runge kutta methods for the multi pantograph delay equation this paper deals with the properties of the numeric solution of the multi pantograph equationu t email protected t email protected 1u q 1t email protected 2u q 2t email protected lu q lt u 0 u 0 where 0

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

on solving systems of multi pantograph equations via spectral tau method abstract the current manuscript focuses on solving systems of multi pantograph equations the spectral tau method is applied for solving systems of multi pantograph equations with shifted jacobi polynomials as basis functions the convergence analysis of the proposed technique is also investigated we introduced the numerical solutions of some test problems and compared the obtained numerical solutions of such problems with those given using different numerical methods

pattern formation in a predator prey diffusion model with stage structure for the predator in this paper we study a neumann boundary value problem in a d dimensional box t d 0 d d 1 2 3 for the predator prey diffusion model u 1 t d 1 u 1 u 1 a u 1 u 2 u 3 u 2 t d 2 u 2 d 4 u 2 u 3 2 k u 1 u 3 u 2 u 3 t d 3 u 3 b u 2 m u 3 with predator stage structure by using the bootstrap technique guo and hwang 2010 and higher order energy estimates we provide a rigorous quantitative characterization for the nonlinear evolution of early spatiotemporal pattern formation on the unstable positive constant equilibrium as a consequence the nonlinear instability occurs

extrinsic radius pinching in space forms of nonnegative sectional curvature we first give new estimates for the extrinsic radius of compact hypersurfaces of the euclidean space mathbb r n 1 and the open hemisphere in terms of high order mean curvatures then we prove pinching results corresponding to these estimates we show that under a suitable pinching condition m is diffeomorphic and almost isometric to an n dimensional sphere

global attractivity of a positive periodic solution for a nonautonomous stage structured population dynamics with time delay and diffusion by employing the continuation theorem of coincidence degree theory the existence of a positive periodic solution for a nonautonomous stage structured population dynamics with time delay and diffusion is established further by constructing a lyapunov functional and using the result of the existence of positive periodic solution the attractivity of a positive periodic solution for above system is obtained

hopf bifurcation of a predator prey system with stage structure and harvesting abstract in this paper a two species predator prey system with stage structure and harvesting is investigated the existence of hopf bifurcations of the system is given and the stability and directions of hopf bifurcations are determined by applying the normal form theory and the center manifold theorem

evans krylov estimates for a nonconvex monge ampere equation we establish evans krylov estimates for certain nonconvex fully nonlinear elliptic and parabolic equations by exploiting partial legendre transformations the equations under consideration arise in part from the study of the pluriclosed flow introduced by streets and tian int math res not 16 3101 3133 2010

dynamics in a periodic competitive model with stage structure in this paper we consider a time delayed periodic system which describes the competition among mature populations by appealing to theories of monotone dynamical systems periodic semiflows and uniform persistence we analyze the evolutionary behavior of the system and establish sufficient conditions for competitive coexistence and exclusion

stability and bifurcation analysis of a stage structured predator prey model with time delay abstract in this paper we proposed and analyzed a prey predator system with stage structured for the predator population a time delay is incorporated due to the gestation for the matured predator all the possible non negative equilibria are obtained and their local as well as global behavior are studied choosing delay as a bifurcation parameter the existence of the hopf bifurcation of the system has been investigated moreover we use the normal form method and the center manifold theorem to examine the direction of the hopf bifurcation and the nature of the bifurcating periodic solution some numerical simulations are given to support the analytical results some interesting conclusions are obtained from our analysis and it is given at the end of the paper

on the t method for delay differential equations with infinite lag abstract this paper discusses the method for the numerical solution of delay differential equations with infinite lag our analysis is based on the test equation y t ay t by t where a b c and 0 1 in order to solve the storage problem involved in studying the long time behaviour of the solution we use a grid with increased stepsizes monotonicity uniform boundedness asymptotic stability algebraic decay and global discretisation error of the numerical solution are investigated

an approximate solution for a neutral functional differential equation with proportional delays in this paper a novel algorithm based on reproducing kernel theory for neutral functional differential equation with proportional delays is proposed the advantages of the presented method are the establishment of complete approximate solution theory and high precision of absolute error some examples are given to demonstrate the validity and applicability of the new method and some comparisons are made with the existing results

the existence of nonoscillatory solutions to a generalized nonautonomous delay logistic equation sufficient conditions for the existence of nonoscillatory solutions to the delay logistic equation x t r t x t 1 x g t k are obtained where 1

stability of a competition model with two stage structure a population model with stage structure which describes the competition of two species for space is considered the model is established under the assumptions that reproductive rates of two species are affected by competition and that the space occupied by one type of juveniles cannot be neglected estimates on the length of delays for which positive equilibrium of the model remains stable are given sufficient conditions are also derived that ensure global stability of positive equilibrium

bifurcation analysis in an approachable haematopoietic stem cells model a haematopoietic stem cells model hsc with one delay is considered at first we investigate the stability and existence of hopf bifurcations by analyzing the distribution of the roots of associated characteristic equation then an explicit formula for determining the stability and the direction of periodic solutions bifurcating from hopf bifurcations is derived using the normal form theory and center manifold argument finally some numerical simulations are carried out for supporting the analytic results

global hopf bifurcation for differential equations with state dependent delay abstract we develop a global hopf bifurcation theory for a system of functional differential equations with state dependent delay the theory is based on an application of the homotopy invariance of s 1 equivariant degree using the formal linearization of the system at a stationary state our results show that under a set of mild conditions the information about the characteristic equation of the formal linearization with frozen delay can be utilized to detect the local hopf bifurcation and to describe the global continuation of periodic solutions for such a system with state dependent delay

optimal harvesting policy for single population with stage structure it is the purpose of this paper to analyze the exploitation of the stage structured singleautonomous population model we analyze the harvesting population equation and obtain existence local and global geometric properties of nonnegative equilibria explicit expressions are obtained for the optimal harvesting effort the maximum sustainable yield and the corresponding population level

global attractivity of nonautonomous stage structured population models with dispersal and harvest the nonautonomous stage structured single species dispersal model with harvesting of mature individuals in an n patch environment is considered in which the individual members of the population have a life history that takes them through two stages immature and mature by using the theory of monotone and concave operators to functional differential equations we establish conditions under which this population dynamical system admits a positive periodic solution which attracts all positive solutions

on eigenvalue pinching in positive ricci curvature we shall show that for manifolds with ric n 1 the radius is close to iff the n 1 st eigenvalue is close to n this extends results of cheng and croke which show that the diameter is close to iff the first eigenvalue is close to n we shall also give a new proof of an important theorem of colding to the effect that if the radius is close to then the volume is close to that of the sphere and the manifold is gromov hausdorff close to the sphere from work of cheeger and colding these conditions imply that the manifold is diffeomorphic to a sphere

higher order mean curvature estimates for bounded complete hypersurfaces abstract we obtain sharp estimates involving the mean curvatures of higher order of a complete bounded hypersurface immersed in a complete riemannian manifold similar results are also given for complete spacelike hypersurfaces in lorentzian ambient spaces

on state dependent delay partial neutral functional integro differential equations in this paper the existence of mild solutions for a class of abstract neutral integro differential equations with state dependent delay is studied

wavefronts and global stability in a time delayed population model with stage structure we formulate and study a one dimensional single species diffusive delay population model the time delay is the time taken from birth to maturity without diffusion the delay differential model extends the well known logistic differential equation by allowing delayed constant birth processes and instantaneous quadratically regulated death processes this delayed model is known to have simple global dynamics similar to that of the logistic equation through the use of a sub supersolution pair method we show that the diffusive delay model continues to generate simple global dynamics this has the important biological implication that quadratically regulated death processes dramatically simplify the growth dynamics we also consider the possibility of travelling wavefront solutions of the scalar equation for the mature population connecting the zero solution of that equation with the positive steady state our main finding here is that our fronts appear to be all monotone regardless of the size of the delay this is in sharp contrast to the frequently reported findings that delay causes a loss of monotonicity with the front developing a prominent hump in some other delay models

stability of nonlinear differential systems with state dependent delayed impulses we consider nonlinear differential systems with state dependent delayed impulses impulses which involve the delayed state of the system for which the delay is state dependent such systems arise naturally from a number of applications and the stability issue is complex due to the state dependence of the delay we establish general and applicable results for uniform stability uniform asymptotic stability and exponential stability of the systems by using the impulsive control theory and some comparison arguments we show how restrictions on the change rates of states and impulses should be imposed to achieve system s stability in comparison with general impulsive delay differential systems with state dependent delay in the nonlinearity or the differential systems with constant delays in our approach the boundedness of the state dependent delay is not required but derives from the stability result obtained examples are given to demonstrate the sharpness and applicability of our general results and the proposed approach

persistence and periodicity of a delayed ratio dependent predator prey model with stage structure and prey dispersal a delayed periodic ratio dependent predator prey model with prey dispersal and stage structure for predator is investigated it is assumed that immature individuals and mature individuals of the predator species are divided by a fixed age and that immature predators don t have the ability to attack prey and that predator species is confined to one of the patches while the prey species can disperse between two patches we first discuss the uniform persistence and impermanence of the model by using gaines and mawhin s continuation theorem of coincidence degree theory a set of easily verifiable sufficient conditions are derived for the existence of positive periodic solutions of the proposed model numerical simulations are presented to illustrate the feasibility of our main results

variational iteration method for solving a generalized pantograph equation the variational iteration method is applied to solve the generalized pantograph equation this technique provides a sequence of functions which converges to the exact solution of the problem and is based on the use of lagrange multipliers for identification of optimal value of a parameter in a functional employing this technique it is possible to find the exact solution or an approximate solution of the problem some examples are given to demonstrate the validity and applicability of the method and a comparison is made with existing results

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

an eigenvalue pinching theorem in 1958 lichnerowicz 7 showed that for a compact n dimensional riemannian manifold m whose ricci curvature is bounded below by n 1 the first non zero eigenvalue 21 of the laplacian satisfies 2 t n if in fact 21 n obata proved that m must be isometric to the standard sphere a natural question is do there exist constants c n 1 depending only on n such that if m is as above and c n n 21 n then m must be diffeomorphic to a sphere here by combining the works of gromov 3 berard and meyer 1 and grove and shiohama 4 we show

stability analysis of theta methods for neutral functional differential equations this paper deals with the subject of numerical stability for the neutral functional differential equation y t ay t by qt cy pt qquad t 0 it is proved that numerical solutions generated by theta methods are convergent if c b if and only if theta geq 1 2 and that the one leg theta method is lambda stable if theta 1 we also find out that inappropriate stepsize causes spurious solution in the marginal case where rm re a 0 and a b

rational approximation method for delay differential equations with proportional delay abstract consider the pantograph differential equation y t ay t by qt f t 0 t t y 0 y 0 with proportional delay qt 0 q m m rational approximation with quasi uniform meshes which corresponds to m th collocation method and establish the global error analysis of o h 2 m on successive mesh points this method is useful than the known collocation method in case that a long time integration is needed that is if t is large then the number of steps in our method is less than that of the collocation method numerical experiments are also contained

global attractivity in concave or sublinear monotone infinite delay differential equations abstract we study the dynamical behavior of the trajectories defined by a recurrent family of monotone functional differential equations with infinite delay and concave or sublinear nonlinearities we analyze different sceneries which require the existence of a lower solution and of a bounded trajectory ordered in an appropriate way for which we prove the existence of a globally asymptotically stable minimal set given by a 1 cover of the base flow we apply these results to the description of the long term dynamics of a nonautonomous model representing a stage structured population growth without irreducibility assumptions on the coefficient matrices

numerical simulation for the space fractional diffusion equations abstract this paper presents a novel semi analytical algorithm based on the chebyshev collocation method for the solution of space fractional diffusion equations the original fractional equation is transformed into a system of ordinary differential equations odes by the chebyshev collocation method a new semi analytical method is then used to approximate the solution of the resulting system to emphasize the reliability of the new scheme a convergence analysis is presented it is shown that highly accurate solutions can be achieved with relatively few approximating terms and absolute errors are rapidly decrease as the number of approximating terms is increased by presenting some numerical examples we show that the proposed method is a powerful and reliable algorithm for solving space fractional diffusion equations and can be extended to solve another space fractional partial differential equations comparison with other methods in the literature demonstrates that the proposed method is both efficient and accurate

symmetries and other transformations of the complex monge ampere equation on definit des transformations tangentes en utilisant la structure symplectique holomorphe du fibre cotangent holomorphe

role of constant prey refuge on stage structure predator prey model with ratio dependent functional response abstract a constant prey refuge has been incorporated in a stage structured predator prey model with ratio dependent functional response sufficient conditions are obtained for the uniform persistence and global asymptotic stability by constructing suitable lyapunov function our model analysis demonstrates that the constant prey refuge plays an important role on the coexistence of stage structured predator prey species

stability and hopf bifurcation analysis of an eco epidemic model with a stage structure abstract in this paper an eco epidemiological model with a stage structure is considered the asymptotical stability of the five equilibria the existence of stability switches about positive equilibrium is investigated it is found that hopf bifurcation occurs when the delay passes though a critical value some explicit formulae determining the stability and the direction of the hopf bifurcation periodic solutions bifurcating from hopf bifurcations are obtained by using the normal form theory and center manifold theory some numerical simulations for justifying the theoretical analysis are also provided finally biological explanations and main conclusions are given

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

extrinsic upper bounds for λ1 si m est compact alors il existe c r tel que 1 m c n m hu 2 volm pour toutes les sous varietes m compactes connexes a n dimensions isometriquement immergees de m 1 m est la premiere valeur propre du laplacien de m

on the attainable order of collocation methods for pantograph integro differential equations for the pantograph integro differential equation pide with nonhomogeneous term y t ay t 01 y q t d q 01 y q t dv q f t t 0 y 0 y0 with proportional delays q t and q t 0 q q 1 0 q 1 we consider the attainable order of m stage implicit collocation based runge kutta methods at the first mesh point t h and give conditions on the collocation polynomials mm t of degree m to u th t 0 1 such that u h y h o h2m 1 where y t is the solution and u t is the collocation solution of pide if m 2 or f t is a polynomial of t whose degree is less than or equal to m then such conditions of mm t are simplified a numerical example is also included

the variational iteration method for solving a neutral functional differential equation with proportional delays in this paper the variational iteration method is applied to neutral functional differential equations with proportional delays illustrative examples are given to show the efficiency of the method we also compare the performance of the method with that of a particular runge kutta method and a one leg q method

a priori estimates for solutions of monge ampere equations on considere les equations de monge ampere elliptiques de la forme ar 2bs ct rt s 2 e pour z z x y definie dans un sous ensemble ouvert du plan x y

stability of runge kutta methods for the generalized pantograph equation this paper deals with stability properties of runge kutta rk methods applied to a non autonomous delay differential equation dde with a constant delay which is obtained from the so called generalized pantograph equation an autonomous dde with a variable delay by a change of the independent variable it is shown that in the case where the rk matrix is regular stability properties of the rk method for the dde are derived from those for a difference equation which are examined by similar techniques to those in the case of autonomous ddes with a constant delay as a result it is shown that some rk methods based on classical quadrature have a superior stability property with respect to the generalized pantograph equation stability of algebraically stable natural rk methods is also considered

approximate solution of multi pantograph equation with variable coefficients this paper deals with the approximate solution of multi pantograph equation with nonhomogenous term in terms of taylor polynomials the technique we have used is based on a taylor matrix method in addition some numerical examples are presented to show the properties of the given method and the results are discussed

h stability of runge kutta methods with general variable stepsize for pantograph equation this paper deals with h stability of runge kutta methods with variable stepsize for pantograph equations it is shown that the runge kutta methods with a regular matrix a are h stable if and only if the modulus of stability function at infinity is less than 1 furthermore the same conclusion can be obtained if the runge kutta methods are stiffly accurate

on abstract differential equations with state dependent non local conditions abstract in this paper we introduce the concept of state dependent nonlocal condition and study the existence and uniqueness of solution for a general class of abstract differential equations with state dependent delay subject to this class of conditions in this type of problem the nonlocal condition can be dependent on times or time intervals determined by the unknown solution some examples on partial differential equations with state dependent delay arising in population dynamics are presented

boundary layer phenomena for differential delay equations with state dependent time lags iii in this paper we begin a study of the differential delay equation varepsilon x t x t f x t r r r x t

large manifolds with positive ricci curvature the main purpose of this paper is to show that an n dimensional manifold with ricci curvature greater or equal to n 1 which is close in the gromov hausdor topology to the unit n sphere has volume close to that of the sphere this shows the converse of the theorem in c1 namely together with c1 it shows that an n manifold with ricci curvature greater or equal to n 1 is close to the sphere if and only if the volume is close to that of the sphere in particular by p such a manifold is homeomorphic to a sphere further as an application of this and the result of c1 we prove a radius theorem saying that if an n manifold with ricci curvature greater or equal to n 1 has radius almost equal to then the volume is close to that of the sphere in order to obtain these results we further develop and apply the estimates of c1 whereas the main concern in c1 were with the large scale geometry the main concern of this paper is with the small scale geometry let n be the volume of the round n sphere sn with sectional curvature one

on neutral functional differential equations with proportional delays abstract in this paper we develop a comprehensive theory on the well posedness of the initial value problem for the neutral functional differential equation y t ay t i 1 b i y q i t i 1 c i y p i t t 0 y 0 y 0 and the asymptotic behaviour of its solutions we prove that the existence and uniqueness of solutions depend mainly on the coefficientsci i 1 2 and on the smoothness of functions in the solution space as far as the asymptotic behaviour of analytic solutions is concerned thecihave little effect we prove that if re a 0 then the solutiony t either grows exponentially or is polynomial the most interesting result is that if re a 0 anda 0 then the asymptotic behaviour of the solution depends mainly on the characteristic equation a i 1 b i q i 0 these results can be generalized to systems of equations finally we present some examples to illustrate the change of asymptotic behaviour in response to the variation of some parameters the main idea used in this paper is to express the solution in either dirichlet or dirichlet taylor series form

stage structured models of intra and inter specific competition within age classes abstract in some species larvae and adults experience competition in completely different ways simple stage structured models without larval competition usually yield a single delay equation for the adults using an age structured system incorporating competition among both larvae and adults we derive a system of distributed delay equations for the numbers of larvae and adults the system is neither cooperative nor reducible to a single equation for either variable positivity boundedness and uniform strong persistence are established linear stability analysis of equilibria is difficult due to the strong coupling but results are proved for small delays using monotone systems theory and exponential ordering for small delay we prove a theorem on generic convergence to equilibria which does not directly follow from standard theory but can be proved indirectly using comparison arguments finally we consider an extension to two strain competition and prove theorems on the linear stability of the boundary equilibria

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

stability of a class of runge kutta methods for a family of pantograph equations of neutral type this paper deals with the stability of runge kutta methods for a class of neutral infinite delay differential equations with different proportional delays under suitable conditions the asymptotic stability of some runge kutta methods with variable stepsize are considered by the stability function at infinity it is proved that the even stage gauss legendre methods are not asymptotically stable but the radau ia methods radau iia methods and lobatto iiic methods are all asymptotically stable furthermore some numerical experiments are given to demonstrate the main conclusions

a higher dimensional partial legendre transform and regularity of degenerate monge ampere equations in dimension n 3 we define a generalization of the classical two dimensional partial legendre transform that reduces interior regularity of the generalized monge ampere equation detd2u k x u du to regularity of a divergence form quasilinear system of special form this is then used to obtain smoothness of c2 1 solutions having n 1 nonvanishing principal curvatures to certain subelliptic monge ampere equations in dimension n 3 a corollary is that if k 0 vanishes only at nondegenerate critical points then a c2 1 convex solution u is smooth if and only if the symmetric function of degree n 1 of the principal curvatures of u is positive and moreover u fails to be c3 1 2n when not smooth

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

a taylor method for numerical solution of generalized pantograph equations with linear functional argument this paper is concerned with a generalization of a functional differential equation known as the pantograph equation which contains a linear functional argument in this paper we introduce a numerical method based on the taylor polynomials for the approximate solution of the pantograph equation with retarded case or advanced case the method is illustrated by studying the initial value problems the results obtained are compared by the known results

a new collocation method for approximate solution of the pantograph functional differential equations with proportional delay the paper presents a new numerical method for solving functional differential equations with proportional delays of the first and higher orders the method consists of replacing the initial equation by an approximate equation which has an exact analytic solution with a set of free parameters these free parameters are determined by the use of the collocation procedure some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results the numerical results show that the proposed method is of a high accuracy and is efficient for solving a wide class of functional differential equations with proportional delays including equations of neutral type the method is applicable to both initial and boundary value problems

the dynamics of a mutual interference age structured predator prey model with time delay and impulsive perturbations on predators in this paper we introduce a mutual interference age structured predator prey natural enemy pest model with constant maturation time delay for the prey and then propose a pest management strategy by constant periodic releasing for the predator we show that there exists a global attractive pest eradication periodic solution when the periodic releasing amount m 1 and m 2 are lager than some critical value further to obtain a more effective pest control strategy we give the conditions involving the estimate of m 1 and m 2 in which the model is uniformly permanent and the pest population is under the economic threshold level we believe that the results will provide reliable tactic basis for the practical pest management