Multiple Stable Steady States Research Articles

Impacts of predation on dynamics of age-structured prey: Allee effects and multi-stability

With a series of mathematical models, we explore impacts of predation on a prey population structured into two age classes, juveniles and adults, assuming generalist, age-specific predators. Predation on any age class is either absent, or represented by types II or III functional responses, in various combinations. We look for Allee effects or more generally for multiple stable steady states in the prey population. One of our key findings is the occurrence of a predator pit (low-density “refuge” state of prey induced by predation; the chance of escaping predation thus increases both below and above an intermediate prey density) when only one age class is consumed and predators use a type II functional response —this scenario is known to occur for an unstructured prey consumed via a type III functional response and can never occur for an unstructured prey consumed via a type II one. In the case where both age classes are consumed by type II generalist predators, an Allee effect occurs frequently, but some parameters give also rise to a predator pit and even three stable equilibria (one extinction equilibrium and two positive ones—Allee effect and predator pit combined). Multiple positive stable equilibria are common if one age class is consumed via a type II functional response and the other via a type III functional response—here, in addition to the behaviours mentioned above one may even observe three stable positive equilibria—“double” predator pit. Some of these results are discussed from the perspective of population management.

Asymptotic and Bifurcation Analysis of Wave-Pinning in a Reaction-Diffusion Model for Cell Polarization

We describe and analyze a bistable reaction-diffusion (RD) model for two interconverting chemical species that exhibits a phenomenon of wave-pinning: a wave of activation of one of the species is initiated at one end of the domain, moves into the domain, decelerates, and eventually stops inside the domain, forming a stationary front. The second ("inactive") species is depleted in this process. This behavior arises in a model for chemical polarization of a cell by Rho GTPases in response to stimulation. The initially spatially homogeneous concentration profile (representative of a resting cell) develops into an asymmetric stationary front profile (typical of a polarized cell). Wave-pinning here is based on three properties: (1) mass conservation in a finite domain, (2) nonlinear reaction kinetics allowing for multiple stable steady states, and (3) a sufficiently large difference in diffusion of the two species. Using matched asymptotic analysis, we explain the mathematical basis of wave-pinning, and predict the speed and pinned position of the wave. An analysis of the bifurcation of the pinned front solution reveals how the wave-pinning regime depends on parameters such as rates of diffusion and total mass of the species. We describe two ways in which the pinned solution can be lost depending on the details of the reaction kinetics: a saddle-node or a pitchfork bifurcation.

Stochastic bifurcation analysis of Rayleigh–Bénard convection

Stochastic bifurcations and stability of natural convection within two-dimensional square enclosures are investigated by different stochastic modelling approaches. Deterministic stability analysis is carried out first to obtain steady-state solutions and primary bifurcations. It is found that multiple stable steady states coexist, in agreement with recent results, within specific ranges of Rayleigh number. Stochastic simulations are then conducted around bifurcation points and transitional regimes. The influence of random initial flow states on the development of supercritical convection patterns is also investigated. It is found that a multi-element polynomial chaos method captures accurately the onset of convective instability as well as multiple convection patterns corresponding to random initial flow states.

Productive consumption and population dynamics in an endogenous growth model: Demographic trends and human development aid in developing economies

We find that by endogenizing the population growth rate, a growth model under the productive consumption hypothesis is more tractable than models in previous studies and has interesting implications for population dynamics. In the zero-saving phase, multiple saddle point stable steady states may exist, and the population growth rate may rise or decline along a transition path. In the positive-saving phase, such a steady state may exist uniquely, and the population growth rate may decline. With phase switching, the population growth rate may follow an inverted U-shaped curve. Human development aid may help an economy escape from an underdevelopment trap.

Non-linear channel?shoal dynamics in long tidal embayments

The dynamics of finite-amplitude bed forms in a tidal channel is studied with the use of an idealized morphodynamic model. The latter is based on depth-averaged equations for the tidal flow over a sandy bottom. The model considers phenomena on spatial scales of the order of the tidal excursion length. Transport of sediment mainly takes place as suspended load. The reference state of this model is characterized by a spatially uniform M2 tidal current over a fixed horizontal bed. The temporal evolution of deviations from this reference state is governed by amplitude equations: these are a set of non-linear equations that describe the temporal evolution of bed forms. These equations are used to obtain new morphodynamic equilibria which may be either static or time-periodic. Several of these bottom profiles show strong similarity with the tidal bars that are observed in natural estuaries. The dependence of the equilibrium solutions on the value of bottom friction and channel width is investigated systematically. For narrow channels (width small compared to the tidal excursion length) stable static equilibria exist if bottom friction is slightly larger than rcr. For channel widths more comparable to the tidal excursion length, multiple stable steady states may exist for bottom friction parameter values below rcr. Regardless of channel width, stable time-periodic equilibria seem to emerge as the bottom friction is increased.

Network Externalities and Dollarization Hysteresis: The Case of Russia

Dollarization in Russia increased rapidly during the early 1990s, but failed to come down in the second half of the 1990s in spite of exchange rate stabilization. To explain this dollarization hysteresis, this paper develops a model in which network externalities in the demand for currency can generate multiple stable steady states for the dollarization ratio. The model is estimated using a new source of data on dollar currency holdings in Russia. On the basis of these estimates, which confirm the existence of network externalities, the paper discusses several policies that could result in a permanent decrease in dollarization.

Network Externalities and Dollarization Hysteresis: The Case of Russia

Dollarization in Russia increased rapidly during the early 1990s, but failed to come down in the second half of the 1990s in spite of exchange rate stabilization. To explain this hysteresis, this paper develops a model in which network externalities in the demand for currency can generate multiple stable steady states for the dollarization ratio. The model is estimated using a new source of data on dollar currency holdings in Russia. On the basis of these estimates, which confirm the existence of network externalities, the paper discusses several policies that could result in a permanent decrease in dollarization.

Positive feedback circuits and memory

The concept of regulatory feedback circuit refers to oriented cyclic interactions between elements of a system. There are two classes of circuits, positive and negative, whose properties are in striking contrast. Positive circuits are a prerequisite for the occurrence of multiple steady states (multistationarity), and hence, they are involved in all processes showing hysteresis or memory. Endogenous or exogenous perturbations can lead the system to exhibit or to evoke one particular stable regime. The role of positive circuits in cell differentiation and in immunology is well documented. Negative circuits are involved in homeostatic regulation, with or without oscillations. The aim of this paper is to show: a) that positive circuits account for many features of memory stricto sensu (i.e., neural memory and mnesic evocation) as well as largo sensu (e.g. differentiation or immunological memory); and b) that simple combinations of positive and negative circuits provide powerful regulatory modules, which can also be associated in batteries. These entities have vast dynamical possibilities in the field of neurobiology, as well as in the fields of differentiation and immunology. Here we consider a universal minimal regulatory module, for which we suggest to adopt the term ‘logical regulon’, which can be considered as an atom of Jacob's integron. It comprises a positive and a negative circuit in its interaction matrix, and we recall the main results related to the simultaneous presence of these circuits. Finally, we give three applications of this type of interaction matrix. The first two deal with the coexistence of multiple stable steady states and periodicity in differentiation and in an immunological system showing hysteretic properties. The third deals with the dual problems of synchronization and desynchronization of a neural model for hippocampus memory evocation processes.

Frequency selectivity, multistability, and oscillations emerge from models of genetic regulatory systems.

To examine the capability of genetic regulatory systems for complex dynamic activity, we developed simple kinetic models that incorporate known features of these systems. These include autoregulation and stimulus-dependent phosphorylation of transcription factors (TFs), dimerization of TFs, crosstalk, and feedback. The simplest model manifested multiple stable steady states, and brief perturbations could switch the model between these states. Such transitions might explain, for example, how a brief pulse of hormone or neurotransmitter could elicit a long-lasting cellular response. In slightly more complex models, oscillatory regimes were identified. The addition of competition between activating and repressing TFs provided a plausible explanation for optimal stimulus frequencies that give maximal transcription. Such optimal frequencies are suggested by recent experiments comparing training paradigms for long-term memory formation and examining changes in mRNA levels in repetitively stimulated cultured cells. In general, the computational approach illustrated here, combined with appropriate experiments, provides a conceptual framework for investigating the function of genetic regulatory systems.

The structure of (linearly) stable double diffusive flow patterns in a laterally heated stratified liquid

Layered double diffusive flow patterns in a laterally heated stably stratified liquid are considered in a configuration which allows for steady states to exist. For the heat/salt system, these flows are characterized by the thermal and solutal Rayleigh numbers RaT and RaS, or equivalently by RaT and the buoyancy ratio Rρ. The bifurcation structure of steady patterns with respect to RaT is computed for two cases: fixed RaS and fixed Rρ. For the first case, results in N. Tsitverblit and E. Kit [Phys. Fluids A 5, 1062 (1993)], are computed and extended, and it is shown that many of the previously found flow patterns are unstable; only in a small interval of RaT, multiple (linearly) stable steady states exist. For the second case, the physical relevance of the unstable steady states with respect to the evolution of the flow toward a stable steady state is demonstrated.

Global stability of a lid-driven cavity with throughflow: Flow visualization studies

Flow visualization studies of a lid-driven cavity (LDC) with a small amount of throughflow reveal multiple steady states at low cavity Reynolds numbers. These results show that the well-known LDC flow, which consists of a primary eddy and secondary corner eddies, is only locally stable, becomes globally unstable, and competes with at least three other steady states before being replaced by a time-periodic flow. The small amount of throughflow present in this system seems to have no qualitative effect on the fluid flow characteristics. These observations suggest that multiple stable steady states may also exist in closed LDC’s. Since stability properties of the closed LDC flows are virtually unexplored, we interpret our flow visualization results by first proposing an expected behavior of an idealized (free-slip end walls) LDC and then treating the problem at hand as a perturbation of the ideal case. The results also suggest that there are nonunique and competing sequences of transitions that lead the flow in a LDC from laminar steady state toward turbulence.

Reflections and calculations on a prey-predator-patch problem

This paper is concerned with models for the interaction of plants, herbivores and their predators. We concentrate on situations in which local colonies of herbivores either over-exploit their host plant or are driven to extinction by predators. Starting from a complicated structured model, in which the local prey and predator density within patches is taken into account, we use time scale arguments to derive a three dimensional system of ordinary differential equations. The simplified system is analysed and the existence of multiple stable steady states is demonstrated.

Oscillations and multiple steady states in a cyclic gene model with repression

In this paper we study the cyclic gene model with repression considered by H. T. Banks and J. M. Mahaffy. Roughly, the model describes a biochemical feedback loop consisting of an integer number G of single gene reaction sequences in series. The model leads to a system of functional differential equations. We show that there is a qualitative difference in the dynamics between even and odd G if the feedback repression is sufficiently large. For even G, multiple stable steady states can coexist while for odd G, periodic orbits exist.

Natural convection in a thin horizontal porous annulus

The possible modes of time-dependent natural convection in a horizontal annulus of finite length are considered. The annulus is filled with porous material and the annular thickness is assumed small in comparison with the mean radius. All boundaries are impermeable and adiabatic; heating is through a circumferentially distributed volumetric heat source. The governing equations reduce to a set of two non-linear ordinary differential equations. Steady non-linear oscillations exist for the special case of infinite Rayleigh number and symmetric heating about the vertical. For lower Rayleigh numbers, damped oscillations are obtained, the degree of damping increasing with the inclination of the line of symmetry and with decreasing Rayleigh number. Multiple stable steady states are obtained for small inclinations. Chaotic motions do not develop for non-inertial Darcy flows.

Mathematical modelling of dynamics and control in metabolic networks. V. Static bifurcations in single biochemical control loops

Here we expand an earlier study of feedback activation in simple linear reaction sequences by searching the parameter space of biologically realistic rate laws for multiple stable steady states. The impetus for this work is to seek the origin of decision making strategies at the metabolic level, with particular emphasis on the switching between the operating conditions needed to meet changing substrate availability and organism requirements. The control loop considered herein is a linear reaction chain in which the end product of the reaction sequence feedback activates the first reaction in the sequence to produce feedback control. It has been found that the criteria for the existence of multiple steady state solutions in such loops involve only the kinetics of the regulatory enzyme controlling the first reaction and that of end product removal. The effects of these kinetics are examined here using two representative models for the regulatory enzyme: the lumped controller, based on Hill-type kinetics, and the symmetry model. The behavior of these two models is qualitatively similar, and both show the characteristics needed for switching between low and high substrate utilization. The removal rate is assumed to be of the Michaelis-Menten type. Judicious scaling of the governing equations permits separation of genetically determined kinetic parameters from concentration dependent ones. This allows us to conclude that, for a fixed set of kinetic parameters, the steady state flux through the loop can be switched between stable steady states by merely varying metabolite or enzyme concentrations. In particular, when the initial substrate exceeds a certain critical level, the loop can be "switched on" (by a discontinuous increase in the flux through the chain), and similarly, when it falls below a critical level, the pathway is shut down. Similar effects can be realized by varying the ratios of enzyme concentrations. It is proposed that by identifying these critical points one can gain significant insight into the objectives of decision making at the metabolic level.

An experimental investigation of stability and multiplicity of steady states in a biological reactor

AbstractThe existence of multiple stable and unstable steady states in a continuous stirred tank biological reactor (CSTBR) was demonstrated experimentally for the first time. These experiments resulted in the determination of growth rate, yield and intermediate metabolite concentration data entirely from continuous culture. This information was used to qualitatively interpret the stability behavior in terms of the biochemistry and bioenergetics of substrate metabolism.

Multiple Stable Steady States and Chemical Hysteresis in a Two-Box Model of the Brusselator

Abstract Explicit Solutions for the steady states of the two-box Brusselator are given. Normalmode analysis and numerical calculations show that multiple stable steady states mayoccur. This suggests a chemical hysteresis. Introduction The two-box model of the Brusselator has been discussed intensively by severalauthors [1—3]. It is a prototype for a chemical system with diffusion, since thecompartimentation allows a spatial differentiation. Therefore it provides qualitativeclues to the behaviour of continuous systems and to the role of diffusion. On theother hand it is still simple enough for an analytical treatment without too muchcomputational expense.The kinetic equations for the steady states of the intermediates X and in theboxes 1 and 2 are (cp. [2]):X 2 = A + Xl Y 2 - (B + 1) X 2 + D X (X! - X 2 ) = 0 ,Y! = - X? Y! + BXi + D y (Y 2 - YI) = 0 ,Y 2 = - Xl Y 2 + BX 2 + D y (Yi - Y 2 ) = 0 .Obviously the steady state of the homogeneous system X = A, Y = B/A is also asteady state of the compartimented system.Furthermore additional inhomogeneous steady states are possible. To our know-ledge these inhomogeneous solutions have been found hitherto by numerical methods.In the following we shall give explicitly the stationary solutions of the system (1 ).

Theory of propagation of discontinuities in kinetic systems with multiple time scales: Fronts, front multiplicity, and pulses

We consider a reaction–diffusion system far from chemical equilibrium, with kinetics on multiple time scales and the capacity of attaining one or multiple homogeneous stable steady states. For such systems we present a theory of the structure (concentration profile) and velocity of fronts based on a matched asymptotic solution of the singular perturbation problem. We analyze single front (wave) propagation as a transition occurs from one stable stationary state to another; show the possibility of multiple mode front propagation; and discuss single pulse propagation in systems with one stable stationary state. For each process we confirm the theory by comparison with numerical solutions of the coupled differential equations for a model system. Finally, by application of the theory we discuss a reduction of the complexity of the stability analysis of these fronts.

An hysteresis loop in the two component Bénard problem

An approximation to the nonlinear theory of the two component Bénard problem, taking into account thermal diffusion, shows that multiple stable steady states are possible. The stability of each steady state is investigated. The nonlinear equations are then integrated numerically. An hysteresis loop in the Rayleigh number-Nusselt number plane is observed.

Substrate inhibition kinetics in assemblages of cells

The collective kinetic behavior of a linear array of cells containing a substrate inhibited enzyme is studied with a model in which each cell is considered a well-stirred compartment surrounded by a semipermeable membrane. At large values of a reaction-permeation modulus, wherein substrate access to interior cells is limited, plots of total reaction rate versus concentration of the external reservoir show sharp projections which correspond to dominant reaction occurring in a pair of symmetrically placed cells. Over prescribed ranges of reservoir concentration, multiple stable steady states can occur, some of which are characterized by asymmetric profiles of concentration and reaction rate across the array. A simple stability criterion is proposed and applied to arrays with arbitrary numbers of cells.