Super-individuals a simple solution for modelling large populations on an individual basis

Modelling populations on an individual-by-individual basis has proven to be a fruitful approach. Many complex patterns that are observed on the population level have been shown to arise from simple interactions between individuals. However, a major problem with these models is that the typically large number of individuals needed requires impractically large computation times. The common solution, reduction of the number of individuals in the model, can lead to loss of variation, irregular dynamics, and large sensitivity to the value of random generator seeds. As a solution to these problems, we propose to add an extra variable feature to each model individual, namely the number of real individuals it actually represents. This approach allows zooming from a real individual-by-individual model to a cohort representation or ultimately an all-animals-are-equal view without changing the model formulation. Therefore, the super-individual concept offers easy possibilities to check whether the observed behaviour is an artifact of following a limited number of individuals or of lumping individuals, and also to verify whether individual variability is indeed an essential ingredient for the observed behaviour. In addition the approach offers arbitrarily large computational advantages. As an example the super-individual approach is applied to a generic model of the dynamics of a size-distributed consumer cohort as well as to an elaborate applied simulation model of the recruitment of striped bass.     

A model for intraguild predation dynamics between immature stages

Dynamical models usually assume that predation occurs between mature stages and/or between mature and immature stages. In this work a stage-structured discrete time model is developed for a system where intraguild predation takes place only in the course of immature stages of predator and its prey. Therefore, the proposed mathematical setup demands functional relations linking predation in immature life stages with survival and fecundity in mature stages. The behavior of the model is examined in order to investigate the interplay among predator attack rate, its satiation of prey consumption and the success of intraguild predator invasion.     

Generation time and the maximum growth rate for populations with age-specific fecundities and unknown juvenile survival

In age-classified population models where all parameters are known, the generation time and growth rate are calculated in a straightforward manner. For many populations, some parameters, such as juvenile survival, are difficult to estimate accurately. In a simplified population model where fecundity and survival are constant from the onset of breeding, it is known that generation time may be calculated given only adult survival, age at first reproduction, and the population growth rate. However, the assumption of constant fecundity from the onset of breeding does not hold for many populations. An extended population model allows calculation of generation time with the additional knowledge of the ratio of age-specific fecundities compared to a maximum fecundity rate. When these relative fecundities are unknown, an ad hoc adjustment to the simplified model performs well.When the study population is in an ideal environment, the optimal generation time and maximum growth rate are linked, and both may be approximated knowing only adult survival, age at first reproduction, and the relative fecundities. The maximum growth rate has important conservation implications, and calculating it correctly is therefore important. Improper use of the simplified population model to calculate the maximum growth rate, combined with a simple decision rule, leads to an average overharvest of 36%, and >60% for three of six bird species studied, compared to the full population model. By comparison, using the approximation from the extended or adjusted models results in average overharvests of only 8% (extended model) and 5% (adjusted model), and <50% for all six species (either model).     

A methodological approach to develop “contaminant migration–population effects” models

The main aim of the present work is to discuss the methodological approaches that underpin the “contaminant migration–population effects” models for the evaluation of the detriment to populations of moving organisms in environmental systems with spatial and time dependent pollution levels. A technique to couple the equations controlling the population dynamics and the pollutant dispersion is described and discussed. The domain of application and the limitations of the methodology are analysed and illustrated by some examples. Possible alternative approaches are briefly presented.     

Modeling changes in the coastal ecosystem of the Pearl River Estuary from 1981 to 1998

The coastal ecosystem of the Pearl River Estuary (PRE) has been overfished and received a high level of combined pollution since the 1980s. Ecopath with Ecosim was used to construct two ecosystem models (for 1981 and 1998) to characterize the food web structure and functioning of the ecosystem. Pedigree work and simple sensitivity analysis were carried out to evaluate the quality of data and the uncertainty of the models. The two models seem reliable with regards to input data of good quality. Comparing the variations of outputs of these two models aimed to facilitate assessment of changes of the ecosystem during the past two decades.The trophic structure of the ecosystem has changed with an increase in the biomass proportion of lower trophic level (TL) organisms and a decrease in top predator biomass proportion. All the indices of ecosystem maturity examined show that the system was in a more mature condition in 1981 than in 1998, although the system has been in a condition of stress due to anthropogenic disturbances, such as environmental pollution and habitat destruction since 1981. The ecosystem was aggregated into six and seven integral TLs in 1981 and 1998, respectively, using the trophic aggregation routine of Ecopath. Most of the total system biomass and catch took place at TL II and III in both years. But the distribution of the total system biomass and catch at different TLs changed with decreasing proportions in higher TLs in 1998. The mean transfer efficiency was 9.1% and 10.2% in 1981 and 1998, respectively.Comparative network analysis allowed quantification of the importance of direct and indirect trophic interactions among functional groups. Moreover, a method derived from the mixed trophic impact (MTI) analysis allowed estimating importance of groups in terms of “keystoneness” and identifying the keystone species in the two models over the past two decades. The results indicate that there were no clear keystone species in 1998 but two keystone species at medium trophic levels were identified in 1981. Moreover, organisms located at low trophic levels such as phytoplankton, zooplankton and benthic invertebrates were identified to have relatively high keystoneness in the ecosystem.     

Modeling individual and population dynamics in a consumer–resource system: Behavior under food limitation and crowding and the effect on population cycling in Daphnia

In population modeling, a considerable level of complexity is often required to provide trustworthy results, comparable with field observations. By assuring sufficient detail at the individual level while preserving the potential to explore the consequences at higher levels, individual-based modeling may thus provide a useful tool to investigate dynamics at different levels of organization. Still, population dynamics resulting from such models are often at odds with observations from the field. This may be partly caused by a lack of focus on the individual dynamics under conditions of food stress and starvation. I developed a physiologically structured, individual-based simulation model to investigate life history of Daphnia and its effect on population dynamics in response to the productivity of the system. In verifying model behavior with available literature data on life history and physiology, I paid special attention to the dynamics of food intake and the verification of individual level results under conditions of food limitation and starvation. I show that the maximum filtering rates under low food levels used in the current model are much closer to measured filtering rates than the ones used in other models. Being consistent with results on physiology and life history from experiments at a wide range of food availability (including starvation), the model generates low amplitude or high amplitude population density cycles depending on the productivity of the system, as observed in field and experimental populations of Daphnia and with the minimum population densities being one to two orders of magnitude lower in the high amplitude than in the low amplitude cycles. To generate results which are not only qualitatively but also quantitatively comparable to experimental and field observations, however, a crowding effect on the filtering response has to be incorporated in the model.     

On the use of correlated beta random variables with animal population modelling

We give reasons why demographic parameters such as survival and reproduction rates are often modelled well in stochastic population simulation using beta distributions. In practice, it is frequently expected that these parameters will be correlated, for example with survival rates for all age classes tending to be high or low in the same year. We therefore discuss a method for producing correlated beta random variables by transforming correlated normal random variables, and show how it can be applied in practice by means of a simple example. We also note how the same approach can be used to produce correlated uniform, triangular, and exponential random variables.     

An inverse model for the analysis of the Venice lagoon food web

A steady-state model of the Venice lagoon food web was constructed, based on a comprehensive set of data, which were collected in the years 2001-2005. Energy flows were estimated by means of an inverse methodology of constrained optimization based on the Minimum Norm criterion, i.e. on the minimization of both the sum of squares of the residuals and of the sum of squares of energy flows. The solution was constrained by a set inequalities, which were derived from general eco-physiological knowledge and site specific data on energy flows. The trophic network was represented by thirty-two nodes, including single-species compartments for the species of high economical or ecological relevance. Mass balance equations were weighted, in order to obtain meaningful results in presence of large differences, up to 5 orders of magnitude, among biomasses. A perturbation technique was applied, with the purpose of reducing the risk of finding solutions heavily affected by the set of constraints and of obtaining a more robust representation of the energy flows. The main patterns of energy flow are consistent with those obtained in previous attempts at modelling the Venice lagoon food web. Micro- and macro-phytobenthos account for the largest fraction of the primary production. Energy is then transferred towards higher trophic levels by means of two main pathways: the recycling of dead biomass through the detritus compartment and the direct consumption by grazers. The first pathway is the most important and accounts for approximately two/thirds of the energy transferred to the second trophic level.     

A model of nematode dynamics in the Westerschelde Estuary

We developed a time dynamic model to investigate the temporal dynamics of nematode community in the brackish zone of the Westerschelde Estuary. The biomass of four nematode feeding groups observed from March 1991 to February 1992 is used to calibrate the model. Using environmental data as the input, the model predicts the temporal modification and interrelation of four nematode feeding groups. Nematodes achieve a dominant position in the community because of their lower loss rate (in respiration, excretion and natural death). Predators which are deposit-feeding macrobenthos control the variations of dominant nematodes, such as omnivores and non-selective deposit feeders. Food availability causes modification only for rare nematodes such as epigrowth feeders and selective deposit feeders. Temperature is a factor affecting both predation death and a loss including respiration, excretion and natural death. Overall, the modification of nematode community by food availability is much lower than by predator. The macrobenthos in the Westerschelde Estuary decrease from upstream to the estuarine mouth. The stability and standing stock of nematode population follow the opposite gradient of their predators. They increase from upstream to the estuarine mouth.     

Lyapunov stability for an age-structured population model

This is a short notice on the McKendrick equation that I actually learned from Yu.M. Svirezhev in the 1990s. This McKendrick equation modelling the evolution in time of an age-structured population has received attention recently from mathematicians. The initial and boundary conditions for the McKendrick equation imposed by the population model are not the standard side conditions one sees in PDE theory for an evolution equation. In the simplest case, the problem reduces to a well-known model in demography, the Lotka integral equation.     

Characterizing cumulative impacts using a brook trout population dynamics model

An individuals-based modelling framework is used to characterize the nature of exploitation and toxaphene stressors acting simultaneously on a population of brook trout (Salvelinus fontinalis) in terms of age 0 + and adult abundance, survivorship and population size-structure. A no-stressor control case was estimated against which exploitation-only, toxaphene-only and cumulative exploitation and toxaphene stressor cases were compared to determine the extent and significance of impacts. Single stressor case results were used to predict cumulative impacts by assuming additivity and predictions compared to modelled cumulative stressor results. Comparisons indicated the inadequacy of assuming additivity in predicting cumulative impacts. A factorial experimental design was used to estimate the size and significance of interactive effects. Effects are substantial and underscore the necessity of interpreting probable impacts of increases in a single stressor in conjunction with knowledge of other stressors acting on a population. A positive functional relationship between variability in population abundance and stress was estimated and is suggested as a potentially useful means of characterizing risks posed to populations by increases in, or additions to, population stressors. Multiple stressors were also demonstrated to effectively eliminate the significance of density-dependent mortality processes in determining age 0 + and adult abundance. Taken together, results indicate the inappropriateness of attempting to predict additional perturbation impacts without considering the sum of population stressors and their associated interactions.     

On the consequences of aggregation and balancing of networks on system properties derived from ecological network analysis

In the ecological network analysis (ENA) of complex flow food webs the assumption is often made that the models characterizing the flows and stocks of ecosystems occur in a steady state where inflows equals outflows. An assessment of the system indices derived from ENA of six balanced and unbalanced system models, respectively, indicate to differences between indices. The aggregation of highly articulated flow models into models with fewer compartments also has drastic effects on the system metrics, particularly on the information indices.     

The behavioural basis of a species replacement: differential aggresssion and predation between the introduced Gammarus pulex and the native G. duebeni celticus (Amphipoda)

Previous studies have shown that differential predation by males on moulted female congenerics may be largely responsible for the elimination and replacement of the native Irish freshwater amphipod Gammarus duebeni celticus by the introduced G. pulex. Predation of moulted females occurs both shortly after their release from precopulatory mate-guarding and whilst they are being guarded by their mates. In the present study, two hypotheses concerning the underlying cause(s) of the differential predation pattern are tested. Firstly, female G. d. celticus may be more vulnerable to predation than female G. pulex due to the former being released from precopula guarding with the new exoskeleton in a less hardened state. Secondly, G. pulex may be an inherently more aggressive species than G. d. celticus during predatory interactions over guarded females. The first experiment indicated that differential predation was not mediated by species differences in the state of the female exoskeleton at the time of release from precopula by guarding males. The second experiment, however, showed that male G. pulex were significantly more aggressive than male G. d. celticus in attacking both guarding male congenerics and guarded moulted female congenerics. In addition, in defence against predatory attacks, paired male and female G. pulex were significantly more aggressive than paired male and female G. d. celticus. These differences in aggressive behaviour led to a significantly higher frequency of predation on G. d. celticus females than on G. pulex females, and also explains this finding in previous studies. It is concluded that differential predation due to differences in aggressive behaviour may explain the pattern of replacement between these species.     

Dynamics of a chemostat with three competitive hydrogen oxidizing denitrifying microbial populations and their efficiency for denitrification

In this work, competition for two nitrogen resources (nitrate-, nitrite-nitrogen) between three hydrogen oxidizing denitrifying populations (Acidovorax sp. strain Ic3 (X₁), Paracoccus sp. strain Ic1 (X₂), and Acinetobacter sp. strain Ic2 (X₃)) was examined. The dynamics of three systems of microbial populations (system I: X₁ − X₃, system II: X₂ − X₃, and system III: X₁ − X₂ − X₃), grown in a chemostat, was studied using bifurcation analysis. The chemostat is the most common type of biological reactor used for the study of microbial growth under controlled conditions. The effect of the operating parameters (i.e., dilution rate and feed nitrate nitrogen concentration) on the long-term behavior of the systems showed that X₃ was the predominant population for a wide range of combinations of dilution rate and feed nitrate nitrogen concentration. Also, coexistence of two populations (X₂X₃, X₁X₃) was observed. The results of the bifurcation analysis were also used to determine the denitrification rate and the nitrite nitrogen accumulation for each of the three systems as a function of the dilution rate (up to 0.17 h⁻¹) and the feed nitrate nitrogen concentration (up to 300 mg/L). The highest denitrification rate was achieved by system I (28 mg/Lh). A comparison between the three systems showed that the nitrite nitrogen concentration in system I was less than the one in system III, while the two systems gave similar denitrification rates. The second system had the greatest accumulation of nitrites with the lowest denitrification rate.     

The influences of climate and hydrology on population dynamics of waterbirds in the lower Murrumbidgee River floodplains in Southeast Australia: Implications for environmental water management

Restoration of waterbird diversity and abundance is a key objective of river system management in Australia. Therefore, understanding the effects of climatic and hydrological variables on waterbird population dynamics is fundamental for successful river restoration programs. We investigated the population dynamics of waterbirds (total abundance) and seven functional waterbird groups in the floodplains of lower Murrumbidgee River. We found a general declining abundance trend from 1983 to 2007, except for the deep water foragers. We modelled the relative contribution of the climatic and hydrological factors to waterbird population decrease using the generalized additive model (GAM) framework after identifying the negative binomial distribution. Most of the seven functional groups were positively related to both annual rainfall and water usage, defined as the total water volume intercepted by the river reach, and the models indicated that rainfall was slightly more important. Temperature also played a role in waterbird abundance: the maximum summer temperature negatively influenced the abundance of dabbling ducks, shoreline foragers and fish eaters, while the minimum winter temperature positively affected the abundance of dabbling ducks and shoreline foragers. Overall, our results support the practice of providing environmental water for sustaining waterbird populations. However, environmental water provision is likely to be most effective when timed to coincide with antecedent rainfall.     

Population as an oscillating system

In this paper we present a new approach describing population dynamics based on the view of a population as an oscillating system. To develop a mathematical model of an oscillating population, we applied a third-order differential equation. Our model describes population dynamics within a parametric-temporal continuum, formed by the relative values of population growth and decrease over time. In this paper we also illustrate how our oscillative model effectively compliments the existing suite of models in population dynamics.     

Sex ratio dynamics and fluctuating selection pressures in natural populations of the Trinidadian guppy,<Emphasis Type="Italic"> Poecilia reticulata</Emphasis>

In many species, population sex ratios have far-reaching consequences for a wide variety of population-level and behavioural processes and can directly influence sexual selection through differential effects on male and female mating behaviour. Although sex ratios are often treated as more or less stable population characteristics, recent theoretical evidence suggests that sex ratios fluctuate under many conditions, and that the amplitude of these fluctuations can be considerable. Few studies have attempted to quantify this variation in systems with prominent, sex ratio-dependent sexual conflict. One of the species with the greatest potential to integrate these factors in the wild is the Trinidadian guppy, Poecilia reticulata. In this study, we quantified natural sex ratio variation both as detailed longitudinal studies of focal guppy populations and as snapshot estimates across a range of freshwater habitats. In line with theoretical predictions, we expected to detect significant sex ratio variation over time. We also investigated the association between juvenile and adult sex ratios to quantify a possible compensatory feedback implied in standard models of sex ratio evolution. Our results confirm that population-level sex ratios in wild guppy populations have a range of dynamic features, with all four focal populations showing significant variation in sex ratio over time. The survey showed that juveniles were generally close to equal (50:50) sex ratios whereas 7 out of 11 adult sex ratios differed significantly from equality. We found no evidence that a surplus of juveniles of the locally rarer sex had been produced. The results indicate that sex ratios and hence the balance between sexual selection and sexual coercion is normally fluctuating in nature, despite juvenile ratios being close to equality.Communicated by J. Krause     

Reconstructing historical marine ecosystems using food web models: Northern British Columbia from Pre-European contact to present

Mass-balance trophic models (Ecopath with Ecosim) are developed for the marine ecosystem of northern British Columbia (BC) for the historical periods 1750, 1900, 1950 and 2000 AD. Time series data are compiled for catch, fishing mortality and biomass using fisheries statistics and literature values. Using the assembled dataset, dynamics of the 1950-based simulations are fitted to agree with observations over 50 years to 2000 through the manipulation of trophic flow parameters and the addition of climate factors: a primary production anomaly and herring recruitment anomaly. The predicted climate anomalies reflect documented environmental series, most strongly sea surface temperature and the Pacific Decadal Oscillation index. The best-fit predator–prey interaction parameters indicate mixed trophic control of the ecosystem. Trophic flow parameters from the fitted 1950 model are transferred to the other historical periods assuming stationarity in density-dependent foraging tactics. The 1900 model exhibited an improved fit to data using this approach, which suggests that the pattern of trophic control may have remained constant over much of the last century. The 1950 model is driven forward 50 years using climate and historical fishing drivers. The resulting ecosystem is compared to the 2000 model, and the dynamics of these models are compared in a predictive forecast to 2050. The models suggest similar restoration trajectories after a hypothetical release from fishing.