bifurcation analysis of a generalized gause model with prey harvesting and a generalized holling response function of type iii abstract in this paper we study a generalized gause model with prey harvesting and a generalized holling response function of type iii p x m x 2 a x 2 b x 1 the goal of our study is to give the bifurcation diagram of the model for this we need to study saddle node bifurcations hopf bifurcation of codimension 1 and 2 heteroclinic bifurcation and nilpotent saddle bifurcation of codimension 2 and 3 the nilpotent saddle of codimension 3 is the organizing center for the bifurcation diagram the hopf bifurcation is studied by means of a generalized lienard system and for b 0 we discuss the potential integrability of the system the nilpotent point of multiplicity 3 occurs with an invariant line and can have a codimension up to 4 but because it occurs with an invariant line the effective highest codimension is 3 we develop normal forms in which the invariant line is preserved for studying of the nilpotent saddle bifurcation for b 0 the reversibility of the nilpotent saddle is discussed we study the type of the heteroclinic loop and its cyclicity the phase portraits of the bifurcations diagram partially conjectured via the results obtained allow us to give a biological interpretation of the behavior of the two species

essentially normal homogeneous quotient modules on the polydisc abstract in the present paper we give a complete characterization of all essentially normal homogeneous quotient modules of the hardy space on the unit polydisc it turns out that our approach also applies to quotients of weighted bergman modules

abelian integrals in unfoldings of codimension 3 singularities with nilpotent linear parts abstract this paper is concerned with the upper bound of the number of limit cycles in unfolding of codimension 3 planar singularities with nilpotent linear parts after making a central rescaling the problem reduces to a perturbation problem of a one parameter family of quadratic reversible systems as the parameter a 1 1 0 is rational except the case a 2 3 based on the chebyshev criterion for abelian integrals and a rationalizing transformation the problem could be solved theoretically to illustrate our approaches two particular cases corresponding to nilpotent codimension 3 saddle and elliptic case respectively are proved where the upper bound of the number of limit cycles is two

estimates for derivatives of holomorphic functions in pseudoconvex domains in this paper we will prove several sobolev type inequalities for holomorphic functions in strictly pseudoconvex domains to a large extent this work was motivated by the following problem let d be a strictly pseudoconvex domain and let m be the intersection with d of a submanifold of a neighborhood o f which intersects 0d transversally let f be a holomorphic function on m lying in some l p class e g a hardy or bergman class we want to find a holomorphic extension of f to d satisfying an optimal l p estimate on d the case p c was solved by henkin 14 who showed that a bounded holomorphic function on m is necessarily the restriction of a bounded holomorphic function on d for certain weighted bergman spaces on m optimal l p estimates for extensions were obtained by cumenge 9 in the case l p o o and by beatrous 4 in the case 0 p c more precisely letting 6 z denote the distance from z to 0d the following result was proved let f be a holomorphic function on m satisfying

bifurcation analysis of a predator prey system involving group defence a class of odes of generalized gause type modeling predator prey interaction is considered the prey are assumed to exhibit a phenomenon called group defence that is predation is decreased or even eliminated due to the ability of the prey to defend or disguise themselves as their numbers increase using the carrying capacity of the environment as the bifurcation parameter it is shown that the model undergoes a sequence of bifurcations that includes a homoclinic bifurcation as well as a hopf bifurcation conditions that hold even in the case of no group defence that ensure a subcritical hopf bifurcation and also the spontaneous appearance of a semistable periodic orbit that splits into a pair one stable and one unstable of periodic orbits are given ecological ramifications are considered unlike the classical model sufficient enrichment of the environment combined with group defence leads to extinction of the predator deterministically for almost all initial conditions providing strong support fo

finite cyclicity of graphics with a nilpotent singularity of saddle or elliptic type abstract in this paper we prove finite cyclicity of several of the most generic graphics through a nilpotent point of saddle or elliptic type of codimension 3 inside c families of planar vector fields in some cases our results are independent of the exact codimension of the point and depend only on the fact that the nilpotent point has multiplicity 3 by blowing up the family of vector fields we obtain all the limit periodic sets we calculate two different types of dulac maps in the blown up family and develop a general method to prove that some regular transition maps have a nonzero higher derivative at a point the finite cyclicity theorems are derived by a generalized derivation division method introduced by roussarie

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

essential normality of polynomial generated submodules hardy space and beyond abstract recently douglas and wang proved that for each polynomial q the submodule q of the bergman module generated by q is essentially normal 11 using improved techniques we show that the hardy space analogue of this result holds and more

toeplitz operators and weighted bergman kernels abstract for a smoothly bounded strictly pseudoconvex domain we describe the boundary singularity of weighted bergman kernels with respect to weights behaving like a power possibly fractional of a defining function and more generally of the reproducing kernels of sobolev spaces of holomorphic functions of any real order this generalizes the classical result of fefferman for the unweighted bergman kernel finally we also exhibit a holomorphic continuation of the kernels with respect to the sobolev parameter to the entire complex plane our main tool are the generalized toeplitz operators of boutet de monvel and guillemin

global analysis in a predator prey system with nonmonotonic functional response a predator prey system with nonmonotonic functional response is considered global qualitative and bifurcation analyses are combined to determine the global dynamics of the model the bifurcation a

bifurcation analysis in a host generalist parasitoid model with holling ii functional response abstract in this paper we study a host generalist parasitoid model with holling ii functional response where the generalist parasitoids are introduced to control the invasion of the hosts it is shown that the model can undergo a sequence of bifurcations including cusp focus and elliptic types degenerate bogdanov takens bifurcations of codimension three and a degenerate hopf bifurcation of codimension at most two as the parameters vary and the model exhibits rich dynamics such as the existence of multiple coexistent steady states multiple coexistent periodic orbits homoclinic orbits etc moreover there exists a critical value for the carrying capacity of generalist parasitoids such that i when the carrying capacity of the generalist parasitoids is smaller than the critical value the invading hosts can always persist despite of the predation by the generalist parasitoids i e the generalist parasitoids cannot control the invasion of hosts ii when the carrying capacity of the generalist parasitoids is larger than the critical value the invading hosts either tend to extinction or persist in the form of multiple coexistent steady states or multiple coexistent periodic orbits depending on the initial populations i e whether the invasion can be stopped and reversed by the generalist parasitoids depends on the initial populations iii in both cases the generalist parasitoids always persist numerical simulations are presented to illustrate the theoretical results

a harmonic analysis approach to essential normality of principal submodules guo and the second author have shown that the closure i in the drury arveson space of a homogeneous principal ideal i in c z1 zn is essentially normal in this note the authors extend this result to the closure of any principal polynomial ideal in the bergman space in particular the commutators and cross commutators of the restrictions of the multiplication operators are shown to be in the schatten p class for p n the same is true for modules generated by polynomials with vector valued coefficients further the maximal ideal space xi of the resulting c algebra for the quotient module is shown to be contained in z i bn where z i is the zero variety for i and to contain all points in bn that are limit points of z i bn finally the techniques introduced enable one to study a certain class of weight bergman spaces on the ball

pp graphics with a nilpotent elliptic singularity in quadratic systems and hilbert s 16th problem abstract this paper is part of the program launched in j differential equations 110 1 1994 86 to prove the finiteness part of hilbert s 16th problem for quadratic system which consists in proving that 121 graphics have finite cyclicity among quadratic systems we show that any pp graphic through a multiplicity 3 nilpotent singularity of elliptic type which does not surround a center has finite cyclicity such graphics may have additional saddles and or saddle nodes altogether we show the finite cyclicity of 15 graphics of j differential equations 110 1 1994 86 in particular we prove the finite cyclicity of a pp graphic with an elliptic nilpotent singular point together with a hyperbolic saddle with hyperbolicity 1 which appears in generic 3 parameter families of vector fields and hence belongs to the zoo of kotova and stanzo concerning the hilbert 16th problem american mathematical society translation series 2 vol 165 american mathematical society providence ri 1995 pp 155 201

an analytic grothendieck riemann roch theorem we extend the boutet de monvel toeplitz index theorem to complex manifolds with isolated singularities following the relative k homology theory of baum douglas and taylor for manifolds with boundary we apply this index theorem to study the arveson douglas conjecture let bmbm be the unit ball in cmcm and i an ideal in the polynomial algebra c z1 zm c z1 zm we prove that when the zero variety zizi is a complete intersection space with only isolated singularities and intersects with the unit sphere s2m 1s2m 1 transversely the representations of c z1 zm c z1 zm on the closure of i in la2 bm and also the corresponding quotient space qiqi are essentially normal furthermore we prove an index theorem for toeplitz operators on qiqi by showing that the representation of c z1 zm c z1 zm on the quotient space qiqi gives the fundamental class of the boundary zi s2m 1zi s2m 1 in the appendix we prove with kai wang that if f la2 bm vanishes on zi bmzi bm then f is contained inside the closure of the ideal i in la2 bm

multiplier algebras of complete nevanlinna pick spaces dilations boundary representations and hyperrigidity abstract we study reproducing kernel hilbert spaces on the unit ball with the complete nevanlinna pick property through the representation theory of their algebras of multipliers we give a complete description of the representations in terms of the reproducing kernels while representations always dilate to representations of the ambient c algebra we show that in our setting we automatically obtain coextensions in fact we show that in many cases this phenomenon characterizes the complete nevanlinna pick property we also deduce operator theoretic dilation results which are in the spirit of work of agler and several other authors moreover we identify all boundary representations compute the c envelopes and determine hyperrigidity for certain analogues of the disc algebra finally we extend these results to spaces of functions on homogeneous subvarieties of the ball

hopf takens bifurcations and centres the paper deals with hopf takens bifurcations both in generic families and in families containing centres attention goes to the relation between normal forms lyapunov coefficients and the bautin ideal in case the bautin ideal has only one generator we pay attention to the first non zero melnikov function

essential normality for quotient modules and complex dimensions abstract for analytic sets m introduced in 9 we show that the corresponding quotient module q of the bergman module is p essentially normal for all p dim c m which verifies the geometric arveson douglas conjecture in this case this result makes it possible to study the helton howe trace invariants on both q and the corresponding submodule r

a method of constructing cycles without contact around a weak focus has a weak focus then limit cycles can be made to appear in its neighborhood by varying slightly the right sides 11 frommer 2 bautin 3 chin et al 4 and tung 5 are examples of excellent expositions of poincare s idea it is necessary to go from numerical examples to general theory in the first part of this paper we present a method of constructing cycles without contact around a weak focus which makes clear poincart s idea used in the above work 2 51 in the second part another method to analyze global phase portraits is used quadratic systems with at least four limit cycles are produced by means of both the above methods this contradicts the results of petrovskii and landis 6

essentially normal hilbert modules and k homology this paper mainly concerns the essential normality of graded submodules essentially all of the basic hilbert modules that have received attention over the years are p essentially normal including the d shift hilbert module the hardy and bergman modules of the unit ball arveson conjectured graded submodules over the unit ball inherit this property and provided motivations to seek an affirmative answer some positive results have been obtained by arveson and douglas however the problem has been resistant in dimensions d 2 3 this paper shows that the arveson s conjecture is true in any dimension the paper also gives an affirmative answer in the case of the graded principal submodule finally the paper is associated with k homology invariants arising from graded quotient modules by which geometry of the quotient modules and geometry of algebraic varieties are connected in dimensions d 2 3 it is shown that k homology invariants determined by graded quotients are nontrivial the paper also establishes results on p smoothness of k homology elements and gives an explicit expression for k homology invariant in dimension d 2

geometric arveson douglas conjecture abstract we prove the arveson douglas essential normality conjecture for graded hilbert submodules that consist of functions vanishing on a given homogeneous subvariety of the ball smooth away from the origin our main tool is the theory of generalized toeplitz operators of boutet de monvel and guillemin

a new index theorem for monomial ideals by resolutions abstract we prove an index theorem for the quotient module of a monomial ideal we obtain this result by resolving the monomial ideal by a sequence of essentially normal hilbert modules each of which is a direct sum of weighted bergman spaces on balls

dixmier trace for toeplitz operators on symmetric domains abstract for toeplitz operators on bounded symmetric domains of higher rank there is no obvious way to define the dixmier trace within the toeplitz c algebra since commutators are in general not compact in this paper we solve this problem by constructing a hilbert quotient module corresponding to partitions of length 1 which leads to commutators in the macaev class l n we also obtain an explicit formula for the dixmier trace of such commutators in terms of the underlying boundary geometry which involves a new type of flag manifold defined in jordan theoretic terms

bifurcations of a ratio dependent predator prey system with constant rate harvesting the ratio dependent predator prey model exhibits rich interesting dynamics due to the singularity of the origin the objective of this paper is to study the dynamical properties of the ratio depend

organizing center for the bifurcation analysis of a generalized gause model with prey harvesting and holling response function of type iii abstract the present note is an addendum to the paper of etoua rousseau 2010 1 which presented a study of a generalized gause model with prey harvesting and a generalized holling response function of type iii p x m x 2 a x 2 b x 1 complete bifurcation diagrams were proposed but some parts were conjectural an organizing center for the bifurcation diagram was given by a nilpotent point of saddle type lying on an invariant line and of codimension greater than or equal to 3 this point was of codimension 3 when b 0 and was conjectured to be of infinite codimension when b 0 this conjecture was in line with a second conjecture that the hopf bifurcation of order 2 degenerates to a hopf bifurcation of infinite codimension when b 0 in this note we prove these two conjectures

on the direct solution of wave propagation for repetitive structures abstract the direct method of solving the wave constants for a repetitive structure with given frequency is developed in this paper the analogy between structural mechanics and optimal control theory is applied the dynamic stiffness matrix approach is used to solve for the eigensolutions of wave propagation the weighted adjoint symplectic orthonormality relationship and the eigenvector expansion theorem are also established for this approach the symplectic eigenproblem is derived for skew symmetric matrices a cell triangular decomposition for skew symmetric matrices is introduced to reduce the generalized eigenproblem and the symplectic householder transformation is introduced to cell tri diagonalize the skew symmetric matrix to solve the eigenequation the problem of a wave impinging on the boundary is considered by the eigenvector expansion method which could also be applied to a wave scattering when passing over an abnormal part numerical examples show that resonance will occur for the localized vibration case

bifurcation analysis of a predator prey system with nonmonotonic functional response we consider a predator prey system with nonmonotonic functional response p x frac mx ax 2 bx 1 by allowing b to be negative b 2 sqrt a p x is concave up for small values of x 0 a

coexistence region and global dynamics of a harvested predator prey system the study of populational dynamics with harvesting is related to the optimal man agement of renewable resources in this paper we consider a predator prey model in which two ecologically interacting species are harvested independently with constant rates the main purpose of the present work is to offer a complete mathematical analysis for the model and to describe some of the significant qualitative results that may be expected to arise from the interplay of biological forces our study shows that the maximum safe harvest may be far less than what would be assumed from a local analysis for the equilibria it is also proved that the harvested predator prey system may exhibit very complicated dynamics such as a spontaneous appearance of a homoclinic orbit and multiple limit cycles