generalized optimal designs and contingency plans in linear systems abstract to select an optimal linear design means to select the best subset from a set of possible products or opportunities for commitment and to construct the corresponding optimal contingency plans for coping with the changes of decision parameters based on lee shi and yu this paper discusses a method for generating generalized good designs from known potentially good designs we describe how to use the multi criteria and multi constraint level mc 2 simplex method to identify the generalized optimal designs and their related optimal contingency plans a procedure for effectively and systematically solving the problem of selecting linear designs is proposed

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

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

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

fuzzy potential solutions in multi criteria and multi constraint level linear programming problems abstract given a multi criteria and multi constraint level mc 2 linear programming problem we can use an mc 2 simplex method to find a set of potential solutions each potential solution optimizes the mc 2 problem under a certain range of decision parameters namely the criteria weight vector and the constraint level weight vector in this paper we extend the potential solution to a fuzzy potential solution for a given fuzzy mc 2 problem with a certain range of the decision parameters a membership function of the mc 2 problem is first constructed then for such a parameter range a fuzzy potential solution is determined in terms of the membership function and a primal potential basis based on the theoretical analysis we propose a heuristic algorithm to systematically and effectively locate all fuzzy potential solutions for all possible changes of the decision parameters a numerical example is used to illustrate the algorithm

computation of feedback control for time optimal state transfer using groebner basis abstract computation of time optimal feedback control law for a controllable linear time invariant system with bounded inputs is considered unlike a recent paper by the authors the target final state is not limited to the origin of state space but is allowed to be in the set of constrained controllable states switching surfaces are formulated as semi algebraic sets using groebner basis based elimination theory using these semi algebraic sets a nested switching logic is synthesized to generate the time optimal feedback control however the optimal control law enforces an unavoidable limit cycle in the time optimal trajectory for most non origin target points the time period of this limit cycle is dependent on the target position this dependence is algebraically characterized and a method to compute the time period of the limit cycle is provided as a natural extension the set of constrained controllable states is also computed

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

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 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

decentralized control via grobner bases and variable elimination we consider the problem of optimal decentralized controller synthesis there are several classes of such problems for which effective solution methods are known including the quadratically invariant one in this technical note we use grobner bases and elimination theory to characterize all closed loop maps achievable by forming a feedback loop with decentralized controllers we show that this approach allows solution of a wide class of optimal decentralized control problems it includes not only quadratically invariant problems under a technical condition but also some other problems which are not quadratically invariant

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

competition in the unstirred chemostat with periodic input and washout the model of an unstirred chemostat is generalized to that of a chemostat with time dependent input washout rates the novelty of the new model is that time periodicity appears in the boundary conditions the asymptotic dynamics of the competition between two microbial populations is determined in terms of the corresponding period map which is shown to preserve the standard competitive ordering it is shown that the dynamics of competition is similar to that of a chemostat with constant boundary conditions simple criteria for coexistence versus competitive exclusion are presented 1 introduction the chemostat represents a basic model of an open system in microbial ecology in its simplest form it consists of three vessels the rst called the feed bottle contains medium with all of the nutrients needed for growth in sur plus except one which hereafter is simply called the nutrient the contents of the feed bottle are pumped at a constant rate into the second vessel called the culture vessel or bioreactor the culture vessel is charged with one or more populations of microorganisms the contents of the culture vessel are pumped into the remaining vessel called the over ow vessel at a constant rate keeping the volume of the reactor constant the organisms compete for the nutrient in a purely exploitative manner basic assumptions include that the vessel is well mixed and that all other parame ters ph temperature etc are strictly controlled the ow rate is assumed to be sucient to preclude wall growth or the accumulation of metabolic products let s t denote the concentration of the nutrient in the culture vessel and xi t i 1 2 denote the concentration of the competitors lets 0 denote the concentration of the input nutrient and let d denote the dilution rate ow rate volume if growth is assumed to be proportional to consumption then the basic equations take the form

positive solutions of a competition model for two resources in the unstirred chemostat this paper deals with a competition model between two species for two growth limiting and perfectly complementary resources in the unstirred chemostat the main purpose is to determine the exact range of the parameters of two species so that the system possesses positive solutions and to investigate multiple positive steady states of the system the main tools used here include the monotone methods and the topological fixed point theory developed by amann

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

generating new designs using union operations abstract in many real world problems we want to generate new designs ideas by taking unions of subsets of a given basic set of potentially good designs primitives this paper proposes a streamlined algorithm of finding distinct unions of subsets of the primitives we show how to use the algorithm to eliminate a number of redundant unions of subsets of the primitives in the process of finding all distinct unions theoretical results are derived to justify the algorithm applications of the algorithm are also discussed

a characterization of convex problems in decentralized control ast we consider the problem of constructing optimal decentralized controllers we formulate this problem as one of minimizing the closed loop norm of a feedback system subject to constraints on the controller structure we define the notion of quadratic invariance of a constraint set with respect to a system and show that if the constraint set has this property then the constrained minimum norm problem may be solved via convex programming we also show that quadratic invariance is necessary and sufficient for the constraint set to be preserved under feedback these results are developed in a very general framework and are shown to hold in both continuous and discrete time for both stable and unstable systems and for any norm this notion unifies many previous results identifying specific tractable decentralized control problems and delineates the largest known class of convex problems in decentralized control as an example we show that optimal stabilizing controllers may be efficiently computed in the case where distributed controllers can communicate faster than their dynamics propagate we also show that symmetric synthesis is included in this classification and provide a test for sparsity constraints to be quadratically invariant and thus amenable to convex synthesis

a fuzzy programming approach for solving a multiple criteria and multiple constraint level linear programming problem abstract this paper presents a solution method to the multiple criteria and multiple constraint level linear program mc2lp using a fuzzy linear programming approach this approach finds a potential solution to the problem by solving a fuzzy linear program which is different than the usual approach of mc2 simplex method

hopf bifurcation analysis of a turbidostat model with discrete delay in this contribution the dynamic behaviors of a turbidostat model with discrete delay are investigated with the delay of digestion chosen to be a bifurcation parameter we find that hopf bifurcations occur when the delay crosses some critical values also by employing the normal form theory and the center manifold theorem we determine the type and stability of the bifurcating periodic solutions finally numerical simulations are offered to support our results

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

a pivotal eigenvalue problem in river ecology this paper studies an eigenvalue problem associated with a linear parabolic equation and a coupled ordinary differential equation the existence and the uniqueness of the principal eigenvalue for this eigenvalue problem is first established then the qualitative dependence of the principal eigenvalue with respect to the several parameters involved in the system is analyzed finally these results are applied to a system in flowing habitats with a hydraulic storage zone and light limitation

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

a food chain model for two resources in un stirred chemostat this paper deals with a multiple food chain model in un stirred chemostat where the prey feeds on two growth limiting nonreproducing resources the conditions for existence of positive steady states are established the global bifurcation of positive solutions is considered also furthermore we obtain the conditions under which the predator possesses uniformly strong persistence or goes to extinction the techniques used in this paper include the degree theory the global bifurcation theory the semigroup theory and the maximum principle

computation of time optimal feedback control using groebner basis the synthesis of time optimal feedback control of a single input continuous time linear time invariant lti system is considered if the control input u t is constrained to obey u t 1 then it is known that the optimal input switches between the extreme values 1 according to some switching surfaces in the state space it is shown that for systems with non zero distinct and rational eigenvalues these switching surfaces are semi algebraic sets and a method to compute them using groebner basis is proposed in the process the null controllable region for such systems is characterized and computed numerical simulations illustrate the proposed computational methods

evolution of dispersal in advective homogeneous environment the effect of boundary conditions abstract we consider a single species model and a two species competition model in a one dimensional advective homogeneous environment one interesting feature in these models concerns the boundary condition at the downstream end where the species can be exposed to a net loss of individuals as tuned by a parameter b which measures the magnitude of the loss we first determine necessary and sufficient conditions for the persistence of a single species for general value of b in terms of the critical habitat size and the critical advection rate then for the competition model we assume that two species are identical except their random diffusion rates we obtain complete understanding when 0 b 1 and our result indicates that larger diffusion rate is selected extending an earlier work 20 b 1 however for b 1 the dynamics can be quite different and particularly we illustrate that some intermediate diffusion rate may be selected when b 3 2

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

convexity of decentralized controller synthesis in decentralized control problems a standard approach is to specify the set of allowable decentralized controllers as a closed subspace of linear operators this then induces a corresponding set of youla kucera parameters previous work has shown that quadratic invariance of the controller set implies that the set of youla kucera parameters is convex in this technical note we prove the converse we thereby show that the only decentralized control problems for which the set of youla kucera parameters is convex are those which are quadratically invariant we further show that under additional assumptions quadratic invariance is necessary and sufficient for the set of achievable closed loop maps to be convex we give two versions of our results the first applies to bounded linear operators on a banach space and the second applies to possibly unstable causal lti systems in discrete or continuous time

the evolution of traveling waves in a simple isothermal chemical system modeling quadratic autocatalysis with strong decay abstract in this paper we study a reaction diffusion system for an isothermal chemical reaction scheme governed by a quadratic autocatalytic step a b 2 b and a decay step b c where a b and c are the reactant the autocatalyst and the inner product respectively previous numerical studies and experimental evidences demonstrate that if the autocatalyst is introduced locally into this autocatalytic reaction system where the reactant a initially distributes uniformly in the whole space then a pair of waves will be generated and will propagate outwards from the initial reaction zone one crucial feature of this phenomenon is that for the strong decay case the formation of waves is independent of the amount of the autocatalyst b introduced into the system it is this phenomenon of kpp type which we would like to address in this paper to study the propagation of reactant and autocatalyst analytically we first use the tail behavior of waves to construct a pair of generalized super sub solutions for the approximate system of the autocatalytic reaction system note that the autocatalytic reaction system does not enjoy comparison principle together with a family of truncated problems we can establish the existence of a family of traveling waves with the minimal speed second we use this pair of generalized super sub solutions to show that the propagation of waves is fully determined by the rate of decay of the initial data at infinity in the sense of aronson weinberger formulation which in turn confirms the aforementioned numerical and experimental results

linear optimal designs and optimal contingency plans in optimal design problems we try to select a subset from the opportunity set for commitment for a period of time as the external parameters can vary with time optimal contingency plans must be constructed to cope with the external conditions we describe the formulation and methods for locating potentially good designs and their corresponding optimal contingency plans using a multiple criteria and multiple constraint level mc2 simplex method to solve the optimal design problems which have been shown to become amenable to the techniques of decision making under uncertainty

a competition model for two resources in un stirred chemostat abstract this paper studies a un stirred chemostat with two species competing for two growth limiting non reproducing resources we determine the conditions for positive steady states of the two species and then consider the global attractors of the model in addition we obtain the conditions under which the two populations uniformly strongly persist or go to extinction since the diffusion mechanism with homogeneous boundary conditions inhibits the growth of the organism species it can be understood that the coexistence will be ensured by proportionally smaller diffusions for the two species in particular it is found that both instability and bi stability subcases of the two semitrivial steady states are included in the coexistence region the two populations will go to extinction when both possess large diffusion rates if just one of them spreads faster with the other one diffusing slower then the related semitrivial steady state will be globally attracting the techniques used for the above results consist of the degree theory the semigroup theory and the maximum principle

traveling waves solutions of isothermal chemical systems with decay this article studies propagating traveling waves in a class of reaction diffusion systems which include a model of microbial growth and competition in a flow reactor proposed by smith and zhao 17 and isothermal autocatalytic systems in chemical reaction of order m with a decay order n where m and n are positive integers and m n a typical system in autocatalysis is a 2b 3b with rate k1ab2 and b c with rate k2b where m 2 and n 1 involving two chemical species a reactant a and an auto catalyst b whose diffusion coefficients da and db are unequal due to different molecular weights and or sizes here a is the concentration density of a b that of b and c an inert chemical species the two constants k1 and k2 are material constants measuring the relative strength of respective reactions it is shown that there exist traveling waves when m 1 and n 1 with suitable relation between the ratio db da traveling speed c and rate constants k1 k2 on the other hand it is proved that there exists no traveling wave when one of the chemical species is immobile db 0 or n m for all choices of other parameters

on a system of reaction diffusion equations arising from competition in an unstirred chemostat a system of reaction diffusion equations modeling two species competition in an unstirred chemostat is considered the asymptotic behavior of solutions is given as a function of the parameters and it is determined when neither one or both competing populations survive techniques include the maximum principle theory of uniform persistence in infinite dimensional dynamical systems and the theory of strongly order preserving semidynamical system

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

dynamics of a periodically pulsed bio reactor model abstract global attractivity and uniform persistence are established for both single species growth and two species competition in a periodically pulsed bio reactor model in terms of principal eigenvalues of the periodic parabolic eigenvalue problem by appealing to the theories of monotone discrete dynamical systems abstract persistence asymptotically periodic semiflows and perturbation of global attractors