Deterministic Growth Model Research Articles

Stochastic modelling of Listeria monocytogenes single cell growth in cottage cheese with mesophilic lactic acid bacteria from aroma producing cultures

A stochastic model was developed for simultaneous growth of low numbers of Listeria monocytogenes and populations of lactic acid bacteria from the aroma producing cultures applied in cottage cheese. During more than two years, different batches of cottage cheese with aroma culture were analysed for pH, lactic acid concentration and initial concentration of lactic acid bacteria. These data and bootstrap sampling were used to represent product variability in the stochastic model. Lag time data were estimated from observed growth data (lactic acid bacteria) and from literature on L. monocytogenes single cells. These lag time data were expressed as relative lag times and included in growth models. A stochastic model was developed from an existing deterministic growth model including the effect of five environmental factors and inter-bacterial interaction [Østergaard, N.B, Eklöw, A and Dalgaard, P. 2014. Modelling the effect of lactic acid bacteria from starter- and aroma culture on growth of Listeria monocytogenes in cottage cheese. International Journal of Food Microbiology. 188, 15-25]. Growth of L. monocytogenes single cells, using lag time distributions corresponding to three different stress levels, was simulated. The simulated growth was subsequently compared to growth of low concentrations (0.4-1.0CFU/g) of L. monocytogenes in cottage cheese, exposed to similar stresses, and in general a good agreement was observed. In addition, growth simulations were performed using population relative lag time distributions for L. monocytogenes as reported in literature. Comparably good predictions were obtained as for the simulations performed using lag time data for individual cells of L. monocytogenes. Therefore, when lag time data for individual cells are not available, it was suggested that relative lag time distributions for L. monocytogenes can be used as a qualified default assumption when simulating growth of low concentrations of L. monocytogenes.

Optimizing continuous cover management of boreal forest when timber prices and tree growth are stochastic

Decisions on forest management are made under risk and uncertainty because the stand development cannot be predicted exactly and future timber prices are unknown. Deterministic calculations may lead to biased advice on optimal forest management. The study optimized continuous cover management of boreal forest in a situation where tree growth, regeneration, and timber prices include uncertainty. Both anticipatory and adaptive optimization approaches were used. The adaptive approach optimized the reservation price function instead of fixed cutting years. The future prices of different timber assortments were described by cross-correlated auto-regressive models. The high variation around ingrowth model was simulated using a model that describes the cross- and autocorrelations of the regeneration results of different species and years. Tree growth was predicted with individual tree models, the predictions of which were adjusted on the basis of a climate-induced growth trend, which was stochastic. Residuals of the deterministic diameter growth model were also simulated. They consisted of random tree factors and cross- and autocorrelated temporal terms. Of the analyzed factors, timber price caused most uncertainty in the calculation of the net present value of a certain management schedule. Ingrowth and climate trend were less significant sources of risk and uncertainty than tree growth. Stochastic anticipatory optimization led to more diverse post-cutting stand structures than obtained in deterministic optimization. Cutting interval was shorter when risk and uncertainty were included in the analyses. Adaptive optimization and management led to 6%–14% higher net present values than obtained in management that was based on anticipatory optimization. Increasing risk aversion of the forest landowner led to earlier cuttings in a mature stand. The effect of risk attitude on optimization results was small.

Centrality Fingerprints for Power Grid Network Growth Models

In our previous work, we have shown that many of the properties of the Florida power grid are reproduced by deterministic network growth models based on the minimization of energy dissipation Ediss. As there is no a priori best Ediss minimizing growth model, we here present a tool, called the “centrality fingerprint,” for probing the behavior of different growth models. The centrality fingerprints are comparisons of the current flow into/out of the network with the values of various centrality measures calculated at every step of the growth process. Finally, we discuss applications to the Maryland power grid.

Did the ever dead outnumber the living and when? A birth-and-death approach

This paper is an attempt to formalize analytically the question raised in “World Population Explained: Do Dead People Outnumber Living, Or Vice Versa?” Huffington Post, Howard (2012). We start developing simple deterministic Malthusian growth models of the problem (with birth and death rates either constant or time-dependent) before running into both linear birth and death Markov chain models and age-structured models.

Preventive maintenance scheduling of multi-component systems with interval costs

We introduce the preventive maintenance scheduling problem with interval costs (PMSPIC), which is to schedule preventive maintenance (PM) of the components of a system over a finite and discretized time horizon, given a common set-up cost and component costs dependent on the lengths of the maintenance intervals. We present a 0-1 integer linear programming (0-1 ILP) model for the PMSPIC; the model is identical to that presented by Joneja (1990) for the joint replenishment problem within inventory management. We study this model from a polyhedral and exact solutions’ point of view, as opposed to previously studied heuristics (e.g. Boctor, Laporte, & Renaud, 2004; Federgruen & Tzur, 1994; Levi, Roundy, & Shmoys, 2006; Joneja, 1990). We show that most of the integrality constraints can be relaxed and that the linear inequality constraints define facets of the convex hull of the feasible set. We further relate the PMSPIC to the opportunistic replacement problem, for which detailed polyhedral studies were performed by Almgren et al. (2012a). The PMSPIC can be used as a building block to model several types of maintenance planning problems possessing deterioration costs. By a careful modeling of these costs, a polyhedrally sound 0-1 ILP model is used to find optimal solutions to realistic-sized multi-component maintenance planning problems. The PMSPIC is thus easily extended by side-constraints or to multiple tiers, which is demonstrated through three applications; these are chosen to span several levels of unmodeled randomness requiring fundamentally different maintenance policies, which are all handled by variations of our basic model.Our first application considers rail grinding. Rail cracks increase with increasing intervals between grinding occasions, implying that more grinding passes must be performed—thus generating higher costs. We optimize the grinding schedule for a set of track sections presuming a deterministic model for crack growth; hence, no corrective maintenance (CM) will occur between the grinding occasions scheduled. The second application concerns two approaches for scheduling component replacements in aircraft engines. The first approach is bi-objective, simultaneously minimizing the cost for the scheduled PM and the probability of unexpected stops. In the second approach the sum of costs for PM and expected CM—without rescheduling—is minimized. When rescheduling is allowed, the 0-1 ILP model is used as a policy by re-optimizing the schedule at a component failure, which then constitutes an opportunity for PM. The policy manages the trade-off between costs for PM and unplanned CM and is evaluated in a simulation of the engine. The third application considers components’ replacement in wind mills in a wind farm, extending the PMSPIC to comprise multiple tiers with joint set-up costs. Due to the large number of components unexpected stops occur frequently, thus calling for a dynamic rescheduling, which is evaluated through a simulation of the system. In each of the three applications, the use of the 0-1 ILP model is compared with age or constant-interval policies; the maintenance costs are reduced by up to 16% as compared with the respective best simple policy. The results are strongest for the first two applications, possessing low levels of unmodeled randomness.

The diffusion model is not a deterministic growth model: comment on Jones and Dzhafarov (2014).

Jones and Dzhafarov (2014) claim that several current models of speeded decision making in cognitive tasks, including the diffusion model, can be viewed as special cases of other general models or model classes. The general models can be made to match any set of response time (RT) distribution and accuracy data exactly by a suitable choice of parameters and so are unfalsifiable. The implication of their claim is that models like the diffusion model are empirically testable only by artificially restricting them to exclude unfalsifiable instances of the general model. We show that Jones and Dzhafarov's argument depends on enlarging the class of "diffusion" models to include models in which there is little or no diffusion. The unfalsifiable models are deterministic or near-deterministic growth models, from which the effects of within-trial variability have been removed or in which they are constrained to be negligible. These models attribute most or all of the variability in RT and accuracy to across-trial variability in the rate of evidence growth, which is permitted to be distributed arbitrarily and to vary freely across experimental conditions. In contrast, in the standard diffusion model, within-trial variability in evidence is the primary determinant of variability in RT. Across-trial variability, which determines the relative speed of correct responses and errors, is theoretically and empirically constrained. Jones and Dzhafarov's attempt to include the diffusion model in a class of models that also includes deterministic growth models misrepresents and trivializes it and conveys a misleading picture of cognitive decision-making research. (PsycINFO Database Record (c) 2014 APA, all rights reserved).

Modeling for Growth and Forecasting of Pulse Production in Bangladesh

The present study was carried out to estimate growth pattern and examine the best ARIMA model to efficiently forecasting pigeon pea, chickpea and field pea pulse production in Bangladesh. It appeared that the time series data for pigeon pea, chickpea and field pea were 1 st order homogenous stationary. Two types of models namely Box-Jenkins type Autoregressive Integrated Moving Average (ARIMA) and deterministic type growth models, are examined to identify the best forecasting models for pigeon pea, chickpea and field pea pulse production in Bangladesh. The study revealed that the best models were ARIMA (1, 1 and 1), ARIMA (0, 1 and 0) and ARIMA (1, 1 and 3) for pigeon pea, chickpea and field pea pulse production, respectively. Among the deterministic type growth models, the cubic model is best for pigeon pea, chickpea and field pea pulse production. The analysis indicated that short-term forecasts were more efficient for ARIMA models compared to the deterministic models. The production uncertainty of pulse could be minimized if production were forecasted well and necessary steps were taken against losses. The findings of this study would be more useful for policy makers, researchers as well as producers in order to forecast future national pulse production more accurately in the short run.

Accounting for variability among individual pigs in deterministic growth models

Inclusion of variation in deterministic nutritional models for growth by repeating simulations using different sets of parameters has been performed in literature without or with only hypothetic consideration of the covariance structure among parameters. However, a description of the structure of links among parameters describing individuals is required to generate realistic sets of parameters. In this study, the mean and covariance structure of model parameters describing feed intake and growth were analyzed from 10 batches of crossbred gilts and barrows. Data were obtained from different crossbreeds, originating from Large White × Landrace sows and nine sire lines. Pigs were group-housed (12 pigs/pen) and performance testing was carried out from 70 days of age to ∼110 kg BW. Daily feed intake (DFI) was recorded using automatic feeding stations and BW was measured at least every 3 weeks. A growth model was used to characterize individual pigs based on the observed DFI and BW. In this model, a Gompertz function was used to describe protein deposition and the resulting BW gain. A gamma function (expressing DFI as multiples of maintenance) was used to express the relationship between DFI and BW. Each pig was characterized through a set of five parameters: BW70 (BW at 70 days of age), BGompertz (a precocity parameter) PDm (mean protein deposition rate) and DFI50 and DFI100 (DFI at 50 and 100 kg BW, respectively). The data set included profiles for 1288 pigs for which no eating or growth disorders were observed (e.g. because of disease). All parameters were affected by sex (except for BW70) and batch, but not by the crossbreed (except for PDm). An interaction between sex and crossbreed was observed for PDm (P < 0.01) and DFI100 (P = 0.05). Different covariance matrices were computed according to the batch, sex, crossbreed, or their combinations, and the similarity of matrices was evaluated using the Flury hierarchy. As covariance matrices were all different, the unit of covariance (subpopulation) corresponded to the combination of batch, sex and crossbreed. Two generic covariance matrices were compared afterwards, with (median matrix) or without (raw matrix) taking into account the size of subpopulations. The most accurate estimation of observed covariance was obtained with the median covariance matrix. The median covariance matrix can be used, in combination with average parameters obtained on-farm, to generate virtual populations of pigs that account for a realistic description of mean performances and their variability.

Novel approaches to improve the particle size distribution prediction of a classical emulsion polymerization model

A recent investigation on the homopolymerization of styrene (Hosseini et al., 2012a) showed that the classical population balance models are incapable of predicting the evolution of the breadth of the experimental particle size distributions correctly when a high resolution discretization method is used to suppress the numerical errors. Also by re-tuning the model parameters the model predictions did not fit the experimental results which points to a structural inadequacy of the conventional deterministic growth models in describing the experimentally observed broadening phenomenon. Two novel approaches are suggested in this work to improve the predictions. In the first approach, a possibly size dependent stochastic term is added to the deterministic growth kernel to account for the inhomogeneities of the growth process. The probability distribution of the resulting stochastic differential equation evolves over time based on the Fokker–Planck equation. The parameters of the (possibly size dependent) dispersion term of the Fokker–Planck equation are used as tuning parameters to fit the model to the experimental results. In the second approach, the growth kernel is extracted from the characteristics of the transient experimental particle size distributions. The extracted growth kernel is described in terms of the states of the system which affect the growth phenomenon. The advantages and disadvantages of both approaches are highlighted.

A statistical description of the structure formation of vinylidene fluoride-hexafluoropropene copolymers according to 19F NMR spectroscopy

The use of the deterministic model of chain growth for the copolymerization of vinylidene fluoride and hexafluoropropene instead of the classical Markov model makes it possible to correctly relate the data of the 19F NMR spectra of copolymers with different compositions to the mechanism of formation of macromolecules. The possible mechanism of copolymerization in the proposed model description involves the coordination of the radical center of the growing chain on the double bond of a monomer with subsequent incorporation of the monomer unit into the polymer chain. Although the proposed model disregards the orientation of monomer units along the chain, it properly describes the experimental data and provides an idea of the alternation of units in the copolymer at different copolymer compositions.

FEM Based Study of Concentration of Proliferating Cell in Brain Tumor

In this chapter we have developed a deterministic model of growth of abnormal cell concentration in a human subject at different positions. The diffusion-reaction Equation has been applied to satisfy growth dynamics .The whole tumor region is divided into layers which, with the growth of tumor form necrotic, quiescent and region of proliferating of tumor cells. Finite element method for one dimension has been employed for solving the Equations. Here we have taken into account the cellular motility along with proliferative growth ,which is particularly required in case of some of the brain tumors, where motility of gliomas cells differ widely in gray and white matter.

Testing the evolution process of prostate-specific antigen in early stage prostate cancer: what is the proper underlying model?

This paper empirically tests a model of stochastic evolutions of prostate-specific antigen (PSA), a trigger for intervention in an early stage prostate cancer surveillance program. It conducts hypothesis testing of the Geometric Browning Motion model based on its attributes of independent increments and linearity of the variance in the increment length versus a wide range of stochastic and deterministic alternatives. These alternatives include the currently accepted deterministic growth model. The paper reports strong empirical evidence in favour of the Geometric Browning Motion model. A model that best describes PSA evolution is a prerequisite to the establishment of decision-making criteria for abandoning active surveillance (i.e. a strategy that involves close monitoring) in early stage prostate cancer. Thus, establishing empirically the type of PSA process is a first step toward the identification of more accurate triggers for abandoning active surveillance and starting treatment while the chances of curing the disease are still high.

Integrating fire risk in stand management scheduling. An application to Maritime pine stands in Portugal

Maritime pine (Pinus pinaster Ait) is a very important forest species in Portugal. Nevertheless, both revenues and timber flows from the pine forests are substantially impacted by forest fires. We present a methodology for integrating fire risk in Maritime pine stand-level management optimization in Portugal. The objective is to determine the optimal prescription for a stand where fire risk is related to its structure and fuel load. The study optimizes the thinning treatments and the rotation length, as well as the fuel treatment schedule, i.e., reduction of understory cover during the rotation. Two components of wildfire risk—occurrence and damage—are considered. Fire damage was treated as an endogenous factor depending on the stand management schedule while fire occurrence was considered exogenous. A preliminary model that relates the expected loss to stand basal area, mean tree diameter and fire severity was used for this purpose. The article demonstrates how a deterministic stand-level growth and yield model may be combined with wildfire occurrence and damage models to optimize stand management. The Hooke-Jeeves direct search method was used to find the optimal prescription. In addition, population-based direct search methods (e.g. differential evolution and particle swarm optimization) were used for further testing purposes. Results are presented for Maritime pine stand management in Leiria National Forest in Portugal. They confirm that fuel treatments improve profitability and reduce the expected damage.

A stochastic approach to optimize Maritime pine (Pinus pinaster Ait.) stand management scheduling under fire risk. An application in Portugal

The paper discusses research aiming at the development of a management scheduling model for even-aged stands that may take into consideration fuel treatments to address the risk of wildfires. A Stochastic dynamic programming (SDP) approach is proposed to determine the policy (e.g. the fuel treatment and thinning schedules and the rotation age) that produces the maximum expected discounted net revenue. Fuel treatment activities encompass shrub cleanings. Emphasis was on combining a deterministic stand-level growth and yield model with wildfire occurrence and damage models to design a SDP network. SDP stages are defined by age and state variables include both the stand basal area and the number of years since the last fuel treatment. Fire occurrence and damage scenarios are addressed at each stage. Results from an application to Maritime pine (Pinus pinaster Ait.) stand management scheduling in Leiria National Forest, Portugal, are presented. Results suggest that the modeling strategy may help assess the impact of wildfire risk on the optimal stand management schedule. They confirm that the maximum expected discounted net revenues decreases. Further, albeit some timber may be salvaged after the wildfire, rotation age also decreases when the risk of fire is considered. Finally, they provide interesting insights about the role of thinning and fuel treatment policies in mitigating risk.

Fatigue crack growth model for assessing reliability of box-girders for cable-stayed bridge combining SHMS with strain data

Enhancement of the computer algorithm developed for the Runyang cable-stayed bridge (RYCB) in China is made by incorporating the fatigue crack growth analysis in contrast to the SN curve approach. Strain data obtained from the structural health monitoring system (SHMS) and finite element calculations are used. This provides the application of a deterministic method in addition to the probabilistic approach with the added feature of crack growth. The choice of selecting the two-parameter fatigue crack growth criterion was based on the definition of reliability index β such that the new results can be compared with those using the SN curves. A gradual drop of the reliability index β with time with an upper limit was obtained for the crack growth model in contrast to the linear time relation for the SN curve model that had no upper limit. This difference is significant and reveals the importance for selecting the fatigue failure criterion. Deterministic and probabilistic crack growth models are used to assess the differences. The results are based on the box-girder component strain history data of the Runyang Cable-stayed Bridge (RYCB) in China, stress history recorded by structural health monitoring system (SHMS) is analyzed using the monitored stress amplitude, mean stress and stress ratio. Finite element calculations are used to supplement data at locations not accessible for measurements. Additional improvement with reference to damage accumulation and the physical meaning of the reliability index will be studies in relation to the fatigue damage of box-girder of long cable-stayed bridges.

The Social Cost of Carbon on an Optimal Balanced Growth Path

This paper derives analytically the growth rate of the social cost of carbon (SSC) on an optimal balanced growth path. More specifically, the paper examines a deterministic Ramsey model of optimal economic growth with carbon emissions. In this model, restrictions on technology and preferences are imposed that guarantee optimal balanced growth, i.e., that guarantee an optimal path with constant and positive economic growth and a constant stock of carbon in the atmosphere. The paper exploits these restrictions to show that the growth rate of the SCC on the optimal balanced growth path is negative, provided the elasticity of marginal utility of consumption with respect to consumption is larger than or equal to one. There seems to be consensus in the literature that this latter requirement is fulfilled in reality. Paper submitted to the special issue The Social Cost of Carbon

The distinct effects of habitat fragmentation on population size

We sought to understand how the separation of habitats into spatially isolated fragments influences the abundance of organisms. Using a simple, deterministic model of population growth, we compared analytically exact solutions predicting abundance of consumers in two isolated patches with abundance of consumers in a single large patch where the carrying capacity of the large patch is the sum of the carrying capacities of the isolated ones. For the deterministic model, the effect of fragmentation was to slow the rate of population growth in the fragmented habitat relative to the intact one. We also analyzed a stochastic version of the model to examine the effect of fragmentation on population abundance when resources vary randomly in time. For the stochastic model, the effect of fragmentation was to reduce population abundance. We proved in closed-form, that for a non-equilibrium population exhibiting logistic population growth, fragmentation will reduce population size even when the total carrying capacity is not affected by fragmentation. We provide a theoretical basis for the prediction that habitat fragmentation amplifies the effect of habitat loss on the abundance of mobile organisms.

Bayesian Inference for Generalized Stochastic Population Growth Models with Application to Aphids

SummaryWe analyse the effects of various treatments on cotton aphids (Aphis gossypii). The standard analysis of count data on cotton aphids determines parameter values by assuming a deterministic growth model and combines these with the corresponding stochastic model to make predictions on population sizes, depending on treatment. Here, we use an integrated stochastic model to capture the intrinsic stochasticity, of both observed aphid counts and unobserved cumulative population size for all treatment combinations simultaneously. Unlike previous approaches, this allows us to explore explicitly and more accurately to assess treatment interactions. Markov chain Monte Carlo methods are used within a Bayesian framework to integrate over uncertainty that is associated with the unobserved cumulative population size and estimate parameters. We restrict attention to data on aphid counts in the Texas High Plains obtained for three different levels of irrigation water, nitrogen fertilizer and block, but we note that the methods that we develop can be applied to a wide range of problems in population ecology.

A climatologically based long-range fire growth model

A long-range fire growth model was developed to predict the potential size or probable extent of a wildfire if it was allowed to grow unimpeded for the course of the fire season. The model combines the probabilities of fire spread and of survival to produce a probable fire extent map. Probability of spread is determined from exponential distributions of potential rates of spread for various fuel types and eight compass directions, based on 30 years of fire weather data and elliptical fire growth. Probability of survival is determined from the probable evolution of the duff moisture code over the fire season predicted with Markov chains and a duff moisture code of extinction. A comparative study was conducted within Wood Buffalo National Park. The long-term model was compared with a distribution of fire perimeters predicted by repeated simulations using an hourly based, deterministic fire growth model. The study was conducted in stages, starting with a homogeneous fuel type and weather from one station. Later, fuels and weather were introduced to determine their effects on the model. The study showed a close agreement between the long-range model and the deterministic model, supporting the probabilistic approach used by the long-range model.

Stochastic approach for the calculation of anti-solvent addition policies in crystallization operations: An application to a bench-scale semi-batch crystallizer

This work discusses a stochastic approach for the calculation of robust anti-solvent addition policies for controlling the mean crystal size (MCS) and prediction of the crystal size distribution (CSD) in semi-batch crystallization operations. The proposed strategy is based on a structured-like population balance where uncertainties associated with the start-up condition and random fluctuations along the semi-batch operation can be taken into account in a very natural way. To include and quantify the effect of the uncertainties, a deterministic crystal growth model is embedded into a Fokker–Planck equation (FPE) resulting in a stochastic model for the MCS dynamics. This approach uses the generalized logistic equation (GLE) that has an adequate mathematical structure that suits the dynamic characteristic of the crystal growth. The numerical solution of the resulting FPE provides the most likely MCS evolution for a given anti-solvent flow-rate. The effects of the anti-solvent are incorporated into the parameters of the FPE and they are computed as linear piece-wise interpolating functions of the anti-solvent flow-rate. The strategy uses a PI-like regulator in closed-loop mode with the FPE to calculate the anti-solvent addition flow-rates for different set-point targets in the MCS. To validate the model and assess the merits of the proposed strategy, the crystallization of sodium chloride in water using ethanol as anti-solvent is performed in an experimental bench-scale semi-batch crystallizer. The implementation of the calculated anti-solvent policies resulted in a good control of the MCS despite modelling mismatch and uncertainties present during the crystallization operation.