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

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

variable threshold concept lattices in this paper the definition of a variable threshold concept lattice is introduced based on a galois connection three kinds of variable threshold concept lattices in which diverse requirements of knowledge discovery can be satisfied by adjusting a threshold are defined the number of formal concepts in a variable threshold concept lattice is far less than that in a fuzzy concept lattice the three kinds of variable threshold concept lattices are constructed between two crisp sets between a crisp set and a fuzzy set and between a fuzzy set and a crisp set their properties are analogous to that of the classical concept lattices and can be induced by the fuzzy concept lattice

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

t prefilter theory abstract in the first part of this paper the notion of t prefilter is introduced and studied as are the applications t and t their properties and relations between them we characterize the minimal prime t prefilters finer than a given t prefilter by using ultrafilters on x and we give equivalents conditions of the t maximality in the second part we define t convergence and we obtain a characterization of t compactness gantner steinlage by means of the maximal t prefilters finally some applications of these concepts related to the continuity separation axioms and fuzzy compactness are studied

a new approach to fuzzy bitopological spaces abstract given a fuzzy bitopological space x 1 2 we have used the operator c12 ix ix defined by c 12 1 cl 2 cl i x which is a supra fuzzy closure operator fuzzy sets and systems 74 1995 353 to generate a family s which is a supra fuzzy topology loc cit 1995 so the space x s is a supra fuzzy topological space associated to the fuzzy bitopological space x 1 2 we extend the notions of fti strong fti and ultra fti due to fuzzy sets and system 43 1991 95 to the space x s the properties of fuzzy bitopological spaces have been studied by using their associated supra fuzzy topological spaces we introduce the notions of fp ti s fp ti and u fp ti by using the level supratopology s of s we investigate the relationship between these notions the notions fp ti loc cit 1995 i 1 2 3 4 and the notions fpti resp s fpti u fpti fuzzy sets and systems 105 1999 459 we give a number of examples which illustrate that these concepts are not equivalent also we extend the notions of f continuous open strong f continuous open and ultra f continuous open due to fuzzy sets and systems 38 1990 115 to the class of supra fuzzy topological spaces associated to the class of fuzzy bitopological spaces we study some properties of fp ti resp s fp ti u fp ti under these mappings

a multi level conceptual data reduction approach based on the lukasiewicz implication starting from fuzzy binary data represented as tables in the fuzzy relational database in this paper we use fuzzy formal concept analysis to reduce the tables size to only keep the minimal rows in each table without losing knowledge i e association rules extracted from reduced databases are identical at given precision level more specifically we develop a fuzzy extension of a previously proposed algorithm for crisp data reduction without loss of knowledge the fuzzy galois connection based on the lukasiewicz implication is mainly used in the definition of the closure operator according to a precision level which makes data reduction sensitive to the variation of this precision level

a fast approach to attribute reduction in incomplete decision systems with tolerance relation based rough sets efficient attribute reduction in large incomplete decision systems is a challenging problem existing approaches have time complexities no less than o c 2 u 2 this paper derives some important properties of incomplete information systems then constructs a positive region based algorithm to solve the attribute reduction problem with a time complexity no more than o c 2 u log u furthermore our approach does not change the size of the original incomplete system numerical experiments show that the proposed approach is indeed efficient and therefore of practical value to many real world problems the proposed algorithm can be applied to both consistent and inconsistent incomplete decision systems

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

strong and ultra separation axioms of fuzzy bitopological spaces given a fuzzy bitopological space x 1 2 we introduce a new notion of fuzzy pairwise separation axioms by using the family of its level bitopologies 1 2 0 1 we prove that these concepts are good extension and we compare them with its corresponding fpt kandil and ei shafee 1991 and fpt1 abu safiya et al 1994 i 0 1 2 3 4 respectively we show that these notions are not equivalent and we give a number of examples which illustrate this fact

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

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

towards the theory of m approximate systems fundamentals and examples the concept of an m approximate system is introduced basic properties of the category of m approximate systems and in a natural way defined morphisms between them are studied it is shown that categories related to fuzzy topology as well as categories related to rough sets can be described as special subcategories of the category of m approximate systems

a heuristic knowledge reduction method for decision formal contexts computing a minimal reduct of a decision formal context by boolean reasoning is an np hard problem thus it is essential to develop some heuristic methods to deal with the issue of knowledge reduction especially for large decision formal contexts in this study we first investigate the relationship between the concept lattice of a formal context and those of its subcontexts in preparation for deriving a heuristic knowledge reduction method then we construct a new framework of knowledge reduction in which the capacity of one concept lattice implying another is defined to measure the significance of the attributes in a consistent decision formal context based on this reduction framework we formulate an algorithm of searching for a minimal reduct of a consistent decision formal context it is proved that this algorithm is complete and its time complexity is polynomial some numerical experiments demonstrate that the algorithm can generally obtain a minimal reduct and is much more efficient than some boolean reasoning based methods

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

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

attribute reduction based on evidence theory in incomplete decision systems attribute reduction is a basic issue in knowledge representation and data mining this paper deals with attribute reduction in incomplete information systems and incomplete decision systems based on dempster shafer theory of evidence the concepts of plausibility reduct and belief reduct in incomplete information systems as well as relative plausibility reduct and relative belief reduct in incomplete decision systems are introduced it is shown that in an incomplete information system an attribute set is a belief reduct if and only if it is a classical reduct and a plausibility consistent set must be a classical consistent set in a consistent incomplete decision system the concepts of relative reduct relative plausibility reduct and relative belief reduct are all equivalent in an inconsistent incomplete decision system an attribute set is a relative plausibility reduct if and only if it is a relative reduct a plausibility consistent set must be a belief consistent set and a belief consistent set is not a plausibility consistent set in general

concept lattices and conceptual knowledge systems abstract concept lattice is the central notion of formal concept analysis a new area of research which is based on a set theoretical model for concepts and conceptual hierarchies this model yields not only a new approach to data analysis but also methods for formal representation of conceptual knowledge these methods are outlined on three levels first basics on concept lattices are explained starting from simple data contexts which consist of a binary relation between objects and attributes indicating which object has which attribute on the second level conceptual relationships are discussed for data matrices which assign attribute values to each of the given objects finally a mathematical model for conceptual knowledge systems is described this model allows us to study mathematically the representation inference acquisition and communication of conceptual knowledge

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

relations of reduction between covering generalized rough sets and concept lattices the reduction theory plays an important role in data analysis this paper studies the relation between the reduction of a covering and the attribute reduction of a concept lattice the reduction of a covering from the perspective of concept lattices is investigated conversely the attribute reduction of a formal context is studied in the framework of covering generalized rough sets the results in this paper show that the reduction of a covering can be viewed as the attribute reduction of a derivative formal context moreover every reduct of a given formal context can be seen as the reduct of an induced covering as an application of the theoretical results an approach to the attribute reduction of concept lattices based on covering generalized rough sets is proposed furthermore experiments are given to show the effectiveness of the proposed method

reduction method for concept lattices based on rough set theory and its application rough set theory and formal concept analysis are two complementary mathematical tools for data analysis in this paper we study the reduction of the concept lattices based on rough set theory and propose two kinds of reduction methods for the above concept lattices first we present the sufficient and necessary conditions for justifying whether an attribute and an object are dispensable or indispensable in the above concept lattices based on the above justifying conditions we propose a kind of multi step attribute reduction method and object reduction method for the concept lattices respectively then on the basis of the defined discernibility functions of the concept lattices we propose a kind of single step reduction method for the concept lattices additionally the relations between the attribute reduction of the concept lattices in fca and the attribute reduction of the information system in rough set theory are discussed in detail at last we apply the above multi step attribute reduction method for the concept lattices based on rough set theory to the reduction of the redundant premises of the multiple rules used in the job shop scheduling problem the numerical computational results show that the reduction method for the concept lattices is effective in the reduction of the multiple rules

on induced fuzzy topological spaces the concept of induced fuzzy topological spaces was introduced by weiss 11 subsequently lowen 2 31 defined two mappings o and i which established the relationship between fuzzy topological spaces and topological spaces and proved some deeper results on induced fuzzy topological spaces since then several other authors continued the investigation of such spaces in this paper we will use the decomposition theorem for fuzzy sets to simplify the proofs of some theorems due to lowen and discuss the closure and interior operators separability ci c and separation properties of induced fuzzy topological spaces

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

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

strong separation and strong countability in fuzzy topological spaces given a fuzzy topological space x we introduce new notions of fuzzy separation and fuzzy countability using the family of its level topologies t t 0 1 we check that these are well defined fuzzy topological concepts and we compare them with the analogous fuzzy ones introduced in the literature we verify that these notions are not equivalent and we give a large number of examples which illustrate this fact

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

reductions in a knowledge base one of the advantages of rough set theory is the fact that an unknown target concept can be approximately characterized based on existing knowledge structures in a knowledge base knowledge reductions are one of the key problems in the study of rough set theory in this paper reductions and their features in a knowledge base are explored moreover communication between knowledge bases is proposed based on the use of homomorphisms between relation information systems and several invariant characterizations under homomorphisms are obtained these results will be significant for establishing a framework for granular computing in knowledge bases

smooth topological spaces abstract in 1986 badard introduced the concept of a smooth topological space in this paper we give some links between smooth topological spaces and the corresponding results in chang s fuzzy topological spaces the concepts of subspace continuity and compactness are studied

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

fuzzy topology fuzzy closure operator fuzzy compactness and fuzzy connectedness abstract in this paper definitions of fuzzy closure operator fuzzy compactness and fuzzy connectedness are given in a redefined fuzzy topological space and then some characteristic properties of fuzzy closure operator product theorems of fuzzy compactness invariance of fuzzy compactness and fuzzy connectedness under gradation preserving maps and invariance of fuzzy connectedness under the fuzzy closure operator are studied

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

gradation of openness fuzzy topology abstract in an earlier paper we have given a new definition of fuzzy topology by introducing a concept of gradation of openness of fuzzy subsets in order to make the concept more appropriate in this paper we modify the definition of gradation function and then study subspace of fuzzy topological spaces gradation preserving maps and category of fuzzy topological spaces and gradation preserving maps

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

a note on variable threshold concept lattices threshold based operators are reducible to classical concept forming operators the present paper deals with formal concept analysis of data with fuzzy attributes we clarify several points of a new approach of s q fan w x zhang variable threshold concept lattice inf sci accepted for publication which is based on using thresholds in concept forming operators we show that the extent and intent forming operators from s q fan w x zhang inf sci accepted for publication can be defined in terms of basic fuzzy set operations and the original operators as introduced and studied e g in r belohlavek fuzzy galois connections math logic quarterly 45 4 1999 497 504 r belohlavek concept lattices and order in fuzzy logic ann pure appl logic 128 2004 277 298 s pollandt fuzzy begriffe springer verlag berlin heidelberg 1997 as a consequence main properties of the new operators from s q fan w x zhang inf sci accepted for publication including the properties studied in s q fan w x zhang inf sci accepted for publication can be obtained as consequences of the original operators from r belohlavek 1999 r belohlavek 2004 s pollandt 1997

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

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

on fuzzy bitopological spaces abstract in this paper we have used the fuzzy topologies 1 and 2 to generate a family s which is a supra fuzzy topology on x using the family s we introduce and study the concepts of separation axioms continuity resp openness closedness of a mapping and compactness for a fuzzy bitopological space x 1 2 our definitions preserve much of the correspondence between concepts of fuzzy bitopological spaces and the associated fuzzy topological spaces we then investigate the relationship between these concepts and their correspondence with the fuzzy bitopological spaces kandil and el shafee 1991

regularity axioms in fuzzy topological spaces and fr i proximities abstract the purpose of this paper is to study the regularity axioms for the proximity notion in fuzzy topology we redefine the notion of cech proximity on i x which has been extended to fuzzy sets in 11 and we study the families m x c and m 1 x also we show that every fr i space can be derived from an r i proximity on i x and vice versa

approaches to knowledge reduction based on variable precision rough set model this paper deals with approaches to knowledge reduction based on variable precision rough set model the concepts of lower distribution reduct and upper distribution reduct based on variable precision rough sets vprs are first introduced their equivalent definitions are then given and the relationships among lower and upper distribution reducts and alternative types of knowledge reduction in inconsistent systems are investigated it is proved that for some special thresholds lower distribution reduct is equivalent to the maximum distribution reduct whereas upper distribution reduct is equivalent to the possible reduct the judgement theorems and discernibility matrices associated with the lower and upper distribution reducts are also established from which we can obtain the approaches to knowledge reduction in vprs