Global behavior of solutions of a periodically forced Sigmoid Beverton-Holt model

Our aim in this paper is to investigate the boundedness, the extreme stability, and the periodicity of positive solutions of the periodically forced Sigmoid Beverton-Holt model: [Formula: see text] where {a ( n )} is a positive periodic sequence with period p and δ>0. In the special case when δ=1, the above equation reduces to the well-known periodic Pielou logistic equation which is known to be equivalent to the periodically forced Beverton-Holt model.     

On the effect of sharp rises in blood pressure in the Shah–Humphrey model for intracranial saccular aneurysms

We consider the model originally proposed by Shah and Humphrey (J Biomech 32:593–599, 1999) for a class of intracranial saccular aneurysms and show that for constant pressure the addition of the viscoelastic term corresponding to the presence of cerebral spinal fluid outside the membrane, no matter how small, does ensure convergence to an equilibrium. Our arguments apply to a general equation of this type, and thus also hold for variations of this model such as that proposed by David and Humphrey (J Biomech 36:1143–1150, 2003). On the other hand, it is known that the presence of damping may destabilize periodic orbits of periodically forced systems or even prevent them from existing altogether. We present numerical simulations showing that for some forcing terms the high-frequency oscillations do not die out with time, and a much more complex behaviour may emerge as a discontinuous forcing term is approached. The key point for this situation to occur is related to the derivative of the forcing term, supporting the hypothesis that sharper rises (or falls) in blood pressure may increase the risk of aneurysm rupture.     

Response to temporal parameter fluctuations in biochemical networks

Metabolic response coefficients describe how variables in metabolic systems, like steady state concentrations, respond to small changes of kinetic parameters. To extend this concept to temporal parameter fluctuations, we define spectral response coefficients that relate Fourier components of concentrations and fluxes to Fourier components of the underlying parameters. It is also straightforward to generalize other concepts from metabolic control theory, such as control coefficients with their summation and connectivity theorems. The first-order response coefficients describe forced oscillations caused by small harmonic oscillations of single parameters: they depend on the driving frequency and comprise the phases and amplitudes of the concentrations and fluxes. Close to a Hopf bifurcation, resonance can occur: as an example, we study the spectral densities of concentration fluctuations arising from the stochastic nature of chemical reactions. Second-order response coefficients describe how perturbations of different frequencies interact by mode coupling, yielding higher harmonics in the metabolic response. The temporal response to small parameter fluctuations can be computed by Fourier synthesis. For a model of glycolysis, this approximation remains fairly accurate even for large relative fluctuations of the parameters.     

Forced vibrations of a circular muscle ring

We consider the small radial displacement of a circular ring of cardiac muscle subjected to periodic forcing. The ring in question is that in the middle layer, at the transverse midsection, of the left ventricle. We show that the ring reacts in a periodic manner when forced in a periodic manner. This is accomplished by writing the differential equation for the ring and solving it for two cases-one for constant and one for variable ring thickness.     

Mechanisms for stabilisation and destabilisation of systems of reaction-diffusion equations

Potential mechanisms for stabilising and destabilising the spatially uniform steady states of systems of reaction-diffusion equations are examined. In the first instance the effect of introducing small periodic perturbations of the diffusion coefficients in a general system of reaction-diffusion equations is studied. Analytical results are proved for the case where the uniform steady state is marginally stable and demonstrate that the effect on the original system of such perturbations is one of stabilisation. Numerical simulations carried out on an ecological model of Levin and Segel (1976) confirm the analysis as well as extending it to the case where the perturbations are no longer small. Spatio-temporal delay is then introduced into the model. Analytical and numerical results are presented which show that the effect of the delay is to destabilise the original system.     

Microbial predation in a periodically operated chemostat: a global study of the interaction between natural and externally imposed frequencies.

Predator-prey systems in continuously operated chemostats exhibit sustained oscillations over a wide range of operating conditions. When the chemostat is operated periodically, the interaction of the natural oscillation frequency with the external forcing gives rise to a wealth of dynamic behavior patterns. Using numerical bifurcation techniques, we perform a detailed computational study of these patterns and the transitions (local and especially global) between them as the amplitude and frequency of the forcing vary. The transition from low-forcing-amplitude quasiperiodicity to entrainment of the chemostat behavior by strong forcing (involving the concerted closing of resonance horns) is analyzed. We concentrate on certain strong resonance phenomena between the two frequencies and provide an extensive atlas of computed phase portraits for our model system. Our observations corroborate recent mathematical results and case studies of periodically forced chemical oscillators. In particular, the existence and relative succession of several distinct types of global bifurcations resulting in chaotic transients and multistability are studied in detail. The location in the operating diagram of several key codimension 2 local bifurcations of periodic solutions is computed, and their interaction with an interesting feature we name "real-eigenvalues horns" is examined.     

The prey-dependent consumption two-prey one-predator models with stage structure for the predator and impulsive effects

In this paper, we consider the prey-dependent consumption two-prey one-predator models with stage structure for the predator and impulsive effects. By applying the Floquet theory of linear periodic impulsive equation, we show that there exists a globally asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value, that is, the pest population can be eradicated totally. But from the point of ecological balance and saving resources, we only need to control the pest population under the economic threshold level instead of eradicating it totally, and thus, we further prove that the system is uniformly permanent if the impulsive period is larger than some critical value, and meanwhile we also give the conditions for the extinction of one of the two preys and permanence of the remaining species. Thus, we can use the stability of the positive periodic solution and its period to control insect pests at acceptably low levels. Considering population communities always are imbedded in periodically varying environments, and the parameters in ecosystem models may oscillate simultaneously with the periodically varying environments, we add a forcing term into the prey population's intrinsic growth rate. The resulting bifurcation diagrams show that with the varying of parameters, the system experiences process of cycles, periodic windows, periodic-doubling cascade, symmetry breaking bifurcation as well as chaos.     

Optimizing the sensitivity of enzyme systems to perturbations from equilibrium

Algebraic derivations and numerical examples illustrate how metabolite pool sizes and enzyme rate constants influence the rate at which a multireactant enzyme system, initially poised in a near-equilibrium steady state, responds to small perturbations in the concentrations of the reactants. Certain enzymes, such as those employing the ordered bi bi catalytic mechanism, become relatively insensitive to perturbations when the reactants are all present at high concentrations. Other enzymes, such as those employing the ping-pong bi bi mechanism, are most sensitive to perturbations at high reactant concentrations. The ratio of the reactant concentrations to one another significantly alters sensitivity to perturbations; equations are presented for calculation of the reactant concentrations yielding maximal sensitivity to perturbations. Natural selection could choose metabolite pool sizes and enzyme rate constants which would optimize the performance of these systems, but changing metabolic loads (naturally or experimentally imposed) constantly alter the sensitivity of these systems to perturbations, changing the relative strengths of various connections in metabolic control networks.     

Spectral methods for parametric sensitivity in stochastic dynamical systems

Stochastic dynamical systems governed by the chemical master equation find use in the modeling of biological phenomena in cells, where they provide more accurate representations than their deterministic counterparts, particularly when the levels of molecular population are small. The analysis of parametric sensitivity in such systems requires appropriate methods to capture the sensitivity of the system dynamics with respect to variations of the parameters amid the noise from inherent internal stochastic effects. We use spectral polynomial chaos expansions to represent statistics of the system dynamics as polynomial functions of the model parameters. These expansions capture the nonlinear behavior of the system statistics as a result of finite-sized parametric perturbations. We obtain the normalized sensitivity coefficients by taking the derivative of this functional representation with respect to the parameters. We apply this method in two stochastic dynamical systems exhibiting bimodal behavior, including a biologically relevant viral infection model.     

Modeling human postural sway using an intermittent control and hemodynamic perturbations

Ground reaction force during human quiet stance is modulated synchronously with the cardiac cycle through hemodynamics [1]. This almost periodic hemodynamic force induces a small disturbance torque to the ankle joint, which is considered as a source of endogenous perturbation that induces postural sway. Here we consider postural sway dynamics of an inverted pendulum model with an intermittent control strategy, in comparison with the traditional continuous-time feedback controller. We examine whether each control model can exhibit human-like postural sway, characterized by its power law behavior at the low frequency band 0.1–0.7 Hz, when it is weakly perturbed by periodic and/or random forcing mimicking the hemodynamic perturbation. We show that the continuous control model with typical feedback gain parameters hardly exhibits the human-like sway pattern, in contrast with the intermittent control model. Further analyses suggest that deterministic, including chaotic, slow oscillations that characterize the intermittent control strategy, together with the small hemodynamic perturbation, could be a possible mechanism for generating the postural sway.     

Pulse coupled oscillators and the phase resetting curve

Limit cycle oscillators that are coupled in a pulsatile manner are referred to as pulse coupled oscillators. In these oscillators, the interactions take the form of brief pulses such that the effect of one input dies out before the next is received. A phase resetting curve (PRC) keeps track of how much an input advances or delays the next spike in an oscillatory neuron depending upon where in the cycle the input is applied. PRCs can be used to predict phase locking in networks of pulse coupled oscillators. In some studies of pulse coupled oscillators, a specific form is assumed for the interactions between oscillators, but a more general approach is to formulate the problem assuming a PRC that is generated using a perturbation that approximates the input received in the real biological network. In general, this approach requires that circuit architecture and a specific firing pattern be assumed. This allows the construction of discrete maps from one event to the next. The fixed points of these maps correspond to periodic firing modes and are easier to locate and analyze for stability compared to locating and analyzing periodic modes in the original network directly. Alternatively, maps based on the PRC have been constructed that do not presuppose a firing order. Specific circuits that have been analyzed under the assumption of pulsatile coupling include one to one lockings in a periodically forced oscillator or an oscillator forced at a fixed delay after a threshold event, two bidirectionally coupled oscillators with and without delays, a unidirectional N-ring of oscillators, and N all-to-all networks.     

Dissipation and maintenance of stable states in an enzymatic system: analysis and simulation

The constraint-based analysis has emerged as a useful tool for analysis of biochemical networks. An essential assumption for constraint-based analysis is the formation of a stable steady state. This work investigates dissipation and maintenance of stable states in a simple reversible enzymatic reaction with substrate inhibition. Under mass-action kinetics, the conditions under which the reaction maintains a stable steady state are analytically derived and numerically confirmed. It is shown that, in order to maintain a steady state in the regulated reaction, maximal enzyme activity must be much higher than input rate. Moreover, it is revealed that requirements for large enzyme activity are due to substrate inhibition. It is suggested that high activities of enzymes may play a vital role in protecting a stable state from its catastrophic collapse, giving an additional explanation to an intriguing problem—why the activities of some enzymes greatly exceed the flux capacity of a pathway. In addition, dissipation of the enzymatic reaction is analysed. It is shown that the collapse of stable states is always associated with a point at which dissipation is the highest. Therefore, in order to maintain a stable state, dissipation of the reaction must be less than a critical value. Moreover, although external forcing may not change net mass flow, it may lead to collapse of stable states. Furthermore, when stable states collapse at a critical forcing amplitude and period, dissipation also reaches a highest value. It is concluded that collapse of stable steady state in the enzyme system with substrate inhibition always corresponds to critical points at which dissipation is highest, regardless if the reaction is forced or not. Therefore, for the substrate inhibited reaction, maintenance of stable states is intrinsically related to level of dissipation.     

Oscillatory population growth in periodic environments

A general first-order nonlinear differential equation is derived for the dynamics of a population in such a way that the inherent growth rate r and the equilibrium “carrying capacity” K appear explicitly as parameters. By means of standard regular perturbation techniques, properties of the periodic asymptotic state of the population are studied under the assumption that r and K suffer periodic perturbations of small amplitude. Specific examples are studied analytically and numerically.     

The effect of periodic habitat fluctuations on a nonlinear insect population model

 Laboratory data show that populations of flour beetles (Tribolium), when grown in a periodically fluctuating volume of flour, can exhibit significant increases in numbers above those attained when grown in a constant volume (of the same average). To analyze and explain this phenomenon a discrete stage-structured model of Tribolium dynamics with periodic environmental forcing is introduced and studied. This model is an appropriately modified version of an experimentally validated model for flour beetle populations growing in a constant volume of flour, in which cannibalism rates are assumed inversely proportional to flour volume. This modeling assumption has been confirmed by laboratory experiments. Theorems implying the existence and stability of periodic solutions of the periodically forced model are proved. The time averages of periodic solutions of the forced model are compared with the equilibrium levels of the unforced model (with the same average flour volume). Parameter constraints are determined for which the average population numbers in the periodic environment are greater than (or less than) the equilibrium population numbers in the associated constant environment. Sample parameter estimates taken from the literature show that these constraints are fulfilled. These theoretical results provide an explanation for the experimentally observed increase in flour beetle numbers as a result of periodically fluctuating flour volumes. More generally, these integrated theoretical and experimental results provide the first convincing example illustrating the possibility of increased population numbers in a periodically fluctuating environment. Received 23 April 1996; received in revised form 28 March 1997     

Control of photosynthetic carbon dioxide fixation by the boundary layer, stomata and ribulose 1,5-biphosphate carboxylase/oxygenase

Abstract Coefficients describing the sensitivity of the rate of photosynthetic carbon dioxide fixation to small changes in the stomatal conductance and boundary layer conductance are derived. These sensitivity or ‘control’ coefficients, together with those for the carboxylase and oxygenase activities of ribulose 1,5-bisphosphate carboxylase/oxygenase, are calculated from standard gas exchange data and apply under conditions where leaf temperature and water vapour concentration at the leaf surface remain largely constant. It is shown that the magnitude of the control coefficients depends on conditions such as photon flux density, ambient CO₂ concentration and relative humidity at the leaf surface. The extension of this analysis to encompass the sensitivity of the photosynthetic fluxes to changes in enzyme concentrations and kinetic properties is also discussed.     

The periodically forced Droop model for phytoplankton growth in a chemostat

 It is proved that the periodically forced Droop model for phytoplankton growth in a chemostat has precisely two dynamic regimes depending on a threshold condition involving the dilution rate. If the dilution rate is such that the sub-threshold condition holds, the phytoplankton population is washed out of the chemostat. If the super-threshold condition holds, then there is a unique periodic solution, having the same period as the forcing, characterized by the presence of the phytoplankton population, to which all solutions approach asymptotically. Furthermore, this result holds for a general class of models with monotone growth rate and monotone uptake rate, the latter possibly depending on the cell quota. Received 10 October 1995; received in revised form 26 March 1996     

Globally attracting attenuant versus resonant cycles in periodic compensatory Leslie models

We use a periodically forced density-dependent compensatory Leslie model to study the combined effects of environmental fluctuations and age-structure on pioneer populations. In constant environments, the models have globally attracting positive fixed points. However, with the advent of periodic forcing, the models have globally attracting cycles. We derive conditions under which the cycle is attenuant, resonant, and neither attenuant nor resonant. These results show that the response of age-structured populations to environmental fluctuations is a complex function of the compensatory mechanisms at different life-history stages, the fertile age classes and the period of the environment.     

A note on the nonautonomous delay Beverton-Holt model

It is well known that the periodic cycle {x(n)} of a periodically forced nonlinear difference equation is attenuant (resonant) if av(x(n)) < av(K(n))(av(x(n)) > av(K(n))),where {K ( n )} is the carrying capacity of the environment and av(t(n)) = (1/p)∑(p?1) (i=0) ti (arithmetic mean of the p-periodic cycle {t ( n )}). In this article, we extend the concept of attenuance and resonance of periodic cycles using the geometric mean for the average of a periodic cycle. We study the properties of the periodically forced nonautonomous delay Beverton-Holt model x(n+1) = r(n)x(n)/1 + (r(n?l) ? 1)x(n?k)/K(n?k), n= 0, 1, . . . , where {K ( n )} and {r ( n )} are positive p-periodic sequences; (K ( n )>0, r ( n )>1) as well as k and l are nonnegative integers. We will show that for all positive solutions {x ( n )} of the previous equation lim sup (n→∞) (∏(n?1)(i=0)xi)(1/n) ≤ ((∏(p?1)(i=0)ri)(1/p) ? 1)(∏(p?1)(i=0)(ri ? 1))(?1/p)(∏(p?1)(i=0)Ki)(1/p). In particular, in the case where {x(n)} is a p-periodic solution of the above equation (assuming that such solution exists) and r ( n )=r>1, the periodic cycle is g-attenuant, that is (∏(p?1)(i=0)x(i))(1/p)<(∏(p?1)(i=0)K(i))(p?1) Surprisingly, the obtained results show that the delays k and l do not play any role.     

Determination of optimal vaccination strategies using an orbital stability threshold from periodically driven systems

We analyse a periodically driven SIR epidemic model for childhood related diseases, where the contact rate and vaccination rate parameters are considered periodic. The aim is to define optimal vaccination strategies for control of childhood related infections. Stability analysis of the uninfected solution is the tool for setting up the control function. The optimal solutions are sought within a set of susceptible population profiles. Our analysis reveals that periodic vaccination strategy hardly contributes to the stability of the uninfected solution if the human residence time (life span) is much larger than the contact rate period. However, if the human residence time and the contact rate periods match, we observe some positive effect of periodic vaccination. Such a vaccination strategy would be useful in the developing world, where human life spans are shorter, or basically in the case of vaccination of livestock or small animals whose life-spans are relatively shorter.     

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.