degenerate variational inequalities of evolution on etudie des inegalites variationnelles de la forme re 0 pour tout v k by 0 bu 0 ou b est lineaire mais peut s annuler a est non lineaire et k est un ensemble convexe ferme

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

on a predator prey system with cross diffusion representing the tendency of predators in the presence of prey species abstract in this paper a predator prey system with cross diffusion representing the tendency of predators to avoid the group defense by a large number of prey or diffuse in the direction of higher concentration of the prey species under homogeneous dirichlet boundary condition is considered using the method of upper and lower solutions developed by pao c v pao strongly coupled elliptic systems and applications to lotka volterra models with cross diffusion nonlinear anal 60 2005 1197 1217 sufficient conditions for the existence of positive solutions are provided when the induced cross diffusion coefficient is sufficiently small furthermore the investigation of non existence of positive solutions is also presented

turing patterns in the lengyel epstein system for the cima reaction the first experimental evidence of the turing pattern was observed by de kepper and her associates 1990 on the cima reaction in an open unstirred gel reactor almost 40 years after turing s prediction lengyel and epstein characterized this famous experiment using a system of reaction diffusion equations in this paper we report some fundamental analytic properties of the lengyel epstein system our result also indicates that if either of the initial concentrations of the reactants the size of the reactor or the effective diffusion rate are not large enough then the system does not admit nonconstant steady states a priori estimates are fundamental to our approach for this nonexistence result the degree theory was combined with the a priori estimates to derive existence of nonconstant steady states

effect of a protection zone in the diffusive leslie predator prey model abstract in this paper we consider the diffusive leslie predator prey model with large intrinsic predator growth rate and investigate the change of behavior of the model when a simple protection zone 0 for the prey is introduced as in earlier work y du j shi a diffusive predator prey model with a protection zone j differential equations 229 2006 63 91 y du x liang a diffusive competition model with a protection zone j differential equations 244 2008 61 86 we show the existence of a critical patch size of the protection zone determined by the first dirichlet eigenvalue of the laplacian over 0 and the intrinsic growth rate of the prey so that there is fundamental change of the dynamical behavior of the model only when 0 is above the critical patch size however our research here reveals significant difference of the model s behavior from the predator prey model studied in y du j shi a diffusive predator prey model with a protection zone j differential equations 229 2006 63 91 with the same kind of protection zone we show that the asymptotic profile of the population distribution of the leslie model is governed by a standard boundary blow up problem and classical or degenerate logistic equations

cyclical interactions with alliance specific heterogeneous invasion rates we study a six species lotka volterra type system on different two dimensional lattices when each species has two superior and two inferior partners the invasion rates from predator sites to a randomly chosen neighboring prey site depend on the predator prey pair whereby cyclic symmetries within the two three species defensive alliances are conserved monte carlo simulations reveal an unexpected nonmonotonous dependence of alliance survival on the difference of alliance specific invasion rates this behavior is qualitatively reproduced by a four point mean field approximation the study addresses fundamental problems of stability for the competition of two defensive alliances and thus has important implications in natural and social sciences

stationary patterns caused by cross diffusion for a three species prey predator model we study a system arising from a three component prey predator model with prey dependent and ratio dependent functional responses where cross diffusion is included in such a way that predator chases the prey and the prey runs away from the predator we prove that the cross diffusion can generate the stationary patterns nonconstant positive steady states

global stability for a class of predator prey systems this paper deals with the question of global stability of the positive locally asymptotically stable equilibrium in a class of predator prey systems the dulac s criterion is applied and liapunov functions are constructed to establish the global stability

non constant positive steady states of a predator prey system with non monotonic functional response and diffusion this paper deals with non constant positive steady state solutions of a predator prey system with non monotonic functional response also called holling type iv interaction terms and diffusion under the homogeneous neumann boundary condition we first establish positive upper and lower bounds for such solutions and then study their non existence global existence and bifurcation

robustifying glejser test of heteroskedasticity verifying that the glejser test for heteroskedasticity is asymptotically invalid unless the error density is symmetric this paper proposes a simple modification to make the test robust to asymmetric disturbances simulation results demonstrate that the size of the modified test is correct for both symmetric and skewed errors

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

turing pattern of the oregonator model abstract the oregonator model is the mathematical dynamics which describes the field koros noyes mechanics of the famous belousov zhabotinskii reaction in this work we establish some fundamental analytic properties of this dynamics and its corresponding steady state under various conditions on the parameters and the size of the reactor we examine the existence and non existence of non constant steady states in particular for some properly chosen parameter ranges we prove the occurrence of the turing pattern generated by this oregonator model our results exhibit interesting and very different roles of the diffusion rates and the reactor in the formation of the turing pattern our mathematical analysis mainly relies on a priori estimates and the topological degree argument

rich dynamics of a predator prey model with spatial motion one of the central issues in ecosystem is the influence of space on the stability of the populations as a result a predator prey model with spatial diffusion and cubic growth rate is investigated asymptotical properties of solutions of such spatial model are obtained based on comparison principle what is more we show the global stability of positive equilibrium by constructing lyapunov function we also show how predator diffuses has impact on the pattern dynamics based on numerical simulations the results may well enrich the findings in predator prey systems with spatial motions

on the dynamics of predator prey models with the beddington deangelis functional response abstract the dynamics of predator prey models with the beddington deangelis functional response are analyzed primarily from the viewpoint of permanence uniform persistence the beddington deangelis functional response is similar to the holling type 2 functional response but contains an extra term describing mutual interference by predators both spatially homogeneous models based on ordinary differential equations and reaction diffusion models are considered criteria for permanence and for predator extinction are derived for systems without diffusion or with no flux boundary conditions criteria are derived for the existence of a globally stable coexistence equilibrium or alternatively for the existence of periodic orbits

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

stationary distribution and extinction of a stochastic ratio dependent predator prey system with stage structure for the predator abstract in this paper we deal with a stochastic predator prey model with stage structure for predator population and ratio dependent functional response the proposed mathematical model consists of a system of three stochastic differential equations to stimulate the interactions between prey population immature predator and mature predator population we first establish sufficient conditions for the existence and uniqueness of the positive solutions by constructing an appropriate lyapunov function then we extend the existence of stationary distribution under certain parametric restrictions we also obtain the sufficient conditions for extinction of the predator populations finally numerical simulations have been carried out to validate our analytical findings

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 note on studentizing a test for heteroscedasticity abstract breusch and pagan 1979 have recently proposed a convenient test for heteroscedasticity in general linear models this note derives the asymptotic distribution of their test under sequences of contiguous alternatives to the null hypothesis of homoscedasticity the test is shown to possess asymptotically incorrect size nominal significance level except in the case of strictly gaussian disturbances a slight modification of the test is proposed which corrects this defect

strategy and stationary pattern in a three species predator prey model abstract in this paper we study a strongly coupled system of partial differential equations which models the dynamics of a two predator one prey ecosystem in which the prey exercises a defense switching mechanism and the predators collaboratively take advantage of the prey s strategy we demonstrate the emergence of stationary patterns for this system and show that it is due to the cross diffusion that arises naturally in the model as far as the authors are aware this is the first example of stationary patterns in a predator prey system arising solely from the effect of cross diffusion

hopf bifurcation and global stability of a diffusive gause type predator prey models abstract this paper mainly provides hopf bifurcation formulas for a general gause type predator prey system with diffusion and neumann boundary condition by using the center manifold theory and normal form method where the spectral and stability analysis around an equilibrium is addressed and our results can be applied to the case without diffusion as an application of these results we give a complete and rigorous analysis of the global dynamics of a diffusive predator prey model with herd behavior especially the hopf bifurcation and its direction and the stability of the bifurcating periodic solutions

a diffusive predator prey model in heterogeneous environment abstract in this paper we demonstrate some special behavior of steady state solutions to a predator prey model due to the introduction of spatial heterogeneity we show that positive steady state solutions with certain prescribed spatial patterns can be obtained when the spatial environment is designed suitably moreover we observe some essential differences of the behavior of our model from that of the classical lotka volterra model that seem to arise only in the heterogeneous case

qualitative analysis of a ratio dependent predator prey system with diffusion ratio dependent predator prey models are favoured by many animal ecologists recently as they better describe predator prey interactions where predation involves a searching process when densities of prey and predator are spatially homogeneous the so called michaelis menten ratio dependent predator prey system which is an ordinary differential system has been studied by many authors the present paper deals with the case where densities of prey and predator are spatially inhomogeneous in a bounded domain subject to the homogeneous neumann boundary condition its main purpose is to study qualitative properties of solutions to this reaction diffusion partial differential system in particular we will show that even though the unique positive constant steady state is globally asymptotically stable for the ordinary differential equation dynamics non constant positive steady states exist for the partial differential equation model this demonstrates that stationary patterns arise as a result of diffusion

stability analysis of diffusive predator prey model with modified leslie gower and holling type iii schemes abstract the stability of a diffusive predator prey model with modified leslie gower and holling type iii schemes is investigated a threshold property of the local stability is obtained for a boundary steady state and sufficient conditions of local stability and un stability for the positive steady state are also obtained furthermore the global asymptotic stability of these two steady states are discussed our results reveal the dynamics of this model system

heteroclinic bifurcation in the michaelis menten type ratio dependent predator prey system the existence of a heteroclinic bifurcation for the michaelis menten type ratio dependent predator prey system is rigorously established limit cycles related to the heteroclinic bifurcation are also discussed it is shown that the heteroclinic bifurcation is characterized by the collision of a stable limit cycle with the origin and the bifurcation triggers a catastrophic shift from the state of large oscillations of predator and prey populations to the state of extinction of both populations it is also shown that the limit cycles related to the heteroclinic bifurcation originally bifurcate from the hopf bifurcation

existence of stationary solutions with transition layers for a class of cross diffusion systems this paper is concerned with the existence of stationary solutions for some cross di usion systems with small parameters using a singular perturbation method we prove the existence of stationary solutions with boundary and interior layers which extends the results of fife and ito to the cross di usion cases

some results on the glejser and koenker tests for heteroskedasticity abstract the glejser and koenker studentized lagrange multiplier tests for heteroskedasticity are considered asymptotic theory and monte carlo experiments are used to investigate the effects of nonnormality under null and alternative hypotheses and also the consequences of using an incorrect alternative a method for using test outcomes to select models of heteroskedasticity is critically examined glejser s test is not in general asymptotically valid under asymmetric disturbances koenker s test is asymptotically valid under asymmetric and other nonnormal disturbances but has estimated finite sample significance levels that are sometimes sensitive to skewness an adjusted version of koenker s statistic is discussed

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

the galerkin method and degenerate evolution equations abstract existence uniqueness and regularity results are obtained for an abstract equation of the form by ay f where b is not 1 1 and may vanish and a is a nonlinear operator the desired solution is obtained as a limit of solutions of finite dimensional ordinary differential equations the method used here generalizes the usual galerkin method by allowing b to be degenerate and by making weaker assumptions of coercivity than are customary in most applications of the galerkin method to evolution equations

mathematical analysis of a parabolic elliptic problem with moving parabolic subdomain through a lagrangian approach abstract the aim of this paper is to study the regularity of the solution of some linear parabolic elliptic problems in which parabolicity region depends on time more specifically this region is the position occupied by a body undergoing a motion a deformation smoothly evolving in time the main tool we introduce is a suitable extension of the motion to the entire spatial domain of the pde this enables us to reduce the original problem to a parabolic elliptic problem with variable coefficients and with a parabolicity region independent of time this problem can be seen as a lagrangian formulation of our original problem next we obtain regularity results for a class of parabolic elliptic problems with variable coefficients and fixed parabolicity region we apply these results to the lagrangian formulation and finally we obtain a regularity result for our original problem

effects of toxin delay on the dynamics of a phytoplankton zooplankton model abstract this paper is concerned with a phytoplankton zooplankton model with toxin delay it is shown that the positive equilibrium of the system is stable when the toxin delay is not included however we find that the incorporation of a discrete toxin delay not only can destabilize the positive equilibrium of the system but also can cause a hopf bifurcation at the positive equilibrium as it crosses some critical values in particular the normal form of the hopf bifurcation arising in the system is determined to investigate the direction and the stability of periodic solutions bifurcating from these hopf bifurcations to verify the obtained conditions a special numerical example is also included these results in this work demonstrate that the toxin delay plays an important role on deriving the complex dynamics

turing patterns of a strongly coupled predator prey system with diffusion effects abstract the main purpose of this work is to investigate the effects of cross diffusion in a strongly coupled predator prey system by a linear stability analysis we find the conditions which allow a homogeneous steady state stable for the kinetics to become unstable through a turing mechanism in particular it is shown that turing instability of the reaction diffusion system can disappear due to the presence of the cross diffusion which implies that the cross diffusion induced stability can be regarded as the cross stability of the corresponding reaction diffusion system furthermore we consider the existence and non existence results concerning non constant positive steady states patterns of the system we demonstrate that cross diffusion can create non constant positive steady state solutions these results exhibit interesting and very different roles of the cross diffusion in the formation and the disappearance of the turing instability

global asymptotic stability of a ratio dependent predator prey system with diffusion this paper is concerned with a predator prey system with diffusion the global asymptotic stability of the unique positive constant equilibrium is obtained under certain conditions the method used here is the upper and lower solutions combined with the monotone iteration and constructing suitable lyapunov functions

glejser s test revisited godfrey 1996 journal of econometrics 72 275 299 has shown that the glejser test for heteroskedasticity is valid only under conditional symmetry here modifications of the glejser test are suggested the proposed test statistics are asymptotically valid even when the disturbances are not symmetrically distributed and can be used to test for heteroskedasticity when conditional location functions other than the conditional mean are estimated

time dependent implicit evolution equations on presente des theoremes d existence et d unicite pour des problemes de cauchy correspondant aux equations abstraites de la forme d b t u t dt a t u t f t ou b t est une famille dependante du temps d operateurs lineaires

asymmetric least squares estimation and testing this paper considers estimation and hypothesis tests for coefficients of linear regression models where the coefficient estimates are based on location measures defined by an asymmetric least squares criterion function these asymmetric least squares estimators have properties which are analogous to regression quantile estimators but are much simpler to calculate as are the corresponding test statistics the coefficient estimators can be used to construct test statistics for homoskedasticity and conditional symmetry of the error distribution and we find these tests compare quite favorably with other commonly used tests of these null hypotheses in terms of local relative efficiency consequently asymmetric least squares estimation provides a convenient and relatively efficient method of summarizing the conditional distribution of a dependent variable given the regressors and a means of testing whether a linear model is an adequate characterization of the typical value for this conditional distribution

global asymptotic stability of a diffusive predator prey model with ratio dependent functional response abstract this paper is concerned with a diffusive leslie gower predator prey system with ratio dependent holling type iii functional response under homogeneous neumann boundary conditions the uniform persistence of the solutions semiflows the existence of global attractors local and global asymptotic stability of the positive constant steady state of the reaction diffusion model are discussed by using comparison principle the linearization method and the lyapunov functional method respectively the global asymptotic stability of the positive constant steady state shows that the prey and predator will be spatially homogeneously distributed as time converges to infinities

a generalized levene s scale test for variance heterogeneity in the presence of sample correlation and group uncertainty we generalize levene s test for variance scale heterogeneity between k groups for more complex data when there are sample correlation and group membership uncertainty following a two stage regression framework we show that least absolute deviation regression must be used in the stage 1 analysis to ensure a correct asymptotic k 12 k 1 distribution of the generalized scale gs test statistic we then show that the proposed gs test is independent of the generalized location test under the joint null hypothesis of no mean and no variance heterogeneity consequently we generalize the recently proposed joint location scale gjls test valuable in settings where there is an interaction effect but one interacting variable is not available we evaluate the proposed method via an extensive simulation study and two genetic association application studies

versal unfoldings of predator prey systems with ratio dependent functional response abstract in this paper we study the versal unfolding of a predator prey system with ratio dependent functional response near a degenerate equilibrium in order to obtain all possible phase portraits for its perturbations we first construct the unfolding and prove its versality and degeneracy of codimension 2 then we discuss all its possible bifurcations including transcritical bifurcation hopf bifurcation and heteroclinic bifurcation give conditions of parameters for the appearance of closed orbits and heteroclinic loops and describe the bifurcation curves phase portraits for all possible cases are presented

controlling bifurcation in a delayed fractional predator prey system with incommensurate orders this paper investigates an issue of bifurcation control for a novel incommensurate fractional order predator prey system with time delay firstly the associated characteristic equation is analyzed by taking time delay as the bifurcation parameter and the conditions of creation for hopf bifurcation are established it is demonstrated that time delay can heavily effect the dynamics of the proposed system and each order has a major influence on the creation of bifurcation simultaneously then a linear delayed feedback controller is introduced to successfully control the hopf bifurcation for such system it is shown that the control effort is markedly influenced by feedback gain it is further found that the onset of the bifurcation can be delayed as feedback gain decreases finally two illustrative examples are exploited to verify the validity of the obtained newly results

a complete analysis of the global dynamics of a diffusive predator and toxic prey model considering many species can release toxic substances to protect themselves against predators a diffusive predator and toxic prey system with spatial heterogeneity in predator and prey populations has been investigated for this system we give a complete and rigorous analysis of the global dynamics with the boundedness globally asymptotical stability transcritical bifurcation hopf bifurcation and its direction and the stability of the bifurcating periodic solutions meanwhile we consider the effects of toxins produced by the prey on the dynamic behavior the consequence of the global stability of the coexistence equilibrium is that the toxin s intrinsic characteristic will not change the stability of the system irreversibly our results show that the toxin produced by the prey phytoplankton may be used as a bio control agent for the harmful algal bloom problems

exhibiting cross diffusion induced patterns for reaction diffusion systems on evolving domains and surfaces the aim of this manuscript is to present for the first time the application of the finite element method for solving reaction diffusion systems with cross diffusion on continuously evolving domains and surfaces furthermore we present pattern formation generated by the reaction diffusion systemwith cross diffusion on evolving domains and surfaces a two component reaction diffusion system with linear cross diffusion in both u and v is presented the finite element method is based on the approximation of the domain or surface by a triangulated domain or surface consisting of a union of triangles for surfaces the vertices of the triangulation lie on the continuous surface a finite element space of functions is then defined by taking the continuous functions which are linear affine on each simplex of the triangulated domain or surface to demonstrate the role of cross diffusion to the theory of pattern formation we compute patterns with model kinetic parameter values that belong only to the cross diffusion parameter space these do not belong to the standard parameter space for classical reaction diffusion systems numerical results exhibited show the robustness flexibility versatility and generality of our methodology the methodology can deal with complicated evolution laws of the domain and surface and these include uniform isotropic and anisotropic growth profiles as well as those profiles driven by chemical concentrations residing in the domain or on the surface