dynamical behavior of lotka volterra competition systems non autonomous bistable case and the effect of telegraph noise this article is concerned with the study of trajectory behavior of lotka volterra competition bistable systems and systems with telegraph noises we proved that for bistable systems there exists a unique solution bounded above and below by positive constants the oscillatory situation of systems with telegraph noises is pointed out

analysis of a predator prey model with modified leslie gower and holling type ii schemes with stochastic perturbation abstract in this paper we consider a predator prey model with modified leslie gower and holling type ii schemes with stochastic perturbation we show there is a unique positive solution to the system with positive initial value and mainly investigate the long time behavior of the system condition for the system to be extinct is given and persistent condition is established at last numerical simulations are carried out to support our results

periodic solution for the stochastic chemostat with general response function abstract this paper addresses a stochastic chemostat model with periodic dilution rate and general class of response functions the general functional response is assumed to satisfy two classifications of conditions and these assumptions on the functional response are relative weak that are valid for many forms of growth response for the chemostat with periodic dilution rate we derive the sufficient criteria for the existence of the stochastic nontrivial positive periodic solution by utilizing khasminskii s theory on periodic markov process

the invariant density for a class of discrete time maps involving an arbitrary monotonic function operator and an integer parameter when xt and xt 1 represent two random variables each belonging to a real interval 0 1 and being related by a first order difference equation xt 1 f xt called a discrete time map the probability density distribution connected with xt can be translated into that associated with xt 1 this yields an evolution equation by means of which one can construct an infinite sequence wt x t n x 0 1 starting from an integrable function w0 x normalized to unity on 0 1 the question of the convergence of the sequence toward a so called invariant density function w x as t and the problem of finding this limit were examined by a number of authors mostly studying isolated cases the present paper solves the problem for a class of discrete time maps characterized by xt 1 f sn l sn 1f 1 xt l 2 3 whereby f is a real continuous monotonically increasing function mapping 0 1 onto itself and sn is the usual symbol for the sinelike jacobian elliptic function with modulus k 0 1 including the

evolution towards an invariant density distribution under a discrete time quadratic map recently nandakumaran 1985 has proved that under the discrete time quadratic map x sub t 1 4x sub t 1 x sub t all the probability density distributions r0 n x x sup n 1 x sup n b n 1 n 1 0 or x or 1 n 0 1 2 converge towards an invariant limit density associated with the map when t tends to infinity the purpose of the present paper is to generalise this result starting from the fourier series expansion of any real single valued function which satisfies the dirichlet conditions in 0 pi 2 a broad class of normalised initial functions w0 x mod 0 or x or 1 is obtained each of which is the sum of a convergent series involving the chebyshev polynomials of both kinds the evolution equation for this class of functions under the map mentioned is found explicitly it is shown that the absolute convergence of the series expansion of the symmetric part in an initial w0 function is a sufficient condition to ensure an evolution for t to infinity towards the same invariant limit density as obtained by nandakumaran with the infinite set of particular initial densities r0 n x some auxiliary results related to the treatment of the considered problem are also presented

invariant density for a class of initial distributions under quadratic mapping for the discrete time quadratic map xt 1 4xt 1 xt the evolution equation for a class of non uniform initial densities is obtained it is shown that in the t to infinity limit all of them approach the invariant density for the map

bifurcation analysis of modified leslie gower predator prey model with michaelis menten type prey harvesting abstract in the present paper we discuss bifurcation analysis of a modified leslie gower prey predator model in the presence of nonlinear harvesting in prey we give a detailed mathematical analysis of the model to describe some significant results that may arise from the interaction of biological resources the model displays a complex dynamics in the prey predator plane the permanence stability and bifurcation saddle node bifurcation transcritical hopf andronov and bogdanov takens of this model are discussed we have analyzed the effect of prey harvesting and growth rate of predator on the proposed model by considering them as bifurcation parameters as they are important from the ecological point of view the local existence and stability of the limit cycle emerging through hopf bifurcation is given the emergence of homoclinic loops has been shown through simulation when the limit cycle arising though hopf bifurcation collides with a saddle point this work reflects that the feasible upper bound of the rate of harvesting for the coexistence of the species can be guaranteed numerical simulations using matlab are carried out to demonstrate the results obtained

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

stochastic persistence and stationary distribution in a holling tanner type prey predator model abstract in this paper we study a stochastic predator prey model with beddington deangelis type functional response and logistic growth for predators the deterministic model is already well studied and we recall some important results here we construct the stochastic model from the deterministic model by introducing multiplicative noise terms into the growth equations of prey and predator populations for the stochastic model we show that the system admits unique positive global solution starting from the positive initial value then we prove that the system is strongly persistent in mean when the intensity of environmental forcing is less than some threshold magnitudes finally we show that the system has a stationary distribution under certain parametric restrictions numerical simulations are carried out to substantiate the analytical results

remarks on stochastic permanence of population models abstract this paper examines the asymptotic behavior of the stochastic lotka volterra model under markovian switching we show that the stochastic lotka volterra model is stochastically permanent moreover we give another type of stochastic permanence and consider the relationship of two types of stochastic permanence under white noise perturbation

stochastic mutualism model under regime switching with levy jumps abstract this paper is concerned with a stochastic mutualism model under regime switching with levy jumps to begin with the existence and uniqueness of the global positive solution is proved with any given positive initial value then the sufficient conditions for stochastic permanence are established the critical value between extinction and persistence in mean is also obtained in addition under some suitable conditions we proved that there is a unique stationary distribution for the system without levy jumps our method relies on the lyapunov function analysis and the fredholm alternative the results demonstrate that regime switching may contribute to the permanence but jump noise may suppress the permanence

analysis of a predator prey model with holling ii functional response concerning impulsive control strategy according to biological and chemical control strategy for pest control we investigate the dynamic behavior of a holling ii functional response predator prey system concerning impulsive control strategy periodic releasing natural enemies and spraying pesticide at different fixed times by using floquet theorem and small amplitude perturbation method we prove that there exists a stable pest eradication periodic solution when the impulsive period is less than some critical value further the condition for the permanence of the system is also given numerical results show that the system we consider can take on various kinds of periodic fluctuations and several types of attractor coexistence and is dominated by periodic quasiperiodic and chaotic solutions which implies that the presence of pulses makes the dynamic behavior more complex finally we conclude that our impulsive control strategy is more effective than the classical one if we take chemical control efficiently

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

dynamics of a stochastic predator prey model with distributed delay and markovian switching abstract in this paper we study a stochastic predator prey model with distributed delay and markovian switching firstly we obtain sufficient criteria for weak persistence of the predator population then in the case of weak persistence we establish sufficient conditions for the existence of positive recurrence of the solutions to the model by constructing a suitable stochastic lyapunov function with regime switching finally we establish sufficient conditions for extinction of the predator population

dynamics of a stochastic cell to cell hiv 1 model with distributed delay abstract in this paper we consider a stochastic cell to cell hiv 1 model with distributed delay firstly we show that there is a global positive solution of this model before exploring its long time behavior then sufficient conditions for extinction of the disease are established moreover we obtain sufficient conditions for the existence of an ergodic stationary distribution of the model by constructing a suitable stochastic lyapunov function the stationary distribution implies that the disease is persistent in the mean finally we provide some numerical examples to illustrate theoretical results

dynamics of a stochastic lotka volterra model perturbed by white noise abstract this paper continues the study of mao et al investigating two aspects of the equation d x t diag x 1 t x n t b a x t d t x t d w t t 0 the first of these is to slightly improve results in x mao s sabais e renshaw asymptotic behavior of stochastic lotka volterra model j math anal 287 2003 141 156 concerning with the upper growth rate of the total quantity i 1 n x i t of species by weakening hypotheses posed on the coefficients of the equation the second aspect is to investigate the lower growth rate of the positive solutions by using lyapunov function technique and using a changing time method we prove that the total quantity i 1 n x i t always visits any neighborhood of the point 0 and we simultaneously give estimates for this lower growth rate

lower bounds on the complexity of some optimal data structures a technique is presented for deriving lower bounds on the complexity of optimal data structures which permit insertions and deletions of records and queries of the form text query region quad text return quad sum begin subarray l text key r in text region end subarray text value r quad text region in gamma where value r the value associated with a record r lies in a commutative semi group s and gamma denotes a set of regions of the space of possible keys this technique is illustrated with several examples

stationary distribution of a regime switching predator prey model with anti predator behaviour and higher order perturbations abstract in this paper we study a regime switching predator prey model with anti predator behaviour and higher order perturbations we obtain the ergodic property by constructing a suitable stochastic lyapunov function with regime switching which provides us a biological perspective of cycling phenomena of a population system and can better describe the stochastic persistence of a population system in practice we find that these restrictive assumptions on the functional response are relative weak and valid for many types of response functions

sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching in this paper we prove that a stochastic logistic population under regime switching controlled by a markov chain is either stochastically permanent or extinctive and we obtain the sufficient and necessary conditions for stochastic permanence and extinction under some assumptions in the case of stochastic permanence we estimate the limit of the average in time of the sample path of the solution by two constants related to the stationary probability distribution of the markov chain and the parameters of the subsystems of the population model finally we illustrate our conclusions through two examples

spatiotemporal dynamics in a delayed diffusive predator model in this paper we investigate the spatiotemporal dynamics of a delayed reaction diffusion leslie gower model based on the stability analysis we demonstrate that delayed feedback may generate hopf and turing instability under some conditions resulting in spatial patterns one of the most interesting findings is that the model exhibits complex pattern replication pure turing instability gives birth to spots spots stripes mixture stripes stripes holes mixture and holes patterns pure hopf instability to spiral wave pattern and hopf turing instability to chaotic wave pattern our results well extend the findings of spatiotemporal dynamics in the delayed reaction diffusion predator prey model and indicate that time delay play an important roles in pattern formation

the schroder functional equation and its relation to the invariant measures of chaotic maps the aim of this paper is to show that the invariant measure for a class of one dimensional chaotic maps t x is an extended solution of the schroder functional equation q t x q x induced by them hence we give a unified treatment of a collection of exactly solved examples worked out in the current literature in particular we show that these examples belong to a class of functions introduced by mira see the text moreover as a new example we compute the invariant densities for a class of rational maps having the weierstrass function as an invariant one also we study the relation between that equation and the well known frobenius perron and koopman s operators

saddle node hopf bifurcation in a modified leslie gower predator prey model with time delay and prey harvesting abstract from the saddle node hopf bifurcation point of view this paper considers a modified leslie gower predator prey model with time delay and the michaelis menten type prey harvesting firstly we discuss the stability of the equilibria obtain the critical conditions for the saddle node hopf bifurcation and give the completion bifurcation set by calculating the universal unfoldings near the saddle node hopf bifurcation point by using the normal form theory and center manifold theorem then we derive the parameter conditions for the existence of monostable coexistence equilibrium and the parameter regions in which both the prey extinction and the coexistence equilibrium or coexistence periodic or quasi periodic solutions are simultaneously stabilized we also investigate the heteroclinic bifurcation and describe the phenomenon that the periodic behavior disappears as through the heteroclinic bifurcation finally some numerical simulations are performed to support our analytic results

asymptotic behaviour of the stochastic lotka volterra model this paper examines the asymptotic behaviour of the stochastic extension of a fundamentally important population process namely the lotka volterra model the stochastic version of this process appears to have far more intriguing properties than its deterministic counterpart indeed the fact that a potential deterministic population explosion can be prevented by the presence of even a tiny amount of environmental noise shows the high level of difference which exists between these two representations

approximations of euler maruyama type for stochastic differential equations with markovian switching under non lipschitz conditions we develop the euler maruyama scheme for a class of stochastic differential equations with markovian switching sdewmss under non lipschitz conditions both l 1 and l 2 convergence are discussed under different non lipschitz conditions to overcome the mathematical difficulties arisen from the markovian switching as well as the non lipschitz coefficients several new analytical techniques have been developed in this paper which should prove to be very useful in the numerical analysis of stochastic systems

permanence of hybrid competitive lotka volterra system with levy noise abstract this paper concerns stochastic permanence of a hybrid competitive lotka volterra system with levy noise sufficient conditions of stochastic permanence are obtained by combining stochastic analytical techniques with m matrix analysis

colored noise induced hopf bifurcations in predator prey communities a broad class of n 1 species ratio dependent predator prey stochastic models which consist of one predator population and n prey populations is considered the effect of a fluctuating environment on the carrying capacities of prey populations is taken into account as colored noise in the framework of the mean field theory approximate self consistency equations for prey populations mean density and for predator population density are derived to the first order in the noise variance in some cases the mean field exhibits hopf bifurcations as a function of noise correlation time the corresponding transitions are found to be reentrant e g the periodic orbit appears above a critical value of the noise correlation time but disappears again at a higher value of the noise correlation time the nonmonotonous dependence of the critical control parameter on the noise correlation time is found and the conditions for the occurrence of hopf bifurcations are presented our results provide a possible scenario for environmental fluctuations induced transitions between the oscillatory regime and equilibrium state of population sizes observed in nature

on the complexity of maintaining partial sums let f bf r i s i 0 leqq i n be a file of n records where bf r i are d dimensional vectors and s i are elements of a commutative semigroup s we are interested in the partial sum problem in which queries of the form sum r i leqq bf a s i are to be answered a space time tradeoff t omega log n log m log n n is established for storing a static two dimensional file it will also be shown that for the one dimensional problem any dynamic algorithm must have a worst case time omega n log n log log n in processing a sequence o n insert and query instructions

qualitative analysis of a stochastic ratio dependent predator prey system a stochastic ratio dependent predator prey model is investigated in this paper by the comparison theorem of stochastic equations and ito s formula we obtain the global existence of a positive unique solution of the ratio dependent model besides a condition for species to be extinct is given and a persistent condition is established we also conclude that both the prey population and the ratio dependent function are stable in time average in the end numerical simulations are carried out to confirm our findings

analysis of autonomous lotka volterra competition systems with random perturbation abstract this paper discusses a randomized n species lotka volterra competition system we show that this system is stable in time average under certain conditions furthermore there is a stationary distribution of this system if extra conditions are satisfied also we give the extinction condition of this system finally numerical simulations are carried out to support our results

on positive recurrence of constrained diffusion processes let g c r k be a convex polyhedral cone with vertex at the origin given as the intersection of half spaces g i i 1 n where n i and d i denote the inward normal and direction of constraint associated with g i respectively stability properties of a class of diffusion processes constrained to take values in g are studied under the assumption that the skorokhod problem defined by the data n i d i i 1 n is well posed and the skorokhod map is lipschitz continuous explicit conditions on the drift coefficient b of the diffusion process are given under which the constrained process is positive recurrent and has a unique invariant measure define c i d i i 0 i 1 n then the key condition for stability is that there exists 0 and a bounded subset a of g such that for all x g a b x c and dist b x c where c denotes the boundary of c

stationary distribution and periodic solutions for stochastic holling leslie predator prey systems abstract the stochastic autonomous and periodic predator prey systems with holling and leslie type functional response are investigated for the autonomous system we prove that there exists a unique stationary distribution which is ergodic by constructing a suitable lyapunov function under relatively small white noise the result shows that stationary distribution doesn t rely on the existence and the stability of the positive equilibrium in the undisturbed system furthermore for the corresponding non autonomous system we show that there exists a positive periodic markov process under relatively weaker condition finally numerical simulations illustrate our theoretical results

pointwise convergence of densities under iteration of ulam and von neumann s map the pointwise convergence of the probability densities induced from an initial density on 0 1 under repeated iteration of the map x to 4s 1 x is established for a large class of initial densities

dynamics of a stochastic holling tanner predator prey model abstract in the paper we study a stochastic holling tanner predator prey system first by constructing a suitable stochastic lyapunov function we establish sufficient conditions for the existence of a unique ergodic stationary distribution of the positive solutions to the system after that sufficient causes for extinction of the predator population is obtained meanwhile we also obtain the threshold between persistence and extinction of the predator population at last some examples and numerical simulations are provided to illustrate our theoretical results

permanence and asymptotic behaviors of stochastic predator prey system with markovian switching and levy noise abstract this paper concerns the dynamics of a stochastic predator prey system with markovian switching and levy noise first the existence and uniqueness of global positive solution to the system is proved then by combining stochastic analytical techniques with m matrix analysis sufficient conditions of stochastic permanence and extinction are obtained furthermore for the stochastic permanence case by means of four constants related to the stationary probability distribution of the markov chain and the parameters of the subsystems both the superior limit and the inferior limit of the average in time of the sample path of the solution are estimated finally our conclusions are illustrated through an example

on pointers versus addresses summary what is the cost of random access to memory we address this fnndamental problem by studying the simulation of random addressing by a machine which lacks it a pointer machine the model we define allows the use of a data type of our choice a ram program of time i and space s can be simulated in o t log a time using a tree however this is not an obvious lower bound since a high level data type may allow us to encode the data in a more economic way our major contribution is the formalization of incompressibility for general data types the definition extends a similar property 01 strings that underlies the theory of kolmogorov complexity the main theorem states that for all incompressible data types an q t log s lower bound holds incompressibility trivially holds for strings but is harder to prove for a powerlul data type we prove incompressibility for the real numbers with a set of primitives which includes all functions which are continuously differentiable except on a countable closed set this may be the richest set of operations considered in a lower bound proof the prooi relies on the implicit functions theorem and baire s category theorem we also show that the integers with arithmetic x and lz zj any boolean operations and left shift are incompressible the inclusion of right shift reverses the situation and we obtain an o tq s upper bound where s is a functional inverse of ackermann s function

strong convergence of euler type methods for nonlinear stochastic differential equations traditional finite time convergence theory for numerical methods applied to stochastic differential equations sdes requires a global lipschitz assumption on the drift and diffusion coefficients in practice many important sde models satisfy only a local lipschitz property and since brownian paths can make arbitrarily large excursions the global lipschitz based theory is not directly relevant in this work we prove strong convergence results under less restrictive conditions first we give a convergence result for euler maruyama requiring only that the sde is locally lipschitz and that the pth moments of the exact and numerical solution are bounded for some p 2 as an application of this general theory we show that an implicit variant of euler maruyama converges if the diffusion coefficient is globally lipschitz but the drift coefficient satisfies only a one sided lipschitz condition this is achieved by showing that the implicit method has bounded moments and may be viewed as an euler maruyama approximation to a perturbed sde of the same form second we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial

optimal harvesting policy of a stochastic food chain population model this note studies the optimal harvesting policy of a stochastic three species food chain model the almost sufficient and necessary conditions for the existence of optimal harvesting policy are established meanwhile the optimal harvesting effort and the maximum of expectation of sustainable yield are obtained

lower bounds for dynamic algebraic problems we consider dynamic evaluation of algebraic functions matrix multiplication determinant convolution fourier transform etc in the model of reif and tate i e if f x1 xn y1 ym is an algebraic problem we consider serving online requests of the form change input xi to value v or what is the value of output yi we present techniques for showing lower bounds on the worst case time complexity per operation for such problems the first gives lower bounds in a wide range of rather powerful models for instance history dependent algebraic computation trees over any infinite subset of a field the integer ram and the generalized real ram model of ben amram and galil using this technique we show optimal n bounds for dynamic matrix vector product dynamic matrix multiplication and dynamic discriminant and an n lower bound for dynamic polynomial multiplication convolution providing a good match with reif and tate s o nlogn upper bound we also show linear lower bounds for dynamic determinant matrix adjoint and matrix inverse and an n lower bound for the elementary symmetric functions the second technique is the communication complexity technique of miltersen nisan safra and wigderson which we apply to the setting of dynamic algebraic problems obtaining similar lower bounds in the word ram model the third technique gives lower bounds in the weaker straight line program model using this technique we show an log n 2 log log n lower bound for dynamic discrete fourier transform technical ingredients of our techniques are the incompressibility technique of ben amram and galil and the lower bound for depth two superconcentrators of radhakrishnan and ta shma the incompressibility technique is extended to arithmetic computation in arbitrary fields 2001 elsevier science

the exact invariant density for a cusp shaped return map the square root iteration xt 1 1 2 mod xt mod 1 2 has a linear invariant density and a characteristic liapunov exponent lambda 1 2

asymptotic properties of hybrid diffusion systems in response to the increasing needs for control and optimization of hybrid systems this work is concerned with such asymptotic properties as recurrence also known as weak stochastic stability in the literature and ergodicity of regime switching diffusions using liapunov functions necessary and sufficient conditions for positive recurrence are developed then ergodicity of positive recurrent regime switching diffusions is obtained by constructing cycles using the associated discrete time markov chains

permanence and asymptotical behavior of stochastic prey predator system with markovian switching in this paper we investigate the stochastic permanence and extinction of a stochastic ratio dependent prey predator model controlled by a markov chain in the permanent case we estimate the superior limit and the inferior limit of the average in time of the sample path of the solution the boundaries are related to the stationary probability distribution of the markov chain and the parameters of the subsystems finally we illustrate our main results by two examples and some numerical experiments

stabilization and destabilization of hybrid systems of stochastic differential equations this paper aims to determine whether or not a stochastic feedback control can stabilize or destabilize a given nonlinear hybrid system new methods are developed and sufficient conditions on the stability and instability for hybrid stochastic differential equations are provided these results are then used to examine stochastic stabilization and destabilization

evolution of predator prey systems described by a lotka volterra equation under random environment abstract in this paper we consider the evolution of a system composed of two predator prey deterministic systems described by lotka volterra equations in random environment it is proved that under the influence of telegraph noise all positive trajectories of such a system always go out from any compact set of int r 2 with probability one if two rest points of the two systems do not coincide in case where they have the rest point in common the trajectory either leaves from any compact set of int r 2 or converges to the rest point the escape of the trajectories from any compact set means that the system is neither permanent nor dissipative

population dynamical behavior of lotka volterra system under regime switching in this paper we investigate a lotka volterra system under regime switching dx t diag x 1 t x n t b r t a r t x t dt s r t db t where b t is a standard brownian motion the aim here is to find out what happens under regime switching we first obtain the sufficient conditions for the existence of global positive solutions stochastic permanence and extinction we find out that both stochastic permanence and extinction have close relationships with the stationary probability distribution of the markov chain the limit of the average in time of the sample path of the solution is then estimated by two constants related to the stationary distribution and the coefficients finally the main results are illustrated by several examples

dynamic normal forms and dynamic characteristic polynomial we present the first fully dynamic algorithm for computing the characteristic polynomial of a matrix in the generic symmetric case our algorithm supports rank one updates in o n2logn randomized time and queries in constant time whereas in the general case the algorithm works in o n2klogn randomized time where k is the number of invariant factors of the matrix the algorithm is based on the first dynamic algorithm for computing normal forms of a matrix such as the frobenius normal form or the tridiagonal symmetric form the algorithm can be extended to solve the matrix eigenproblem with relative error 2 b in additional o nlog2nlogb time furthermore it can be used to dynamically maintain the singular value decomposition svd of a generic matrix together with the algorithm the hardness of the problem is studied for the symmetric case we present an n2 lower bound for rank one updates and an n lower bound for element updates

rich dynamics of a stochastic michaelis menten type ratio dependent predator prey system abstract stochastic predator prey systems with different functional responses have been studied but the dynamics of a michaelis menten type ratio dependent predator prey system with stochastic perturbation is not investigated systematically in this paper we give the asymptotic behavior of this system sufficient criteria for the existence of a stationary distribution and ergodicity are obtained which means the species are permanent besides we show the situations in which the species are non persistence finally examples and simulations are carried on to verify these results

dynamic matrix rank we consider maintaining information about the rank of a matrix under changes of the entries for nxn matrices we show an upper bound of o n 1 5 7 5 arithmetic operations and a lower bound of w n arithmetic operations per element change the upper bound is valid when changing up to o n 0 5 7 5 entries in a single column of the matrix we also give an algorithm that maintains the rank using o n 2 arithmetic operations per rank one update these bounds appear to be the first nontrivial bounds for the problem the upper bounds are valid for arbitrary fields whereas the lower bound is valid for algebraically closed fields the upper bound for element updates uses fast rectangular matrix multiplication and the lower bound involves further development of an earlier technique for proving lower bounds for dynamic computation of rational functions