Periodic orbits near heteroclinic cycles in a cyclic replicator system

A species is semelparous if every individual reproduces only once in its life and dies immediately after the reproduction. While the reproduction opportunity is unique per year and the individual’s period from birth to reproduction is just n years, the individuals that reproduce in the ith year (modulo n) are called the ith year class, i = 1, 2, . . . , n. The dynamics of the n year-class system can be described by a differential equation system of Lotka–Volterra type. For the case n = 4, there is a heteroclinic cycle on the boundary as shown in previous works. In this paper, we focus on the case n = 4 and show the existence, growth and disappearance of periodic orbits near the heteroclinic cycle, which is a part of the conjecture by Diekmann and van Gils (SIAM J Appl Dyn Syst 8:1160–1189, 2009). By analyzing the Poincaré map near the heteroclinic cycle and introducing a metric to measure the size of the periodic orbit, we show that (i) when the average competitive degree among subpopulations (year classes) in the system is weak, there exists an asymptotically stable periodic orbit near the heteroclinic cycle which is repelling; (ii) the periodic orbit grows in size when some competitive degree increases, and converges to the heteroclinic cycle when the average competitive degree tends to be strong; (iii) when the average competitive degree is strong, there is no periodic orbit near the heteroclinic cycle which becomes asymptotically stable. Our results provide explanations why periodic solutions expand and disappear and why all but one subpopulation go extinct.     

Mathematical modeling of bacterial virulence and host-pathogen interactions in the Dictyostelium/Pseudomonas system

We present some studies on the mechanisms of pathogenesis based on experimental work and on its interpretation through a mathematical model. Using a collection of clinical strains of the opportunistic human pathogen Pseudomonas aeruginosa, we performed co-culture experiments with Dictyostelium amoebae, to investigate the two organisms’ interaction, characterized by a cross action between amoeba, feeding on bacteria, and bacteria exerting their pathogenic action against amoeba. In order to classify bacteria virulence, independently of this cross interaction, we have also performed killing experiments of bacteria against the nematode Caenorhabditis elegans.A mathematical model was developed to infer how the populations of the amoeba-bacteria system evolve according to a number of parameters, taking into account the specific features underlying the interaction. The model does not fall within the class of traditional prey-predator models because not only does an amoeba feed on bacteria, but also it is in turn attacked by them; thus the model must include a feedback term modeling this further interaction aspect. The model shows the existence of multiple steady states and the resulting behavior of the solutions, showing bi-stability of the system, gives a qualitative explanation of the co-culture experiments.     

Global dynamics of an SIRS epidemic model with saturation incidence

In this paper, the dynamical behavior of an SIRS epidemic model with birth pulse, pulse vaccination, and saturation incidence is studied. By using a discrete map, the existence and stability of the infection-free periodic solution and the endemic periodic solution are investigated. The conditions required for the existence of supercritical bifurcation are derived. A threshold for a disease to be extinct or endemic is established. The Poincaré map and center manifold theorem are used to discuss flip bifurcation of the endemic periodic solution. Moreover, numerical simulations for bifurcation diagrams, phase portraits and periodic solutions, which are illustrated with an example, are in good agreement with the theoretical analysis.     

The information capacity of hypercycles

Hypercycles are information integration systems which are thought to overcome the information crisis of prebiotic evolution by ensuring the coexistence of several short templates. For imperfect template replication, we derive a simple expression for the maximum number of distinct templates n_m that can coexist in a hypercycle and show that it is a decreasing function of the length L of the templates. In the case of high replication accuracy we find that the product n_mL tends to a constant value, limiting thus the information content of the hypercycle. Template coexistence is achieved either as a stationary equilibrium (stable fixed point) or a stable periodic orbit in which the total concentration of functional templates is nonzero. For the hypercycle system studied here we find numerical evidence that the existence of an unstable fixed point is a necessary condition for the presence of periodic orbits.     

A Malaria Transmission Model with Temperature-Dependent Incubation Period

Malaria is an infectious disease caused by Plasmodium parasites and is transmitted among humans by female Anopheles mosquitoes. Climate factors have significant impact on both mosquito life cycle and parasite development. To consider the temperature sensitivity of the extrinsic incubation period (EIP) of malaria parasites, we formulate a delay differential equations model with a periodic time delay. We derive the basic reproduction ratio \(R_0\) and establish a threshold type result on the global dynamics in terms of \(R_0\), that is, the unique disease-free periodic solution is globally asymptotically stable if \(R_0<1\); and the model system admits a unique positive periodic solution which is globally asymptotically stable if \(R_0>1\). Numerically, we parameterize the model with data from Maputo Province, Mozambique, and simulate the long-term behavior of solutions. The simulation result is consistent with the obtained analytic result. In addition, we find that using the time-averaged EIP may underestimate the basic reproduction ratio.     

Periodic Solutions of Piecewise Affine Gene Network Models with Non Uniform Decay Rates: The Case of a Negative Feedback Loop

This paper concerns periodic solutions of a class of equations that model gene regulatory networks. Unlike the vast majority of previous studies, it is not assumed that all decay rates are identical. To handle this more general situation, we rely on monotonicity properties of these systems. Under an alternative assumption, it is shown that a classical fixed point theorem for monotone, concave operators can be applied to these systems. The required assumption is expressed in geometrical terms as an alignment condition on so-called focal points. As an application, we show the existence and uniqueness of a stable periodic orbit for negative feedback loop systems in dimension 3 or more, and of a unique stable equilibrium point in dimension 2. This extends a theorem of Snoussi, which showed the existence of these orbits only.     

Cell cycle dynamics in a response/signalling feedback system with a gap

We consider a dynamical model of cell cycles of n cells in a culture in which cells in one specific phase (S for signalling) of the cell cycle produce chemical agents that influence the growth/cell cycle progression of cells in another phase (R for responsive). In the case that the feedback is negative, it is known that subpopulations of cells tend to become clustered in the cell cycle; while for a positive feedback, all the cells tend to become synchronized. In this paper, we suppose that there is a gap between the two phases. The gap can be thought of as modelling the physical reality of a time delay in the production and action of the signalling agents. We completely analyse the dynamics of this system when the cells are arranged into two cell cycle clusters. We also consider the stability of certain important periodic solutions in which clusters of cells have a cyclic arrangement and there are just enough clusters to allow interactions between them. We find that the inclusion of a small gap does not greatly alter the global dynamics of the system; there are still large open sets of parameters for which clustered solutions are stable. Thus, we add to the evidence that clustering can be a robust phenomenon in biological systems. However, the gap does effect the system by enhancing the stability of the stable clustered solutions. We explain this phenomenon in terms of contraction rates (Floquet exponents) in various invariant subspaces of the system. We conclude that in systems for which these models are reasonable, a delay in signalling is advantageous to the emergence of clustering.     

The Beverton–Holt q-difference equation

The Beverton–Holt model is a classical population model which has been considered in the literature for the discrete-time case. Its continuous-time analogue is the well-known logistic model. In this paper, we consider a quantum calculus analogue of the Beverton–Holt equation. We use a recently introduced concept of periodic functions in quantum calculus in order to study the existence of periodic solutions of the Beverton–Holt q-difference equation. Moreover, we present proofs of quantum calculus versions of two so-called Cushing–Henson conjectures.     

Seasonal dynamics in an SIR epidemic system

We consider a seasonally forced SIR epidemic model where periodicity occurs in the contact rate. This periodical forcing represents successions of school terms and holidays. The epidemic dynamics are described by a switched system. Numerical studies in such a model have shown the existence of periodic solutions. First, we analytically prove the existence of an invariant domain $D$ containing all periodic (harmonic and subharmonic) orbits. Then, using different scales in time and variables, we rewrite the SIR model as a slow-fast dynamical system and we establish the existence of a macroscopic attractor domain $K$ , included in $D$ , for the switched dynamics. The existence of a unique harmonic solution is also proved for any value of the magnitude of the seasonal forcing term which can be interpreted as an annual infection. Subharmonic solutions can be seen as epidemic outbreaks. Our theoretical results allow us to exhibit quantitative characteristics about epidemics, such as the maximal period between major outbreaks and maximal prevalence.     

Degenerate Hopf bifurcation and isolated periodic solutions of the Hodgkin-Huxley model with varying sodium ion concentration

Points of degenerate Hopf bifurcation in the Hodgkin-Huxley model are found as parameters temperature T and voltage level of sodium VNa are varied. Local techniques of degenerate Hopf bifurcation analysis are used to show the existence of families of periodic solutions of the model: isolated branches of periodic solutions (i.e. branches not connected to the stationary branch) are found in addition to Hopf branches. Purely numerical techniques are used to show that the isolas persist for VNa up to a value slightly greater than 114 mV. Under some conditions there are multiple stable periodic solutions, so "jumping" between action potentials of different amplitudes might be observed.     

具有庇护所的三种群捕食者-食饵模型

讨论了一类具有庇护所的自治三种群捕食者一食饵模型,运用Liapunov函数方法,得到了该模型持久性的充分条件．对于该模型的周期系统,在一定的条件下,将产生唯一一个全局渐近稳定的周期正解．对更具普遍意义的概周期现象,也得出了概周期正解唯一存在且全局渐近稳定性的充分条件．     

Effects of density dependent sex allocation on the dynamics of a simultaneous hermaphroditic population: Modelling and analysis

In this work we present a mathematical model describing the dynamics of a population where sex allocation remains flexible throughout adult life and so can be adjusted to current environmental conditions. We consider that the fractions of immature individuals acquiring male and female sexual roles are density dependent through nonlinear functions of a weighted total population size. The main goal of this work is to understand the role of life-history parameters on the stabilization or destabilization of the population dynamics.The model turns out to be a nonlinear discrete model which is analysed by studying the existence of fixed points as well as their stability conditions in terms of model parameters. The existence of more complex asymptotic behaviours of system solutions is shown by means of numerical simulations.Females have larger fertility rate than males. On the other hand, increasing population density favours immature individuals adopting the male role. A positive equilibrium of the system exists whenever fertility and survival rates of one of the sexual roles, if shared by all adults, allow population growing while the opposite happens with the other sexual role. In terms of the female inherent net reproductive number, η_F, it is shown that the positive equilibria are stable when η_F is larger and closed to 1 while for larger values of η_F a certain asymptotic assumption on the investment rate in the female function implies that the population density is permanent. Depending on the other parameters values, the asymptotic behaviour of solutions becomes more complex, even chaotic. In this setting the stabilization/destabilization effects of the abruptness rate in density dependence, of the survival rates and of the competition coefficients are analysed.     

State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences

A state-dependent impulsive model is proposed for integrated pest management (IPM). IPM involves combining biological, mechanical, and chemical tactics to reduce pest numbers to tolerable levels after a pest population has reached its economic threshold (ET). The complete expression of an orbitally asymptotically stable periodic solution to the model with a maximum value no larger than the given ET is presented, the existence of which implies that pests can be controlled at or below their ET levels. We also prove that there is no periodic solution with order larger than or equal to three, except for one special case, by using the properties of the LambertW function and Poincare map. Moreover, we show that the existence of an order two periodic solution implies the existence of an order one periodic solution. Various positive invariant sets and attractors of this impulsive semi-dynamical system are described and discussed. In particular, several horseshoe-like attractors, whose interiors can simultaneously contain stable order 1 periodic solutions and order 2 periodic solutions, are found and the interior structure of the horseshoe-like attractors is discussed. Finally, the largest invariant set and the sufficient conditions which guarantee the global orbital and asymptotic stability of the order 1 periodic solution in the meaningful domain for the system are given using the Lyapunov function. Our results show that, in theory, a pest can be controlled such that its population size is no larger than its ET by applying effects impulsively once, twice, or at most, a finite number of times, or according to a periodic regime. Moreover, our theoretical work suggests how IPM strategies could be used to alter the levels of the ET in the farmers' favour.     

Coal Fly Ash Impairs Airway Antimicrobial Peptides and Increases Bacterial Growth

Air pollution is a risk factor for respiratory infections, and one of its main components is particulate matter (PM), which is comprised of a number of particles that contain iron, such as coal fly ash (CFA). Since free iron concentrations are extremely low in airway surface liquid (ASL), we hypothesize that CFA impairs antimicrobial peptides (AMP) function and can be a source of iron to bacteria. We tested this hypothesis in vivo by instilling mice with Pseudomonas aeruginosa (PA01) and CFA and determine the percentage of bacterial clearance. In addition, we tested bacterial clearance in cell culture by exposing primary human airway epithelial cells to PA01 and CFA and determining the AMP activity and bacterial growth in vitro. We report that CFA is a bioavailable source of iron for bacteria. We show that CFA interferes with bacterial clearance in vivo and in primary human airway epithelial cultures. Also, we demonstrate that CFA inhibits AMP activity in vitro, which we propose as a mechanism of our cell culture and in vivo results. Furthermore, PA01 uses CFA as an iron source with a direct correlation between CFA iron dissolution and bacterial growth. CFA concentrations used are very relevant to human daily exposures, thus posing a potential public health risk for susceptible subjects. Although CFA provides a source of bioavailable iron for bacteria, not all CFA particles have the same biological effects, and their propensity for iron dissolution is an important factor. CFA impairs lung innate immune mechanisms of bacterial clearance, specifically AMP activity. We expect that identifying the PM mechanisms of respiratory infections will translate into public health policies aimed at controlling, not only concentration of PM exposure, but physicochemical characteristics that will potentially cause respiratory infections in susceptible individuals and populations.     

A Stochastic Model Correctly Predicts Changes in Budding Yeast Cell Cycle Dynamics upon Periodic Expression of CLN2

In this study, we focus on a recent stochastic budding yeast cell cycle model. First, we estimate the model parameters using extensive data sets: phenotypes of 110 genetic strains, single cell statistics of wild type and cln3 strains. Optimization of stochastic model parameters is achieved by an automated algorithm we recently used for a deterministic cell cycle model. Next, in order to test the predictive ability of the stochastic model, we focus on a recent experimental study in which forced periodic expression of CLN2 cyclin (driven by MET3 promoter in cln3 background) has been used to synchronize budding yeast cell colonies. We demonstrate that the model correctly predicts the experimentally observed synchronization levels and cell cycle statistics of mother and daughter cells under various experimental conditions (numerical data that is not enforced in parameter optimization), in addition to correctly predicting the qualitative changes in size control due to forced CLN2 expression. Our model also generates a novel prediction: under frequent CLN2 expression pulses, G1 phase duration is bimodal among small-born cells. These cells originate from daughters with extended budded periods due to size control during the budded period. This novel prediction and the experimental trends captured by the model illustrate the interplay between cell cycle dynamics, synchronization of cell colonies, and size control in budding yeast.     

Comparative evaluation of anaerobic bacterial communities associated with roots of submerged macrophytes growing in marine or brackish water sediments

Sediment microbial communities are important for seagrass growth and carbon cycling, however relatively few studies have addressed the composition of prokaryotic communities in seagrass bed sediments. Selective media were used enumerate culturable anaerobic bacteria associated with the roots of the seagrass, Halodule wrightii, the fresh to brackish water plant, Vallisneria americana, and the respective vegetated and unvegetated sediments. H. wrightii roots and sediments had high numbers of sulfate-reducing bacteria whereas iron-reducing bacteria appeared to have a more significant role in V. americana roots and sediments. Numbers of glucose-utilizing but not acetate-utilizing iron reducers were higher on the roots of both plants relative to the vegetated sediments indicating a difference within the iron reducing bacterial community. H. wrightii roots had lower glucose-utilizing iron reducers, and higher acetogenic bacteria than did V. americana roots suggesting different aquatic plants support different anaerobic microbial communities. Sulfur-disproportionating and sulfide-oxidizing bacteria were also cultured from the roots and sediments. These results provide evidence of the potential importance of sulfur cycle bacteria, in addition to sulfate-reducing bacteria, in seagrass bed sediments.     

Hopf bifurcations in multiple-parameter space of the Hodgkin-Huxley equations I. Global organization of bistable periodic solutions

 The Hodgkin-Huxley equations (HH) are parameterized by a number of parameters and shows a variety of qualitatively different behaviors depending on the parameter values. We explored the dynamics of the HH for a wide range of parameter values in the multiple-parameter space, that is, we examined the global structure of bifurcations of the HH. Results are summarized in various two-parameter bifurcation diagrams with I _ext (externally applied DC current) as the abscissa and one of the other parameters as the ordinate. In each diagram, the parameter plane was divided into several regions according to the qualitative behavior of the equations. In particular, we focused on periodic solutions emerging via Hopf bifurcations and identified parameter regions in which either two stable periodic solutions with different amplitudes and periods and a stable equilibrium point or two stable periodic solutions coexist. Global analysis of the bifurcation structure suggested that generation of these regions is associated with degenerate Hopf bifurcations. Received: 23 April 1999 / Accepted in revised form: 24 September 1999     

Travelling wave solutions of diffusive Lotka-Volterra equations

We establish the existence of travelling wave solutions for two reaction diffusion systems based on the Lotka-Volterra model for predator and prey interactions. For simplicity, we consider only 1 space dimension. The waves are of transition front type, analogous to the travelling wave solutions discussed by Fisher and Kolmogorov et al. for a scalar reaction diffusion equation. The waves discussed here are not necessarily monotone. For any speed c there is a travelling wave solution of transition front type. For one of the systems discussed here, there is a distinguished speed c^* dividing the waves into two types, waves of speed c < c^* being one type, waves of speed c ? c^* being of the other type. We present numerical evidence that for this system the wave of speed c^* is stable, and that c^* is an asymptotic speed of propagation in some sense. For the other system, waves of all speeds are in some sense stable. The proof of existence uses a shooting argument and a Lyapunov function. We also discuss some possible biological implications of the existence of these waves.     

Enzymatic release of iron from sideramines in fungi. NADH: Sideramine oxidoreductase in Neurospora crassa

Young mycelia of the fungus Neurospora crassa contain a soluble NADH-linked sideramine reductase, which may be responsible for liberating iron in vivo from accumulated sideramines during iron-deficient cultivation. The enzymes can be assayed using a soluble supernatant fraction, EDTA, and an atmosphere of pure nitrogen. The enzyme is stable without loss of activity up to 45°C and has an optimum of activity at pH 7.0. Besides coprogen (K_m = 100 μM, V = 2.8 nmol/min. per mg protein), some other ferrichrome-type compounds are reduced. However, ferrichrome, ferrirubin, coprogen B and ferrioxamine are poor substrates. When the mucelia were grown in a medium containing 10^?5 M ferric iron, the activity of the reductase was found to be only 30% of that found under low iron conditions. The enzyme is inhibited by oxygen, SH-alkylating agents and partly by some detergents. Unlike the reductase of N. crassa, the corresponding enzyme from Aspergillus fumigatus revealed low reduction of coprogen and high reduction of ferrichrome, indicating genus-dependent specificities of sideramine reduction enzymes in fungi. The participation of acids of the citric acid cycle as natural iron acceptors during strong iron deficiency is studied and confirmed by iron uptake measurements on isolated mitochondria.     

Discovery of chromene compounds as inhibitors of PvdQ acylase of Pseudomonas aeruginosa

The acquisition of iron is a crucial mechanism for the survival of pathogenic bacteria such as Pseudomonas aeruginosa in eukaryotic hosts. The key iron chelator in this organism is the siderophore pyoverdine, which was shown to be crucial for iron homeostasis. Pyoverdine is a non-ribosomal peptide with several maturation steps in the cytoplasm and others in the periplasmatic space. A key enzyme for its maturation is the acylase PvdQ. The inhibition of PvdQ stops the maturation of pyoverdine causing a significant imbalance in the iron homeostasis and hence can negatively influence the survival of P. aeruginosa. In this work, we successfully synthesized chromene-derived inhibitory molecules targeting PvdQ in a low micromolar range. In silico modeling as well as kinetic evaluations of the inhibitors suggest a competitive inhibition of the PvdQ function. Further, we evaluated the inhibitor in vivo on P. aeruginosa cells and report a dose-dependent reduction of pyoverdine formation. The compound also showed a protecting effect in a Galleria mellonella infection model.