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

how many species can two essential resources support a chemostat model of n species of microorganisms competing for two perfectly complementary growth limiting nutrients is considered sufficient conditions are given for there to be a single winning species and for two species to coexist driving the others to extinction in the case when n 3 it is shown that every solution converges to one of the single species or two species steady states and hence the dynamics of the model is completely determined the results generalize those of hsu cheng and hubbell siam j appl math 41 1981 pp 422 444 as well as butler and wolkowicz math biosci 83 1987 pp 1 48 who considered two species

on the spectra of some weighted rooted trees and applications abstract let t be a weighted rooted tree of k levels such that 1 the vertices in level j have a degree equal to d k j 1 for j 1 2 k and 2 the edges joining the vertices in level j with the vertices in level j 1 have a weight equal to w k j for j 1 2 k 1 we give a complete characterization of the eigenvalues of the laplacian matrix and adjacency matrix of t they are the eigenvalues of leading principal submatrices of two nonnegative symmetric tridiagonal matrices of order k k moreover we give some results concerning their multiplicities by application of the above mentioned results we derive upper bounds on the largest eigenvalue of any weighted tree and the spectra of some weighted bethe trees

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

a mathematical model of competition for two essential resources in the unstirred chemostat a mathematical model of competition between two species for two growth limiting essential complementary resources in the unstirred chemostat is considered the existence of a positive steady state solution and some of its properties are established analytically techniques include the maximum principle the fixed point index and numerical simulations the simulations also seem to indicate that there are regions in parameter space for which a globally stable positive equilibrium occurs and that there are other regions for which the model admits bistability and even multiple positive equilibria

controlled growth of competing species a mathematical model of a chemostat treated as a control process is considered several microorganisms compete for a single essential growth limiting nutrient earlier theory supported by experiments predicts that if the nutrient input and washout rates are constant at most one competitor survives and that if either the nutrient input rate or the washout rate is periodic coexistence is possible under certain conditions here the rate of input of the nutrient is used as the control function necessary as well as sufficient conditions for the existence of admissible controls producing coexistence of all the species and sufficient conditions for the existence of admissible controls for selective extinction of some species are derived

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

mathematical assessment of the role of non linear birth and maturation delay in the population dynamics of the malaria vector abstract a delay ordinary deterministic differential equation model for the population dynamics of the malaria vector is rigorously analysed subject to two forms of the vector birth rate function the verhulst pearl logistic growth function and the maynard smith slatkin function it is shown that for any birth rate function satisfying some assumptions the trivial equilibrium of the model is globally asymptotically stable if the associated vectorial reproduction number is less than unity further the model has a non trivial equilibrium which is locally asymptotically stable under a certain condition the non trivial equilibrium bifurcates into a limit cycle via a hopf bifurcation it is shown by numerical simulations that the amplitude of oscillating solutions increases with increasing maturation delay numerical simulations suggest that the maynard smith slatkin function is more suitable for modelling the vector population dynamics than the verhulst pearl logistic growth model since the former is associated with increased sustained oscillations which in our view is a desirable ecological feature since it guarantees the persistence of the vector in the ecosystem

new result on the model based biological control of the chemostat abstract we investigate a known competition model of the chemostat with general nonmonotone response functions and distinct removal rates based on the competitive exclusion principle rapaport and harmand 2008 13 established the concept of the so called biological control the proof of the latter result is based on a theorem of li 1998 11 here we first propose a modification of li s theorem and then present an extension of the biological control concept

on the spectral radius of weighted trees with fixed diameter and weight set abstract the spectrum of weighted graphs are often used to solve the problems in the design of networks and electronic circuits we first give some perturbational results on the spectral radius of weighted graphs when some weights of edges are modified then we derive the weighted tree with the largest spectral radius in the set of all weighted trees with fixed diameter and weight set furthermore an open problem of spectral radius on weighted paths is solved h z yang g z hu y hong bounds of spectral radii of weighted tree tsinghua science and technology 8 2003 517 520

global dynamics of a state dependent delay model with unimodal feedback abstract we obtain the global attractivity of nonnegative stationary states for a parameterized state dependent delay equation with unimodal feedback under certain mild conditions we show that the equation with unimodal type nonlinearity can generate rich dynamics as the parameter varies to be specific global attractivity of the positive stationary state is obtained in a set of nonnegative bounded continuous functions when the stationary state is less than a value implicitly determined by a condition on the unimodal feedback the general results of global attractivity are illustrated through two examples arising from population dynamics moreover hopf bifurcations are demonstrated in the examples when the positive stationary states lose global attractivity

a group of non uniform exponential growth locally isomorphic to img z 2 i we prove that a sequence of marked three generated groups isomorphic to the iterated monodromy group of z 2 i converges to a group of non uniform exponential growth which is an extension of the infinite direct sum of cyclic groups of order 4 by a grigorchuk group

global stability for non monotone delay equations with application to a model of blood cell production abstract a set of sufficient conditions for the global stability of the positive equilibrium is established for a large class of non monotone time delayed differential equations by the method of fluctuations and the exponential ordering approach then these analytic results are applied to a mathematical model of white blood cell production numerical simulations are also presented to illustrate the global stability regions of parameters and the long term behavior of solutions

a mathematical model of the chemostat with a general class of functions describing nutrient uptake a model of the chemostat involving n microorganisms competing for a single essential growth limiting substrate is considered instead of assuming the familiar michaelis menten kinetics for nutrient uptake a general class of functions is used which includes all monotone increasing uptake functions but which also allows uptake functions that describe inhibition by the substrate at high concentrations the qualitative behaviour of this generalized model is determined analytically it is shown that the behaviour depends intimately upon certain parameters provided that all the parameters are distinct which is a biologically reaonable assumption at most one competitor survives the substrate and the surviving competitor if one exists approach limiting values thus there is competitive exclusion however unlike the standard model in certain cases the outcome is initial condition dependent

global stability and wavefronts in a cooperation model with state dependent time delay abstract this article deals with a diffusive cooperative model with state dependent delay which is assumed to be an increasing function of the population density with lower and upper bounds for the cooperative dde system the positivity and boundedness of solutions are firstly given using the comparison principle of the state dependent delay equations obtained the stability criterion of model is analyzed both from local and global points of view when the diffusion is properly introduced the existence of traveling waves is obtained by constructing a pair of upper lower solutions and schauder s fixed point theorem calculating the minimum wave speed shows that the wave is slowed down by the state dependent delay finally the traveling wavefront solutions for large wave speed are also discussed and the fronts appear to be all monotone regardless of the state dependent time delay this is an interesting property since many findings are frequently reported that delay causes a loss of monotonicity with the front developing a prominent hump in some other delay models

on the laplacian spectral radius of weighted trees with fixed matching number q and a positive weight set abstract denote by t n q w 1 w 2 w n 1 the set of n vertex weighted trees with matching number q and fixed positive weight set w n 1 w 1 w 2 w n 1 where w 1 w 2 w n 1 0 tan s w tan on the sharp upper bound of spectral radius of weighted trees j math res exposition 29 2009 293 301 determined the weighted tree in t n q w 1 w 2 w n 1 with the largest adjacent spectral radius whereas in s w tan on the laplacian spectral radius of weighted trees with a positive weight set discrete math 310 2010 1026 1036 tan determined the weighted tree in t n q w 1 w 2 w n 1 with the largest laplacian spectral radius in this paper we use a unified approach to identify the unique weighted tree in t n q w 1 w 2 w n 1 with the largest adjacent spectral radius and largest laplacian spectral radius respectively

global asymptotic behavior of the chemostat general response functions and different removal rates in this paper we consider a competition model between n species in a chemostat that incorporates both monotone and nonmonotone general response functions and distinct removal rates we show that only the species with the lowest break even concentration survives provided that the variation of distinct removal rates relative to the flow rate of the chemostat can be controlled by either the difference between the two lowest break even concentrations or by a parameter based on the structure of response functions lasalle s extension theorem of the lyapunov stability theory and fluctuation lemma are the main tools

analysis of a delayed monod type chemostat model with impulsive input the polluted nutrient in this paper a monod type chemostat model with delayed response in growth and impulsive input the polluted nutrient is considered using the discrete dynamical system determined by the stroboscopic map we obtain a microorganism extinction periodic solution further it is globally attractive the permanent condition of the investigated system is also obtained by the theory of impulsive delay differential equation our results reveal that the delayed response in growth plays an important role on the outcome of the chemostat

global asymptotic behavior of a chemostat model with discrete delays this paper studies the global asymptotic behavior of an exploitative competition model between n species in a chemostat the model incorporates discrete time delays to describe the delay in the conversion of nutrient consumed to viable biomass and hence includes delays simultaneously in variables of nutrient and species concentrations in the case where only two species are engaged in competition it is shown that competitive exclusion holds for any monotone growth response functions sufficient conditions are also obtained for the model to exhibit competitive exclusion in the n species case in regard to the delay effects on the qualitative outcome of competition it is demonstrated that when the delays are relatively small the predictions of the model are identical with the predictions given by corresponding models without time delays however including large delays in the model may alter the predicted outcome of competition the techniques used also work when different removal rates are permitted an

further results on stabilization of periodic trajectories for a chemostat with two species we discuss an important class of problems involving the tracking of prescribed trajectories in the chemostat model we provide new tracking results for chemostats with two species and one limiting substrate based on lyapunov function methods in particular we use a linear feedback control of the dilution rate and an appropriate time varying substrate input concentration to produce a locally exponentially stable oscillatory behavior this means that all trajectories for the nutrient and corresponding species concentrations in the closed loop chemostat that stay near the oscillatory reference trajectory are attracted to the reference trajectory exponentially fast we also obtain a globally stable oscillatory reference trajectory for the species concentrations using a nonlinear feedback control depending on the dilution rate and the substrate input concentration this guarantees that all trajectories for the closed loop chemostat dynamics are attracted to the reference trajectory finally we construct an explicit lyapunov function for the corresponding global error dynamics we demonstrate the efficacy of our method in a simulation

on the spectra of some graphs like weighted rooted trees abstract let g be a weighted rooted graph of k levels such that for j 2 k 1 each vertex at level j is adjacent to one vertex at level j 1 and all edges joining a vertex at level j with a vertex at level j 1 have the same weight where the weight is a positive real number 2 if two vertices at level j are adjacent then they are adjacent to the same vertex at level j 1 and all edges joining two vertices at level j have the same weight 3 two vertices at level j have the same degree 4 there is not a vertex at level j adjacent to others two vertices at the same level we give a complete characterization of the eigenvalues of the laplacian matrix and adjacency matrix of g they are the eigenvalues of leading principal submatrices of two nonnegative symmetric tridiagonal matrices of order k k and the roots of some polynomials related with the characteristic polynomial of the referred submatrices by application of the above mentioned results we derive an upper bound on the largest eigenvalue of a graph defined by a weighted tree and a weighted triangle attached by one of its vertices to a pendant vertex of the tree

global dynamics of a mathematical model of competition in the chemostat general response functions and differential death rates a model of exploitative competition of n species in a chemostat for a single essential nonreproducing growth limiting resource is considered s b hsu siam j appl math 34 1978 pp 760 763 applies lasalle s extension theorem of lyapunov stability theory to study the asymptotic behavior of solutions in the special case that the response functions are modeled by michaelis menten dynamics g j butler and g s k wolkowicz siam j appl math 45 1985 pp 138 151 on the other hand allow more general response functions including monotone and nonmonotone functions but their analysis requires the assumption that the death rates of all the species are negligible in comparison with the washout rate and hence can be ignored by means of lyapunov stability theory the global dynamics of the model for a large class of response functions are studied including both monotone and nonmonotone functions though it is not as general as the class studied by butler and wolkowicz and the results in

pullback invariants of thurston maps associated to a thurston map f s2 s2 with postcritical set p are several di erent invariants obtained via pullback a relation sp f sp on the set sp of free homotopy classes of curves in s2 p a linear operator f r sp r sp on the free r module generated by sp a virtual endomorphism f pmod s2 p 99k pmod s2 p on the pure mapping class group an analytic self map f t s2 p t s2 p of an associated teichmuller space and an analytic self correspondence x y 1 m s2 p m s2 p of an associated moduli space viewing these associated maps as invariants of f we investigate relationships between their properties 1

oscillatory coexistence in the chemostat a codimension two unfolding a nontrivial branch of stable oscillatory solutions of a two predator one prey competition model is found in the positive octant by a multiparameter bifurcation analysis the branch of solutions is global in that it is a continuous branch connecting the oscillations in each of the one predator one prey planes this result gives analytical verification of the numerical observations of other authors that a branch of nonplanar oscillations crosses the positive octant in certain parameter ranges

traveling wave solutions in partially degenerate cooperative reaction diffusion systems abstract we study the existence of traveling wave solutions for partially degenerate cooperative reaction diffusion systems that can have three or more equilibria we show via integral systems that there exist traveling wave solutions in a partially degenerate reaction diffusion system with speeds above two well defined extended real numbers we prove that the two numbers are the same and may be characterized as the spreading speed as well as the slowest speed of a class of traveling wave solutions provided that the linear determinacy conditions are satisfied we demonstrate our theoretical results by examining a partially degenerate lotka volterra competition model with advection terms

global asymptotic behavior of a chemostat model with delayed response in growth this paper studies the global asymptotic behavior of solutions of a competition model between n species in the chemostat the model incorporates time delays that allow the competing organisms to be stored the nutrient so that the free nutrient concentration does not reflect the nutrient available for growth and allow growth response functions to be nonmonotone by introducing three auxiliary functions and using a lemma it is shown that at most one competitor survives and the substrate and the surviving competitor if two exist approach limiting values

renormalizations and wandering jordan curves of rational maps we realize a dynamical decomposition for a post critically finite rational map which admits a combinatorial decomposition we split the riemann sphere into two completely invariant subsets one is a subset of the julia set consisting of uncountably many jordan curve components most of them are wandering the other consists of components that are pullbacks of finitely many renormalizations together with possibly uncountably many points the quotient action on the decomposed pieces is encoded by a dendrite dynamical system we also introduce a surgery procedure to produce post critically finite rational maps with wandering jordan curves and prescribed renormalizations

designing optimal multiplex networks for certain laplacian spectral properties we discuss the design of interlayer edges in a multiplex network under a limited budget with the goal of improving its overall performance we analyze the following three problems separately first we maximize the smallest nonzero eigenvalue also known as the algebraic connectivity second we minimize the largest eigenvalue also known as the spectral radius and finally we minimize the spectral width maximizing the algebraic connectivity requires identical weights on the interlayer edges for budgets less than a threshold value however for larger budgets the optimal weights are generally nonuniform the dual formulation transforms the problem into a graph realization embedding problem that allows us to give a fuller picture namely before the threshold budget the optimal realization is one dimensional with nodes in the same layer embedded to a single point while beyond the threshold the optimal embeddings generally unfold into spaces with dimension bounded by the multiplicity of the algebraic connectivity finally for extremely large budgets the embeddings again revert to lower dimensions minimizing the largest eigenvalue is driven by the spectral radius of the individual networks and its corresponding eigenvector before a threshold the total budget is distributed among interlayer edges corresponding to the nodal lines of this eigenvector and the optimal largest eigenvalue of the laplacian remains constant for larger budgets the weight distribution tends to be almost uniform in the dual picture the optimal graph embedding is one dimensional and nonhomogeneous at first with the nodes corresponding to the layer with the largest spectral radius distributed on a line according to its eigenvector while the other layer is embedded at the origin beyond this threshold the optimal embedding expands to be multidimensional and for larger values of the budget the two layers fill the embedding space finally we show how these two problems are connected to minimizing the spectral width

population dynamics under selection and mutation long time behavior for differential equations in measure spaces abstract we study the long time behavior of solutions to a measure valued selection mutation model that we formulated in 14 we establish permanence results for the full model and we study the limiting behavior even when there is more than one strategy of a given fitness a case that arises in applications we show that for the pure selection case the solution of the dynamical system converges to a dirac measure centered at the fittest strategy class provided that the support of the initial measure contains a fittest strategy thus we term this dirac measure an asymptotically stable strategy we also show that when the strategy space is discrete the selection mutation model with small mutation has a locally asymptotically stable equilibrium that attracts all initial conditions that are positive at the fittest strategy

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

relaxed lyapunov criteria for robust global stabilisation of non linear systems the notion of the restricted robust control lyapunov function rclf is introduced and is exploited for the design of robust feedback stabilisers for non linear systems particularly it is shown for systems with input constraints that restricted rclfs can be easily obtained while rclfs are not available moreover it is shown that the use of restricted rclfs usually results in different feedback designs from the ones obtained by the use of the standard rclf methodology using the restricted rclfs feedback design methodology a simple controller that guarantees robust global stabilisation of a perturbed chemostat model is provided

extinction in two dimensional nonautonomous lotka volterra systems with the effect of toxic substances in this paper we consider a nonautonomous competitive lotka volterra system of two species with the effect of toxic substances it is shown that toxic substances play an important role in the extinction of species we prove that one of components will be driven to extinction while the other will stabilize at a certain solution of a logistic equation under some conditions

a competition model for size structured species the asymptotic dynamics of a system of ordinary differential equations describing the dynamics of nsize structured species competing for a single unstructured resource are studied the system is

global attractivity in a periodic chemostat with general uptake functions a system of ode s is considered as a model of two populations competing for a nutrient where nutrient inputs and chemostat washouts are periodic as well general nutrient uptake functions which may also be time dependent and periodic are considered criteria are derived for there to exist a globally attracting positive periodic solution and a theorem on global monotonicity of attracting solutions between different systems is proved

a stochastic chemostat model with an inhibitor and noise independent of population sizes abstract in this paper a stochastic chemostat model with an inhibitor is considered here the inhibitor is input from an external source and two organisms in chemostat compete for a nutrient firstly we show that the system has a unique global positive solution secondly by constructing some suitable lyapunov functions we investigate that the average in time of the second moment of the solutions of the stochastic model is bounded for a relatively small noise that is the asymptotic behaviors of the stochastic system around the equilibrium points of the deterministic system are studied however the sufficient large noise can make the microorganisms become extinct with probability one although the solutions to the original deterministic model may be persistent finally the obtained analytical results are illustrated by computer simulations

competition in the chemostat global asymptotic behavior of a model with delayed response in growth the asymptotic behavior of solutions of a model for competition between two microorganisms in the chemostat is studied the model incorporates time delays that allow for the cellular components of each competing species to be structured to include unassimilated or stored nutrient sufficient conditions on the parameters of the model that guarantee i competitive exclusion and ii persistence of at least one of the competing organisms are determined the results extend earlier results on the corresponding model for a single species and also extend these to include competition

sensitivity analysis and feedback control of noise induced extinction for competition chemostat model with mutualism abstract we consider a stochastic chemostat model involving two obligate mutualistic species feeding on a limiting substrate it is shown that the spatial arrangement of the random states near the deterministic coexistence equilibrium is enlarged as noise intensity increases more precisely in this paper based on the technique of stochastic sensitivity functions a confidence ellipse of the random states is constructed and a critical value for the noise intensity is established it is shown that a new dynamic phenomenon of noise induced extinction can be observed when the noise intensity passes the critical value we then propose some feedback control strategies which can reduce the size of the confidence ellipse so that to prevent the noise induced extinction

thurston equivalence of topological polynomials we answer hubbard s question on determining the thurston equivalence class of twisted rabbits i e composita of the rabbit polynomial with nth powers of the dehn twists about its ears the answer is expressed in terms of the 4 adic expansion of n we also answer the equivalent question for the other two families of degree 2 topological polynomials with three post critical points in the process we rephrase the questions in group theoretical language in terms of wreath recursions

competition in the chemostat a distributed delay model and its global asymptotic behavior in this paper we propose a two species competition model in a chemostat that uses a distributed delay to model the lag in the process of nutrient conversion and study the global asymptotic behavio

minimum spectral radius of a weighted graph abstract m fiedler recently introduced the following question what is the optimal distribution of nonnegative weights with total sum one among the edges of a given graph so that the spectral radius of the resulting adjacency matrix is minimum he himself has shown that the optimum solution is achieved by some decomposition of the given graph g into a collection of mutually vetex disjoint odd cycles and balanced bipartite subgraphs which maximizes a certain objective function we present a polynomial time algorithm which finds this decomposition our approach is related to matching theory

asymptotic behavior of a stochastic delayed chemostat model with nonmonotone uptake function abstract in this paper a stochastic delay differential equations chemostat model with nonmonotone uptake function is considered and the nutrient conversion process involves time delay first we verify that there is a unique global positive solution of the stochastic system second we find that the solutions of stochastic system will oscillate around the equilibria of the corresponding deterministic model moreover results show that time delay has critical effects on the extinction and persistence of the microorganism that is to say under small noise when the time delay is small microorganism is persistent when the time delay is large microorganism will be extinct in addition we can find by the computer simulation that large noise may lead to microorganism become extinct although microorganism is persistent in the deterministic systems when the time delay is small finally computer simulations are carried out to illustrate the obtained results and the existence of bistability is observed

a delayed chemostat model with general nonmonotone response functions and differential removal rates a chemostat model with general nonmonotone response functions is considered the nutrient conversion process involves time delay we show that under certain conditions when n species compete in the chemostat for a single resource that is allowed to be inhibitory at high concentrations the competitive exclusion principle holds in the case of insignificant death rates the result concerning the attractivity of the single species survival equilibrium already appears in the literature several times see h m el owaidy m ismail asymptotic behavior of the chemostat model with delayed response in growth chaos solitons fractals 13 2002 787 795 h m el owaidy a a moniem asymptotic behavior of a chemostat model with delayed response growth appl math comput 147 2004 147 161 s yuan m han z ma competition in the chemostat convergence of a model with delayed response in growth chaos solitons fractals 17 2003 659 667 however the proofs are all incorrect in this paper we provide a correct proof that also applies in the case of differential death rates in addition we provide a local stability analysis that includes sufficient conditions for the bistability of the single species survival equilibrium and the washout equilibrium thus showing the outcome can be initial condition dependent moreover we show that when the species specific death rates are included damped oscillations may occur even when there is no delay thus the species specific death rates might also account for the damped oscillations in transient behavior observed in experiments

traveling waves in a bio reactor model with stage structure abstract in this paper the existence and non existence of traveling wave solutions are established for a bio reactor model with stage structure the method to prove the existence of wave solutions is to construct an invariant cone of initial functions defined in a large but bounded domain to apply a fixed point theorem on this cone and then extend to the unbounded spatial domain r by a limiting argument

sharp conditions for the existence of a stationary distribution in one classical stochastic chemostat abstract this paper studies the asymptotic behaviors of one classical chemostat model in a stochastic environment based on the feller property sharp conditions are derived for the existence of a stationary distribution by using the mutually exclusive possibilities known in 11 12 see lemma 2 4 for details which closes the gap left by the lyapunov function further we obtain a sufficient condition for the extinction of the organism based on two noise induced parameters an analogue of the feed concentration s and the break even concentration results indicate that both noises have negative effects on persistence of the microorganism

uniform and strict persistence in monotone skew product semiflows with applications to non autonomous nicholson systems abstract we determine sufficient conditions for uniform and strict persistence in the case of skew product semiflows generated by solutions of non autonomous families of cooperative systems of odes or delay fdes in terms of the principal spectrums of some associated linear skew product semiflows which admit a continuous separation our conditions are also necessary in the linear case we apply our results to a noncooperative almost periodic nicholson system with a patch structure whose persistence turns out to be equivalent to the persistence of the linearized system along the null solution

teichmuller theory and critically finite endomorphisms abstract we present a systematic way to generate critically finite endomorphisms of p n these maps arise in the context of teichmuller theory specifically in thurston s topological characterization of rational maps the dynamical objects for the endomorphisms correspond to central objects from thurston s theorem our theorems build infinitely many of these endomorphisms in fact a large number of examples of critically finite endomorphisms of p n found in the literature arise from this construction

simple food chain in a chemostat with distinct removal rates abstract in this paper we consider a model describing predator prey interactions in a chemostat that incorporates both general response functions and distinct removal rates in this case the conservation law fails to overcome this difficulty we make use of a novel way of constructing a lyapunov function in the study of the global stability of a predator free steady state local and global stability of other steady states persistence analysis as well as numerical simulations are also presented our findings are largely in line with those of an identical removal rate case

extinction in periodic competitive stage structured lotka volterra model with the effects of toxic substances in this paper we consider a periodic competitive stage structured lotka volterra model with the effects of toxic substances it is shown that toxic substances play an important role in the extinction of species we obtain a set of sufficient conditions which guarantee that one of the components is driven to extinction while the other is globally attractive the numerical simulation of an example verifies our main results

global asymptotic behavior of a chemostat model with two perfectly complementary resources and distributed delay a model of the chemostat involving two species of microorganisms competing for two perfectly complementary growth limiting nutrients is considered the model incorporates distributed time delay in the form of integral differential equations in order to describe the time involved in converting nutrient to biomass the delays are included in the nutrient and species concentrations simultaneously a general class of monotone increasing functions is used to describe nutrient uptake sufficient conditions based on biologically meaningful parameters in the model are given that predict competitive exclusion for certain parameter ranges and coexistence for others we prove that the global asymptotic attractivity of steady states of the model is similar to that of the corresponding model without time delays however our results indicate that when the inherent delays are in fact large ignoring them may result in incorrect predictions

stability of the boundary solution of a nonautonomous predator prey system with the beddington deangelis functional response abstract the dynamics of a nonautonomous predator prey system with the beddington deangelis functional response is studied from the perspective of extinction of the predator with the help of a fluctuation lemma we obtain a set of new sufficient conditions on the global asymptotic stability of the boundary solution which means the extinction of the predator the result not only improves but also complements some existing ones moreover the result indicates that the effect of the parameters is accumulative rather than pointwise

coexistence in a model of competition in the chemostat incorporating discrete delays a model of two species competing for a single nutrient in the chemostat is proposed incorporating general monotone uptake functions with discrete time delays after transforming the model an anal

stabilization in a two species chemostat with monod growth functions we design feedback controllers for two species chemostats so that an equilibrium with arbitrary prescribed positive species concentrations becomes globally asymptotically stable we use a new global explicit strict lyapunov function construction which allows us to quantify the effects of disturbances using the input to state stability paradigm we assume that only a linear combination of the species concentrations is known we illustrate our approach using a numerical example

global stability of the endemic equilibrium in infinite dimension lyapunov functions and positive operators abstract the global stability of the endemic equilibrium is shown for an endemic model with infinite dimensional population structure using a volterra like lyapunov function and the krein rutman theorem

permanence for an integrodifferential model of mutualism abstract sufficient conditions are obtained for the permanence of the following integrodifferential model of mutualism d n 1 t d t r 1 t n 1 t k 1 t 1 t 0 j 2 s n 2 t s d s 1 0 j 2 s n 2 t s d s n 1 t 1 t d n 2 t d t r 2 t n 2 t k 2 t 2 t 0 j 1 s n 1 t s d s 1 0 j 1 s n 1 t s d s n 2 t 2 t where r i k i i and i i 1 2 are continuous functions bounded above and below by positive constants i k i i 1 2 j i c 0 0 and 0 j i s d s 1 i 1 2

competitive exclusion in delayed chemostat models with differential removal rates competitive exclusion is frequently observed in the chemostat when multiple species compete exploitatively for a single growth limiting nutrient however rigorous mathematical proof for competitive exclusion is very limited especially for delayed chemostat models with differential removal rates in this paper by employing the method of liapunov functionals and placing two assumptions on the response functions we prove that competitive exclusion holds under a generic condition for chemostat models with differential removal rates and discrete also finitely distributed delays for delayed chemostat models with a large variety of widely used response functions including holling types i ii and iii and even some nonmonotone response functions our results show that the competition outcome is completely determined by the species break even concentrations it is the species with the lowest break even concentration that survives in the chemostat and drives other species to extinction

analysis of a model of the glucose insulin regulatory system with two delays we continue a recent attempt to better understand the glucose insulin regulatory system via a mathematical model of delay differential equations with two discrete time delays with explicit delays the model is more realistic in physiology more accurate in mathematics and more robust in applications we study this model analytically and perform carefully designed numerical simulations by allowing two parameters to vary our analytical and numerical results confirm most current existing physiological observations and reveal more insightful information the following factors are critical for ensuring the sustained oscillatory regulation and insulin secretion 1 the time lag for insulin secretion stimulated by glucose and the newly synthesized insulin becoming remote insulin theorem 4 2 b and theorem 5 6 2 the delayed effect of hepatic glucose production theorem 4 2 c and theorem 5 6 3 moderate insulin clearance rate theorem 5 6 and simulations in section 6 4 and 4 nonoverwhelming glucose infusion simulations in section 6 2 6 3 and 6 4

bifurcation analysis of a chemostat model with a distributed delay a chemostat model of a single species feeding on a limiting nutrient supplied at a constant rate is proposed the model incorporates a general nutrient uptake function and a distributed delay the delay indicates that the growth of the species depends on the past concentration of nutrient using the average time delay as a bifurcation parameter it is proven that the model undergoes a sequence of hopf bifurcations stability criteria for the bifurcating periodic solutions are derived it is also found that the periodic solutions become unstable if the dilution rate is increased computer simulations illustrate the results

stability analysis of delay differential equation models of hiv 1 therapy for fighting a virus with another virus considering two kinds of delays accounting respectively for i a latent period between the time target cells are contacted by the virus particles and the time the virions enter the cells and ii a virus production period for new virions to be produced within and released from the infected cells we develop and analyze a mathematical model for hiv 1 therapy by fighting a virus with another virus for the different values of the basic reproduction number and the second basic reproduction number we investigate the stability of the infection free equilibrium the single infection equilibrium and the double infection equilibrium we conclude that increasing delays will decrease the values of the basic reproduction number and the second basic reproduction number our results have potential applications in hiv 1 therapy the approach we use here is a combination of analysis of characteristic equations fluctuation lemma and lyapunov function

a proof of thurston s topological characterization of rational functions the criterion proved in this paper was stated by thurston in november 1982 thurston lectured on its proof on several occasions notably at the nsf summer conference in duluth 1983 where one of the authors jhh was present using the notes of various attendants at these lectures we have reconstructed a proof that we have made as precise as we could since this required a certain amount of work on our part we thought it might be of some use to present this proof to the reader we thank dennis sullivan for useful conversations and fritz yon haesseler and especially ben wittner for help with the writing and valuable suggestions after the first version was written clifford earle pointed out that better estimates than what we had were to be found in b

existence of traveling wave solutions for influenza model with treatment abstract to investigate the spreading speed of influenza and the influence of treatment on the spreading speed a reaction diffusion influenza model with treatment is established the existence of traveling wave solutions is shown by introducing an auxiliary system and applying the schauder fixed point theorem the non existence of traveling wave solutions is proved by a two sided laplace transform which needs a new approach for the prior estimate of exponential decay of traveling wave solutions

the interaction of steady state and hopf bifurcations in a two predator one prey competition model the existence of a stable periodic solution in the positive octant of a two predator one prey competition model is shown to result from the interaction of a steady state bifurcation with a hopf bifurcation this result confirms the conjectures based on numerical simulations of several authors that coexistence in the form of periodic oscillations in all three population sizes takes place for some parameter values of the model

stability analysis of a bioreactor model for biodegradation of xenobiotics we consider an ecological model for biodegradation of toxic substances in aquatic and atmospheric biotic systems the model which is described by a nonlinear system of four ordinary differential equations is known to be experimentally validated we compute the equilibrium points of the model and study their asymptotic stability basic properties of the solutions like uniform boundedness and uniform persistence are established global asymptotic results are also developed numerical simulation results are presented to demonstrate the theoretical studies