Modelling Bacterial Growth of Lactobacillus curvatus as a Function of Acidity and Temperature

Models that describe the effect of acidity, temperature, and the combined effect of these variables on the growth parameters of Lactobacillus curvatus are developed and validated. Growth parameters (lag time, specific growth rate, and maximum population density) were calculated from growth data at different temperature-acidity combinations. Experiments were set up to assess the quantitative effects of temperature and acidity on the growth parameters rather than for parameter estimation solely. The effect of acidity is monitored at several constant temperature values. Models are set up and fitted to the data. The same procedure is used at constant acidity values to model the effect of temperature. For lag time, specific growth rate, and maximum population density, the effect of temperature could be multiplied with the effect of acidity to obtain combinatory models that describe the effect of both controlling factors on the growth parameters. Lag time measurements showed large deviations, and therefore the lag time models developed can only be used to estimate the order of magnitude of lag time.     

Interspecific competition between insect herbivores: asymmetric competition between cinnabar moth and the ragwort seed-head fly

ABSTRACT.

• 1 Removal field experiments and observational studies have been undertaken to determine whether feeding by cinnabar moth Tyria jacobaeae L. on the flower heads of ragwort Senecio jacobaea L. affects the abundance of the fly Pegohylemyia seneciella (Meade) that feeds in the flower heads as a larva.
• 2 Correlations between the population density of cinnabar moth and the population density of the fly were suggestive of habitat separation, but provided little evidence of exploitation competition.
• 3 Removal of cinnabar moth by hand from replicated plots over two years shows that, in years when ragwort flower production is consumed by cinnabar moth caterpillars, the fly may show no recruitment at all.
• 4 Fly populations persist in refugia, exploiting ragwort plants that grow in areas where there are no cinnabar moth.
• 5 Recruitment of ragwort is not seed limited, so the reduction in seed production caused by P. seneciella (maximum about 30%) has no impact on ragwort abundance, or on the abundance of cinnabar moth.
• 6 We conclude that there is strong interspecific competition between these two species, and that the competition is highly asymmetric. The cinnabar moth had a substantial effect on the recruitment of the fly in 1986, but the fly has no measurable impact on the recruitment of the moth. In six years out of seven in our long-term study, cinnabar moth reduced flower production to levels comparable to those measured in 1986, and we infer that strong competition with the fly was likely in six years out of seven.
• 7 One reason why there are so few published examples of asymmetric interspecific competition may be simply that the experiments are thought too obvious to be worth doing. We argue that this is not a good reason for eschewing manipulative field experiments, and that few processes in ecology are at all obvious when investigated in detail.

     

Modelling the Dynamical Components of the Sugar Beet Crop

Many models have been constructed to describe the growth ofthe sugar beet crop up to harvesting. In general, these modelshave a complex physiological basis, requiring a large numberof parameters yet relying on empirical functions with no mechanisticbasis to partition assimilates within the crop. An importantfactor in considering the growth of the crop, both from an economicand environmental point of view, is the response of the cropto varying amounts of available nitrogen in the soil. In thispaper, a model is described for crop growth using soil nitrogencontent and solar radiation as driving functions. The parsimoniousapproach to construction resulted in a 14 parameter model, sevenof which are associated with the driving variables. This issubstantially fewer than for other crop models. The model containsa new dynamical way of describing partitioning of assimilatesbetween shoot, storage root and fibrous roots. The partitioningmodel is derived from observations on the effect of soil nitrogenon crop growth. Interception of light is determined by foliagecover, which makes the model suitable for use with data collectedfrom satellite imaging. The model fits well to three independentdata sets with estimated parameters lying within biologicallyreasonable bounds. The model is used to test the sensitivityof yield to changes in soil nitrogen. Modelling; partitioning; parameter estimation; sugar beet; Beta vulgarisL.; nitrogen; crop growth dynamics     

Dynamics of a plant-herbivore model

We formulate a simple host-parasite type model to study the interaction of certain plants and herbivores. Our two-dimensional discrete-time model utilizes leaf and herbivore biomass as state variables. The parameter space consists of the growth rate of the host population and a parameter describing the damage inflicted by herbivores. We present insightful bifurcation diagrams in that parameter space. Bistability and a crisis of a strange attractor suggest two control strategies: reducing the population of the herbivore under some threshold or increasing the growth rate of the plant leaves.     

Statistical analysis of nonlinear parameter estimation for Monod biodegradation kinetics using bivariate data

A nonlinear regression technique for estimating the Monod parameters describing biodegradation kinetics is presented and analyzed. Two model data sets were taken from a study of aerobic biodegradation of the polycyclic aromatic hydrocarbons (PAHs), naphthalene and 2-methylnaphthalene, as the growth-limiting substrates, where substrate and biomass concentrations were measured with time. For each PAH, the parameters estimated were: q(max), the maximum substrate utilization rate per unit biomass; K(S), the half-saturation coefficient; and Y, the stoichiometric yield coefficient. Estimating parameters when measurements have been made for two variables with different error structures requires a technique more rigorous than least squares regression. An optimization function is derived from the maximumlikelihood equation assuming an unknown, nondiagonal covariance matrix for the measured variables. Because the derivation is based on an assumption of normally distributed errors in the observations, the error structures of the regression variables were examined. Through residual analysis, the errors in the substrate concentration data were found to be distributed log-normally, demonstrating a need for log transformation of this variable. The covariance between ln C and X was found to be small but significantly nonzero at the 67% confidence level for NPH and at the 94% confidence level for 2MN. The nonlinear parameter estimation yielded unique values for q(max), K(S), and Y for naphthalene. Thus, despite the low concentrations of this sparingly soluble compound, the data contained sufficient information for parameter estimation. For 2-methylnaphthalene, the values of q(max) and K(S) could not be estimated uniquely; however, q(max)/K(S) was estimated. To assess the value of including the relatively imprecise biomass concentration data, the results from the bivariate method were compared with a univariate method using only the substrate concentration data. The results demonstrated that the bivariate data yielded a better confidence in the estimates and provided additional information about the model fit and model adequacy. The combination of the value of the bivariate data set and their nonzero covariance justifies the need for maximum likelihood estimation over the simpler nonlinear least squares regression.     

Parametric analysis of the errors associated with the Michaelis-Menten equation

Based on the well-known mechanism describing Michaelis-Menten kinetics, three rate expressions may be developed: the exact solution (Model 1), a rate equation resulting from the pseudo-steady-state assumption (Model 2), and Model 2 with the additional assumption that the amount of free substrate is approximately equal to the total amount of substrate (Model 3). Although Model 1 is the most precise, it must be integrated numerically and it requires three experimentally determined parameters. Models 2 and 3, however, are simpler and require only two parameters. Using dimensionless forms of the three models, we have evaluated the errors in the two simplified models relative to the exact solution using a wide range of parameter values. The choice of model for reactor design depends on the initial substrate to enzyme ratio (alpha(0)), and on the ratio of the Michaelis-Menten constant to the enzyme concentration (sigma). Based on a 2% model error criteria, when alpha(0) > 15 or sigma >/= 100, Model 3 is adequate; if 5 < alpha(0) < 15, or if sigma >/= 10, then Model 2 may be used; and if alpha(0) < 5 and sigma < 10, then the exact solution (Model 1) is required.     

Fitting parameters of stochastic birth-death models to metapopulation data

Populations that are structured into small local patches are a common feature of ecological and epidemiological systems. Models describing this structure are often referred to as metapopulation models in ecology or household models in epidemiology. Small local populations are subject to demographic stochasticity. Theoretical studies of household disease models without resistant stages (SIS models) have shown that local stochasticity can be ignored for between patch disease transmission if the number of connected patches is large. In that case the distribution of the number of infected individuals per household reaches a stationary distribution described by a birth-death process with a constant immigration term. Here we show how this result, in conjunction with the balancing condition for birth-death processes, provides a framework to estimate demographic parameters from a frequency distribution of local population sizes. The parameter estimation framework is applicable to estimate parameters of disease transmission models as well as metapopulation models.     

Climate and landscape correlates of African lion (Panthera leo) demography

Although many studies have documented aspects of lion ecology, they have generally focused on single sites, leaving broader-scaled factors unanalysed. We assessed range-wide effects of eight biotic and 26 abiotic variables on lion distribution and ecology, based on data compiled from published sources on lion population ecology in 27 protected areas in Africa. Lion pride size and composition were independent of lion density; lion density and home range size were inversely related; and lion density was positively related to rainfall, soil nutrients and annual mean temperature, with some interactive effects between rainfall and soil nutrients. Lion demography was associated most strongly with rainfall, temperature and landscape features. Herbivore biomass and lion density were correlated in univariate regression analyses. However, because herbivore biomass was also related to rainfall and temperature, hierarchical partitioning (HP) allowed us to evaluate independent effects of each variable on lion demography revealing that herbivore biomass had negligible independent contributions. HP indicated that climatic parameters explained 62% of overall variance in demographic parameters, whereas landscape features explained only 32%; climatic parameters were fairly balanced between effects of temperature (34%) and rainfall (28%). Prey (herbivore) biomass is important for lion survival, but its effects appear secondary to environmental factors.     

Growth of newly established alien populations: comparison of North American gypsy moth colonies with invasion theory

A common characteristic observed in many biological invasions is the existence of a lag between the time of arrival by the alien population and the time when established populations are noticed. Considerable advances have been made in modeling the expansion of invading species, and there is often remarkable congruence between the behavior of these models with spread of actual populations. While these models have been used to characterize expansion of very newly founded colonies, there have been few attempts to compare the behavior predicted from theory with spread in actual newly founded populations, largely due to the difficulty of sampling sparse populations. Models predict that time lags in the radial expansion of newly invaded populations may be due to time requirements for the population to grow from founding to detectable levels. Models also indicate that these time lags can be predicted based upon population parameters such as the intrinsic rate of population growth and diffusion coefficient. In this paper, we compared the behavior of these models with historical data on gypsy moth, Lymantria dispar, establishment and spread to show similarities between model predictions and observed population spread, both of which exhibited temporal lags of expansion. However, actual populations exhibited certain behaviors that were not predicted, and this could be due, in part, to the existence of Allee effects and stochasticity. Further work that incorporates these effects is needed to more fully understand the growth of incipient colonies of invading species. Ultimately, this information can be of critical importance in the selection of effective strategies for their detection and eradication.     

The kinetics of cometabolism

Experimental observations indicate that the rates of cometabolic transformation are linked to the consumption of growth substrate during growth and to the consumption of cell mass and/or energy substrate in the absence of growth substrate. Three previously proposed models (models 1 through 3) describing the kinetics of cometabolism by resting cells are compared, and the interrelationships and underlying assumptions for these models are explored. Models 1 to 3 are shown to converge at high concentrations of the nongrowth substrate. An expression describing nongrowth substrate transformation in the presence of growth substrate is proposed, and this expression is integrated with an expression for cell growth to give a single unstructured model (model 4) that encompasses models 1 to 3 and describes cometabolism by both resting and growing cells. Model 4 couples transformation of nongrowth substrate to consumption of growth substrate and biomass, and predicts that cometabolism will result, and decreased specific growth rates for a cometabolizing population. Competitive inhibition can also be incorporated in the model. Experimental aspects of model calibration and verification are discussed. The need for models that distinguish between the exhaustion of cell activity and cell death is emphasized. (c) 1993 Wiley & Sons, Inc.     

Experimentally Verified Parameter Sets for Modelling Heterogeneous Neocortical Pyramidal-Cell Populations

Models of neocortical networks are increasingly including the diversity of excitatory and inhibitory neuronal classes. Significant variability in cellular properties are also seen within a nominal neuronal class and this heterogeneity can be expected to influence the population response and information processing in networks. Recent studies have examined the population and network effects of variability in a particular neuronal parameter with some plausibly chosen distribution. However, the empirical variability and covariance seen across multiple parameters are rarely included, partly due to the lack of data on parameter correlations in forms convenient for model construction. To addess this we quantify the heterogeneity within and between the neocortical pyramidal-cell classes in layers 2/3, 4, and the slender-tufted and thick-tufted pyramidal cells of layer 5 using a combination of intracellular recordings, single-neuron modelling and statistical analyses. From the response to both square-pulse and naturalistic fluctuating stimuli, we examined the class-dependent variance and covariance of electrophysiological parameters and identify the role of the h current in generating parameter correlations. A byproduct of the dynamic I-V method we employed is the straightforward extraction of reduced neuron models from experiment. Empirically these models took the refractory exponential integrate-and-fire form and provide an accurate fit to the perisomatic voltage responses of the diverse pyramidal-cell populations when the class-dependent statistics of the model parameters were respected. By quantifying the parameter statistics we obtained an algorithm which generates populations of model neurons, for each of the four pyramidal-cell classes, that adhere to experimentally observed marginal distributions and parameter correlations. As well as providing this tool, which we hope will be of use for exploring the effects of heterogeneity in neocortical networks, we also provide the code for the dynamic I-V method and make the full electrophysiological data set available.     

A new approach of fitting biomass dynamics models to data

A non-traditional approach of fitting dynamic resource biomass models to data is developed in this paper. A variational adjoint technique is used for dynamic parameter estimation. In the variational formulation, a cost function measuring the distance between the model solution and the observations is minimized. The data assimilation method provides a novel and computationally efficient procedure for combining all available information, i.e., the data and the model in the analysis of a resource system. This technique will be used to analyze data for the North-east Arctic cod stock. Two alternative population growth models: the logistic and the Gompertz model are used for estimating parameters of simple bioeconomic models by the method of constrained least squares. Estimates of the parameters of the models dynamics are reasonable and can be accepted. The main inference from the work is that the average fishing mortality is found to be significantly above the maximum sustainable yield value.     

Approximate reduction of linear population models governed by stochastic differential equations: application to multiregional models

In this work we develop approximate aggregation techniques in the context of slow-fast linear population models governed by stochastic differential equations and apply the results to the treatment of populations with spatial heterogeneity. Approximate aggregation techniques allow one to transform a complex system involving many coupled variables and in which there are processes with different time scales, by a simpler reduced model with a fewer number of ‘global’ variables, in such a way that the dynamics of the former can be approximated by that of the latter. In our model we contemplate a linear fast deterministic process together with a linear slow process in which the parameters are affected by additive noise, and give conditions for the solutions corresponding to positive initial conditions to remain positive for all times. By letting the fast process reach equilibrium we build a reduced system with a lesser number of variables, and provide results relating the asymptotic behaviour of the first- and second-order moments of the population vector for the original and the reduced system. The general technique is illustrated by analysing a multiregional stochastic system in which dispersal is deterministic and the rate growth of the populations in each patch is affected by additive noise.     

Estimating the susceptibility of tree species to attack by the gypsy moth, Lymantria dispar

ABSTRACT.

• 1 Numbers of gypsy moth larvae feeding on each of 922 randomly sampled trees in a Quercus—Acer—Fraxinus forest in southwestern Quebec, Canada were counted in 1979 and in 1980 to quantify the larval feeding preferences as observed in the field for eighteen deciduous and one coniferous tree species at the northern range limit of the gypsy moth.
• 2 Both the diameter at breast height (dbh) and the estimated foliage biomass of the sampled trees were used to calculate the relative proportions of foliage represented by each of the nineteen tree species in the forest canopy. With these data on availability and utilization of the tree species by the gypsy moth larvae an Ivlev-type electivity index was used to quantify the larval feeding preferences. These preferences observed in the field define the susceptibility of a tree species to attack by the gypsy moth.
• 3 The feeding preferences calculated using estimated foliage biomass were comparable to the simpler calculation based on dbh (Spearman's rho = 0.79; P= 0.0001). The dbh-based feeding preferences remained almost unchanged in 1979 and 1980 (Spearman's rho = 0.83; P= 0.0001).
• 4 The composite 1979—80, dbh-based feeding preferences show Quercus rubra, Populus grandidentata, Ostrya virginiana, Amelanchier spp. and Acer saccharum were preferentially attacked by gypsy moth. Prunus serotina, Betula lutea, Acer rubrum, A. pensylvanicum, Fraxinus americana, Ulmus rubra, P. pensylvanicum and B. papyrifera were avoided. All nineteen tree species were, however, utilized to at least some degree by gypsy moth larvae.
• 5 These results quantitatively affirm and clarify earlier reports of gypsy moth feeding preferences in North America and Eurasia. The advantages and limitations of using an electivity index to estimate the susceptibility of different tree species to attack by folivores like the gypsy moth are discussed.

     

Modelling Asymmetrical Growth Curves that Rise and then Fall: Applications to Foliage Dynamics of Sugar Beet (Beta vulgaris L.)

This paper discusses the derivation and fitting of three empiricalmodels with turning points for describing the growth of plantcomponents, such as shoot weight, leaf area and root length,that typically rise and then fall during the course of the growingseason. The models (Models I, II and III) have analytical solutionsand may be viewed as extensions of the Gompertz, Richards andChanter growth equations. They differ by having an additionalparameter which, following a sigmoidal rise of the dependentvariable, determines subsequent net rate of decline. The modelswere fitted to sequential measures of foliage cover of sugarbeet crops grown in the UK during 1980–1991. It was importantthat this could be done with relative ease using standard statisticalprocedures. Partial linear transformations of two of the models,with one non-linear parameter remaining, are described; thesewere useful for estimating initial values for the parameters.All three models described the data well, although the fittingof Model II invariably failed to converge. For Models I andIII common and separate parameters, amongst years, were estimatedrelating to date of emergence, initial relative growth rate,maximum cover attained and rate of late season decline of foliagecover. The reduction in the residual mean square on fittingseparate, rather than common, parameters was usually significant.The models accommodate several biological processes that yieldsimilar shapes. This is demonstrated for Model I, in relationto its formulation and to effects of small perturbations inthe values of the parameters on the shape of the curves. ModelI, the simplest of the three models tested, has good fittingproperties, and in practice was best suited to describing foliagecover dynamics of sugar beet. Beta vulgaris L.; sugar beet; foliage cover; senescence; models; parameter estimation; growth functions     

Bayesian state-space modeling of metapopulation dynamics in the Glanville fritillary butterfly

The complexity of mathematical models of ecological dynamics varies greatly, and it is often difficult to judge what would be the optimal level of complexity in a particular case. Here we compare the parameter estimates, model fits, and predictive abilities of two models of metapopulation dynamics: a detailed individual-based model (IBM) and a population-based stochastic patch occupancy model (SPOM) derived from the IBM. The two models were fitted to a 17-year time series of data for the Glanville fritillary butterfly (Melitaea cinxia) inhabiting a network of 72 small meadows. The data consisted of biannual counts of larval groups (IBM) and the annual presence or absence of local populations (SPOM). The models were fitted using a Bayesian state-space approach with a hierarchical random effect structure to account for observational, demographic, and environmental stochasticities. The detection probability of larval groups (IBM) and the probability of false zeros of local populations (SPOM) in the observation models were simultaneously estimated from the time-series data and independent control data. Prior distributions for dispersal parameters were obtained from a separate analysis of mark–recapture data. Both models fitted the data about equally, but the results were more precise for the IBM than for the SPOM. The two models yielded similar estimates for a random effect parameter describing habitat quality in each patch, which were correlated with independent empirical measures of habitat quality. The modeling results showed that variation in habitat quality influenced patch occupancy more through the effects on movement behavior at patch edges than on carrying capacity, whereas the latter influenced the mean population size in occupied patches. The IBM and the SPOM explained 63% and 45%, respectively, of the observed variation in the fraction of occupied habitat area among 75 independent patch networks not used in parameter estimation. We conclude that, while carefully constructed, detailed models can have better predictive ability than simple models, this advantage comes with the cost of greatly increased data requirements and computational challenges. Our results illustrate how complex models can be helpful in facilitating the construction of effective simpler models.