Synchronism in a metapopulation model

We consider a spatially explicit metapopulation model with interaction among the two nearest neighbors to relate, with a simple mathematical expression, chaos in the local, uncoupled, populations, the degree of interaction among patches, size of the metapopulation, and the stability of the synchronized attractor. Since synchronism is strongly correlated with extinction, our results can provide useful information on factors leading to population extinction.     

Monitoring in a Lotka-Volterra model

The problem of monitoring arises when in an ecosystem, in particular in a system of several populations, observing some components, we want to recover the state of the whole system as a function of time. Due to the difficulty to construct exactly this state process, we look for an auxiliary system called an observer. This system reproduces this process with a certain approximation. This means that the solution of the observer tends to that of the original system. An important concept for this work is observability. This means that from the observation it is possible to recover uniquely the state process, however, without determining a constructive method to obtain it. If observability holds for the original system, it guarantees the existence of an auxiliary matrix that makes it possible to construct an observer of the system. The considered system of populations is described by the classical Lotka-Volterra model with one predator and two preys and the construction of its observer is illustrated with a numerical example. Finally, it is shown how the observer can be used for the estimation of the level of an abiotic effect on the population system.     

Three-species ecosystems in a solvable model

A detailed discussion of the three-species ecosystems is presented in an exactly solvable model with interactions of the Gompertz form. Three different possibilities, namely, a one-prey-two-predator system, a two-prey-one-predator system and a three-step prey-predator food chain are considered. These systems are studied not only when they include their basic prey-predator interactions, but also when various self-interactions as well as competition between like species, in different possible combinations, are included. It is then inferred, by obtaining and examining the exact solutions, as to when these systems possess stable equilibrium and when not, or when they are purely oscillatory, etc. We also study, within our model, the two-species versus three-species situation. It is seen that there are situations when the three-species system possesses stable equilibrium even under circumstances under which the corresponding two-species system is unstable. We also come across cases when the addition of the third species destroys the possibility of stable equilibrium which the initial two-species system possessed. Some other results also follow. Of particular interest is the one where the initial two-species system is purely oscillatory but the enlarged system, which is a three-step prey-predator chain, has the first and the last populations of the chain rising indefinitely and the middle population remains oscillatory. A comparison of our results with results of other authors, wherever possible, has also been made.     

Oscillations in a size-structured prey-predator model

This article introduces a predator–prey model with the prey structured by body size, based on reports in the literature that predation rates are prey-size specific. The model is built on the foundation of the one-species physiologically structured models studied earlier. Three types of equilibria are found: extinction, multiple prey-only equilibria and possibly multiple predator–prey coexistence equilibria. The stabilities of the equilibria are investigated. Comparison is made with the underlying ODE Lotka–Volterra model. It turns out that the ODE model can exhibit sustain oscillations if there is an Allee effect in the net reproduction rate, that is the net reproduction rate grows for some range of the prey’s population size. In contrast, it is shown that the structured PDE model can exhibit sustain oscillations even if the net reproductive rate is strictly declining with prey population size. We find that predation, even size-non-specific linear predation can destabilize a stable prey-only equilibrium, if reproduction is size specific and limited to individuals of large enough size. Furthermore, we show that size-specific predation can also destabilize the predator–prey equilibrium in the PDE model. We surmise that size-specific predation allows for temporary prey escape which is responsible for destabilization in the predator–prey dynamics.     

Forced oscillations in a windkessel model

Volume elasticity of the arterial system and its component parts is developed starting from a Windkessel^*-model, which is defined in 4 points. Emphasis is laid on the simplicity of the derived equations and accessibility to experimental verification. The theory is an extension of earlier work achieved by Wetterer and Pieper (1953), who introduced an essentially physical method for the indirect determination of volume elasticity in situ, by creating forced sinusoidal oscillations in the arterial system, using a special pump operated at a considerably lower frequency than the mean heart frequency. The elegance of both experimental technique and the derived equations incited us to investigate the mathematical foundation and possible generalization of the method.     

Bifurcations in a white-blood-cell production model

We study the dynamics of a model of white-blood-cell (WBC) production. The model consists of two compartmental differential equations with two discrete delays. We show that from normal to pathological parameter values, the system undergoes supercritical Hopf bifurcations and saddle-node bifurcations of limit cycles. We characterize the steady states of the system and perform a bifurcation analysis. Our results indicate that an increase in apoptosis rate of either hematopoietic stem cells or WBC precursors induces a Hopf bifurcation and an oscillatory regime takes place. These oscillations are seen in some hematological diseases.     

What is in a model?

After reading contradictory claims of model status for some insect species, we feel a brief discussion of the topic may be useful. Here, we document a few examples where clarity on model status seems to be lacking, briefly review work on widely recognized models, and offer criteria for including any given species as a model organism.     

Instability in a generalized Keller-Segel model

We present a generalized Keller-Segel model where an arbitrary number of chemical compounds react, some of which are produced by a species, and one of which is a chemoattractant for the species. To investigate the stability of homogeneous stationary states of this generalized model, we consider the eigenvalues of a linearized system. We are able to reduce this infinite dimensional eigenproblem to a parametrized finite dimensional eigenproblem. By matrix theoretic tools, we then provide easily verifiable sufficient conditions for destabilizing the homogeneous stationary states. In particular, one of the sufficient conditions is that the chemotactic feedback is sufficiently strong. Although this mechanism was already known to exist in the original Keller-Segel model, here we show that it is more generally applicable by significantly enlarging the class of models exhibiting this instability phenomenon which may lead to pattern formation.     

Traveling waves in a chemotactic model

A model for chemotaxis in a bacteria-substrate mixture introduced by Keller and Segel, which is described by nonlinear partial differential equations, is studied analytically. The existence of traveling waves is shown for the system in which the substrate diffusion is taken into account and the chemotactic coefficient is greater than the motility one, and the instability of traveling waves is discussed.     

Choking phenomena in a lung-like model

A simple, continuous, one-dimensional model for the geometry and structure of the bronchial airways is used for the analysis of fluid flow patterns which have been observed in forced expiration maneuvers. Various phenomena within the conducting system associated with flow limitation are investigated: the conditions in which a "choke" (flow limitation) can occur in a compliant system; theoretical flows that are physically impossible; the possibility of having elastic jumps downstream of the choke point; perturbations in the physical parameters of the conducting system.     

Estimation in a semi-Markov transformation model

Semi-Markov and modulated renewal processes provide a large class of multi-state models which can be used for analysis of longitudinal failure time data. In biomedical applications, models of this kind are often used to describe evolution of a disease and assume that patient may move among a finite number of states representing different phases in the disease progression. Several authors proposed extensions of the proportional hazard model for regression analysis of these processes. In this paper, we consider a general class of censored semi-Markov and modulated renewal processes and propose use of transformation models for their analysis. Special cases include modulated renewal processes with interarrival times specified using transformation models, and semi-Markov processes with with one-step transition probabilities defined using copula-transformation models. We discuss estimation of finite and infinite dimensional parameters and develop an extension of the Gaussian multiplier method for setting confidence bands for transition probabilities and related parameters. A transplant outcome data set from the Center for International Blood and Marrow Transplant Research is used for illustrative purposes.     

Forced oscillations in a windkessel model

Volume elasticity of the arterial system and its component parts is developed starting from a Windkessel^*-model, which is defined in 4 points. Emphasis is laid on the simplicity of the derived equations and accessibility to experimental verification. The theory is an extension of earlier work achieved by Wetterer and Pieper (1953), who introduced an essentially physical method for the indirect determination of volume elasticity in situ, by creating forced sinusoidal oscillations in the arterial system, using a special pump operated at a considerably lower frequency than the mean heart frequency. The elegance of both experimental technique and the derived equations incited us to investigate the mathematical foundation and possible generalization of the method.     

Sensitivity of a Hill-based muscle model to perturbations in model parameters

Musculoskeletal simulations of human movement commonly use Hill muscle models to predict muscle forces, but their sensitivity to model parameter values is not well understood. The purpose of this study was to evaluate muscle model sensitivity to perturbations in 14 Hill muscle model parameters in forward dynamic simulations of running and walking by varying each by +/-50%. Three evaluations of the muscle model were performed based on: (1) calculating the sensitivity of the muscle model only, (2) determining the continuous partial derivatives of the muscle equations with respect to each parameter, and (3) evaluating the effects on the running and walking simulations. Model evaluations were found to be very sensitive (percent change in outputs greater than parameter perturbation) to parameters defining the series elastic component (tendon), force-length curve of the contractile element and maximum isometric force. For some parameters, the range of literature values was larger than the model sensitivity. Model evaluations were insensitive to parameters defining the parallel elastic element, force-velocity curve of the contractile element and muscle activation time constants. The derivative method provided similar results, but also provided a generic, continuous equation that can easily be applied to other motions. The sensitivities of the running and walking simulations were reduced compared to the sensitivity of the muscle model alone. Results demonstrate the importance of evaluating sensitivity of a musculoskeletal simulation in a controlled manner and provide an indication of which parameters must be selected most carefully based on the sensitivity of a given movement.     

Flow pattern analysis in a highly stenotic patient-specific carotid bifurcation model using a turbulence model

The aim of this study is to investigate the blood flow pattern in carotid bifurcation with a high degree of luminal stenosis, combining in vivo magnetic resonance imaging (MRI) and computational fluid dynamics (CFD). A newly developed two-equation transitional model was employed to evaluate wall shear stress (WSS) distribution and pressure drop across the stenosis, which are closely related to plaque vulnerability. A patient with an 80% left carotid stenosis was imaged using high resolution MRI, from which a patient-specific geometry was reconstructed and flow boundary conditions were acquired for CFD simulation. A transitional model was implemented to investigate the flow velocity and WSS distribution in the patient-specific model. The peak time-averaged WSS value of approximately 73 Pa was predicted by the transitional flow model, and the regions of high WSS occurred at the throat of the stenosis. High oscillatory shear index values up to 0.50 were present in a helical flow pattern from the outer wall of the internal carotid artery immediately after the throat. This study shows the potential suitability of a transitional turbulent flow model in capturing the flow phenomena in severely stenosed carotid arteries using patient-specific MRI data and provides the basis for further investigation of the links between haemodynamic variables and plaque vulnerability. It may be useful in the future for risk assessment of patients with carotid disease.     

Mappings between a macroscopic neural-mass model and a reduced conductance-based model

We present two alternative mappings between macroscopic neuronal models and a reduction of a conductance-based model. These provide possible explanations of the relationship between parameters of these two different approaches to modelling neuronal activity. Obtaining a physical interpretation of neural-mass models is of fundamental importance as they could provide direct and accessible tools for use in diagnosing neurological conditions. Detailed consideration of the assumptions required for the validity of each mapping elucidates strengths and weaknesses of each macroscopic model and suggests improvements for future development.     

Simplifying a physiologically structured population model to a stage-structured biomass model

We formulate and analyze an archetypal consumer-resource model in terms of ordinary differential equations that consistently translates individual life history processes, in particular food-dependent growth in body size and stage-specific differences between juveniles and adults in resource use and mortality, to the population level. This stage-structured model is derived as an approximation to a physiologically structured population model, which accounts for a complete size-distribution of the consumer population and which is based on assumptions about the energy budget and size-dependent life history of individual consumers. The approximation ensures that under equilibrium conditions predictions of both models are completely identical. In addition we find that under non-equilibrium conditions the stage-structured model gives rise to dynamics that closely approximate the dynamics exhibited by the size-structured model, as long as adult consumers are superior foragers than juveniles with a higher mass-specific ingestion rate. When the mass-specific intake rate of juvenile consumers is higher, the size-structured model exhibits single-generation cycles, in which a single cohort of consumers dominates population dynamics throughout its life time and the population composition varies over time between a dominance by juveniles and adults, respectively. The stage-structured model does not capture these dynamics because it incorporates a distributed time delay between the birth and maturation of an individual organism in contrast to the size-structured model, in which maturation is a discrete event in individual life history. We investigate model dynamics with both semi-chemostat and logistic resource growth.     

Bayesian model selection validates a biokinetic model for zirconium processing in humans

ABSTRACT: BACKGROUND: In radiation protection, biokinetic models for zirconium processing are of crucial importance in dose estimation and further risk analysis for humans exposed to this radioactive substance. They provide limiting values of detrimental effects and build the basis for applications in internal dosimetry, the prediction for radioactive zirconium retention in various organs as well as retrospective dosimetry. Multi-compartmental models are the tool of choice for simulating the processing of zirconium. Although easily interpretable, determining the exact compartment structure and interaction mechanisms is generally daunting. In the context of observing the dynamics of multiple compartments, Bayesian methods provide efficient tools for model inference and selection. RESULTS: We are the first to apply a Markov chain Monte Carlo approach to compute Bayes factors for the evaluation of two competing models for zirconium processing in the human body after ingestion. Based on in vivo measurements of human plasma and urine levels we were able to show that a recently published model is superior to the standard model of the International Commission on Radiological Protection. The Bayes factors were estimated by means of the numerically stable thermodynamic integration in combination with a recently developed copula based Metropolis-Hastings sampler. CONCLUSIONS: In contrast to the standard model the novel model predicts lower accretion of zirconium in bones. This results in lower levels of noxious doses for exposed individuals. Moreover, the Bayesian approach allows for retrospective dose assessment, including credible intervals for the initially ingested zirconium, in a significantly more reliable fashion than previously possible. All methods presented here are readily applicable to many modeling tasks in systems biology.