From individual behavior to metapopulation dynamics: unifying the patchy population and classic metapopulation models

Spatially structured populations in patchy habitats show much variation in migration rate, from patchy populations in which individuals move repeatedly among habitat patches to classic metapopulations with infrequent migration among discrete populations. To establish a common framework for population dynamics in patchy habitats, we describe an individual-based model (IBM) involving a diffusion approximation of correlated random walk of individual movements. As an example, we apply the model to the Glanville fritillary butterfly (Melitaea cinxia) inhabiting a highly fragmented landscape. We derive stochastic patch occupancy model (SPOM) approximations for the IBMs assuming pure demographic stochasticity, uncorrelated environmental stochasticity, or completely correlated environmental stochasticity in local dynamics. Using realistic parameter values for the Glanville fritillary, we show that the SPOMs mimic the behavior of the IBMs well. The SPOMs derived from IBMs have parameters that relate directly to the life history and behavior of individuals, which is an advantage for model interpretation and parameter estimation. The modeling approach that we describe here provides a unified framework for patchy populations with much movements among habitat patches and classic metapopulations with infrequent movements.     

Relating individual behaviour to population dynamics

How do the behavioural interactions between individuals in an ecological system produce the global population dynamics of that system? We present a stochastic individual-based model of the reproductive cycle of the mite Varroa jacobsoni, a parasite of honeybees. The model has the interesting property in that its population level behaviour is approximated extremely accurately by the exponential logistic equation or Ricker map. We demonstrated how this approximation is obtained mathematically and how the parameters of the exponential logistic equation can be written in terms of the parameters of the individual-based model. Our procedure demonstrates, in at least one case, how study of animal ecology at an individual level can be used to derive global models which predict population change over time.     

Socially informed random walks: incorporating group dynamics into models of population spread and growth

Simple correlated random walk (CRW) models are rarely sufficient to describe movement of animals over more than the shortest time scales. However, CRW approaches can be used to model more complex animal movement trajectories by assuming individuals move in one of several different behavioural or movement states, each characterized by a different CRW. The spatial and social context an individual experiences may influence the proportion of time spent in different movement states, with subsequent effects on its spatial distribution, survival and fecundity. While methods to study habitat influences on animal movement have been previously developed, social influences have been largely neglected. Here, we fit a 'socially informed' movement model to data from a population of over 100 elk (Cervus canadensis) reintroduced into a new environment, radio-collared and subsequently tracked over a 4-year period. The analysis shows how elk move further when they are solitary than when they are grouped and incur a higher rate of mortality the further they move away from the release area. We use the model to show how the spatial distribution and growth rate of the population depend on the balance of fission and fusion processes governing the group structure of the population. The results are briefly discussed with respect to the design of species reintroduction programmes.     

Emergence of individual behaviour at the population level. Effects of density-dependent migration on population dynamics

The aim of this work is to study the effects of different individual behaviours on the overall growth of a spatially distributed population. The population can grow on two spatial patches, a source and a sink, that are connected by migrations. Two time scales are involved in the dynamics, a fast one corresponding to migrations and a slow one associated with the local growth on each patch. Different scenarios of density-dependent migration are proposed and their effects on the population growth are investigated. A general discussion on the use of aggregation methods for the study of integration of different ecological levels is proposed.     

Density-structured models for plant population dynamics

Density-structured models are structured population models in which the state variable is the proportion of populations or sites in a small number of discrete density states. Although such models have rarely been used, they have the advantage that they are straightforward to parameterize, make few assumptions about population dynamics, and permit rapid data collection using coarse density assessment. In this article, we highlight their use in relating population dynamics to environmental variation and their robustness to measurement error. We show that density-structured models are able to accurately represent population dynamics under a wide range of conditions. We look at the effects of including a persistent seedbank and describe numerical approximations for the mean and variance of population size. For simulated data, we determine the extent to which the underlying continuous process may be inferred from density-structured data. Finally, we discuss issues of parameter estimation and applications for which these types of models may be useful.     

From inducible defences to population dynamics: modelling refuge use and life history changes in Daphnia

We investigated the relative importance of a behavioural defence (refuge use through diel vertical migration) and a life history change (a reduced size at first reproduction) that are used by daphnids to decrease the risk of predation by visually hunting fish. We used an individual based model of a Daphnia population in a stratified lake to quantify the effects of these inducible defences on Daphnia predation-mortality and the resulting Daphnia population dynamics. Our analysis shows that diel vertical migration (DVM) confers a much stronger protection against fish predation than a reduced size at first reproduction (SFR). DVM allows daphnids to withstand a higher predation pressure in the epilimnion and it decelerates a Daphnia population decline more strongly than a reduced SFR. DVM effectively reduces the (P/B) flow of carbon from daphnids to fish.
Many theoretical studies have only considered the fitness benefits of DVM above 'staying up' in the epilimnion of a lake. Our results suggest that 'staying down' in the hypolimnion would confer an even stronger fitness benefit to Daphnia than DVM at times of peak predation risk. Daphnids that remain in the hypolimnion avoid the predation suffered by migrating daphnids around dusk and dawn. Staying down could prevent a Daphnia population decline, while DVM and a reduced SFR can only decelerate the decrease of Daphnia population densities under heavy fish predation. Staying down at high concentrations of fish infochemicals has in fact been observed within a variety of Daphnia clones and species, both in the laboratory and in stratified lakes.     

Linking frugivore behavior to plant population dynamics

Despite the acknowledged importance of frugivores as seed dispersal agents we still lack a general understanding of the mechanisms by which these animals could shape plant populations and communities. We used a spatially explicit stochastic simulation to explore how frugivore movement decisions interact with landscape properties, thus affecting plant population dynamics through dispersal. The model simulated bird movement, foraging, seed deposition and plant recruitment. We assumed that plants lived only for one season and that recruitment was a function of local seed density. We also considered the effect of perches as non‐food landscape features. Our simulation experiments consisted in varying the parameters governing bird foraging decisions in relation to 1) how fruit abundance biased their movement, and 2) how the willingness to visit a plant or perch decreased with distance to current location. Simulated plant population dynamics was strongly influenced by bird behavior. The scale of foraging decisions had a much stronger effect on plant dynamics than biases due to fruit abundance. Birds tended to concentrate their activities in the center of the landscape where plants became more abundant, increasing local competition. The presence of perches reduced this tendency resulting in larger population sizes. The importance of perches highlights the fact that behaviors other than foraging can have a strong impact on the patterns of seed deposition and hence on plant population dynamics. Several recent studies have combined animal movement data with seed retention time in order to predict seed dispersal kernels. These studies usually emphasize the ecological implications of the scale and shape of such kernels. However, our simulation results reveal that movement directionality and the fact that birds moved mostly among plants and perches can have a major impact on plant population dynamics.     

Adaptive dynamics for physiologically structured population models

We develop a systematic toolbox for analyzing the adaptive dynamics of multidimensional traits in physiologically structured population models with point equilibria (sensu Dieckmann et al. in Theor. Popul. Biol. 63:309–338, 2003). Firstly, we show how the canonical equation of adaptive dynamics (Dieckmann and Law in J. Math. Biol. 34:579–612, 1996), an approximation for the rate of evolutionary change in characters under directional selection, can be extended so as to apply to general physiologically structured population models with multiple birth states. Secondly, we show that the invasion fitness function (up to and including second order terms, in the distances of the trait vectors to the singularity) for a community of N coexisting types near an evolutionarily singular point has a rational form, which is model-independent in the following sense: the form depends on the strategies of the residents and the invader, and on the second order partial derivatives of the one-resident fitness function at the singular point. This normal form holds for Lotka–Volterra models as well as for physiologically structured population models with multiple birth states, in discrete as well as continuous time and can thus be considered universal for the evolutionary dynamics in the neighbourhood of singular points. Only in the case of one-dimensional trait spaces or when N = 1 can the normal form be reduced to a Taylor polynomial. Lastly we show, in the form of a stylized recipe, how these results can be combined into a systematic approach for the analysis of the (large) class of evolutionary models that satisfy the above restrictions.      

From individual behaviour to population models: a case study using swimming algae

Trajectories of swimming algae are analysed, and two random-walk models developed to link the individual-level behaviour of cells to population-level advection-diffusion models for the spatial-temporal distribution of cells. The models are both of the advection-diffusion form but are based on two different sets of assumptions about the underlying random-walk behaviours, a velocity jump behaviour and a velocity diffusion behaviour. The mathematical models were developed to allow for an arbitrary (non-weak) bias in the random walk and a variable swimming speed in order to represent the experimental data. For the algal species considered, Heterosigma akashiwo, the mean upward swimming speed was computed as , and the calculated diffusion constants ranged from 2×10³ to depending on the details of the models. That two widely used modelling approaches yield substantially different population-level predictions when applied to the same empirical data suggests that better theoretical tools are needed for identifying adequate approximations for behavioural characteristics.     

Neighborhood models of plant population dynamics. 2. Multi-species models of annuals

Models are developed for the dynamics of multi-species communities of annual plants that lack seed dormancy. These models explicitly include plastic plant growth, the spatial distribution of individuals, and the fact that individuals interact primarily with nearby individuals. Because the models are based on submodels of individual plants (fecundity, survivorship and dispersal, and how these are affected by inter-individual interactions), they provide explanations of community-level phenomena in terms of the biology of individuals. All model parameters and functional forms may be estimated from data obtained in simple experiments of a single years's duration. The models are used to examine the community-level consequences of some types of inter-individual interactions that have been reported in the ecological literature. In addition, the models are used to demonstrate that dispersal may markedly influence the outcome of competition among plant species, even in a physically homogeneous environment, due to an effect of dispersal on the spatial distribution of individuals.     

Stability results for delayed-recruitment models in population dynamics

This paper relates the stability properties of a class of delay-difference equations to those of an associated scalar difference equation. Simple but powerful conditions for testing global stability are presented which are independent of the length of the time delay involved. For models which do not have globally stable equilibria, estimates of stability regions are obtained. Some well known baleen whale models are used to illustrate the results.     

From individual interactions to population dynamics: individual resource partitioning simulation exposes the causes of nonlinear intra-specific competition

Intra-specific competition defines the relationship between population density and the performance of individual organisms (R-function). Observation of this relationship in nature shows it to be frequently nonlinear, and it has been argued, on intuitive grounds, that this nonlinearity is due to the type of competition (scramble or contest) being expressed. Here, we use an individual-based simulation model to investigate the effects of three resource partitioning schemes, representing different types of competition, on the form of the R-function. Results indicate that all resource partitioning schemes can give rise to concave or convex functions depending on the balance between maximum individual birth rate, maintenance cost, and demand for resources. Given high growth rates and maintenance costs, contest competitors tend to exhibit less concavity than scramblers. Therefore, population stability can be strongly affected by the strategy of resource partitioning. Life histories and environmental conditions that encourage the homogeneous distribution of resources among individuals lead to complex and unstable dynamics. Stable dynamics is fostered by heterogeneous resource distribution, which could result from such things as social hierarchies, individual and environmental variability, and large, indivisible resource packets.     

The Krylov-Bogoliubov-Mitropolskii method applied to models of population dynamics

In this study the possibility of applying the asymptotic method of Krylov-Bogoliubov-Mitropolskii to problems of population dynamics is shown. Especially a general Volterra-Gause-Witt type model for prey-predator interaction is investigated. A discussion on the results obtained is given for the general model and for a particular case as well.     

Diffusion models for animals in complex landscapes: incorporating heterogeneity among substrates, individuals and edge behaviours

1. Animals move commonly through a variety of landscape elements and edges in search of food, mates and other resources. We developed a diffusion model for the movement of an insect herbivore, the planthopper Prokelisia crocea, that inhabits a landscape composed of patches of its host plant, prairie cordgrass Spartina pectinata, embedded in a matrix of mudflat or smooth brome Bromus inermis. 2. We used mark-release-resight experiments to quantify planthopper movements within cordgrass-brome and cordgrass-mudflat arenas. A diffusion model was then fitted that included varying diffusion rates for cordgrass and matrix, edge behaviour in the form of a biased random walk and heterogeneity among planthoppers (sessile vs. mobile). The model parameters were estimated by maximum likelihood using the numerical solution of the diffusion model as a probability density. Akaike's information criterion (AIC) values were used to compare models with different subsets of features. 3. There was clear support for models incorporating edge behaviour and both sessile and mobile insects. The most striking difference between the cordgrass-brome and cordgrass-mudflat experiments involved edge behaviour. Planthoppers crossed the cordgrass-brome edge readily in either direction, but traversed the cordgrass-mudflat edge primarily in one direction (mudflat to cordgrass). Diffusion rates were also significantly higher on mudflat than for cordgrass and brome. 4. The differences in behaviour for cordgrass-brome vs. cordgrass-mudflat edges have implications for the connectivity of cordgrass patches as well as their persistence. Higher dispersal rates are expected between cordgrass patches separated by brome relative to mudflat, but patches surrounded by mudflat appear more likely to persist through time. 5. The experimental design and diffusion models used here could potentially be extended to any organism where mass mark-recapture experiments are feasible, as well as complex natural landscapes.     

Probabilistic lattice models of collective motion and aggregation: from individual to collective dynamics

We construct lattice gas automaton models to study collective dynamics and aggregation processes in biological systems. Corresponding transport equations are derived in appropriate space and time scaling under mean-field assumption and complemented with automation simulations.     

Rapidly converging numerical algorithms for models of population dynamics

We propose algorithms for the approximation of the age distributions of populations modeled by the McKendrick-von Foerster and the Gurtin-MacCamy systems both in one- and two-sex versions. For the one-sex model methods of second and fourth order are proposed. For the two-sex model a second order method is described. In each case the convergence is demonstrated. Several numerical examples are given.     

Bayesian experimental design for nonlinear mixed-effects models with application to HIV dynamics

Bayesian experimental design is investigated for Bayesian analysis of nonlinear mixed-effects models. Existence of the posterior risk for parameter estimation is shown. When the same prior distribution is used for both design and inference, existence of the preposterior risk for design is also proven. If the prior distribution used in design is different from that used for inference, sufficient conditions are established for existence of the preposterior risk for design. A case study of design for an experiment in population HIV dynamics is provided.     

Integrating individual behavior and population ecology: the potential for habitat-dependent population regulation in a reef fish

We used the predictions of the ideal free and ideal despoticdistributions (IFD and IDD, respectively) as a basis to evaluatethe link between spatial heterogeneity, behavior, and populationdynamics in a Caribbean coral reef fish. Juvenile three-spotdamselfish (Stegastes planifrons) were more closely aggregatedin patch reef habitat than on continuous back reef. Agonisticinteractions were more frequent but feeding rates were lowerin the patch versus the continuous reef habitat. Growth rateswere lower in patch reef habitat than on the continuous reef,but mortality rates did not differ. A separate experiment usingstandard habitat units demonstrated that the patterns observedin natural habitat were the result of the spatial distributionof the habitat patches rather than resource differences between habitats. Our results do not follow the predictions of simpleIFD or IDD models. This deviation from IFD and IDD predictionsmay be the result of a number of factors, including lack ofperfect information about habitat patches, high movement costs,and higher encounter rates of dispersed patches. Our resultsdemonstrate that behavioral interactions are an integral partof population dynamics and that it is necessary to considerthe spatial organization of the habitat in both behavioraland ecological investigations.