competition for a single limiting resource in continuous culture the variable yield model the global asymptotic behavior of solutions of the so called variable yield model of competition between two microbial populations for a single growth limiting nutrient is determined this model also referred to as the variable internal stores or caperon droop model is a generalization of the classical monod model it assumes that the growth rate of a population depends on the amount of nutrient stored by the organism taken to be the same for all organisms rather than the concentration of ambient nutrient in the culture it is shown that competitive exclusion holds the winner being the organism that can grow at the lower ambient nutrient concentration

exclusion and persistence in deterministic and stochastic chemostat models abstract we first introduce and analyze a variant of the deterministic single substrate chemostat model in this model microbe removal and growth rates depend on biomass concentration with removal terms increasing faster than growth terms using a comparison principle we show that persistence of all species is possible in this scenario then we turn to modelling the influence of random fluctuations by setting up and analyzing a stochastic differential equation in particular we show that random effects may lead to extinction in scenarios where the deterministic model predicts persistence on the other hand we also establish some stochastic persistence results

stability analysis for lotka volterra systems based on an algorithm of real root isolation based on the theory of monotone flows of solutions of systems of differential equations the routh hurwitz theorem and a real root isolation algorithm of multivariate polynomials are applied to a class of lotka volterra diffusion systems an algorithm to determine the location of equilibria and the stability of the nonlinear dynamics systems is implemented in maple

flocking of the cucker smale model on general digraphs in the literature the study of the cucker smale c s flocking model is all restricted to connected undirected graphs or digraphs of some special types containing only a single root in this technical note we establish the flocking behavior of the c s model on general digraphs which contain all existing interaction topologies in relation to the c s model as special cases in particular the topology with multiple leaders is included the flocking results are guaranteed under some conditions on the initial state of the flock only the critical exponent below which unconditional convergence holds is given depending only on the interaction topology it is consistent with the ones obtained under some known special topologies in the literature

vibration of a two member open frame abstract the dynamics of a two member open frame structure undergoing both in and out of plane motion is examined the frames are modelled using the euler bernoulli beam theory and are further generalized by permitting an arbitrary angle between the beams and the attachment of a payload at the end of the second beam the equations of motion are derived using hamilton s principle and the orthogonality conditions are presented it is shown that the in and out of plane motions can be decoupled by including the axial deformation components in the assumed displacement fields the natural frequencies of the system and the contribution of each member into the system potential energy are examined via numerical examples

nonmonotone systems decomposable into monotone systems with negative feedback motivated by the work of angeli and sontag 1 and enciso and sontag 7 in control theory we show that certain flnite and inflnite dimensional semi dynamical systems with negative feedback can be decomposed into a monotone open loop system with inputs and a decreasing output function the original system is reconstituted by plugging the output into the input employing a technique of gouz 9 and cosner 5 of imbedding the system into a larger symmetric monotone system we are able to obtain information on the asymptotic behavior of solutions including existence of positively invariant sets and global convergence

competition in an n vessel gradostat a mathematical model of the exploitative competition between two microbial species in a general multivessel gradostat is considered sufficient conditions for the two species to coexist in the grad

on the asymptotic behavior of a class of deterministic models of cooperating species in this paper the global behavior of solutions of a class of ordinary differential equations modeling n cooperating biological species is determined the class of systems includes the classical lot

limit cycles for competitor competitor mutualist lotka volterra systems abstract it is known that a limit cycle or periodic coexistence can occur in a competitor competitor mutualist lotka volterra system x 1 x 1 r 1 a 11 x 1 a 12 x 2 a 13 x 3 x 2 x 2 r 2 a 21 x 1 a 22 x 2 a 23 x 3 x 3 x 3 r 3 a 31 x 1 a 32 x 2 a 33 x 3 where r i a i j are positive real constants x liang j jiang the dynamical behavior of type k competitive kolmogorov systems and its applications to 3 dimensional type k competitive lotka volterra systems nonlinearity 16 2003 785 801 in this paper we shall construct an example with at least two limit cycles and furthermore we will show that the number of periodic orbits and hence a fortiori of limit cycles is finite it is also shown that contrary to three dimensional competitive lotka volterra systems nontrivial periodic coexistence does occur even if none of the three species can resist invasion from either of the others in this case new amenable conditions are given on the coefficients under which the system has no nontrivial periodic orbits these conditions imply that the positive equilibrium if it exists is globally asymptotically stable

oscillations in i o monotone systems under negative feedback oscillatory behavior is a key property of many biological systems the small gain theorem sgt for input output monotone systems provides a sufficient condition for global asymptotic stability of an equilibrium and hence its violation is a necessary condition for the existence of periodic solutions one advantage of the use of the monotone sgt technique is its robustness with respect to all perturbations that preserve monotonicity and stability properties of a very low dimensional in many interesting examples just one dimensional model reduction this robustness makes the technique useful in the analysis of molecular biological models in which there is large uncertainty regarding the values of kinetic and other parameters however verifying the conditions needed in order to apply the sgt is not always easy this paper provides an approach to the verification of the needed properties and illustrates the approach through an application to a classical model of circadian oscillations as a nontrivial case study and provides a theorem in the converse direction of predicting oscillations when the sgt conditions fail

on cooperative systems with respect to arbitrary orderings we derive conditions for cooperativity of ordinary differential equations with respect to an arbitrary order cone and generalize the classical results of m muller and e kamke to this situation some examples and applications are presented

sequential vs parallel complexity in simple gene assembly we investigate some differences between the general intramolecular model for gene assembly and its restricted simple model although both models satisfactorily sort all current experimental data we show that the general model offers assembly strategies for a given string that vary in both assembly length and the operations used while the simple model will always use the same number of each type of operation to sort a gene when simple operations are applied in parallel this is given a new twist we prove that for any n 1 there exists a string having maximally parallel assemblies of any length between n and 2n

the classification for dynamics of an std model in this paper a 3 n 1 dimensional sexually transmitted disease model is studied the model involves two competing strains 1 and 2 where females are divided into n different groups based on their susceptibility to two distinct pathogenic strains the first and second reproduction numbers r i r i which have their biological meaning are defined for strain i i 1 2 respectively with the non degeneracy assumption h a complete classification for dynamics of this model in terms of r i r i i 1 2 is presented which is based on the theory of type k monotone dynamical systems

competing subcommunities of mutualist and a generalized kamke theorem a model of a biological community of species is considered the community consists of two competing subcommunities each of which has the property that each pair of species of the subcommunity interact in a mutually beneficial or noninterfering manner sufficient conditions are given that the two subcommunities can coexist the coexistence state being a stable equilibrium the domains of attraction of stable equilibria are considered the sharpest results are for the special case of lotka volterra systems mathematically the fundamental tool is a generalized kamke theorem

cooperative systems of differential equations with concave nonlinearities on considere le systeme dynamique discret engendre par une application monotone concave t r n r n on etudie ensuite des systemes d equations differentielles cooperatifs periodiques x f t x a non linearites concaves

vibrations of elastically restrained frames abstract this paper deals with the determination of eigenfrequencies of a frame which consists of a beam supported by a column and is submitted to intermediate elastic constraints the ends of the frame are elastically restrained against rotation and translation the individual members of the frame are assumed to be governed by the transverse and axial vibration theory of an euler bernoulli beam the boundary and eigenvalue problem which governs the dynamical behavior of the frame structure is derived using the techniques of calculus of variations exact values of eigenfrequencies are determined by the application of the separation of variables method also results are obtained by the use of the finite element method the natural frequencies and mode shapes are presented for a wide range of values of the restraint parameters several particular cases are presented and some of these have been compared with those available in the literature

saddle point behavior for monotone semiflows and reaction diffusion models the saddle point behavior is established for monotone semiflows with weak bistability structure and then these results are applied to three reaction diffusion systems modeling man environment man epidemics single loop positive feedback and two species competition respectively

slow integral manifolds for lagrangian fluid dynamics in unsteady geophysical flows abstract the authors consider lagrangian motion of fluid particles in unsteady gravity currents in geophysical flows the vertical motion of fluid particles especially the induced vertical mixing in these currents is partially responsible for the ocean thermohaline circulation and thus plays a role in the global climate dynamics first a reduced dynamic system for slow variables is derived for a nonautonomous multiscale system the reduced system still nonautonomous is the original system restricted to a centre like nonautonomous invariant manifold so called slow manifold which holds slow motions of the system an algorithm is also presented to obtain an approximation of the nonautonomous slow manifold a novelty here is that the reduction principle applies to nonautonomous multiscale systems which satisfy conditions that are true only locally in space as in many physical cases this makes the reduction principle applicable to real physical systems then this invariant manifold reduction principle is applied to an approximate conceptual lagrangian model of gravity currents and a reduced nonautonomous system for slow vertical motion is obtained this reduced system may be useful as a conceptual tractable tool for understanding some features of vertical mixing in unsteady gravity currents

discrete infinite dimensional type k monotone dynamical systems and time periodic reaction diffusion systems abstract the asymptotic behavior of discrete type k monotone dynamical systems and reaction diffusion equations is investigated the studying content includes the index theory for fixed points permanence global stability convergence everywhere and coexistence it is shown that the system has a globally asymptotically stable fixed point if every fixed point is locally asymptotically stable with respect to the face it belongs to and at this point the principal eigenvalue of the diagonal partial derivative about any component not belonging to the face is not one a nice result presented is the sufficient and necessary conditions for the system to have a globally asymptotically stable positive fixed point it can be used to establish the sufficient conditions for the system to persist uniformly and the convergent result for all orbits applications are made to time periodic lotka volterra systems with diffusion and sufficient conditions for such systems to have a unique positive periodic solution attracting all positive initial value functions are given for more general time periodic type k monotone reaction diffusion systems with spatial homogeneity a simple condition is given to guarantee the convergence of all positive solutions

some non existence results for a class of stochastic partial differential equations abstract consider the following stochastic partial differential equation t u t x l u t x u t x f t x t 0 and x r d the operator l is the generator of a strictly stable process and f is the random forcing term which is assumed to be gaussian under some additional conditions most notably on and the initial condition we show non existence of global random field solutions our results complement those of p chow in 5 and 6 among others

the coexistence of a community of species with limited competition abstract this paper is a study of a system modeling a biological community of species with limited competition the community consists of two competing subcommunities all species of which cooperate and some species of one subcommunity can invade the steady state of another subcommunity whereas others cannot sufficient conditions are given that all species can coexist for lotka volterra systems we can improve this result by showing that there is a unique globally asymptotically stable steady state

formal systems for gene assembly in ciliates dna processing in ciliates a very ancient group of organisms is among the most sophisticated dna processing in living organisms it has a quite clear computational structure and even uses explicitly the linked list data structure particularly interesting from the computational point of view is the process of gene assembly from its micronuclear to its macronuclear form we investigate here the string rewriting and the graph rewriting models of this process involving three molecular operations which together form a universal set of operations in the sense that they can assembly any macronuclear gene from its micronuclear form in particular we prove that although the graph rewriting system is more abstract than the string rewriting system no essential information is lost in the sense that one can translate assembly strategies from one system into the other

systems of ordinary differential equations which generate an order preserving flow a survey of results this article consists of a survey of results concerning the qualitative behavior of solutions of systems of ordinary differential equations which generate an order preserving flow we restrict our consideration to partial orderings on r n induced by any one of its orthants a flow preserves ordering if any two solutions x t and y t are ordered x t leqq y t for all t 0 whenever x 0 leqq y 0 many of the important results for such systems have only recently been obtained principally by m w hirsch who pointed out the tendency of their solutions to converge to equilibrium less well known are some global geometric constraints on the stable manifold of an equilibrium and the existence of heteroclinic orbits connecting ordered equilibria a particularly striking result for this class of systems is the easily computable necessary and sufficient condition for stability of an equilibrium one of our main goals is to show that by allowing partial orderings on r n generated by orthants

vibration of arbitrarily oriented two member open frames with tip mass abstract a generalized two member open frame structure is modelled using euler bernoulli beam theory the generalization is approached in two ways an arbitrary angle between the beams and the attachment of a payload at the end of the second beam the linearized equations of motion are derived using hamilton s principle and the general frequency equation mode shapes and orthogonality condition are presented along with some numerical examples

dynamics of the periodic type k competitive kolmogorov systems abstract for time periodic dissipative and irreducible type k competitive kolmogorov systems it is proved that there is a canonically defined countable family f of unordered disjoint invariant sets with the property that for every persistent trajectory whose limit set is not a cycle there exists a unique trajectory in some element of f such that these two trajectories are asymptotic and the corresponding points in these two trajectories are k related

singularity and existence for a wave system of nematic liquid crystals abstract in this paper we prove the global existence and singularity formation for a wave system modeling nematic liquid crystals in one space dimension in our model although the viscous damping term is included the solution with smooth initial data still has gradient blowup in general even when the initial energy is arbitrarily small

the complete strategic classification for a cooperation competition model in the www market abstract wang and wu physica a 339 2004 609 620 proposed a cooperation competition model for the www market under the assumptions that the intrinsic growth rates and maximum fractions are equal respectively and that the interaction is symmetric they provided the strategies for the small sites to win and the powerful site to be extinct this paper gives the complete strategic classification for this model without any hypothesis

new generic quasi convergence principles with applications in this paper essentially strongly order preserving and conditionally set condensing semiflows are considered obtained is a new type of generic quasi convergence principles implying the existence of an open and dense set of stable quasi convergent points when the state space is order bounded the generic quasi convergence principles are then applied to essentially cooperative and irreducible systems in the forms of ordinary differential equations and delay differential equations giving some results of theoretical and practical significance

periodic orbits of competitive and cooperative systems abstract in this paper we consider the existence location and stability type of periodic orbits of competitive and cooperative systems of autonomous ordinary differential equations particular attention is given to the existence of invariant manifolds related to periodic orbits and these results are used to improve a result of hirsch for three dimensional irreducible competitive and cooperative systems in particular the poincare bendixson theorem holds for such three dimensional systems

on the finite dimensional dynamical systems with limited competition the asymptotic behavior of dynamical systems with limited competition is investigated we study index theory for fixed points permanence global stability convergence everywhere and coexistence it is shown that the system has a globally asymptotically stable fixed point if every fixed point is hyperbolic and locally asymptotically stable relative to the face it belongs to a nice result is the necessary and sufficient conditions for the system to have a globally asymptotically stable positive fixed point it can be used to establish the sufficient conditions for the system to persist uniformly and the convergence result for all orbits applications are made to time periodic ordinary differential equations and reaction diffusion equations

reducibility of gene patterns in ciliates using the breakpoint graph gene assembly in ciliates is one of the most involved dna processings going on in any organism this process transforms one nucleus the micronucleus into another functionally different nucleus the macronucleus we continue the development of the theoretical models of gene assembly and in particular we demonstrate the use of the concept of the breakpoint graph known from another branch of dna transformation research more specifically 1 we characterize the intermediate gene patterns that can occur during the transformation of a given micronuclear gene pattern to its macronuclear form 2 we determine the number of applications of the loop recombination operation the most basic of the three molecular operations that accomplish gene assembly needed in this transformation 3 we generalize previous results and give elegant alternatives for some proofs concerning characterizations of the micronuclear gene patterns that can be assembled using a specific subset of the three molecular operations

the dynamical systems approach to differential equations bulletin new series of the american mathematical society volume 11 number 1 july 1984 the dynamical systems approach to differential equations by morris w hirsch 1 this harmony that human intelligence believes it discovers in nature does it exist apart from that intelligence no without doubt a reality completely independent of the spirit which conceives it sees it or feels it is an impossibility a world so exterior as that even if it existed would be forever inaccessible to us but what we call objective reality is in the last analysis that which is common to several thinking beings and could be common to all this common part we will see can be nothing but the harmony expressed by mathematical laws h poincare la valeur de la science p 9 ignorance of the roots of the subject has its price no one denies that modern formulations are clear elegant and precise it s just that it s impossible to comprehend how any one ever thought of them m spivak a comprehensive introduction to differential geometry introduction the mathematical subject we call dynamical systems was fathered by poin care developed sturdily under birkhoff and has enjoyed a vigorous new growth for the last twenty years as i try to show in chapter i it is interesting to look at this mathematical development as the natural outcome of a much broader theme which is as old as science itself the universe is a sytem which changes in time every mathematician has a particular way of thinking about mathematics but rarely makes it explicit yet such perspectives in any particular field can be of great value to nonexperts who must apply the results of the field or who want to learn them in some fields the points of view of its practitioners cluster around several comparatively clearly defined concepts and insights dynamical systems theory is one of them the basic concepts go back to poincare the main thrust of chapters i and ii is to explain rather discursively some of received by the editors december 14 1983 1980 mathematics subject classification primary 00a25 00a99 01a45 01a55 01a60 34a40 34c35 34d10 34d30 35b05 35b35 35b40 35b50 35b30 35k55 54h20 58f25 58f40 58f12 58f10 58d25 58d07 46a40 46e10 47h07 47h20 92a15 92a17 secondary 06f30 35b65 58f13 35j60 46e05 92 03 90 03 90a16 supported in part by national science foundation and institute for advanced study 1984 american mathematical society 0273 0979 84 1 00 25 per page

combining overlap and containment for gene assembly in ciliates abstract gene assembly in ciliates is an intricate biological process that has been studied formally and modeled through string and graph rewriting systems recently a restriction of the general intramolecular model called simple gene assembly has been introduced this restriction has subsequently been defined as a string rewriting system we show that by extending the notion of overlap graph to also represent containment it is possible to define an equivalent graph rewriting system for two of the three types of rules that make up simple gene assembly it turns out that this graph rewriting system is often less involved to study than its corresponding string rewriting system we illustrate this by giving characterizations of the power of both types of graph rewriting rules also a first step is made towards a characterization of the extended overlap graphs

systems of differential equations that are competitive or cooperative v convergence in 3 dimensional systems abstract for 3 dimensional systems that are competitive or cooperative we present new criteria for existence of equilibria and cycles convergence of all trajectories global stability and structural stability a certain type of positive feedback loop is shown to be structurally stable

one sided maximal inequalities for a stock process abstract let x x t t 0 be a stock process assumed a geometric brownian motion with drift 0 and volatility 0 starting at x 0 for any stopping time for x we establish an upper moment bound for e x max 0 t x t under certain restrictions our method of proof employs the gronwall inequality and a comparison principle for a system of first order nonlinear differential equations the bound obtained in this paper extends an existing result and the method of proof employed is quite new

qualitative theory of compartmental systems dynamic models of many processes in the biological and physical sciences which depend on local mass balance conditions give rise to systems of ordinary differential equations many nonlinear that are called compartmental systems in this paper the authors define compartmental systems specify their relations to other nonnegative systems and discuss examples of applications the authors review the qualitative results on linear and nonlinear compartmental systems including their relation to cooperative systems they review the results for linear compartmental systems and then integrate and expand the results on nonlinear compartmental systems providing a framework for unifying them under a few general theorems in the course of that they complete the solution of a problem posed by bellman and show that closed nonlinear autonomous n compartment systems can show the full gamut of possible behaviors of systems of odes finally to provide additional structure to this study the authors show how to partiti

competing subcommunities of mutualists abstract persistence and convergence to equilibria are established for a general system of kolmogorov type modeling an ecosystem in which two subcommunities of mutualists interact competitively the equations allow for time delays in interspecies interactions