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

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

chemotaxis driven pattern formation for a reaction diffusion chemotaxis model with volume filling effect in this paper we analytically and numerically investigate the emerging process of pattern formation for a reaction diffusion chemotaxis model with volume filling effect we first apply globally asymptotic stability analysis to show that the chemotactic flux is the key mechanism for pattern formation then by weakly nonlinear analysis with multiple scales and the adjoint system theory we derive the cubic and the quintic stuart landau equations to describe the evolution of the amplitude of the most unstable mode and thus the analytical approximate solutions of the patterns are obtained next we present the selection law of principal wave mode of the emerging pattern by considering the competition of the growing modes and for this we deduce the change rule of the most unstable mode and the coupled ordinary differential equations that indicates the significant nonlinear interaction of two competing modes finally in the subcritical case we clarify that there exists the phenomenon of hysteresis which implies the existence of large amplitude pattern for the bifurcation parameter values smaller than the first bifurcation point therefore we answer the open problems proposed in the known references and improve some of results obtained there all the theoretical results are tested against the numerical results showing excellent qualitative and good quantitative agreement

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

global bifurcations of limit and separatrix cycles in a generalized lienard system abstract we construct a canonical cubic dynamical system of kukles type and carry out the global qualitative analysis of its special case corresponding to a generalized lienard equation in particular we prove that the foci of such a lienard system can be at most of second order and that this particular system can have at least three limit cycles in the whole phase plane moreover unlike all previous works on the kukles type systems we study global bifurcations of limit and separatrix cycles using arbitrary including as large as possible field rotation parameters of our canonical system as a result we have obtained a classification of all possible types of separatrix cycles for this generalized lienard system and also all possible distributions of its limit cycles

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

bifurcation of limit cycles from centers and separatrix cycles of planar analytic systems this paper presents a survey of results on the bifurcation of limit cycles from centers and separatrix cycles of perturbed planar analytic systems and contributes some new results on the bifurcation of multiple limit cycles from centers and on the multiplicity of separatrix cycles of such systems the basic theme throughout the paper is that the number positions and multiplicities of the limit cycles that bifurcate under perturbations are related to the number positions and multiplicities of the zeros of the melnikov function for the system the general theory is illustrated by a number of examples from the literature some of which are extended to include new results

global attractor for a chemotaxis model with prevention of overcrowding abstract chemotaxis model with volume filling introduced by painter and hillen is studied under no flux or dirichlet boundary conditions existence of global in time solution to a full model is proved for some cases existence of a global attractor in the space w 1 p r 2 p n r n is shown

computing centre conditions for certain cubic systems abstract we present necessary and sufficient conditions for a critical point of certain two dimensional cubic differential systems to be a centre extensive use of the computer algebra system reduce is involved the search for necessary and sufficient conditions for a centre has long been of considerable interest in the theory of nonlinear differential equations it has proved to be a difficult problem and full conditions are known for very few classes of systems such conditions are also required in the investigation of hilbert s sixteenth problem concerning the number of limit cycles of polynomial systems

on maximal abelian subgroups in finite 2 groups we determine here up to isomorphism the structure of any finite nonabelian 2 group g in which every two distinct maximal abelian subgroups have cyclic intersection we obtain five infinite classes of such 2 groups theorem 1 1 this solves for p 2 the problem nr 521 stated by berkovich in preparation the more general problem nr 258 stated by berkovich in preparation about the structure of finite nonabelian p groups g such that a b z g for every two distinct maximal abelian subgroups a and b is treated in theorems 3 1 and 3 2 in corollary 3 3 we get a new result for an arbitrary finite 2 group as an application of theorems 3 1 and 3 2 we solve for p 2 a problem of heineken mann problem nr 169 stated in berkovich in preparation classifying finite 2 groups g such that a z g is cyclic for each maximal abelian subgroup a theorem 4 1

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

basic mechanisms driving complex spike dynamics in a chemotaxis model with logistic growth recently wang and hillen chaos 17 3 2007 037108 037108 introduced a modified keller segel model with logistic growth that exhibits complex spatiotemporal dynamics of spikes these dynamics are driven by merging of spikes on one hand and spike insertion on the other in this paper we analyse the basic mechanisms that initiate and sustain these events we identify two distinguished regimes in the first regime a single interior spike drifts towards a boundary this instability is responsible for spike merging the same regime further exhibits spike insertion we identify a fold point bifurcation which is precursor to spike insertion event in the second regime we show that it is possible to stabilize a single interior spike and we compute analytically a critical threshold which is responsible for spike stabilization in particular our calculation characterizes a stable spike in the keller segel model with logistic growth this is in contrast to the classical keller segel model where the interior spike is known to be unstable

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

perturbation from an elliptic hamiltonian of degree four iv figure eight loop the paper deals with lienard equations of the form x y y p x yq x with p and q polynomials of degree respectively 3 and 2 attention goes to perturbations of the hamiltonian vector fields with an elliptic hamiltonian of degree four exhibiting a figure eight loop it is proved that the least upper bound of the number of zeros of the related elliptic integral is five and this is a sharp bound multiplicity taken into account moreover if restricting to the level curves inside a saddle loop or outside the figure eight loop the sharp upper bound is respectively two or four also the multiplicity of the zeros is at most four this is the last one in a series of papers on this subject the results of this paper together with j differential equations 176 2001 114 j differential equations 175 2001 209 j differential equations to be published largely finish the study of the cubic perturbations of the elliptic hamiltonians of degree four and presumably provide a complete description of the number and the possible configurations of limit cycles for cubic lienard equations with small quadratic damping as a special case we obtain a configuration of four limit cycles surrounding three singularities together with a small limit cycle which surrounds one of the singularities

global qualitative analysis of a quartic ecological model abstract in this paper we complete the global qualitative analysis of a quartic ecological model in particular studying global bifurcations of singular points and limit cycles we prove that the corresponding dynamical system has at most two limit cycles

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

higher order bifurcations of limit cycles abstract this paper shows that asymmetrically perturbed symmetric hamiltonian systems of the formx y y x x3 1y 2x2 3xy 4x2y with analytic j e o e have at most two limit cycles that bifurcate for smalle 0 from any period annulus of the unperturbed system this fact agrees with previous results of petrov dangelmayr and guckenheimer and chicone and iliev but shows that the result of three limit cycles for the asymmetrically perturbed exterior duffing oscillator recently obtained by jebrane and oladek is incorrect the proofs follow by deriving an explicit formula for thekth order melnikov function mk h and using a picard fuchs analysis to show that in each case mk h has at most two zeros moreover the method developed in this paper for determining the higher order melnikov functions also applies to more general perturbations of these systems

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

multiple limit cycle bifurcation surfaces and global families of multiple limit cycles abstract sufficient conditions are given for the local exitence of multiplicity m limit cycle bifurcation surfaces cm of planar analytic systems depending on n parameters with n m 2 in the generic case the surfaces c2 c3 and c4 are the familiar saddle node cusp and swallow tail bifurcation surfaces respectively the author s termination principle for global one parameter families of simple limit cycles of relatively prime planar analytic systems is generalized to a termination principle for global one parameter families of multiple limit cycles which implies that the boundary of a global limit cycle bifurcation surface typically consists of hopf bifurcation surfaces and or homoclinic or heteroclinic loop bifurcation surfaces

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

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

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

pattern formation for a two dimensional reaction diffusion model with chemotaxis abstract this paper is devoted to study the formation of stationary patterns for a chemotaxis model with nonlinear diffusion and volume filling effect over a bounded rectangular domain by using linear stability analysis around the homogeneous steady states we establish conditions for the existence of unstable mode bands that lead to the formation of spatial patterns we derive the stuart landau equations for the pattern amplitudes by means of weakly nonlinear multiple scales analysis and fredholm theory in particular we find asymptotic expressions for a wide range of patterns sustained by the system these patterns include mixed mode square hexagonal and roll stationary configurations our analytical results are corroborated by direct simulations of the underlying chemotaxis system

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

multiple limit cycle bifurcations of the fitzhugh nagumo neuronal model abstract in this paper we complete the global qualitative analysis of the well known fitzhugh nagumo neuronal model in particular studying global limit cycle bifurcations and applying the wintner perko termination principle for multiple limit cycles we prove that the corresponding dynamical system has at most two limit cycles

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

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

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