ⅱ Functional Response 认领 引用
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作者 Shuping CAO Li-na ZHANG 《Journal of Mathematical Research with Applications》 CSCD 2021年第4期383-392,共10页
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
关键词 predator-prey model Holling typeⅡfunctional response cross-diffusion coexistence states
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Global Existence of Solutions to the Prey-predator System of Three Species with Cross-diffusion 认领 引用 被引量：1
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作者 CHEN Zhi-hui CHEN Xia 《Chinese Quarterly Journal of Mathematics》 CSCD 2011年第1期16-20,共5页
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
关键词 prey-predator system cross-diffusion self-diffusion
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Some qualitative analyses on a vegetation-water model with cross-diffusion and internal competition 认领 引用
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作者 Gaihui Guo Anna Niu +1 位作者 Qian Cao Lixin Yang 《International Journal of Biomathematics》 SCIE 2025年第3期221-249,共29页
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
关键词 Vegetation-water model cross-diffusion turing instability steady-state bifurcation double eigenvalues
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Spatial patterns of a reaction-diffusion population system with cross-diffusion and habitat complexity 认领 引用 被引量：1
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作者 Weiyu Li Yanfeng Li Ruizhi Yang 《International Journal of Biomathematics》 SCIE 2025年第8期69-90,共22页
In this paper,a population system with cross-diffusion and habitat complexity is selected as study object.We investigate that how cross-diffusion and habitat complexity destabilize the otherwise stable periodic soluti... In this paper,a population system with cross-diffusion and habitat complexity is selected as study object.We investigate that how cross-diffusion and habitat complexity destabilize the otherwise stable periodic solutions of the ODEs to generate the new abundant spatial Turing patterns.By utilizing the local Hopf bifurcation theorem and perturbation theory,we establish a formula to determine the Turing instability of periodic solutions of the population system withcross-diffusion and habitat complexity.Finally,numerical simulations are performed to verify theoretical analysis,simultaneously,we verify the formation process of spatial Turing patterns when the cross-diffusion coefficients and habitat complexity change. 展开更多
关键词 Population system cross-diffusion habitat complexity periodic solution Turing instability
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Positive solutions for a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-Ⅱ functional response 认领 引用 被引量：8
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作者 ZHOU Jun KIM Chan-Gyun 《Science China Mathematics》 SCIE 2014年第5期991-1010,共20页
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
关键词 Lotka-Volterra prey-predator model Holling type-II functional response cross-diffusion positive solutions coexistence uniqueness degree theory
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TURING PATTERNS OF A PREDATOR-PREY MODEL WITH A MODIFIED LESLIE-GOWER TERM AND CROSS-DIFFUSIONS 认领 引用 被引量：5
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作者 GUANG-PING HU XIAO-LING LI 《International Journal of Biomathematics》 SCIE 2012年第6期203-219,共17页
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 ... 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
关键词 Cross-diffusion Turing patterns priori estimates non-constant positivesteady state.
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The existence and stability of traveling waves with transition layers for the S-K-T competition model with cross-diffusion 认领 引用 被引量：3
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作者 Wu YaPing Zhao Ye 《Science China Mathematics》 SCIE 2010年第5期124-147,共24页
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
关键词 traveling waves existence stability cross-diffusion spectral analysis unstable bundle
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CROSS-DIFFUSION INDUCED TURING PATTERNS IN A SEX-STRUCTURED PREDATOR-PREY MODEL 认领 引用 被引量：2
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作者 JIA LIU HUA ZHOU LAI ZHANG 《International Journal of Biomathematics》 SCIE 2012年第4期39-61,共23页
In this paper, we consider a sex-structured predator prey model with strongly coupled nonlinear reaction diffusion. Using the Lyapunov functional and Leray Schauder degree theory, the existence and stability of both h... In this paper, we consider a sex-structured predator prey model with strongly coupled nonlinear reaction diffusion. Using the Lyapunov functional and Leray Schauder degree theory, the existence and stability of both homogenous and heterogenous steady-states are investigated. Our results demonstrate that the unique homogenous steady-state is locally asymptotically stable for the associated ODE system and PDE system with self-diffusion. With the presence of the cross-diffusion, the homogeneous equilibrium is destabilized, and a heterogenous steady-state emerges as a consequence. In addition, the conditions guaranteeing the emergence of Turing patterns are derived. 展开更多
关键词 Predator-prey model cross-diffusion turing pattern sex structure.
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Spatial patterns created by cross-diffusion for a three-species food chain model 认领 引用 被引量：1
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作者 Canrong Tian Zhi Ling Zhigui Lin 《International Journal of Biomathematics》 SCIE 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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The Global Existence of Solutions for a Cross-diffusion System 认领 引用 被引量：1
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作者 Yi Wang 《Acta Mathematicae Applicatae Sinica》 2005年第3期519-528,共10页
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
关键词 Cross-diffusion system self-diffusion global existence a priori estimates
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Stationary Patterns of a Ratio-dependent Prey-predator Model with Cross-diffusion 认领 引用 被引量：3
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作者 Jing-fu ZHAO Hong-tao ZHANG Jing YANG 《Acta Mathematicae Applicatae Sinica》 SCIE CSCD 2017年第2期497-504,共8页
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
关键词 existence predator-prey system cross-diffusion positive steady state ratio-dependence topolog-ical degree
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Coexistence in a mutualistic model with cross-diffusion in a heterogeneous environment 认领 引用 被引量：2
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作者 Badradeen Adam Zhigui Lin Abdelrazig K. Tarboush 《International Journal of Biomathematics》 SCIE 2018年第6期59-75,共17页
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
关键词 Mutualistic model strongly-coupled cross-diffusion coexistence.
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