unimodal dynamical systems comparison principles spreading speeds and travelling waves abstract reaction diffusion equations with delayed nonlinear reaction terms are used as prototypes to motivate an appropriate abstract formulation of dynamical systems with unimodal nonlinearity for such non monotone dynamical systems we develop a general comparison principle and show how this general comparison principle coupled with some existing results for monotone dynamical systems can be used to establish results on the asymptotic speeds of spread and travelling waves we illustrate our main results by an integral equation which includes a nonlocal delayed reaction diffusion equation and a nonlocal delayed lattice differential system in an unbounded domain with the non monotone nonlinearities including the ricker birth function and the mackey glass hematopoiesis feedback

endemic bubbles generated by delayed behavioral response global stability and bifurcation switches in an sis model during infectious disease outbreaks people may reduce their contact numbers or take other precautions to prevent transmission the change in their behavior can be directly or indirectly triggered by the density of infected individuals in the population in this paper we investigate an sis susceptible infected susceptible model where the transmission rate is a decreasing function of the prevalence of the disease determined by a reduction function h with the assumption that such a change in the transmission rate occurs with some time delay we prove that if the basic reproduction number r 0 is less than one then the disease free equilibrium is globally asymptotically stable while for r 0 1 a unique endemic equilibrium exists and the disease uniformly persists regardless of the delay or the specific form of h however characterized by the shape of the response function h various dynamics are possible if r 0 1 roughly speaking if h is decreasing slowly weak response then the end

global dynamics in a commodity market model a b s t r a c t we study the global behavior of the price dynamics in a commodity market governed by a balance between demand and supply while the dependence of demand on price is considered instantaneous the supply term contains a delay leading to a delay differential equation a discrete model is naturally defined as a limit case of this equation we provide a thorough study of the discrete case and use these results to get new sufficient conditions for the global convergence of the solutions to the positive equilibrium in the continuous case for when the equilibrium is unstable we provide some bounds for the amplitude of the oscillations that are quite sharp when the delay is large

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

map dynamics versus dynamics of associated delay reaction diffusion equations with a neumann condition in this paper we consider a class of delay reaction diffusion equations drdes with a parameter e 0 a homogeneous neumann boundary condition and non negative initial functions are posed to the equation by letting such an equation is formally reduced to a scalar difference equation or map dynamical system the main concern is the relation of the absolute or delay independent global stability of a steady state of the equation and the dynamics of the nonlinear map in the equation by employing the idea of attracting intervals for solution semiflows of the drdes we prove that the globally stable dynamics of the map indeed ensures the delay independent global stability of a constant steady state of the drdes we also give a counterexample to show that the delay independent global stability of drdes cannot guarantee the globally stable dynamics of the map finally we apply the abstract results to the diffusive delay nicholson blowfly equation and the diffusive mackey glass haematopoiesis equation the resulting criteria for both model equations are amazingly simple and are optimal in some sense although there is no existing result to compare with for the latter

dynamics of a class of delayed reaction diffusion systems with neumann boundary condition abstract this paper considers a class of delayed reaction diffusion systems under the neumann boundary condition which arise in epidemiology and can describe the temporal and spatial evolutionary phenomena for the bacteria population and the human infective population with the help of the iterative properties of interval mapping and dynamical system approaches some positively invariant sets and attractive basins of the considered systems are analyzed detailedly in addition combining the global attractivity of interval mapping we provide some sufficient conditions to ensure local or global attractivity of steady states of the systems finally we apply these theoretical results to some models with different nonlinearity demonstrating force of infection and then obtain some sufficient conditions about force of infection to describe the survival and extinction of bacteria and infective populations

the poincare bendixson theorem for monotone cyclic feedback systems with delay abstract we consider cyclic nearest neighbor systems of differential delay equations in which the coupling between neighbors possesses a monotonicity property using a discrete integer valued lyapunov function we prove that the poincare bendixson theorem holds for such systems we also obtain results on piecewise monotonicity and stability of periodic solutions of such systems

domain decomposition method for the global dynamics of delay differential equations with unimodal feedback x t zkmx t c f x tkt with unimodal feedback is studied the existence of the global attractor is shown and bounds of the attractor are given we find attractive invariant intervals and give sufficient conditions that guarantee that all solutions enter the domain where f 0 is negative with respect to a positive equilibrium so the results for delayed monotone feedback can be applied to describe the asymptotic behaviour of solutions in particular the existence of heteroclinic orbits from the trivial equilibrium to a periodic orbit oscillating around the positive equilibrium is established numerical examples using nicholson s blowflies equation and the mackey glass equation are provided to illustrate the main results

mackey glass model of hematopoiesis with non monotone feedback stability oscillation and control for the blood cell production model with a unimodal hump feedback email protected t email protected q ny email protected q n y n email protected we review the known results and investigate generalizations of this equation permanence oscillation and stability of the positive equilibrium are studied for non autonomous equations including equations with a distributed delay in addition a linear control is introduced and possibilities to stabilize an otherwise unstable positive equilibrium are explored

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

domain decomposition methods for a class of spatially heterogeneous delayed reaction diffusion equations abstract we derive and study a class of delayed reaction diffusion equations with spatial heterogeneity which models the population of a single species with different habitats for mature and immature individuals we introduce new solid cones obtain spectral bounds of several spatial heterogeneous operators and establish limiting non negativeness property for the whole space and the eventual comparison principle for bounded domains as a result we develop new domain decomposition methods so that one can compare solutions with those to associated equations from a suitable bounded spatial domain to the whole space then by employing domain decomposition methods and dynamical system approaches we obtain threshold results under the supremum norm these results are greatly different from the existing ones of other evolution equations in unbounded domains or the whole space the main results are applied to two examples with the ricker birth function and with the mackey glass birth function it reveals that the size of the immature habitat can affect the reproduction and spread of the population

attractivity multistability and bifurcation in delayed hopfieldʼs model with non monotonic feedback abstract for a system of delayed neural networks of hopfield type we deal with the study of global attractivity multistability and bifurcations in general we do not assume monotonicity conditions in the activation functions for some architectures of the network and for some families of activation functions we get optimal results on global attractivity our approach relies on a link between a system of functional differential equations and a finite dimensional discrete dynamical system for it we introduce the notion of strong attractor for a discrete dynamical system which is more restrictive than the usual concept of attractor when the dimension of the system is higher than one our principal result shows that a strong attractor of a discrete map gives a globally attractive equilibrium of a corresponding system of delay differential equations our abstract setting is not limited to applications in systems of neural networks we illustrate its use in an equation with distributed delay motivated by biological models we also obtain some results for neural systems with variable coefficients

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

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

symmetric functional differential equations and neural networks with memory we establish an analytic local hopf bifurcation theorem and a topological global hopf bifurcation theorem to detect the existence and to describe the spatial temporal pattern the asymptotic form and the global continuation of bifurcations of periodic wave solutions for functional differential equations in the presence of symmetry we apply these general results to obtain the coexistence of multiple large amplitude wave solutions for the delayed hopfield cohen grossberg model of neural networks with a symmetric circulant connection matrix

global dynamics of a delay differential equation with spatial non locality in an unbounded domain abstract in this paper we study the global dynamics of a class of differential equations with temporal delay and spatial non locality in an unbounded domain adopting the compact open topology we describe the delicate asymptotic properties of the nonlocal delayed effect and establish some a priori estimate for nontrivial solutions which enables us to show the permanence of the equation combining these results with a dynamical systems approach we determine the global dynamics of the equation under appropriate conditions applying the main results to the model with ricker s birth function and mackey glass s hematopoiesis function we obtain threshold results for the global dynamics of these two models we explain why our results on the global attractivity of the positive equilibrium in c 0 under the compact open topology becomes invalid in c 0 with respect to the usual supremum norm and we identify a subset of c 0 in which the positive equilibrium remains attractive with respect to the supremum norm

instabilities in threshold diffusion equations with delay the introduction of delays into ordinary or partial differential equation models is well known to facilitate the production of rich dynamics ranging from periodic solutions through to spatio temporal chaos in this paper we consider a class of scalar partial differential equations with a delayed threshold nonlinearity which admits exact solutions for equilibria periodic orbits and travelling waves importantly we show how the spectra of periodic and travelling wave solutions can be determined in terms of the zeros of a complex analytic function using this as a computational tool to determine stability we show that delays can have very different effects on threshold systems with negative as opposed to positive feedback direct numerical simulations are used to confirm our bifurcation analysis and to probe some of the rich behaviour possible for mixed feedback

sis model on homogeneous networks with threshold type delayed contact reduction we study the dynamics of a delayed sis epidemic model on homogeneous networks where it is assumed that individuals modify their contact patterns upon realizing the risk of infection this decision is made with some time delay and it is threshold type when the density of infected nodes reaches a critical value the number of links is reduced by a given factor such assumptions lead to a delay differential equation with discontinuous right hand side we show that if the basic reproduction number r 0 1 then the disease will be eradicated while it persists for r 0 1 in the latter case there is a globally asymptotically stable endemic equilibrium except for a crucial interval of reproduction numbers where the system shows oscillations we construct explicitly the unique slowly oscillatory periodic solution which has strong attractivity properties and show the existence of rapidly oscillatory periodic solutions with any frequency the amplitude of the oscillations is determined by the time delay our results indicate that with such information delays the link density of a network has an important effect on the qualitative dynamics of infectious diseases

stable coexistence and bi stability for competitive systems on ordered banach spaces abstract autonomous and asymptotically autonomous semiflows modeling two species competition are studied which are strongly order preserving on a convex subset of a banach space with competitive order conditions are derived for stable coexistence bi stability and competitive exclusion a complete classification of all possible outcomes is obtained in case that there is at most one positive steady state representing coexistence of both species

mackey glass type delay differential equations near the boundary of absolute stability for an equation x t x t f x t h x r f 0 1 0 with c 3 nonlinearity f which has a negative schwarzian derivative and satisfies xf x 0a nd h 1 1 8 are less than some constant independent on h this result gives additional insight to the conjecture about the equivalence between local and global asymptotical stabilities in the mackey glass type delay differential equations 2002 elsevier science usa all rights reserved

uniqueness of fast travelling fronts in reaction diffusion equations with delay we consider positive travelling fronts u t x xct 0 of the equation ut t x u t x u t x g u th x xm this equation is assumed to have exactly two non negative equilibria u10 and u2 0

linear and nonlinear diffusion and reaction diffusion equations from discrete velocity kinetic models two velocity kinetic models are used to derive in the appropriate limit the equations which govern the macroscopic density of fluid systems such equations are obtained from an asymptotic expansion in powers of a small parameter related to the microscopic mean free path it is shown that the density of a fluid interacting with a non equilibrium background satisfies a linear diffusion equation and that the hierarchy of equations arising from the asymptotic expansion can be completely solved by a recursive scheme for a system of interacting particles a nonlinear diffusion equation is obtained and some of its solutions are analysed finally the density of a system of particles undergoing chemical reactions is shown to satisfy a nonlinear equation which formally coincides with the reaction diffusion equation proposed ad hoc at the macroscopic level

hopf bifurcation analysis in a delayed nicholson blowflies equation abstract the dynamics of a nicholson s blowflies equation with a finite delay are investigated we prove that a sequence of hopf bifurcations occur at the positive equilibrium as the delay increases explicit algorithm for determining the direction of the hopf bifurcations and the stability of the bifurcating periodic solutions are derived using the theory of normal form and center manifold global existence of periodic solutions are established using a global hopf bifurcation result of wu trans amer math soc 350 1998 4799 and a bendixson criterion for higher dimensional ordinary differential equations due to li and muldowney j differential equations 106 1994 27