A kinetic model for the activated sludge process which considers diffusion and reaction in the microbial floc

A mathematical model has been developed which describes substrate removal, oxygen utilization, and biomass production in an aggregated microbial suspension containing the substrate as a soluble biodegradable material and a uniform floc size. It is applicable to both steady-state and transient conditions. The model, consisting of three partial differential equations and two ordinary differential equations, takes into account the flow pattern in the reactor, intraparticle mass transport of oxygen and substrate, and biochemical reaction by individual cells embedded in the floc. Efficient numerical solution of the coupled nonlinear equations is obtained using an implicit finite difference approach for both the reactor and floc equations. A convergent solution is realized through block interation utilizing the tridiagonal algorithm. Results indicate that a unifying theory of activated sludge dynamics will have to consider coupling between floc chemical kinetics and changes in the bulk liquid characteristics. Floc size emerges as an important influence on system performance. It appears necessary to distinguish between a system response caused by diffuslonal resistances and nutrient limitations within the floc and a response caused by physiological adaption when analyzing the transient behavior of an activated sludge process. Future research should be devoted to rigorous laboratory determinations of model parameters along with extensions to include limitations of nutrients other than orgabnic carbon and oxygen.     

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.     

Modelling and estimation of infectious diseases in a population with heterogeneous dynamic immunity

The paper presents a model for the evolution of an infectious disease in a population with individual-specific immunity. The immune state of an individual varies with time according to its own dynamics, depending on whether the individual is infected or not. The model involves a system of size-structured (first-order) PDEs that capture both the dynamics of the immune states and the transition between compartments consisting of infected, susceptible, etc. individuals. Due to the unavailability of precise data about the immune states of the individuals, the main focus in the paper is on developing a technique for set-membership estimations of aggregated quantities of interest. The technique involves solving specific optimization problems for the underlying PDE system and is developed up to a numerical method. Results of numerical simulations are presented for a benchmark model of SIS-type, potentially applicable to diseases like influenza and to various sexually transmitted diseases.     

Physiologically structured models – from versatile technique to ecological theory

A ubiquitous feature of natural communities is the variation in size that can be observed between organisms, a variation that to a substantial degree is intraspecific. Size variation within species by necessity implies that ecological interactions vary both in intensity and type over the life cycle of an individual. Physiologically structured population models (PSPMs) constitute a modelling approach especially designed to analyse these size‐dependent interactions as they explicitly link individual level processes such as consumption and growth to population dynamics. We discuss two cases where PSPMs have been used to analyse the dynamics of size‐structured populations. In the first case, a model of a size‐structured consumer population feeding on a non‐structured prey was successful in predicting both qualitative (mechanisms) and quantitative (individual growth, survival, cycle amplitude) aspects of the population dynamics of a planktivorous fish population. We conclude that single generation cycles as a result of intercohort competition is a general outcome of size‐structured consumer–resource interactions. In the second case, involving both cohort competition and cannibalism, we show that PSPMs may predict double asymptotic growth trajectories with individuals ending up as giants. These growth trajectories, which have also been observed in field data, could not be predicted from individual level information, but are emergent properties of the population feedback on individual processes. In contrast to the size‐structured consumer–resource model, the dynamics in this case cannot be reduced to simpler lumped stage‐based models, but can only be analysed within the domain of PSPMs. Parameter values used in PSPMs adhere to the individual level and are derived independently from the system at focus, whereas model predictions involve both population level processes and individual level processes under conditions of population feedback. This leads to an increased ability to test model predictions but also to a larger set of variables that is predicted at both the individual and population level. The results turn out to be relatively robust to specific model assumptions and thus render a higher degree of generality than purely individual‐based models. At the same time, PSPMs offer a much higher degree of realism, precision and testing ability than lumped stage‐based or non‐structured models. The results of our analyses so far suggest that also in more complex species configurations only a limited set of mechanisms determines the dynamics of PSPMs. We therefore conclude that there is a high potential for developing an individual‐based, size‐dependent community theory using PSPMs.     

Coexistence of individual and social learners during range expansion

Individual learning and social learning are two primary abilities supporting cultural evolution. Conditions for their evolution have mostly been studied by investigating gene frequency dynamics, which essentially implies constant population size. Predictions from such “static” models may only be of partial relevance to the evolution of advanced individual learning in modern humans, because modern humans have experienced rapid population growth and range expansion during “out-of-Africa.” Here we model the spatial population dynamics of individual and social learners by a reaction–diffusion system. One feature of our model is the inclusion of the possibility that social learners may fail to find an exemplar to copy in regions where the population density is low. Due to this attenuation effect, the invasion speed of social learners is diminished, and various kinds of invasion dynamics are observed. Our primary findings are: (1) individual learners can persist indefinitely when invading environmentally homogeneous infinite space; (2) the occurrence of individual learners at the front may inhibit the spread of social learners. These results suggest that “out-of-Africa” may have driven the evolution of advanced individual learning ability in modern humans.     

Coupled land use and ecological models reveal emergence and feedbacks in socio‐ecological systems

Understanding the dynamics of socio‐ecological systems is crucial to the development of environmentally sustainable practices. Models of social or ecological sub‐systems have greatly enhanced such understanding, but at the risk of obscuring important feedbacks and emergent effects. Integrated modelling approaches have the potential to address this shortcoming by explicitly representing linked socio‐ecological dynamics. We developed a socio‐ecological system model by coupling an existing agent‐based model of land‐use dynamics and an individual‐based model of demography and dispersal. A hypothetical case‐study was established to simulate the interaction of crops and their pollinators in a changing agricultural landscape, initialised from a spatially random distribution of natural assets. The bi‐directional coupled model predicted larger changes in crop yield and pollinator populations than a unidirectional uncoupled version. The spatial properties of the system also differed, the coupled version revealing the emergence of spatial land‐use clusters that neither supported nor required pollinators. These findings suggest that important dynamics may be missed by uncoupled modelling approaches, but that these can be captured through the combination of currently‐available, compatible model frameworks. Such model integrations are required to further fundamental understanding of socio‐ecological dynamics and thus improve management of socio‐ecological systems.     

Discrete Model of Opinion Changes Using Knowledge and Emotions as Control Variables

We present a new model of opinion changes dependent on the agents emotional state and their information about the issue in question. Our goal is to construct a simple, yet nontrivial and flexible representation of individual attitude dynamics for agent based simulations, that could be used in a variety of social environments. The model is a discrete version of the cusp catastrophe model of opinion dynamics in which information is treated as the normal factor while emotional arousal (agitation level determining agent receptiveness and rationality) is treated as the splitting factor. Both variables determine the resulting agent opinion, which itself can be in favor of the studied position, against it, or neutral. Thanks to the flexibility of implementing communication between the agents, the model is potentially applicable in a wide range of situations. As an example of the model application, we study the dynamics of a set of agents communicating among themselves via messages. In the example, we chose the simplest, fully connected communication topology, to focus on the effects of the individual opinion dynamics, and to look for stable final distributions of agents with different emotions, information and opinions. Even for such simplified system, the model shows complex behavior, including phase transitions due to symmetry breaking by external propaganda.     

Dynamics of Mechanically Coupled Hair-Cell Bundles of the Inner Ear

The high sensitivity and effective frequency discrimination of sound detection performed by the auditory system rely on the dynamics of a system of hair cells. In the inner ear, these acoustic receptors are primarily attached to an overlying structure that provides mechanical coupling between the hair bundles. Although the dynamics of individual hair bundles has been extensively investigated, the influence of mechanical coupling on the motility of the system of bundles remains underdetermined. We developed a technique of mechanically coupling two active hair bundles, enabling us to probe the dynamics of the coupled system experimentally. We demonstrated that the coupling could enhance the coherence of hair bundles’ spontaneous oscillation, as well as their phase-locked response to sinusoidal stimuli, at the calcium concentration in the surrounding fluid near the physiological level. The empirical data were consistent with numerical results from a model of two coupled nonisochronous oscillators, each displaying a supercritical Hopf bifurcation. The model revealed that a weak coupling can poise the system of unstable oscillators closer to the bifurcation by a shift in the critical point. In addition, the dynamics of strongly coupled oscillators far from criticality suggested that individual hair bundles may be regarded as nonisochronous oscillators. An optimal degree of nonisochronicity was required for the observed tuning behavior in the coherence of autonomous motion of the coupled system.     

Quantification of the spatial aspect of chaotic dynamics in biological and chemical systems

The need to study spatio-temporal chaos in a spatially extended dynamical system which exhibits not only irregular, initial-value sensitive temporal behavior but also the formation of irregular spatial patterns, has increasingly been recognized in biological science. While the temporal aspect of chaotic dynamics is usually characterized by the dominant Lyapunov exponent, the spatial aspect can be quantified by the correlation length. In this paper, using the diffusion-reaction model of population dynamics and considering the conditions of the system stability with respect to small heterogeneous perturbations, we derive an analytical formula for an ‘intrinsic length’ which appears to be in a very good agreement with the value of the correlation length of the system. Using this formula and numerical simulations, we analyze the dependence of the correlation length on the system parameters. We show that our findings may lead to a new understanding of some well-known experimental and field data as well as affect the choice of an adequate model of chaotic dynamics in biological and chemical systems.     

Individual differences in metabolomics: individualised responses and between-metabolite relationships

Many metabolomics studies aim to find ‘biomarkers’: sets of molecules that are consistently elevated or decreased upon experimental manipulation. Biological effects, however, often manifest themselves along a continuum of individual differences between the biological replicates in the experiment. Such differences are overlooked or even diminished by methods in standard use for metabolomics, although they may contain a wealth of information on the experiment. Properly understanding individual differences is crucial for generating knowledge in fields like personalised medicine, evolution and ecology. We propose to use simultaneous component analysis with individual differences constraints (SCA-IND), a data analysis method from psychology that focuses on these differences. This method constructs axes along the natural biochemical differences between biological replicates, comparable to principal components. The model may shed light on changes in the individual differences between experimental groups, but also on whether these differences correspond to, e.g., responders and non-responders or to distinct chemotypes. Moreover, SCA-IND reveals the individuals that respond most to a manipulation and are best suited for further experimentation. The method is illustrated by the analysis of individual differences in the metabolic response of cabbage plants to herbivory. The model reveals individual differences in the response to shoot herbivory, where two ‘response chemotypes’ may be identified. In the response to root herbivory the model shows that individual plants differ strongly in response dynamics. Thereby SCA-IND provides a hitherto unavailable view on the chemical diversity of the induced plant response, that greatly increases understanding of the system.     

Robustness of circadian rhythms in the presence of molecular fluctuations: an investigation based on a mechanistic, statistical theory and a simulation algorithm

After a very brief introduction to a mechanistic and statistical theory of molecular fluctuations in chemical reactions developed by Joel Keizer, we explore the robustness of a circadian rhythm model by using the theory and the exact stochastic simulation (ESS). The comparative study shows that the theory reflects the effects of the dynamics of the model on the robustness more than ESS does. Even though the theory is a macroscopic one, the robustness of the model compares well with that computed from the ESS when the system size is larger than 50. The robustness increases nonlinearly with the system size and it reaches an asymptotic value at higher system sizes. As we can expect from the dynamics of the system, the robustness is minimum near the bifurcation point and as the most sensitive parameter increases away from the bifurcation point the robustness according to the theory as well as the ESS increases and then reaches to a steady value.     

Individual variability and population regulation: an individual-based model

To study the influence of individual variability on population dynamics an individual-based model of the dynamics of a single population consisting of different individuals is constructed. The model is based on differences in individual assimilation rates due to intraspecific competition and variability of initial weights. The model exhibits "imperfect regulation", i.e., the number of individuals in the population oscillates and sooner or later the population becomes extinct. When individual variability is included, the model produces longer population extinction times than without individual variability. The average extinction time is not however a monotonic function of the degree of individual variability.     

First Principles Modeling of Dissociative Adsorption at Crystal Surfaces: Hydrogen on Pt(111)

Multiscale simulation has the potential of becoming the new modeling paradigm in chemical sciences. An important class of multiscale models involves the mapping of a finer scale model into an approximate surface that is used by a coarser scale model. As a specific example of this class we present the case of the adsorption dynamics of diatomic molecules on single crystal catalyst surfaces. The prototype system studied is the dissociative adsorption of H₂ on Pt(111). The finer scale model consists of density functional theory (DFT) periodic slab calculations that provide a small dataset for training an atomistic scale potential energy surface. The coarser scale model uses a semi-classical molecular dynamics (MD) algorithm to obtain the sticking coefficient as a function of the incident energy. Comparison to experimental data and published simulation work is presented. Finally, major challenges in multiscale modeling of chemical reactivity in coupled DFT/MD simulations are discussed, specifically the need for a systematic method of assessing the accuracy of the coarse graining process.     

Oscillations in the lateral pressure of lipid monolayers induced by nonlinear chemical dynamics of the second messengers MARCKS and protein kinase C

The binding of the MARCKS peptide to the lipid monolayer containing PIP₂ increases the lateral pressure of the monolayer. The unbinding dynamics modulated by protein kinase C leads to oscillations in lateral pressure of lipid monolayers. These periodic dynamics can be attributed to changes in the crystalline lipid domain size. We have developed a mathematical model to explain these observations based on the changes in the physical structure of the monolayer by the translocation of MARCKS peptide. The model indicates that changes in lipid domain size drives these oscillations. The model is extended to an open system that sustains chemical oscillations.     

Combining field experiments and individual-based modeling to identify the dynamically relevant organizational scale in a field system

Community ecologists continually strive to build analytical models that realistically describe long‐term dynamics of the systems they study. A key step in this process is identifying which details are relevant for predicting dynamics. Currently, this remains a limiting step in development of analytical theory because experimental field ecology, which provides the key empirical insight, and theoretical ecology, which translates empirical knowledge into analytical theory, remain weakly linked. I illustrate how an individual‐based computational model of species interactions is a useful way to bridge the gulf between empirical research and theory development. I built a computational model that reproduced key natural history and biological detail of an old‐field interaction web composed of a predator species, a herbivore species and two plant groups that had been the subject of extensive previous field research. I examined, using simulation experiments, how individual behavior of herbivores in response to changing resource and predator abundance scaled to long‐term population‐level and community‐level dynamics. The simulation experiments revealed that the long‐term community dynamics could be highly predictable because of two counterintuitive reasons. First, seasonality was a strong forcing variable on the system that removed the possibility of serial dependence in population abundance over time. Second, because of seasonality, short‐term behavioral responses of herbivores played a much stronger role in shaping community structure than longer‐term processes such as density responses. So, simply knowing the short‐term responses of herbivores at the evolutionary ecological level was sufficient to forecast the long‐term outcome of experimental manipulations. This study shows that an individual‐based model, once it is calibrated to the real‐world field system, can provide key insight into the biological detail that analytical models should include to predict long‐term dynamics.     

Autocatalysis in cultural ecology: model ecosystems and the dynamics of biocultural evolution

Using a well-known mathematical model frequently applied in theoretical population dynamics, certain ecological mechanisms are investigated that are inherent in the organic evolution of cultural capacities in man. Culture is argued to involve ecological interactions exhibiting analogies to the interaction of chemical species in autocatalytic biomolecular reactions. In the model, biocultural evolution proceeds by more and more broadening ecological niches and, thus, releasing competitive selection pressure on the populations involved. This, in turn, facilitates the maintenance of polymorphism in these populations as well as the individual acquisition of organic traits through learning and cultural transmission. The result is that the genetic variance in phenotypic expressions decreases at an accelerated rate.     

Use of a simple mathematical model to predict the behavior of Escherichia coli overproducing beta-lactamase within continuous single- and two-stage reactor systems

A simple mathematical model is developed to help explain the complex population dynamics of an Escherichia coli host-plasmid expression/excretion system for beta-lactamase within single- and two-stage reactors. The model successfully integrates the individual regulatory (tac promoter induction), genetic (runaway plasmid replication), and population dynamics (culture instability) aspects of the system. The model predicts, and experiment confirms, that high-level beta-lactamase production and excretion cannot be easily maintained in single-stage reactors using the current plasmid construction. Stable target protein production and excretion is mathematically predicted, and experimentally confirmed, within two-stage reactors. The model is used to provide insight into engineering a more stable host-vector expression/excretion system for use in single-stage reactors. (c) 1993 John Wiley & Sons, Inc.     

A physical analogy of the genetic toggle switch.

We show that there is a physical analogy between a stochastic model of a genetic toggle switch system and a thermostated particle moving in a potential field, derived from the probability distribution of the toggle switch. This result suggests that one can actually simulate the dynamics of a more complex gene network by considering an ensemble of thermostated particles moving in a potential field, derived from the stationary distribution of the chemical stochastic model describing the gene network.