Showing papers in "Journal of Mathematical Biology in 2005"

Analysis of the periodically fragmented environment model: I--species persistence.

Abstract: This paper is concerned with the study of the stationary solutions of the equation where the diffusion matrix A and the reaction term f are periodic in x. We prove existence and uniqueness results for the stationary equation and we then analyze the behaviour of the solutions of the evolution equation for large times. These results are expressed by a condition on the sign of the first eigenvalue of the associated linearized problem with periodicity condition. We explain the biological motivation and we also interpret the results in terms of species persistence in periodic environment. The effects of various aspects of heterogeneities, such as environmental fragmentation are also discussed.

An interacting particle system modelling aggregation behavior: from individuals to populations

Abstract: In this paper we investigate the stochastic modelling of a spatially structured biological population subject to social interaction. The biological motivation comes from the analysis of field experiments on a species of ants which exhibits a clear tendency to aggregate, still avoiding overcrowding. The model we propose here provides an explanation of this experimental behavior in terms of "long-ranged" aggregation and "short-ranged" repulsion mechanisms among individuals, in addition to an individual random dispersal described by a Brownian motion. Further, based on a "law of large numbers", we discuss the convergence, for large N, of a system of stochastic differential equations describing the evolution of N individuals (Lagrangian approach) to a deterministic integro-differential equation describing the evolution of the mean-field spatial density of the population (Eulerian approach).

Derivation of hyperbolic models for chemosensitive movement

Abstract: A Chapman-Enskog expansion is used to derive hyperbolic models for chemosensitive movements as a hydrodynamic limit of a velocity-jump process. On the one hand, it connects parabolic and hyperbolic chemotaxis models since the former arise as diffusion limits of a similar velocity-jump process. On the other hand, this approach provides a unified framework which includes previous models obtained by ad hoc methods or methods of moments. Numerical simulations are also performed and are motivated by recent experiments with human endothelial cells on matrigel. Their movements lead to the formation of networks that are interpreted as the beginning of a vasculature. These structures cannot be explained by parabolic models but are recovered by numerical experiments on hyperbolic models. Our kinetic model suggests that some kind of local interactions might be enough to explain them.

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

Abstract: 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.

An Analysis of Vegetation Stripe Formation in Semi-Arid Landscapes

Abstract: Vegetation stripes (“tiger bush”) are a characteristic feature of semi-arid environments. The stripes typically lie along the contours of gentle slopes, and some authors report a gradual uphill migration. A previous mathematical model (Klausmeier, Science, 284:1826, 1999) has shown that this phenomenon can be explained relatively simply by the downhill flow of rainwater coupled with the diffusive spread of the plant population. This paper presents a detailed analysis of pattern formation in the Klausmeier model. The author derives formulae for the wavelength and migration speed of the predicted patterns, and systematically investigates how these depend on model parameters. The results make new predictions and suggest possible approaches to testing the model.

Kinetic models for chemotaxis: hydrodynamic limits and spatio-temporal mechanisms.

Abstract: We study kinetic models for chemotaxis, incorporating the ability of cells to assess temporal changes of the chemoattractant concentration as well as its spatial variations. For prescribed smooth chemoattractant density, the macroscopic limit is carried out rigorously. It leads to a drift equation with a chemotactic sensitivity depending on the time derivative of the chemoattractant density. As an application it is shown by numerical experiments that the new model can resolve the chemotactic wave paradox. For this purpose, the macroscopic equation is coupled to a simple activation-inhibition model for the chemoattractant which produces the chemoattractant waves typical for the slime mold Dictyostelium discoideum.

Spatial effects in discrete generation population models.

Abstract: A framework is developed for constructing a large class of discrete generation, continuous space models of evolving single species populations and finding their bifurcating patterned spatial distributions. Our models involve, in separate stages, the spatial redistribution (through movement laws) and local regulation of the population; and the fundamental properties of these events in a homogeneous environment are found. Emphasis is placed on the interaction of migrating individuals with the existing population through conspecific attraction (or repulsion), as well as on random dispersion. The nature of the competition of these two effects in a linearized scenario is clarified. The bifurcation of stationary spatially patterned population distributions is studied, with special attention given to the role played by that competition.

Stability analysis of pathogen-immune interaction dynamics.

Abstract: The paper considers models of dynamics of infectious disease in vivo from the standpoint of the mathematical analysis of stability. The models describe the interaction of the target cells, the pathogens, and the humoral immune response. The paper mainly focuses on the interior equilibrium, whose components are all positive. If the model ignores the absorption of the pathogens due to infection, the interior equilibrium is always asymptotically stable. On the other hand, if the model does consider it, the interior equilibrium can be unstable and a simple Hopf bifurcation can occur. A sufficient condition that the interior equilibrium is asymptotically stable is obtained. The condition explains that the interior equilibrium is asymptotically stable when experimental parameter values are used for the model. Moreover, the paper considers the models in which uninfected cells are involved in the immune response to pathogens, and are removed by the immune complexes. The effect of the involvement strongly affects the stability of the interior equilibria. The results are shown with the aid of symbolic calculation software.

Bounding the number of hybridisation events for a consistent evolutionary history.

Abstract: Evolutionary processes such as hybridisation, lateral gene transfer, and recombination are all key factors in shaping the structure of genes and genomes. However, since such processes are not always best represented by trees, there is now considerable interest in using more general networks instead. For example, in recent studies it has been shown that networks can be used to provide lower bounds on the number of recombination events and also for the number of lateral gene transfers that took place in the evolutionary history of a set of molecular sequences. In this paper we describe the theoretical performance of some related bounds that result when merging pairs of trees into networks.

Resident-invader dynamics and the coexistence of similar strategies.

Abstract: We study the resident-invader dynamics for a given class of models of unstructured populations of finite-dimensional strategies. We prove various results on the existence and uniqueness of ω-limit sets in the interior of the resident-invader population state space, and we classify the generically possible types of dynamics in terms of the invasion conditions when the resident and invader strategies are similar to one another.

On the impossibility of coexistence of infinitely many strategies.

Abstract: We investigate the possibility of coexistence of pure, inherited strategies belonging to a large set of potential strategies. We prove that under biologically relevant conditions every model allowing for coexistence of infinitely many strategies is structurally unstable. In particular, this is the case when the "interaction operator" which determines how the growth rate of a strategy depends on the strategy distribution of the population is compact. The interaction operator is not assumed to be linear. We investigate a Lotka-Volterra competition model with a linear interaction operator of convolution type separately because the convolution operator is not compact. For this model, we exclude the possibility of robust coexistence supported on the whole real line, or even on a set containing a limit point. Moreover, we exclude coexistence of an infinite set of equidistant strategies when the total population size is finite. On the other hand, for infinite populations it is possible to have robust coexistence in this case. These results are in line with the ecological concept of "limiting similarity" of coexisting species. We conclude that the mathematical structure of the ecological coexistence problem itself dictates the discreteness of the species.

Harvesting in seasonal environments

Abstract: Most harvest theory is based on an assumption of a constant or stochastic environment, yet most populations experience some form of environmental seasonality. Assuming that a population follows logistic growth we investigate harvesting in seasonal environments, focusing on maximum annual yield (M.A.Y.) and population persistence under five commonly used harvest strategies. We show that the optimal strategy depends dramatically on the intrinsic growth rate of population and the magnitude of seasonality. The ordered effectiveness of these alternative harvest strategies is given for different combinations of intrinsic growth rate and seasonality. Also, for piecewise continuous-time harvest strategies (i.e., open / closed harvest, and pulse harvest) harvest timing is of crucial importance to annual yield. Optimal timing for harvests coincides with maximal rate of decline in the seasonally fluctuating carrying capacity. For large intrinsic growth rate and small environmental variability several strategies (i.e., constant exploitation rate, linear exploitation rate, and time-dependent harvest) are so effective that M.A.Y. is very close to maximum sustainable yield (M.S.Y.). M.A.Y. of pulse harvest can be even larger than M.S.Y. because in seasonal environments population size varies substantially during the course of the year and how it varies relative to the carrying capacity is what determines the value relative to optimal harvest rate. However, for populations with small intrinsic growth rate but subject to large seasonality none of these strategies is particularly effective with M.A.Y. much lower than M.S.Y. Finding an optimal harvest strategy for this case and to explore harvesting in populations that follow other growth models (e.g., involving predation or age structure) will be an interesting but challenging problem.

A model of intracellular transport of particles in an axon.

Abstract: In this paper we develop a model of intracellular transport of cell organelles and vesicles along the axon of a nerve cell. These particles are moving alternately by processive motion along a microtubule with the aid of motor proteins, and by diffusion. The model involves a degenerate system of diffusion equations. We prove a maximum principle and establish existence and behavior of a unique solution. Numerical results show how the transportation of mass depends on the relevant parameters of the model.

A multilocus analysis of intraspecific competition and stabilizing selection on a quantitative trait.

Abstract: The equilibrium structure of an additive, diallelic multilocus model of a quantitative trait under frequency- and density-dependent selection is derived. The trait is under stabilizing selection and mediates intraspecific competition as induced, for instance, by differential resource utilization. It is assumed that stabilizing selection is weak, but the strength of competition may be arbitrary relative to it. Density dependence is caused by population regulation, which may be of a very general kind. The number and effects of loci are arbitrary, and stabilizing selection is not necessarily symmetric with respect to the range of phenotypic values. All previously studied models of intraspecific competition for a continuum of resources known to the author reduce to a special case of the present model if overall selection is weak. Therefore, in this case our results are applicable as approximations to all these models. Our central result is the (nearly) complete characterization of the equilibrium and stability structure in terms of all parameters. It is derived under the sole assumption that selection is weak enough relative to recombination to ignore linkage disequilibrium. In particular, necessary and sufficient conditions on the strength of competition relative to stabilizing selection are found that ensure the maintenance of multilocus polymorphism and the occurrence of disruptive selection. In this case, explicit formulas for the number of polymorphic loci at equilibrium, the allele frequencies, the genetic variance, and the strength of disruptive selection are obtained. For two loci, the effects of linkage are investigated analytically; for several loci, they are studied numerically.

Differential susceptibility epidemic models

Abstract: We formulate compartmental differential susceptibility (DS) susceptible-infective-removed (SIR) models by dividing the susceptible population into multiple subgroups according to the susceptibility of individuals in each group. We analyze the impact of disease-induced mortality in the situations where the number of contacts per individual is either constant or proportional to the total population. We derive an explicit formula for the reproductive number of infection for each model by investigating the local stability of the infection-free equilibrium. We further prove that the infection-free equilibrium of each model is globally asymptotically stable by qualitative analysis of the dynamics of the model system and by utilizing an appropriately chosen Liapunov function. We show that if the reproductive number is greater than one, then there exists a unique endemic equilibrium for all of the DS models studied in this paper. We prove that the endemic equilibrium is locally asymptotically stable for the models with no disease-induced mortality and the models with contact numbers proportional to the total population. We also provide sufficient conditions for the stability of the endemic equilibrium for other situations. We briefly discuss applications of the DS models to optimal vaccine strategies and the connections between the DS models and predator-prey models with multiple prey populations or host-parasitic interaction models with multiple hosts are also given.

Modelling Human Granulopoiesis under Poly-chemotherapy with G-CSF Support

Abstract: Cytotoxic drugs administered in polychemotherapy cause a characteristic neutropenic period depending on the schedule of the drugs, which can partly be prevented by G-CSF growth factor support. To quantify these effects and to gain a deeper insight into the dynamics of bone marrow recovery after such suppressing and stimulating disturbances, we construct a biomathematical compartment model of human granulopoiesis under polychemotherapy with G-CSF support. The underlying assumptions and mathematical techniques used to obtain the model are explained in detail. A large variety of biological and clinical data as well as knowledge from a model of murine haematopoiesis are evaluated to construct a physiological model for humans. Particular emphasis is placed on estimating the influence of chemotherapeutic drugs on the granulopoietic system. As a result, we present an innovative method to estimate the bone marrow damage caused by cytotoxic drugs with respect to single identifiable cell stages only on the basis of measured peripheral blood leukocyte dynamics. Conversely, our model can be used in a planning phase of a clinical trial to estimate the haematotoxicity of regimens based on new combinations of drugs already considered and with or without growth factor support.

The competitive dynamics between tumor cells, a replication-competent virus and an immune response

Abstract: Replication-competent viruses have been used as an alternative therapeutic approach for cancer treatment. However, new clinical data revealed an innate immune response to virus that may mitigate the effects of treatment. Recently, Wein, Wu and Kirn have established a model which describes the interaction between tumor cells, a replication-competent virus and an immune response (Cancer Research 63 (2003):1317–1324). The purpose of this paper is to extend their model from the viewpoints of mathematics and biology and then prove global existence and uniqueness of solution to this new model, to study the dynamics of this novel therapy for cancers, and to explore a explicit threshold of the intensity of the immune response for controlling the tumor. We also study a time-delayed version of the model. We analytically prove that there exists a critical value τ0 of the time-delay τ such that the system has a periodic solution if τ>τ0. Numerical simulations are given to verify the analytical results. Furthermore, we numerically study the spatio-temporal dynamics of the model. The effects of the diffusivity of the immune response on the tumor growth are also discussed.

A cardiovascular-respiratory control system model including state delay with application to congestive heart failure in humans.

Abstract: This paper considers a model of the human cardiovascular-respiratory control system with one and two transport delays in the state equations describing the respiratory system. The effectiveness of the control of the ventilation rate is influenced by such transport delays because blood gases must be transported a physical distance from the lungs to the sensory sites where these gases are measured. The short term cardiovascular control system does not involve such transport delays although delays do arise in other contexts such as the baroreflex loop (see [46]) for example. This baroreflex delay is not considered here. The interaction between heart rate, blood pressure, cardiac output, and blood vessel resistance is quite complex and given the limited knowledge available of this interaction, we will model the cardiovascular control mechanism via an optimal control derived from control theory. This control will be stabilizing and is a reasonable approach based on mathematical considerations as well as being further motivated by the observation that many physiologists cite optimization as a potential influence in the evolution of biological systems (see, e.g., Kenner [29] or Swan [62]). In this paper we adapt a model, previously considered (Timischl [63] and Timischl et al. [64]), to include the effects of one and two transport delays. We will first implement an optimal control for the combined cardiovascular-respiratory model with one state space delay. We will then consider the effects of a second delay in the state space by modeling the respiratory control via an empirical formula with delay while the the complex relationships in the cardiovascular control will still be modeled by optimal control. This second transport delay associated with the sensory system of the respiratory control plays an important role in respiratory stability. As an application of this model we will consider congestive heart failure where this transport delay is larger than normal and the transition from the quiet awake state to stage 4 (NREM) sleep. The model can be used to study the interaction between cardiovascular and respiratory function in various situations as well as to consider the influence of optimal function in physiological control system performance.

Modelling antibiotic- and anti-quorum sensing treatment of a spatially-structured Pseudomonas aeruginosa population.

Abstract: The bacterial cell to cell signalling system known as quorum sensing (QS) is essential for the regulation of virulence in many pathogens and offers a specific biochemical target for novel antibacterial therapies. Expanding on earlier work, in which consideration was given to the primary QS system (lasR system) in a homogeneous population of the common human pathogen Pseudomonas aeruginosa, we build a simple spatial model of an early-stage P. aeruginosa biofilm subject to treatment with topically applied anti-QS drugs (of two specific kinds) and conventional antibiotics. In the case of a slowly growing biofilm we show that both kinds of anti-quorum sensing drug are effective in reducing the level of the relevant signal molecule (3-oxo-C12-homoserine lactone; henceforth AHL), in each case obtaining an explicit bound on the steady-state AHL profile in terms of a prescribed surface drug concentration. Using numerical methods, we are also able to reproduce the hysteretic phenomena exhibited by the homogeneous model, in particular showing that for each kind of anti-QS drug there is a parameter regime in which a catastrophic collapse occurs in the steady-state AHL concentration as the surface drug concentration passes some critical value; an alternative way of interpreting this result is to say that, for a prescribed surface drug concentration, there is a critical biofilm depth such that treatment is successful until this depth is reached, but fails thereafter. In the thick-biofilm limit we show that the critical concentration of each drug increases exponentially with the biofilm thickness (or, conversely, that the critical depth increases logarithmically with surface drug concentration); this is dramatically different to the behaviour observed in the corresponding homogeneous model, where the critical concentrations grow linearly with bacterial carrying capacity, and thus highlights the relative difficulty of treating a large, spatially-structured population with diffusing antibacterials.

Transient oscillations induced by delayed growth response in the chemostat

Abstract: In this paper, in order to try to account for the transient oscillations observed in chemostat experiments, we consider a model of single species growth in a chemostat that involves delayed growth response. The time delay models the lag involved in the nutrient conversion process. Both monotone response functions and nonmonotone response functions are considered. The nonmonotone response function models the inhibitory effects of growth response of certain nutrients when concentrations are too high. By applying local and global Hopf bifurcation theorems, we prove that the model has unstable periodic solutions that bifurcate from unstable nonnegative equilibria as the parameter measuring the delay passes through certain critical values and that these local periodic solutions can persist, even if the delay parameter moves far from the critical (local) bifurcation values. When there are two positive equilibria, then positive periodic solutions can exist. When there is a unique positive equilibrium, the model does not have positive periodic oscillations and the unique positive equilibrium is globally asymptotically stable. However, the model can have periodic solutions that change sign. Although these solutions are not biologically meaningful, provided the initial data starts close enough to the unstable manifold of one of these periodic solutions they may still help to account for the transient oscillations that have been frequently observed in chemostat experiments. Numerical simulations are provided to illustrate that the model has varying degrees of transient oscillatory behaviour that can be controlled by the choice of the initial data.

Lattice-Boltzmann model for bacterial chemotaxis

Abstract: We present a new numerical approach for modeling bacterial chemotaxis and the fate and transport of a chemoattractant in bulk liquids. This Lattice-Boltzmann method represents the microorganisms and the chemoattractant by quasi-particles that move, collide, and react with each other on a two-dimensional numerical lattice. We use the model to simulate traveling bands of bacteria along self-generated gradients in substrate concentration in bulk liquids. Particularly, we simulate Pseudomonas putida that respond chemotactically to naphthalene dissolved in water. We find that only a fraction of a bacterial slug injected into a domain containing the chemoattractant at constant concentration forms a traveling band as the slug length exceeds a critical value. An expanding bacterial ring forms as one injects a droplet of bacteria into a two-dimensional domain.

Gene algebra from a genetic code algebraic structure

Abstract: By considering two important factors involved in the codon-anticodon interactions, the hydrogen bond number and the chemical type of bases, a codon array of the genetic code table as an increasing code scale of interaction energies of amino acids in proteins was obtained. Next, in order to consecutively obtain all codons from the codon AAC, a sum operation has been introduced in the set of codons. The group obtained over the set of codons is isomorphic to the group (Z64, +) of the integer module 64. On the Z64-algebra of the set of 64 N codon sequences of length N, gene mutations are described by means of endomorphisms f:(Z64) N →(Z64) N . Endomorphisms and automorphisms helped us describe the gene mutation pathways. For instance, 77.7% mutations in 749 HIV protease gene sequences correspond to unique diagonal endomorphisms of the wild type strain HXB2. In particular, most of the reported mutations that confer drug resistance to the HIV protease gene correspond to diagonal automorphisms of the wild type. What is more, in the human beta-globin gene a similar situation appears where most of the single codon mutations correspond to automorphisms. Hence, in the analyses of molecular evolution process on the DNA sequence set of length N, the Z64-algebra will help us explain the quantitative relationships between genes.

Analysis of the periodically fragmented environment model : I - Influence of periodic heterogeneous environment on species persistence.

Abstract: This paper is concerned with the study of the stationary solutions of the equation $$u_t- abla\cdot(A(x) abla u)=f(x,u),\ \ x\in{\mathbb{R}}^N,$$ where the diffusion matrix $A$ and the reaction term $f$ are periodic in $x$. We prove existence and uniqueness results for the stationary equation and we then analyze the behaviour of the solutions of the evolution equation for large times. These results are expressed by a condition on the sign of the first eigenvalue of the associated linearized problem with periodicity condition. We explain the biological motivation and we also interpret the results in terms of species persistence in periodic environment. The effects of various aspects of heterogeneities, such as environmental fragmentation are also discussed.

On the (dis) advantages of cannibalism

Abstract: Cannibalism is an interaction between individuals that can produce counter- intuitive effects at the population level. A striking effect is that a population may persist under food conditions such that the non-cannibalistic variant is doomed to go extinct. This so-called life boat mechanism has received considerable attention. Implicitly, such studies sometimes suggest, that the life boat mechanism procures an evolutionary advantage to the cannibalistic trait. Here we compare, in the context of a size structured population model, the conditions under which the life boat mechanism works, with those that guarantee, that a cannibalistic mutant can invade successfully under the steady environmental conditions as set by a non-cannibalistic resident. We find qualitative agreement and quantitative difference. In particular, we find that a prerequisite for the life boat mechanism is, that cannibalistic mutants are successful invaders. Roughly speaking, our results show that cannibalism brings advantages to both the individuals and the population when adult food is limiting.

Dynamical behaviour of epidemiological models with sub-optimal immunity and nonlinear incidence

Abstract: In this paper we analyze the dynamics of two families of epidemiological models which correspond to transitions from the SIR (susceptible-infectious-resistant) to the SIS (susceptible-infectious-susceptible) frameworks. In these models we assume that the force of infection is a nonlinear function of density of infectious individuals, I. Conditions for the existence of backwards bifurcations, oscillations and Bogdanov-Takens points are given.

On the effect of population heterogeneity on dynamics of epidemic diseases

Abstract: The paper investigates a class of SIS models of the evolution of an infectious disease in a heterogeneous population The heterogeneity reflects individual differences in the susceptibility or in the contact rates and leads to a distributed parameter system, requiring therefore, distributed initial data, which are often not available It is shown that there exists a corresponding homogeneous (ODE) population model that gives the same aggregated results as the distributed one, at least in the expansion phase of the disease However, this ODE model involves a nonlinear "prevalence-to-incidence" function which is not constructively defined Based on several established properties of this function, a simple class of approximating function is proposed, depending on three free parameters that could be estimated from scarce data How the behaviour of a population depends on the level of heterogeneity (all other parameters kept equal) - this is the second issue studied in the paper It turns out that both for the short run and for the long run behaviour there exist threshold values, such that more heterogeneity is advantageous for the population if and only if the initial (weighted) prevalence is above the threshold

Eye movement instabilities and nystagmus can be predicted by a nonlinear dynamics model of the saccadic system

Abstract: The study of eye movements and oculomotor disorders has, for four decades, greatly benefitted from the application of control theoretic concepts. This paper is an example of a complementary approach based on the theory of nonlinear dynamical systems. Recently, a nonlinear dynamics model of the saccadic system was developed, comprising a symmetric piecewise-smooth system of six first-order autonomous ordinary differential equations. A preliminary numerical investigation of the model revealed that in addition to generating normal saccades, it could also simulate inaccurate saccades, and the oscillatory instability known as congenital nystagmus (CN). By varying the parameters of the model, several types of CN oscillations were produced, including jerk, bidirectional jerk and pendular nystagmus. The aim of this study was to investigate the bifurcations and attractors of the model, in order to obtain a classification of the simulated oculomotor behaviours. The application of standard stability analysis techniques, together with numerical work, revealed that the equations have a rich bifurcation structure. In addition to Hopf, homoclinic and saddlenode bifurcations organised by a Takens-Bogdanov point, the equations can undergo nonsmooth pitchfork bifurcations and nonsmooth gluing bifurcations. Evidence was also found for the existence of Hopf-initiated canards. The simulated jerk CN waveforms were found to correspond to a pair of post-canard symmetry-related limit cycles, which exist in regions of parameter space where the equations are a slow-fast system. The slow and fast phases of the simulated oscillations were attributed to the geometry of the corresponding slow manifold. The simulated bidirectional jerk and pendular waveforms were attributed to a symmetry invariant limit cycle produced by the gluing of the asymmetric cycles. In contrast to control models of the oculomotor system, the bifurcation analysis places clear restrictions on which kinds of behaviour are likely to be associated with each other in parameter space, enabling predictions to be made regarding the possible changes in the oscillation type that may be observed upon changing the model parameters. The analysis suggests that CN is one of a range of oculomotor disorders associated with a pathological saccadic braking signal, and that jerk and pendular nystagmus are the most probable oscillatory instabilities. Additionally, the transition from jerk CN to bidirectional jerk and pendular nystagmus observed experimentally when the gaze angle or attention level is changed is attributed to a gluing bifurcation. This suggests the possibility of manipulating the waveforms of subjects with jerk CN experimentally to produce waveforms with an extended foveation period, thereby improving visual resolution.

A Mathematical Model of Cerebral Circulation and Oxygen Supply

Abstract: A compartment model of cerebral circulation and oxygen supply including an autoregulation mechanism is presented. The model is focused on the analysis of slow dynamical variations of long term neurophysiological parameters like the partial oxygen pressure of brain tissue or the cerebral blood flow. The circulatory part of the model is built up of seven compartments including arteries, capillaries, veins, brain tissue, cerebrospinal fluid, the sagittal sinus and an artificial compartment for the simulation of brain swelling. The description of oxygen supply is based on a Krogh model. Numerical calculations reproduce the experimentally well established connection between arterial blood pressure and the production of cerebrospinal fluid. Furthermore we found an approximately linear correlation of the partial oxygen pressure of brain tissue to the mean arterial blood pressure in the case of an impaired autoregulation mechanism. In a first test such a linear dependence could also be detected in clinical data from the neurosurgical intensive care monitoring.

A delay recruitment model of the cardiovascular control system

Abstract: We develop a nonlinear delay-differential equation for the human cardiovascular control system, and use it to explore blood pressure and heart rate variability under short-term baroreflex control. The model incorporates an intrinsically stable heart rate in the absence of nervous control, and allows us to compare the baroreflex influence on heart rate and peripheral resistance. Analytical simplifications of the model allow a general investigation of the roles played by gain and delay, and the effects of ageing.

Global analysis of competition for perfectly substitutable resources with linear response

Abstract: We study a model of the chemostat with two species competing for two perfectly substitutable resources in the case of linear functional response. Lyapunov methods are used to provide sufficient conditions for the global asymptotic stability of the coexistence equilibrium. Then, using compound matrix techniques, we provide a global analysis in a subset of parameter space. In particular, we show that each solution converges to an equilibrium, even in the case that the coexistence equilibrium is a saddle. Finally, we provide a bifurcation analysis based on the dilution rate. In this context, we are able to provide a geometric interpretation that gives insight into the role of the other parameters in the bifurcation sequence.