pseudospectral method based on prolate spheroidal wave functions for frequency domain electromagnetic simulations we apply prolate spheroidal wave functions of order zero as basis functions in the pseudospectral method for frequency domain electromagnetic simulation problems like the traditional pseudospectral frequency domain psfd methods based on chebyshev and legendre polynomial series the prolate psfd method yields exponential order of accuracy in terms of the number of samples utilized per wavelength the prolate expansion is superior to the chebyshev and legendre polynomial series by a factor of spl pi 2 in addition the prolate psfd method employs a more uniform spatial grid achieving better resolution near the center of the domain

on hp convergence of prolate spheroidal wave functions and a new well conditioned prolate collocation scheme the first purpose of this paper is to provide further illustrations from both theoretical and numerical perspectives for the nonconvergence of h refinement in hp approximation by the prolate spheroidal wave functions pswfs a surprising convergence property that was first discovered by boyd et al 2013 3 the second purpose is to offer a new basis that leads to prolate collocation systems with condition numbers independent of c n the intrinsic bandwidth parameter and the number of collocation points we highlight that the collocation scheme together with a very practical rule for pairing up c n significantly outperforms the legendre polynomial based method and likewise other jacobi polynomial based methods in approximating highly oscillatory bandlimited functions

bifurcations for a predator prey system with two delays in this paper a predator prey system with two delays is investigated by choosing the sum of two delays as a bifurcation parameter we show that hopf bifurcations can occur as crosses some critical values by deriving the equation describing the flow on the center manifold we can determine the direction of the hopf bifurcations and the stability of the bifurcating periodic solutions in addition special attention is paid to the global continuation of local hopf bifurcations using a global hopf bifurcation result of j wu symmetric functional differential equations and neural networks with memory trans amer math soc 350 1998 4799 4838 we may show the global existence of periodic solutions

semilinear elliptic equations on unbounded domains on presente une nouvelle approche variationnelle a l existence des solutions pour le probleme aux valeurs limites u u r x f u x x x h 0 1 sur des domaines non bornes

stability and bifurcation for a delayed predator prey model and the effect of diffusion abstract we consider a predator prey system with one or two delays and a unique positive equilibrium e its dynamics are studied in terms of the local stability of e and of the description of the hopf bifurcation that is proven to exist as one of the delays taken as a parameter crosses some critical values we also consider a reaction diffusion system with neumann conditions resulting from adding one spatial variable and diffusion terms in the previous model the spectral and bifurcation analysis in the neighborhood of e now as a stationary point of this latter system is addressed and the results obtained for the case without diffusion are applied

fast computation of the spectral differentiation by the fast multipole method abstract this paper is concerned with the application of the fast multipole method fmm to the pseudospectral methods the spectral differentiation in pseudospectral methods involves dense matrix vector products which if evaluated directly entail o n 2 operations the core of this work lies in the fact that the fmm leads to a significant speed up in cpu time with a major reduction in the amount of computer memory needed when performing matrix vector products the fmm is exploited to estimate the barycentric weights and the pseudospectral differentiation which reduces the complexity to o n furthermore we show the convergence analysis and present the applications to the second order boundary value problems numerical examples confirm the advantages of the method

periodic solutions of a predator prey system with stage structures for predator and prey abstract by using the continuation theorem of coincidence degree theory the existence of a positive periodic solution for the predator prey system with stage structures for predator and prey is established

eigenfunctions of the airyʼs integral transform properties numerical computations and asymptotic behaviors abstract this paper is devoted to the study of the spectrum of the integral operator with airy s kernel we provide the reader with some efficient methods of computing the eigenfunctions and the eigenvalues of this operator the first method is based on the use of a differential operator which commutes with this integral operator the second method is based on a gaussian quadrature method applied to the finite airy s transform the asymptotic behavior of the eigenfunctions is studied by the use of a wkb method some numerical examples are given to illustrate results of this work

global existence of positive periodic solutions of periodic predator prey system with infinite delays abstract a set of easily verifiable sufficient conditions is derived for the global existence of periodic solutions with strictly positive components for a periodic predator prey system with infinite delays by using the method of coincidence degree

asymptotic behaviors and numerical computations of the eigenfunctions and eigenvalues associated with the classical and circular prolate spheroidal wave functions abstract for a fixed bandwidth c the classical prolate spheroidal wave functions pswfs n c and the circular ones cpswfs n c form remarkable hilbertian bases for the spaces of fourier and hankel band limited functions with bandwidth c respectively the prolate spheroidal wave functions have already found many applications from various scientific area such as signal processing numerical analysis physics and random matrix theory these concrete applications of the pswfs and cpswfs require the accurate computation of these special functions to this end we develop in this work two approximate procedures the first procedure is based on pushing forward the wkb method and provide accurate approximations of the pswfs and the cpswfs in terms of some classical special functions the second procedure is based on efficient and accurate quadrature methods for the computation of the values and the eigenvalues associated with pswfs and the cpswfs both procedures are valid for small as well as for large values of the frequency band width c also we provide the reader with some numerical examples that illustrate the different results of this work

optimal control of harvest and bifurcation of a prey predator model with stage structure abstract this paper describes a prey predator model with stage structure for prey the adult prey and predator populations are harvested in the proposed system the dynamic behavior of the model system is discussed it is observed that singularity induced bifurcation phenomenon is appeared when variation of the economic interest of harvesting is taken into account state feedback controller is incorporated to stabilize the model system in case of positive economic interest harvesting of prey and predator population are used as controls 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 the pontryagin s maximum principle is used to characterize the optimal controls the optimality 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

permanence and stability of a stage structured predator prey model abstract a predator prey model with a stage structure for the predator which improves the assumption that each individual predator has the same ability to capture prey is proposed it is assumed that immature individuals and mature individuals of the predator are divided by a fixed age and that immature predators do not have the ability to attack prey we obtain conditions that determine the permanence of the populations and the extinction of the populations furthermore we establish necessary and sufficient conditions for the local stability of the positive equilibrium of the model by exploiting the monotonicity of one equation of the model we obtain conditions for the global attractivity of the positive equilibrium which allow for long delay as long as the predator birth rate is large or the death rate of immature predators is small by constructing liapunov functionals we also obtain conditions under which the positive equilibrium is globally stable when the delay is small

global stability of a lotka volterra type predator prey model with stage structure and time delay a delayed lotka volterra type predator prey model with stage structure for predator is investigated it is assumed in the model that the individuals in the predator population may belong to one of two classes the immatures and the matures the age to maturity is presented by a time delay and that the immature predators do not have the ability to prey by analyzing characteristic equations and using an iterative technique a set of easily verifiable sufficient conditions are derived for the local and global stability of the nonnegative equilibria of the model numerical simulations are carried out to illustrate the validity of our results

spectral methods based on prolate spheroidal wave functions for hyperbolic pdes we examine the merits of using prolate spheroidal wave functions pswfs as basis functions when solving hyperbolic pdes using pseudospectral methods the relevant approximation theory is reviewed and some new approximation results in sobolev spaces are established an optimal choice of the band limit parameter for pswfs is derived for single mode functions our conclusion is that one might gain from using the pswfs over the traditional chebyshev or legendre methods in terms of accuracy and efficiency for marginally resolved broadband solutions

large mode number eigenvalues of the prolate spheroidal differential equation we show that the eigenvalues of the prolate spheroidal differential equation of zeroth order g n c where c is the so called bandwidth or ellipticity parameter are well approximated for large mode number n by a single function w in the form g n c email protected c c where c p 2 n 1 2 w is defined implicitly as the root of an algebraic equation e min 1 m 1 2 m 1 2 1 w where e is the usual incomplete integral of the second kind and email protected 2 w with c c c w has a weak singularity at c 1 proportional to c 1 log c 1 plus iterated logarithms we give chebyshev series for w accurate for email protected 0

stability analysis on a predator prey system with distributed delays abstract this paper concerns the local and global dynamical properties of the nonnegative and positive equilibria of a lotka volterra predator prey system with distributed delays it is shown that while the positive equilibrium does not exist the nonnegative equilibrium is globally asymptotically stable or globally attractive as long as the delays are small enough if the positive equilibrium exists it is shown that it is locally asymptotically stable when the delays are suitably small furthermore an explicit asymptotic stability region for the positive equilibrium is also obtained based on a liapunov functional

hopf bifurcation and global stability for a delayed predator prey system with stage structure for predator abstract in this paper a delayed predator prey system with stage structure for predator is proposed and studied it is found that the time delay is harmless for permanence of the stage structured system if 1 we show that the unique positive equilibrium is locally asymptotically stable when time delay is sufficiently small while loss of stability by a hopf bifurcation can occur as the delay increases

rearrangement inequalities and duality theory for a semilinear elliptic variational problem on a domain q in r n 3 2 throughout we assume that 52 is steinersymmetric with respect to a finite family of linear subspaces j j of w such that jf jf are mutually orthogonal and that q has the cone property so the sobolev embedding theorems hold for r we take f to be odd and increasing and r to be a bounded positive measurable function that is symmetric decreasing with respect to j j let 3 0 be the greatest number satisfying

periodic solutions of a nonautonomous predator prey system with stage structure and time delays a nonautonomous lotka volterra type predator prey model with stage structure and time delays is investigated it is assumed in the model that the individuals in each species may belong to one of two classes the immatures and the matures the age to maturity is presented by a time delay and that the immature predators do not feed on prey and do not have the ability to reproduce by some comparison arguments we first discuss the permanence of the model by using the continuation theorem of coincidence degree theory sufficient conditions are derived for the existence of positive periodic solutions to the model by means of a suitable lyapunov functional sufficient conditions are obtained for the uniqueness and global stability of the positive periodic solutions to the model

permanence of variable coefficients predator prey system with stage structure abstract we studied a finite delay predator prey model with stage structure for predator by analyzing right hand of function and the standard comparison theorem some new sufficient conditions are derived for the permanence of population and some biological explanations are made

hopf bifurcation analysis for a ratio dependent predator prey system with two delays and stage structure for the predator the ratio dependent theory is favored by researchers since it is more suitable for describing the relationship between predator and its prey in this paper a ratio dependent predator prey system with holling type ii functional response two time delays and stage structure for the predator is investigated firstly by choosing the two time delays as the bifurcation parameter the sufficient conditions for the local stability and the existence of hopf bifurcation with respect to both delays are established furthermore based on the normal form method and center manifold theorem explicit formulas are derived to determine the direction of hopf bifurcation and stability of the bifurcating periodic solution finally numerical simulations are given to verify the theoretical analysis

permanence of a predator prey system with stage structure and time delay abstract a stage structured predator prey system stage structure for both predator and prey with discrete time delay has been presented in this paper by analyzing the model and using the standard comparison theorem the sufficient conditions are derived for permanence of the system and non permanence of predators

vortex rings in a cylinder and rearrangements on demontre l existence d une solution faible au probleme de valeur limite pour un tourbillon annulaire stationnaire dans un fluide parfait s ecoulant le long d un tube infini de section circulaire

permanence for the michaelis menten type discrete three species ratio dependent food chain model with delay abstract a discrete three trophic level food chain model with ratio dependent michaelis menten type functional response is investigated it is shown that under some appropriate conditions the system is permanent the results indicate that to make the species coexist in the long run it is a surefire strategy to keep the death rate of the predator and top predator rather small and the intrinsic growth rate of the prey relatively large

an improved estimate of pswf approximation and approximation by mathieu functions abstract in this paper an error estimate of spectral approximations by prolate spheroidal wave functions pswfs with explicit dependence on the bandwidth parameter and optimal order of convergence is derived which improves the existing result in chen et al spectral methods based on prolate spheroidal wave functions for hyperbolic pdes siam j numer anal 43 5 2005 1912 1933 the underlying argument is applied to analyze spectral approximations of periodic functions by mathieu functions which leads to new estimates featured with explicit dependence on the intrinsic parameter

stability and hopf bifurcation in a delayed ratio dependent holling tanner type model in this study a delayed ratio dependent holling tanner type predator prey model is investigated first the local stability of a positive equilibrium is studied and then the existence of hopf bifurcations is established by using the normal form theory and center manifold theorem the explicit algorithm determining the stability direction of the bifurcating periodic solutions are derived finally we perform the numerical simulations for justifying the theoretical results

rapid prolate pseudospectral differentiation and interpolation with the fast multipole method pseudospectral methods utilizing prolate spheroidal wave functions as basis functions have been shown to possess advantages over the conventional pseudospectral methods based on trigonometric and orthogonal polynomials however the spectral differentiation and interpolation steps of the prolate pseudospectral method involve matrix vector products which if evaluated directly entail o n2 memory requirement and computational complexity where n is the number of unknowns utilized for discretization and interpolation in this work we show that the fast multipole method fmm can be used to reduce the memory requirement and computational complexity of the prolate pseudospectral method to o n example simulation results demonstrate the speed and accuracy of the resulting fast prolate pseudospectral solver

analysis of a model representing stage structured population growth with state dependent time delay a stage structured model of population growth is proposed where the time to ma turity is itself state dependent it is shown that under appropriate assumptions all solutions are positive and bounded criteria for the existence of positive equilibria and further conditions for the uniqueness of the equilibria are given the stability of the equilibria are also discussed in addition an attracting region is determined for solutions such that this region collapses to the unique positive equilibrium in the state independent case

analysis of spectral approximations using prolate spheroidal wave functions in this paper the approximation properties of the prolate spheroidal wave functions of order zero pswfs are studied and a set of optimal error estimates are derived for the pswf approximation of non periodic functions in sobolev spaces these results serve as an indispensable tool for the analysis of pswf spectral methods a pswf spectral galerkin method is proposed and analyzed for elliptic type equations illustrative numerical results consistent with the theoretical analysis are also presented

stability and traveling fronts in lotka volterra competition models with stage structure this paper is concerned with a delay differential equation model for the interaction between two species the adult members of which are in competition the competitive effects are of the lotka volterra kind and in the absence of competition it is assumed that each species evolves according to the predictions of a simple age structured model which reduces to a single equation for the total adult population for each of the two species the model incorporates a time delay which represents the time from birth to maturity of that species thus the time delays appear in the adult recruitment terms the dynamics of the model are determined and global stability results are established for each equilibrium the equilibria of the model involve the maturation delays the criteria for global convergence to each equilibrium are sharp and involve these delays a reaction diffusion extension of the model is also studied for the case when only the adult members of each species can diffuse we prove the existence of a

a predator prey system with stage structure for predator abstract this paper studies the asymptotoc behavior of a predator prey model with stage structure it is found that an orbitally asymptotically stable periodic orbit exists in that model when time delay due to gestation of predator and time delay from crowding effect of prey are incorporated we establish the condition for the permanence of populations and sufficient conditions under which positive equilibrium of the model is globally stable

persistence of some stage structured ecosystems with finite and infinite delay abstract the persistence of some stage structured predator prey model with finite and infinite delay was studied in this paper by analyzing right hand functions and using the comparison theorem some sufficient conditions for the persistence of populations with finite and infinite delay were derived

stability and delays in a predator prey system abstract sufficient conditions for both local and global stability of the positive equilibrium in a predator prey system with time delays are obtained by constructing suitable lyapunov functionals a remark on the mistake by k n murty and m a s srinivas j math anal appl 158 1991 333 341 is also included

prolate spheroidal wavefunctions as an alternative to chebyshev and legendre polynomials for spectral element and pseudospectral algorithms prolate spheroidal functions of order zero are generalizations of legendre polynomials which when the bandwidth parameter c 0 oscillate more uniformly on x 1 1 than either chebyshev or legendre polynomials this suggests that compared to these polynomials prolate functions give more uniform spatial resolution further when used as the spatial discretization for time dependent partial differential equations in combination with explicit time marching prolate functions allow a longer stable timestep than legendre polynomials we show that these advantages are real and further that it is almost trivial to modify existing pseudospectral and spectral element codes to use the prolate basis however improvements in spatial resolution are at most a factor of 2 approached slowly as n the timestep can be lengthened by several times but not by a factor that grows rapidly with n the prolate basis is not likely to radically expand the range of problems that can be done on a workstation however for production runs on the bleeding edge edge of arithmurgy such as numerical weather prediction the rewards for switching to a prolate basis may be considerable

hopf bifurcation and global periodic solutions in a delayed predator prey system this paper is concerned with a delayed predator prey system with same feedback delays of predator and prey species to their growth respectively we prove that a sequence of hopf bifurcations occur at the positive equilibrium as the delay increases monotonously from zero by using the theory of normal norm and center manifold reduction an explicit algorithm for determining the direction of hopf bifurcations and the stability of bifurcating periodic solutions is derived in addition the global existence results of periodic solutions bifurcating from hopf bifurcations are established by using a global hopf bifurcation result due to wu j wu symmetric functional differential equations and neural networks with memory trans amer math soc 350 1998 4799 4838 finally a numerical example supporting our theoretical prediction is also given our findings are contrasted with recent studies on a delayed predator prey system with the feedback time delay of prey species to its growth by song and wei y song j wei local hopf bifurcation and global periodic solutions in a delayed predator prey system j math anal appl 301 2005 1 21 as the feedback time delay increases monotonously from zero the positive equilibrium of the latter switches k times from stability to instability to stability in contrast the positive equilibrium of our system appears to lose the above property

extinction and permanence in nonautonomous competitive system with stage structure in this paper the n species nonautonomous stage structured competitive system is constructed and considered sufficient conditions for its extinction and permanence are obtained results here generalize and unify some previous ones moreover it is concluded that stage structure in this system is one of the important factors that effect the extinction and permanence of species

global convergence of a reaction diffusion predator prey model with stage structure for the predator abstract a lotka volterra type reaction diffusion predator prey model with stage structure for the predator and time delay due to maturity is investigated by successively modifying the coupled lower upper solution pairs sufficient conditions independent of the effect of spatial diffusion are derived for the global convergence of positive solutions to the proposed problem

permanence and stability of a predator prey system with stage structure for predator a stage structured predator prey system with delays for prey and predator respectively is proposed and analyzed mathematical analysis of the model equations with regard to boundedness of solutions permanence and stability are analyzed some sufficient conditions which guarantee the permanence of the system and the global asymptotic stability of the boundary and positive equilibrium respectively are obtained

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

existence uniqueness and global asymptotic stability of positive solutions of a predator prey system with holling ii functional response with random perturbation abstract this paper discusses a randomized two species predator prey system with holling ii functional response we show that the positive solution of the associated stochastic delay differential equation does not explode to infinity in a finite time in addition the existence uniqueness and global asymptotic stability of the positive solutions are studied