A general random walk model for the leptokurtic distribution of organism movement: Theory and application

The movement of organisms is usually leptokurtic in which some individuals move long distances while the majority remains at or near the area they are released. There has been extensive research into the origin of such leptokurtic movement, but one important aspect that has been overlooked is that the foraging behaviour of most organisms is not Brownian as assumed in most existing models. In this paper we show that such non-Brownian foraging indeed gives rise to leptokurtic distribution. We first present a general random walk model to describe the organism movement by breaking the foraging of each individual into events of active movement and inactive stationary period; its foraging behaviour is therefore fully characterized by a joint probability of how far the individual can move in each active movement and the duration it remains stationary between two consecutive movements. The spatio-temporal distribution of the organism can be described by a generalized partial differential equation, and the leptokurtic distribution is a special case when the stationary period is not exponentially distributed. Empirical observations of some organisms living in different habitats indicated that their rest time shows a power-law distribution, and we speculate that this is general for other organisms. This leads to a fractional diffusion equation with three parameters to characterize the distributions of stationary period and movement distance. A method to estimate the parameters from empirical data is given, and we apply the model to simulate the movement of two organisms living in different habitats: a stream fish (Cyprinidae: Nocomis leptocephalus) in water, and a root-feeding weevil, Sitona lepidus in the soil. Comparison of the simulations with the measured data shows close agreement. This has an important implication in ecology that the leptokurtic distribution observed at population level does not necessarily mean population heterogeneity as most existing models suggested, in which the population consists of different phenotypes; instead, a homogeneous population moving in homogeneous habitat can also lead to leptokurtic distribution.     

Joint modelling of breeding and survival in the kittiwake using frailty models

Assessment of population dynamics is central to population dynamics and conservation. In structured populations, matrix population models based on demographic data have been widely used to assess such dynamics. Although highlighted in several studies, the influence of heterogeneity among individuals in demographic parameters and of the possible correlation among these parameters has usually been ignored, mostly because of difficulties in estimating such individual-specific parameters. In the kittiwake (Rissa tridactyla), a long-lived seabird species, differences in survival and breeding probabilities among individual birds are well documented. Several approaches have been used in the animal ecology literature to establish the association between survival and breeding rates. However, most are based on observed heterogeneity between groups of individuals, an approach that seldom accounts for individual heterogeneity. Few attempts have been made to build models permitting estimation of the correlation between vital rates. For example, survival and breeding probability of individual birds were jointly modelled using logistic random effects models by [Cam, E., Link, W.A., Cooch, E.G., Monnat, J., Danchin, E., 2002. Individual covariation in life-history traits: seeing the trees despite the forest. Am. Naturalist, 159, in press]. This is the only example in wildlife animal populations we are aware of. Here we adopt the survival analysis approaches from epidemiology. We model the survival and the breeding probability jointly using a normally distributed random effect (frailty). Conditionally on this random effect, the survival time is modelled assuming a lognormal distribution, and breeding is modelled with a logistic model. Since the deaths are observed in year-intervals, we also take into account that the data are interval censored. The joint model is estimated using classic frequentist methods and also MCMC techniques in Winbugs. The association between survival and breeding attempt is quantified using the standard deviation of the random frailty parameters. We apply our joint model on a large data set of 862 birds, that was followed from 1984 to 1995 in Brittany (France). Survival is positively correlated with breeding indicating that birds with greater inclination to breed also had higher survival.     

Escape from the cell: Spatially explicit modelling with and without grids

This paper is concerned with the representation of individuals embedded in a two- (or three-) dimensional environment, and with the techniques that can be used to simulate the evolution of the spatial patterns both of the populations of those individuals and of their environment. Its scope is therefore that of individual based or agent based modelling, of a general type, including herbivore populations, predator-prey models or any other type that is concerned with the spatial patterning evolving from recruitment, interaction and/or movement of discrete individuals. The aim is to discuss a modelling technique that allows more flexibility in the representation of the positions of individuals than is typically the case for cellular automata (CA), but which also deals efficiently with the problem of searching for neighbours when individual positions can vary nearly continuously. A scaling problem is discussed that arises when the range over which individuals interact is much smaller than the size of the domain. It is argued that validation of CA models involving discrete individuals is made more difficult when the system scale exceeds the size of individuals by a large factor. However, even when the domain size is small, if interaction between individuals is mediated by their size, imposition of a fixed grid upon the dynamics may cause important phenomena to be misrepresented or missed altogether. We suggest that cellular automata, as usually formulated, do not deal adequately with this type of problem, and introduce a particle-in-cell (PIC) method to deal with it in intermediate cases. Alternative data structures are discussed for dealing with more extreme cases, including the possibility of modelling an indefinitely large domain using a changing set of cells (PIC:SI).     

Closed population estimation models and their extensions in Program MARK

Program MARK provides > 65 data types in a common configuration for the estimation of population parameters from mark-encounter data. Encounter information from live captures, live resightings, and dead recoveries can be incorporated to estimate demographic parameters. Available estimates include survival (S or ϕ), rate of population change (λ), transition rates between strata (Ψ), emigration and immigration rates, and population size (N). Although N is the parameter most often desired by biologists, N is one of the most difficult parameters to estimate precisely without bias for a geographically and demographically closed population. The set of closed population estimation models available in Program MARK incorporate time (t) and behavioral (b) variation, and individual heterogeneity (h) in the estimation of capture and recapture probabilities in a likelihood framework. The full range of models from M ₀ (null model with all capture and recapture probabilities equal) to M _tbh are possible, including the ability to include temporal, group, and individual covariates to model capture and recapture probabilities. Both the full likelihood formulation of Otis et al. (1978) and the conditional model formulation of Huggins (1989, 1991) and Alho (1990) are provided in Program MARK, and all of these models are incorporated into the robust design (Kendall et al. 1995, 1997; Kendall and Nichols 1995) and robust-design multistrata (Hestbeck et al. 1991, Brownie et al. 1993) data types. Model selection is performed with AICc (Burnham and Anderson 2002) and model averaging (Burnham and Anderson 2002) is available in Program MARK to provide estimates of N with standard error that reflect model selection uncertainty.     

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.     

Development of the gap model ZELIG-CFS to predict the dynamics of North American mixed forest types with complex structures

When the development of gap models began about three decades ago, they became a new category of forest productivity models. Compared with traditional growth and yield models, which aim at deriving empirical relationships that best fit data, gap models use semi-theoretical relationships to simulate biotic and abiotic processes in forest stands, including the effects of photosynthetic active radiation interception, site fertility, temperature and soil moisture on tree growth and seedling establishment. While growth and yield models are appropriate to predict short-term stemwood production, gap models may be used to predict the natural course of species replacement for several generations. Because of the poor availability of historical data and knowledge on species-specific allometric relationships, species replacement and death rate, it has seldom been possible to develop and evaluate the most representative algorithms to predict growth and mortality with a high degree of accuracy. For this reason, the developers of gap models focused more on developing simulation tools to improve the understanding of forest succession than predicting growth and yield accurately.In a previous study, the predictions of simulations in two southeastern Canadian mixed ecosystem types using the ZELIG gap model were compared with long-term historical data. This exercise highlighted model components that needed modifications to improve the predictive capacity of ZELIG. The updated version of the model, ZELIG-CFS, includes modifications in the modelling of crown interaction effects, survival rate and regeneration. Different algorithms representing crown interactive effects between crowns were evaluated and species-specific model components that compute individual-tree mortality probability rate were derived. The results of the simulations were compared using long-term remeasurement data obtained from sample plots located in La Mauricie National Park of Canada in Quebec. In the present study, three forest types were studied: (1) red spruce-balsam fir-yellow birch, (2) yellow birch-sugar maple-balsam fir, and (3) red spruce-balsam fir-white birch mixed ecosystems. Among the seven algorithms that represented individual crown interactions, two better predicted the changes in basal area and individual-tree growth: (1) the mean available light growing factor (ALGF), which is computed from the proportion of light intercepted at different levels of individual crowns adjusted by the species-specific shade tolerance index, and (2) the ratio of mean ALGF to crown width. The long-term predicted patterns of change in basal area were consistent with the life history of the different species.     

An individual-based model to study the reproduction of egg bearing copepods: Application to Eurytemora affinis (Copepoda Calanoida) from the Seine estuary,France

Limited empirical studies have elucidated the daily egg production and associated reproductive processes of egg bearing copepod. Herein, we present an individual-based model which constitutes a realistic representation of the reproduction in egg bearing copepods. The model has been parameterized using an extensive set of experimental data obtained from the literature and from the laboratory and field experiments on the estuarine copepod Eurytemora affinis. The proposed model takes into account the adult female longevity, the clutch size and interclutch duration, which is a function of egg maturation time and latency time required by the female after egg hatching to produce a new clutch. The embryonic development time and hatching success are also taken into account. The effect of temperature on the means and variances of above-mentioned reproductive parameters has been also incorporated. A multi agent system based generic platform “Mobidyc” has been used to generate and calibrate the model. The model demonstrates the reproductive parameters of females of E. affinis which is validated through individual based experiments. Temperature specific simulations provide a dynamical explanation of temperature effect on the cumulative egg production. The daily survival principally affects the number of clutches produced per female during its life span. The results obtained in the present study by combining temperature and survival effects reveal the relatively greater importance of the first factor on the daily egg production of egg-carrying copepods. The present model is generic and hence easily applicable to other animals with comparable reproductive strategy.     

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).     

Modelling effects of time-variable exposure to the pyrethroid beta-cyfluthrin on rainbow trout early life stages

Background

Available literature and regulatory studies show that the severity of effects of beta-cyfluthrin (a synthetic pyrethroid) on fish is influenced by the magnitude and duration of exposure. To investigate how the exposure pattern to beta-cyfluthrin (constant vs peak) may influence the response of the fish, we used a mechanistic effect model to predict the survival and growth of the rainbow trout over its early life stages (i.e. egg, alevin and swim-up fry). We parameterized a toxicokinetic–toxicodynamic (TKTD) module in combination with a dynamic energy budget model enabling us to describe uptake and elimination, as well as to predict the threshold concentration for survival and sublethal effects (feeding behaviour and growth). This effect model was calibrated using data from an early life stage experiment where trout was exposed to a constant concentration of cyfluthrin. The model was validated by comparing model predictions to independent data from a pulsed-exposure study with early life stages of rainbow trout.

Results

The co-occurrence of effects on behaviour and growth raised the possibility that these were interrelated, i.e. impairment of feeding behaviour may have led to reduced food intake and slower growth. We, therefore, included ‘effect on feeding’ as mode of action in the TKTD module. At higher concentrations, the constant exposure led to death. The model was able to adequately capture this effect pattern in the calibration. The model was able to adequately predict the response of fish eggs, alevins and swim-up fry, from both the qualitative (response pattern) and quantitative points of view.

Conclusions

Since the model was successfully validated, it can be used to predict survival and growth of early life stages under various realistic time-variable exposure profiles (e.g. profiles from FOCUS surface water modelling) of beta-cyfluthrin.

     

On the Estimation of Dispersal Kernels from Individual Mark-Recapture Data

We present a new method for estimating a distribution of dispersal displacements (a dispersal kernel) from mark-recapture data. One conventional method of calculating the dispersal kernel assumes that the distribution of displacements are Gaussian (e.g. resulting from a diffusion process) and that individuals remain within sampled areas. The first assumption prohibits an analysis of dispersal data that do not exhibit the Gaussian distribution (a common situation); the second assumption leads to underestimation of dispersal distance because individuals that disperse outside of sampling areas are never recaptured. Our method eliminates these two assumptions. In addition, the method can also accommodate mortality during a sampling period. This new method uses integrodifference equations to express the probability of spatial mark-recapture data; associated dispersal, survival, and recapture parameters are then estimated using a maximum likelihood method. We examined the accuracy of the estimators by applying the method to simulated data sets. Our method suggests designs for future mark-recapture experiments. Received: January 2004 / Revised: July 2005     

Hierarchical demographic approaches for assessing invasion dynamics of non-indigenous species: An example using northern snakehead (Channa argus)

Models of species’ demographic features are commonly used to understand population dynamics and inform management tactics. Hierarchical demographic models are ideal for the assessment of non-indigenous species because our knowledge of non-indigenous populations is usually limited, data on demographic traits often come from a species’ native range, these traits vary among populations, and traits are likely to vary considerably over time as species adapt to new environments. Hierarchical models readily incorporate this spatiotemporal variation in species’ demographic traits by representing demographic parameters as multi-level hierarchies. As is done for traditional non-hierarchical matrix models, sensitivity and elasticity analyses are used to evaluate the contributions of different life stages and parameters to estimates of population growth rate. We applied a hierarchical model to northern snakehead (Channa argus), a fish currently invading the eastern United States. We used a Monte Carlo approach to simulate uncertainties in the sensitivity and elasticity analyses and to project future population persistence under selected management tactics. We gathered key biological information on northern snakehead natural mortality, maturity and recruitment in its native Asian environment. We compared the model performance with and without hierarchy of parameters. Our results suggest that ignoring the hierarchy of parameters in demographic models may result in poor estimates of population size and growth and may lead to erroneous management advice. In our case, the hierarchy used multi-level distributions to simulate the heterogeneity of demographic parameters across different locations or situations. The probability that the northern snakehead population will increase and harm the native fauna is considerable. Our elasticity and prognostic analyses showed that intensive control efforts immediately prior to spawning and/or juvenile-dispersal periods would be more effective (and probably require less effort) than year-round control efforts. Our study demonstrates the importance of considering the hierarchy of parameters in estimating population growth rate and evaluating different management strategies for non-indigenous invasive species.     

Effects of attached data-logging devices on little penguins (Eudyptula minor)

Data-logging devices are commonly used to study the foraging behaviour of individual seabirds. Such studies need to examine the potential effects of using devices on instrumented individuals, not only for ethical reasons but also to ensure the validity of data gathered. We studied the effects of two types of device (time-depth recorder and global positioning system) on little penguins (Eudyptula minor) during the 2010 and 2011 breeding season at Oamaru, New Zealand. Mixed-effect models were used to test for effects of devices by comparing changes in body weight, chick growth and breeding performance between instrumented and control individuals. We found no detectable effects of the attached devices on body weight change, hatching success, fledging success, chick growth parameters or adult survival. We conclude that it is possible to attach data-logging devices to adult little penguins for extended periods during the breeding season with minimal impacts.     

Individual deviation from behavioural correlations: a simple approach to study the evolution of behavioural syndromes

The study of correlations between different behaviours in a population—referred to as behavioural syndromes—has begun to flourish during recent years. However, the evolutionary mechanisms that cause behavioural traits to vary non-independently from each other are still poorly understood. Here, we bring behavioural syndromes into a new perspective, in which the phenomenon is regarded at the individual level and on a continuous scale instead of as a population-level presence/absence trait. As the correlation between behaviours is never perfect (i.e. r < 1), individuals are likely to vary in how consistently they behave. Therefore, we can predict that if behavioural syndromes at the population level are results of natural selection, the consistency in a suite of behaviours—and not the behavioural configuration per se—should be heritable and involve fitness advantages at the individual level. We define a variable that describes the individual deviation from the hypothetical perfect correlation predicted by the syndrome. The use of such a variable depicting the consistency of behaviours of individuals allows us to make solid evolutionary inferences about correlated behaviours from patterns of individual instead of population variation. We suggest that, by adopting the concept of syndrome deviation, understanding the evolution of behavioural syndromes and, in particular, testing competing evolutionary hypotheses about the origin of behavioural syndromes becomes possible in a more rigorous manner than before.     

Estimating dyad association probability under imperfect and heterogeneous detection

In animal behaviour studies, association indices estimate the proportion of time two individuals (i.e. a dyad) spend in association. In terms of dyads, all association indices can be interpreted as estimators of the probability that a dyad is associated. However, traditional indices rely on the assumptions that the probability to detect a particular individual (p) is either approximately one and/or homogeneous between associated and not associated individuals. Based on marked individuals we develop a likelihood based model to estimate the probability a dyad is associated (ψ) accounting for p < 1 and possibly varying between associated and not associated individuals. The proposed likelihood based model allows for both individual and dyadic missing observations. In addition, the model can easily be extended to incorporate covariate information for modeling p and ψ. A simulation study showed that the likelihood based model approach yield reasonably unbiased estimates, even for low and heterogeneous individual detection probabilities, while, in contrast, traditional indices showed moderate to strong biases. The application of the proposed approach is illustrated using a real data set collected from a population of Commerson's dolphin (Cephalorhynchus commersonii) in Patagonia Argentina. Finally, we discuss possible extensions of the proposed model and its applicability in animal behaviour and ecological studies.     

Estimating age from recapture data: integrating incremental growth measures with ancillary data to infer age-at-length

Estimating the age of individuals in wild populations can be of fundamental importance for answering ecological questions, modeling population demographics, and managing exploited or threatened species. Significant effort has been devoted to determining age through the use of growth annuli, secondary physical characteristics related to age, and growth models. Many species, however, either do not exhibit physical characteristics useful for independent age validation or are too rare to justify sacrificing a large number of individuals to establish the relationship between size and age. Length-at-age models are well represented in the fisheries and other wildlife management literature. Many of these models overlook variation in growth rates of individuals and consider growth parameters as population parameters. More recent models have taken advantage of hierarchical structuring of parameters and Bayesian inference methods to allow for variation among individuals as functions of environmental covariates or individual-specific random effects. Here, we describe hierarchical models in which growth curves vary as individual-specific stochastic processes, and we show how these models can be fit using capture-recapture data for animals of unknown age along with data for animals of known age. We combine these independent data sources in a Bayesian analysis, distinguishing natural variation (among and within individuals) from measurement error. We illustrate using data for African dwarf crocodiles, comparing von Bertalanffy and logistic growth models. The analysis provides the means of predicting crocodile age, given a single measurement of head length. The von Bertalanffy was much better supported than the logistic growth model and predicted that dwarf crocodiles grow from 19.4 cm total length at birth to 32.9 cm in the first year and 45.3 cm by the end of their second year. Based on the minimum size of females observed with hatchlings, reproductive maturity was estimated to be at nine years. These size benchmarks are believed to represent thresholds for important demographic parameters; improved estimates of age, therefore, will increase the precision of population projection models. The modeling approach that we present can be applied to other species and offers significant advantages when multiple sources of data are available and traditional aging techniques are not practical.     

Quantifying site quality in a heterogeneous landscape: recruitment of a reef fish

Heterogeneity in site quality can play an important role in patterns of abundance and population dynamics. Yet, estimating site quality in natural systems can be problematic because site quality can (1) vary through ontogeny for a focal organism, leading to shifts in site quality with age, (2) be confounded with (or masked by) variation in traits of individuals populating the sites, and (3) be correlated with local density. For example, if high-quality sites attract more individuals but vital rates are density dependent, then observed vital rates will be relatively homogeneous in space despite strong heterogeneity in site quality. Here, we operationally define site quality for a reef fish as the mean survival time of juveniles transplanted to sites at a common density and size structure, with random assignment of individuals to sites to remove potential confounding effects of local variation in individual quality and density. Our assays using juvenile age classes of the six-bar wrasse (Thalassoma hardwicke) showed that site quality varied in space (i.e., among patch reefs) but was constant through time. Site quality increased with availability of the branching coral Pocillopora (which is used as a refuge), but decreased with density of a predator, the arc-eye hawkfish, Paracirrhites arcatus (which also uses Pocillopora). We experimentally added colonies of Pocillopora to reefs and (1) increased site quality, (2) enhanced natural settlement rates of six-bar wrasse, but (3) attracted more hawkfish predators, and (4) did not increase survival of juvenile fish under ambient densities. Our results suggest that Pocillopora increases site quality, but attracts greater densities of settlers and predators, resulting in increased density dependence and predation, which mask the underlying effects of Pocillopora on site quality (supporting the hypothesis of "cryptic density dependence"). Variation in site quality and the possible confounding effects of density and individual traits warrant more experimental study.     

Modeling survival and mark loss in molting animals: recapture, dead recoveries, and exuvia recoveries

Capture-mark-recapture (CMR) analyses aim primarily at estimating relevant life history parameters, despite the fact that some individuals are not always recaptured, even if alive on the study site. Applying such approaches to species with a complex life cycle, such as insects, remains challenging because each change of stage tends to cause mark loss through molting. We developed a multistate model based on three exclusive events ("dead", "surviving and molting", and "surviving and staying in the same larval stage") to estimate probabilities of survival and mark loss. Estimates of biologically relevant parameters were derived from those of the probabilities of transition between these states. The model was applied to data from radio-tracking diodes glued on grasshoppers. The estimates of recapture probabilities decreased throughout the season for animals remaining alive, while the detection of dead animals and lost diodes was exhaustive. The survival probability was higher for larvae than for adults (0.98 vs. 0.96), and mark loss was stronger in larvae than in adults (0.09 vs. 0.06). We show that the survival rate of a species with a high rate of mark loss can be estimated using multistate models, provided that marks can be recovered after being lost. These models are flexible enough to test for several effects that potentially affect survival and mark loss probabilities.     

Effects of Multiple Levels of Social Organization on Survival and Abundance

Abstract: Identifying how social organization shapes individual behavior, survival, and fecundity of animals that live in groups can inform conservation efforts and improve forecasts of population abundance, even when the mechanism responsible for group‐level differences is unknown. We constructed a hierarchical Bayesian model to quantify the relative variability in survival rates among different levels of social organization (matrilines and pods) of an endangered population of killer whales (Orcinus orca). Individual killer whales often participate in group activities such as prey sharing and cooperative hunting. The estimated age‐specific survival probabilities and survivorship curves differed considerably among pods and to a lesser extent among matrilines (within pods). Across all pods, males had lower life expectancy than females. Differences in survival between pods may be caused by a combination of factors that vary across the population's range, including reduced prey availability, contaminants in prey, and human activity. Our modeling approach could be applied to demographic rates for other species and for parameters other than survival, including reproduction, prey selection, movement, and detection probabilities.