hopf bifurcation and stability for a differential algebraic biological economic system abstract in this paper we analyze the stability and hopf bifurcation of the biological economic system based on the new normal form and the hopf bifurcation theorem the basic model we consider is owed to a ratio dependent predator prey system with harvesting compared with other researches on dynamics of predator prey population this system is described by differential algebraic equations due to economic factor here as bifurcation parameter it is found that periodic solutions arise from stable stationary states when the parameter increases close to a certain limit finally numerical simulations illustrate the effectiveness of our results

a symplectic fixed point theorem for cpn two symplectic diffeomorphisms 0 1 of a symplectic manifold x are said to be homologous if there exists a smooth homotopy 1 t 0 1 of symplectic diffeomorphisms between them such that the timedependent vector field t defined byd dt t t o t is a globally hamiltonian vector field for allt i e there exists a smooth real valued timedependent hamiltonian functionh x t onx x 0 1 such that t dh t whereht h x t

multiple periodic solutions of hamiltonian systems with prescribed energy abstract consider the periodic solutions of autonomous hamiltonian systems x j h x on the given compact energy hypersurface h 1 1 if is convex or star shaped there have been many remarkable contributions for existence and multiplicity of periodic solutions it is a hard problem to discuss the multiplicity on general hypersurfaces of contact type in this paper we prove a multiplicity result for periodic solutions on a special class of hypersurfaces of contact type more general than star shaped ones

multiplicity of periodic solutions for a delayed ratio dependent predator prey model with monotonic functional response and harvesting terms abstract the existence of four positive periodic solutions for a delayed ratio dependent predator prey model with monotonic functional response x t x t a t b t x t c t g x t y t y t h 1 t y t y t f t g x t t y t t d t h 2 t is established by using the generalized continuation theorem where a t b t c t d t f t t h 1 t and h 2 t are all nonnegative periodic continuous functions with period 0 when h 1 t h 2 t 0 the conditions that guarantee the existence of four positive periodic solutions reduces exactly to that in fan et al 2004 1 and our main result also improves the corresponding results in zhang and hou 2010 2 and fan and wang 2001 3

a three species dynamical system involving prey predation competition and commensalism in this paper a three species dynamical system is explored the system consisting of two logistically growing competing species and the third species acts as a predator as well as host it is predating over second species with holling type ii functional response while first species is benefited from the third species in addition the prey species move into a refuge to avoid high predation the essential mathematical features of the proposed model are studied in terms of boundedness persistence local stability and bifurcation the existence of transcritical bifurcations have been established about two axial points it has been observed that survival of all three species may be possible due to commensalism numerical simulations have been performed to show the hopf bifurcation about interior equilibrium point the existence of period 2 solution is observed further the bifurcations of codimension 2 have also been investigated

multiplicity of lagrangian orbits on symmetric star shaped hypersurfaces abstract in this paper the multiplicity of lagrangian orbits on c 2 smooth compact symmetric star shaped hypersurfaces with respect to the origin in r 2 n is studied we prove an infinitely many existence result via z 2 index theory this is a multiplicity result about the arnold chord in some sense

on the periodic fucik spectrum and a superlinear sturm liouville equation in the first part of the paper a variational characterisation of the periodic eigenvalues the so called fucik spectrum of a semilinear positive homogeneous sturm liouville equation is given the proof relies on the s 1 invariance of the equation in the second part a nonlinear sturm liouville equation with typically an exponential nonlinearity is considered it is proved that under certain conditions this equation is solvable for arbitrary forcing terms the proof uses a comparison of the minimax levels of the functional associated to this equation with suitable values in the fucik spectrum

a matrix nearness problem related to iterative methods we consider a matrix nearness problem arising from an analysis of the speed of convergence of gmres for solving a linear system ax b with a in bbb c n times n and b in bbb c n more precisely denoting by cal f k the set of matrices of rank k at most we solve min s in cal s f k in cal f k vert a s f k vert end equation where cal s subset bbb c n times n denotes the set of matrices of the form e i theta h lambda i with theta in 0 2 pi lambda in bbb c and h belonging to the set of hermitian matrices as to iterative methods the set cal s is of interest in a larger context to give an example having a regular splitting a s fk of a the system ax b can be solved using a k 3 term recurrence with inner outer iterations

a perturbation theory near convex hamiltonian systems soit he x p une famille de fonctions sur r n r n dependant du parametre e r d pour e petit on considere des solutions periodiques aux equations de hamilton dx i dt he x p p i dp i dt he x p x i 1 i et on cherche a les relier aux solutions periodiques du systeme non perturbe dx i dt h 0 x p p i dp i dt h 0 x p x i 1 i

a borsuk ulam theorem for orthogonal tk and zpr actions and applications abstract a version if the borsuk ulam theorem is proved for the torus tk and p torus zpr we deal with the case of a nonempty fixed point set of groups on the sphere of representation as an application a geometrical index theory is introduced for these groups the index is used to obtain the existence of multiple critical points for functions invariant under tk or zpr action

existence of multiple periodic solutions for a natural hamiltonian system in a potential well we prove under a pinching hypothesis a theorem of existence of at least n periodic orbits for a closed regular energy hypersurface in r2n which is the level set of a natural i e of the type potential kinetic energy hamiltonian function and projects onto a potential well in the configuration space

bifurcation and chaotic behavior of a discrete singular biological economic system abstract in this paper the complex dynamics of a discrete singular biological economic system are first investigated firstly we establish a discrete singular biological economic system which is based on the discretization of a differential algebraic equations that is described by a ratio dependent predator prey system with harvesting and economic factor then it is shown that the system undergoes flip bifurcation and neimark sacker bifurcation in the interior of r 3 by using the new normal form of discrete singular systems the center manifold theorem and the bifurcation theory as varying the economic profit in some range numerical simulations are presented not only to illustrate our results with the theoretical analysis but also to exhibit the complex dynamic behaviors such as cascades of period doubling bifurcation in orbits of period 2 4 8 and chaotic sets these results reveal far richer dynamics of the discrete model compared with the continuous model the lyapunov exponents are numerically computed to confirm further the complexity of the dynamical behaviors

qualitative behavior of n dimensional ratio dependent predator prey systems abstract this paper deals with the qualitative properties of an n dimensional autonomous system of differential equations modeling the general ratio dependent predator prey interaction

on critical point theory for indefinite functionals in the presence of symmetries we consider functionals which are not bounded from above or from below even modulo compact perturbations and which exhibit certain symmetries with respect to the action of a compact lie group we develop a method which permits us to prove the existence of multiple critical points for such functionals the proofs are carried out directly in an infinite dimensional hilbert space and they are based on minimax arguments the applications given here are to hamiltonian systems of ordinary differential equations where the existence of multiple time periodic solutions is established for several classes of hamiltonians symmetry properties of these hamiltonians such as time translation invariancy or evenness are exploited introduction many variational problems which arise from physics or mathematical physics are indefinite in the sense that the functionals involved are not bounded from below or from above however some of these functionals defined in an appropriate function space are semidefinite in the sense that they are bounded from below or from above if perturbed with a weakly continuous functional this paper deals with functionals which are not semidefinite usually problems involving indefinite functionals are more difficult to handle and only very recently a method has been developed which permits us to treat such problems directly in an infinite dimensional function space 7 in 7 some theorems have been proved which establish the existence of at least one nontrivial critical point for such functionals in many physical situations there are problems which have symmetries with respect to the action of some lie group in this case we expect the existence of many critical points this has been established for semidefinite functionals cf e g 8 and 3 for even functionals 9 for a zp action with p prime numbers 15 16 10 and 6 for an s action in this paper we develop a method which allows us to estimate the number of critical points of indefinite functionals in the presence of symmetries more precisely received by the editors july 16 1980 and in revised form june 26 1981 1980 mathematics subject classification primary 58e05 70h05 secondary 34c25

positive periodic solutions for neutral delay ratio dependent predator prey model with holling type iii functional response abstract by using a continuation theorem based on coincidence degree theory some new and interesting sufficient conditions are obtained for the existence of positive periodic solutions for neutral delay ratio dependent predator prey model with holling type iii functional response x t x t r t a t x t 1 x t 2 c t x 2 t m 2 y 2 t x 2 t y t y t y t d t h t x 2 t t m 2 y 2 t t x 2 t t an example is represented to illustrate the feasibility of our main results

closed characteristics on compact convex hypersurfaces in r 2n for any given compact c 2 hypersurface in r 2n bounding a strictly convex set with nonempty interior in this paper an invariant n is defined and satisfies n n 2 1 where a denotes the greatest integer which is not greater than a r the following results are proved in this paper there always exist at least n geometrically distinct closed characteristics on if all the geometrically distinct closed characteristics on are nondegenerate then n n if the total number of geometrically distinct closed characteristics on is finite there exists at least an elliptic one among them and there exist at least n 1 of them possessing irrational mean indices if this total number is at most 2 n 2 there exist at least two elliptic ones among them

global dynamics and bifurcation in a stage structured prey predator fishery model with harvesting abstract this paper describes a prey predator model with stage structure for predator and selective harvesting effort on predator population the holling type ii functional response function is taken into consideration all the equilibria of the proposed system are determined and the behavior of the system is investigated near them local stability of the system is analyzed geometric approach is used to derive the sufficient conditions for global stability of the system the occurrence of hopf bifurcation of the model system in the neighborhood of the co existing equilibrium point is shown through considering maximal relative increase of predation as bifurcation parameter fishing effort used to harvest predator population is considered as a control to develop a dynamic framework to investigate the optimal utilization of the resource sustainability properties of the stock and the resource rent earned from the resource pontryagin s maximum principle is used to characterize the optimal control the optimal system is derived and then solved numerically using an iterative method with runge kutta fourth order scheme simulation results show that the optimal control scheme can achieve sustainable ecosystem

hopf pitchfork bifurcation in van der pol s oscillator with nonlinear delayed feedback abstract first we identify the critical values for hopf pitchfork bifurcation second we derive the normal forms up to third order and their unfolding with original parameters in the system near the bifurcation point by the normal form method and center manifold theory then we give a complete bifurcation diagram for original parameters of the system and obtain complete classifications of dynamics for the system furthermore we find some interesting phenomena such as the coexistence of two asymptotically stable states two stable periodic orbits and two attractive quasi periodic motions which are verified both theoretically and numerically

on a theorem by ekeland and lasry concerning the number of periodic hamiltonian trajectories this note deals with the existence of iz distinct modes of vibration for hamihonian systems with n degrees of freedom on a prescribed energy surface this problem has been studied in a recent remarkable paper by ekeland and lasry 5 who have given an important contribution to the complete extension in the large of the local results of liapunov ri weinstein 111 and moser s the purpose of the present paper is to give a new proof of the result of 5 jointly with a slight improvement

existence of multiple periodic orbits of hamiltonian systems on positive type hypersurfaces in r2n abstract this paper deals with the fixed energy problem of hamiltonian systems in r 2n the main result is an existence theorem of multiple periodic orbits for hamiltonian vector fields on a class of contact type but not necessarily star shaped hypersurfaces this is the first multiplicity result for hypersurfaces that are more general than star shaped ones the main result is achieved by using critical point theory for a special hamiltonian function and central gravity of the proof relies on a delicate construction of such hamiltonian function

an index theory and existence of multiple brake orbits for star shaped hamiltonian systems soit h r 2n r une fonction continument differentiable et j o i ou i est la matrice identite n n on considere le systeme hamiltonien x jh x on etudie l existence de solutions periodiques sur une hypersurface compacte donnee h 1 1

on the fucik spectrum of the laplacian on a torus abstract we study the fucik spectrum of the laplacian on a two dimensional torus t 2 exploiting the invariance properties of the domain t 2 with respect to translations we obtain a good description of large parts of the spectrum in particular for each eigenvalue of the laplacian we will find an explicit global curve in the fucik spectrum which passes through this eigenvalue these curves are ordered and we will show that their asymptotic limits are positive on the other hand using a topological index based on the mentioned group invariance we will obtain a variational characterization of global curves in the fucik spectrum also these curves emanate from the eigenvalues of the laplacian and we will show that they tend asymptotically to zero thus we infer that the variational and the explicit curves cannot coincide globally and that in fact many curve crossings must occur we will give a bifurcation result which partially explains these phenomena

traveling wavefronts of a prey predator diffusion system with stage structure and harvesting from a biological point of view we consider a prey predator type free diffusion fishery model with stage structure and harvesting first we study the stability of the nonnegative constant equilibria in particular the effect of harvesting on the stability of equilibria is discussed and supported with numerical simulation then employing the upper and lower solution method we show that when the wave speed is large enough there exists a traveling wavefront connecting the zero solution to the positive equilibrium of the system numerical simulation is also carried out to illustrate the main result

stochastic periodic solution for a perturbed non autonomous predator prey model with generalized nonlinear harvesting and impulses abstract in this paper stochastic non autonomous predator prey models with and without impulses are investigated the effects of generalized nonlinear harvesting for prey and predator populations are considered for the stochastic system without impulses the existence and uniqueness of the positive solution is proven and sufficient conditions that guarantee the extinction and persistence of the population in the mean are achieved we show the existence of a nontrivial positive periodic solution by constructing appropriate lyapunov functions and using khasminskii s theory moreover the global attractiveness and stochastic persistence in probability of the stochastic model are discussed results show that the stronger noises and nonlinear harvesting component can significantly influence the dynamics of the system and lead to the extinction of the predator population additionally for the stochastic predator prey system with impulsive effect we prove that there exists a positive periodic solution numerical simulations are conducted to show the effectiveness and feasibility of the obtained results

solutions of minimal period for a class of nonconvex hamiltonian systems and applications to the fixed energy problem on considere le systeme hamiltonien jz h z ou j x y y x x y r n r n h est non convexe a un bon comportement superquadratique et satisfait une condition simple reliant ses coefficients de superquadraticite et l exposant dans ce cas on demontre que pour t 0 l existence de solutions pour le systeme hamiltonien de periode minimale t

equivariant morse theory for starshaped hamiltonian systems soit une hypersurface en forme d etoile dans r 2n le probleme de trouver des caracteristiques fermees de peut se reduire classiquement a un probleme variationnel ceci conduit a etudier une fonctionnelle s 1 equivariante sur un espace de hilbert

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

borsuk ulam theorems for arbitrary s1 actions and applications abstract an s to the first power version of the borsuk ulam theorem is proved for a situation where fix s to the first power may be nontrivial the proof is accomplished with the aid of a new relative index theory applications are given to intersection theorems and the existence of multiple critica points is established for a class of functionals invariant under an s to the first power symmetry minimax arguments from the calculus of variations serve as an important tool in establishing the existence of nonlinear vibrations of discrete mechanical systems as modelled by hamilton s equations in these arguments one obtains the solutions of the differential equations as critical points of an associated lagrangian by minimaxing the lagrangian over appropriate classes of sets intersection theorems such as are proved in this paper play a crucial role in this process in addition to obtaining some intersection theorems this report illustrates their use by proving an existence theorem for multiple critical points of the functional invarianet under an s to the first power symmetry group

minimal decompositions and iterative methods summary we study how singular values and approximation numbers by algebraic operators of a bounded linear operator a on a separable hilbert space h can be related to the speed of convergence of iterative methods in infinite dimensional h these two quantities lead us to consider those a that can be split as a n k with n normal andk compact then the spectrum of n via a classical polynomial approximation problem bounds below the speed of convergence of gmres in finite dimensional h we combine these two approximation numbers to approximation numbers by normal operators of a matrix a that lead us to consider splittings ofa as a n f with n normal andf of smallest possible rank we show analogously to infinite dimensions that the eigenvalues of n can be used via a new polynomial approximation problem in assessing the speed of convergence of gmres for upper bounds we obtain estimates that can be combined with the results of 28

on the number of periodic orbits of hamiltonian systems on positive type hypersurfaces in r2n abstract in this paper we prove some existence theorems of multiple periodic orbits of hamiltonian systems on a class of hypersurfaces in r 2n that are more general than star shaped ones our results extend some earlier works on this subject

periodic orbits of hamiltonian systems on symmetric positive type hypersurfaces this paper discusses the existence and multiplicity of periodic orbits of hamiltonian systems on symmetric positive type hypersurfaces we prove that each such energy hypersurface carries at least one symmetric periodic orbit under some suitable pinching conditions we also get an existence result of multiple symmetric periodic orbits

convex hamiltonian energy surfaces and their periodic trajectories in this paper we introduce symplectic invariants for convex hamiltonian energy surfaces and their periodic trajectories and show that these quentities satisfy several nontrivial relations in particular we show that they can be used to prove multiplicity results for the number of periodic trajectories

complex dynamics of a diffusive predator prey model with strong allee effect and threshold harvesting abstract in this paper complex dynamics of a diffusive predator prey model is investigated where the prey is subject to strong allee effect and threshold harvesting the existence and stability of nonnegative constant steady state solutions are discussed the existence and nonexistence of nonconstant positive steady state solutions are analyzed to identify the ranges of parameters of pattern formation spatially homogeneous and nonhomogeneous hopf bifurcation and discontinuous hopf bifurcation are proved these results show that the introduction of strong allee effect and threshold harvesting increases the system spatiotemporal complexity finally numerical simulations are presented to validate the theoretical results

interspecies competition between prey and two different predators with holling iv functional response in diffusive system this paper deals with a prey middle predator top predator ecosystem model with holling type iv predator response in the unreserved zone the model system is studied analytically and the threshold values for the existence and stability of various steady states are worked out the global stability analysis is carried out it is observed that if the intrinsic growth rate of prey population crosses a certain critical value the system enters into hopf bifurcation the existence of bionomic equilibrium of the system has been discussed further we study a path of optimal harvesting policy by introducing the pontryagins maximum principle moreover we have found out a condition for diffusive instability of a locally stable equilibrium finally some numerical simulations are performed to justify analytical findings

dynamics of generalist predator in a stochastic environment modified ratio dependent holling tanner model with prey refuge is considered delays are taken in logistic growth of prey as feedback mechanism and gestation period of predator model system exhibits hopf bifurcation when the delay parameters cross their critical values stochastic analysis of the model system is carried out by incorporating gaussian white noise using fourier transform technique fluctuation and stability of the stochastic model is studied in this paper an attempt has been made to understand the dynamics of a prey predator system with multiple time delays where the predator population is regarded as a generalist type in this regard we consider a modified holling tanner prey predator system where a constant time delay is incorporated in the logistic growth of the prey to represent a delayed density dependent feedback mechanism and the second time delay is considered to account for the length of the gestation period of the predator predator s interference in prey predator relationship provides better descriptions of predator s feeding over a range of prey predator abundances so the predator s functional response is considered to be type ii ratio dependent and foraging efficiency of predator largely varies with the refuge strategy of prey population in accordance with previous studies it is observed that delay destabilizes the system in general and stability loss occurs via hopf bifurcation in particular we show that there exists critical values of the delay parameters below which the coexistence equilibrium is stable and above which it is unstable hopf bifurcation occurs when the delay parameters cross their critical values also environmental stochasticity in the form of gaussian white noise plays a significant role to describe the system and its values numerical computation is also performed to validate and visualize different theoretical results presented the analysis and results in this work are interesting both in mathematical and biological point of views

resonance identities for closed characteristics on compact star shaped hypersurfaces in r2n resonance relations among periodic orbits on given energy hypersurfaces are very important for getting deeper understanding of the dynamics of the corresponding hamiltonian systems in this paper we establish two new resonance identities for closed characteristics on every compact star shaped hypersurface in r2n when the number of geometrically distinct closed characteristics on is finite which extend those identities established by c viterbo in 1989 for star shaped hypersurfaces assuming in addition that all the closed characteristics and their iterates are non degenerate and that by w wang x hu and y long in 2007 for strictly convex hypersurfaces in r2n

multiplicity of characteristics with lagrangian boundary values on symmetric star shaped hypersurfaces abstract in this paper the multiplicity of lagrangian orbits on c 2 smooth compact symmetric star shaped hypersurfaces with respect to the origin in r 2 n is studied these lagrangian orbits begin from one lagrangian subspace and end on another an infinitely many existence result is proved via z 2 index theory this is a multiplicity result about the arnold chord conjecture in some sense and is a generalization of the problem about the multiplicity of lagrangian orbits beginning from and ending on the same lagrangian subspace which was considered in the authors previous paper f guo c liu multiplicity of lagrangian orbits on symmetric star shaped hypersurfaces nonlinear anal 69 4 2008 1425 1436

local stability analysis of ratio dependent predator prey models with predator harvesting rates we study the local stability of ratio dependent predator prey models with nonconstant predator harvesting rates we provide more general conditions on the parameters involved in the models under which the models are locally stable or unstable at the two positive equilibria we show that the predator free positive equilibrium can be a stable node or saddle node and the positive interior equilibrium can be a stable or unstable node or focus or a center point depending on the ranges of the parameters involved our results generalize and improve some known results and show that the models have more rich dynamics than those ratio dependent predator prey models without any harvesting rates

dynamics in a ratio dependent predator prey model with predator harvesting abstract the objective of this paper is to study systematically the dynamical properties of a ratio dependent predator prey model with nonzero constant rate predator harvesting it is shown that the model has at most two equilibria in the first quadrant and can exhibit numerous kinds of bifurcation phenomena including the bifurcation of cusp type of codimension 2 i e bogdanov takens bifurcation the subcritical and supercritical hopf bifurcations these results reveal far richer dynamics compared to the model with no harvesting and different dynamics compared to the model with nonzero constant rate prey harvesting in d xiao l jennings bifurcations of a ratio dependent predator prey system with constant rate harvesting siam appl math 65 2005 737 753 biologically it is shown that nonzero constant rate predator harvesting can prevent mutual extinction as a possible outcome of the predator prey interaction and remove the singularity of the origin which was regarded as pathological behavior for a ratio dependent predator prey model in p yodzis predator prey theory and management of multispecies fisheries ecological applications 4 2004 51 58

dynamics of prey threshold harvesting and refuge the dynamics of a predator prey model with continuous threshold prey harvesting and prey refuge is studied one central question is how harvesting and refuge could directly affect the dynamics of the ecosystem such as the stability properties of some coexistence equilibria and periodic solutions theoretical and numerical methods are used to investigate boundedness of solutions existence of bionomic equilibria as well as the existence and stability properties of equilibrium points and periodic solutions several bifurcations are also studied

existence of multiple periodic orbits on star shaped hamiltonian surfaces consider the hamiltonian system hs i 1 n here h c2 2n in this paper we investigate the existence of periodic orbits of hs on a given energy surface z 2n h z c c o is a constant the surface is required to verify certain geometric assumptions bounds a star shaped compact region and for some ellipsoid 2n o o depending in an explicit fashion on the lengths of the main axes of and one other geometrical parameter of such that if furthermore 2 2 1 then hs has at least n distinct geometric orbits on this result is shown to extend and unify several earlier works on this subject among them works by weinstein rabinowitz ekeland lasry and ekeland in proving this result we construct index theories for an s1 action from which we derive abstract critical point theorems for s1 invariant functionals we also derive an estimate for the minimal period of solutions to differential equatious