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
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
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
The homogeneity-breaking instability of the periodic solutions triggered by Hopf bifurcations of a diffusive Gierer-Meinhart system is studied in this paper.Sufficient conditions on the diffusion coefficients and the ...The homogeneity-breaking instability of the periodic solutions triggered by Hopf bifurcations of a diffusive Gierer-Meinhart system is studied in this paper.Sufficient conditions on the diffusion coefficients and the cross diffusion coefficients were derived to guarantee the occurrence of the aforementioned homogeneity-breaking instability.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
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 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
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Sponsored by the National Natural Science Foundation of China(Grant Nos.12061033,2020GG0130,2020MS04007,2020BS11,and NJZZ22286).
文摘The homogeneity-breaking instability of the periodic solutions triggered by Hopf bifurcations of a diffusive Gierer-Meinhart system is studied in this paper.Sufficient conditions on the diffusion coefficients and the cross diffusion coefficients were derived to guarantee the occurrence of the aforementioned homogeneity-breaking instability.
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 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.
