non constant positive steady states of a prey predator system with cross diffusions abstract in this paper we study a strongly coupled elliptic system arising from a lotka volterra prey predator system where cross diffusions are included in such a way that the prey runs away from the predator and the predator moves away from a large group of preys we establish the existence and non existence of its non constant positive solutions our results show that if m 1 b a 2 m 1 b 1 m 1 m 2 when 0 m 1 m 2 1 or a m 1 b when m 1 m 2 1 0 d 1 m 1 v u 1 d 2 0 d 3 0 and d 4 1 m 1 v u then there exists d 1 d 2 d 3 d 4 such that the stationary problem admits non constant positive solutions otherwise the stationary problem has no non constant positive solution in particular the results indicate that its non constant positive solutions are mainly created by the cross diffusion d 4

generation of complex bipartite graphs by using a preferential rewiring process it is important in computer science sociology and so on to investigate complex bipartite graphs from a viewpoint of statistical physics we propose a model to generate complex bipartite graphs without growing the bipartite graphs are assumed to have two sets of the fixed numbers of nodes and a fixed number of edges between nodes belonging to different sets of nodes in this model essential ingredients are a preferential rewiring process and a fitness distribution function by using the preferential rewiring process we confirm that a bipartite graph reaches a stationary state after a sufficiently long time has passed we find that the obtained bipartite graph has a scale free like property when a suitable fitness distribution is used it turns out that a condensation of edges takes place in the cases of certain fitness distributions

pattern formation in the brusselator system in the paper we deal with a reaction diffusion system well known as the brusselator model and some improved results for the steady states of this model are presented we first give an a priori estimates positive upper and lower bounds of positive steady states then we obtain the non existence and existence of positive non constant steady states as the parameters and b are varied which means some certain conditions under which the pattern formation occurs or not

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

a priori bounds for reaction diffusion systems arising in chemical and biological dynamics the authors investigate reaction diffusion equations which arise in chemical and biological dynamics it is shown that several common systems share a useful property a structure on the non linearity which arises from conservation of mass or population this conservation property is used to demonstrate a priori bounds for the parabolic problems and the associated elliptic problem the types of systems included in the analysis are the gray scott system sir model and the selkov model of glycolysis

mesa type patterns in the one dimensional brusselator and their stability the brusselator is a generic reaction diffusion model for a tri molecular chemical reaction we consider the case when the input and output reactions are slow in this limit we show the existence of k periodic spatially bi stable structures mesas and study their stability using singular perturbation techniques we find a threshold for the stability of k mesas this threshold occurs in the regime where the exponentially small tails of the localized structures start to interact by comparing our results with turing analysis we show that in the generic case a turing instability is followed by a slow coarsening process whereby logarithmically many mesas are annihilated before the system reaches a steady equilibrium state we also study a breather type instability of a mesa which occurs due to a hopf bifurcation full numerical simulations are shown to confirm the analytical results in 1952 turing proposed that the formation of spatial patterns during morphogenesis could be explained in terms of the instability of a homogeneous steady state solution of a reaction diffusion network describing the evolution of a set of morphogens 31 turing himself illustrated his ideas on two chemical models turing s original work is primarily concerned with the stability analysis of the homogeneous steady state solution of the rate equations for the interacting morphogens 22 the main point of biological interest however is whether stable spatial structures may be generated beyond the instability i e whether the rate equations admit stable and positive inhomogeneous solutions exhibiting the most characteristic features of morphogenetic patterns this point has been taken up seriously in the early seventies more systematic numerical studies of turing s model have been performed showing irregular spatial structures 4 this led to serious reservations about the relevance of turing s theory in developmental biology particularly its ability to generate regular patterns but these criticisms originate from a somewhat unfortunate choice of turing s example and they do not touch the essential points of turing s theory more specifically the obvious requirement that the rate equations must admit positive and bounded solutions is not satisfied in turing s example 11

the stochastic hopf bifurcation analysis in brusselator system with random parameter hopf bifurcation in brusselator system with random parameter is explored in this paper firstly according to the orthogonal polynomial approximation in hilbert space the brusselator system with random parameter can be reduced into the deterministic equivalent system then the hopf bifurcation in deterministic equivalent system is discussed by the mathematical analysis and the first lyapunov coefficient method the studies discovered that different from the deterministic system the critical value of stochastic hopf bifurcation is determined not only by deterministic parameters in stochastic system but also by the intensity of random parameter as the intensity of random parameter is increased the critical value of stochastic hopf bifurcation is decreased at last theoretical results are verified by numerical simulations

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

turing space in reaction diffusion systems with density dependent cross diffusion reaction diffusion systems with cross diffusion terms that depend linearly on density are studied via linear stability analysis and weakly nonlinear analysis we obtain and analyze the conditions for the turing instability and derive a universal form of these conditions we discuss the features of the pattern forming regions in parameter space for a cross activator inhibitor system the brusselator model and for a pure activator inhibitor system the two variable oregonator model the supercritical or subcritical character of the turing bifurcation for the brusselator is determined by deriving an amplitude equation for patterns near the instability threshold

bifurcation analysis of an enzyme catalyzed reaction diffusion system abstract in this paper we concern about the dynamics of a diffusive enzyme catalyzed system arising from glycolysis describing a biochemical reaction in which a substrate is converted into a product with positive feedback and into a branched sink the temporal and spatiotemporal dynamics of the system under homogeneous neumann boundary conditions are studied preliminary analysis on the local asymptotic stability and hopf bifurcation of the spatially homogeneous model based on ordinary differential equation is presented for the reaction diffusion model firstly the parameter regions for the stability or instability of the unique constant steady state are discussed finally bifurcations of spatially homogeneous and nonhomogeneous periodic solutions as well as nonconstant steady state solutions are studied numerical simulations are presented to verify and illustrate the theoretical results

turing pattern formation in the brusselator system with nonlinear diffusion in this work we investigate the effect of density dependent nonlinear diffusion on pattern formation in the brusselator system through linear stability analysis of the basic solution we determine the turing and the oscillatory instability boundaries a comparison with the classical linear diffusion shows how nonlinear diffusion favors the occurrence of turing pattern formation we study the process of pattern formation both in one dimensional and two dimensional spatial domains through a weakly nonlinear multiple scales analysis we derive the equations for the amplitude of the stationary patterns the analysis of the amplitude equations shows the occurrence of a number of different phenomena including stable supercritical and subcritical turing patterns with multiple branches of stable solutions leading to hysteresis moreover we consider traveling patterning waves when the domain size is large the pattern forms sequentially and traveling wave fronts are the precursors to patterning we derive the ginzburg landau equation and describe the traveling front enveloping a pattern which invades the domain we show the emergence of radially symmetric target patterns and through a matching procedure we construct the outer amplitude equation and the inner core solution

pattern formation in a general two cell brusselator model abstract in this article we are concerned with the following general coupled two cell brusselator type system d 1 u a b 1 u f u v c w u in d 2 v b u f u v in d 1 w a b 1 w f w z c u w in d 2 z b w f w z in u v w z 0 on here r n n 1 is a smooth and bounded domain a b c d 1 d 2 are positive constants and f c 1 0 c 0 is a nonnegative and nondecreasing function such that f 0 in 0 when f u u 2 this system corresponds to the coupled two cell brusselator model in the present work we exhibit the crucial role played by the nonlinearity f in generating the stationary patterns our conclusions show that if f has a sublinear growth then no stationary patterns occur while if f has a superlinear growth stationary patterns may exist

multiscale adjoint waveform difference tomography using wavelets abstractfull waveform seismic inversions based on minimizing the distance between observed and predicted seismograms are in principle able to yield better resolved earth models than those minimizing misfits derived from traveltimes alone adjoint based methods provide an efficient way of calculating the gradient of the misfit function via a sequence of forward modeling steps which using spectral element codes can be carried out in realistically complex media convergence and stability of full waveform difference adjoint schemes are greatly improved when data and synthetics are progressively presented to the algorithms in a constructive multiscale approximation using a bi orthogonal wavelet transform wavelets provide the nonredundant spectral decomposition that paves the way for the inversion to proceed successively from long wavelength fitting to detailed exploration of the phases in the seismogram the choice of wavelet class and type the initial depth of the multiscale decomposition and the min

stationary solutions of a volume filling chemotaxis model with logistic growth and their stability in this paper we derive the conditions for the existence of stationary solutions i e nonconstant steady states of a volume filling chemotaxis model with logistic growth over a bounded domain subject to homogeneous neumann boundary conditions at the same time we show that the same system without the chemotaxis term does not admit pattern formations moreover based on an explicit formula for the stationary solutions which is derived by asymptotic bifurcation analysis we establish the stability criteria and find a selection mechanism of the principal wave modes for the stable stationary solution by estimating the leading term of the principal eigenvalue we show that all bifurcations except the one at the first location of the bifurcation parameter are unstable and if the pattern is stable then its principal wave mode must be a positive integer which minimizes the bifurcation parameter for a special case where the carrying capacity is one half we find a necessary and sufficient condition for th

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

growing network with local rules preferential attachment clustering hierarchy and degree correlations the linear preferential attachment hypothesis has been shown to be quite successful in explaining the existence of networks with power law degree distributions it is then quite important to determine if this mechanism is the consequence of a general principle based on local rules in this work it is claimed that an effective linear preferential attachment is the natural outcome of growing network models based on local rules it is also shown that the local models offer an explanation for other properties like the clustering hierarchy and degree correlations recently observed in complex networks these conclusions are based on both analytical and numerical results for different local rules including some models already proposed in the literature

local rewiring rules for evolving complex networks abstract the effects of link rewiring are considered for the class of directed networks where each node has the same fixed out degree we model a network generated by three mechanisms that are present in various networked systems growth global rewiring and local rewiring during a rewiring phase a node is randomly selected one of its out going edges is detached from its destination then re attached to the network in one of two possible ways either globally to a randomly selected node or locally to a descendant of a descendant of the originally selected node although the probability of attachment to a node increases with its connectivity the probability of detachment also increases the result is an exponential degree distribution with a small number of outlying nodes that have extremely large degree we explain these outliers by identifying the circumstances for which a set of nodes can grow to very high degree

positive steady state solutions of the noyes field model for belousov zhabotinskii reaction abstract in the paper we investigate the noyes field model for belousov zhabotinskii reaction and study positive steady state solutions of this model with the homogeneous neumann boundary condition we obtain the existence and non existence of non constant positive steady state solutions

on steady state solutions of the brusselator type system abstract in this article we shall be concerned with the following brusselator type system u 1 b 1 u b u m v in v a 2 u u m v in under the homogeneous neumann boundary conditions this system was recently investigated by m ghergu in nonlinearity 21 2008 2331 2345 here r n n 1 is a smooth and bounded domain and a b m and are positive constants when m 2 this system corresponds to the well known stationary brusselator model which has received extensive studies analytically as well as numerically in the present work we derive some further results for the general system our conclusions show that there is no non constant positive steady state for large a while small a may produce non constant positive steady states if 1 n 3 and 1 m 3 we particularly determine the asymptotic behavior of non constant positive steady states as a converges to zero thereby solving an open problem left in ghergu work

organization of growing random networks the organizational development of growing random networks is investigated these growing networks are built by adding nodes successively and linking each to an earlier node of degree k with an attachment probability a k when a k grows more slowly than linearly with k the number of nodes with k links n k t decays faster than a power law in k while for a k growing faster than linearly in k a single node emerges which connects to nearly all other nodes when a k is asymptotically linear n k t ensuremath sim mathrm tk ensuremath ensuremath nu with ensuremath nu dependent on details of the attachment probability but in the range 2l ensuremath nu l ensuremath infty the combined age and degree distribution of nodes shows that old nodes typically have a large degree there is also a significant correlation in the degrees of neighboring nodes so that nodes of similar degree are more likely to be connected the size distributions of the in and out components of the network with respect to a given node namely its descendants and ancestors are also determined the in component exhibits a robust s ensuremath 2 power law tail where s is the component size the out component has a typical size of order mathrm ln t and it provides basic insights into the genealogy of the network

computing traveltime and amplitude sensitivity kernels in finite frequency tomography the efficient computation of finite frequency traveltime and amplitude sensitivity kernels for velocity and attenuation perturbations in global seismic tomography poses problems both of numerical precision and of validity of the paraxial approximation used we investigate these aspects using a local model parameterization in the form of a tetrahedral grid with linear interpolation in between grid nodes the matrix coefficients of the linear inverse problem involve a volume integral of the product of the finite frequency kernel with the basis functions that represent the linear interpolation we use local and global tests as well as analytical expressions to test the numerical precision of the frequency and spatial quadrature there is a trade off between narrowing the bandpass filter and quadrature accuracy and efficiency using a minimum step size of 10km for s waves and 30km for ss waves relative errors in the quadrature are of the order of 1 for direct waves such as s and a few percent for ss waves which are below data uncertainties in delay time or amplitude anomaly observations in global seismology larger errors may occur wherever the sensitivity extends over a large volume and the paraxial approximation breaks down at large distance from the ray this is especially noticeable for minimax phases such as ss waves with periods 20s when kernels become hyperbolic near the reflection point and appreciable sensitivity extends over thousands of km errors becomes intolerable at epicentral distance near the antipode when sensitivity extends over all azimuths in the mantle effects of such errors may become noticeable at epicentral distances140 we conclude that the paraxial approximation offers an efficient method for computing the matrix system for finite frequency inversions in global tomography though care should be taken near reflection points and alternative methods are needed to compute sensitivity near the antipode

stability of turing patterns in the brusselator model the selection and competition of turing patterns in the brusselator model are reviewed the stability of stripes and hexagons towards spatial perturbations is studied using the amplitude equation formalism for hexagonal patterns these equations include both linear and nonpotential spatial terms enabling distorted solutions the latter modify substantially the stability diagrams and select patterns with wave numbers quite different from the critical value the analytical results from the amplitude formalism agree with direct simulations of the model moreover we show that slightly squeezed hexagons are locally stable in a full range of distortion angles the stability regions resulting from the phase equation are similar to those obtained numerically by other authors and to those observed in experiments

spatio temporal chaos in a chemotaxis model abstract in this paper we explore the dynamics of a one dimensional keller segel type model for chemotaxis incorporating a logistic cell growth term we demonstrate the capacity of the model to self organise into multiple cellular aggregations which according to position in parameter space either form a stationary pattern or undergo a sustained spatio temporal sequence of merging two aggregations coalesce and emerging a new aggregation appears this spatio temporal patterning can be further subdivided into either a time periodic or time irregular fashion numerical explorations into the latter indicate a positive lyapunov exponent sensitive dependence to initial conditions together with a rich bifurcation structure in particular we find stationary patterns that bifurcate onto a path of periodic patterns which prior to the onset of spatio temporal irregularity undergo a periodic doubling sequence based on these results and comparisons with other systems we argue that the spatio temporal irregularity observed here describes a form of spatio temporal chaos we discuss briefly our results in the context of previous applications of chemotaxis models including tumour invasion embryonic development and ecology

long time behaviour of solutions to a chemotaxis model with volume filling effect the lyapunov functional is constructed for a quasilinear parabolic system which models chemotaxis and takes into account a volume filling effect for some typical case it is proved that the limit set of any trajectory consists of regular stationary solutions some lower and upper bounds on the stationary solutions are found for a given range of parameters there are stationary solutions which are inhomogeneous in space

scaling of weighted spectral distribution in deterministic scale free networks abstract scale free networks are abundant in the real world in this paper we investigate the scaling properties of the weighted spectral distribution in several deterministic and stochastic models of evolving scale free networks first we construct a new deterministic scale free model whose node degrees have a unified format using graph structure features we derive a precise formula for the spectral metric in this model this formula verifies that the spectral metric grows sublinearly as network size i e the number of nodes grows additionally the mathematical reasoning of the precise formula theoretically provides detailed explanations for this scaling property finally we validate the scaling properties of the spectral metric using some stochastic models the experimental results show that this scaling property can be retained regardless of local world node deleting and assortativity adjustment

continuation and bifurcations numerical techniques and applications bifurcation to rotating waves from non trivial steady states use of approximate inertial manifolds in bifurcation calculations understanding steady state bifurcation diagrams for a model reaction diffusion system bifurcations chaos and self organization in reaction diffusion systems eigenvalue problems with the symmetry of a group and bifurcations steady state steady state mode interaction in nonlinear equations with z2 symmetry symbolic computation and bifurcation methods bifurcation analysis a combined numerical and analytical approach to the numerical solution of symmetry breaking bifurcation problems a computational method and path following for periodic solutions with symmetry global bifurcations and their numerical computation computation of invariant manifold bifurcations a method for homoclinic and heteroclinic continuation in two and three dimensions the global attractor under discretisation the numerical detection of hopf bifurcation points a newton like method for simple bifurcation problems with application to large sparse systems aspects of continuation software interactive system for studies in nonlinear dynamics linlbf a program for continuation and bifurcation analysis of equilibria up to codimension three on the topology of three dimensional separations a guide for classification the construction of cubature formulae using continuation and bifurcation software determining an organizing center for passive optical systems optimization by continuation stability of marangoni convection in a microgravity environment computing with reaction diffusion systems applications in image processing bifurcation of codimension 2 for a discrete map bifurcation of periodic solutions in pde s numerical techniques and applications abstract bifurcation and chaos in chua s circuit family abstract continuation and collocation for parameter dependent boundary value problems abstract porous medium combustion abstract block elimination and the computation of simple turning points abstract bifurcation into gaps in the essential spectrum abstract application of numerical continuation in aerospace problems abstract application of a reduced basis method in structural analysis abstract some applications of bifurcation theory in engineering abstract higher order predictors in numerical path following schemes abstract

multiscale adjoint waveform tomography for surface and body waves abstractwe have developed a wavelet multiscale adjoint scheme for the elastic full waveform inversion of seismic data including body waves bws and surface waves sws we start the inversion on the sw portion of the seismograms to avoid cycle skipping and reduce the dependence on the initial model of these dispersive waves we commence by minimizing an envelope based misfit function subsequently we proceed to the minimization of a waveform difference wd metric applied to the sws only after that we fit bws and sws indiscriminately using a wd misfit metric in each of these three steps we guide the iterative inversion through a sequence of nested subspace projections in a wavelet basis sw analysis preserves a wealth of near surface features that would be lost in conventional bw tomography we used a toy model to illustrate the dispersive and cycle skipping behavior of the sws and to introduce the two ways by which we combat the nonlinearity of waveform inversions involving sws the first is the

sensitivity kernels for seismic fresnel volume tomography fresnel volume tomography fvt offers higher resolution and better accuracy than conventional seismic raypath tomography a key problem in fvt is the sensitivity kernel we propose amplitude and traveltime sensitivity kernels expressed directly with green s functions for transmitted waves for 2d 3d homogeneous heterogeneous media the green s functions are calculated with a finite difference operator of the full wave equation in the frequency space domain in the special case of homogeneous media we analyze the properties of the sensitivity kernels extensively and gain new insight into these properties according to the constructive interference of waves the spatial distribution ranges of the monochromatic sensitivity kernels in fvt differ from each other greatly and are 1 8 2 8 3 8 and 4 8 periods of seismic waves respectively for 2d amplitude 3d amplitude 2d traveltime and 3d traveltime conditions we also have a new understanding of the relationship between raypath tomography and fvt within the first fresnel volume of the dominant frequency the band limited sensitivity kernels of fvt in homogeneous media or smoothly heterogeneous media are very close to those of the dominant frequency thus it is practical to replace the band limited sensitivity kernel with a few selected frequencies or even the single dominant frequency to save computation when performing band limited fvt the numerical experiment proves that fvt using our sensitivity kernels can achieve more accurate results than traditional raypath tomography

memory and burstiness in dynamic networks a discrete time random process is described which can generate bursty sequences of events a bernoulli process where the probability of an event occurring at time t is given by a fixed probability x is modified to include a memory effect where the event probability is increased proportionally to the number of events that occurred within a given amount of time preceding t for small values of x the interevent time distribution follows a power law with exponent 2 x we consider a dynamic network where each node forms and breaks connections according to this process the value of x for each node depends on the fitness distribution x from which it is drawn we find exact solutions for the expectation of the degree distribution for a variety of possible fitness distributions and for both cases where the memory effect either is or is not present this work can potentially lead to methods to uncover hidden fitness distributions from fast changing temporal network data such as online social communications and fmri scans

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

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

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

minimum structural controllability problems of complex networks abstract controllability of complex networks has been one of the attractive research areas for both network and control community and has yielded many promising and significant results in minimum inputs and minimum driver vertices however few studies have been devoted to studying the minimum controlled vertex set through which control over the network with arbitrary structure can be achieved in this paper we prove that the minimum driver vertices driven by different inputs are not sufficient to ensure the full control of the network when the associated graph contains the inaccessible strongly connected component which has perfect matching and propose an algorithm to identify a minimum controlled vertex set for network with arbitrary structure using convenient graph and mathematical tools and the simulation results show that the controllability of network is correlated to the number of inaccessible strongly connected components which have perfect matching and these results promote us to better understand the relationship between the network s structural characteristics and its control

l2 energy stability via new dependent variables for circumventing strongly nonlinear reaction terms abstract the paper concerns a pair of reaction diffusion system of p d es which have strongly nonlinear reaction terms in the dependent variables and which do not involve cross diffusion terms under robin boundary conditions the introduction of new dependent variables suitable linear combinations of the original ones which effectively induce cross diffusion terms allows one to i characterize the high nonlinearities with hidden dissipation ii determine a procedure for bypassing these nonlinearities in the l 2 energy stability analysis iii obtain liapunov functionals guaranteeing the same conditions of linear stability as an application the l 2 energy stability analysis of the brusselator system is performed

bifurcation and stability analysis of steady states to a brusselator model abstract in this paper we consider the structure of the nonnegative steady state solutions of a brusselator model we prove the existence and boundedness of continua of the steady state solutions by applying the theorems of global bifurcation theory the concentration of one intermediary reactant is treated as a bifurcation parameter through the asymptotic bifurcation analysis we derive an explicit formula for the non constant steady states i e stationary patterns using it as a base we establish the stability criteria and find a selection mechanism of the wave modes for the stable patterns by computing the leading term of the principal eigenvalue our results demonstrate that all of the bifurcations other than the one at the first bifurcation location are unstable and if the pattern is stable then its mode must be a positive integer which minimizes the bifurcation parameter a specific example is presented to illustrate our analytical results

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