the instability of spiky steady states for a competing species model with cross diffusion this paper is concerned with the stability instability of a class of positive spiky steady states for a quasi linear cross diffusion system describing two species competition by detailed spectral analysis it is proved that the spiky steady states for the related shadow system are linearly unstable and the spiky steady states for the original cross diffusion system are non linearly unstable

pattern analysis of a modified leslie gower predator prey model with crowley martin functional response and diffusion abstract in this paper we study a modified leslie gower predator prey model with crowley martin functional response and spatial diffusion under homogeneous neumann boundary condition and obtain some important qualitative properties including the existence of the global positive solution the dissipation and persistence of the two species the local and global asymptotic stability of the constant equilibria and hopf bifurcation around the interior constant equilibrium in addition we establish the existence and nonexistence of nonconstant positive steady states

effect of cross diffusion in the diffusion prey predator model with a protection zone ii abstract in the current paper we continue the mathematical analysis studied in li and wu 2017 15 and oeda 2011 22 and further study the effect of cross diffusion for the predator on the stationary problem the existence of positive solutions is first established by the bifurcation theory we next discuss the limiting behavior of positive solutions when the intrinsic growth rate of the predator species tends to infinity moreover as the prevention coefficient tends to infinity we obtain two shadow systems and give the complete limiting characterization of positive solutions

global solutions to a class of multi species reaction diffusion systems with cross diffusions arising in population dynamics in this paper an n species strongly coupled cooperating diffusive system is considered in a bounded smooth domain subject to homogeneous neumann boundary conditions employing the method of energy estimates we obtain some conditions on the diffusion matrix and inter specific cooperatives to ensure the global existence and uniform boundedness of a nonnegative solution the globally asymptotical stability of the constant positive steady state is also discussed as a consequence all the results hold true for multi species lotka volterra type competition model and prey predator model

dynamics in a diffusive predator prey system with ratio dependent predator influence abstract a diffusive predator prey system with predator functional response subject to neumann boundary conditions is considered existence nonexistence and boundedness of positive steady state solutions are shown to identify the ranges of parameters of spatial pattern formation hopf and steady state bifurcation analyses are carried out in detail these results provide theoretical evidences to the complex spatio temporal dynamics found by numerical simulation

positive steady states of the holling tanner prey predator model with diffusion this paper is concerned with the holling tanner prey predator model with diffusion subject to the homogeneous neumann boundary condition we obtain the existence and non existence of positive non constant steady states

smooth solutions to a quasi linear system of diffusion equations for a certain population model abstract the system of diffusion equations proposed by kawasaki shigesada and teramoto describes a population model of two competing species with self and cross population pressures the densities of the two species are denoted by u and v in this paper the author studies the initial boundary value problem associated with the neumann boundary condition corresponds to the case where the flux is zero at the boundary many investigators have considered nonlinear diffusion systems arising from various physical and biological problems these equations however have a special structure the highest order derivatives are not coupled or at least the coefficient matrix for the highest order derivatives is positive definite this is not the case for the system and hence some of the techniques which are effective for the system are no longer applicable to our case nevertheless we can still use sobolevski s method see reference 2 to establish the local existence of solutions under the special assumption c1 c2 in we can also prove the global existence of solutions by energy estimates the unusual structure of seems to make it difficult to settle the question of asymptotic stability of solutions

qualitative analysis of a holling tanner predator prey model with mutual interference a reaction diffusion system arising from the holling tanner predator prey model is proposed and mutual interference among predators is considered conditions for the local and global asymptotical stability of equilibrium solutions are given by means of analyzing eigenvalue spectrum as well as lyapunov functionals some non existence results of non constant positive steady state solutions are obtained also the numerical simulation is given to verify the theoretical results

coexistence in the three species predator prey model with diffusion the three species food chain model is discussed in which the third species is the predator of the second one and the second species is the predator of the first one we consider coexistence states of the associated weakly coupled elliptic problem under the homogeneous neumann boundary conditions it is shown that there are no non constant solutions if the diffusion rates of species are strong or if the intrinsic growth rate of a prey is slow and the intrinsic drop rates of predators are fast it is also shown that the weakly coupled parabolic system has a unique global solution for any non negative initial data

positive steady states for a nonlinear diffusion beddington deangelis model the main purpose of this work is to investigate the effects of nonlinear diffusion on positive steady states in a beddington deangelis model by the energy method and leray schauder degree theory we consider the non existence and existence of concerning non constant positive steady states of the model we demonstrate that nonlinear diffusion can create non constant positive steady state solutions even when the random diffusion fails to do so

traveling waves for a class of cross diffusion systems with small parameters abstract this paper is concerned with the existence of the travelling wave solutions for some cross diffusion systems with small parameters by using singular perturbation method we prove the existence of the travelling wave solutions with a speed c which extend the results of 22 and 23

an elliptic cross diffusion system describing two species models on a bounded domain with different natural conditions this paper is concerned with an elliptic cross diffusion system describing two species models on a bounded domain where consists of a finite number of subdomains i i 1 m separated by interfaces j j 1 m 1 and the natural conditions of the subdomains i are different this system is strongly coupled and the coefficients of the equations are allowed to be discontinuous on interfaces j the main goal is to show the existence of nonnegative solutions for the system by schauder s fixed point theorem furthermore as applications the existence of positive solutions for some lotka volterra models with cross diffusion self diffusion and discontinuous coefficients are also investigated

spatiotemporal patterns in a reaction diffusion model with the degn harrison reaction scheme abstract spatial and temporal patterns generated in ecological and chemical systems have become a central object of research in recent decades in this work we are concerned with a reaction diffusion model with the degn harrison reaction scheme which accounts for the qualitative feature of the respiratory process in a klebsiella aerogenes bacterial culture we study the global stability of the constant steady state existence and nonexistence of nonconstant steady states as well as the hopf and steady state bifurcations in particular our results show the existence of turing patterns and inhomogeneous periodic oscillatory patterns while the system parameters are all spatially homogeneous these results also exhibit the critical role of the system parameters in leading to the formation of spatiotemporal patterns

dynamics in a parabolic elliptic two species population competition model with cross diffusion for one species abstract this work deals with a simplified parabolic elliptic population competition model with cross diffusion for one species we first claim that this model has at least a non trivial global smooth solution and then we detect explicit parameter ranges under which the coexistence steady state and the two semi trivial steady states are globally and asymptotically stable

nonexistence of nonconstant steady state solutions in a triangular cross diffusion model abstract in this paper we study the shigesada kawasaki teramoto model for two competing species with triangular cross diffusion we determine explicit parameter ranges within which the model exclusively possesses constant steady state solutions

qualitative analysis for a diffusive predator prey model with a transmissible disease in the prey population in this paper we study a diffusive predator prey model with a transmissible disease in the prey population we offer a complete discussion of the dynamical properties under the homogeneous neumann boundary condition we analyze local and global stabilities of nonnegative constant steady states and long time behaviors of the positive solutions moreover we study the existence nonexistence and bifurcation of nonconstant positive stationary solutions finally some numerical simulations are presented to support and strengthen our analytical analysis

stationary patterns of strongly coupled prey predator models abstract we study some elliptic systems arising from 3 component predator prey models where cross diffusions are included in such a way that predator chases the prey and the prey runs away from the predator we establish the existence and non existence of non constant positive solutions our results show that the cross diffusions can create the stationary patterns

on stability of two degenerate reaction diffusion systems abstract in this paper we construct a partially degenerate reaction diffusion equation subject to the neumann boundary condition and show that the zero solution is asymptotically stable but not exponentially asymptotically stable in this way we solve an open problem proposed by casten and holland 1977 4 moreover we give the exponential asymptotic stability of the zero solution to a totally degenerate system with cross diffusion effects which cannot be determined by a simple spectral analysis based on the well developed semigroup theory

spatio temporal complexity of a delayed diffusive model for plant invasion abstract in this paper we develop a diffusive plant invasion model with delay under the homogeneous neumann boundary condition first the existence and uniqueness of a non negative solution persistence property and local asymptotic stability of the constant steady states are established then by analyzing the associated characteristic equation the stability of steady states and the existence of hopf bifurcation are demonstrated under special circumstance the discontinuous hopf bifurcation is also investigated furthermore the existence and non existence of nonconstant positive steady states of this model are studied through considering the effect of large diffusivity finally in order to verify our theoretical results some numerical simulations are also included it shows that the numerically observed behaviors are in good agreement with the theoretically proposed results on the basis of numerical simulations we provide some useful comparisons for the readers between our results and the existing ones and discuss the effects of the delay and the diffusion term on dynamic behaviors numerical results indicate that diffusion can make the system unstable and increasing delay may cause the plant to go extinct

spatiotemporal pattern formation of a diffusive bimolecular model with autocatalysis and saturation law in this paper a reaction diffusion bimolecular model with autocatalysis and saturation law is investigated firstly we provide some conditions for the stability turing instability of the constant positive solution then we mainly consider hopf bifurcations and steady state bifurcations which bifurcate from the unique constant positive solution of the system our results suggest the existence of spatially non homogeneous periodic orbit and non constant positive steady state solutions which implies the possibility of rich spatiotemporal patterns in this diffusive bimolecular model numerical examples are presented to support our theoretical analysis furthermore non existence of non constant steady state is investigated in terms of parameters

diffusion driven instability in reaction diffusion systems abstract for a stable matrix a with real entries sufficient and necessary conditions for a d to be stable for all non negative diagonal matrices d are obtained implications of these conditions for the stability and instability of constant steady state solutions to reaction diffusion systems are discussed and an example is given to show applications

a strongly coupled predator prey system with non monotonic functional response abstract the predator prey system with non monotonic functional response is an interesting field of theoretical study in this paper we consider a strongly coupled partial differential equation model with a non monotonic functional response a holling type iv function in a bounded domain with no flux boundary condition we prove a number of existence and non existence results concerning non constant steady states patterns of the underlying system in particular we demonstrate that cross diffusion can create patterns when the corresponding model without cross diffusion fails

a strongly coupled predator prey system with modified holling tanner functional response in this paper a strongly coupled system of partial differential equations in a bounded domain with the homogeneous neumann boundary condition which models a predator prey system with modified holling tanner functional response is considered first the authors study the stability of the positive constant solution sufficient conditions are derived for the global stability of the positive equilibrium by constructing a suitable lyapunov function by using the leray schauder theorem the authors prove a number of existence and non existence results about the non constant steady states of the system

pathwise mild solutions for quasilinear stochastic partial differential equations abstract stochastic partial differential equations spdes have become a key modeling tool in applications yet there are many classes of spdes where the existence and regularity theory for solutions is not completely developed here we contribute to this aspect and prove the existence of mild solutions for a broad class of quasilinear cauchy problems including among others cross diffusion systems as a key application our solutions are local in time and are derived via a fixed point argument in suitable function spaces the key idea is to combine in a suitable way the classical theory of deterministic quasilinear parabolic partial differential equations pdes with recent theory of evolution semigroups we also show how to apply our theory to the shigesada kawasaki teramoto skt model furthermore we provide examples of blow up and ill posed operators which can occur after finite time showing that solutions can only be local in time for general quasilinear spdes while they might be global in time for special subclasses of problems

qualitative analysis for a diffusive predator prey model in this paper we study the qualitative properties of a diffusive predator prey model subject to the homogeneous neumann boundary condition by a comparison argument and iteration technique under some hypotheses we prove that the positive constant steady state is globally asymptotically stable we also establish the existence and non existence of non constant positive steady states stationary patterns by use of the degree theory and the a priori estimates

a new approach toward stabilization in a two species chemotaxis model with logistic source abstract this paper aims at providing an alternative approach to study global dynamic properties for a two species chemotaxis model with the main novelty being that both populations mutually compete with the other on account of the lotka volterra dynamics more precisely we consider the following neumann initial boundary value problem u t d 1 u 1 u w 1 u 1 u a 1 v x t 0 v t d 2 v 2 v w 2 v 1 a 2 u v x t 0 0 d 3 w w u v x t 0 in a bounded domain r n n 1 with smooth boundary where d 1 d 2 d 3 1 2 1 2 a 1 a 2 are positive constants when a 1 0 1 and a 2 0 1 it is shown that under some explicit largeness assumptions on the logistic growth coefficients 1 and 2 the corresponding neumann initial boundary value problem possesses a unique global bounded solution which moreover approaches a unique positive homogeneous steady state u v w of above system in the large time limit the respective decay rate of this convergence is shown to be exponential when a 1 1 and a 2 0 1 if 2 is suitable large for all sufficiently regular nonnegative initial data u 0 and v 0 with u 0 0 and v 0 0 the globally bounded solution of above system will stabilize toward 0 1 1 as t in algebraic

a spatial sis model in advective heterogeneous environments abstract we study the effects of diffusion and advection for a susceptible infected susceptible epidemic reaction diffusion model in heterogeneous environments the definition of the basic reproduction number r 0 is given if r 0 1 the unique disease free equilibrium dfe is globally asymptotically stable asymptotic behaviors of r 0 for advection rate and mobility of the infected individuals denoted by d i are established and the existence of the endemic equilibrium when r 0 1 is studied the effects of diffusion and advection rates on the stability of the dfe are further investigated among other things we find that if the habitat is a low risk domain there may exist one critical value for the advection rate under which the dfe changes its stability at least twice as d i varies from zero to infinity while the dfe is unstable for any d i when the advection rate is larger than the critical value these results are in strong contrast with the case of no advection where the dfe changes its stability at most once as d i varies from zero to infinity

a fitness driven cross diffusion system from population dynamics as a gradient flow we consider a fitness driven model of dispersal of n interacting populations which was previously studied merely in the case n 1 n 1 based on some optimal transport distance recently introduced we identify the model as a gradient flow in the metric space of radon measures we prove existence of global non negative weak solutions to the corresponding system of parabolic pdes which involves degenerate cross diffusion under some additional hypotheses and using a new multicomponent poincare beckner functional inequality we show that the solutions converge exponentially to an ideal free distribution in the long time regime

turing instability and pattern induced by cross diffusion in a predator prey system with allee effect to have spatial patterns form small allee effect admits relatively large range of predator mortality rate found cross diffusion is the key mechanism of spatial pattern formation obtained amplitude equations suggest both supercritical and subcritical bifurcation may occur weak allee effect admits supercritical bifurcation while strong one allows subcritical bifurcation in this paper we first propose a mathematical model for a spatial predator prey system with allee effect and then by using the proposed model we investigate the turing instability and the phenomena of pattern formation we show how cross diffusion destabilizes the spatially uniform steady state the method of multiple time scales is employed to derive the amplitude equations which is the cubic stuart landau equation in the supercritical case and the quintic in the subcritical case based on the amplitude equations we obtain the asymptotic solutions of the model close to the onset of instability

dynamics of a general prey predator model with prey stage structure and diffusive effects in this paper we propose and study the dynamics of a diffusive prey predator model with general functional response and stage structure for the prey firstly we consider the asymptotical stability of equilibrium points and hopf bifurcation for the reduced ode system secondly the existence and uniform boundedness of global solutions and stability of equilibrium points for the corresponding reaction diffusion system are discussed finally we establish the existence and the nonexistence of nonconstant positive steady states of this reaction diffusion system which indicates the effect of large diffusivity our results shows the importance of the diffusion rate of the predator species i e d 3 the large diffusion rate of the predator alone will help the generation of patterns however a large diffusion rate of the immature prey species or a large diffusion rate of the mature prey species can lead to the nonexistence of spatial patterns

stationary problem of a predator prey system with nonlinear diffusion effects in this paper we are concerned with the positive solution set of the following quasilinear elliptic system 1 v u u a u c v 1 m u x 1 1 u v v b v d u 1 m u x u v 0 x where is a bounded smooth domain in r n are nonnegative constants a c d m are positive constants b is a real constant this elliptic system is the stationary problem of a predator prey model in which u and v denote the population densities of the prey and predator respectively regarding b as the bifurcation parameter the global bifurcation structure of the positive solution set is shown then the nonlinear effect of either large or on the bifurcation point is deduced moreover as is large we show that the positive solution set is only of one type whereas as is large the positive solution set is of two types additionally in certain circumstances we also show that which one of the two types can characterize the positive solution set

stationary patterns of the holling tanner prey predator model with diffusion and cross diffusion this paper is concerned with the holling tanner prey predator model subject to the homogeneous neumann boundary condition we will show that under certain hypotheses even though the unique positive constant steady state is uniformly asymptotically stable for the dynamics with diffusion non constant positive steady states can exist due to the emergence of cross diffusion the biological implication of cross diffusion means that the prey species exercise a self defense mechanism to protect themselves from the attack of the predator our result also demonstrates that cross diffusion can create stationary patterns

non existence of non constant positive steady states of two holling type ii predator prey systems strong interaction case abstract we prove the non existence of non constant positive steady state solutions of two reaction diffusion predator prey models with holling type ii functional response when the interaction between the predator and the prey is strong the result implies that the global bifurcating branches of steady state solutions are bounded loops

diffusion self diffusion and cross diffusion u1 x 0 u 1 0 x u2 x 0 u 2 0 x i n 0 where 2 n 1 2 x 2 is the laplace operator 0 is a bounded smooth domain of r n with n1 0 and 0 are the boundary and the closure of 0 respectively 0t 0 0 t and 0t 0 0 t for some t 0 is the outward unit normal vector on 0 di ai bi ci i 1 2 are all positive constants while ij i j 1 2 denote non negative constants the initial values u1 0 and u2 0 are non negative smooth func tions which are not identically zero the mathematical model 1 1 was proposed by shigesada et al 27 in their study of spatial segregation of interacting species where u1 and u2 represent the densities of two competing species d1 and d2 are their diffu sion rates a1 and a2 denote the intrinsic growth rates b1 and c2 account for intra specific competitions b2 and c1 are the coefficients of inter specific competitions 11 and 22 are usually referred as self diffusion pressures and 12 and 21 are cross diffusion pressures an interesting feature of 1 1 is that the movements of these two competing species are affected by the population pressures due to the mutual interference between the individuals and it was hoped that the nonlinear dispersive force due to the

optimality for a diffusive predator prey system in a spatially heterogeneous environment incorporating a prey refuge abstract in this paper we consider the optimal harvesting controls for a diffusive predator prey system which is formulated by prey refuge and modified leslie gower functional response the locally and globally asymptotic stability of unique positive constant steady state is discussed under homogeneous neumann boundary condition the nonexistence and existence of nonconstant steady states are obtained by leray schauder degree theory also the optimal harvesting strategies to get maximum sustainable yield msy and maximum economic yield mey are correspondingly given

global existence of solutions for a strongly coupled quasilinear parabolic system we are interested in existence uniqueness and a priori estimates of non negative solutions to the following problem begin gathered u t delta left left 1 alpha v right u right uf u v hfill v t delta v vg u v quad in 0 infty times omega hfill u v 0 quad in 0 infty times partial omega hfill u 0 u 0 v 0 v 0 quad in omega hfill end gathered 1 0

analysis of ratio dependent food chain model abstract in this paper a food chain model with ratio dependent functional response is studied under homogeneous neumann boundary conditions the large time behavior of all non negative equilibria in the time dependent system is investigated i e conditions for the stability at equilibria are found moreover non constant positive steady states are studied in terms of diffusion effects namely turing patterns arising from diffusion driven instability turing instability are demonstrated the employed methods are comparison principle for parabolic problems and leray schauder theorem

on the finite dimensional dynamical systems with limited competition the asymptotic behavior of dynamical systems with limited competition is investigated we study index theory for fixed points permanence global stability convergence everywhere and coexistence it is shown that the system has a globally asymptotically stable fixed point if every fixed point is hyperbolic and locally asymptotically stable relative to the face it belongs to a nice result is the necessary and sufficient conditions for the system to have a globally asymptotically stable positive fixed point it can be used to establish the sufficient conditions for the system to persist uniformly and the convergence result for all orbits applications are made to time periodic ordinary differential equations and reaction diffusion equations

coexistence and asymptotic periodicity in a competitor competitor mutualist model in this paper the competitor competitor mutualist three species lotka volterra model is discussed firstly by schauder fixed point theory the coexistence state of the strongly coupled system is given applying the method of upper and lower solutions and its associated monotone iterations the true solutions are constructed our results show that this system possesses at least one coexistence state if cross diffusions and cross reactions are weak secondly the existence and asymptotic behavior of t periodic solutions for the periodic reaction diffusion system under homogeneous dirichlet boundary conditions are investigated sufficient conditions which guarantee the existence of t periodic solution are also obtained

qualitative analysis of a prey predator model with stage structure for the predator in this paper we propose a diffusive prey predator model with stage structure for the predator we first analyze the stability of the nonnegative steady states for the reduced ode system and then study the same question for the corresponding reaction diffusion system with homogeneous neumann boundary conditions we find that a hopf bifurcation occurs in the ode system but no turing pattern happens in the reaction diffusion system however when a natural cross diffusion term is included in the model we can prove the emergence of stationary patterns i e nonconstant positive stationary solutions for this system moreover these stationary patterns do not exist in the considered parameter regime when there is no cross diffusion

uniform holder bounds for a strongly coupled elliptic system with strong competition abstract this article is concerned with a strongly coupled elliptic system modeling the steady state of populations that compete in some region we prove that the solutions are uniformly holder bounded as the competition rate tends to infinity the proof relies on the blow up technique and the monotonicity formula

large diffusivity and asymptotic behavior in parabolic systems abstract for systems of reaction diffusion equations with neumann boundary conditions it is shown that the solutions are asymptotic to the solutions of an ordinary differential equation if the diffusivity is large the methods apply also to reaction diffusion systems with time delays

stationary patterns for a prey predator model with prey dependent and ratio dependent functional responses and diffusion abstract we study a reaction diffusion system arising from a three species prey predator model with prey dependent and ratio dependent functional responses the main aim of this paper is to study the global existence and bifurcation of non constant positive steady states in particular we will show that even though the unique positive constant steady state is stable for the ode dynamics non constant positive steady states exist for the reaction diffusion system this demonstrates that stationary patterns arise as a result of diffusion

analysis of a population model with strong cross diffusion in unbounded domains we study a parabolic population model in the full space and prove the global in time existence of a weak solution this model consists of two strongly coupled diffusion equations describing the population densities of two competing species the system features intrinsic growth inter and intra specific competition of the species as well as diffusion cross diffusion and self diffusion and drift terms related to varying environment quality the cross diffusion terms can be large making the system non parabolic for large initial data for the first time solutions in unbounded domains are studied the method of our proof is a combination of a time semi discretization a transformation of the system guaranteeing the positivity of the solution a special entropy symmetrizing the system and compactness arguments 2000 mathematics subject classification 35k55 primary 92d25 35d05 secondary

pattern formation of a coupled two cell brusselator model abstract in this paper we study the stationary problems for the coupled two cell brusselator model as follows d 1 u 1 b 1 u b u 2 v c w u in d 2 v u u 2 v in d 1 w 1 b 1 w b w 2 z c u w in d 2 z w w 2 z in u v w z 0 on we first study the stability of the unique positive constant solution u v w z 1 1 1 1 then we give a priori estimate positive upper and lower bounds to the positive solution at last we obtain the non existence and existence of positive non constant solutions as parameters d 1 d 2 and b varied

positive solutions of a diffusive predator prey model with modified leslie gower and holling type ii schemes abstract in this paper we investigate the existence multiplicity and stability of positive solutions to a prey predator model with modified leslie gower and holling type ii schemes p u u a 1 b u c 1 v u k 1 in v v a 2 c 2 v u k 2 in u 0 v 0 in u v 0 on where r n n 1 is a bounded domain with a smooth boundary the parameters a i b c i k i i 1 2 are positive numbers u and v are the respective populations of prey and predator here we say u v with u v 0 is a positive solution of problem p if u v is a solution of p and u v 0 in

global bifurcation analysis and pattern formation in homogeneous diffusive predator prey systems abstract the dynamics of a general diffusive predator prey system is considered existence and nonexistence of non constant positive steady state solutions are shown to identify the ranges of parameters of spatial pattern formation bifurcations of spatially homogeneous and nonhomogeneous periodic solutions as well as non constant steady state solutions are studied

qualitative analysis of steady states to the sel kov model in this work we are concerned with a reaction diffusion system well known as the sel kov model which has been used for the study of morphogenesis population dynamics and autocatalytic oxidation reactions we derive some further analytic results for the steady states to this model in particular we show that no nonconstant positive steady state exists if 0 1 and large where nonconstant positive steady states can occur thus these conclusions indicate that the parameter p plays a crucial role in leading to spatially nonhomogeneous distribution of the two reactants the a priori estimates are fundamental to our mathematical approaches

bayesian error estimation in density functional theory we present a practical scheme for performing error estimates for density functional theory calculations the approach which is based on ideas from bayesian statistics involves creating an ensemble of exchange correlation functionals by comparing with an experimental database of binding energies for molecules and solids fluctuations within the ensemble can then be used to estimate errors relative to experiment on calculated quantities such as binding energies bond lengths and vibrational frequencies it is demonstrated that the error bars on energy differences may vary by orders of magnitude for different systems in good agreement with existing experience

non constant positive steady states of the sel kov model abstract this paper deals with the reaction diffusion system known as the sel kov model with the homogeneous neumann boundary condition this model has been applied to various problems in chemistry and biology we first give a priori estimates positive upper and lower bounds of positive steady states and then study the non existence bifurcation and global existence of non constant positive steady states as the parameters and are varied

an initial boundary value problem for a certain density dependent diffusion system on considere le systeme parabolique quasilineaire u t x c 1 d 1 v v e 1 a 1 v b 1 u u v t x c 2 d 2 u v e 2 a 2 u b 2 o v dans o t u x s 0 v x s sur o t u x o w 1 x v x o w 2 x sur ou r n est un domaine borne a frontiere lisse