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It is well known that for a large class of coagulation kernels, Smoluchowski coagulation equations have particular power law solutions which yield a constant flux of mass along all scales of the system. In this paper, we prove that for some choices of the coagulation kernels there are solutions with a constant flux of mass along all scales which are not power laws. The result is proved by means of a bifurcation argument. [ABSTRACT FROM AUTHOR]

Spatial memory is key in animal movement modeling, but it has been challenging to explicitly model learning to describe memory acquisition. In this paper, we study novel cognitive consumer-resource models with different consumer learning mechanisms and investigate their dynamics. These models consist of two PDEs in composition with one ODE such that the spectrum of the corresponding linearized operator at a constant steady state is unclear. We describe the spectra of the linearized operators and analyze the eigenvalue problems to determine the stability of the constant steady states. We then perform bifurcation analysis by taking the perceptual diffusion rate as the bifurcation parameter. It is found that steady-state and Hopf bifurcations can both occur in these systems, and the bifurcation points are given so that the stability region can be determined. Moreover, rich spatial and spatiotemporal patterns can be generated in such systems via different types of bifurcation. Our effort establishes a new approach to tackling a hybrid model of PDE-ODE composition and provides a deeper understanding of cognitive movement-driven consumer-resource dynamics. [ABSTRACT FROM AUTHOR]

The bidomain model is the standard model for cardiac electrophysiology. This paper investigates the instability and asymptotic behavior of planar fronts and planar pulses of the bidomain Allen-Cahn equation and the bidomain FitzHugh--Nagumo equation in two spatial dimensions. Previous work showed that planar fronts of the bidomain Allen--Cahn equation could become unstable in contrast to the classical Allen--Cahn equation. After the planar front is destabilized, a rotating zigzag front develops whose shape can be explained by simple geometric arguments using a suitable Frank diagram. We also show that the Hopf bifurcation through which the front becomes unstable can be either supercritical or subcritical by demonstrating a parameter regime in which a stable planar front and zigzag front can coexist. Our computational studies of the bidomain FitzHugh--Nagumo pulse solution show that the pulses can also become unstable, like the bidomain Allen--Cahn fronts. However, unlike the bidomain Allen--Cahn case, the destabilized pulse does not necessarily develop into a zigzag pulse. For certain choices of parameters, the destabilized pulse can disintegrate entirely. These studies are made possible by developing a numerical scheme that allows for the accurate computation of the bidomain equation in a two-dimensional strip domain of an infinite extent. [ABSTRACT FROM AUTHOR]

Glucose, insulin, and glucagon together form a complete endocrine metabolic feedback loop. Most existing modeling work pertaining to this loop focuses mainly on the regulations between glucose and insulin. The role of glucagon in these works is summarized by hepatic glucose production (HGP) dependent solely on insulin levels. In this paper, we carefully analyze the secretion of glucagon, its role in HGP, and its interactions with insulin. We then formulate a dynamical system model to further analyze its effects in balancing blood sugar levels. The presence of glucagon allows us to examine the impacts of HGP governed by insulin and glucagon separately. This reveals insight about how a very recent concept, glucagon resistance, can affect blood glucose levels and lead to rampant glucagon production and consequently blood sugar levels. We also show that a form of insulin resistance in hepatic cells can lead to excess insulin secretion, a characteristic of type 2 diabetes mellitus. An analytical result indicates that in the equilibrium point range, the HGP caused by glucagon should not be severely outweighed by glucose uptake so that a balance between HGP and overall glucose uptake must be met for oscillations to occur. Furthermore, this model provides a guideline for the design of a dual-pump artificial pancreas system, which is believed to be the ideal treatment for type 1 diabetes mellitus. In addition, the method employed to study the instability of the equilibrium point can be extended to study general delay differential equations. [ABSTRACT FROM AUTHOR]

In this paper we present a general approach to rigorously validate Hopf bifurcations as well as saddle-node bifurcations of periodic orbits in systems of ODEs. By a combination of analytic estimates and computer-assisted calculations, we follow solution curves of cycles through folds, checking along the way that a single nondegenerate saddle-node bifurcation occurs. Similarly, we rigorously continue solution curves of cycles starting from their onset at a Hopf bifurcation. We use a blowup analysis to regularize the continuation problem near the Hopf bifurcation point. This extends the applicability of validated continuation methods to the mathematically rigorous computational study of bifurcation problems. [ABSTRACT FROM AUTHOR]

Singular Hopf bifurcation occurs in generic families of vector fields with two slow variables and one fast variable. Prototypes for this bifurcation depend upon several parameters, and the dynamics displayed by these systems is intricate. The core of this paper is a numerical investigation of the bifurcations in one prototypical family. It presents extensive diagrams of bifurcations of equilibrium points and periodic orbits that are close to singular Hopf bifurcation. In addition, parameters are determined where there is a tangency between invariant manifolds that are important in the appearance of mixed-mode oscillations in systems near singular Hopf bifurcation. One parameter of the prototype is identified as the primary bifurcation parameter, and the paper presents a catalogue of bifurcation sequences that occur as the primary bifurcation parameter is varied. These results are applied to estimating the parameters for the onset of mixed-mode oscillations in a model of chemical oscillations. [ABSTRACT FROM AUTHOR]

We consider a widely used form of models for ship maneuvering, whose nonlinearities entail continuous but nonsmooth second-order modulus terms. For such models bifurcations of straight motion are not amenable to standard center manifold reduction and normal forms. Based on a recently developed analytical approach, we nevertheless determine the character of local bi-furcations when stabilizing the straight motion course with standard proportional control. For a specifc model class we perform a detailed analysis of the linearization to determine the location of these bifurcations in the control parameter space and its dependence on selected design parameters. By computing the analytically derived characteristic parameters, we fnd that "safe" supercritical Andronov-Hopf bifurcations are typical. Through numerical continuation we provide a more global bifurcation analysis, which identifes the arrangement and relative location of stable and unstable equilibria and periodic orbits. [ABSTRACT FROM AUTHOR]

This paper deals with the qualitative analysis of the solutions to a model that refers to the competition between the immune system and an aggressive host such as a tumor. The model which describes this competition is governed by a system of differential equations with one delay. It is shown that the dynamics depends crucially on the time delay parameter. By using the time delay as a parameter of bifurcation, the analysis is focused on the Hopf bifurcation problem to predict the occurrence of a limit cycle bifurcating from the nontrivial steady state. The obtained results depict the oscillations, given by simulations (see [M. Galach, Int. J. Appl. Math. Comput. Sci., 13 (2003), pp. 395–406]), which are observed in reality (see [D. Kirschner and J. C. Panetta, J. Math. Biol., 37 (1998), pp. 235–252]). It is suggested to examine by laboratory experiments how to employ these results for control of tumor growth. [ABSTRACT FROM AUTHOR]

A hybrid asymptotic-numerical approach is developed to study the existence and linear stability of steady-state hotspot patterns for a three-component one-dimensional reaction-diffusion (RD) system that models urban crime with police intervention. Our analysis is focused on a new scaling regime in the RD system where there are two distinct competing mechanisms of hotspot annihilation and creation that, when coincident in a parameter space, lead to complex spatio-temporal dynamics of hotspot patterns. Hotspot annihilation events are shown numerically to be triggered by an asynchronous oscillatory instability of the hotspot amplitudes that arises from a secondary instability on the branch of periodic solutions that emerges from a Hopf bifurcation of the steady-state solution. In addition, hotspots can be nucleated from a quiescent background when the criminal diffusivity is below a saddle-node bifurcation threshold of hotspot equilibria, which we estimate from our asymptotic analysis. To investigate instabilities of hotspot steady states, the spectrum of the linearization around a two-boundary hotspot pattern is computed, and instability thresholds due to either zero-eigenvalue crossings or Hopf bifurcations are shown. The bifurcation software pde2path is used to follow the branch of periodic solutions and detect the onset of the secondary instability. Overall, these results provide a phase diagram in parameter space where distinct types of dynamical behaviors occur. In one region of this phase diagram, where the police diffusivity is small, a two-boundary hotspot steady state is unstable to an asynchronous oscillatory instability in the hotspot amplitudes. This instability typically triggers a nonlinear process leading to the annihilation of one of the hotspots. However, for parameter values where this instability is coincident with the nonexistence of a one-hotspot steady state, we show that hotspot patterns undergo complex "nucleation-annihilation" dynamics that are characterized by large-scale persistent oscillations of the hotspot amplitudes. In this way, our results identify parameter ranges in the three-component crime model where the effect of police intervention is to simply displace crime between adjacent hotspots and where new crime hotspots regularly emerge "spontaneously" from regions that were previously free of crime. More generally, it is suggested that when these annihilation and nucleation mechanisms are coincident for other multihotspot patterns, the problem of predicting the spatial-temporal distribution of crime is largely intractable. [ABSTRACT FROM AUTHOR]

This paper deals with the motion of a multilegged modular robot. The robot consists of a set of homogenous modules, each of which has a body and two legs and is connected to the others through a three-degree-of-freedom rotary joint. The leg joints are manipulated to follow periodic desired trajectories, and the joints between the modules act like a passive spring with a damper. This robot has characteristic dynamic properties. Specifically, a straight walk naturally turns into a meandering walk by changing the compliance of the joints between the modules without incorporation of any oscillatory inputs. We first show that this transition is excited due to a Hopf bifurcation, based on a numerical simulation and Floquet analysis. Following that, we examine whether the maneuverability and agility of the robot increase by utilizing the dynamic characteristics inherent in the robot. In particular, we conduct an experiment in which the robot pursues a target moving across the floor. We propose a simple controller to accomplish the task and achieve high maneuverability and agility by making the most of the robot's dynamic features. [ABSTRACT FROM AUTHOR]

In this paper, control systems with two uncontrollable modes on the imaginary axis are studied. The main contributions include the local orientation control of periodic solutions and center manifolds, the quadratic normal form of systems with two imaginary uncontrollable modes, the stabilization of the Hopf bifurcation by state feedback, and the quadratic invariants that characterize the nonlinearity of a system and its Hopf bifurcation. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in a mechanistic model describing the anaerobic mineralization of chlorophenol in a three-step food web. The model is a six-dimensional system of ordinary differential equations. In our study, we take into account the phenol and the hydrogen input concentrations as well as the maintenance terms. Moreover, we consider the case of a large class of growth rates, instead of specific kinetics. In this general case, a recent study shows that the system can have up to eight steady states, and their existence conditions were analytically determined. We focus here on the necessary and sufficient conditions of the local stability of the steady states, according to the four operating parameters of the process, which are the dilution rate and the chlorophenol, phenol, and hydrogen input concentrations. In previous studies, this stability analysis was performed only numerically. Using the Lienard-Chipart stability criterion, we show that the positive steady state can be unstable and we give numerical evidence for a supercritical Hopf bifurcation with the appearance of a stable periodic orbit. We give two bifurcation diagrams with respect to the dilution rate, first, and then to the chlorophenol input concentration as the bifurcating parameters, showing that the system can present rich behavior, including bistability, coexistence, and occurrence of limit cycle. [ABSTRACT FROM AUTHOR]

In view of the molecular biological mechanism of the cytotoxic T lymphocytes proliferation induced by hepatitis B virus infection in vivo, a novel dynamical model with interval delay is proposed. The interval delay is determined by two delay parameters, namely delay center and delay radius. We derive the basic reproduction number R0 for the viral infection and obtain that the virus-free equilibrium (VFE) is globally asymptotically stable if R0 < 1. When R0 >1, besides VFE, the unique virus-present equilibrium (VPE) exists and the conditions of its asymptotical stability are obtained. Moreover, we study the Hopf bifurcations induced by the two delay parameters. Although there is no mitotic term in the target-cell dynamics, the results indicate that both these delay parameters can lead to periodic fluctuations at VPE, but only the smaller delay radius will destabilize the system, which is different from the classical discrete delay or distributed delay. Numerical simulations indicate that the proposed model can capture the profiles of the clinical data of two untreated chronic hepatitis B patients. The ability of interval delay to destabilize the system is between discrete delay and distributed delay, and the delay center plays the primary role. Pharmaceutical treatment can affect the stability of VPE and induce the fast-slow periodic phenomenon. [ABSTRACT FROM AUTHOR]

We consider small amplitude periodic forcing (with forcing frequency ?F) of a system of differential equations near a point of Hopf bifurcation (with Hopf frequency ?H). We follow Zhang and Golubitsky [SIAM J. Appl. Dyn. Syst., 10 (2011), pp. 1272-1306] and consider only those small amplitude periodic solutions to the forced system whose frequency is ?F ; that is, the 1:1 phase-locked or entrained solutions. These authors assume sinusoidal forcing of a normal form Hopf system when ?F is close to ?H, and they classify the existence and multiplicity of the entrained solutions. The forced response curve is a bifurcation diagram showing the amplitude of the entrained solutions as the forcing frequency is varied. Zhang and Golubitsky showed that there are six kinds of forced response curves with distinguished bifurcation parameter = H -F. In this paper we show that there are 41 possible bifurcation diagrams when stability in addition to multiplicity of the periodic solutions in the forced response curves is included. [ABSTRACT FROM AUTHOR]

This paper analyzes the stability and bifurcation criteria of cyclic genetic regulatory networks with mixed time delays (discrete and Gamma-type distributed delays). It is of more realistic biological background to use Gamma-type distributed delay kernels to describe the delays in the process between genes. The key aim of our research is to reveal more about the dynamic functions played by biochemical parameters and different mechanisms of discrete and distributed time delays. The existence of positive equilibria in this kind of genetic network is verified. With the mathematical tools of subharmonic function and complex theory, exact conditions of biochemical parameters for stability and bifurcations in cyclic genetic regulatory networks with both positive and negative gains are deduced, respectively. Finally, several simulation examples are adopted to deepen our analysis. [ABSTRACT FROM AUTHOR]

In this paper, we consider a two-dimensional delay differential system with two delays. By analyzing the distribution of eigenvalues, linear stability of the equilibria and existence of Hopf, Bautin, and Hopf-Hopf bifurcations are obtained in which the time delays are used as the bifurcation parameter. General formula for the direction, period, and stability of the bifurcated periodic solutions are given for codimension one and codimension two bifurcations, including Hopf bifurcation, Bautin bifurcation, and Hopf–Hopf bifurcation. As an application, we study the dynamical behaviors of a model describing the interaction between tumor cells and effector cells of the immune system. Numerical examples and simulations are presented to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In some applications of delay differential equations in population dynamics, the need for incorporation of distributed delay is often the result of the existence of some processes or structures. The main objective of this paper is to provide some information on the stability and Hopf bifurcation analysis in a general functional differential equation with distributed delay. The local stability parameter regions related to time delay are given and compared for a general distribution delay function and three frequently used distributed delays including Dirac, uniform, and Gamma distributions. With the loss of stability at the boundary of these regions, we discuss the Hopf bifurcation using a normal-form method; there the computation of the coefficients is given explicitly in the form of the corresponding characteristic equation. Using the theoretical results, we study two examples of white blood cell models and address the effect of the distributed time delay on the physiological oscillations. Numerical simulations are illustrated to verify our theoretical predictions. [ABSTRACT FROM AUTHOR]

In this paper, the effects of harvesting and time delay on two different types of predator-prey systems with delayed predator specific growth and Holling type II functional response are studied by applying the normal form theory of retarded functional differential equations developed by Faria andMagalhãaes [ J. Differential Equations, 122 (1995), pp. 181-200, J. Differential Equations, 122 (1995), pp. 201-224]. Hopf bifurcations are demonstrated in models with harvesting of the prey at a constant rate by taking the delay as a bifurcation parameter, and numerical examples supporting our theoretical prediction are also given. Furthermore, bifurcation analysis indicates that delayed predator-prey systems with predator harvesting exhibit Bogdanov-Takens bifurcation. The versal unfoldings of the models at the Bogdanov-Takens singularity are obtained, and numerical simulations and bifurcation diagrams are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

This paper studies a parsimonious model of landslide motion, which consists of the one-dimensional diffusion equation (for pore pressure) coupled through a boundary condition to a first-order ODE (Newton's second law). Velocity weakening of sliding friction gives rise to nonlinearity in the model. Analysis shows that solutions of the model equations exhibit a subcritical Hopf bifurcation in which stable, steady sliding can transition to cyclical, stick-slip motion. Numerical computations confirm the analytical predictions of the parameter values at which bifurcation occurs. The existence of stick-slip behavior in part of the parameter space is particularly noteworthy because, unlike stick-slip behavior in classical models, here it arises in the absence of a reversible (elastic) driving force. Instead, the driving force is static (gravitational), mediated by the effects of pore-pressure diffusion on frictional resistance. [ABSTRACT FROM AUTHOR]

In this paper we define a class of formal neuron models being computationally efficient and biologically plausible, i.e., able to reproduce a wide range of behaviors observed in in vivo or in vitro recordings of cortical neurons. This class includes, for instance, two models widely used in computational neuroscience, the Izhikevich and the Brette-Gerstner models. These models consist of a 4-parameter dynamical system. We provide the full local bifurcation diagram of the members of this class and show that they all present the same bifurcations: an Andronov-Hopf bifurcation manifold, a saddle-node bifurcation manifold, a Bogdanov-Takens bifurcation, and possibly a Bautin bifurcation, i.e., all codimension two local bifurcations in a two-dimensional phase space except the cusp. Among other global bifurcations, this system shows a saddle homoclinic bifurcation curve. We show how this bifurcation diagram generates the most prominent cortical neuron behaviors. This study leads us to introduce a new neuron model, the quartic model, able to reproduce among all the behaviors of the Izhikevich and Brette-Gerstner models self-sustained subthreshold oscillations, which are of great interest in neuroscience. [ABSTRACT FROM AUTHOR]

In this paper, we develop a criterion to calculate the multiplicity of a multiple focus for general predator-prey systems. As applications of this criterion, we calculate the largest multiplicity of a multiple focus in a predator-prey system with nonmonotonic functional response p(x) = x / ax²+bx+1 studied by Zhu, Campbell, and Wolkowicz [SIAM J. Appl. Math., 63 (2002), pp. 636-682] and prove that the degenerate Hopf bifurcation is of codimension two. Furthermore, we show that there exist parameter values for which this system has a unique positive hyperbolic stable equilibrium and exactly two limit cycles, the inner one unstable and outer one stable. Numerical simulations for the existence of the two limit cycles bifurcated from the multiple focus are also given in support of the criterion. [ABSTRACT FROM AUTHOR]

Border zones between normal and ischemic tissue have been implicated as a cause of arrhythmic cardiac activity. A variety of experiments with coupled cells and strips of tissue has been designed to understand the arrhythmogenic effects of border zone currents. In this paper, we use systems of differential equations to model an ischemic (depolarized) cell (or region) coupled to a normal cell under a variety of conditions. For two ionic models (reduced Hodgkin--Huxley and Luo--Rudy I), we find the boundary in parameter space between oscillatory and nonoscillatory solutions. We find that there are regions in parameter space for which the ischemic cell (region) is stable and inexcitable when uncoupled, but when coupled to a normal, excitable cell the cells oscillate. We state a general principle that relates the oscillation of a forced single cell to oscillations of the coupled system. Furthermore, in modeling drug-modified experimental dynamics we are able to reproduce early after depolarization (EAD)--like phenomena, which has implications in locating oscillations in the drug-free experiment. Finally, we describe a mechanism by which oscillations in the transmembrane potential may be encountered during reperfusion. [ABSTRACT FROM AUTHOR]

We develop the principle of linearized stability and a Hopf bifurcation theorem as elements of a geometric theory for fully nonlinear parabolic-hyperbolic problems. Crucial steps in our work are showing the differentiability of the time-t map, showing that the admissible initial data form a manifold (whose failure to be linear is due to the general boundary conditions we study), and analyzing the spectrum of the generator of the linearized semigroup. This paper provides the abstract framework for the study of a class of concrete problems of self-sustained oscillations of nonlinearly viscoelastic bodies like that treated by Antman and Koch [SIAM J. Appl. Math., 60 (2000), pp. 1357-1387]. Our equations are intrinsically interesting: They provide an example of a new kind of semiflow that combines properties of ordinary differential equations and parabolic equations in a novel way. [ABSTRACT FROM AUTHOR]

Increasing evidence shows the importance of cellular communication between human arterial cells in relation to cellular physiology and, in some cases, pathophysiology. Therefore, the main goal of this paper is to show an effective method of studying an unrevealed complex dynamics of a homocellular system by applying multiscale analysis and homogenization. For this, we study similar dynamics of a system of smooth muscle cells (SMCs) signaling in the macroscopic arterial domain. We use a modified version of FitzHugh{Nagumo nonlinear models, and the possibility of using these models in the modeling of SMCs is described as local dynamical similarities between our models and the existing dynamical results of SMCs. We show Hopf bifurcations to be followed by membrane potential oscillations in a ring of discrete cells (RDC). RDC is then homogenized along the arterial axis and the multiscale structure of the cellular system is studied. Moreover, by developing a suitable numerical example, we show the effective diffusion to be more significant in cells of thin membrane than that of thick membrane. Spatial patterning of membrane potential illustrates forward and backward waves moving along the arterial axis, under the action of a spatially varying external driver in the form of a first-order spline. Furthermore, we discover a new dynamics,"spatial dynamical silence"(spatiotemporal propagation failure of waves) based on notions, both "point of initiation" and "point of interference" of waves, which is then discussed in the context of arterial vasomotion. Finally, the existence of a traveling wave solution is proved by constructing a homoclinic orbit corresponding to a critical wave speed. [ABSTRACT FROM AUTHOR]

We develop a mathematical model that describes concentration dynamics of PrPC (Prion Protein Cellular) and PrPSc (Prion Protein Scrapie) prion proteins at the neuron scale and includes the effect of the unfolded protein response (UPR). We first introduce a single neuron model taking the UPR mechanism into account. We investigate it and propose a stability study among which a bifurcation analysis with respect to three of its parameters. Then, we generalize it to two neurons showing PrPSc proteins interaction. Stability results are given when neurons exhibit identical parameters but interact differently (strong, weak, or no interaction). [ABSTRACT FROM AUTHOR]

We study the Becker-Döring bubblelator, a variant of the Becker-Döring coagulationfragmentation system that models the growth of clusters by gain or loss of monomers. Motivated by models of gas evolution oscillators from physical chemistry, we incorporate the injection of monomers and depletion of large clusters. For a wide range of physical rates, the Becker-Döring system itself exhibits a dynamic phase transition as mass density increases past a critical value. We connect the Becker-Döring bubblelator to a transport equation coupled with an integrodifferential equation for the excess monomer density by formal asymptotics in the near-critical regime. For suitable injection/depletion rates, we argue that time-periodic solutions appear via a Hopf bifurcation. Numerics confirm that the generation and removal of large clusters can become desynchronized, leading to temporal oscillations associated with bursts of large-cluster nucleation. [ABSTRACT FROM AUTHOR]

We analyze a class of cell-bulk coupled PDE-ODE models, motivated by quorum-sensing and diffusion-mediated behavior in microbial systems, that characterize communication between localized spatially segregated dynamically active signaling compartments or "cells" that have a permeable boundary. In this model, the cells are disks of a common radius ε ≪ 1 and they are spatially coupled through a passive extracellular bulk diffusion field with diffusivity D in a bounded 2-D domain. Each cell secretes a signaling chemical into the bulk region at a constant rate and receives a feedback of the bulk chemical from the entire collection of cells. This global feedback, which activates signaling pathways within the cells, modifies the intracellular dynamics according to the external environment. The cell secretion and global feedback are regulated by permeability parameters across the cell membrane. For arbitrary reaction-kinetics within each cell, the method of matched asymptotic expansions is used in the limit ε ≪ 1 of small cell radius to construct steady-state solutions of the PDE-ODE model and to derive a nonlinear globally coupled eigenvalue problem (GCEP) that characterizes the linear stability properties of the steady-states. The analysis and computation of the nullspace of the GCEP as parameters are varied is central to the linear stability analysis. In the limit of large bulk diffusivity D = D0/ν ≫ 1, where ν ≡ -1/log ε, an asymptotic analysis of the PDE-ODE model leads to a limiting ODE system for the spatial average of the concentration in the bulk region that is coupled to the intracellular dynamics within the cells. Results from the linear stability theory and ODE dynamics are illustrated for Sel'kov reaction-kinetics, where the kinetic parameters are chosen so that each cell is quiescent when uncoupled from the bulk medium. For various specific spatial configurations of cells, the linear stability theory is used to construct phase diagrams in parameter space characterizing where a switch-like emergence of intracellular oscillations can occur through a Hopf bifurcation. The effect of the membrane permeability parameters, the reaction-kinetic parameters, the bulk diffusivity, and the spatial configuration of cells on both the emergence and synchronization of the oscillatory intracellular dynamics, as mediated by the bulk diffusion field, is analyzed in detail. The linear stability theory is validated from full numerical simulations of the PDE-ODE system, and from the reduced ODE model when D is large. [ABSTRACT FROM AUTHOR]

Pseudospectral approximation reduces delay differential equations (DDE) to ordinary differential equations (ODE). Next one can use ODE tools to perform a numerical bifurcation analysis. By way of an example we show that this yields an efficient and reliable method to qualitatively as well as quantitatively analyze certain DDE. To substantiate the method, we next show that the structure of the approximating ODE is reminiscent of the structure of the generator of translation along solutions of the DDE. Concentrating on the Hopf bifurcation, we then exploit this similarity to reveal the connection between DDE and ODE bifurcation coefficients and to prove the convergence of the latter to the former when the dimension approaches infinity. [ABSTRACT FROM AUTHOR]

We analyze a coupled bulk-membrane PDE model in which a scalar linear 2-D bulk diffusion process is coupled through a linear Robin boundary condition to a two-component 1-D reaction-diffusion (RD) system with Gierer--Meinhardt (nonlinear) reaction kinetics defined on the domain boundary. For this coupled model, in the singularly perturbed limit of a long-range inhibition and short-range activation for the membrane-bound species, asymptotic methods are used to analyze the existence of localized steady-state multispike membrane-bound patterns and to derive a nonlocal eigenvalue problem (NLEP) characterizing the \scrO (1) time-scale instabilities of these patterns. A central, and novel, feature of this NLEP is that it involves a membrane Green's function that is coupled nonlocally to a bulk Green's function. When the domain is a disk, or in the well-mixed shadow-system limit corresponding to an infinite bulk diffusivity, this Green's function problem is analytically tractable, and as a result we will use a hybrid analytical-numerical approach to determine unstable spectra of this NLEP. This analysis characterizes how the 2-D bulk diffusion process and the bulk-membrane coupling modify the well-known linear stability properties of steady-state spike patterns for the 1-D Gierer--Meinhardt model in the absence of coupling. In particular, phase diagrams in parameter space for our coupled model characterizing either oscillatory instabilities due to Hopf bifurcations or competition instabilities due to zero-eigenvalue crossings are constructed. Finally, linear stability predictions from the NLEP analysis are confirmed with full numerical finite element simulations of the coupled PDE system. We remark that our approach is valid in more general settings than the disk or the well-mixed shadow system, with the key hurdle being the computation of the relevant Green's functions. By restricting our detailed analysis to these two specialized cases, we can bypass the computational challenges of calculating the Green's functions and therefore focus instead on the novel effects of coupling on the construction and stability of multispike solutions. [ABSTRACT FROM AUTHOR]

A network is introduced to describe the spatiotemporal dynamics of two-species competitive and allelopathic plankton models, where the network structure represents the movement directions be- tween every two patches. Time delay is also incorporated to describe the time required to produce stimulatory effect of one species on the growth of the other species. The model is described by a system of discrete-space and continuous-time equations with time delay in a network. Using the time delay as a bifurcation parameter, it is shown that a Hopf bifurcation occurs in the system. The stability of the Hopf bifurcation is also considered by applying the center manifold theory. Numerical simulations reveal that the stability of Hopf bifurcation leads to the emergence of planktonic blooms. Moreover, it is found that the network structure can switch the types of spatiotemporal patterns, a new feature observed only in delay differential equations with network structure. [ABSTRACT FROM AUTHOR]

Internet and mobile services often provide waiting time or queue length information to customers. This information allows a customer to determine whether to remain in line or, in the case of multiple lines, better decide which line to join. Unfortunately, there is usually a delay associated with waiting time information. Either the information itself is stale, or it takes time for the customers to travel to the service location after having received the information. Recent empirical and theoretical work uses functional dynamical systems as limiting models for stochastic queueing systems. This work has shown that if information is delayed long enough, a Hopf bifurcation can occur and cause unwanted oscillations in the queues. However, it is not known how large the oscillations are when a Hopf bifurcation occurs. To answer this question, we model queues with functional differential equations and implement two methods for approximating the amplitude of these oscillations. The first approximation is analytic and yields a closed-form approximation in terms of the model parameters. The second approximation uses a statistical technique, and delivers highly accurate approximations over a wider range of parameters. [ABSTRACT FROM AUTHOR]

We analyze the existence and linear stability of steady-state localized hotspot patterns for a 1-D three-component singularly perturbed reaction-diffusion (RD) system modeling urban crime in the presence of police intervention. Our three-component RD model augments the two-component system for an attractiveness field and criminal density, as introduced by Short et al. [Math. Models Methods Appl. Sci., 18 (2008), suppl., pp. 1249-1267], by including the effect of a police deployment that exhibits a biased random walk toward maxima of the attractiveness field. In our model, the rate at which criminals are introduced is decreased by the total level of police deployment, and the strength of the bias in the police random walk toward the maxima of the attractiveness field is modeled by a patrol focus parameter, q > 0. For our three-component model, hotspot steady-state patterns are constructed asymptotically and, from a detailed derivation and analysis of certain nonlocal eigenvalue problems (NLEPs), phase diagrams in parameter space are obtained that characterize regions of linear stability of the steady-state pattern. In certain parameter regimes, we show that the police intervention leads to a rapid annihilation of some hotspots, whereas in other parameter regimes, notably when the police diffusivity is below a threshold value, the police intervention only displaces crime periodically to neighboring spatial regions (at least on short time-scales). Mathematically, we show that this crime displacement effect arises due to a Hopf bifurcation in the NLEP associated with certain asynchronous modes of instability of the steady-state hotspot pattern. Such robust asynchronous temporal oscillations of the hotspot amplitudes in our three-component system is a new phenomenon, which does not typically occur in two-component RD systems. The effect of a ''cops-on-the-dots" patrol strategy, corresponding to q = 2, in which the police mimic the bias of the criminals toward spatial maxima of the attractiveness, is examined through a combination of rigorous spectral results and a numerical parameterization of any Hopf bifurcation threshold. For the special choice q = 3, we show that explicit linear stability results can be readily obtained from the NLEP. Our linear stability results are validated through full numerical PDE simulations of the three-component RD system. [ABSTRACT FROM AUTHOR]

For three specific singularly perturbed two-component reaction diffusion systems in a bounded 2-D domain admitting localized multi-spot patterns, we provide a detailed analysis of the parameter values for the onset of temporal oscillations of the spot amplitudes. The two key bifurcation parameters in each of the RD systems are the reaction-time parameter τ and the inhibitor diffusivity D. In the limit of large diffusivity D = D0/ν ≫ 1, with D0 = O(1), ν ≡ -1/log ε and ε denoting the activator diffusivity, a leading-order-in-ν analysis shows that the linear stability of multi-spot patterns is determined by the spectrum of a class of nonlocal eigenvalue problems (NLEPs). The specific form for these NLEPs depends on whether τ = O(1) or τ ≫ 1. For D0 < D0c, where D0c > 0 is some critical threshold, we show from a new parameterization of the NLEP that no Hopf bifurcations leading to temporal oscillations in the spot amplitudes can occur for any O(1) value of the reaction-time parameter τ. This resolves a long-standing open problem in NLEP theory (see [33] [J. Wei, M. Winter Mathematical aspects of pattern formation in biological systems, Applied Mathematical Science Series, Vol. 189, Springer, (2014)]). Instead, by deriving a new modified NLEP appropriate to the regime τ ≫ 1, we show for the range D0 < D0c that a Hopf bifurcation will occur at some τ = τH ≫ 1, where τH has the anomalous scaling law τH ∼ ν-1εc ≫ 1, for some τc satisfying 0 < τc < 2. The anomalous exponent τc is calculated from the modified NLEP for each of the three RD systems. [ABSTRACT FROM AUTHOR]

Periodic traveling waves (wavetrains) have been an invaluable tool in the understanding of spatiotemporal oscillations observed in ecological data sets. Various mechanisms are known to trigger this behavior, but here we focus on invasion, resulting in a predator--prey-type interaction. Previous work has focused on the normal form reduction of PDE models to the well-understood λ-ω equations near a Hopf bifurcation, though this is valid only when assuming an equal rate of dispersion for both predators and prey--an unrealistic assumption for many ecosystems. By relaxing this constraint, we obtain the complex Ginzburg--Landau normal form equation, which has a one-parameter family of periodic traveling wave solutions, parametrized by the amplitude. We derive a formula for the wave amplitude selected by invasion before investigating the stability of the solutions. This gives us a complete description of small-amplitude periodic traveling waves in the governing model ecosystem. [ABSTRACT FROM AUTHOR]

A linear stability analysis of localized spike solutions to the singularly perturbed two-component Gierer--Meinhardt (GM) reaction-diffusion (RD) system with a fixed time delay T in the nonlinear reaction kinetics is performed. Our analysis of this model is motivated by the computational study of Lee, Gaffney, and Monk [Bull. Math. Bio., 72 (2010), pp. 2139--2160] on the effect of gene expression time delays on spatial patterning for both the GM model and some related RD models. It is shown that the linear stability properties of such localized spike solutions are characterized by the discrete spectra of certain nonlocal eigenvalue problems (NLEP). Phase diagrams consisting of regions in parameter space where the steady-state spike solution is linearly stable are determined for various limiting forms of the GM model in both 1-dimensional and 2-dimensional domains. On the boundary of the region of stability, the spike solution is found to undergo a Hopf bifurcation. For a special range of exponents in the nonlinearities for the 1-dimensional GM model, and assuming that the time delay only occurs in the inhibitor kinetics, this Hopf bifurcation boundary is readily determined analytically. For this special range of exponents, the challenging problem of locating the discrete spectrum of the NLEP is reduced to the much simpler problem of locating the roots to a simple transcendental equation in the eigenvalue parameter. By using a hybrid analytical-numerical method, based on a parametrization of the NLEP, it is shown that qualitatively similar phase diagrams occur for general GM exponent sets and for the more biologically relevant case where the time delay occurs in both the activator and inhibitor kinetics. Overall, our results show that there is a critical value T* of the delay for which the spike solution is unconditionally unstable for T>T*, and that the parameter region where linear stability is assured is, in general, rather limited. A comparison of the theory with full numerical results computed from the RD system with delayed reaction kinetics for a particular parameter set suggests that the Hopf bifurcation can be subcritical, leading to a global breakdown of a robust spatial patterning mechanism. [ABSTRACT FROM AUTHOR]

Spatial-temporal dynamics associated with a class of coupled membrane-bulk PDE-ODE models in one spatial dimension is analyzed using a combination of linear stability theory, numerical bifurcation software, and full time-dependent simulations. In our simplified one-dimensional setting, the mathematical model consists of two dynamically active membranes, separated spatially by a distance 2L, that are coupled together through a linear bulk diffusion field, with a fixed bulk decay rate. The coupling of the bulk and active membranes arises through both nonlinear flux boundary conditions for the bulk diffusion field and from feedback terms, depending on the local bulk concentration, to the dynamics on each membrane. For this class of models, it is shown both analytically and numerically that bulk diffusion can trigger a synchronous oscillatory instability in the temporal dynamics associated with the two active membranes. For the case of a single active component on each membrane, and in the limit L? ∞, rigorous spectral results for the linearization around a steady-state solution, characterizing the possibility of Hopf bifurcations and temporal oscillations in the membranes, are obtained. For finite L, a weakly nonlinear theory, accounting for eigenvalue-dependent boundary conditions appearing in the linearization, is developed to predict the local branching behavior near the Hopf bifurcation point. The analytical theory, together with numerical bifurcation results and full numerical simulations of the PDE-ODE system, is undertaken for various coupled membrane-bulk systems, including two specific biologically relevant applications. Our results show the existence of a wide parameter range where stable synchronous oscillatory dynamics in the two membranes can occur. [ABSTRACT FROM AUTHOR]

Neural field equations are integro-differential systems describing the macroscopic activity of spatially extended pieces of cortex. In such cortical assemblies, the propagation of information and the transmission machinery induce communication delays, due to the transport of information (propagation delays) and to the synaptic machinery (constant delays). We investigate the role of these delays on the formation of structured spatio-temporal patterns for these systems in arbitrary dimensions. We focus on localized activity, either induced by the presence of a localized stimulus (pulses) or by transitions between two levels of activity (fronts). Linear stability analysis allows us to reveal the existence of Hopf bifurcation curves induced by the delays, along different modes that may be symmetric or asymmetric. We show that instabilities strongly depend on the dimension, and in particular may exhibit transversal instabilities along invariant directions. These instabilities yield pulsatile localized activity, and, depending on the symmetry of the destabilized modes, produce either spatio-temporal breathing or sloshing patterns. [ABSTRACT FROM AUTHOR]

We propose a model to analyze the dynamics of interacting proliferating and quiescent cell populations. The model includes age dependence of cell division, transitions between the two subpopulations, and regulation of the recruitment of quiescent cells. We formulate the model as a pair of renewal equations and apply a rather recent general result to prove that (in)stability of equilibria can be analyzed by locating roots of characteristic equations. We are led to a parameter plane analysis of a characteristic equation, which has not been analyzed in this way so far. We conclude with how quiescence of cells as well as two submodels for cell division may influence the possibility of destabilization via oscillations. [ABSTRACT FROM AUTHOR]

In the singularly perturbed limit of an asymptotically small diffusivity ratio ε2, the existence and stability of localized quasi-equilibrium multispot patterns is analyzed for the Brusselator reaction-diffusion model on the unit sphere. Formal asymptotic methods are used to derive a nonlinear algebraic system that characterizes quasi-equilibrium spot patterns and to formulate eigenvalue problems governing the stability of spot patterns to three types of "fast" O(1) time-scale instabilities: self-replication, competition, and oscillatory instabilities of the spot amplitudes. The nonlinear algebraic system and the spectral problems are then studied using simple numerical methods, with emphasis on the special case where the spots have a common amplitude. Overall, the theoretical framework provides a hybrid asymptotic-numerical characterization of the existence and stability of spot patterns that is asymptotically correct to within all logarithmic correction terms in powers of v = -1/log ε. From a leading-order-in-v analysis, and with an asymptotically large inhibitor diffusivity, some rigorous results for competition and oscillatory instabilities are obtained from an analysis of a new class of nonlocal eigenvalue problem (NLEP). Theoretical results for the stability of spot patterns are confirmed with full numerical computations of the Brusselator PDE system on the sphere using the closest point method. [ABSTRACT FROM AUTHOR]

The identification of instability in large-scale dynamical systems caused by Hopf bifurcation is difficult because of the problem of identifying the rightmost pair of complex eigenvalues of large sparse generalized eigenvalue problems. A new method developed in [K. Meerbergen and A. Spence, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1982-1999] avoids this computation, instead performing an inverse iteration for a certain set of real eigenvalues that requires the solution of a large-scale Lyapunov equation at each iteration. In this study, we refine the Lyapunov inverse iteration method to make it more robust and efficient, and we examine its performance on challenging test problems arising from fluid dynamics. Various implementation issues are discussed, including the use of inexact inner iterations and the impact of the choice of iterative solution for the Lyapunov equations, and the effect of eigenvalue distribution on performance. Numerical experiments demonstrate the robustness of the algorithm. [ABSTRACT FROM AUTHOR]

Using an equation-free analysis approach we identify a Hopf bifurcation point and perform a two-parameter continuation of the Hopf point for the macroscopic dynamical behavior of an interacting particle model. Due to the nature of systems with a moderate number of particles and noise, the quality of the available numerical information requires the use of very robust numerical algorithms for each of the building blocks of the equation-free methodology. As an example, we consider a particle model of a crowd of pedestrians where particles interact through pairwise "social forces." The pedestrians move along a corridor where they are constrained by the walls of the corridor, and two crowds are aiming, from opposite directions, to pass through a narrowing doorway perpendicular to the corridor. We focus our investigation on the collective behavior of the model. As the width of the doorway is increased, we observe an onset of oscillations of the net pedestrian flux through the doorway, described by a Hopf bifurcation. An equation-free continuation of the Hopf point in the two parameters, door width and ratio of the pedestrian velocities of the two crowds, is performed. [ABSTRACT FROM AUTHOR]

We analyze the weakly nonlinear stability of a stationary pulse undergoing a Hopf bifurcation in a neural field model with an excitatory or Mexican hat synaptic weight function and Heaviside firing rate nonlinearity. The presence of a spatially localized input inhomogeneity I(x) precludes the 0 eigenvalue related to translation invariance of the pulse. Consequently, in the spectral analysis of the linearization about the stationary pulse U(x), there are two spatial modes, either of which can undergo a Hopf bifurcation in the Mexican hat network to produce a periodic orbit that either expands/contracts (breather) or moves side-to-side (slosher). We derive the normal form for each mode becoming critical in the Hopf bifurcation by (I) the method of amplitude equations and (ii) center manifold reduction, which are shown to agree. Importantly, the critical third order coefficient of the normal form is found to be in strong agreement with numerical simulations of the full model, particularly when the bifurcation switches from super-to subcritical. The motivation of this work is to establish the framework for a perturbative analysis of the breather in a neighborhood of the Hopf bifurcation point to study weakly interacting breathers analytically. [ABSTRACT FROM AUTHOR]

The dynamics and stability of multispot patterns to the Gray--Scott (GS) reaction-diffusion model in a two-dimensional domain is studied in the singularly perturbed limit of small diffusivity ε of one of the two solution components. A hybrid asymptotic-numerical approach based on combining the method of matched asymptotic expansions with the detailed numerical study of certain eigenvalue problems is used to predict the dynamical behavior and instability mechanisms of multispot quasi-equilibrium patterns for the GS model in the limit ε → 0. For ε → 0, a quasi-equilibrium κ-spot pattern is constructed by representing each localized spot as a logarithmic singularity of unknown strength Sj for j =1,…,k at an unknown spot location xj ∈ Ω for j =1,…,κ. A formal asymptotic analysis is then used to derive a differential algebraic ODE system for the collective coordinates Sj and xj for j =1,…,κ, which characterizes the slow dynamics of a spot pattern. Instabilities of the multispot pattern due to the three distinct mechanisms of spot self-replication, spot oscillation, and spot annihilation are studied by first deriving certain associated eigenvalue problems by using singular perturbation techniques. From a numerical computation of the spectrum of these eigenvalue problems, phase diagrams in the GS parameter space corresponding to the onset of spot instabilities are obtained for various simple spatial configurations of multispot patterns. In addition, it is shown that there is a wide parameter range where a spot instability can be triggered only as a result of the intrinsic slow motion of the collection of spots. The construction of the quasi-equilibrium multispot patterns and the numerical study of the spectrum of the eigenvalue problems relies on certain detailed properties of the reduced-wave Green's function. The hybrid asymptotic-numerical results for spot dynamics and spot instabilities are validated from full numerical results computed from the GS model for various spatial configurations of spots. [ABSTRACT FROM AUTHOR]

We propose a mathematical model describing the dynamics of a hematopoietic stem cell population. The method of characteristics reduces the age-structured model to a system of differential equations with a state-dependent delay. A detailed stability analysis is performed. A sufficient condition for the global asymptotic stability of the trivial steady state is obtained using a Lyapunov-Razumikhin function. A unique positive steady state is shown to appear through a transcritical bifurcation of the trivial steady state. The analysis of the positive steady state behavior, through the study of a first order exponential polynomial characteristic equation, concludes the existence of a Hopf bifurcation and gives criteria for stability switches. A numerical analysis confirms the results and stresses the role of each parameter involved in the system on the stability of the positive steady state. [ABSTRACT FROM AUTHOR]

Hopf bifurcations have been studied intensively in two dimensional vector fields with one slow and one fast variable [É. Benoît et al., Collect. Math., 31 (1981), pp. 37-119; F. Dumortier and R. Roussarie, Mem. Amer. Math. Soc., 121 (577) (1996); W. Eckhaus, in Asymptotic Analysis II, Lecture Notes in Math. 985, Springer-Verlag, Berlin, 1983, pp. 449-494; M. Krupa and P. Szmolyan, SIAM J. Math. Anal., 33 (2001), pp. 286-314; J. Guckenheimer, in Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, NATO Sci. Ser. II Math. Phys. Chem. 137, Kluwer, Dordrecht, The Netherlands, 2004, pp. 295-316]. Canard explosions are associated with these singular Hopf bifurcations [S. M. Baer and T. Erneux, SIAM J. Appl. Math., 46 (1986), pp. 721-739; S. M. Baer and T. Erneux, SIAM J. Appl. Math., 52 (1992), pp. 1651-1664; B. Braaksma, J. Nonlinear Sci., 8 (1998), pp. 457-490; Y. Lijun and Z. Xianwu, J. Differential Equations, 206 (2004), pp. 30-54], manifested by a very rapid growth in the amplitude of periodic orbits. There has been less analysis of Hopf bifurcations in slow-fast systems with two slow variables where singular Hopf bifurcation occurs simultaneously with type II folded saddle-nodes [A. Milik and P. Szmolyan, in Multiple-Time-Scale Dynamical Systems, IMA Vol. Math. Appl. 122, Springer-Verlag, New York, 2001, pp. 117-140; M. Wechselberger, SIAM J. Appl. Dyn. Syst., 4 (2005), pp. 101-139]. This work contributes to our understanding of these Hopf bifurcations in five ways: (1) it computes the first Lyapunov coefficient of the bifurcation in terms of a normal form, (2) it describes global features of the flow that constrain the types of trajectories found in the system near the bifurcation, (3) it identifies codimension two bifurcations that occur as coefficients in the normal form vary, (4) it exhibits complex solutions that occur in the vicinity of the bifurcation for some values of the normal form coefficients, and (5) it identifies singular Hopf bifurcation as a mechanism for the creation of mixed-mode oscillations. A subtle aspect of the normal form is that terms of higher order contribute to the first Lyapunov coefficient of the bifurcation in an essential way. [ABSTRACT FROM AUTHOR]

Stable periodic solutions of a system of two nonlocal coupled complex Ginzburg-Landau (CGL) equations describing the dynamics of a subcritical Hopf bifurcation in a spatially extended system are found analytically in the limit of large dispersion coefficients. The domains in the parameter space where these solutions exist and are stable are determined. It is shown that the existence and stability depend on the sign of the coupling parameter and on the ratio of the dispersion coefficients. Numerical simulations of the system of nonlocal coupled CGL equations confirm the analytical results and exhibit other bounded dynamic regimes, such as standing waves and spatio-temporal chaos. [ABSTRACT FROM AUTHOR]

We show that the perturbation theory for dual semigroups (sun-star-calculus) that has proved useful for analyzing delay-differential equations is equally efficient for dealing with Volterra functional equations. In particular, we obtain both the stability and instability parts of the principle of linearized stability and the Hopf bifurcation theorem. Our results apply to situations in which the instability part has not been proved before. In applications to general physiologically structured populations even the stability part is new. [ABSTRACT FROM AUTHOR]

The vestibular system in almost all vertebrates, and in particular in humans, controls balance by employing a set of six semicircular canals, three in each inner ear, to detect angular accelerations of the head in three mutually orthogonal coordinate planes. Signals from the canals are transmitted to eight (groups of) neck motoneurons, which activate the eight corresponding muscle groups. These signals may be either excitatory or inhibitory, depending on the direction of head acceleration. McCollum and Boyle have observed that in the cat the relevant network of neurons possesses octahedral symmetry, a structure that they deduce from the known innervation patterns (connections) from canals to muscles. We rederive the octahedral symmetry from mathematical features of the probable network architecture, and model the movement of the head in response to the activation patterns of the muscles concerned. We assume that connections between neck muscles can be modeled by a ‘coupled cell network,’ a system of coupled ODEs whose variables correspond to the eight muscles, and that this network also has octahedral symmetry. The network and its symmetries imply that these ODEs must be equivariant under a suitable action of the octahedral group. It is observed that muscle motoneurons form natural ‘push-pull pairs’ in which, for given movements of the head, one neuron produces an excitatory signal, whereas the other produces an inhibitory signal. By incorporating this feature into the mathematics in a natural way, we are led to a model in which the octahedral group acts by signed permutations on muscle motoneurons. We show that with the appropriate group actions, there are six possible spatiotemporal patterns of time-periodic states that can arise by Hopf bifurcation from an equilibrium representing an immobile head. Here we use results of Ashwin and Podvigina. Counting conjugate states, whose physiological interpretations can have significantly different features, there are 15 patterns of periodic oscillation, not counting left-right reflections or time-reversals as being different. We interpret these patterns as motions of the head, and note that all six types of pattern appear to correspond to natural head motions. [ABSTRACT FROM AUTHOR]

Several recent experiments on DNA-damage-induced signaling networks in mammalian cells have shown interesting dynamics in p53 protein expression during the repair cycle. Pulses of p53 are produced, whose frequency and amplitude are fairly independent of the amount of damage, but the probability of a cell exhibiting this pulsatile behavior increases with damage. This phenomenon has been described as a ‘digital oscillator.’ We present here a simple model oscillator comprising two species, p53 and Mdm2, which is activated by the Atm kinase. The Atm kinase exhibits bistable switch-like behavior. The network dynamics essentially consists of the core p53 oscillator, which is turned ON/OFF by the Atm switch, which is in turn activated by DNA damage. The complex dynamics are thus explained by the modular nature of the network and are fairly independent of the biological details. A stochastic model of the network dynamics reveals that the pulsatile behavior is robust to intrinsic noise of the protein components and extrinsic noise which arises due to noisy damage signals. The robustness is due to the bistable switch, which makes the system more resilient to stochastic fluctuations in its components. However, the system is more susceptible to noise in the Mdm2 protein production rate. [ABSTRACT FROM AUTHOR]

In this article, we consider parameter-dependent systems of reaction-diffusion equations on the sphere, which are equivariant under the group SO(3) of all rigid rotations on the sphere. It is known that the transition from rotating waves to modulated rotating waves on the sphere can be explained via a supercritical Hopf bifurcation from a rotating wave, SO(3)-symmetry, and finite-dimensional equivariant center manifold reduction. Using Floquet theory, it is easy to get the decomposition of these modulated rotating waves into the primary frequency vector part and the periodic part. Going further, we use the Baker-Campbell-Hausdor. (BCH) formula in the Lie algebra so(3) to get the closed form of the reduced differential equations on so(3), and then closed formulas for the primary frequency vectors and for the periodic parts associated to the bifurcating modulated rotating waves. As a consequence, we get the explicit characterization of these modulated rotating waves and all possible tip motions on the sphere: quasi-periodically meandering and slowly drifting. This approach does not treat separately the resonant and nonresonant Hopf bifurcation of a rotating wave on the sphere; the change that appears in the resonant case in the Taylor expansions for the primary frequency vectors is implicitly captured by using the BCH formula in so(3), and it is independent of the normal forms theory developed in [B. Fiedler and D. Turaev, Arch. Ration. Mech. Anal., 145 (1998), pp. 129-159]. When systems with two parameters are involved and the norm of the frequency vector of the initial rotating wave undergoing Hopf bifurcation is a multiple integer of the critical eigenvalue leading to Hopf bifurcation, we give a shorter and more intuitive proof for the following fact stated in [C. Wulff, Doc. Math., 5 (2000), pp. 227-274] for a general Lie group: the primary frequency vectors of a branch of these modulated rotating waves are generically orthogonal to the frequency vector of the initial rotating wave undergoing Hopf bifurcation, and their tip motions are slowly drifting along the equator of the sphere. Due to the computational nature of the BCH formula in so(3), this approach can be translated in a computer-implemented method which will allow us to better control the tip motions of the modulated rotating waves on the sphere. [ABSTRACT FROM AUTHOR]