The spatial structure of populations

Authors


Centre for Biodiversity and Conservation, School of Biology, University of Leeds, Leeds, LS2 9JT, UK

Abstract

1. Studies of the spatio-temporal dynamics and structure of populations have identified many categories of population type. However, recognized categories intergrade, making it difficult to assign empirical population systems to single categories.

2. We suggest that most population categories can be arranged along two axes that combine per capita birth (B), death (D), emigration (E) and immigration (I) rates. The ‘Compensation Axis’ describes the source-sink component of population structure, with source populations exporting individuals (B > D, E > I) and sinks and pseudosinks consuming individuals (B < D, E < I). The ‘Mobility Axis’ describes the involvement of a local population in regional (I + E) rather than local (B + D) processes, running from separate populations, through metapopulations, to patchy populations.

3. Each sample area within a spatially structured population system can potentially be assigned to a position along each of these axes, with individual sample areas weighted by local population size. The positions of these sample areas and their relative weightings allow the relative importance of different types of process to be judged. A worked exampled is provided, using the butterfly Hesperia comma. This approach shifts the emphasis from pattern (categories that real population systems do not fit) onto process.

4. In many systems, continuous variation in habitat quality and demographic parameters make clear distinctions between ‘habitat’ and ‘non-habitat’ difficult to sustain. In such cases, we advocate the use of a spatial grid system, with effects of patch size and isolation combined into a single, weighted distance function (neighbourhood).

5. The relative importance of different processes depends on the spatial scale at which the system is observed. This again emphasizes the value of a process-based approach.

Introduction

Research on spatially structured populations has produced both theoretical and empirical evidence for a range of possible types of populations, ranging from classical closed populations through various types of interacting systems of subpopulations (e.g. in Hanski & Gilpin 1997). Such population systems have generally been divided into discrete categories, and have spawned a diverse jargon of descriptive terms; for example, populations may be described as ‘open’ or ‘closed’, may be deemed to be ‘sources’, ‘sinks’ or ‘pseudosinks’ (Pulliam 1988; Watkinson & Sutherland 1995), and together may be classified as networks of ‘separate’ populations, as ‘metapopulations’ or as single ‘patchy’ populations (Harrison 1991, 1994; Harrison & Taylor 1997). These categories have provided an essential conceptual framework, but they have also generated confusion. Time and effort has been wasted trying to pigeon-hole particular systems into one or another category, when real systems may truly be intermediate between defined categories, or show elements of several population processes simultaneously. We can now see that many of the labels currently used to describe population structure might better be considered as points on various continua, rather than as clearly definable categories. The purposes of this paper are (i) to illustrate how existing population categories map onto two axes describing the relationships between birth, death, emigration and immigration; and (ii) to describe the effects of spatial scale on our perception of where a particular system falls on these continua. We begin by discussing the properties of individual subpopulations, and then broaden our analysis to incorporate unoccupied areas and then systems of multiple subpopulations. We restrict our attention to single-species population systems; multi-species interactions raise additional issues beyond the scope of this paper. Likewise, we avoid discussion of the evolutionary causes and consequences of the patterns and processes we describe.

Single sample areas with populations

A spatially structured population can typically be divided into a series of subsets, which may represent discrete habitat patches, population clusters, or arbitrary sampling areas. We will call these subsets ‘population units.’ The changes in population within any such unit can be described in terms of inputs (birth, immigration) and outputs (death, emigration). Here, we operationally define immigrants as individuals arriving into a sample area, and emigrants as individuals leaving the area within some defined time interval (preferably a generation), even though some immigrants may subsequently leave again and some emigrants may eventually return. Immigration and emigration rates can be expressed per capita, by dividing the number of individuals coming and going per unit time by the local population size. Local birth and death rates during the time interval can similarly be expressed on a per capita basis.

The net inputs and outputs for a population unit can be separated into internal processes (birth minus death, B–D) on the one hand, and external processes (immigration minus emigration, I–E) on the other. The balance of these four processes differentiates many of the types of population units that have been described: classical populations, sources, sinks and pseudosinks (Table 1). Indeed, by using net internal (B–D) and external (I–E) demographic processes as axes, we can define what we might term ‘demographic space’ into which these same population categories can simply be arranged (Fig. 1). Thus, at equilibrium, classical populations fall at the intersection of the B–D and I–E axes, source populations sit in the lower right, and sink and pseudosink population units are situated in the upper left (sinks and pseudosinks may overlap in position). Note that the categories lie neatly along a diagonal in demographic space, along the line defined by (B + I) – (D + E) = 0. This occurs because, if a population is to be at equilibrium, the factors increasing population size (birth and immigration) must balance the forces decreasing it (death and emigration). This diagonal line, which we call the ‘Compensation Axis,’ captures much of the variation between population categories, with net demographic generators of individuals on one side and net consumers on the other. The compensation axis serves as an attractor in demographic space; assuming even weak density-dependence, any population unit that is temporarily away from this line will return towards it. For example, if the immigration rate were to be increased, either the birth rate would decline in compensation, or the emigration or death rates would increase, pulling the population towards the line.

Table 1.  Population and habitat units; mainly after Pulliam (1988) and Watkinson & Sutherland (1995)
 B, D
(actual or potential)
I, E
I → zero
  • Effect on local population size of reducing immigration rate to zero.

  • Allee effects may result in B < D at very low densities for populated areas (converting these areas into true sinks), and B > D at higher densities, leading to alternative stable states. These could be regarded as four new categories.

  • §

    When colonized, these habitats may become either net consumers or exporters.

Places with populations
SourcesB > DI < EUsually limited impact
Classical populationsB = DI = EUsually limited impact
PseudosinksB < DI > EDecline
(B > D at lower density)  
SinksB < DI > EGo extinct
Places without populations
Potential sinksB < D  
Empty habitat§B > DI ≈0 
Sieves§B > DI < E 
and (E–I) > (B–D)  
Figure 1.

The Compensation Axis. B = birth; D = death; I = immigration; E = emigration; per capita rates.

Actual population units, of course, need not be at equilibrium, and so may not fall directly on the compensation axis at any one time. The degree of scatter around the diagonal should depend on the relative strength of density-dependence on one hand, and of stochastic variation in demographic rates (B, D, I and E) on the other. Population units with low demographic or environmental stochasticity, and with strong density-dependence, will usually sit tightly on the line (except when there are deterministic population cycles or chaotic behaviour). The more stochastic variation exists in the system, or the weaker the density-dependence, the less precisely will the population sit on the line (Fig. 2a). Even in the presence of very strong density-dependence, however, environmental stochasticity may move populations along the compensation axis with impunity, changing from net exporters to net importers or vice versa. Thus, a single population unit over time can be represented as a cloud of points. The trajectories of population cycles and chaotic dynamics may be represented as lines, circles of points, or as more complex shapes. In all cases, the distributions of points will sit astride the compensation axis.

Figure 2.

Hypothetical distribution of population units on B–D, I–E, compensation, and mobility axes, in high (a, c) and low (b, d) stochasticity systems. Population units are weighted by size; larger circles represent larger populations, at one snapshot in time. Over time, each population unit will move around in demographic space, the amount of movement depending on the levels of stochasticity and density-dependence. Darker shading indicates symbols relatively deeply embedded in the three-dimensional plot (c). Because population units are close to the compensation axis in the low stochasticity system (b), only two axes are required to plot low stochasticity habitat networks (d).

In a stochastic world, where B, D, I and E may all vary by an order of magnitude or more, many population units could produce sizeable clouds of points in demographic space. Accepting this, it can immediately be seen that many populations near the centre of the compensation axis (Fig. 1) may regularly appear to switch between sources and sinks. It makes much more sense, then, to define population units by the distribution of B, D, I and E, than to struggle to place them in population categories from which they may regularly stray. Similarly, differences between population units in the strength of the net export or import of individuals may be as interesting as the qualitative separation between sources and sinks.

Three further issues deserve consideration in this section. The first is the implication of instantaneously removing the source of immigrants (Table 1), which could easily occur as a result of habitat destruction. All population units receiving any immigration at all will immediately be pushed below the compensation axis, and also below the line I–E = 0; i.e. to the bottom half of the graph (Fig. 1). Density-dependent feedback mechanisms should then pull each population unit back towards the compensation axis, but in the absence of immigration the consumer half of the axis will no longer be feasible. Only population units that return to the remaining exporter half of the compensation axis (the portion below and to the right of the I–E and B–D intersection) will persist. Former consumer population units that follow trajectories to this section of the compensation axis, and survive, were pseudosinks (B > [D + E] at reduced density), as opposed to true sinks that become extinct (B < [D + E] at low density; Holt 1985; Pulliam 1988; Watkinson & Sutherland 1995; Thomas, Singer & Boughton 1996; Table 1). However, there is a complete intergrade between pseudosinks and true sinks. Stochastic variation in B or D could easily result in population units acting as pseudosinks in some years and as true sinks in others. As the distinction between sinks and pseudosinks rests on hypothetical trajectories which could take some time to play out in a stochastic world, the categories may be extremely difficult to differentiate in practice.

The second major complication is the Allee effect, in which inverse density-dependence takes place at very low density (e.g. finding a mate becomes progressively harder at low densities), but positive density-dependence can still take place at moderate and high densities. Thus, some population units could potentially occupy two stable conditions (Sinclair 1989), either as consumers or exporters of individuals. At low densities, immigrants would suffer Allee effects, and the population unit could have a locally stable equilibrium at some point along the consumer half of the compensation axis, or at extinction (below). At moderate or high densities, however, there may be another equilibrium on the exporter half of the compensation axis (two equilibria within the consumer or exporter portions of the axis are also possible). The frequency with which a population unit would flip from one equilibrium point to the other would depend mainly on how frequently stochastic variation resulted in the population unit entering the domain of attraction of the other equilibrium point.

Finally, we should note that the position of particular population units on the compensation axis may change through time; for example, a species inhabiting some kind of mid-successional vegetation might initially have to overcome Allee effects during colonization, become a pseudosink at early stages of succession, move along the axis and become a source in mid-successional vegetation, become a pseudosink again, then a true sink, before becoming extinct as succession progresses to later stages. In such a system, it seems more sensible to describe time-dependent changes in position on the compensation axis, rather than to assign the unit to a complex sequence of distinct categories.

Single sample areas without populations

The existence of unoccupied but potentially habitable areas may also be important in the dynamics of many spatially structured populations (e.g. Gilpin & Hanski 1991; Hanski & Gilpin 1997). We will refer to any sample areas where we do not observe our organisms of interest as ‘habitat units’. These unoccupied areas fall into a number of categories with differing properties (Table 1).

For most species, most unoccupied areas in the world are unsuitable, either naturally or because of human activities. However, there is no clear distinction between these unsuitable habitats and true population sinks; indeed, we can consider all such habitats to be ‘potential sinks’. Unless it is instantaneously fatal, any habitat could eventually become populated if the immigration rate were to be sufficiently high. Under current conditions, these habitats are not populated, but they may become so if immigration rates increase. If population units do establish in these areas, they become net consumers of individuals (true sinks); in Fig. 1 these habitats would fall to the upper-left, were they to be populated (Fig. 1).

The second category of unoccupied habitat units consists of sites that are intrinsically suitable for the species in question, but which are currently vacant. In a stochastic world, populations may go locally extinct even in perfectly suitable habitats, and the inevitable delays between extinction and recolonization should leave some fraction of suitable habitat areas unoccupied at any given time. In addition, new habitats may be created (e.g. through disturbance or succession) and will initially be unoccupied. Once colonists arrive at such a site, however, they should show net population growth and establish a population (B–D > 0 at low density). Once populated, these units will normally, but not always, become net exporters of individuals.

A variant of the above scenario may develop where Allee effects are important (Table 1), providing a third category of unoccupied sites. This is also a quantitative variant of population units with alternative stable equilibria, described above. If Allee effects exist, such that B–D < 0 at very low numbers or densities, occasional immigrants will fail to establish a population. This may be a common condition in sexually reproducing species that mate after dispersal. Where Allee effects are important, unoccupied habitat units will only become populated if sufficient numbers of immigrants arrive to overcome the initial barrier (e.g. as a result of stochastic variation in I), or if stochastic variation in B or D is sufficiently large to enable immigrants to overcome Allee effects in particularly favourable years. Once colonized and over the Allee hurdle, a population may establish and persist, and will usually become a net exporter of individuals. Nonetheless, stochastic variation in B, D, I or E may eventually cause some fraction of these population units to slip back into their former domain of attraction–extinction.

A fourth category of habitat units without populations, we call ‘sieves,’ to indicate that they are too leaky to be populated. These units are relatively isolated habitats, where any local breeding population would suffer a drain of emigrants, which would not be replaced by immigrants. If the emigration fraction is too high to be replaced by density-dependent increases in local birth rate (or decreased death), the population would crash to extinction and the habitat would be empty, despite a potential B > D (Thomas & Hanski 1997; Thomas & Singer 1998; see also Tilman, Lehman & Kareiva 1997). Immigrants to such habitats would not exhibit a positive population growth rate. These habitat units could become populated if, for example, they become less isolated (immigrants would replace some of the emigrants), or if conservation managers were able to construct barriers to dispersal around a habitat unit, reducing emigration.

As with populated units, these categories of empty habitat units merge into one another, depending on the stochastic distribution of potential B, D, I and E in empty habitat, and where they would lie on the compensation axis if they were to become populated. Therefore, it generally seems more useful to examine the relative contributions of potential B, D, I and E when considering the absence of a species from a particular habitat unit, rather than attempting to force a particular area of empty habitat into one of the four categories. Of course, there are great practical difficulties in identifying where an unoccupied habitat unit would fall along these axes, but the same problem must be faced when attempting to place unoccupied units into categories. The best approach is generally to compare the characteristics of habitat units (e.g. area, isolation, resource density) with the same variables in populated units, where B, D, I and E can be estimated. The potential population parameters can then be predicted for each unoccupied habitat unit: experimental introductions should be made to a subset of these units to test the accuracy of the predictions.

Multiple sample areas

We have so far considered each site (whether occupied or not) in isolation. Yet by referring to immigration and emigration, we have acknowledged the importance of surrounding areas. Networks of habitat and population units have been the subject of a great deal of attention in recent years (Hanski & Gilpin 1997), and have been divided into various categories of population structure (Harrison 1991, 1994; Harrison & Taylor 1997). This has provided a helpful conceptual framework for researchers since 1991 but, unfortunately, it has often been very difficult to assign habitat and population networks (hereafter ‘networks’) to simple categories. This is not surprising. As we have outlined above, even single population or habitat units are potentially complex in their properties, and there is no reason to assume that the many such units making up a network will be similar to one another. Empirical networks may contain populations and empty habitats representing all of the categories listed in Table 1! Attempting to apply tight definitions of population structure to such networks is ultimately doomed to failure, both because of the diversity of population units in each system and because of the scaling issues described below.

If a discrete typology of network categories seems inappropriate, how then are we to proceed? Again, we suggest that it is helpful to describe the relative importance of different processes within habitat networks. We can plot each sample unit within a network onto axes chosen to reflect important aspects of the processes governing their behaviour. In determining the behaviour of networks, Harrison & Taylor (1997) describe the importance of mobility relative to patch size and spacing. We consequently add a ‘Mobility Axis’, defined as (I + E) – (B + D). It describes the extent to which dynamics within an individual unit are dominated by local events, or by interactions with other units.

We can then consider where different categories of population structure might fall on the mobility axis. At the high migration end of the mobility axis (high positive values), individuals move in and out of habitat units quite frequently. Here, behaviour is a key determinant of local distribution, and foraging theory can be used to predict the distribution of individuals. These fit Harrison's (1991) definition of ‘patchy-population’ systems. At the opposite extreme of very low migration, there is essentially no movement between what can be described as separate populations. In separate populations, where mobility values are negative, local birth and death patterns are the predominant cause of differences in population size. In the middle, with intermediate movement relative to patch spacing, come metapopulations, where both behaviour and demography are key elements determining variation in local densities. Because metapopulations fall in the middle along this mobility continuum, it is impossible to draw a clear line that separates population structure into three categories, nor would we want to.

Although the mobility axis also contains combinations of B, D, I and E, it is nonetheless orthogonal to the compensation axis. The mobility axis passes through the intersection of the B–D and I–E axes shown in Fig. 1 (Fig. 2c), at right angles to the compensation axis. This three-dimensional approach may be necessary in many field systems, where stochastic variation and weak density-dependence may cause a significant fraction of patches to wander far from the equilibrium plane, now described by the compensation and mobility axes. In less noisy systems, however, virtually all population units may rest very close to the diagonal compensation axis, allowing us to collapse our three dimensions to two: compensation and mobility (Fig. 2d)

In this scheme, no conceptual difficulty is posed by the existence of a wide variety of types of population and habitat units within one, interconnected network. We simply define the nature of networks by the frequencies of units in different locations along both the compensation and mobility axes. Frequencies can be weighted by population size in each unit, so as to reflect the importance of each type of unit, in terms of the proportion of individuals in the entire system (Fig. 3; see Schoener 1991).

Figure 3.

Hypothetical distribution of population units (size of symbol proportional to population size) on compensation and mobility axes: (a) mainland-island metapopulation system; (b) source-sink system; (c) patchy population system; and (d) mixed system with a range of compensation and mobility values. On the compensation axis, +ve indicates net exporters of individuals, –ve indicates net importers. On the mobility axis, +ve indicates relatively high importance of migration, –ve indicates relatively low importance of migration.

Many possible population patterns may not fit neatly into any of the conventionally defined categories of population structure. We expect that natural variation in the size, spacing and quality of habitat patches will ensure that many, and perhaps most, spatially structured populations will include some population units scattered quite widely across both compensation and mobility axes, most closely resembling the ‘mixed’ structure illustrated in Fig. 3(d) (within the constraints imposed by population parameters contributing to both axes). The Appendix (Fig. 4; Table 2) considers a worked example, and illustrates some short-cut approaches which may be applied when it is impractical to measure all parameters in all patches.

Figure 4.

Figure 4.

Surrey and East Sussex (England) population systems of Hesperia comma, plotted against compensation and mobility axes. Details of approach, calculations, assumptions and conclusions are given in the Appendix. Symbol sizes are proportional to local population size (Surrey, 1–600 adult individuals; E. Sussex, 3–3720 individuals). Overall, the Surrey system has many small patches that are close together, characterized by higher emigration and immigration rates, with some patches acting as consumers. The Sussex system has fewer, larger patches, that are more widely spaced, leading to lower emigration rates, and much lower immigration rates for most patches.

Figure 4.

Figure 4.

Surrey and East Sussex (England) population systems of Hesperia comma, plotted against compensation and mobility axes. Details of approach, calculations, assumptions and conclusions are given in the Appendix. Symbol sizes are proportional to local population size (Surrey, 1–600 adult individuals; E. Sussex, 3–3720 individuals). Overall, the Surrey system has many small patches that are close together, characterized by higher emigration and immigration rates, with some patches acting as consumers. The Sussex system has fewer, larger patches, that are more widely spaced, leading to lower emigration rates, and much lower immigration rates for most patches.

Table 2.  Summary statistics for analysis of migration in Surrey and East Sussex population systems of Hesperia comma
 (a) Surrey(b) East Sussex
  • Calculated as the percentage of emigrants that are not expected to find another patch, from negative power function.

  • Calculated as the percentage of the entire population in each system that emigrates, but fails to find another patch.

Number of breeding patches 51 22
Population size (total)64008660
Overall emigration (%) 52 29
Mortality of migrating individuals(%) 62 96
System-wide mortality due to migration (%) 33 28

Defining sample areas

Before describing the properties of a spatially structured population system, it is first necessary to decide how to subdivide the landscape. When the distinction between habitat and non-habitat is relatively clear (e.g. on true islands or discrete resources), species can be described as inhabiting distinct patches, and population systems can be described in terms of the properties of these patches. For many species, however, there is no absolute distinction between habitat and non-habitat (an implication of the compensation axis), making it difficult or impossible to define patch boundaries precisely. We then have to proceed without making arbitrary decisions as to the level of habitat quality, or the density of individuals, required to separate habitat from non-habitat.

An alternative approach is to grid the landscape into cells. Every habitat or population unit is then identical in area. Each unit can potentially be described in terms of local population size, position on the compensation and mobility axes, and its neighbourhood (location with respect to other population units). Neighbourhood is used as a single parameter to summarize aspects of both habitat area and isolation. Following Hanski (1994; Hanski & Thomas 1994; Hanski, Kuussaari & Nieminen 1994), neighbourhood can be measured as the distances from one sample unit to every other sample unit. Although Hanski and colleagues predefined habitat patches before calculating neighbourhood indices, the approach can be applied equally to systems of grid cells. Thus, the neighbourhood (H) of population unit i can be represented as:

image

where Nj is the population size in habitat unit j, dij is the distance between habitat units i and j, and the sum is taken across all habitat units. The function, f (dij ), weights the effect of distance, typically using a negative exponential or some other function that relates dispersal probability to distance. The index describes the level of population surrounding the focal unit, weighted by distance, irrespective of whether the surrounding population is evenly spaced across the landscape, or occurs in several distinct patches (a classical patch approach will work better if there are strong habitat boundary effects).

We think that there can be conceptual and practical advantages to this kind of approach. Operational definitions of ‘habitat’ and ‘patch’ by different researchers may be sensible in the context of their specific studies, but the results of different studies may then be difficult or impossible to compare directly. In contrast, the scale of a grid can be compared between separate research projects quickly and objectively. For practical reasons, researchers will still have to select grid sizes that are appropriate to the question and study system, but at least the spatial context is then made explicit. When it is possible to generate a reasonably fine grid, we can always reconstruct habitat areas later, by grouping contiguous units together (Burgman, Ferson & Akçakaya 1993; Akçakaya 1994). We can also deal with expansions and contractions in contiguously populated areas, which might otherwise blur or obliterate patch distinction (see Thomas & Harrison 1992). The approach is also amenable to grid-based modelling (e.g. Hassell, Comins & May 1991; Hastings & Higgins 1994), allows distributional data to be related to satellite-derived vegetation maps using GIS raster-based approaches, meshes well with existing conservation mapping schemes, and may provide a common approach through which to unite metapopulation biology and landscape ecology (Wiens 1997).

Scale

One of the commonest questions about the structure of population networks is: ‘what proportion of species can be described as having metapopulations (or some other category of spatial structure)?’ As we have argued above, this question is unanswerable, both because the categories are arbitrarily defined along a continuum of mobility and because different units within a network may behave differently. There is, however, a third difficulty: the perceived behaviour of a spatially structured population depends on the spatial scale at which one examines the system (e.g. Wiens 1989; Morris 1995; Pickett & Cadenasso 1995). Scale can be considered to have two components; resolution (the size of grid cell) and extent (total area of the region included in a study).

If we are to impose a grid onto our landscape, we are forced to make decisions as to the appropriate resolution and extent at which to view the system. This choice will have implications for the way we interpret population structure (and evolution), as has been described for the Californian butterfly Euphydryas editha Boisduval (Thomas & Singer 1998). If we lay out a 2-m resolution grid across a 100 × 100 m extent of habitat, an egg-laying female of this species will readily move in and out of individual sample units, some of which contain host plants. Females will sample potential host plant individuals, and make behavioural decisions as to whether to lay eggs on each plant (Mackay 1985; Parmesan 1991). If our entire horizon was the 100 × 100 m extent of habitat, then we could describe the system as being at the high mobility end of the migration axis, with a regular flow of individuals in and out of each sample unit (patchy population). Moving up a scale, Thomas et al. (1996) described a habitat network for E. editha in an area 10 × 10 km in extent. Laying a 200-m resolution grid across this landscape, each sample unit could be described as a panmictic local population, with some movement between them. Behavioural decisions impact immigration and emigration rates for each unit, but local birth and death rates also have major impacts on differences in local densities. Described at this resolution and extent, the system could be placed at an intermediate position on the mobility axis, and be described as a metapopulation (Thomas et al. 1996). Each such ‘metapopulation’ is often widely separated from other metapopulations of the same species. If we take a 1000 × 1000 km extent of the western USA, and impose a 20-km resolution grid across it, most of these metapopulations would occupy just one or two distributional units, with virtually no exchanges between them (Thomas & Singer 1998). They could be described as separate populations, with birth and death the predominant processes. The mortality of migrants moving between habitats could then simply be added to other forms of mortality within each 20 km unit.

This one species could be described as conforming to completely different population structural categories, depending on the scale at which the system was examined. Yet, there is only one set of individuals in one world. Although we have just described a scale hierarchy, differences in size, isolation and quality of patches at each level quickly blur the distinctions, allowing a seamless slide between structural categories with incremental changes in the resolution of analysis. Indeed, this is the implication of the mobility axis. Some empirical systems may exhibit stepped scale hierarchies, but others will not.

Changing resolution and extent can also generate differences in position along the compensation axis. Hochberg et al. (1994) and Clarke et al. (1997) have shown that different parts of a single field may effectively act as local sources and sinks for a butterfly population. This occurred because the butterflies were unable to select just the best locations for egg-laying, and survival varied from high to zero in different parts of the same field. In this case, source-sink dynamics were probably stronger within fields than between them. In the E. editha system, just described, some habitats generated individuals and colleagues consumed them, such that the system could be described as exhibiting source-pseudosink dynamics at 200 m resolution (Thomas et al. 1996), but would not have been described as doing so at either finer or coarser resolution.

Single-species dynamics may be dominated by different blends of processes at different scales, so it is not surprising that different species (which differ both in mobility and in the spatial patterns of potential resources) appear to differ if compared at the same scale as one another. It is important to compare like with like; for example: (i) the plant Melampyrum pratense L. distributes its seeds over centimetres to a few metres, and shows variation along the compensation axis when the world is examined at a resolution of metres (L. Bastin & C. D. Thomas, unpublished data); the butterfly E. editha moves tens to hundreds of metres, and shows variation in position on this axis at a resolution of a few hundred metres (Thomas et al. 1996); (ii) the host-specialist arctic aphid (Acyrthosiphon svalbardicum Heikinheimo) has no winged individuals, and is likely to show metapopulation dynamics at the scale of extinction and colonization from individual host plants within one area of vegetation (Strathdee & Bale 1995); some butterflies show metapopulation dynamics at the scale of fields within a landscape (Thomas & Jones 1993; Hanski et al. 1995); mountain sheep (Ovis canadensis Shaw) do so at the scale of mountain ranges within southern California (Bleich, Wehausen & Holl 1990); and bearded vultures [Gypaetus barbatus (L.)] might show comparable dynamics at the scale of mountain ranges within continents (Frey & Bijleveld van Lexmond 1994; Mingozzi & Esteve 1997).

The differences among these species are largely ones of physical scale. Described with respect to the mobility of each species (the mobility axis), many of the apparent differences disappear. Consideration of scale is absolutely critical in comparisons of different systems, and in assigning importance to different population processes. We strongly urge researchers to consider multiple scales of resolution and extent in any study system.

Conclusion

We advocate relating observed population processes and patterns to continuous axes, rather than attempting to consign each population unit or network into some defined category, which it is unlikely to fit. We appreciate the practical difficulty involved in attempting to quantify B, D, I and E in multiple habitats across a landscape, but exactly the same difficulties must be confronted when one attempts to place particular population units and networks into existing categories of population structure. Furthermore, scaling issues must be addressed seriously in studies of spatial population structure and dynamics.

Overall, conceptual approaches to spatio-temporal population dynamics must shift in emphasis from pattern to process. The names in Table 1 may sometimes be useful labels, but category-based schemes encourage unnecessary and unproductive dispute: two authors could examine a single data set, and yet reach different conclusions as to whether it represents a metapopulation or a patchy population. That decision would lead them to draw quite different conclusions about which processes are important. Were they both to consider the empirical data in relation to continuous variables, we believe that their perceptions of the important processes would be much more comparable.

Acknowledgements

We would particularly like to thank David Boughton, Ilkka Hanski, Susan Harrison, Bob Holt, John Lawton, Bryan Shorrocks, Mike Singer and Jeremy Thomas for sharing their thoughts over the years. Thanks also to Jens Krause, John Lawton and Bryan Shorrocks for comments on the manuscript. CDT was supported by NERC grant GST/04/1211 and a University of Leeds Fellowship.

Received 25 February 1998;revisionreceived 9 November 1998

Appendix

Appendix 1

It will rarely be possible to measure I, E, B, D and local population size in every patch in a population system, and each researcher will have to develop their own practical approaches to simplify the problem. In some systems, it may be possible to measure these four population parameters directly, or deduce them from other demographic measurements. However, the general strategy often has to be to establish environmental correlates of these parameters in a subsample of patches, and then extrapolate over the entire system. Our worked example concerns two population systems of the silver-spotted skipper butterfly, Hesperia comma L., that inhabits grassland habitat fragments in southern England. For this species, we have been able to estimate patch-specific I, E and local population size, but the data were not collected specifically to test the conceptual framework outlined in this paper, and we lack patch-specific information on B and D. Therefore, we had to make the simplifying assumption that the number of skippers in each local breeding area was at equilibrium. This is unlikely to be strictly true, and the results are intended to illustrate the approach, rather than to provide a definitive description of population structure in this butterfly.

Survey methods for measuring local population size in systems with many patches have been developed and applied to H. comma (Thomas 1983; Thomas & Jones 1993; Thomas et al. 1996). For a subset of patches, one develops a regression equation to describe the relationship between actual densities (from mark-release-recapture) and transect counts. Transect counts at every other site are then converted into estimates of local population size on the date surveyed (with knowledge of the butterfly's phenology, these can be corrected to give estimates of numbers likely to be present throughout the entire adult emergence period—H. comma has one generation per year). This approach provided estimates of local population size (N) in 1991, shown as differences in the size of symbols in Fig. 4.

Estimates of emigration fractions have already been published, using a combination of empirical mark-release-recapture and simulation modelling (Hill, Thomas & Lewis 1996; Thomas et al. 1998). Small patches [patch area A (ha)] have higher per capita emigration rates (E), and can be described by the equation:

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Because we knew habitat patch areas in 1991 (Thomas & Jones 1993), this equation could be applied to every patch in the system.

Per capita immigration was estimated as follows. The number of individuals emigrating from patch i (Mi) was estimated as the product of Ni and Ei. The proportions of emigrants (Zi) achieving different distances (Dij, in km, between origin patch i and target patch j) were measured empirically by mark-release-recapture (Hill et al. 1996), and were described by the negative power function:

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Using this equation, the number of individuals immigrating into patch j was ΣZi Mi, summing across all other patches in the system. Per capita, Ij was calculated by dividing this value by Nj.

I–E was calculated for each patch as the estimated number of immigrants, minus the estimated number of emigrants, divided by local population size.

There are various quantitative uncertainties in these calculations, but they give qualitatively the right result; for example, large patches with large populations have relatively low I, especially if isolated. Small patches that are close to large populations receive more immigrants than they generate emigrants.

Given the lack of direct information on B–D, we assumed that each local sample area was at equilibrium; thus (B–D) = –(I–E) for each patch. Position on the compensation axis could then be measured (and given the same sign as B–D; net importers were negative, and exporters positive). These are the values plotted against the compensation axis in Fig. 4.

It would have been preferable to measure B and D independently, but it will rarely be possible to do so in every patch of a system. In resource-limited systems, a useful approach may be to calculate B and D for a subset of patches, to establish how B and D vary with parameters that are relatively easy to measure (e.g. resource density relative to population density). The same parameters could then be measured in every other patch in the system, estimating relative B and D throughout a system. For predator-limited systems, the ratio of predators to prey might be used to predict patch-specific variation in D.

To estimate position on the mobility axis, I + E was calculated, giving a possible range of values between 0 and +2. Assuming equilibrium conditions, then (I + B) + (E + D) = 2, because each individual in a patch is born there or immigrates, and then either dies there or emigrates. Thus (I + E) was subtracted from 2 to estimate (B + D), from which we calculated (I + E) – (B + D). Position on the mobility axis potentially ranged from –2 (all local birth and death) to +2 (all immigration and emigration).

Using this approach, we can conclude that a range of local population types exist within both H. comma systems (considerable variation on the mobility axis, and some on the compensation axis; Fig. 4), although there are quantitative differences. The Surrey system contains relatively small populations with moderate migration rates, and has a considerable number of patches that are predicted to be net consumers of individuals (small patches close to larger populations), whereas the East Sussex system is characterized by larger populations with lower migration rates (in large, widely spaced patches).

In both systems, the system-wide mortality of migrants (emigrants that fail to find other patches) results in fewer patches consuming individuals than exporting them (Fig. 4). This mortality runs at about one-third of all adult individuals emerging in both systems (bottom line of Table 2). The break-down shows a different balance of processes in the two systems, with higher emigration but lower mortality of migrants in Surrey, compared to Sussex (Table 2); this difference is generated by relatively small patch areas and short distances between patches in Surrey. If more than two-thirds of all individuals in an H. comma population system were to die during migration, the system could be expected to collapse because those that remain will be unable to breed fast enough to replace these losses (given estimates of r, although a few large local populations might survive; Thomas & Hanski 1997; Thomas et al. 1998). Losses to migration would only have to double for this to happen in these systems. Thus, by moving away from a simple metapopulation caricature of extinctions and colonizations, we have revealed a potentially very important additional role of migration in the persistence or collapse of H. comma population systems.

This provides an interesting possible re-interpretation of the rapid decline of H. comma in Britain, following declines in habitat quantity. The standard explanation is that the species’ habitat was completely eliminated from large parts of England, driving populations and metapopulations extinct (Thomas et al. 1986; Thomas & Jones 1993). However, the current analyses (and Thomas & Hanski 1997; Thomas et al. 1998) suggest that the butterfly would disappear just as surely if patch areas declined rather than disappeared completely (increasing emigration rate), and if distances between patches increased (increasing mortality of migrants), to the point at which system-wide mortality resulting from migration exceeded two-thirds. This may explain why H. comma refuge areas were either very large habitat patches (with low emigration rates) or networks of smaller patches (many emigrants reach other patches unsuccessfully) (Thomas & Jones 1993).

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