Narrow Results

In this paper, we investigate the dynamical behaviors of a modified Bazykin-type two predator-one prey model involving the intra-specific and inter-specific competition among predators. A Caputo fractional-order derivative is utilized to include the influence of the memory on the constructed mathematical model. The mathematical validity is ensured by showing the model always has a unique, non-negative and bounded solution. Four kinds of equilibria are well identified which represent the condition when all populations are extinct, both two predators are extinct, only the first predator is extinct, only the second predator is extinct, and all populations are extinct. The Matignon condition is given to identify the dynamics around equilibrium points. The Lyapunov direct method, the Lyapunov function, and the generalized LaSalle invariant principle are also provided to show the global stability condition of the model. To explore the dynamics of the model more deeply, we utilize the predictor–corrector numerical scheme. Numerically, we find the forward bifurcation and the bistability conditions by showing the bifurcation diagram, phase portraits, and the time series. The biological interpretation of the analytical and numerical results is described explicitly when an interesting phenomenon occurs.

This paper centers on the analysis of the dynamics of a modified Leslie–Gower predator–prey model employing Holling-type II schemes, with the prey exhibiting pure random diffusion and the predator undergoing a mixed form of movement. The extinction of species and uniform persistence of this system are explored, and several conditions for the stability, uniqueness and multiplicity of positive steady-state solutions are derived. In contrast to the specialist and generalist predator–prey systems in open advective environments, the dynamics of this system are more intricate. It emerges that multiple positive steady-state solutions and the bistable phenomenon exist for this system when a small advection rate and a moderate predation rate are imposed. Numerical simulations reveal that the increase of diffusion rate for prey disadvantages the survival of itself and has no impact on predator invasion, while the increase of diffusion rate for predators favors the invasion of itself.

In this paper, we study a prey–predator–top predator food chain model with nonlinear harvesting of top predator. We have derived two important thresholds: the top predator extinction threshold and the coexistence threshold. We found that the top predator will die out if the nonlinear harvesting from predator to top predator is larger than the top predator extinction threshold. On the other hand, the prey, predator and top predator coexist if the nonlinear harvesting from predator to top predator is less than the coexistence threshold. While the parameter value of nonlinear harvesting from predator to top predator is between two critical thresholds, the system displays bistability phenomena, implying that the top predator species either die out or exist with the prey and predator species, which largely depend on the initial condition. Thus, a bistable interval exists between two critical thresholds, which is a significant phenomenon for the model. Meanwhile, we performed bifurcation analysis for the model, showing that the system would arise backward/forward bifurcation and saddle-node bifurcation and Hopf bifurcation. Finally, we performed numerical simulations to verify the theoretical analysis.

Quantitative models may exhibit sophisticated behaviour that includes having multiple steady states, bistability, limit cycles, and period-doubling bifurcation. Such behaviour is typically driven by the numerical dynamics of the model, where the values of various numerical parameters play the crucial role. We introduce in this paper natural correspondents of these concepts to reaction systems modelling, a framework based on elementary set theoretical, forbidding/enforcing-based mechanisms. We construct several reaction systems models exhibiting these properties.

In this paper, we study potentials of positive feedback in spatial phosphoprotein signal propagation. For this, we consider a signaling pathway of four-tiered protein kinase cascades with each tier involving single (de)phosphorylation reactions only. In the case of a small cell, we propose a short positive feedback for short-range cell signaling, which can generate bistability to facilitate the phosphoprotein signal propagation from the plasma membrane to the periphery of cell nucleus. In contrast, in the case of a large cell for which the long-range signaling cannot be achieved by the short feedback, we propose a long positive feedback, and find that it can facilitate the propagation of phosphoprotein signal over a long distance. These results imply that positive-feedback mechanisms would be employed by living organisms for spatial information transfer and cellular decision-making processing.

In population biology, the Allee dynamics refer to negative growth rates below a critical population density. In this paper, we study a reaction-diffusion (RD) model of popoulation growth and dispersion in one dimension, which incorporates the Allee effect in both the growth and mortatility rates. In the absence of diffusion, the bifurcation diagram displays regions of both finite population density and zero population density, i.e. extinction. The early signatures of the transition to extinction at the bifurcation point are computed in the presence of additive noise. For the full RD model, the existence of traveling wave solutions of the population density is demonstrated. The parameter regimes in which the traveling wave advances (range expansion) and retreats are identified. In the weak Allee regime, the transition from the pushed to the pulled wave is shown as a function of the mortality rate constant. The results obtained are in agreement with the recent experimental observations on budding yeast populations.

In this paper, we study the coevolving behavior of mixed games when agents have a relationship represented by a fully connected network or a square lattice. Under the imitation update rule, whether the system will evolve to a state of pure game or mixed games and what the level of cooperation of the population will finally be are dependent on the initial fraction of mixed games, the game parameters and the network structures. We find that agents prefer to afford the prisoner’s dilemma (PD) game than the snowdrift game in the full connected network or in the square lattice and thus the cooperation is greatly suppressed. When the PD game mixes with the stag hunt game initially, they will coexist during evolution and a bistable phenomenon is observed. Meanwhile, the fraction of cooperation is enhanced when agents compete in a square lattice by comparison with the case of a fully connected network. If the PD game mixes with the harmony game (HG) initially, which one will dominate the other is related to the game parameters. The cooperation prevails in the population if the HG dominates the PD game. We also analyze the case of a fully connected network by a theory and the theoretical results are in good agreement with the simulation data.

We investigate two three-dimensional Ising models with non-Hamiltonian Glauber dynamics. The transition probabilities of these models can, just as in the case of equilibrium models, be expressed in terms of Boltzmann factors depending only on the interacting spins and the bond strengths. However, the bond strength associated with each lattice edge assumes different values for the two spins involved. The first model has cubic symmetry and consists of two sublattices at different temperatures. In the second model a preferred direction is present. These two models are investigated by Monte Carlo simulations on the Delft Ising System Processor. Both models undergo a phase transition between an ordered and a disordered state. Their critical properties agree with Ising universality. The second model displays magnetization bistability.

We identify a scheme in which we can control the behavior of an atomic medium to switch between optical bistability (OB) and optical multistability (OM) at multiple frequencies. The scheme relies on the absorption and dispersion characteristics of the optical medium at different frequencies, which can be modulated significantly by changing the magnetic field $B$ for tuning the energy difference between Zeeman sublevels. The result shows that the transition from OB to OM or vice versa can be easily realized not only at single frequency channel but also between multiple frequency channels. Furthermore, the optical steady-state behavior also varies with the ratio of the applied two coupling fields at any frequency channel. Such controllable switching between multiple frequency channels could also be used as some applications in all-optical switching and quantum information processing.

The electromagnetic induction induced by neuron membrane potential plays an important role in regulating its electrical activities, bistability and hidden dynamics. In this paper, a nonsmooth threshold control strategy is proposed, in which the membrane potential is utilized as the threshold to determine the switching function corresponding to the electromagnetic induction intensity and external stimulus current. Consequently, a four-dimensional Filippov-type HR neuron model is established. First, the existence, stability and global bifurcation behaviors of equilibrium points of two subsystems are discussed by using stability theory and numerical simulation. Then, the bistable behavior and evolution modes of subsystems are investigated based on the two-parameter bifurcation analysis. Further, the existence of various equilibrium points and sliding dynamics of the system are analyzed by Filippov convex method and Utkin’s equivalent control method. Finally, the sliding firing modes and multistable features under the control of the threshold are revealed by the method of fast-slow variable dissection. These results will provide useful theoretical support for understanding the hidden dynamic mechanism of neurons and constructing functional neural networks.

We study a delayed system with feedback modulation of the nonlinear parameter. Study of the system as a function of nonlinearity and modulation parameters reveals complex dynamical phenomena: different types of coexisting attractors, local or global bifurcations and transitions. Bistability and dynamical attractors can be distinguished in some parameter-space regions, which may be useful to drive chaotic dynamics, unstable attractors or bistability towards regular dynamics. At the bifurcation to bistability, two striking features are that they lead to fundamentally unpredictable behavior of orbits and crisis of attractors as system parameters are varied slowly through the critical curve. Unstable attractors are also investigated in bistable regions, which are easily mistaken for true multi-periodic orbits judging merely from zero measure local basins. Lyapunov exponents and basins of attraction are also used to characterize the phenomenon observed.

As is well known, Fano resonance originates from the interference between a continuum energy band and an embedded discrete energy level. We study transmission properties of the discrete chain-structure of additional defects with an isolated ring composed of N defect states, and obtain the analytical transmission coefficient of similar Fano formula. Using the formula, we reveal conditions for perfect reflections and transmissions due to either destructive or constructive interferences. It is found that a nonlinear Kerr-like response leads to bistable transmission, and for either linear cases or nonlinear ones, the defects in main arrays have a major impact on perfect reflections, but has no effect on perfect transmission.

Electromagnetic induction and autapse play important roles in regulating the electric activities, excitability, and bistable structure of neurons. The firing activities and global bifurcation patterns of a four-dimensional (4D) hybrid neuron model that combines the fast dynamic variables of the Wilson model and the slow feedback variables of the Hindmarsh–Rose (HR) model and magnetic flux are investigated based on the Matcont software and numerical calculation. The effect of electrical autapse on the dynamic evolution of the system is also discussed emphatically. Upon encountering electromagnetic induction, the hybrid neuron model exhibits complex global stability, Hopf bifurcation, and saddle-node bifurcation. Intriguingly, the system presents initial sensitivity and a bistable structure consisting of quiescent and period-1 spiking near the Hopf bifurcation point. It is worth noting that the feedback type of electrical autapse, including positive and negative feedback, has completely different effects on this bistable structure. Notably, the negative feedback autapse can expand and change the bistable region, so that the system generates a new bistable structure consisting of quiescent and periodic bursting states, and its bursting activities are also promoted. Moreover, extensive numerical results show that the system generally maintains a comb-shaped chaotic structure, abundant bifurcation patterns, and multistability. It should be noted that electrical autapse feedback types and time delays do not change the regular bifurcation structures but operate a complex regulatory mechanism for the coexistence of multiple attractors. These results will provide useful insights into the neuron’s dynamics under the atmosphere of electromagnetic induction and also electrical autapse.

Relaxation oscillations appear in processes which involve transitions between two states characterized by fast and slow time scales. When a relaxation oscillator is coupled to an external periodic force its entrainment by the force results in a response which can include multiple periodicities and bistability. The prototype of these behaviors is the harmonically driven van der Pol equation which displays regions in the parameter space of the driving force amplitude where stable orbits of periods 2n ± 1 coexist, flanked by regions of periods 2n + 1 and 2n - 1. The parameter regions of such bistable orbits are derived analytically for the closely related harmonically driven Stoker–Haag piecewise discontinuous equation. The results are valid over most of the control parameter space of the system. Also considered are the reasons for the more complicated dynamics featuring regions of high multiple periodicity which appear like noise between ordered periodic regions. Since this system mimics in detail the less analytically tractable forced van der Pol equation, the results suggest extensions to situations where forced relaxation oscillations are a component of the operating mechanisms.

The estimation of the domain of stability of fixed points of nonlinear differential systems constitutes a practical problem of much interest in engineering. The procedures based on Lyapunov's second method configures an alternative worth consideration. It has the appeal of reducing calculation complexity and is time-saving with respect to the direct, computer crunching approach which requires a systematic numerical integration of the evolution equations from a gridlike pattern of initial conditions. However, it is not devoid of problems inasmuch as the Lyapunov function itself is problem-dependent and relies too much on presumptions. Additionally, the evaluation of its corresponding domain is produced in terms of a nonlinear programming problem with inequality constraints the resolution of which may sometimes require a large investment in computer time. These problems are in part avoided by restricting to quadratic Lyapunov functions, with the possible obvious consequence of limiting the estimation of the domain. In order to simplify the estimation of domains we exploit here a novel formulation of the issue of stability of invariant surfaces within Lyapunov's direct method [Díaz-Sierra et al., 2001]. The resulting method addresses directly the optimization problem associated to the evaluation of the stability domain. The problem is recast in a new, simpler form by playing both on the Lyapunov function itself and on the constraints. The gains from the procedure permit to conceive increased returns in the application of Lyapunov's direct method once it is realized that it is not prohibitive from a computational point of view to depart from the limited quadratic Lyapunov functions.

The coalescence of a Hopf bifurcation with a codimension-two cusp bifurcation of equilibrium points yields a codimension-three bifurcation with rich dynamic behavior. This paper presents a comprehensive study of this cusp-Hopf bifurcation on the three-dimensional center manifold. It is based on truncated normal form equations, which have a phase-shift symmetry yielding a further reduction to a planar system. Bifurcation varieties and phase portraits are presented. The phenomena include all four cases that occur in the codimension-two fold–Hopf bifurcation, in addition to bistability involving equilibria, limit cycles or invariant tori, and a fold–heteroclinic bifurcation that leads to bursting oscillations. Uniqueness of the torus family is established locally. Numerical simulations confirm the prediction from the bifurcation analysis of bursting oscillations that are similar in appearance to those that occur in the electrical behavior of neurons and other physical systems.

The results of analytical study of vibrational resonance (VR) occurring in overdamped bistable system driven by two periodic signals with very different frequencies are presented. Approximate solutions for responses at the low-frequency as a function of the amplitude, and the frequency of the additional high frequency modulation which describe well the main features of vibrational resonance are obtained. Scaling laws for the gain factor and the switching threshold in VR are also found. Analytical results are compared with results of the numerical simulation, showing a good agreement.

Bazykin proposed a Lotka–Volterra-type ecological model that accounts for simplified territoriality, which neither depends on territory size nor on food availability. In this study, we describe the global dynamics of the Bazykin model using analytical and numerical methods. We specifically focus on the effects of mutual predator interference and the prey carrying capacity since the variability of each could have especially dramatic ecological repercussions. The model displays a broad array of complex dynamics in space and time; for instance, we find the coexistence of a limit cycle and a steady state, and bistability of steady states. We also characterize super- and subcritical Poincaré–Andronov–Hopf bifurcations and a Bogdanov–Takens bifurcation. To illustrate the system's ability to naturally shift from stable to unstable dynamics, we construct bursting solutions, which depend on the slow dynamics of the carrying capacity. We also consider the stabilizing effect of the intraspecies interaction parameter, without which the system only shows either a stable steady state or oscillatory solutions with large amplitudes. We argue that this large amplitude behavior is the source of chaotic behavior reported in systems that use the MacArthur–Rosenzweig model to describe food-chain dynamics. Finally, we find the sufficient conditions in parameter space for Turing patterns and obtain the so-called "back-eye" pattern and localized structures.

A degeneration in a two-parameter setting for the focus-center-limit cycle bifurcation of symmetric 3D piecewise linear systems with three zones is studied. The analysis gives a rigorous mathematical explanation for the appearance in piecewise linear oscillators of up to two limit cycles along with a related hysteresis phenomenon.

A piecewise linear oscillator, which is a member of the generic Chua's circuit family, is shown to exhibit bistability, that is, coexistence of stable oscillations with a stable steady state. Also, the corresponding hysteresis in the appearance of periodic oscillations is explained. The analytical results included in the paper predict accurately the observed behavior.

This review summarizes recent studies on the catalytic CO oxidation on Iridium(111) surfaces. This was investigated experimentally under ultrahigh vacuum (UHV) conditions using mass spectroscopy to detect gaseous products and photoelectron emission microscopy (PEEM) to visualize surface species. The underlying reaction–diffusion system based on the Langmuir–Hinshelwood mechanism was analyzed numerically.

The existence of bistability for this surface reaction was shown in experiment. For the first time the effect of noise on a bistable surface reaction was examined. In a surface science experiment the effects on product formation and the development of spatio-temporal patterns on the surface were explored.

Steady state CO₂ rates were measured under constant flux of the CO + O mixture as a function of sample temperature (360 K < T < 700 K) and gas composition, characterized by the molar fraction of CO in the feed gas (0 ≤ Y ≤ 1). The reaction reveals bistability in a limited region of Y and T. A rate hysteresis with two steady state rates was observed for cycling Y up and down, one of high reactivity (upper rate, oxygen covered surface) and one of low reactivity (lower rate, CO covered surface). The position of the hysteresis loop shifts to higher Y values and decreases in width with increasing temperature. For small CO content in the feed gas the CO₂ rate is proportional to Y^3/2. At 500 K extremely slow Y cycling measurements (about 100 hours per direction) were done and showed that bistability still exists and no slowly changing transients were observed. The requirements for the speed with which experiments can be executed without producing experimental artifacts were explored. Since over-sampling alters the measured hysteresis loop, a maximum rate for changing the gas composition in Y cycling experiments was determined.

The influence of noise on the reaction rates and the formation of spatio-temporal patterns on the surface was explored by superposing noise of Gaussian white type on Y and on T. Noisy Y (deviation Δ Y) represents multiplicative and additive noise, noisy T (deviation Δ T) multiplicative noise only. Noisy T enters the reaction via the rate-determining step, the observed CO₂ rates become noisy for low temperatures (around 420 K) when the surface is dominantly oxygen covered (CO + O reaction step is rate-limiting) and for higher temperatures (around 500 K) when the surface is dominantly CO covered (CO desorption step is rate-limiting).

The effect of noisy Y was examined for a sample temperature of 500 K and is dependent on the selected average gas composition. In the regions with one steady state CO₂ rate (outside the hysteresis) the recorded rates were noisy. The probability distribution of the rates is Gaussian shaped for the upper rate (below hysteresis) and asymmetric for the lower rate (above hysteresis). For large noise strength bursts, short-time excursions to and above the upper rate, were observed.

Inside the hysteresis small noise made the steady state rates noisy, but above a Y-dependent Δ Y transients from the locally stable to the globally stable rate branch were observed. These transients take up to several ten thousand seconds and become faster with increasing noise. For larger Y noise strength bursts and switching between both steady state rates were detected.

Photoelectron emission microscopy (PEEM) was done to visualize spatio-temporal adsorbate patterns on the surface as expected from the observations in the CO₂ rate measurements. CO- and oxygen-covered regions on the Ir(111) surface are visible in PEEM images as gray and black areas as a consequence of their work function contrast. Islands of the adsorbate, corresponding to the globally stable branch, are formed in a background of the other one. The long transient times are the result of the extremely slow domain wall motion of these islands (around 0.05 μm s^-1). In the hysteresis region CO oxidation on Iridium(111) surfaces is dominated by domain formation and wall motion for small to moderate noise strength. The island density increases with noise, but the wall velocity is independent of applied Δ Y. For larger noise amplitudes, fast switching between oxygen- and CO-dominated surfaces is observed as well as nucleation and growth of the minority phase in the majority phase.

In the numerically analyzed reaction–diffusion system all parameters were taken from the experiment. Modeling the reaction–diffusion system shows qualitative up to quantitative agreement with the experimental observations. The length scale for the modeling grid is determined from wall velocity seen in the experiments.