Using a signature function to determine resonant and attenuant 2-cycles in the Smith–Slatkin population model

We study the responses of discretely reproducing populations to periodic fluctuations in three parameters: the carrying capacity and two demographic characteristics of the species. We prove that small 2-periodic fluctuations of the three parameters generate 2-cyclic oscillations of the population. We develop a signature function for predicting the responses of populations to 2-periodic fluctuations. Our signature function is the sign of a weighted sum of the relative strengths of the oscillations of the three parameters. Periodic environments are deleterious for populations when the signature function is negative, while positive signature functions signal favorable environments. We compute the signature function for the Smith–Slatkin model, and use it to determine regions in parameter space that are either favorable or detrimental to the species.     

Dynamically Active Compartments Coupled by a Stochastically Gated Gap Junction

We analyze a one-dimensional PDE-ODE system representing the diffusion of signaling molecules between two cells coupled by a stochastically gated gap junction. We assume that signaling molecules diffuse within the cytoplasm of each cell and then either bind to some active region of the cell’s membrane (treated as a well-mixed compartment) or pass through the gap junction to the interior of the other cell. We treat the gap junction as a randomly fluctuating gate that switches between an open and a closed state according to a two-state Markov process. This means that the resulting PDE-ODE is stochastic due to the presence of a randomly switching boundary in the interior of the domain. It is assumed that each membrane compartment acts as a conditional oscillator, that is, it sits below a supercritical Hopf bifurcation. In the ungated case (gap junction always open), the system supports diffusion-induced oscillations, in which the concentration of signaling molecules within the two compartments is either in-phase or anti-phase. The presence of a reflection symmetry (for identical cells) means that the stochastic gate only affects the existence of anti-phase oscillations. In particular, there exist parameter choices where the gated system supports oscillations, but the ungated system does not, and vice versa. The existence of oscillations is investigated by solving a spectral problem obtained by averaging over realizations of the stochastic gate.     

An Asymptotic Analysis of a 2-D Model of Dynamically Active Compartments Coupled by Bulk Diffusion

A class of coupled cell–bulk ODE–PDE models is formulated and analyzed in a two-dimensional domain, which is relevant to studying quorum-sensing behavior on thin substrates. In this model, spatially segregated dynamically active signaling cells of a common small radius \(\epsilon \ll 1\) are coupled through a passive bulk diffusion field. For this coupled system, the method of matched asymptotic expansions is used to construct steady-state solutions and to formulate a spectral problem that characterizes the linear stability properties of the steady-state solutions, with the aim of predicting whether temporal oscillations can be triggered by the cell–bulk coupling. Phase diagrams in parameter space where such collective oscillations can occur, as obtained from our linear stability analysis, are illustrated for two specific choices of the intracellular kinetics. In the limit of very large bulk diffusion, it is shown that solutions to the ODE–PDE cell–bulk system can be approximated by a finite-dimensional dynamical system. This limiting system is studied both analytically, using a linear stability analysis and, globally, using numerical bifurcation software. For one illustrative example of the theory, it is shown that when the number of cells exceeds some critical number, i.e., when a quorum is attained, the passive bulk diffusion field can trigger oscillations through a Hopf bifurcation that would otherwise not occur without the coupling. Moreover, for two specific models for the intracellular dynamics, we show that there are rather wide regions in parameter space where these triggered oscillations are synchronous in nature. Unless the bulk diffusivity is asymptotically large, it is shown that a diffusion-sensing behavior is possible whereby more clustered spatial configurations of cells inside the domain lead to larger regions in parameter space where synchronous collective oscillations between the small cells can occur. Finally, the linear stability analysis for these cell–bulk models is shown to be qualitatively rather similar to the linear stability analysis of localized spot patterns for activator–inhibitor reaction–diffusion systems in the limit of long-range inhibition and short-range activation.     

Bifurcation analysis of a delayed mathematical model for tumor growth

In this study, we present a modified mathematical model of tumor growth by introducing discrete time delay in interaction terms. The model describes the interaction between tumor cells, healthy tissue cells (host cells) and immune effector cells. The goal of this study is to obtain a better compatibility with reality for which we introduced the discrete time delay in the interaction between tumor cells and host cells. We investigate the local stability of the non-negative equilibria and the existence of Hopf-bifurcation by considering the discrete time delay as a bifurcation parameter. We estimate the length of delay to preserve the stability of bifurcating periodic solutions, which gives an idea about the mode of action for controlling oscillations in the tumor growth. Numerical simulations of the model confirm the analytical findings.     

Chaos in a bienzymatic cyclic model with two autocatalytic loops

A general bienzymatic cyclic system including two autocatalytic loops is studied and used as a basic design principle for modelling extracellular matrix turnover. Using classical enzyme kinetic rates, the model is described by a set of four ordinary differential equations and numerically studied by bifurcation diagrams and Poincaré sections. We observe limit-cycle oscillations and chaotic behaviors arising from period-doubling cascades or intermittency. Chaotic oscillations originate from distinct strange attractors that undergo boundary and internal crisis. For some parameter values, the system presents several bistable areas, where a limit cycle coexists with another one or with a strange attractor. The dynamics are qualitatively modified when the weight of the autocatalytic loops on the system varies, resulting in the change in the number of attractors.     

An ecogenetic model

A model for the effects of a predator on a genetically distinguished prey population is formulated and investigated. The predator-free system settles at an equilibrium which can be destabilized by the predators if a suitably defined parameter, the predator invasion number, exceeds a threshold. The system can then coexist at a stable equilibrium or via persistent oscillations.     

Pacemaker dynamics in the full Morris–Lecar model

This article reports the finding of pacemaker dynamics in certain region of the parameter space of the three-dimensional version of the Morris–Lecar model for the voltage oscillations of a muscle cell. This means that the cell membrane potential displays sustained oscillations in the absence of an external electrical stimulation. The development of this dynamic behavior is shown to be tied to the strength of the leak current contained in the model. The approach followed is mostly based on the use of linear stability analysis and numerical continuation techniques. In this way it is shown that the oscillatory dynamics is associated to the existence of two Hopf bifurcations, one subcritical and other supercritical. Moreover, it is explained that in the region of parameter values most commonly studied for this model such pacemaker dynamics is not displayed because of the development of two fold bifurcations, with the increase of the strength of the leak current, whose interaction with the Hopf bifurcations destroys the oscillatory dynamics.     

Multiple Timescales, Mixed Mode Oscillations and Canards in Models of Intracellular Calcium Dynamics

A range of representative models of intracellular calcium dynamics are surveyed, with the aim of determining which model characteristics are qualitatively unchanged by changes to details of the model components. Techniques from geometric singular perturbation theory are used to investigate the role of separation of timescales in determining model dynamics, with particular emphasis on identifying parameter regimes in which mixed mode oscillations are present as a result of the separation of timescales. We find that the number of distinct timescales and the number of variables evolving on each timescale varies between models and depends on both the model assumptions and on the parameter regime of interest within the model, but in all cases, the presence of canards and associated mixed mode oscillations provides a mechanism by which the models can robustly exhibit complex oscillations, with the frequency of oscillation depending sensitively on parameter values. We find that analysis of the number and nature of the distinct timescales in a model allows us to make useful predictions about the dynamics associated with the model, and that this may give us more information about the model dynamics than a classification according to the modelling assumptions made about different cellular mechanisms in deriving the models.     

Numerical Analysis of the Synchronization of Coupled Chemical Oscillators

A reaction–diffusion model describing a system of coupled oscillators is constructed and investigated. The oscillators in this study are chemical oscillators that represent an oscillatory heterogeneous catalytic reaction in a granular catalyst layer. The oscillators are arranged serially in the reagent stream and are coupled through the gaseous phase. The dynamic behavior of the system is investigated as a function of the main external parameter — the partial pressure of one of the reagents in the gaseous phase. Existence regions of regular and chaotic oscillations are identified. Synchronization conditions are established for the oscillations in such a chain of coupled chemical oscillators.     

Autoresonance in a dynamic system

Results on the asymptotic analysis of a model of autoresonance using a natural small parameter, the amplitude of forcing oscillations, are presented. The given forcing perturbation is rapidly oscillating oscillations with small amplitude and slow varying frequency. The goal of the paper is to reveal conditions under which the trajectory of the system goes away from the initial equilibrium point by a distance of order 1 under the influence of such a perturbation.Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 5, Asymptotic Methods, 2003.     

Forced oscillations with a sliding regime range of a two-mass system interacting with a fixed stop : PMM vol. 37, n6, 1973, pp. 999–1006

We investigate, by the method developed in [1]. the forced oscillations with a sliding regime range of a two-mass system with elastic connection between the elements, impacting a fixed stop. The system being considered is a dynamic model for a number of vibrational mechanisms. Forced oscillations with a sliding regime range of a system with shock interactions are periodic motions accompanied by a period of an infinite succession of instantaneous collisions of two fixed elements of the model [2]. Within the framework of conditions of roughness of the parameter space [3], in this paper we study by the method of [1] periodic motions with a sliding regime range of a two-mass system with a stop. This problem was posed because in real systems the velocity recovery factor R changes from shock to shock, mainly taking small values (0, 0.2). At the same time, the regions of realizability of one-impact oscillations, in practice the most essential ones among motions with a finite number of interactions over a period, narrow down sharply as R decreases and becomes very small even for R < 0.6 [4]. Thus, the stability of the given operation can be ensured by a law of motion which is independent or weakly dependent on R (^*) (see footnote on the next page). By virtue of what has been said above, finite-impact periodic modes are little suitable for this purpose. Regions, delineated in the parameter space of the model being considered, of existence of stable periodic motions with a sliding regime range have proved to be sufficiently broad. By virtue of the adopted approximation of the sliding regime, the dynamic characteristics of these motions do not depend upon R. The circumstances mentioned confirm the practical value of motions with a sliding regime range in dynamic systems with impact interactions.     

Traveling waves in a coarse-grained model of volume-filling cell invasion: Simulations and comparisons

Many reaction–diffusion models produce traveling wave solutions that can be interpreted as waves of invasion in biological scenarios such as wound healing or tumor growth. These partial differential equation models have since been adapted to describe the interactions between cells and extracellular matrix (ECM), using a variety of different underlying assumptions. In this work, we derive a system of reaction–diffusion equations, with cross-species density-dependent diffusion, by coarse-graining an agent-based, volume-filling model of cell invasion into ECM. We study the resulting traveling wave solutions both numerically and analytically across various parameter regimes. Subsequently, we perform a systematic comparison between the behaviors observed in this model and those predicted by simpler models in the literature that do not take into account volume-filling effects in the same way. Our study justifies the use of some of these simpler, more analytically tractable models in reproducing the qualitative properties of the solutions in some parameter regimes, but it also reveals some interesting properties arising from the introduction of cell and ECM volume-filling effects, where standard model simplifications might not be appropriate.     

Mechanisms for oscillations in a biological competition model

A four-dimensional dynamical system that models perceptual bistability in the brain is analyzed. Two variables represent the activity of two competing neural populations and they evolve in fast time; other two variables are slow and they are associated with an intrinsic negative feedback to each population. The external stimulus strength I is the bifurcation parameter. We construct the normal form and prove that oscillations occur in the system through supercritical Hopf bifurcations: as I decreases from large to moderate values a limit cycle is born; then it disappears for lower values of I. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling and simulation of cell populations interaction

Regenerative medicine and cell therapy provide great hopes for the use of adult and stem cells. The latter are far less present in tissue than the former and must be expanded using cell culture. Stem cells culture requires the conservation of their proliferation and self-renewal capabilities. Still, the complex interaction between cell populations, for example in primary cell cultures, are not well-known and may account for part of the variability of such cultures. In order to represent and understand the evolution of cultured stem cells, we present here a mathematical model of cell proliferation and differentiation. Based on the formalism of cellular automata, this model simulates the evolution of several cell classes (which may represent either different levels of differentiation or different cell types) in an environment modeling the growth medium. We model the cell cycle as on the one hand a quiescence phase during which a cell rests, and on the other hand a division phase during which the cell starts the division process. In order to represent cell–cell interaction, the transition probability between those phases depends on the local composition of the growth medium depending itself on neighboring cells. An interaction between cellular populations is represented by a quantitative parameter which has a direct impact on cellular proliferation. Differentiation results in a change of the cell class and depends on the biological model studied : it may result from an asymmetric division or be a consequence of the local composition of the growth medium. This mathematical model aims at a better understanding of the interactions between cell populations in a culture. By defining constraints on the potential or the type of the cells at the end of a culture, it will then be possible to find optimal experimental conditions for cell production.     

The period on a family of non-linear oscillations and periodic motions of a perturbed system at a critical point of the family

Single-frequency oscillations of a reversible mechanical system are considered. It is shown that the oscillation period of a non-linear system usually only depends on a single parameter and it is established that, at a critical point of the family, at which the derivative of the period with respect to the parameter vanishes, due to the action of perturbations two families of symmetrical resonance periodic motions are produced. The oscillations of a satellite in an elliptic orbit, due to the action of gravitational and aerodynamic moments, are considered as an example. The operations in a circular orbit are investigated in detail initially, and then in an elliptical orbit of small eccentricity.     

A qualitative analysis of the subharmonic oscillations of a parametrically excited flexible rod

A non-autonomous non-linear dynamical system with a small parameter that describes the parametric oscillations of a flexible rod with three static equilibrium positions is obtained. The generating equation of this model is a dynamical system in a plane with a separatrix loop. The qualitative analysis presented includes an investigation of the stability and bifurcation of subharmonic motions at resonance energy levels.     

A general framework for modeling tumor-immune system competition at the mesoscopic level

We propose a class of nonlinear integro-differential equations that at the mesoscopic level models the competition between a tumor and the immune system. The model describes the evolution of a distribution function of the microscopic parameter referred to as activity of cells. The idea is somehow similar to the Enskog theory in kinetic theory. By averaging with respect to the parameter, the mesoscopic class of models reduces to the general class of macroscopic models introduced by A. d’Onofrio that may assess the effect of delays in stimulation of the immune system by tumor cells. The existence and uniqueness theory is developed.     

Dynamical consequences of predator interference in a tri-trophic model food chain

A model food chain involving a specialist and a generalist predator is proposed and studied. One of the salient features of this model food chain is that it combines both the schemes (Volterra and Leslie) of modeling predator–prey interaction in one system in such a way that the demerits of these individual formulations are suppressed and the resulting model system represents a common unit of real world food webs. The stability analysis of the proposed model is carried out. The Hopf bifurcation conditions of the positive equilibrium point are established. Our numerical computations show that chaotic dynamics is sensitive to changes in values of parameters measuring attributes of either interacting populations or their environments. Two dimensional parameter scans suggest that the model food chain displays short-term recurrent chaos. This can be regarded as a plausible explanation for why it has been so difficult to detect deterministic chaos in natural populations.     

A singular perturbation approach for a reaction diffusion system arising in nuclear reactor theory

The qualitative behavior of the solutions of a reaction-diffusion system arising in the theory of nuclear reactors is investigated by means of singular perturbation techniques, the small parameter ? representing the inverse of neutron velocity. At the lowest approximation for small ? the solutions exhibit oscillations about the unique strictly positive equilibrium, much as in the case of the associated lumped parameter system. A two-times representation for solutions of small amplitudes is also given.