Monotone semiflows generated by functional differential equations

Abstract : This obtains sufficient condition for an autonomous functional differential equation to generate a strongly monotone semiflow on a suitable state space. This allows the application to functional differential equations of very powerful recent results on strongly monotone semiflows due to Hirsch and Matano. In addition, a very striking relationship is established between such functional differential equations and corresponding ordinary differential equations. An example, involving a biochemical feedback loop is considered. Keywords: stability; steady states.

• Publication year

  1987

• Venue

  Journal of Differential Equations

• Publication date

  1987-03-15

• Fields of study

  Mathematics

• Identifiers
  DOI 10.1016/0022-0396(87)90027-1
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• A relationship is established between these functional differential equations and corresponding ordinary differential equations.

  Confidence 0.93

  박진우 (dztg5apj7m) extractionB (s683577b42) reviewAnonymous (12632b8b5f) reviewAK (4715169a40) review
• The strongly monotone semiflow setting makes it possible to apply Hirsch and Matano results to functional differential equations.

  Confidence 0.91

  박진우 (dztg5apj7m) extractionB (s683577b42) reviewAnonymous (12632b8b5f) reviewAK (4715169a40) review
• A sufficient condition is obtained for an autonomous functional differential equation to generate a strongly monotone semiflow on a suitable state space.

  Confidence 0.97

  박진우 (dztg5apj7m) extractionB (s683577b42) reviewAnonymous (12632b8b5f) reviewAK (4715169a40) review

• autonomous functional differential equation

  equation

  A functional differential equation with no explicit time dependence, where the derivative depends on the current state and past history.

  Aliases: autonomous FDE

  박진우 (dztg5apj7m) extractionB (s683577b42) reviewAnonymous (12632b8b5f) reviewAK (4715169a40) review
• functional differential equation

  equations

  A differential equation whose evolution depends on the present state and on past states or histories.

  Aliases: FDE

  박진우 (dztg5apj7m) extractionB (s683577b42) reviewAnonymous (12632b8b5f) reviewAK (4715169a40) review
• hirsch and matano results

  theorems

  Recent theorems on strongly monotone semiflows that provide dynamical conclusions for ordered systems.

  Aliases: Hirsch results, Matano results

  박진우 (dztg5apj7m) extractionB (s683577b42) reviewAnonymous (12632b8b5f) reviewAK (4715169a40) review
• ordinary differential equation

  equations

  A differential equation involving derivatives with respect to a single independent variable and no functional history term.

  Aliases: ODE

  박진우 (dztg5apj7m) extractionB (s683577b42) reviewAnonymous (12632b8b5f) reviewAK (4715169a40) review
• strongly monotone semiflow

  dynamical system

  A semiflow on an ordered state space that preserves order and is strictly order-improving for comparable distinct states.

  Aliases: strongly monotone flow

  박진우 (dztg5apj7m) extractionB (s683577b42) reviewAnonymous (12632b8b5f) reviewAK (4715169a40) review
• suitable state space

  state space

  The function space on which the semiflow is defined and where the order structure needed for monotonicity is available.

  박진우 (dztg5apj7m) extractionB (s683577b42) reviewAnonymous (12632b8b5f) reviewAK (4715169a40) review

• Systems of ordinary differential equations which generate an order preserving flow. A survey of results

  1988cited by this paper
• Oscillations and multiple steady states in a cyclic gene model with repression

  1987cited by this paper
• Invariant curves for mappings

  1986cited by this paper
• Positive operators and elliptic eigenvalue problems

  1984cited by this paper
• The dynamical systems approach to differential equations

  1984cited by this paper
• Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems

  1984cited by this paper
• Review: Abraham Berman and Robert J. Plemmons,Nonnegative matrices in the mathematical sciences

  1982cited by this paper
• A Hopf bifurcation in single-loop positive-feedback systems

  1982cited by this paper
• Systems of di erential equations which are competitive or cooperative I: limit sets

  1982cited by this paper
• Qualitative analysis of nonlinear quasi-monotone dynamical systems described by functional-differential equations

  1981cited by this paper
• Asymptotic behavior of solutions to single loop positive feedback systems

  1980cited by this paper
• Some recent results on dissipative processes

  1980cited by this paper
• Asymptotic Behavior and Stability of Solutions of Semilinear Diffusion Equations

  1979cited by this paper
• Theory of Functional Differential Equations

  1977cited by this paper
• A global stability criterion for simple control loops

  1977cited by this paper
• Fixed Point Equations and Nonlinear Eigenvalue Problems in Ordered Banach Spaces

  1976cited by this paper
• Positively invariant closed sets for systems of delay differential equations

  1976cited by this paper
• Dissipative periodic processes

  1971cited by this paper
• Temporal organization in cells

  1963cited by this paper

• Nonautonomous Control Systems and Skew Product Flows

  2025cites this paper
• Resistance in oncolytic viral therapy for solid tumors

  2024cites this paper
• Spreading dynamics of a biological invasion model with free boundaries and time delay

  2023cites this paper
• Global stability of delayed genetic regulatory networks with wider hill functions: A mixing monotone semiflows approach

  2023cites this paper
• A coupled algebraic-delay differential system modeling tick-host interactive behavioural dynamics and multi-stability

  2023cites this paper
• Quasimonotone random and stochastic functional differential equations with applications

  2023cites this paper
• Poisson stable motions of monotone and strongly sub-linear non-autonomous dynamical systems

  2022cites this paper
• Modeling the dynamics of rumor diffusion over complex networks

  2021cites this paper
• Global Dynamics and Asymptotic Spreading Speeds for a Partially Degenerate Epidemic Model with Time Delay and Free Boundaries

  2021cites this paper
• Structured Tick Population Dynamics

  2020cites this paper
• Monotony and Comparison Principle in Non Autonomous Size Structured Models

  2020cites this paper
• Criteria of global attraction in systems of delay differential equations with mixed monotonicity

  2020cites this paper
• Oscillations Due To Diapause

  2020cites this paper
• Implications of vector attachment and host grooming behaviour for vector population dynamics and distribution of vectors on their hosts

  2020cites this paper
• Nonlinear dynamical analysis and control strategies of a network-based SIS epidemic model with time delay

  2019cites this paper
• Structured Population Models for Vector-Borne Infection Dynamics

  2019cites this paper
• Global convergence to equilibria in non-monotone delay differential equations

  2019cites this paper
• A continuous semiflow on a space of Lipschitz functions for a differential equation with state-dependent delay from cell biology

  2019cites this paper
• Poisson stable motions of monotone nonautonomous dynamical systems

  2019cites this paper
• Asymptotic behavior of bounded solutions to a system of neutral functional differential equations in critical case

  2018cites this paper
• Global dynamics of an epidemiological model with age-of-infection dependent treatment rate

  2018cites this paper
• Delay-induced switched states in a slow–fast system

  2018cites this paper
• Homogeneous Herz spaces with variable exponents and regularity results

  2018cites this paper
• Dynamical behaviours and control measures of rumour-spreading model with consideration of network topology

  2017cites this paper
• A Meshless Modelling Framework for Simulation and Control of Nonlinear Synthetic Biological Systems

  2017influential citation
• Bifurcation analysis of a spruce budworm model with diffusion and physiological structures

  2017cites this paper
• CRITICAL CONTACT RATE FOR VECTOR–HOST–PATHOGEN OSCILLATION INVOLVING CO-FEEDING AND DIAPAUSE

  2017cites this paper
• Mathematical modelling of interacting fibre structures and non-woven materials

  2017cites this paper
• Poisson stable motions of monotone nonautonomous dynamical systems

  2017cites this paper
• I. U. Bronshtein's conjecture for monotone nonautonomous dynamical systems

  2017cites this paper
• A SHORT SURVEY ON DELAY DIFFERENTIAL SYSTEMS WITH PERIODIC COEFFICIENTS

  2017cites this paper
• Comparison theorems for neutral stochastic functional differential equations

  2016cites this paper
• Monotone dynamical systems: Reflections on new advances & applications

  2016cites this paper
• Cell-free infection and cell-cell transmission HIV-1 dynamics model with cure rate

  2016cites this paper
• Complexity in a predator-prey-parasite model with nonlinear incidence rate and incubation delay

  2015cites this paper
• Epidemic spreading and global stability of an SIS model with an infective vector on complex networks

  2015cites this paper
• Trend to Equilibrium for a Delay Vlasov-Fokker-Planck Equation and Explicit Decay Estimates

  2015cites this paper
• EPIDEMIC SPREADING AND GLOBAL STABILITY OF A NEW SIS MODEL WITH DELAY ON HETEROGENEOUS NETWORKS

  2015cites this paper
• Global analysis of HIV-1 dynamics with Hill type infection rate and intracellular delay☆

  2014cites this paper
• POSITIVELY INVARIANT SETS , MONOTONE SOLUTIONS , AND CONTRACTING RECTANGLES IN NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS

  2014cites this paper
• Asymptotics of delay differential equations via polynomials

  2014cites this paper
• Stability Criteria for Delay Differential Equations

  2014cites this paper
• Global Dynamics of a Predator-Prey Model with Stage Structure and Delayed Predator Response

  2013cites this paper
• Global dynamics of Nicholsonʼs blowflies equation revisited: Onset and termination of nonlinear oscillations

  2013cites this paper
• Permanence of a stage-structured predator-prey system with the Beddington-DeAngelis functional response

  2013influential citation
• Transmission dynamics for vector-borne diseases in a patchy environment

  2013cites this paper
• Non-autonomous Functional Differential Equations and Applications

  2013cites this paper
• Monotone Dynamical Systems

  2013cites this paper
• Relaxation oscillation and attractive basins of a two-neuron Hopfield network with slow and fast variables

  2012cites this paper
• A DISCRETE TIME DELAYED NEURAL NETWORK WITH POTENTIAL FOR ASSOCIATIVE MEMORY REVISITED

  2012cites this paper
• Monotonicity of semiflows generated by cooperative delayed full-range CNNs

  2012cites this paper
• Relaxation oscillation and attractive basins of a two-neuron Hopfield network with slow and fast variables

  2012cites this paper
• Stability Analysis of Positive Systems with Applications to Epidemiology

  2012cites this paper
• Limit Set Dichotomy and Multistability for a Class of Cooperative Neural Networks With Delays

  2012cites this paper
• Global dynamics of a predator-prey system with Holling type II functional response

  2011cites this paper
• The Interaction of Migratory Birds and Domestic Poultry and Its Role in Sustaining Avian Influenza

  2011influential citation
• Spatiotemporal Variation of Mistletoes: a Dynamic Modeling Approach

  2011cites this paper
• ω limit sets of solutions for a class of neutral functional differential equations

  2010cites this paper
• The asymptotical behavior of cyclic genetic regulatory networks

  2010cites this paper
• New generalization of the two-dimensional Bernfeld–Haddock conjecture and its proof☆

  2010cites this paper
• Fluctuations in a SIS epidemic model with variable size population

  2010cites this paper
• A note on the dichotomy of limit sets for cooperative CNNs with delays

  2010cites this paper
• Modelling and analysis of gene regulatory network using feedback control theory

  2010cites this paper
• Positive periodic solutions for nonautonomous impulsive neutral functional differential systems with time-varying delays on time scales.

  2010cites this paper
• Extremal equilibria for monotone semigroups in ordered spaces with application to evolutionary equations

  2010cites this paper
• Global stability of a class of non-autonomous delay differential systems

  2010cites this paper
• ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO A SYSTEM OF DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT DELAYS

  2010influential citation
• Metastable Periodic Patterns in Singularly Perturbed Delayed Equations

  2010cites this paper
• Asymptotic behavior of bounded solutions for a system of neutral functional differential equations

  2010cites this paper
• Global stability of a class of delay differential systems

  2009cites this paper
• Dynamic behavior for a system of neutral functional differential equations

  2009cites this paper
• Unstable sets, heteroclinic orbits and generic quasi-convergence for essentially strongly order-preserving semiflows

  2009cites this paper
• RETRACTED: Asymptotic behavior of solutions to a system of differential equations with state-dependent delays

  2009cites this paper
• A generalization of the three-dimensional Bernfeld-Haddock conjecture and its proof

  2009cites this paper
• The Periodic Solution of a Class of Two Neurons Hopfield Network with Distributed Delay

  2009cites this paper
• New generic quasi-convergence principles with applications

  2009cites this paper
• Dynamic behavior for a class of neutral functional differential equations

  2009cites this paper
• RETRACTED: Asymptotic constancy for a differential equation with multiple state-dependent delays

  2009cites this paper
• BLOW-UP IN SOME ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS WITH TIME-DELAY

  2009cites this paper
• RETRACTED: Asymptotic behavior of solutions to a differential equation with state-dependent delay

  2009cites this paper
• Generic Quasi-Convergence for Essentially Strongly Order-Preserving Semiflows

  2009cites this paper
• Asymptotic constancy for a differential equation with multiple state-dependent delays

  2009cites this paper
• Dynamics of smooth essentially strongly order-preserving semiflows with application to delay differential equations

  2008cites this paper
• Permanence and existence of a positive periodic solution to a periodic stage-structured system with infinite delay

  2008cites this paper
• Asymptotic Behavior of a Predator-Prey Population Model with Stage-Structure

  2007cites this paper
• Global exponential periodicity and global exponential stability of a class of recurrent neural networks with various activation functions and time-varying delays

  2007cites this paper
• Understanding the somitogenesis clock: what's missing?

  2007cites this paper
• Asymptotic Behavior of Solutions to a Class of Systems of Delay Differential Equations

  2007influential citation
• Periodic solutions of neutral differential models with state‐dependent delay and feedback control

  2007cites this paper
• Periodic solutions for general nonlinear state-dependent delay logistic equations

  2007cites this paper
• Permanence in delayed ratio-dependent predator–prey models with monotonic functional responses☆

  2007cites this paper
• Chapter 4 Monotone Dynamical Systems

  2006cites this paper
• Convergence and stability for essentially strongly order-preserving semiflows

  2006cites this paper
• Global attractivity of positive periodic solution to periodic Lotka-Volterra competition systems with pure delay

  2006cites this paper
• Permanence of a class of delay differential equations with an application to multi-species competition systems

  2006cites this paper
• 4. Monotone Dynamical Systems

  2005cites this paper
• Convergence of Solutions to a Class of Systems of Delay Differential Equations

  2005influential citation
• Permanence extinction and global asymptotic stability in a stage structured system with distributed delays

  2005cites this paper
• Global exponential periodicity of a class of recurrent neural networks with oscillating parameters and time-varying delays

  2005cites this paper
• Convergence for pseudo monotone semiflows on product ordered topological spaces

  2005cites this paper