Modelling dispersal with diffusion and habitat selection: Analytical results for highly fragmented landscapes

Quantifying dispersal is crucial both for understanding ecological population dynamics, and for gaining insight into factors that affect the genetic structure of populations. The role of dispersal becomes pronounced in highly fragmented landscapes inhabited by spatially structured populations. We consider a landscape consisting of a set of habitat patches surrounded by unsuitable matrix, and model dispersal by assuming that the individuals follow a random walk with parameters that may be specific to the habitat type. We allow for spatial variation in patch quality, and account for edge-mediated behavior, the latter meaning that the individuals bias their movement towards the patches when close to an edge between a patch and the matrix. We employ a diffusion approximation of the random walk model to derive analytical expressions for various characteristics of the dispersal process. For example, we derive formulae for the time that an individual is expected to spend in its current patch i, and for the time that it will spend in the matrix, both conditional on the individual hitting next a given patch j before hitting any of the other patches or dying. The analytical formulae are based on the assumptions that the landscape is infinitely large, that the patches are circularly shaped, and that the patches are small compared to interpatch distances. We evaluate the effect of these assumptions by comparing the analytical results to numerical results in a real patch network that violates all of the three assumptions. We then consider a landscape that fulfills the assumptions, and show that in this case the analytical results are in a very good agreement with the numerical results. The results obtained here allow the construction of computationally efficient dispersal models that can be used as components of metapopulation models.     

Incorporating the perceptual range of animals into connectivity models

The perceptual range of an animal towards different landscape elements affects its movements through heterogeneous landscapes. However, empirical knowledge and modeling tools are lacking to assess the consequences of variation in the perceptual range for movement patterns and connectivity. In this study we tested how changes in the assumed perception of different landscape elements affect the outcomes of a connectivity model. We used an existing individual-based, spatially explicit model for the dispersal of Eurasian lynx (Lynx lynx). We systematically altered the perceptual range in which animals recognize forest fragments, water bodies or cities, as well as the probability that they respond to these landscape elements. Overall, increasing the perceptual range of the animals enhanced connectivity substantially, both qualitatively and quantitatively. An enhanced range of attraction to forests had the strongest impact, doubling immigration success; an enhanced range of attraction to rivers had a slightly lower impact; and an enhanced range of avoidance of cities had the lowest impact. Correcting the enhancement in connectivity by the abundance of each of the landscape elements in question reversed the results, indicating the potential sensitivity of connectivity models to rare landscape elements (in our case barriers such as cities). Qualitatively, the enhanced perception resulted in strong changes in movement patterns and connectivity. Furthermore, model results were highly parameter-specific and patch-specific. These results emphasize the need for further empirical research on the perceptual capabilities of different animals in different landscapes and conditions. They further indicate the usefulness of spatially explicit individual-based simulation models for recognizing consistent patterns that emerge, despite uncertainty regarding animals’ movement behavior. Altogether, this study demonstrates the need to extend the concept of ‘perceptual ranges’ beyond patch detection processes, to encompass the wide range of elements that can direct animal movements during dispersal through heterogeneous landscapes.     

Establishment patterns in a secondary tree line ecotone

On semi-open pre-alpine fen pastures Alder encroachment creates a dynamic mosaic of grassland and woodland, which is rich in ecotones from fen to Carr. The structural diversity in colonisation patterns of Alder on fens suggests a dependency on multiple environmental drivers. Unidirectional progressive ecotone development provides an opportunity to address a current deficit in understanding successional patterns, i.e. process-pattern relationships in a multiple factor regime.We developed an individual-based model of Alder establishment on fen grassland to investigate the dependency of encroachment patterns upon seed production, dispersal distances and safe site availability. The purpose of the model is to provide a causal understanding of establishment patterns of Alder. In the model, all life processes of Alder individuals were parameterised with field data. This allowed us to strictly perform bottom-up simulations and successfully check plausibility by comparing simulated establishment patterns of cohorts with observed ecotone structures.Simulation results show that establishment patterns strongly depend on environmental drivers. Spatial progression of Alder encroachment and width of ecotones, respectively, mainly depend on wind speed during seed dispersal. Dense establishment of Alder leading to community change from fen grassland to Carr, requires windows of opportunity, which are defined by the rare coincidence of widespread dispersal, high seed production and favorable establishment conditions. Life-history traits of Alder (mast year cycle, high seed weight, weak establishment in fen) spatially and temporarily constrain the encroachment process. The structural diversity of long-term encroachment patterns is explained by the event-driven character of encroachment.Modelling individual establishment pathways of seedlings starting from germination revealed an endogenous stochasticity in establishment patterns emerging from low seed densities in the tail of the dispersal function. We conclude an inherent stochastic structure of dispersal-limited tree line ecotones, which limits reconstruction of processes from patterns.In order to describe long-term successional patterns of Alder encroachment at landscape scale, we propose the combination of two concepts: deterministic “patch-movement” of Alder woodland driven by continuous ecotone migration together with rare and stochastic “infiltration” of single Alder trees into open fen grasslands. Conservation management can control predictable “patch-movement” by cutting off maturing saplings around existing Alder woods. But the preservation of the actual large proportion of open grassland in fen pastures from infiltrating Alder seedlings and from the subsequent shift of the pasture to a densely wooded state would require mowing additionally to extensive grazing.     

Using circuit theory to model connectivity in ecology, evolution, and conservation

Connectivity among populations and habitats is important for a wide range of ecological processes. Understanding, preserving, and restoring connectivity in complex landscapes requires connectivity models and metrics that are reliable, efficient, and process based. We introduce a new class of ecological connectivity models based in electrical circuit theory. Although they have been applied in other disciplines, circuit-theoretic connectivity models are new to ecology. They offer distinct advantages over common analytic connectivity models, including a theoretical basis in random walk theory and an ability to evaluate contributions of multiple dispersal pathways. Resistance, current, and voltage calculated across graphs or raster grids can be related to ecological processes (such as individual movement and gene flow) that occur across large population networks or landscapes. Efficient algorithms can quickly solve networks with millions of nodes, or landscapes with millions of raster cells. Here we review basic circuit theory, discuss relationships between circuit and random walk theories, and describe applications in ecology, evolution, and conservation. We provide examples of how circuit models can be used to predict movement patterns and fates of random walkers in complex landscapes and to identify important habitat patches and movement corridors for conservation planning.     

Animal dispersal modelling: Handling landscape features and related animal choices

Animal dispersal in a fragmented landscape depends on the complex interaction between landscape structure and animal behavior. To better understand how individuals disperse, it is important to explicitly represent the properties of organisms and the landscape in which they move. A common approach to modelling dispersal includes representing the landscape as a grid of equal sized cells and then simulating individual movement as a correlated random walk. This approach uses a priori scale of resolution, which limits the representation of all landscape features and how different dispersal abilities are modelled.We develop a vector-based landscape model coupled with an object-oriented model for animal dispersal. In this spatially explicit dispersal model, landscape features are defined based on their geographic and thematic properties and dispersal is modelled through consideration of an organism's behavior, movement rules and searching strategies (such as visual cues). We present the model's underlying concepts, its ability to adequately represent landscape features and provide simulation of dispersal according to different dispersal abilities. We demonstrate the potential of the model by simulating two virtual species in a real Swiss landscape. This illustrates the model's ability to simulate complex dispersal processes and provides information about dispersal such as colonization probability and spatial distribution of the organism's path.     

The effect of patch constellation on the exchange of individuals between habitat-islands

Metapopulation models assume that inter-patch dispersal dominantly depends on distance between patches and the dispersal capability of organisms in question. We used a spatially explicit, individual-based model to investigate the potential effect of patch constellation on the exchange of individuals between patches. We simulated migration of individuals from a start- into a target-patch with both patches having the same size and shape. Simulation experiments were carried out for four patch constellations and two different movement patterns. Our results demonstrate a substantial effect of patch constellation on the exchange of individuals. They also show that the magnitude and even the direction of this effect crucially depends on their movement pattern. We conclude that particularly for highly correlated movement patterns patch shape and constellation can not readily be ignored when modelling inter-patch dispersal between habitat-islands.     

Old models explain new observations of butterfly movement at patch edges

Understanding movement in heterogeneous environments is central to predicting how landscape changes affect animal populations. Several recent studies point out an intriguing and distinctive looping behavior by butterflies at habitat patch edges and hypothesize that this behavior requires a new framework for analyzing animal movement. We show that this looping behavior could be caused by a longstanding movement model, biased correlated random walk, with bias toward habitat patches. The ability of this longstanding model to explain recent observations reinforces the point that butterflies respond to habitat heterogeneity and do not move randomly through heterogeneous environments. We discuss the implications of different movement models for predicting butterfly responses to landscape change, and our rationale for retaining longstanding movement models, rather than developing new modeling frameworks for looping behavior at patch edges.     

An individual-based stochastic hazard model of eastern king prawn (Melicertus plebejus) migration with spatially and temporally varying fishing effort

This paper describes an individual-based stochastic model of eastern king prawn migration along the eastern Australian coast. Migration is treated as one-dimensional diffusion with drift. Capture of a prawn is seen as a failure event driven by movement through a spatially and temporally variable fishing mortality hazard. This hazard is combined with a uniform natural mortality hazard. We use a Monte Carlo method to estimate parameters by comparing expected numbers of tag-returns predicted from the model with previously published tag-release data. As the previous study used a discrete compartmental model, with compartments corresponding to zones of constant fishing effort, we used the same zones and fishing effort in our comparison. The marginal distribution of yield in each zone per single recruit is determined, providing more information compared with the deterministic approach to yield-per-recruit. Using our model we also derive the constant fishing mortality rate equivalent to a spatially variable fishing mortality rate in its impact on the proportion of prawns surviving the migration to reach spawning grounds. Determination of this proportion could contribute significantly to a sustainability assessment of the fishery. It is demonstrated using the AIC that better fits to the data of the previous study and greater parsimony are obtained using our model than were found in the deterministic compartmental analysis of that study. This improvement results from the ability of our model to account separately for average speed of movement and average dispersal rate, whereas in the previous deterministic compartmental model, movement is governed by just one parameter. Our individual-based model includes a parameter that explicitly accounts for dispersal of prawns in migration, so it can distinguish between speed effects and dispersal effects in the data. It also models both types of mortality as processes distinct from those of movement. This enables it to better separate movement and mortality effects compared to the compartmental approach, in which movement and mortality are treated as similar departure processes from a compartment. This separation reduces confounding of movement and mortality effects when parameters are estimated.     

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.     

Unveiling human-assisted dispersal mechanisms in invasive alien insects: Integration of spatial stochastic simulation and phenology models

Capturing the spread of biological invasions in heterogeneous landscapes is a complex modelling task where information on both dispersal and population dynamics needs to be integrated. Spatial stochastic simulation and phenology models have rarely been combined to assist in the study of human-assisted long-distance dispersal events.Here we develop a process-based spatially explicit landscape-extent simulation model that considers the spread and detection of invasive insects. Natural and human-assisted dispersal mechanisms are modelled with an individual-based approach using negative exponential and negative power law dispersal kernels and gravity models. The model incorporates a phenology sub-model that uses daily temperature grids for the prediction and timing of the population dynamics in each habitat patch. The model was applied to the study of the invasion by the important maize pest western corn rootworm (WCR) Diabrotica virgifera ssp. virgifera in Europe. We parameterized and validated the model using maximum likelihood and simulation methods from the historical invasion of WCR in Austria.WCR was found to follow stratified dispersal where international transport networks in the Danube basin played a key role in the occurrence of long-distance dispersal events. Detection measures were found to be effective and altitude had a significant effect on limiting the spread of WCR. Spatial stochastic simulation combined with phenology models, maximum likelihood methods and predicted versus observed regression showed a high degree of flexibility that captured the salient features of WCR spread in Austria. This modelling approach is useful because it allows to fully exploit and the often limited and heterogeneous information available regarding the population dynamics and dispersal of alien invasive insects.     

The effect on dispersal from complex correlations in small-scale movement

Calculations of large-scale displacement distances were made to evaluate the combined effect of small-scale movement pattern of a Collembola, Protaphorura armata. The effect of presence of food and conspecific density on turning angle, step length and activity/motility was investigated. Calculations of net square displacement were made both by assuming correlated random walk (CRW) and by resampling data to account for correlation structures in movement patterns that violate the assumptions of CRW.     

Modelling pheromone anemotaxis for biosecurity surveillance: Moth movement patterns reveal a downwind component of anemotaxis

Insect pheromone traps are becoming an increasingly important tool in biosecurity and pest surveillance, alerting managers to the presence of unwanted organisms. To expand the role of these traps beyond their present sentinel role, it is necessary to develop reliable operational models of local insect dispersal. Following the detection of an insect incursion using a pheromone trap, such models could simulate the dispersal of the insect from its emergence site to the point of detection, enabling biosecurity managers to estimate the most likely proximal source of the incursion. An individual-based moth movement model was developed to simulate observed patterns of moth movement in response to the presence or absence of a pheromone. Using parameters derived from a genetic algorithm, it was possible to fit a model based on the three behavioural components (upwind, upwind with zigzags and casting) described in insect anemotaxis theory to a subset of observed movement patterns (0-135° to the wind), but not to the whole spectrum of movement patterns. It appears that current insect anemotaxis theory is missing a downwind flight component. Whilst the frequency of downwind movements is small; their ground speed could lead to significant downwind displacement, having a disproportionately strong influence on a moth movement model, and hence projections of the likely source or target locations.     

Modelling the effect of in-stream and overland dispersal on gene flow in river networks

Modelling gene flow across natural landscapes is a current challenge of population genetics. Models are essential to make clear predictions about conditions that cause genetic differentiation or maintain connectivity between populations. River networks are a special case of landscape matrix. They represent stretches of habitat connected according to a branching pattern where dispersal is usually limited to upstream or downstream movements. Because of their peculiar topology, and the increasing concern about conservation issues in hydrosystems, there has been a recent revival of interest in modelling dispersal in river networks. Network complexity has been shown to influence global population differentiation. However, geometric characteristics are likely to interact with the way individuals move across space. Studies have focused on in-stream movements. None of the work published so far took into consideration the ability of many species to disperse overland between branches of the same network though. We predicted that the relative contribution of these two dispersal modalities (in-stream and overland) would affect the overall genetic structure. We simulated dispersal in synthetic river networks using an individual-based model. We tested the effect of dispersal modalities, i.e. the ratio of overland/in-stream dispersal, and two geometric parameters, bifurcation angle between branches and network complexity. Data revealed that if geometrical parameters affected population differentiation, dispersal parameters had the strongest effect. Interestingly, we observed a quadratic relationship between p the proportion of overland dispersers and population differentiation. We interpret this U-shape pattern as a balance between isolation by distance caused by in-stream movements at low values of p and intense migrant exchanges within the same branching unit at high values of p. Our study is the first attempt to model out-of-network movements. It clearly shows that both geometric and dispersal parameters interact. Both should be taken into consideration in order to refine predictions about dispersal and gene flow in river network.     

Group navigation and the "many-wrongs principle" in models of animal movement

Traditional studies of animal navigation over both long and short distances have usually considered the orientation ability of the individual only, without reference to the implications of group membership. However, recent work has suggested that being in a group can significantly improve the ability of an individual to align toward and reach a target direction or point, even when all group members have limited navigational ability and there are no leaders. This effect is known as the "many-wrongs principle" since the large number of individual navigational errors across the group are suppressed by interactions and group cohesion. In this paper, we simulate the many-wrongs principle using a simple individual-based model of movement based on a biased random walk that includes group interactions. We study the ability of the group as a whole to reach a target given different levels of individual navigation error, group size, interaction radius, and environmental turbulence. In scenarios with low levels of environmental turbulence, simulation results demonstrate a navigational benefit from group membership, particularly for small group sizes. In contrast, when movement takes place in a highly turbulent environment, simulation results suggest that the best strategy is to navigate as individuals rather than as a group.     

Nonrandom movement behavior at habitat boundaries in two butterfly species: implications for dispersal

We observed meadow brown (Maniola jurtina) and gatekeeper (Pyronia tithonus) butterflies at habitat boundaries and observed spontaneous movements out of suitable habitat in order to investigate such movements in relation to dispersal. We found that butterflies of both species were aware of the position of a highly permeable habitat boundary without needing to cross it. Nevertheless, a considerable proportion of butterflies close to the boundary left their habitat (25-43%). Butterflies that crossed the boundary, and moved substantial distances into unsuitable habitat (up to 350 m in M. jurtina and 70 m in P. tithonus), usually returned to their original habitat patch (98-100%). Movement trajectories, at least in M. jurtina, were significantly different from, and more directed and systematic than, a correlated random walk. Approximately 70-80% of spontaneous movements into unsuitable habitat in both species were "foray" loops comparable to those described in mammals and birds. We conclude that, since migrants seemed to have considerable control over leaving their patch and over their subsequent movement trajectories, chance encounter rates with habitat boundaries, and indeed habitat leaving rates, might be less crucial in determining dispersal rates than is usually assumed. In addition, random dispersal trajectories should not be taken for granted in population or evolution models.     

The concepts of emergent and collective properties in individual-based models—Summary and outlook of the Bornhöved case studies

Ecology requires the conceptual and technical ability to analyse complex and dynamic systems consisting of a high and variable number of components and relations. These components are part of a variable interaction structure in a spatially heterogeneous context. The components of ecological interaction networks can give rise to self-organised, and scale-dependent interaction patterns and processes, which are the underlying causes of the overall ecological systems states.The individual-based modelling approach provides a widely applicable simulation framework based on a ‘hierarchy theory’ view of ecological systems.Here, we summarise and generalise the theoretical implications of the modelling studies presented in this volume in the field of terrestrial and aquatic, animal and plant ecology. The case studies cover a representative profile of processes related to ecological applications, such as food web interactions, population dynamics, dispersal, energy physiology, nutrient allocation and mutual impact of morphological and physiological development. The generic approach applied in this context allows a hierarchical representation of ecological systems and their components. Model results are obtained as self-organised structural relation networks and as aggregated quantitative states. In order to address different model characteristics we distinguish collective and emergent properties. Collective properties are those that are attributed equally to different organisation levels of the system. Emergent properties result from the activities of lower level entities on a higher organisation level, while not being present on the lower level. They can be subdivided into aggregational and connective properties. Emergent properties that are aggregational are those which emerge as a result of an aggregation procedure by an observer on the higher level which does not make sense or is not applicable on lower levels. Emergent properties that are connective, however, are based on an interaction network of lower level entities, which brings about the specific system characteristic.This classification of model results will allow to generalise the achievements and potential of the individual-based modelling approach in ecology.     

Diffusion about the mean drift location in a biased random walk

Random walks are used to model movement in a wide variety of contexts: from the movement of cells undergoing chemotaxis to the migration of animals. In a two-dimensional biased random walk, the diffusion about the mean drift position is entirely dependent on the moments of the angular distribution used to determine the movement direction at each step. Here we consider biased random walks using several different angular distributions and derive expressions for the diffusion coefficients in each direction based on either a fixed or variable movement speed, and we use these to generate a probability density function for the long-time spatial distribution. We demonstrate how diffusion is typically anisotropic around the mean drift position and illustrate these theoretical results using computer simulations. We relate these results to earlier studies of swimming microorganisms and explain how the results can be generalized to other types of animal movement.     

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.     

Probabilistic movement model with emigration simulates movements of deer in Nebraska, 1990–2006

Movements of deer can affect population dynamics, spatial redistribution, and transmission and spread of diseases. Our goal was to model the movement of deer in Nebraska in an attempt to predict the potential for spread of chronic wasting disease (CWD) into eastern Nebraska. We collared and radio-tracked >600 white-tailed deer (Odocoileus virginianus) and mule deer (Odocoileus hemionus) in Nebraska during 1990–2006. We observed large displacements (>10 km) for both species and sexes of deer, including migrations up to 100 km and dispersals up to 50 km. Average distance traveled between successive daily locations was 166 m for male and 173 for female deer in eastern Nebraska, and 427 m for male and 459 for female deer in western Nebraska. Average daily displacement from initial capture point was 10 m for male and 14 m for female deer in eastern Nebraska, and 27 m for male and 28 m for female deer in western Nebraska. We used these data on naturally occurring movements to create and test 6 individual-based models of movement for white-tailed deer and mule deer in Nebraska, including models that incorporated sampling from empirical distributions of movement lengths and turn angles (DIST), correlated random walks (CRW), home point fidelity (FOCUS), shifting home point (SHIFT), probabilistic movement acceptance (MOVE), and probabilistic movement with emigration (MOVEwEMI). We created models in sequence in an attempt to account for the shortcomings of the previous model(s). We used the Kolmogrov–Smirnov goodness-of-fit test to verify improvement of simulated annual displacement distributions to empirical displacement distributions. The best-fit model (D = 0.07 and 0.08 for eastern and western Nebraska, respectively) included a probabilistic movement chance with emigration (MOVEwEMI) and resulted in an optimal daily movement length of 350 m (maximum daily movement length of 2800 m for emigrators) for eastern Nebraska and 370 m (maximum of 2960 m) for western Nebraska. The proportion of deer that moved as emigrators was 0.10 and 0.13 for eastern and western Nebraska, respectively. We propose that the observed spread of CWD may be driven by large movements of a small proportion of deer that help to establish a low prevalence of the disease in areas east of the current endemic area. Our movement models will be used in a larger individual-based simulation of movement, survival, and transmission of CWD to help determine future surveillance and management actions.