effects of random motility on microbial growth and competition in a flow reactor the authors investigate the effects of random motility on the ability of a microbial population to survive in pure culture and to be a good competitor for scarce nutrient in mixed culture in a flow reactor model consisting of a nonlinear parabolic system of partial differential equations for pure culture a single population a sharp condition is derived which distinguishes between the two outcomes 1 washout of the population from the reactor or 2 persistence of the population and the existence of a unique single population steady state the simulations suggest that this steady state is globally attracting for the case of two populations competing for scarce nutrient they obtain sufficient conditions for the uniform persistence of the two populations for the existence of a coexistence steady state and for the ability of one population to competitively exclude a rival extensive simulations are reported which suggest that 1 all solutions approach some steady state solution 2 all possible outcomes exhibited by the classical competitive lotka volterra ode model can occur in the model and 3 the outcome of competition between two bacterial strains can depend rather subtly on their respective random motility coefficients

factorization and approximation problems for matrix functions the purpose of this paper is to develop a new approach to certain approximation and factorization problems for matrix valued functions the approach is based on studying properties of maximizing vectors of hankel operators with matrix valued symbols and on the solution of the so called recovery problem for unitary valued matrix functions in the case of scalar functions such problems were studied in detail in pk it turns out however that the case of matrix functions is considerably more complicated than the scalar case in this paper we solve the recovery problem with the help of hankel operators this solution also allows us to give a new approach to wiener hopf factorization problems and obtain some new results another application we consider here is the problem of finding a superoptimal analytic approximation to a matrix function on the unit circle we improve earlier results on hereditary properties of such approximations see py1 and on continuity properties of the operator of superoptimal approximation see py3 recall that for a bounded m x n matrix function b on the unit circle tf the hankel operator hp is defined on the hardy class h2 cn of cn valued functions by hdf padf f e h cn

global attractors and steady state solutions for a class of reaction diffusion systems abstract we show that weak l p dissipativity implies strong l dissipativity and therefore implies the existence of global attractors for a general class of reaction diffusion systems this generalizes the results of alikakos and rothe the results on positive steady states especially for systems of three equations in our earlier work j differential equations 130 1996 59 91 are improved

instability of the boundary in the billiard ball problem we consider the billiard ball problem in the interior of a plane closed convexc 1 curve which is piecewisec 2 if the curvature has a discontinuity then the boundary is unstable i e no caustics exist near the boundary however in the interior there can exist caustics as we show by an example

superoptimal analytic approximations of matrix functions we study the approximation of a bounded matrix valued function g on the unit circle by functions q bounded and analytic in the unit disc we show that if g is continuous then there is a unique q for which the error g q has a strong minimality property involving not only the l norm of g q but also the suprema of its subsequent singular values we obtain structural properties of the error g q and show that certain smoothness properties of g are inherited by q e g membership of besov spaces

traveling fronts of the volume filling chemotaxis model with general kinetics this paper is devoted to the study of traveling fronts for the volume filling chemotaxis model with general kinetics by applying the perturbation method the proof relies on the fredholm theory and the banach contractive mapping principle the obtained results can easily detect the existence of traveling fronts for some models investigated in the known references

analytical study of a triple hopf bifurcation in a tritrophic food chain model we provide an analytical proof of the existence of a stable periodic orbit contained in the region of coexistence of the three species of a tritrophic chain the method used consists in analyzing a triple hopf bifurcation for some values of the parameters three limit cycles born via this bifurcation one is contained in the plane where the top predator is absent another one is not contained in the domain of interest where all variables are positive the third one is contained where the three species coexist the techniques for proving these results have been introduced in previous articles by the second author and are based on the averaging theory of second order existence of this triple hopf bifurcation has been previously discovered numerically by kooij et al

billiards with pesin region of measure one we give a description of a large class of plane billiards with pesin region of measure one open conditions including properly those founded by wojtkowski w1 forc4 focusing boundaries are obtained lyapunov s forms introduced by lewowicz are used

a transversal fundamental theorem for semi dispersing billiards for billiards with a hyperbolic behavior fundamental theorems ensure an abundance of geometrically nicely situated and sufficiently large stable and unstable invariant manifolds a transversal fundamental theorem has recently been suggested by the present authors to proveglobal ergodicity and then as an easy consequence the k property of semidispersing billiards in particular the global ergodicity of systems ofn 3 elastic hard balls conjectured by the celebratedboltzmann sinai ergodic hypothesis in fact the suggested transversal fundamental theorem has been successfully applied by the authors in the casesn 3 and 4 the theorem generalizes the fundamental theorem of chernov and sinai that was really the fundamental tool to obtainlocal ergodicity of semi dispersing billiards our theorem however is stronger even in their case too since its conditions are simpler and weaker moreover a complete set of conditions is formulated under which the fundamental theorem and its consequences like the zig zag theorem are valid for general semi dispersing billiards beyond the utmost interesting case of systems of elastic hard balls as an application we also give conditions for the ergodicity and consequently the k property of dispersing billiards transversality means the following instead of the stable and unstable foliations occurring in the chernov sinai formulation of the stable version of the fundamental theorem we use the stable foliation and an arbitrary nice one transversal to the stable one

principles for the design of billiards with nonvanishing lyapunov exponents we introduce a large class of billiards with convex pieces of the boundary which have nonvanishing lyapunov exponents

food chain length of a ecological model with prey taxis abstract in this paper we construct a ecological system with prey taxis from the dynamic point of view we analytically and numerically investigate the steady states our results show how the combined action of the death rates chemotaxis diffusivity and conversion efficiency to influence the survival ability of multiple consumer levels in a ecological system with self produced nutriment furthermore the food chain length of our model can be determined by the co existence conditions of all consumer levels numerical simulation corroborates the analytical results

an exhaustive criterion for the non existence of invariant circles for area preserving twist maps we present a necessary and sufficient condition for the non existence of rotational invariant circles for area preserving twist maps of the cyclinder or annulus based on the cone crossing and killends criteria of mackay and percival 1985 given a number of technical restrictions on the implementation of these criteria this condition leads to a proof of mackay and percival s finite computation conjecture

chaotic properties of the elliptical stadium the elliptical stadium is a curve constructed by joining two half ellipses with half axesa 1 andb 1 by two straight segments of equal length 2h

unitary interpolants factorization indices and infinite hankel block matrices abstract the existence uniqueness and construction of unitary n n matrix valued functions j j j in wiener like algebras on the circle with prescribed matrix fourier coefficients j j for j 0 are studied in particular if j then such an exists with j if and only if 0 1 where v denotes the infinite block hankel matrix j k v j k 0 1 acting in the sequence space l n 2 one of the main results is that the nonnegative factorization indices of every such are uniquely determined by the given data in terms of the dimensions of the kernels of i v v whereas the negative factorization indices are arbitrary it is also shown that there is a unique such if and only if the data forces all the factorization indices to be nonnegative and simple conditions for that and a formula for in terms of certain schmidt pairs of 0 are given the results depend upon a fine analysis of the structure of the kernels of i v v and of the one step extension problem of adamjan arov and krein funct anal appl 2 1968 1 18 isometric interpolants for the nonsquare case are also considered

converse kam theory and practice we unify extend reinterpret and apply criteria of birkhoff 1 herman 9 mather 2 3 aubry et al 4 5 and newman and percival 6 for the nonexistence of invariant circles for area preserving twist maps the criteria enable one to establish regions of phase space through which no rotational invariant circles pass for families of maps the same can be done for regions of the combined space of phase points and parameters the criteria can be implemented rigorously on a computer and give a practical method of proving quite strong results as an example we present a computer program which proved that the standard map has no rotational invariant circles for any parameter value k 63 64

dynamical and numerical analyses of a generalized food chain model this paper focuses on the qualitative and quantitative analysis of a generalized three species food chain model the conditions for the existence and stability of the boundary and positive equilibria of the model are established by constructing an appropriate bifurcation function it is shown that the model undergoes a hopf bifurcation from the positive equilibrium for certain parameter values the root of this bifurcation function gives the bifurcation value a robust non standard numerical method is constructed and used to obtain the solution of the model numerical simulations using this method show that the hopf bifurcation is supercritical with a stable limit cycle in the positive octant

using integrability to produce chaos billiards with positive entropy a new open class of convex 2 dimensional planar billiards with positive lyapunov exponent almost everywhere is constructed we introduce the notion of a focusing arc and show that such arcs can be used to build billiard systems with positive lyapunov exponents we prove that under smallc6 perturbations focusing arcs remain focusing and thereby show that perturbations of the bunimovich stadium billiard have positive lyapunov exponents

a theorem on ergodicity of two dimensional hyperbolic billiards ergodicity of two dimensional billiards which satisfy some general conditions is proved this theorem is applied to one concrete class of billiards that contains in particular billiards in the stadium

effect of time delay on a food chain model this paper aims to study the effect of discrete time delay on a tritrophic food chain model with holling type ii functional responses dynamical behaviours such as boundedness stability persistence and bifurcation of the model are studied our analytical findings are illustrated through computer simulation biological implications of our analytical findings are addressed critically

on the bernoulli property of planar hyperbolic billiards we consider billiards in non polygonal domains of the plane with boundary consisting of curves of three different types straight segments strictly convex inward curves and strictly convex outward curves of a special kind the billiard map for these domains is known to have non vanishing lyapunov exponents a e provided that the distance between the curved components of the boundary is sufficiently large and the set of orbits having collisions only with the flat part of the boundary has zero measure under a few additional conditions we prove that there exists a full measure set of the billiard phase space such that each of its points has a neighborhood contained up to a zero measure set in one bernoulli component of the billiard map using this result we show that there exists a large class of planar hyperbolic billiards that have the bernoulli property this class includes the billiards in convex domains bounded by straight segments and strictly convex inward arcs constructed by donnay

a lower bound for chaos on the elliptical stadium abstract the elliptical stadium is a plane region bounded by a curve constructed by joining two half ellipses with half axes a 1 and b 1 by two parallel segments of equal length 2 h donnay comm math phys 141 1991 225 257 proved that if 1 2 and if h is large enough then the corresponding billiard map has non vanishing lyapunov exponents almost everywhere moreover h as a 2 in a previous paper markarian et al comm math phys 174 1996 661 679 we found a bound for h assuring the k property for these billiards for values of a very close to 1 in this work we study the stability of a particular family of periodic orbits obtaining a new bound for the chaotic zone for any value of a 2

on superoptimal approximation by analytic and meromorphic matrix valued functions abstract it is a well known fact that for any continuous scalar valued function on the unit circle there is a unique best approximation in the hardy class h of bounded analytic functions however in the matrix valued case a best approximation by bounded analytic functions is almost never unique to make it unique one has to impose additional assumptions in 1986 n j young introduced the so called superoptimal solution of the nehari problem the word superoptimal means that we are seeking f h matrix valued to minimize not only sup f 1 sup s0 f 1 but also the suprema of further singular values sj f j 1 it was proved by v v peller and n j young that for matrix valued function in h c such superoptiminal solution is unique in this paper an alternative geometric approach to the problem is presented it allows us to obtain another proof of the uniqueness and some new results uniqueness of the superoptimal approximation by meromorphic functions a generalization of the adamyan arov krein result on approximation of scalar valued functions inequalities between s numbers of a hankel operator and superoptimal singular values our approach works well for operator valued functions as for matrix valued ones

scaling of dynamical properties of the fermi ulam accelerator abstract we investigate numerically the chaotic sea of the complete fermi ulam model fum and of its simplified version sfum we perform a scaling analysis near the integrable to non integrable transition to describe the average energy as function of time t and as function of iteration or collision number n when t is employed as independent variable the exponents of fum and sfum are different however when n is used the exponents are the same for both fum and sfum in the collision number analysis we present analytical arguments supporting the values of the exponents related to the control paramenter and to the initial velocity we describe also how the scaling exponents obtained by using t as independent variable are related to the ones obtained with n in contrast to sfum the average energy in fum saturates for long times we discuss the origin of the observed differences and similarities between fum and its simplified version

hyperbolic billiards with nearly flat focusing boundaries i abstract the standard wojtkowski markarian donnay bunimovich technique for the hyperbolicity of focusing or mixed billiards in the plane requires the diameter of a billiard table to be of the same order as the largest ray of curvature along the focusing boundary this is due to the physical principle that is used in the proofs the so called defocusing mechanism of geometrical optics in this paper we construct examples of hyperbolic billiards with a focusing boundary component of arbitrarily small curvature whose diameter is bounded by a constant independent of that curvature our proof employs a nonstandard cone bundle that does not solely use the familiar dispersing and defocusing mechanisms

many dimensional nowhere dispersing billiards with chaotic behavior abstract we consider some examples of domains in the d dimensional euclidean space d 3 that generate billiards with chaotic behavior time correlations in these billiards tend to zero in the limit t the rate of decay of the correlations is a power law for billiards in discrete time and could be a power law or quasiexponential for billiards with continuous time

approximation by analytic matrix functions the four block problem abstract the four block problem is a generalization of nehari s problem for matrix functions it plays an important role in h optimal control theory it is well known that nehari s problem for a continuous scalar function has a unique solution however in the matrix case the situation is quite different v v peller and n j young 1994 j funct anal 120 300 343 studied superoptimal solutions of nehari s problem they minimize not only the l norm of the corresponding matrix function but also the essential suprema of all further singular values it was shown that for h c matrix functions nehari s problem has a unique superoptimal solution in this paper we study superoptimal solutions of the four block problem and we find a natural condition under which such a superoptimal solution is unique our result is new even in the case of nehari s problem we study some related problems such as thematic factorizations invariance of indices and inequalities between the singular values of the four block operator and the superoptimal singular values

unitary interpolants and factorization indices of matrix functions abstract for an n n bounded matrix function we study unitary interpolants u i e unitary valued functions u such that u j j j u for which the toeplitz operator t u is fredholm we give a new approach based on superoptimal singular values and thematic factorizations we describe wiener hopf factorization indices of u in terms of superoptimal singular values of and thematic indices of f where f is a superoptimal approximation of by bounded analytic matrix functions the approach essentially relies on the notion of a monotone thematic factorization introduced in ap in the last section we discuss hereditary properties of unitary interpolants in particular for matrix functions of class h c we study unitary interpolants u of class qc