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).     

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.     

A simple population theory for mutualism by the use of lattice gas model

The population dynamics of species interactions provides valuable information for life sciences. Lotka-Volterra equations (LVEs) are known to be the most popular model, and they are mainly applied to the systems of predation and competition. However, LVEs often fail to catch the population dynamics of mutualism; the population sizes of species increase infinitely under certain condition (divergence problem). Furthermore, LVEs never predicts the Allee effect in the systems of obligate mutualism. Instead of LVEs, several models have been presented for mutualism; unfortunately, they are rather complicated. It is, therefore, necessary to introduce a simpler theory for mutualism. In the present paper, we apply the lattice gas model which corresponds to the mean-field theory of the usual lattice model. The derived equations are cubic and contain only essential features for mutualism. In the case of obligate mutualism, the dynamics exhibits the Allee effect, and it is almost the same as in the male-female systems. In our model, the population sizes never increase infinitely, because our model contains not only intra- but also interspecific competitions. If the density of one species increases disproportionately in respect of its mutual partners, then this might imply downward pressure on the population abundance of the mutual partner species and such feedback would eventually act as a controlling influence on the population abundance of either species. We discuss several assumptions in our model; in particular, if both species can occupy in each cell simultaneously, then the interspecific competition disappears.     

A hydrologically driven model of swamp water mosquito population dynamics

We develop a swamp water mosquito population model that is forced solely by environmental variability. Measured temperature and land surface wetness conditions are used to simulate Anopheles walkeri population dynamics in a northern New Jersey habitat. Land surface wetness conditions, which represent oviposition habitat availability, are derived from simulations using a dynamic hydrology model. Using only these two density-independent effects, population model simulations of biting Anoph. walkeri correlate significantly with light trap collections. These results suggest that prediction of mosquito populations and the diseases they transmit could be better constrained by inclusion of environmental variability.     

A survival-analysis-based simulation model for Russian wheat aphid population dynamics

A simulation model for Russian wheat aphid (RWA), Diuraphis noxia (Mordvilko), populations is built by integrating survival-analysis-based development and survivor functions and the same-shape reproduction distribution model in the framework of Leslie [Leslie, P.H., 1945. On the use of matrices in certain population mathematics. Biometrika 33, 183–212] matrix structure. Survival analysis is utilized to model both the development and survival of RWA populations, and the Cox (1972) proportional hazards model is fitted with the data sets from our laboratory observation of 1800 RWA individuals under 25 factorial combinations of five temperature regimes and five barley plant-growth stages. Rather than using simple age-specific survivor rates as in the traditional Leslie matrix, the survivor functions based on survival analysis describe age-specific, temperature and plant stage-dependent RWA survival probabilities. Similarly, a probability model from survival analysis to estimate the probability that an individual will reach mature adult stage is utilized to describe the development process; this makes the transition from nymphal stage to mature adult stage dependent on RWA age as well as temperature and plant-growth stage.Inspired by the same-shape distribution and rate-summation approach for modeling insect development, a similar approach for modeling insect reproduction under variable temperature is developed. This new same-shape reproduction distribution model incorporates individual variation in reproduction capability, as well as the effects of RWA age, temperature and plant-growth stage. Consequently, the same-shape reproduction distribution model replaces the simple age-specific fecundities in Leslie matrix model. To the best of our knowledge, this work is the first to introduce survival analysis to simulation modeling in entomology and ecology and also the first to integrate our newly developed same-shape reproduction distribution model into application.     

Population dynamics of splitnose rockfish (Sebastes diploproa) in the Northeast Pacific Ocean

We developed an age-structured population model of splitnose rockfish, Sebastes diploproa, in the Northeast Pacific Ocean. Splitnose rockfish is a bycatch species that co-occurs with several commercially important species that are currently declared overfished. Bycatch species are typically not the focus of stock assessment efforts because of their limited economic importance, but they may suffer the same population declines as species with which they co-occur. To examine the dynamics of splitnose rockfish for the first time, we analyzed data from three groundfish fisheries and four research surveys conducted in the Northeast Pacific Ocean. To develop a model, we used Stock Synthesis, a statistical framework for the construction of a population dynamics models utilizing both fishery-dependent and fishery-independent data. In the model, we reconstructed the total catch of the species back to 1900, estimated the dynamics of the stock spawning output and recruitment and evaluated biomass depletion relative to the stock's unfished state, as well as sources of uncertainty in model outputs. The results indicate that the splitnose rockfish is currently not overfished even though it has experienced several periods of abrupt decline in its biomass. Revisiting age data from earlier years, monitoring fishery discard, and investigating the spatial dynamics of splitnose rockfish is important to further improve the understanding of this species’ population dynamics, and decrease uncertainty in model results.     

Zooplankton population model coupled to a biogeochemical model of the North Western Mediterranean Sea ecosystem

A stage structured population (SSP) model based on Fennel's [Fennel, W., 2001. Modelling copepods with links to circulation models. Journal of Plankton Research, 23, 1217–1232] equations is applied to Centropages typicus (Kröyer), a dominant copepod species of the North Western Mediterranean Sea (NWMS) and a prey of small pelagic fish. The model considers five groups of stages and development rates are represented by a mechanistic formulation depending on individual specific growth in each stage. Individual growth is calculated from the individual energy budget depending on food availability and temperature.     

A finite element method to solve a population dynamics stage-structured model of intertidal barnacles

Many marine organisms are fixed or highly sedentary as adults but the adult population may be strongly dependent on the oceanic transport of planktonic larvae. In order to assess interactions between oceanographic and biological processes that determine the population dynamics of marine organisms with a sessile adult phase restricted to the coastline and a planktonic larval phase, we present a stage-structured finite element model for the barnacle Balanus glandula that inhabits the rocky intertidal zone of central California, USA.     

Individual based model of slug population and spatial dynamics

The slug, Deroceras reticulatum, is one of the most important pests of agricultural and horticultural crops in UK and Europe. In this paper, a spatially explicit individual based model (IbM) is developed to study the dynamics of a population of D. reticulatum. The IbM establishes a virtual field within which slug spatial dynamics and changes in abundance were simulated. The strong dependence of slug behaviour on environmental conditions is built into the model, which is based upon previous work on the environmental dependence of slug population dynamics. The simulation results show that the IbM described well changes in the slug population. The IbM proved capable of describing slug populations over 3.5 years, including the presence, magnitude and duration of D. reticulatum population crashes within this period. Moreover, the model was capable of reproducing slug population dynamics at two sites, with distinct weather and some 100 km apart, with minor changes in initialisation values but no change in model structure and parameter values. A study of field heterogeneity, which might simulate various field designs, indicated the importance of spatial structuring to slug population dynamics and the utility of the IbM for simulating a range of potential spatial management treatments for slug control to maximise crop yield. This IbM system performs well and is currently being used as part of an integrated approach to predict slug population dynamics and control in the UK.     

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.     

Modelling crayfish population dynamics using catch data: A size-structured model

In this study we explore a rather unique time series (1979–2002) of catch data of the crayfish Astacus astacus in Lake Steinsfjorden (SE Norway) in combination with temperature data and data on Canadian pondweed Elodea canadensis coverage. In 1977, E. canadensis was for the first time observed in the lake. Over the following years, the plant established dense covers over large parts of the shallow areas, excluding the crayfish from these areas and causing a sudden drop in population size. A size-structured model with bi-stability including a range of observed stage-specific life-cycle attributes (e.g. growth, fecundity, fertility, sex-ratio), population specific parameters and density-dependant (shelters, cannibalism, unspecified predators, competition between individuals, catch, number of traps) as well as density independent factors (temperature and Elodea coverage) were constructed to evaluate the various drivers for the population dynamics, and as a predictive tool for assessing the effects of future changes. Our model revealed that the decline primarily was due to density-dependant effect of the Elodea expansion with reduced number of hides and thus increased risk for predation and cannibalism, but also that temperature played an important role related to recruitment. The model should be relevant for crayfish stock management in general, and by demonstrating the major role of temperature, it is also relevant for predicting population responses under a changing climate. The model should also be applicable to other crustaceans and species with discrete growth and late maturation.     

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 and validation of an individual based Daphnia magna population model: The influence of crowding on population dynamics

An individual-based model was developed to predict the population dynamics of Daphnia magna at laboratory conditions from individual life-history traits observed in experiments with different feeding conditions. Within the model, each daphnid passes its individual life cycle including feeding on algae, aging, growing, developing and – when maturity is reached – reproducing. The modelled life cycle is driven by the amount of ingested algae and the density of the Daphnia population. At low algae densities the population dynamics is mainly driven by food supply, when the densities of algae are high, the limiting factor is “crowding” (a density-dependent mechanism due to chemical substances released by the organisms or physical contact, but independent of food competition).     

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.     

A Voronoi diagram based population model for social species of wildlife

A population model is presented that accounts for spatial structure within habitat patches. It is designed for social species of wildlife that form social group home ranges that are much smaller than patch size. The model represents social group home ranges by Voronoi regions that tessellate a patch to form a Voronoi diagram. Neighbouring social groups are linked with habitat-confined shortest paths and form a dispersal network. The model simulates population dynamics and makes use of Voronoi diagrams and dispersal networks as a spatial component. It then produces density maps as outputs. These are maps that show predicted animal densities across the patches of a landscape. A construction procedure for the particular Voronoi diagram type used by the model is described. As a test case, the model is run for the squirrel glider (Petaurus norfolcensis), a small arboreal marsupial native to Australia. A time series of density maps are produced that show squirrel glider density changing across a landscape through time.     

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.     

Integrating farming techniques in an ecological matrix model: Implementation on the primrose (Primula vulgaris)

Several studies have proven the importance of field margins in sustaining biodiversity and other work has been done on the effect of field management on field margin flora. However few models have been built to predict the effects of field management on the flora. Our project addresses this need for a model capable of predicting the effect of cropping techniques and their timing on the flora of field margins. Primula vulgaris is a biodiversity indicator, characteristic of undisturbed flora and found in field margins and woodlands: its population has been declining for several years. We created a temporal matrix model of P. vulgaris populations on field margins, taking into account the effects of field, field margin and roadside management based on literature and expert knowledge. We then analysed its sensitivity to demographic parameters by comparing lambda (growth rate) sensitivity and elasticity. We compared the management parameter effect using the relative growth rate of the population after 6 years of simulation. Sensitivity analysis to biological parameters showed the importance of adult survival and seed production and germination. Results show that P. vulgaris is particularly sensitive to broad-spectrum herbicides and that other management techniques like early mowing, scything and scrub-killer (diluted broad-spectrum herbicide or specific herbicide) are less aggressive. Our simulations show that management of cash crops in Brittany is too aggressive for P. vulgaris populations and that 4-5 years of grassland in the adjacent field are necessary to maintain populations.     

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.     

Choosing classes for size projection matrix models

Projection matrix models are intensely used in ecology to model the dynamics of structured populations. When dealing with size-structured populations, there is no satisfactory algorithm to partition size into discrete classes. We show that the Vandermeer-Moloney algorithm for choosing classes is inconsistent with the Usher model, and systematically selects the finest classes. Considering that the matrix model is a discrete approximation of a continuous model, we define an approximation error as the sum of a distribution error (the difference between the discrete distribution and its continuous counterpart), and a sample error. The optimal partition of size into classes is the one that minimizes the approximation error. This method for choosing classes also shows that the choice of the class width cannot be disconnected from the choice of the time step. When applied to 520 trees of Dicorynia guianensis in French Guiana, this algorithm identified 8 classes of 11.4 cm in width, which is in agreement with the empirical choice of foresters.