creating and relating three dimensional integrable maps we show how some integrable third order difference equations recently given in the literature are related to one another by the process of interchanging parameters and integrals using the same process we then create a 21 parameter family of integrable third order difference equations that contains the previous examples as special cases our methodology illustrates that the combination of finding 2 integrals i e integrals of the second iterate of the map exploiting linear parameter dependence and using the interchange process provides a powerful way to relate and create higher dimensional discrete integrable systems

asymptotically periodic solutions of n competing species problem with time delays abstract a general nonautonomous lotka volterra system of integro differential equations with infinite delay is considered and sufficient conditions for uniform stability and asymptotic stability of solutions are investigated these limiting equations are used to obtain the existence of asymptotically periodic solutions of the corresponding system of equations where lower limit in the integration is zero

interchanging parameters and integrals in dynamical systems the mapping case we consider dynamical systems with discrete time maps that possess one or more integrals depending upon parameters we show that integrals can be used to replace parameters in the original map so as to construct a different map with different integrals we also highlight a process of reparametrization that can be used to increase the number of parameters in the original map prior to using integrals to replace them properties of the original map and the new map are compared the theory is motivated by and illustrated with examples of a three dimensional trace map and some four dimensional maps previously shown to be integrable

a new family of four dimensional symplectic and integrable mappings abstract we investigate the generalisations of the quispel roberts and thompson qrt family of mappings in the plane leaving a rational quadratic expression invariant to the case of four variables we assume invariance of the rational expression under a cyclic permutation of variables and we impose a symplectic structure with poisson brackets of the weyl type all mappings satisfying these conditions are shown to be integrable either as four dimensional mappings with two explicit integrals which are in involution with respect to the symplectic structure and which can also be inferred from the periodic reductions of the double discrete versions of the modified korteweg devries mkdv and sine gordon sg equations or by reduction to two dimensional mappings with one integral of the symmetric qrt family

dynamic analysis of stochastic virus infection model with delay effect abstract this paper mainly investigates the dynamic behavior of virus infection model with delay effect influenced by external noise first a mathematical model describing the evolution of virus under the influences of stochastic noise and delay effect is developed and necessary conditions for extinction and weak persistence are derived by ito s formula a new basic reproduction number r 0 for virus infection model with delay effect to determine whether extinction or persistence is given it is found that the delay parameter affects the new basic reproduction number r 0 and the behavior of extinction and survival of virus besides the threshold for extinction of stochastic model is smaller than the one of deterministic model in the end stochastic simulations are applied to illustrate and test the theoretical conclusions

the spectrum of the chebyshev collocation operator for the heat equation it is shown that the eigenvalues of the pseudospectral chebyshev second derivative operator with separated boundary conditions are real negative and distinct an application to the full potential equation is also presented

an algorithmic introduction to numerical simulation of stochastic differential equations a practical and accessible introduction to numerical methods for stochastic differential equations is given the reader is assumed to be familiar with euler s method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable however no knowledge of advanced probability theory or stochastic processes is assumed the article is built around 10 matlab programs and the topics covered include stochastic integration the euler maruyama method milstein s method strong and weak convergence linear stability and the stochastic chain rule

integrable fourth order difference equations in this paper an attempt is made to find four dimensional analogs of two dimensional quispel roberts and thompson mappings and identified four distinct cases have been identified the obtained mappings are measure preserving the integrability of the isolated mappings is examined by constructing a sufficient number of integrals and their symplectic structure wherever possible

persistence attractivity and delay in facultative mutualism abstract relations between persistence and ultimate boundedness of the solutions of the lotka volterra system dx t dt x t r 1 a 11 x t a 22 y t dy t dt y t r 2 a 21 x t a 22 y t modelling facultative mutualism with delayed responses are established and sufficient conditions are obtained for the global attractivity of the positive equilibrium of the delay system

third order integrable difference equations generated by a pair of second order equations we show that the third order difference equations proposed by hirota kimura and yahagi are generated by a pair of second order difference equations in some cases the pair of the second order equations are equivalent to the quispel robert thomson qrt system but in the other cases they are irrelevant to the qrt system we also discuss an ultradiscretization of the equations

bilinear trilinear forms and exact solution of certain fourth order integrable difference equations a systematic investigation of finding bilinear or trilinear representations of fourth order autonomous ordinary difference equation x n 4 f x n x n 1 x n 2 x n 3 or xn 4 f xn xn 1 xn 2 xn 3 is made as an illustration we consider fourth order symplectic integrable difference equations reported by capel and sahadevan physica a 289 86 2001 and derived their bilinear or trilinear forms also it is shown that the obtained bilinear representations admit exact solution of rational form

stochastic differential delay equations of population dynamics in this paper we investigate the almost surely asymptotic stability for the nonlinear stochastic differential delay equations with markovian switching some sufficient criteria on the controllability and robust stability are also established for linear stochastic differential delay equations with markovian switching

higher dimensional integrable mappings we construct higher dimensional analogues of two dimensional integrable mappings preserving one biquadratic integral we also provide their non autonomous forms

global stability for infinite delay lotka volterra type systems abstract in this paper sufficient conditions are established for the global stability of the saturated equilibrium of an infinite delay nonautonomous lotka volterra type system the present work is distinguished from previous work principally be allowing the system to be nonautonomous and relaxing the traditional requirement that the undelayed intraspecific competition dominates both the delayed intraspecific competition as well as the interspecific interactions we require the undelayed intraspecific competition to dominate the latter but not the former

integrable mappings and soliton equations ii abstract some simple solutions of discrete soliton equations are shown to satisfy 2d mappings we show that these belong to a recently introduced 18 parameter family of integrable reversible mappings of the plane thus lending weight to a previous conjecture we also give an example of an integrable mapping occuring in an exactly solvable model in statistical mechanics finally we discuss the notion of generalized reversibility

integrable mappings derived from soliton equations abstract we derive a hierarchy of ibtegrable mappings integrable ordinary difference equations corresponding to solutions of the initial value problem of an integrable partial difference equation with periodic initial data for each n n this hierarchy contains at least one integrable mapping r n r n the integrals of these mappings are constructed using the lax pair of the underlying partial difference equation our approach is illustrated for the integrable partial difference analogues of the sine gordon and the modified korteweg de vries equations

direct method to construct integrals for nth order autonomous ordinary difference equations a direct method to construct integrals for an nth order autonomous ordinary difference equation oe wnnf wn wnn1 is presented as an illustration we first consider third order autonomous oe wn3

some properties of the k dimensional lyness map this paper is devoted to study some properties of the k dimensional lyness map f x1 xk x2 xk a ki 2xi x1 our main result presents a rational vector field that gives a lie symmetry for f this vector field is used for k 5 to give information about the nature of the invariant sets under f when k is odd we also present a new as far as we know first integral for ff which allows us to deduce in a very simple way several properties of the dynamical system generated by f in particular for this case we prove that except on a given codimension one algebraic set none of the positive initial conditions can be a periodic point of odd period

closed form expressions for integrals of mkdv and sine gordon maps we present closed form expressions for approximately n integrals of 2n dimensional maps the maps are obtained by travelling wave reductions of the modified korteweg de vries equation and of the sine gordon equation respectively we provide the integrating factors corresponding to the integrals moreover we show how the integrals and the integrating factors relate to the staircase method

studying discrete dynamical systems through differential equations abstract in this paper we consider dynamical systems generated by a diffeomorphism f defined on u an open subset of r n and give conditions over f which imply that their dynamics can be understood by studying the flow of an associated differential equation x x x also defined on u in particular the case where f has n 1 functionally independent first integrals is considered in this case x is constructed by imposing that it shares with f the same set of first integrals and that the functional equation f x det d f x x x u has some non zero solution several examples for n 2 3 are presented most of them coming from several well known difference equations

a preconditioning matrix for the chebyshev differencing operator an efficient preconditioner for the chebyshev differencing operator is considered the corresponding preconditioned eigenvalues are real and positive and lie between 1 and pi 2 an eixpelicit formula for these eigenvalues and the corresponding eigenfunctions is given the results are generalized to the case of operators related to chebyshev discretizations of systems of linear differential equations

uniform persistence in functional differential equations abstract in this paper we investigate the question of uniform persistence for retarded functional differential equations by utilizing liapunov like functions razumikhin techniques and differential inequalities we are able to establish criteria for uniform persistence analogous to those obtained by others for ordinary differential equations difference equations and reaction diffusion equations we apply these criteria to some well known biological models with delay our results indicate that the conditions which guarantee the existence of an interior equilibrium are enough to ensure uniform persistence moreover these conditions are equivalent to uniform persistence for the cases without delay as well

k integrals and k lie symmetries in discrete dynamical systems abstract we generalize the concept of symplectic maps to that of k symplectic maps maps whose kth iterates are symplectic similarly k symmetries and k integrals are symmetries resp integrals of the kth iterate of the map it is shown that k symmetries and k integrals are related by the k symplectic structure as in the k 1 continuous case noether s theorem examples are given of k integrals and their related k symmetries for k 1 4

super integrable four dimensional autonomous mappings a systematic investigation of the complete integrability of a fourth order autonomous difference equation of the type w n 4 w n f w n 1 w n 2 w n 3 is presented we identify seven distinct families of four dimensional mappings which are super integrable and have three independent integrals via a duality relation as introduced in a recent paper by quispel capel and roberts 2005 j phys a math gen 38 3965 80 it is observed that these seven families can be related to the four dimensional symplectic mappings with two integrals including all the four dimensional periodic reductions of the integrable double discrete modified korteweg devries and sine gordon equations treated in an earlier paper by two of us capel and sahadevan 2001 physica a 289 86 106

a new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations a new method to impose boundary conditions for pseudospectral approximations to hyperbolic equations is suggested this method involves the collocation of the equation at the boundary nodes as well as satisfying boundary conditions stability and convergence results are proven for the chebyshev approximation of linear scalar hyperbolic equations the eigenvalues of this method applied to parabolic equations are shown to be real and negative

stochastic permanence of an siqs epidemic model with saturated incidence and independent random perturbations abstract this article discusses a stochastic siqs epidemic model with saturated incidence we assume that random perturbations always fluctuate at the endemic equilibrium the existence of a global positive solution is obtained by constructing a suitable lyapunov function under some suitable conditions we derive the stochastic boundedness and stochastic permanence of the solutions of a stochastic siqs model some numerical simulations are carried out to check our results

third order difference equations with two rational integrals a systematic investigation to derive three dimensional analogs of two dimensional quispel roberts and thompson qrt mappings is presented the question of integrability of the obtained three dimensional mappings with two independent integrals is also analyzed it is also shown that there exist three dimensional qrt maps with three n dependent integrals

error estimates for spectral and pseudospectral approximations of hyperbolic equations the linear one dimensional advection equation with variable coefficients and nonperiodic boundary conditions is considered we investigate some spectral and pseudospectral collocation approximations of jacobi and legendre type in the latter case the boundary condition is implicitly treated stability and convergence results in weighted norms are given some precise estimates of the rate of convergence in terms of the regularity of the continuous solution are derived

the threshold of a stochastic delayed sir epidemic model with temporary immunity abstract this paper is concerned with the asymptotic properties of a stochastic delayed sir epidemic model with temporary immunity sufficient conditions for extinction and persistence in the mean of the epidemic are established the threshold between persistence in the mean and extinction of the epidemic is obtained compared with the corresponding deterministic model the threshold affected by the white noise is smaller than the basic reproduction number r 0 of the deterministic system

how to find the conserved quantities of nonlinear discrete equations a method of finding conserved quantities of nonlinear ordinary difference equations is briefly discussed the method is based on the hypothesis that the conserved quantities are expressed by a ratio of two polynomials of dependent variables we obtain the conserved quantities of the discrete equation of motion of an anharmonic oscillator and a nonlinear difference equation of third order

persistence and extinction in stochastic non autonomous logistic systems this paper studies two widely used stochastic non autonomous logistic models for the first system sufficient conditions for extinction non persistence in the mean weak persistence and stochastic permanence are established the critical number between weak persistence and extinction is obtained for the second system sufficient criteria for extinction non persistence in the mean weak persistence in the mean strong persistence in the mean and stochastic permanence are established the critical number between weak persistence in the mean and extinction is obtained it should be pointed out that this research is systematical and complete in fact the behaviors of the two models in every coefficient cases are cleared up by the results obtained in this paper

dynamics and bifurcations of a three dimensional piecewise linear integrable map in this paper we consider a four parameter family of piecewise linear ordinary difference equations o es in r 3 this system is obtained as a limit of another family of three dimensional integrable systems of o es we prove that the limiting procedure sends integrals of the original system to integrals of the limiting system we derive some results for the solutions such as boundedness of solutions and the existence of periodic solutions we describe all topologically different shapes of the integral manifolds and present all possible scenarios of transitions as we vary the natural parameters in the system i e the values of the integrals

global stability in a viral infection model with lytic and nonlytic immune responses this paper investigates the global stability of a viral infection model with lytic and nonlytic immune responses if the basic reproductive ratio of the virus is less than or equal to one by the lasalle s invariance principle and center manifold theorem the disease free steady state is globally asymptotically stable if the basic reproductive ratio of the virus is greater than one then the virus persists in the host and the disease steady state is locally asymptotically stable furthermore by the method of lyapunov function the global stability of the disease steady state is established at the same time if we neglect the efficacy of the lytic component using a geometrical approach we obtain a different type of conditions for the global stability of the disease steady state

some new results of nonautonomous lotka volterra competitive systems with delays abstract in this paper we improve and generalize the results of montes de oca and zeeman j math anal appl 192 1995 360 370 proc amer math soc 124 1996 3677 3687 to general n species nonautonomous lotka volterra type competitive systems with finite or infinite delays we establish a series of criteria under which a part of the n species of the systems is driven to extinction while the remaining part of the n species is persistent or uniformly persistent or coexists globally asymptotically stable