Monotone Dynamics Research Articles

Dynamic Logics of Diffusion and Link Changes on Social Networks

AbstractThis paper introduces a comprehensive logical framework to reason about threshold-driven diffusion and threshold-driven link change in social networks. It considers both monotonic dynamics, where agents can only adopt new features and create new connections, and non-monotonic dynamics, where agents may also abandon features or cut ties. Three types of operators are combined: one capturing diffusion only, one capturing link change only, and one capturing both at the same time. We first characterise the models on which diffusion of a unique feature and link change stabilise, whilst discussing salient properties of stable models with multiple spreading features. Second, we show that our operators (and any combination of them) are irreplaceable, in the sense that the sequences of model updates expressed by a combination of operators cannot always be expressed using any other operators. Finally, we analyse classes of models on which some operators can be replaced.

Convergence of Bi-Virus Epidemic Models With Non-Linear Rates on Networks—A Monotone Dynamical Systems Approach

We study convergence properties of competing epidemic models of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Susceptible-Infected-Susceptible</i> ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$SIS$</tex-math> </inline-formula> ) type. The SIS epidemic model has seen widespread popularity in modelling the spreading dynamics of contagions such as viruses, infectious diseases, or even rumors/opinions over contact networks (graphs). We analyze the case of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">two</i> such viruses spreading on overlaid graphs, with non-linear rates of infection spread and recovery. We call this the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">non-linear bi-virus model</i> and, building upon recent results, obtain precise conditions for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">global convergence</i> of the solutions to a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">trichotomy</i> of possible outcomes: a virus-free state, a single-virus state, and to a coexistence state. Our techniques are based on the theory of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">monotone dynamical systems (MDS)</i> , in contrast to Lyapunov based techniques that have only seen partial success in determining convergence properties in the setting of competing epidemics. We demonstrate how the existing works have been unsuccessful in characterizing a large subset of the model parameter space for bi-virus epidemics, including all scenarios leading to coexistence of the epidemics. To the best of our knowledge, our results are the first in providing complete convergence analysis for the bi-virus system with non-linear infection and recovery rates on general graphs.

Comparison Theorems from the Theory of Monotone Dynamical Systems for Linear Systems of Functional-Differential Equations with Linearly Transformed Argument

Comparison Theorems from the Theory of Monotone Dynamical Systems for Linear Systems of Functional-Differential Equations with Linearly Transformed Argument

Fixed point equations for superlinear operators with strong upper or strong lower solutions and applications

&lt;abstract&gt;&lt;p&gt;It is well known that sublinear operators and superlinear operators are two classes of important nonlinear operators in nonlinear analysis and dynamical systems. Since sublinear operators have only weak nonlinearity, this advantage makes it easy to deal with them. However, superlinear operators have strong nonlinearity, and there are only a few results about them. In this paper, the convergence of Picard iteration for the superlinear operator $ A $ is obtained based on the conditions that the fixed point equation $ Ax = x $ has a strong upper solution and a lower solution (or alternatively, an upper solution and a strong lower solution). Besides, the uniqueness of the fixed point of strongly increasing operators as well as the global attractivity of strongly monotone dynamical systems are also discussed. In addition, the main results are applied to monotone dynamics of superlinear operators and nonlinear integral equations. The method used in our work develops the traditional method of upper and lower solutions. Since a strong upper (upper) solution and a lower (strong lower) solution are easily checked, the obtained results are effective and practicable in the study of nonlinear equations and dynamical systems. The main novelty is that this paper provides new fixed point results for increasing superlinear operators and the obtained results are applied to strongly monotone systems to investigate their global attractivity.&lt;/p&gt;&lt;/abstract&gt;

Quantum Communication with Polarization-Encoded Qubits under Majorization Monotone Dynamics

Quantum communication can be realized by transmitting photons that carry quantum information. Due to decoherence, the information encoded in the quantum state of a single photon can be distorted, which leads to communication errors. In particular, we consider the impact of majorization monotone dynamical maps on the efficiency of quantum communication. The mathematical formalism of majorization is revised with its implications for quantum systems. The discrimination probability for two arbitrary orthogonal states is used as a figure of merit to track the quality of quantum communication in the time domain.

Excess payoff dynamics in games

We present the family of Excess Payoff Dynamics for normal-form games, where the growth of a strategy depends only on its current proportion and the excess payoff, i.e., the payoff advantage of the strategy over the average population payoff. Requiring dependence only on the own excess payoff and a natural sign-preserving condition, the class essentially reduces to aggregate monotonic dynamics, a functional generalization of the Replicator Dynamics. However, Excess Payoff Dynamics also include a different subclass which contains the Replicator Dynamics, the Brown-von Neumann-Nash Dynamics, and other interesting examples as, e.g., satisficing dynamics. We also clarify the relation to excess demand dynamics from microeconomics.

Invasion into remnant instability: a case study of front dynamics

We study the invasion of an unstable state by a propagating front in a peculiar but generic situation where the invasion process exhibits a remnant instability. Here, remnant instability refers to the fact that the spatially constant invaded state is linearly unstable in any exponentially weighted space in a frame moving with the linear invasion speed. Our main result is the nonlinear asymptotic stability of the selected invasion front for a prototypical model coupling spatio-temporal oscillations and monotone dynamics. We establish stability through a decomposition of the perturbation into two pieces: one that is bounded in the weighted space and a second that is unbounded in the weighted space but which converges uniformly to zero in the unweighted space at an exponential rate. Interestingly, long-time numerical simulations reveal an apparent instability in some cases. We exhibit how this instability is caused by round-off errors that introduce linear resonant coupling of otherwise non-resonant linear modes, and we determine the accelerated invasion speed.

Quantum quench and coherent-incoherent dynamics of Ising chains interacting with dissipative baths.

The modular path integral methodology is used to extend the well-known spin-boson dynamics to finite-length quantum Ising chains, where each spin is coupled to a dissipative harmonic bath. The chain is initially prepared in the ferromagnetic phase where all spins are aligned, and the magnetization is calculated with spin-spin coupling parameters corresponding to the paramagnetic phase, mimicking a quantum quench experiment. The observed dynamics is found to depend significantly on the location of the tagged spin. In the absence of a dissipative bath, the time evolution displays irregular patterns that arise from multiple frequencies associated with the eigenvalues of the chain Hamiltonian. Coupling of each spin to a harmonic bath leads to smoother dynamics, with damping effects that are stronger compared to those observed in the spin-boson model and more prominent in interior spins, a consequence of additional damping from the spin environment. Interior spins exhibit a transition from underdamped oscillatory to overdamped monotonic dynamics as the temperature, spin-bath, or spin-spin coupling is increased. In addition to these behaviors, a new dynamical pattern emerges in the evolution of edge spins with strong spin-spin coupling at low and intermediate temperatures, where the magnetization oscillates either above or below the equilibrium value.

Nonlinear dynamics and noise actuated by the cycle of gene inactivation in stochastic transcription

Gene transcription is a stochastic process with random switching between gene off and on states. In this paper, we established a cycle of gene inactivation consisting of two off states. External signals trigger the random transition from the “ground off” (I1) state to the “excited off” (I2) state, which in turn either jumps forward to the on state or recycles back to I1 state. By integrating inactivation cycle with mRNA birth and death processes, we focus on how the recycling efficiency λ21 from I2 to I1 influences dynamical mRNA mean m(t) and the stationary noise strength Φ*. We first reveal a bidirectional regulation of λ21 on m(t): Increase of λ21 can either induce the monotonic dynamics of m(t) to the non-monotonic behavior or suppress non-monotonic dynamics to monotonicity for m(t). For Φ*, we observe two regulation scenarios that the increase of λ21 can significantly up-regulate Φ* and weaken the non-monotonic dependence of Φ* on the other parameter rates. The robustness of these results is further tested against experimental data in E.coli, yeast and mammalian cells. Taken together, we show that the cycle of gene inactivation can provide a more comprehensive mechanism on gene regulation.

Specification-Guided Verification and Abstraction Refinement of Mixed Monotone Stochastic Systems

This article addresses the problem of verifying discrete-time stochastic systems against omega-regular specifications using finite-state abstractions. Omega-regular properties allow specifying complex behavior and encompass, for example, linear temporal logic. We focus on a class of systems with mixed monotone dynamics. This class is shown to be amenable to efficient reachable set computation and models a wide range of physically relevant systems. In general, finite-state abstractions of continuous state stochastic systems give rise to augmented Markov chains wherein the probabilities of transition between states are restricted to an interval. We present a procedure to compute a finite-state interval-valued Markov chain (IMC) abstraction of discrete-time, mixed monotone stochastic systems subject to affine disturbances given a rectangular partition of the state space. Then, we suggest an algorithm for performing verification against omega-regular properties in IMCs. Specifically, we aim to compute bounds on the probability of satisfying a specification from any initial state in the IMC. This is achieved by solving a reachability problem on the sets of so-called winning and losing components in the Cartesian product between the IMC and a Rabin automaton representing the specification. Next, the verification of IMCs may yield a set of states whose acceptance status is undecided with respect to the specification, requiring a refinement of the abstraction. We describe a specification-guided approach that compares the best and worst case behaviors of accepting paths in the IMC and targets the appropriate states accordingly. Finally, we show a case study.

Asymmetric diffusion in a two-patch mutualism system characterizing exchange of resource for resource

This paper considers a two-patch mutualism system derived from exchange of resource for resource, where the obligate mutualist can diffuse asymmetrically between patches. First, we give a complete analysis on dynamics of the system without diffusion, which exhibit how resource production of the obligate mutualist leads to its survival/extinction. Using monotone dynamics theory, we show global stability of a positive equilibrium in the three-dimensional system with diffusion. A novel finding of this work is that the obligate species' final abundance is explicitly expressed as a function of the diffusion rate and asymmetry, which demonstrates precise mechanisms by which the diffusion and asymmetry lead to the abundance higher than if non-diffusing, even though the facultative species declines. It is shown that for a fixed diffusion rate, intermediate asymmetry is favorable while extremely large asymmetry is unfavorable; For a fixed asymmetry, small diffusion is favorable while extremely large asymmetry is unfavorable. Initial densities of the species are also shown to be important in species' persistence and abundance. Numerical simulations confirm and extend our results.

Monotone dynamics and global behaviors of a West Nile virus model with mosquito demographics.

In this paper a mathematical model is formulated to study transmission dynamics of West Nile virus (WNv), which incorporates mosquito demographics including pair formation, metamorphic stages and intraspecific competition. The global behaviors of the model are obtained from a geometric approach and theory of monotone dynamics, even though bistability is present due to backward bifurcation. It turns out that the model can be investigated through two auxiliary subsystem, which are cooperative and K-competitive, respectively. Together with implement of compound matrices and Poincaré-Bendixson theorem, a thorough classification of dynamics of the full model is characterized by mosquito reproduction number [Formula: see text], WNv reproduction number [Formula: see text] and a bistability subthreshold [Formula: see text]. The theoretical results show that if [Formula: see text] is not greater than 1, mosquitoes will not survive, and the WNv will die out; if [Formula: see text] is greater than 1, then mosquitoes will persist, and disease may prevail or vanish depending on basin of attraction of the local attractors which are singletons. Our method in this paper can be applied to other mosquito-borne diseases such as malaria, dengue fever which have a similar monotonicity.

Poisson stable motions of monotone nonautonomous dynamical systems

In this paper, we study the Poisson stability (in particular, stationarity, periodicity, quasi-periodicity, Bohr almost periodicity, almost automorphy, recurrence in the sense of Birkhoff, Levitan almost periodicity, pseudo periodicity, almost recurrence in the sense of Bebutov, pseudo recurrence, Poisson stability) of motions for monotone nonautonomous dynamical systems and of solutions for some classes of monotone nonautonomous evolution equations (ODEs, FDEs and parabolic PDEs). As a byproduct, some of our results indicate that all the trajectories of monotone systems converge to the above mentioned Poisson stable trajectories under some suitable conditions, which is interesting in its own right for monotone dynamics.

Adaptive dispersal effect on the spread of a disease in a patchy environment.

During outbreaks of a communicable disease, people intensely follow the media coverage of the epidemic. Most people attempt to minimize contact with others, and move themselves to avoid crowds. This dispersal may be adaptive regarding the intensity of media coverage and the population numbers in different patches. We propose an epidemic model with such adaptive dispersal rates to examine how appropriate adaption can facilitate disease control in connected groups or patches. Assuming dependence of the adaptive dispersal on the total population in the relevant patches, we derived an expression for the basic reproduction number to be related to the intensity of media coverage, and we show that the disease-free equilibrium is globally asymptotically stable if and it becomes unstable if . In the unstable case, we showed a uniform persistence of disease by using a perturbation theory and the monotone dynamics theory. Specifically, when the disease mildly affects the dispersal of infectious individuals and rarely induces death, a unique endemic equilibrium exists in the model, which is globally asymptotically stable in positive states. Moreover, we performed numerical calculations to explain how the intensity of media coverage causes competition among patches, and influences the final distribution of the population.

Monotone dynamics or not? Dynamical consequences of various mechanisms for delayed logistic growth

In this paper we interpret the global stability properties of the delayed single species chemostat in terms of monotone dynamics on an asymptotically invariant hyperplane in the state space. The co ...

A nosocomial-pathogens-infections model with impulsive antibiotics treatment on multiple bacteria

A nosocomial-pathogens-infections model with impulsive antibiotics treatment on multiple bacteria and time-dependent drug efficacy is proposed in this paper to describe the patients infected by the bacterial populations of both antibiotic-wild-type and antibiotic-resistant strains during the course of combination treatment. The purposes of this paper are to investigate the efficacies of periodic input of antibiotic dosage on bacterial populations with impulsive drug effects and to preserve or restore antibiotic effectiveness. Two antibiotics are used to induce instantaneous antibiotic efficacies at fixed times and antibiotic concentrations decay exponentially. Using the theories of asymptotic periodic systems, uniform persistence theory of discrete dynamical systems and monotone dynamics, sufficient conditions for treatment success as well as for treatment failure are established via the basic reproduction ratio of periodic compartment models. In particular, the results show that if any basic reproduction ratio for the patients infected by wild-type bacteria, resistant bacteria or those infected by both strains is larger than unity, then there will be persistent treatment failure for patients infected by resistant bacteria. This study indicates the significance of exploring a more effective therapeutic regimen for nosocomial infection. Numerical simulations have been performed to verify/extend our analytical results.

Rich transitional dynamics, physical capital, and technology intensity

This paper develops a non-scale growth model with physical capital and two types of R&D under a lab-equipment specification, where both the intensive and extensive growth margins are fully endogenous. We study analytically the long-run equilibrium and transitional dynamics properties of the model, and establish meaningful sufficient conditions for saddle-path stability. We relate the different combinations of initial conditions of the dynamical system with the observation of monotonic versus non-monotonic transitional dynamics. Our model is able to predict monotonic, hump-shaped, and inverted hump-shaped trajectories, therefore encompassing the diverse convergence behaviour observed in the empirical data on modern growth experiences.

Quorum-Sensing Synchronization of Synthetic Toggle Switches: A Design Based on Monotone Dynamical Systems Theory.

Synthetic constructs in biotechnology, biocomputing, and modern gene therapy interventions are often based on plasmids or transfected circuits which implement some form of “on-off” switch. For example, the expression of a protein used for therapeutic purposes might be triggered by the recognition of a specific combination of inducers (e.g., antigens), and memory of this event should be maintained across a cell population until a specific stimulus commands a coordinated shut-off. The robustness of such a design is hampered by molecular (“intrinsic”) or environmental (“extrinsic”) noise, which may lead to spontaneous changes of state in a subset of the population and is reflected in the bimodality of protein expression, as measured for example using flow cytometry. In this context, a “majority-vote” correction circuit, which brings deviant cells back into the required state, is highly desirable, and quorum-sensing has been suggested as a way for cells to broadcast their states to the population as a whole so as to facilitate consensus. In this paper, we propose what we believe is the first such a design that has mathematically guaranteed properties of stability and auto-correction under certain conditions. Our approach is guided by concepts and theory from the field of “monotone” dynamical systems developed by M. Hirsch, H. Smith, and others. We benchmark our design by comparing it to an existing design which has been the subject of experimental and theoretical studies, illustrating its superiority in stability and self-correction of synchronization errors. Our stability analysis, based on dynamical systems theory, guarantees global convergence to steady states, ruling out unpredictable (“chaotic”) behaviors and even sustained oscillations in the limit of convergence. These results are valid no matter what are the values of parameters, and are based only on the wiring diagram. The theory is complemented by extensive computational bifurcation analysis, performed for a biochemically-detailed and biologically-relevant model that we developed. Another novel feature of our approach is that our theorems on exponential stability of steady states for homogeneous or mixed populations are valid independently of the number N of cells in the population, which is usually very large (N ≫ 1) and unknown. We prove that the exponential stability depends on relative proportions of each type of state only. While monotone systems theory has been used previously for systems biology analysis, the current work illustrates its power for synthetic biology design, and thus has wider significance well beyond the application to the important problem of coordination of toggle switches.

Dynamics of a generalized Ricker–Beverton–Holt competition model subject to Allee effects

In this article, we propose and study a generalized Ricker–Beverton–Holt competition model subject to Allee effects to obtain insights on how the interplay of Allee effects and contest competition affects the persistence and the extinction of two competing species. By using the theory of monotone dynamics and the properties of critical curves for non-invertible maps, our analysis show that our model has relatively simple dynamics, i.e. almost every trajectory converges to a locally asymptotically stable equilibrium if the intensity of intra-specific competition intensity exceeds that of inter-specific competition. This equilibrium dynamics is also possible when the intensity of intra-specific competition intensity is less than that of inter-specific competition but under conditions that the maximum intrinsic growth rate of one species is not too large. The coexistence of two competing species occurs only if the system has four interior equilibria. We provide an approximation to the basins of the boundary attractors (i.e. the extinction of one or both species) where our results suggests that contest species are more prone to extinction than scramble ones are at low densities. In addition, in comparison to the dynamics of two species scramble competition models subject to Allee effects, our study suggests that (i) Both contest and scramble competition models can have only three boundary attractors without the coexistence equilibria, or four attractors among which only one is the persistent attractor, whereas scramble competition models may have the extinction of both species as its only attractor under certain conditions, i.e. the essential extinction of two species due to strong Allee effects; (ii) Scramble competition models like Ricker type models can have much more complicated dynamical structure of interior attractors than contest ones like Beverton–Holt type models have; and (iii) Scramble competition models like Ricker type competition models may be more likely to promote the coexistence of two species at low and high densities under certain conditions: At low densities, weak Allee effects decrease the fitness of resident species so that the other species is able to invade at its low densities; While at high densities, scramble competition can bring the current high population density to a lower population density but is above the Allee threshold in the next season, which may rescue a species that has essential extinction caused by strong Allee effects. Our results may have potential to be useful for conservation biology: For example, if one endangered species is facing essential extinction due to strong Allee effects, then we may rescue this species by bringing another competing species subject to scramble competition and Allee effects under certain conditions.

Monotone Dynamical Systems with Polyhedral Order Cones and Dense Periodic Points

Let $X\subset \mathbb{R}^{n}$ be a set whose interior is connected and dense in $X$, ordered by a closed convex cone $K\subset \mathbb{R}^{n}$ having nonempty interior. Let $T: X\approx X$ be an order-preserving homeomorphism. The following result is proved: Assume the set of periodic points of $T$ is dense in $X$, and $K$ is a polyhedron. Then $T$ is periodic.