A new class of non-linear stochastic population models with mass conservation

We study the effects of random feeding, growing and dying in a closed nutrient-limited producer/consumer system, in which nutrient is fully conserved, not only in the mean, but, most importantly, also across random events. More specifically, we relate these random effects to the closest deterministic models, and evaluate the importance of the various times scales that are involved. These stochastic models differ from deterministic ones not only in stochasticity, but they also have more details that involve shorter times scales. We tried to separate the effects of more detail from that of stochasticity. The producers have (nutrient) reserve and (body) structure, and so a variable chemical composition. The consumers have only structure, so a constant chemical composition. The conversion efficiency from producer to consumer, therefore, varies. The consumers use reserve and structure of the producers as complementary compounds, following the rules of Dynamic Energy Budget theory. Consumers die at constant specific rate and decompose instantaneously. Stochasticity is incorporated in the behaviour of the consumers, where the switches to handling and searching, as well as dying are Poissonian point events. We show that the stochastic model has one parameter more than the deterministic formulation without time scale separation for conversions between searching and handling consumers, which itself has one parameter more than the deterministic formulation with time scale separation for these conversions. These extra parameters are the contributions of a single individual producer and consumer to their densities, and the ratio of the two, respectively. The tendency to oscillate increases with the number of parameters. The focus bifurcation point has more relevance for the asymptotic behaviour of the stochastic model than the Hopf bifurcation point, since a randomly perturbed damped oscillation exhibits a behaviour similar to that of the stochastic limit cycle particularly near this bifurcation point. For total nutrient values below the focus bifurcation point, the system gradually becomes more confined to the direct neighbourhood of the isocline for which the producers do not change.     

Relating damped oscillations to sustained limit cycles describing real and ideal batch fermentation processes

A batch fermentation model is presented in which the specific growth rate and yield functions are chosen such that sustained oscillations in both the cell and substrate concentration occur. This phenomenon is shown to be a Hopf bifurcation in the underlying system of non-linear ordinary differential equations which comprises the model. It is shown that for oscillations in the substrate concentration to occur it is necessary for the yield term to depend on both the cell and substrate levels.     

Delays in nutrient cycling and plant population oscillations

It is well known that delay-differential and delay-difference equations can produce plausible simulations of population oscillations, but many of these equations lack a specific mechanism responsible for the delay. We suggest that delays in release of nitrogen from decomposing litter, caused by microbial uptake, could produce oscillations in populations when the delay in the release of nitrogen is longer than the characteristic time scale of nitrogen uptake. We present a model which captures these dynamics. As the parameter controlling microbial uptake of nitrogen during litter decay increases, the model solutions bifurcate from fixed point equilibria, to periodic orbits (oscillations) which remain bounded for ecologically very long times, and finally to extinction of the plant population because of rapid increases in the amplitude of the oscillations. We suggest that such a mechanism may be especially important for annual plants which do not store nitrogen in perennial tissues to buffer delays. Natural populations of wild rice ( Zizania palustris ), an annual plant, oscillate with approximately four-year periods. Our model qualitatively mimics the period and shape of population oscillations in wild rice with parameter values in the range of those determined by experiments. The model therefore demonstrates a logical and experimentally plausible link between plant population dynamics and the ecosystem processes delaying the cycling of limiting nutrients.     

Heterogeneity of glycolytic oscillatory behaviour in individual yeast cells

There are many examples of oscillations in biological systems and one of the most investigated is glycolytic oscillations in yeast. These oscillations have been studied since the 1950s in dense, synchronized populations and in cell-free extracts, but it has for long been unknown whether a high cell density is a requirement for oscillations to be induced, or if individual cells can oscillate also in isolation without synchronization. Here we present an experimental method and a detailed kinetic model for studying glycolytic oscillations in individual, isolated yeast cells and compare them to previously reported studies of single-cell oscillations. The importance of single-cell studies of this phenomenon and relevant future research questions are also discussed.     

Interfering with Wnt signalling alters the periodicity of the segmentation clock

Somites are embryonic precursors of the ribs, vertebrae and certain dermis tissue. Somite formation is a periodic process regulated by a molecular clock which drives cyclic expression of a number of clock genes in the presomitic mesoderm. To date the mechanism regulating the period of clock gene oscillations is unknown. Here we show that chick homologues of the Wnt pathway genes that oscillate in mouse do not cycle across the chick presomitic mesoderm. Strikingly we find that modifying Wnt signalling changes the period of Notch driven oscillations in both mouse and chick but these oscillations continue. We propose that the Wnt pathway is a conserved mechanism that is involved in regulating the period of cyclic gene oscillations in the presomitic mesoderm.     

Stability of discrete age-structured and aggregated delay-difference population models

Sufficiency conditions for local stability are derived for a class of density dependent Leslie matrix models. Four of the recruitment functions in common use in fisheries management are then considered. In two of these oscillating instability can never occur (Beverton and Holt and Cushing forms). In the other two (Deriso-Schnute and Shepherd forms) undamped oscillations are possible within the region of parameter space described here. An algorithm is developed for calculating necessary and sufficient local stability conditions for a simplified form of the general age-structured model. The complete spectrum of stability states (monotonic stability; monotonic instability; oscillating-stable; oscillating-unstable) and the bifurcation periods are given for selected examples of this model. The examples cover a large portion of the parameter space of interest in resource management. It is shown that in perfectly deterministic systems which are observed with error, oscillating instabilities may be missed, and such systems could be erroneously assumed to be stable.     

Emergent oscillations in networks of stochastic spiking neurons

Networks of neurons produce diverse patterns of oscillations, arising from the network's global properties, the propensity of individual neurons to oscillate, or a mixture of the two. Here we describe noisy limit cycles and quasi-cycles, two related mechanisms underlying emergent oscillations in neuronal networks whose individual components, stochastic spiking neurons, do not themselves oscillate. Both mechanisms are shown to produce gamma band oscillations at the population level while individual neurons fire at a rate much lower than the population frequency. Spike trains in a network undergoing noisy limit cycles display a preferred period which is not found in the case of quasi-cycles, due to the even faster decay of phase information in quasi-cycles. These oscillations persist in sparsely connected networks, and variation of the network's connectivity results in variation of the oscillation frequency. A network of such neurons behaves as a stochastic perturbation of the deterministic Wilson-Cowan equations, and the network undergoes noisy limit cycles or quasi-cycles depending on whether these have limit cycles or a weakly stable focus. These mechanisms provide a new perspective on the emergence of rhythmic firing in neural networks, showing the coexistence of population-level oscillations with very irregular individual spike trains in a simple and general framework.     

Bifurcations in a mathematical model for circadian oscillations of clock genes

Circadian oscillations with a period of about 24h are observed in nearly all living organisms as conspicuous biological rhythms. In this paper, we investigate various kinds of bifurcation phenomena produced in a circadian oscillator model of Drosophila. In Drosophila, it is known that circadian oscillations in the levels of two proteins, PER and TIM, result from the negative feedback exerted by a PER-TIM complex on the expression of the per and tim genes that code for the two proteins. For studying circadian oscillations of proteins in Drosophila, a mathematical model has been proposed. The model cannot only account for regular circadian oscillations in environmental conditions such as constant darkness, but also give rise to more complex oscillatory phenomena including chaos and birhythmicity. By calculating bifurcations using Kawakami's method, we obtain detailed bifurcation diagrams related to stable and unstable invariant sets, and identify parameter regions in which the model generates complex oscillations as well as regular circadian oscillations. Moreover, we study bifurcations observed in the model incorporating the effect on a light-dark (LD) cycle and show that the waveform of the periodic variation in the light-induced parameter has a marked influence on the global bifurcation structure or the type of dynamic behavior resulting from the forcing term of the circadian oscillator by the LD cycles.     

Gain-induced oscillations in blood pressure

 “Mayer waves” are long-period (6 to 12 seconds) oscillations in arterial blood pressure, which have been observed and studied for more than 100 years in the cardiovascular system of humans and other mammals. A mathematical model of the human cardiovascular system is presented, incorporating parameters relevant to the onset of Mayer waves. The model is analyzed using methods of Liapunov stability and Hopf bifurcation theory. The analysis shows that increase in the gain of the baroreflex feedback loop controlling venous volume may lead to the onset of oscillations, while changes in the other parameters considered do not affect stability of the equilibrium state. The results agree with clinical observations of Mayer waves in human subjects, both in the period of the oscillations and in the observed age-dependence of Mayer waves. This leads to a proposed explanation of their occurrence, namely that Mayer waves are a gain-induced oscillation. Received: 15 September 1997/Revised version: 15 March 1998     

Oscillatory dynamics in rock-paper-scissors games with mutations

We study the oscillatory dynamics in the generic three-species rock-paper-scissors games with mutations. In the mean-field limit, different behaviors are found: (a) for high mutation rate, there is a stable interior fixed point with coexistence of all species; (b) for low mutation rates, there is a region of the parameter space characterized by a limit cycle resulting from a Hopf bifurcation; (c) in the absence of mutations, there is a region where heteroclinic cycles yield oscillations of large amplitude (not robust against noise). After a discussion on the main properties of the mean-field dynamics, we investigate the stochastic version of the model within an individual-based formulation. Demographic fluctuations are therefore naturally accounted and their effects are studied using a diffusion theory complemented by numerical simulations. It is thus shown that persistent erratic oscillations (quasi-cycles) of large amplitude emerge from a noise-induced resonance phenomenon. We also analytically and numerically compute the average escape time necessary to reach a (quasi-)cycle on which the system oscillates at a given amplitude.     

Self-sustained oscillations in epidemic models with infective immigrants

A relevant issue related to eco-epidemiological studies concerns the demographic mechanisms that can lead to self-sustained oscillations in the composition of a host population subject to infection. In particular, why does the prevalence of some contagious diseases oscillate over time? Here, we address this question by using susceptible-infective-recovered-empty models including migration of infective foreigners and variable population size. These models are described in terms of ordinary differential equations (ODE) and also in terms of probabilistic cellular automaton (PCA), in which each cell is connected to others either by a regular lattice or by a random graph favoring local contacts. Each cell in the PCA model can be either empty or occupied by a single individual. The amount of neighbors per cell affects the value of the basic reproduction number R₀, which is, in fact, a bifurcation parameter. We show that, by varying the amount of neighbors per cell (and consequently R₀), the number of infective individuals can start to exhibit periodic behavior, which corresponds to a Hopf bifurcation in the ODE model. This bifurcation gives rise to a self-sustained oscillation and it can only occur if the immigration rate of infective individuals is above a critical value. We also investigate how the sum of new infections, within the considered time window, depends on the number of neighbors per cell.     

Modeling partitioning of Min proteins between daughter cells after septation in Escherichia coli

Ongoing sub-cellular oscillation of Min proteins is required to block minicelling in Escherichia coli. Experimentally, Min oscillations are seen in newly divided cells and no minicells are produced. In model Min systems many daughter cells do not oscillate following septation because of unequal partitioning of Min proteins between the daughter cells. Using the 3D model of Huang et al, we investigate the septation process in detail to determine the cause of the asymmetric partitioning of Min proteins between daughter cells. We find that this partitioning problem arises at certain phases of the MinD and MinE oscillations with respect to septal closure and it persists independently of parameter variation. At most 85% of the daughter cells exhibit Min oscillation following septation. Enhanced MinD binding at the static polar and dynamic septal regions, consistent with cardiolipin domains, does not substantially increase this fraction of oscillating daughters. We believe that this problem will be shared among all existing Min models and discuss possible biological mechanisms that may minimize partitioning errors of Min proteins following septation.     

Bifurcation phenomena appearing in the Lotka-Volterra competition equations: a numerical study

Bifurcation phenomena appearing in the Lotka-Volterra competition equations with periodically varying coefficients are studied numerically. We assume sinusoidal oscillations of the coefficients and use phase differences between them as free parameters. We are mainly concerned with the case where a pair of stable and unstable positive periodic solutions exists, although one of the trivial periodic solutions is stable and the other is unstable. We obtain a very curious bifurcation diagram in which two branches of stable and unstable positive periodic solutions are connected at both ends, but are connected with no other branches. We show how this unusual diagram can be viewed as a cross-section of a multidimensional bifurcation diagram. The region in a 3-dimensional parameter space where a pair of stable and unstable positive periodic solutions exists is shown in an example, and the ecological meaning of the phase differences necessary for stable coexistence of two species is considered. Finally, a bifurcation problem with the average intrinsic growth rate as a parameter is also dealt with numerically, in relation with Cushing's result.     

Synergism and antagonism of neurons caused by an electrical synapse

To investigate the role of electrical junctions in the nervous system, a model system consisting of two nearly identical neurons electrotonically coupled is studied. We assume that each neuron discharges a train of impulses or bursts either spontaneously or under constant stimulus via chemical synapses. It is known that not only an electric current but also chemical substances whose molecular weight is about 1000 can pass through the junction of an electrical synapse (gap junction). So, our model system is regarded as a set of non-linear oscillators coupled by diffusion, and it may be described by a system of ordinary differential equations. Neurons are excited constantly when they are stimulated by an electric current above the threshold level. Therefore, we expect Hopf bifurcation to occur at the critical magnitude of a stimulating electric current in the system of differential equations which describes the dynamics of a single neuron. Studying our model system according to the theory of Hopf bifurcation, we found regions of diffusion constants of the electrical junction which give two kinds of periodic solutions. One is the solution where two neurons oscillate in phase synchrony. The other is where two neurons oscillate 180° out of phase. In the case where one neuron is described by the BVP model, the following was found by computer simulation. When the initial difference between the phase of two neurons is small, the two neurons come to oscillate synchronously. If the initial difference is large, however, the two come to be excited alternately. The physiological implications of these results are discussed.Department of Behaviorology, Faculty of Human Sciences     

Feedback-mediated dynamics in a model of a compliant thick ascending limb

The tubuloglomerular feedback (TGF) system in the kidney, which is a key regulator of filtration rate, has been shown in physiologic experiments in rats to mediate oscillations in tubular fluid pressure and flow, and in NaCl concentration in the tubular fluid of the thick ascending limb (TAL). In this study, we developed a mathematical model of the TGF system that represents NaCl transport along a TAL with compliant walls. The model was used to investigate the dynamic behaviors of the TGF system. A bifurcation analysis of the TGF model equations was performed by deriving and finding roots of the characteristic equation, which arises from a linearization of the model equations. Numerical simulations of the full model equations were conducted to assist in the interpretation of the bifurcation analysis. These techniques revealed a complex parameter region that allows a variety of qualitatively different model solutions: a regime having one stable, time-independent steady-state solution; regimes having one stable oscillatory solution only; and regimes having multiple possible stable oscillatory solutions. Model results suggest that the compliance of the TAL walls increases the tendency of the model TGF system to oscillate.     

The struggle for life: III. A predator-prey chain

Ann species predator-prey chain is analyzed to determine what oscillations occur in population sizes. It is found that only the populations of the first and second species in the chain must necessarily oscillate around the point of equilibrium if they do not come to equilibrium. The other species may or may not oscillate.     

Emergent properties of electrically coupled smooth muscle cells

Asynchronous and synchronous calcium oscillations occur in a variety of cells. A well-established pathway for intercellular communication is provided by gap junctions which connect adjacent cells and can mediate electrical and chemical coupling. Several experimental studies report that cells presenting only a transient increase when freshly dispersed may oscillate when they are coupled. Such observations suggest that the role of gap junctions is not only to coordinate calcium oscillations of adjacent cells. Gap junctions may also be important to generate oscillations. Here we illustrate the emergent properties of electrically coupled smooth muscle cells using a model that we recently proposed. A bifurcation analysis in the case of two cells reveals that synchronous and asynchronous calcium oscillations can be induced by electrical coupling. In a larger population of smooth muscle cells, electrical coupling may result in the creation of groups of cells presenting synchronous calcium oscillations. The elements of one group may be distant from each other. Moreover, our results highlight a general mechanism by which gap junctional electrical coupling can give rise to out of phase calcium oscillations in smooth muscle cells that are non-oscillating when uncoupled. All these observations remain true in the case of non-identical cells, except that the solution corresponding to synchronous calcium oscillations disappears and that the formation of groups is sensitive to the degree of heterogeneity. The first two authors contributed equally to this work.     

Elimination of the initial value parameters when identifying a system close to a Hopf bifurcation

One of the biggest problems when performing system identification of biological systems is that it is seldom possible to measure more than a small fraction of the total number of variables. If that is the case, the initial state, from where the simulation should start, has to be estimated along with the kinetic parameters appearing in the rate expressions. This is often done by introducing extra parameters, describing the initial state, and one way to eliminate them is by starting in a steady state. We report a generalisation of this approach to all systems starting on the centre manifold, close to a Hopf bifurcation. There exist biochemical systems where such data have already been collected, for example, of glycolysis in yeast. The initial value parameters are solved for in an optimisation sub-problem, for each step in the estimation of the other parameters. For systems starting in stationary oscillations, the sub-problem is solved in a straight-forward manner, without integration of the differential equations, and without the problem of local minima. This is possible because of a combination of a centre manifold and normal form reduction, which reveals the special structure of the Hopf bifurcation. The advantage of the method is demonstrated on the Brusselator.     

Hopf bifurcation in three-species food chain models with group defense.

Three-species food-chain models, in which the prey population exhibits group defense, are considered. Using the carrying capacity of the environment as the bifurcation parameter, it is shown that the model without delay undergoes a sequence of Hopf bifurcations. In the model with delay it is shown that using a delay as a bifurcation parameter, a Hopf bifurcation can also occur in this case. These occurrences may be interpreted as showing that a region of local stability (survival) may exist even though the positive steady states are unstable. A computer code BIFDD is used to determine the stability of the bifurcation solutions of a delay model.     

Periodic metabolic systems: Oscillations in multiple-loop negative feedback biochemical control networks

Summary For a general multiple loop feedback inhibition system in which the end product can inhibit any or all of the intermediate reactions it is shown that biologically significant behaviour is always confined to a bounded region of reaction space containing a unique equilibrium. By explicit construction of a Liapunov function for the general n dimensional differential equation it is shown that some values of reaction parameters cause the concentration vector to approach the equilibrium asymptotically for all physically realizable initial conditions. As the parameter values change, periodic solutions can appear within the bounded region. Some information about these periodic solutions can be obtained from the Hopf bifurcation theorem. Alternatively, if specific parameter values are known a numerical method can be used to find periodic solutions and determine their stability by locating a zero of the displacement map. The single loop Goodwin oscillator is analysed in detail. The methods are then used to treat an oscillator with two feedback loops and it is found that oscillations are possible even if both Hill coefficients are equal to one.