Linking movement behaviour, dispersal and population processes: is individual variation a key?

Authors


Correspondence author. E-mail: colin2.hawkes@uwe.ac.uk

Summary

  • 1Movement behaviour has become increasingly important in dispersal ecology and dispersal is central to the development of spatially explicit population ecology. The ways in which the elements have been brought together are reviewed with particular emphasis on dispersal distance distributions and the value of mechanistic models.
  • 2There is a continuous range of movement behaviours and in some species, dispersal is a clearly delineated event but not in others. The biological complexities restrict conclusions to high-level generalizations but there may be principles that are common to dispersal and other movements.
  • 3Random walk and diffusion models when appropriately elaborated can provide an understanding of dispersal distance relationships on spatial and temporal scales relevant to dispersal. Leptokurtosis in the relationships may be the result of a combination of factors including population heterogeneity, correlation, landscape features, time integration and density dependence. The inclusion in diffusion models of individual variation appears to be a useful elaboration. The limitations of the negative exponential and other phenomenological models are discussed.
  • 4The dynamics of metapopulation models are sensitive to what appears to be small differences in the assumptions about dispersal. In order to represent dispersal realistically in population models, it is suggested that phenomenological models should be replaced by those based on movement behaviour incorporating individual variation.
  • 5The conclusions are presented as a set of candidate principles for evaluation. The main features of the principles are that uncorrelated or correlated random walk, not linear movement, is expected where the directions of habitat patches are unpredictable and more complex behaviour when organisms have the ability to orientate or navigate. Individuals within populations vary in their movement behaviour and dispersal; part of this variation is a product of random elements in movement behaviour and some of it is heritable. Local and metapopulation dynamics are influenced by population heterogeneity in dispersal characteristics and heritable changes in dispersal propensity occur on time-scales short enough to impact population dynamics.

Introduction

The linking of movement behaviour with the quantitative description of dispersal has become increasingly important in the development of dispersal ecology. In parallel, dispersal has taken a key position in the development of spatially explicit theoretical concepts and models of populations. This review explores the ways in which the different elements have been brought together and how mechanistic analysis has aided understanding. The emphasis is on the distribution of dispersal distances and how random walk models have and could explain the observed distributions. The evolution of dispersal distances is considered in relation to landscape dynamics and the consequences for population dynamics.

The development of metapopulation ecology has been dependent upon a move away from the simple and unrealistic assumptions about dispersal in classical metapopulation theory and has required the quantitative description of dispersal in relation to landscape features and dynamics (Hanski 1999). The way in which dispersal has been incorporated in metapopulation models has however remained relatively simple and despite or possibly because of that, metapopulation thinking has made major contributions to the development of ecology and its applications, particularly in conservation biology. The substantial body of work on spatially realistic models was initiated by Hanski (1994) with the development of the incidence function model. Dispersal was represented by an emigration rate parameter and a dispersal distance distribution based on the negative exponential. The Virtual Migration (VM) model (Hanski, Alho & Moilanen 2000) has continued to utilize these features and has been used to estimate survival and dispersal parameters from empirical data in butterflies (Petit et al. 2001; Wahlberg et al. 2002; Schtickzelle & Baguette 2003; Mennechez et al. 2004; Wang et al. 2004; Cizek & Konvicka 2005; Keyghobadi et al. 2005; Baguette & Van Dyck 2007) and the red milkweed beetle Tetraopes tetraophthalmus, (Matter 2006) and the connectivity measure of the model has been applied to the chinook salmon Oncorhynchus tshawytscha, (Isaak et al. 2007). Stochastic simulation of metapopulation dynamics and evolution has also used the negative exponential as a dispersal function (Ronce, Perret & Olivieri 2000; Murrell, Travis & Dytham 2002; Hanski & Heino 2003; Hanski & Ovaskainen 2003; Ovaskainen & Hanski 2003; Travis 2003; Moilanen 2004; Ozgul et al. 2006) although more recently, diffusion approximations of random walk have been used (Ovaskainen 2004; Hanski, Saastamoinen & Ovaskainen 2006).

The derivation of the negative exponential was based on the assumption that movement is linear (Buechner 1987) but the use has been as a phenomenological model without test of the assumption. The list of species where the negative exponential underestimates frequencies in the tail and sometimes overestimates at short distances is growing (Buechner 1987; Hill, Thomas & Lewis 1996; Kot, Lewis & van den Driessche 1996; Thomas & Hanski 1997; Baguette, Petit & Quéva 2000; Roslin 2000; Baguette 2003; Kuras et al. 2003). Chapman, Dytham & Oxford (2007a) concluded on the basis of goodness-of-fit that the negative exponential should not be considered the function of automatic choice in describing distance distributions and modelling metapopulations. A more general criticism has been made of the way in which metapopulation theory has developed by Bowler & Benton (2005) who concluded that dispersal has been treated too simplistically and that emigration, interpatch movement and immigration should be considered as a series of condition-dependent processes. They also concluded that dispersal cannot be collapsed into a single parameter derived from a simple function. Given the centrality of dispersal in metapopulation models, unsurprisingly the dispersal function has been found to affect dynamics (Casagrandi & Gatto 2006; Heinz, Wissel & Frank 2006).

The debate about the way in which dispersal has been considered in metapopulation theory chimes with a more general debate about the limitations of phenomenological models and the value of mechanistic ones. Turchin (1998) and Wein (2001) concluded that an understanding of the fundamentals of dispersal required the study of movement behaviour. The value of mechanistic over phenomenological models is that they may permit the comparison of different concepts of dispersal processes and the rejection of inappropriate models on the basis of data collected in the field (Turchin 1998). Mechanistic home-range analysis was developed by Moorcroft & Lewis (2006) because phenomenological models have limited predictive capability. In developing an understanding of what causes oscillations in populations, Turchin (2003) used a combination of mechanistic and phenomenological approaches.

There is a need for quantitative descriptions of dispersal which fit distance data well and incorporate realistic assumptions about the underlying movement behaviour. Much work has been carried out on behaviourally based analytical models of movement (Skellam 1951a; Okubo 1980; Turchin 1998; Okubo & Levin 2001) which potentially could bridge the gap between behaviour and the quantitative description of dispersal. The starting point of this review is that an understanding of movement behaviour can improve the way in which dispersal is described and increase understanding of process. Mechanistic approaches may not only inform dispersal ecology but also facilitate the linkage of dispersal and population dynamics.

The review starts with a clarification of what is usefully included under the heading of dispersal. Random walk models are then described and their potential value in explaining distance distributions assessed. The causes of leptokurtosis in distance data is explored as a means of establishing how to elaborate diffusion models to provide better descriptions of dispersal. Some current models of distance distributions are then compared and conclusions drawn about which models have potential to provide new insights about dispersal ecology and population processes. The possible outcomes of individual variation in dispersal propensity are considered in the context of the evolution of dispersal and population dynamics. In conclusion, a set of candidate principles for dispersal ecology are proposed and suggestions made for the further development of research.

What is usefully labelled dispersal?

There is increasing clarity about which behaviours and ecological events are usefully labelled dispersal while there remain some issues and inconsistencies. Movements between habitat patches and breeding sites are now generally described as dispersal and are considered to be different from movements within patches or foraging areas. Within a metapopulation, habitat patches support local populations and dispersal occurs between patches across a nonhabitat matrix (Hanski 1999). In vertebrate ecology, metapopulation concepts may not be used explicitly and there are some differences in approach (South et al. 2002). In particular, some vertebrates are considered to have a breeding area supporting a breeding colony (e.g. Austin, Brown & McMillan 2004; Matthiopoulos et al. 2004; Pinaud & Weimerskirch 2007), while in others the basic spatial breeding unit is considered to be the home range supporting a breeding pair and siblings (e.g. Lande 1988; Winkler et al. 2005). The differences between the approaches do not indicate the need for different conceptual frameworks. This is well exemplified by Lande's (1988) work on the development of metapopulation models for the northern spotted owl Strix occidentalis caurina where the smallest population unit was the breeding pair instead of the local population and the home range was the habitat patch. The implication is that a habitat patch may be defined in a number of ways on the basis that it is useful and informative to do so.

Dispersal had been conceptualized as an event which occurs largely as a juvenile or early in the life of an adult and which if successful results in a move from the natal to a new patch or site (Johnson 1969; Greenwood & Harvey 1982). This framework may apply to some animals but there is variation between species. In some vertebrates, dispersal may be open ended and spatially ill defined (Boyd 2002) and home ranges may gradually change in size and shift across the landscape (e.g. Doncaster & Macdonald 1991; Gautestad & Mysterud 1995; Walls & Kenward 1998). European hedgehogs Erinaceus euroapeus have no clearly defined dispersal phase, do not defend a feeding territory (Reeve 1994) and do not have a persistent breeding site (Doncaster, Rondinini & Johnson 2001). For some vertebrates, dispersal cannot be associated in a simple way with movement between clearly delineated spatial units.

A parallel can be drawn with insects as shown for example by crucifer feeders. Cabbage aphid Brevicoryne brassicae is polymorphic and the production of alates is conditional upon high conspecific density and/or low quality of the host plant (Hughes 1963). Alates are obligatory dispersers and engage once in a high-altitude flight soon after developing into the winged form. In the cabbage root fly Delia radicum, females may move away from the natal habitat patch because they only become responsive to the volatiles of the larval host plant when mated and gravid (Hawkes 1975; Hawkes & Coaker 1979). After oviposition, females again become unresponsive to the host plant, may move away from the first habitat patch encountered and may need to relocate a patch if a second batch of eggs is laid. Males do not respond to host plants and may move away from the natal patch and feed and mate in the matrix. In the cabbage butterfly Pieris rapae, females move throughout their lives and can lay eggs among many host plants and patches (Jones et al. 1980). They do not appear to detect host plants except at short distances using visual cues (Bukovinszky et al. 2005) and readily move away from host plants (Jones1977; Fahrig & Paloheimo 1987). For species like P. rapae, there is no clearly defined phase of dispersal within the life of the adult and the concept of the habitat patch may not be meaningful.

Movement during dispersal has been found or assumed to be qualitatively and quantitatively different to that associated with activities such as foraging, mate finding and location of refuges within the habitat patch. In particular, it has been expected that movement in the matrix may be faster and more direct than in the patch (see reviews, Van Dyck & Baguette 2005; Fahrig 2007). Some studies have confirmed this expectation but there appears to be considerable variation between species as some examples show. The movements of the bog fritillary butterfly Proclossiana eunomia (Schtickzelle et al. 2007) and the tansy leaf beetle Chrysolina graminis (Chapman, Dytham & Oxford 2007b) were found to be straighter and longer in the matrix than in habitat patch. Morales et al. (2004) identified two types of movement in the elk Cervus canadensis with short step lengths and large turning angles in areas where forage could be obtained and with large step lengths and small turning angles in other habitats. When released at the border between suitable habitat and a crop field, the bush cricket Platycleis albopunctata did not show a consistent preference for the suitable habitat and movement in the matrix was not always different from that in the habitat (Hein et al. 2003). There were no significant differences in turn angle, step length and dispersal rate of the alpine butterfly Parnassius smintheus when comparing suitable and unsuitable meadows (Fownes & Roland 2002). For some species, dispersal may not be as behaviourally distinct as originally hypothesized.

In behavioural and spatial terms, dispersal is sharply differentiated from foraging and patch-based activities in some species but not in others. This makes the theoretical framework for movement behaviour and dispersal more complex and the creation of general principles more difficult. However, Dennis, Shreeve & Van Dyck (2003) have questioned the validity of the metapopulation concept of habitat patch and matrix on the basis of butterfly biology. They consider that habitat and matrix are not clearly delineated and that the practice of defining the habitat patch on the basis of larval host plant is too narrow. It is concluded that the habitat should be considered to be an area comprising a set of resources, consumables and utilities and that a resource-based approach is needed to avoid an unrealistic view of the nature and dynamics of environments and populations. It is not possible to reject the central tenet of the Dennis et al. argument and greater complexity may be required and with it much more data. However, given the biological complexity it may only be possible to make high-level generalizations about movement behaviour, dispersal and population processes.

The debate about what is meant by dispersal echoes the debate about migration which ended in a reduction of the use of the term except for seasonal migrations, e.g. between breeding site and wintering area. Baker (1978) came to the view that definitions of migration based on habitat or behaviour were unsatisfactory and considered that there was a continuous range of movement patterns which could be understood within a common set of evolutionary principles. As discussed above there are, at least in some animals, important discontinuities in movement behaviour but in seeking generalizations, there may be value in recognizing that some of the principles underlying dispersal may be common to all movement behaviours.

Random walk and diffusion models

Random walk concepts, the associated diffusion equations and approximate analytical solutions to those equations have been used in a variety of biological contexts (Codling, Plank & Benhamou 2008). The derivation of diffusion equations from random walk models of movement behaviour and their use in the quantification of dispersal is also described by Turchin (1998) and Okubo & Levin (2001). The early development of models of dispersal assumed random walk behaviour (Skellam 1951a). The simplest form is an uncorrelated random walk in which movement is described as a series of straight line steps between which are random turns in direction. The assumption was not based on a detailed analysis of movement behaviour but on mark and recapture experiments which showed patterns indicative of uncorrelated random walk including dispersal in all directions, maximum density remaining at the release point and progressive declines in the rates of dispersal with time after release. Skellam showed that uncorrelated random walk movement would lead to a density-distance relationship which could be approximated by the Gaussian curve thus establishing a link between behaviour and distance distributions.

This opens the possibility that movement behaviour could be inferred from density or frequency distance relationships but there are limitations to this approach. Van Dyck & Baguette (2005) suggest that mark and release experiments may underestimate dispersal and the tails of distributions particularly where there is individual variation in dispersal propensity. Where there is good agreement between a distance distribution and the predictions of a behavioural model, it cannot be assumed that there are no alternative behaviours which could produce the same outcome (Turchin 1998). A comparison between distance distributions and model predictions can be useful in rejecting inappropriate models.

There are therefore a number of ways of assessing whether dispersal can be described using a particular behavioural model. In addition to the direct observation of movement, distance distributions and other features of dispersal can been used. The simple diffusion equation for movement in two dimensions yields the following form of the Gaussian (half-normal) curve (Turchin 1998):

u(r, t) = (1/4πDt) exp(–r2/4Dt)((eqn 1))

where u(r, t) is the probability density at r, distance from the release point or natal patch, at t, time from release or start of movement and D, diffusion coefficient (see Fig. 1). The expected frequency distribution of dispersal distances can be obtained from the cumulative probability distribution derived from equation 1 (Inoue 1978):

Figure 1.

Comparisons of the density- (a) and (c), and frequency- (b) and (d), distance relationships for the Gaussian (half-normal) curve (solid lines) and diffusion with heterogeneity model (Skellam 1951b) and the negative exponential (dashed lines). Gaussian, D = 40 distance unit2 time unit−1, t = 1 time unit; diffusion with heterogeneity, α = 0·33, λ = 11; negative exponential, k = 0·1, density →∞ as r → 0. Parameters chosen so that the distance within which 50% of a population of 1000 individuals remains is coincident for all models.

image((eqn 2))

where p is the proportion of the population within the distance rp (see Fig. 1).

Uncorrelated random walk or simple diffusion has been observed or deduced in a range of taxonomic groups, the sea urchin Strongylocentrotus droebachiensis (Dumont, Himmelman & Robinson 2007), insects (Ito & Miyashita 1965; Rogers 1977; Kareiva 1983; Cronin, Hyland & Abrahamson 2001; Reeve, Cronin & Haynes 2008), the northern spotted owl S. o. caurina in part of juvenile dispersal (Turchin 1998) and the European hedgehog E. euroapeus in favourable areas (Doncaster et al. 2001). However, field studies have generally yielded distributions with densities or frequencies higher than expected close to source and in the tail and lower than expected at intermediate distances, i.e. leptokurtic distributions (Okubo & Levin 2001). The movement behaviour most commonly found by direct observation is correlated random walk (CRW) in which the direction of one step is correlated with that of the previous step or steps. Insect examples of CRW include P. rapae (Root & Kareiva 1984), P. eunomia, (Schtickzelle et al. 2007) and C. graminis (Chapman et al. 2007b).

The movement behaviour of vertebrates during dispersal is a neglected area (Andreassen, Stenseth & Ims 2002), but there have been a number of detailed studies of movement which may be useful in understanding dispersal. During the dispersal of the buzzard Buteo buteo, there may be a series of transition movements to separate home ranges before arrival at what becomes the breeding site (Walls & Kenward 1998; Kenward, Walls & Hodder 2001). The movement between home ranges is de facto a part of the dispersal process and contributes to the redistribution. The study of movement of animals between foraging areas can therefore be informative about some aspects of dispersal. Morales et al. (2004) described elk C. canadensis behaviour as a mixture of two CRWs, one termed ‘encampment’ with short step lengths and large turning angles, the other ‘exploratory’ with large step lengths and small turning angles. The slower movement was associated with open habitats in which elk could obtain forage and faster movement with a variety of habitats. Haydon et al. (2008) further developed this approach by elaborating the two CRWs model to include the effects of social interactions. Johnson et al. (2002) identified small- and large-scale movements from movement paths of woodland caribou Rangifer tarandus caribou and associated the small-scale movements with foraging (but see also Nams 2006 and Johnson et al. 2006).

Austin et al. (2004) have found an additional important feature in the movement of the grey seal Halichoerus grypus that is persistent individual variation in behaviour. Using satellite telemetry, the positions of adult seals were determined at 1 or 2-day intervals over 3–10 months and modelled using CRW. It was concluded that the population studied was not homogeneous and individuals were described as directed movers, some of which did not return to the breeding site throughout the study, residents and an intermediate group. The directions of the movements of directed movers were positively correlated while those of residents were biased and negatively correlated, i.e. there was a tendency for reversals in direction. It appears that some intermediate individuals were performing an uncorrelated random walk.

The derivation of distance distributions for CRW has no general analytical solution (Turchin 1998). Kareiva & Shigesada (1983) produced a relationship for mean squared distance dispersed and time or number of steps which can be derived from CRW movement parameters. The use of the mean squared distance statistic is, however, a blunt tool as information on the distribution of dispersal distances is not used in the analysis. An important feature of CRW is that the frequency with which the movement path is sampled is an important determinant of the strength of correlation observed between movement directions. If an animal moves in a series of straight-line steps, sampling during a step produces strong correlation between directions even if there are random turns between steps. Such over-sampling is unproductive in describing movement (Turchin 1998). As sampling is carried out less frequently or after a large number of steps, CRW increasingly comes to resemble uncorrelated random walk (Okubo 1980). That is relevant in exploring the implications of the correlation identified in movement behaviour because CRW may generate distance distributions similar to or the same as those of uncorrelated random walk over time-scales relevant to dispersal. In P. rapae, CRW or biased movement has been observed (Jones 1977; Root & Kareiva 1984), but the preferred direction during 1 day changes randomly from day to day (Jones et al. 1980) and uncorrelated random walk is an adequate description on large spatial and temporal scales.

There is another feature of movement behaviour which has important consequences for distance distributions and that is bias in direction. This may be the result of a behavioural response to an environmental stimulus or the movement of the medium, air or water, in which the animal is moving. Biased random walks have been modelled using the advection-diffusion equation (Okubo 1980; Turchin 1998) and the solution of the equation yields a density-distance curve of the same shape as that of equation 1 but with the whole distribution moving progressively from the source. The dispersal of freshwater invertebrates and fish has been modelled using this framework (e.g. Skalski & Gilliam 2000; Petty & Grossman 2004).

Moorcroft, Lewis & Crabtree (2006) have shown in the coyote Canis latrans that even when landscape heterogeneity and social cues influence behaviour, the complexities of movement within a home range can be modelled successfully by combining correlated random walk and advection-diffusion.

Why leptokurtosis?

This section considers the factors that could give rise to leptokurtic relationships and first assesses the role of movement behaviours. The general finding that there is correlation between the movement directions of an individual clearly infringes the assumptions underpinning the half-normal but does not provide an explanation of leptokurtic distance distributions. On the contrary, CRW can generate hollow distributions when there is strong positive correlation between movement directions and in less extreme cases Holmes (1993) derived platykurtic distributions. Where correlation is not strong or movement involves a large number of steps, the relationship may be described approximately by equation 1.

A number of studies have demonstrated how individual variation could contribute to the leptokurtic distributions observed in movement and dispersal studies. Yamamura (2002) and Hapca, Crawford & Young (2009) found continuous variation while bimodal variation has been assumed or demonstrated (Dobzhansky & Wright 1943; Inoue 1978; Skalski & Gilliam 2000; Rodríguez 2002; Egli & Babcock 2004; Winkler et al. 2005; Reeve et al. 2008). Given the potential importance of individual variation for the evolution of dispersal, there has been little development of analytical solutions of diffusion equations incorporating population heterogeneity (Turchin 1998; Okubo & Levin 2001). In a paper that appears to have gone largely unnoticed, Skellam (1951b) developed a model for continuous variation which has an exactly equivalent basis to that of equation 1. The assumptions that match those of Skellam's model are that the diffusion coefficient, Di, is an individual rather than a population characteristic and 1/Di is a gamma variate. Yamamura (2002) and Skalski & Gilliam (2003) modelled variation between individuals by assuming that the time spent in dispersal states varies. Using a CRW model, Hapca et al. (2009) derived diffusion coefficients for individuals and described the variation using a gamma distribution. Bimodal variation was modelled by Inoue (1978) by assuming different D values for each of the subpopulations.

The effect of population heterogeneity on density and frequency distributions is illustrated (see Fig. 1) using Skellam's model:

u(r) = (α/πλ)/(1 +r2/λ)(α+1)((eqn 3))

where α is the shape and λ is the rate parameter of the gamma distribution and other symbols are as equation 1.

A cumulative probability distribution was also derived by Skellam, a modified form of which is:

image((eqn 4))

where symbols are as equations 2 and 3.

A number of methods have been used to model bimodal variation (see Yamamura et al. 2007). A study of the tree swallow Tachycineta bicolor (Winkler et al. 2005) provides an interesting example because of the way in which seasonal migration is separated from dispersal between breeding sites. The swallow shows very high breeding site fidelity despite the long-distance migration but dispersal distances from the natal site to the first breeding site are relatively short. The distribution of dispersal distances was best described by a half Cautchy distribution, that is a distribution combining two normal distributions.

It is clear from the above that distance distributions have the potential to provide estimates of population heterogeneity. Random walk leads to substantial variation in the distances dispersed by individuals in homogeneous populations and it is the additional variation that could provide estimates of heterogeneity. However, there are a number of other factors that could cause leptokurtosis or moderate it complicating the estimation of heterogeneity.

Skellam (1951b) considered how changes in the behaviour of individuals during dispersal could moderate leptokurtosis and concluded that if heterogeneity declined, leptokurtosis would also decline and could disappear if individual differences were smoothed out over time. Yamamura (2004) and Zhang et al. (2007) found that leptokurtosis occurred when there was variation within individuals in random walk parameters. In mark and recapture experiments where traps are used, estimates of density are being made while distributions are changing. This time integration can produce a leptokurtic relationship (Hawkes 1972). The behaviour at habitat boundaries can also produce leptokurtosis as shown in a simulation model (Morales 2002). Landscape features are likely to have a major effect and the clustering of patches and the associated increased probability of locating and settling in a habitat patch close to the natal patch could increase leptokurtosis (Turchin 1998). Density-dependent dispersal could produce platykurtic distributions if it results in relatively high rates of dispersal close to source. Observed distance distributions and the extent of leptokurtosis are potentially the result of the combined effects of many factors and processes including population heterogeneity, correlation, landscape features, time integration and density dependence. Some knowledge of movement behaviour is essential if the effects of these factors are to be identified, heterogeneity estimated and dispersal understood.

Interpretation of dispersal distance relationships

A more comprehensive interpretation of dispersal distance relationships can now be attempted. When traps are used in mass mark and release experiments, density-distance relationships have been used in analysis. The phenomenological relationship most commonly used has the general form (Turchin 1998):

u= exp(abrc)((eqn 5))

where u is the number recaptured (assumed to be an estimate of relative density), r, distance from the release point (midpoint of the effective area of a trap), a is determined by numbers released, trapping efficiency and mortality, b is a rate and c a shape parameter. With c = 2, the form of equation 5 is the same as that of equation 1 and it can therefore describe the density-distance relationship generated by uncorrelated random walk, while c < 2 describes a leptokurtic distribution. The form can be used to describe density-distance relationships such as those arising from a random walk and individual variation (Hawkes 1972).

The negative exponential may appear to be related to equation 5 but its derivation and main use has been for frequency distributions of dispersal distances. The expected frequency distribution can be obtained from the cumulative probability distribution (Hill et al. 1996):

p= 1 – exp(–krp)((eqn 6))

where k is a rate parameter and other symbols are as equation 2. When compared to the half-normal distribution it can be described as a leptokurtic relationship (Fig. 1). However, the assumptions underlying the use of the negative exponential to describe frequency distributions of distances are several and specific and not based on diffusion (Buechner 1987). They are that movement is linear (i.e. straight movement progressively away from the starting point) or that the total distance moved is proportional to linear distance and that dispersal leads to settlement, in a home range or patch, at a rate which is constant and proportional to distance moved. Time is not explicitly included in the relationship but time cannot be ignored in its interpretation. If observations take place a part of the way through the dispersal of a sizeable proportion of a dispersing population, a variety of distributions could be expected depending on whether dispersers and settlers can be distinguished. If settlers could be identified there would be too few in the tail and the negative exponential would not fit their frequency distribution.

There is another important feature that is required to obtain the negative exponential and that concerns the distribution of habitat patches. For the probability of settling to be approximately constant in relation to linear distance, the distribution of habitat patches must be uniform or random. The settlement process has been shown to have a key influence on the consequences of dispersal and on spatial distributions (Broadbent & Kendall 1953; Yamamura, Moriya & Tanaka 2003). Stamps, Krishnan & Reid (2005) have explored how habitat patch quality could affect settlement behaviour and considered the consequences of a decline in acceptance threshold after a period of dispersal. The tail of a distance distribution would be reduced by such a decline in comparison with the situation where there was no change in behaviour over time. Some of the interactions between settlement and habitat patch distribution can be inferred from a simulation study (Heinz et al. 2005) in which random walk and more systematic search represented by looping and spiralling were compared. The effect of competition between patches for dispersers can be expected to result in relative increases in the numbers of settlers close to source and may increase leptokurtosis in distance distributions. Heinz et al. also compared the outcomes of the simulations with the expectations of the negative exponential and concluded that the best model differed from the negative exponential in a number of ways.

The problems in the use of the negative exponential can be illustrated with an example, the study of a butterfly, the brown argus Aricia agestis (Wilson & Thomas 2002) which is selected because it is sufficiently comprehensive to provide an understanding of the key features of dispersal. Unusually but informatively, the mean distances dispersed were related to the time from release and for females, distances increased throughout the study period. For males, dispersal distances did not increase with time and it was concluded that they were moving continuously within a home range. The distances involved however were large enough to span more than one habitat patch implying repeated movements between patches. The ability of the negative exponential to describe the data is, therefore, unexpected. The question that then remains is why does the negative exponential give a good description of frequency distributions in such cases?

A comparison of the distributions obtained with the negative exponential and a diffusion with heterogeneity model (Skellam 1951b) shows that there are broad similarities (Fig. 1). The main difference is that the diffusion with heterogeneity model predicts lower frequencies than the negative exponential close to source. In some examples in which the negative exponential provides a poor fit to data, the observed frequencies have been lower than expected close to source (Buechner 1987; Baguette 2003; Kuras et al. 2003). The method used to fit the negative exponential to distance data has been applied to a frequency distribution generated from the diffusion with heterogeneity model (Fig. 2). This demonstrates that the negative exponential and the diffusion with heterogeneity model can be closely coincident. It is possible that a diffusion with heterogeneity model can provide an adequate description of data sets that are well fitted by the negative exponential. This does not imply that these are the only two models that can describe such data well. Random walk models coupled with settlement could also be expected to provide a good description.

Figure 2.

A comparison of the negative exponential and a diffusion with heterogeneity model (Skellam 1951b). Negative exponential (solid line) fitted (for method see Wilson & Thomas 2002) to expected log inverse cumulative proportions (solid circles) derived from the diffusion and heterogeneity model; fitted k = 0·083 ± 0·001; diffusion with heterogeneity, parameters as Fig. 1.

It is now clear from studies of movement behaviour that the assumptions underlying the use of the negative exponential cannot be supported. Movement behaviour over the spatial and temporal scales relevant to dispersal cannot generally be described as linear. In the case of CRW, the strength of correlation would have to be very high and persistent throughout dispersal and sustained even in heterogeneous matrices. Bias could lead readily to linear movements but it has not generally been observed in dispersal and occurs in particular circumstances. The high altitude dispersal flights of aphids could result in linear movement if wind direction is constant. Seasonal migration may be linear where individuals are able to orientate or navigate to maintain a persistent direction of movement.

On the basis of current knowledge, the use of the negative exponential to describe dispersal is no longer sustainable. It should not be replaced by better phenomenological models such as the inverse power law (see Chapman et al. 2007a), but by models based on movement behaviour. Simple diffusion models based on uncorrelated random walk behaviour are not adequate to describe the varied repertoires of behaviours that occur during dispersal in heterogeneous landscapes. However, random walk and diffusion models continue to be useful in the analysis and description of dispersal when appropriately elaborated. One of the elaborations that may in general be necessary for the understanding dispersal is the inclusion of individual variation in dispersal propensity. The following section illustrates how a consideration of heterogeneity could provide opportunities for better linkage of dispersal and population processes.

Consequences of variation in dispersal propensity

The substantial body of work on models of metapopulations has included both the effects of emigration rate and dispersal type and distance on dynamics and viability. The emphasis here will be on models which have explored the effects of the magnitude of dispersal distances and their distributions and the type of movement. Johst, Brandl & Eber (2002) showed that long dispersal distances could increase metapopulation viability using a negative exponential dispersal function. This outcome was not, however, general and was dependent on both the local population growth and patch regeneration rates. No enhancement of persistence by long-distance dispersal was found where patches were aggregated. Etienne et al. (2002) showed that dispersal could have stabilizing effects on metapopulations and also that persistence could be facilitated by population heterogeneity in comparison with a simple diffusion model.

In a model exploring the evolution of dispersal distance using the negative exponential function, Murrell, Travis & Dytham (2002) showed that local dynamics were critical to outcomes. With the cost of dispersal increasing linearly with distance and with stable and damped local dynamics, selection was for short distances but longer distances were favoured as local dynamics became more complex. Hiebeler (2004) showed that the distribution of habitat patches was important in determining whether short or long dispersers were selected but long-distance dispersal was sometimes advantageous even where the aggregation of suitable patches would otherwise favour short-distance dispersal.

The success in the location of habitat patches of random walk, with various degrees of correlation, and of systematic searching, looping and spiralling, has been explored by Zollner & Lima (1999) using simulation. Only under high mortality and low energy reserves in a uniform landscape did linear movement perform better than any random walk. Systematic search was superior to the best-correlated random walk with low mortality risks and high energy reserves. The most successful movement type was generally found to be correlated random walk with relatively high correlation and the simulations did not therefore indicate why some dispersers perform an uncorrelated random walk. Where habitat patches were aggregated and persisted in one location for more than a generation, uncorrelated random walk was found to give the highest success rate (see Hawkes in Feeny 1982).

The implication of this modelling work is that dispersal can be critical to population dynamics and that the evolution of dispersal is in turn dependent on landscape configuration and dynamics. It also shows that a variety of outcomes can be obtained from simulation and that models are sensitive to what appear to be small differences in assumptions. If the use of the negative exponential is to be replaced by more realistic representations of dispersal, as in some recent models (Ovaskainen 2004; Hanski et al. 2006), how could our understanding of metapopulation dynamics and dispersal evolution change? The importance of fatter tails has now been established in understanding the rate of spread of invading species (Shigesada & Kawasaki 2002; but see also Lindstrom et al. 2008). More modelling work will be necessary to determine other consequences of replacing the negative exponential function. A diffusion with heterogeneity model has the potentially important feature that the distance dispersed is not simply a product of dispersal propensity. The expected density, irrespective of the dispersal propensity of individuals, remains highest at source throughout dispersal and this could be expected to have impact in modelling the evolution of dispersal.

Studies of the Glanville fritillary Melitaea cinxia have demonstrated how individual variation in dispersal propensity can be an important part of the dynamics of metapopulations (Hanski et al. 2002; Hanski et al. 2004; Haag et al. 2005; Hanski et al. 2006). At isolated patches, female mobility was high in newly established local populations and low in old populations while at patches with high connectivity, old and new populations did not differ and showed intermediate levels of mobility. The differences in individual dispersal propensity were associated with a specific allele of a flight metabolic enzyme. It was concluded that there were heritable differences between individuals and that variation in dispersal propensity was maintained by landscape dynamics.

It appears that the selection of dispersal traits can be disruptive, operating in different directions simultaneously depending on the history of local populations and position of patches relative to others. This can have impacts on dynamics with for example new populations at isolated patches having relatively high emigration and low growth rates in comparison with old isolated populations. It is concluded that the exploration of the implications of dispersal for population processes will be facilitated by the use of models of dispersal explicitly incorporating heterogeneity.

Why random walk?

In uncorrelated and correlated random walk, the directions in which an animal moves can be described as having a random element. These types of movement result in paths that are not straight and some can be tortuous. This section considers the ultimate cause of random walk and why such movement predominates in dispersal. A simple assumption is made about proximate causes and that is that both intrinsic and extrinsic factors generate the random components of random walk. The evidence presented here demonstrates that dispersal traits can be variable. Indeed it would be surprising to find that populations are homogeneous in all the characteristics that might affect dispersal and reasonable to assume that heterogeneity is ubiquitous. It has also been shown in a range of examples from different taxonomic groups that dispersal traits can be heritable (Roff & Fairbairn 2001).

Fahrig (2007) suggested that the form of movement behaviour during dispersal is affected by habitat type and the quality of the matrix and that with low-quality matrix, linear movement is to be expected. The evidence presented here suggests that, while movement may be straighter in the matrix than in the habitat patch, dispersal cannot be described as linear. Random walk is an adequate description of movement behaviour on the spatial and temporal scales relevant to dispersal. It can result in a relatively efficient search strategy when coupled with step lengths and turning angles that produce an intensive search in the area where resources are concentrated. Short dispersal distances may be advantageous where habitat patches are aggregated (Hiebeler 2004). It is concluded that random walk is an appropriate search strategy when there is a high probability that a habitat patch is close but the direction of the patch from the current location is unpredictable. In this context, ‘close’ is not measured by absolute distance but in relation to the mobility of the animal over the period during which dispersal occurs. Within that specific context, it is clear that searching for a patch may be analogous to searching for resources and that dispersal strategies may have parallels with foraging strategies. When resources have to be found repeatedly, a Lévy distribution, a combination of short steps mixed with less frequent longer steps conforming to an inverse power-law, may provide an optimum strategy for animals performing a random walk and with randomly distributed resources (Viswanathan et al. 1999; Bartumeus et al. 2005). The identification of Lévy patterns in animal movements has, however, proved to be controversial (Benhamou 2007; Edwards et al. 2007; Sims, Righton & Pitchford 2007; Benhamou 2008; Edwards 2008; Reynolds 2008).

There are other possible ultimate reasons for random walk. There may be spreading of risk where there is variation in the distances between habitat patches. Higher relative fitness may be achieved when random walk has a higher mean success rate than that achieved with other forms of movement. In addition, there is the possibility that mortality risk from predation may be reduced when movement paths change unpredictably.

Random walk can result in some areas being searched more than once and there is likely to be selection pressure to improve efficiency. This could be achieved by systematic search or the detection of a habitat patch from distance. Meadow brown butterflies Maniola jurtina perform looping movements which may improve efficiency and move towards habitat patches from at least 125 m away (Conradt et al. 2000). D. radicum orientate and move upwind in the presence of host plant volatiles when downwind of a habitat patch up to 25–40 m away (Hawkes 1974, 1975; Hawkes, Patton & Coaker 1978).

There may also be mechanisms for assessing the suitability of habitat patches before dispersal occurs. Before the natal dispersal of the rabbit Oryctolagus cuniculus, there are periods of extensive movements beyond the foraging area which appear to be exploratory (Richardson et al. 2002). Spatially realistic population models of the Thomson's gazelle Gazella thomsoni thomsoni showed that it could not persist under Serengeti conditions if it dispersed in a random walk and it was concluded that it was able to move selectively towards favourable areas (Fryxell et al. 2005).

Finally, there appears to be an alternative evolutionary response to the occurrence of long distances between habitat patches which involves the avoidance of dispersal in such circumstances. Hawkes, Kowalski & Brindle (1987) concluded that the emergence of D. radicum was delayed in the spring when the distances between habitat patches was high relative to those later in the year.

Conclusions

The conclusions of this review are presented as a set of candidate principles for test, modification and rejection. The principles emphasize those features which may be the most useful in linking movement behaviour, dispersal and population processes and are not intended to capture all the elements that contribute to the proximate and ultimate causes of dispersal. It is intended that they are applied broadly and not confined just to dispersal between habitat patches or breeding sites. Movement from one home range to another is included.

  • 1Movement behaviour and dispersal evolve in relation to landscapes features and dynamics which include the positions and quality of habitat patches. The phenotypes selected are those that are the most successful on the spatial and temporal scales relevant to the location of patches.
  • 2The basic mode of movement behaviour is random walk, uncorrelated or correlated. It occurs where the directions of habitat patches are unpredictable but patches are close, defined in relation to the organism's movement distances.
  • 3Behaviour more complex than random walk may occur where the position of habitat patches is predictable and when organisms have the ability to orientate or navigate even inaccurately. The ways in which information about the position of suitable habitat patches is obtained includes sampling and the use of social contacts.
  • 4Individuals within populations vary in their movement behaviour and dispersal and this may result in local populations differing from one another. Part of the individual variation is a product of random elements in movement behaviour and some of it is heritable.
  • 5Local and metapopulation dynamics are influenced by the heterogeneity within populations in dispersal characteristics. Heritable changes in dispersal propensity occur on time-scales short enough to impact population dynamics.

The evolution of dispersal including distance and other key features such as response to patch boundaries is encompassed in Principle 1. The phenotypes that are selected are those that provide relatively high fitness in relation to landscape features and dynamics. The expected outcomes arising from Principle 2 are that with unpredictable patch directions and close patches, linear movement would be inappropriate and random walk could be relatively efficient. Interpatch movement may include relatively long movements accompanied by changes in direction which can be described as having a random element. Principle 3 implies that random walk would be inappropriate where the position of a habitat patch is predictable and when an animal has the capability of moving to a patch or moving to an area where the probability of location is relatively high. Under such circumstances, a diversity of outcomes is expected.

In landscapes where the mean distance between habitat patches is high relative to the animal's movement potential and direction is unpredictable, metapopulations may not be viable. Linear movement may be the only possible dispersal strategy but in such challenging circumstances, evolution is expected to drive dispersal in one of a number of possible directions. Movement behaviour may become more complex if there is the potential for increases in the efficiency of location of patches. Dispersal distances may decline and then local populations may diverge. Dispersal distances may increase until patches become close at which point linear movement would be inappropriate and random walk movement behaviour would be expected. Alternatively, reproduction rate may increase while dispersal related mortality remains high. Under such circumstances linear movement may persist.

The dispersal distances of animals are expected to match the distances between the habitat patches of their recent ancestors. The evolution of distance could be conceived to be the acquisition of genetic information about the distances between patches. Variability in patch distances is expected to result in heritable variation in dispersal propensity and differences in local populations and this is encompassed in Principle 4.

The emphasis of Principle 5 is on the connection between dispersal and population dynamics and genetics. The development of population dynamics has assumed, no doubt as a means of making progress, that dispersal can be regarded as neutral and that genetic change takes place on a sufficiently long time-scale that it does not affect dynamics. Principle 5 implies that population dynamics can only be understood if dispersal and population genetics are considered explicitly because changes in dispersal traits and genetic composition of populations can impact dynamics on short time-scales.

This review has identified a number of directions for the further development of research in dispersal ecology. Taking these together, a change in emphasis is indicated and this would be needed in order to test the principles. In movement behaviour research, data have commonly been pooled to estimate population parameters, implicitly assuming that populations are homogeneous. That unrealistic assumption could be avoided simply by structuring analyses to include individuals. It will be increasingly important to observe the influence of landscape on movement behaviour in order to identify proximate causes and to lay firmer foundations for understanding pattern in spatial distributions or dispersal distances. There will need to be a move away from phenomenological to mechanistic models in the description of dispersal so that movement behaviour can be more readily related to the dispersal process. The use in dispersal studies of the statistic, mean squared distance, should be supplemented by using data on distance distributions to yield information on population heterogeneity. An understanding of key aspects of movement behaviour is required to determine how to elaborate simple diffusion models and it is likely that the inclusion of individual variation will generally be required. Metapopulation models appear to be sensitive to small changes in the assumptions concerning dispersal and cannot therefore provide unambiguous conclusions about the evolution of dispersal and dynamics until there is greater clarity about how to model dispersal and to estimate parameters from field data. The estimation of dispersal propensity will need to be linked with descriptions of landscapes and their histories in order to understand variation in local populations. The list above indicates that dispersal ecology will be further developed by making appropriate linkages between behaviour, dispersal and population processes and recognizing the potential importance of individual variation. One of the obstacles to progress is the availability of field data, but some existing data may yield more information if there is the change in emphasis proposed here.

Acknowledgements

I would like to express special thanks to Tom Coaker for encouragement and many productive discussions. Thanks also to Neil Gilbert, Tony Solomonides and Iain Weir for constructive criticism and comment on some aspects and to two anonymous reviewers for helpful suggestions for improvement.

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