Dr G. D. Ruxton, Division of Environmental & Evolutionary Biology, Graham Kerr Building, University of Glasgow G12 8QQ, UK. Tel: 0141 330 6617. Fax: 0141 330 5971. E-mail: email@example.com
1. We present a novel metapopulation model where dispersal is fitness dependent: the strength of migration from a site is dependent on the expected reproductive fitness of individuals there. Furthermore, individuals continue to migrate until they reach a suitable habitat where their expected fitness is above a threshold value.
2. Fitness-dependent dispersal has a very strong stabilizing effect on population dynamics, even when the intrinsic dynamics of populations in the absence of dispersal exhibit complex high-amplitude oscillations. This stabilizing effect is much stronger than that of the density-independent dispersal normally considered in metapopulation models.
3. Even when fitness-dependent dispersal does not stabilize the dynamics in a formal sense, it generally leads to simplification, with complex or even chaotic fluctuations being reduced to simple cycles.
4. This form of dispersal also has a strong tendency to synchronize local population dynamics across the spatial extent of the metapopulation.
5. These conclusions are robust to the addition of strong stochasticity in the migration threshold.
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The influence of dispersal on population dynamics is a field of intense current interest to both empirical and theoretical ecologists (Kareiva & Wennergren 1995). One particularly intensely studied group of models are collectively termed coupled map lattices (CMLs). Here, a group of distinct populations are each represented by simple equations which convert the current local population density to that at the next generation. Conceptually, these populations are placed on the nodes of a (normally square) lattice. The equations are linked by transport terms which represent dispersal of individuals between the local populations in the system (Allen 1975; Hassell, Comins & May 1991; Hastings 1992; Rohani, May & Hassell 1995; Ruxton & Rohani 1996; Bascompte & Sole 1998). Despite the proliferation of spatial models in recent years, dispersal is generally modelled in a very simple way. In the overwhelming majority of cases, dispersal is assumed to be density-independent, with a constant fraction of each population dispersing at each generation (but see McCauley, Wilson & de Roos 1993; Scheuring & Janosi 1996 and Ruxton 1996a for exceptions). From an evolutionary viewpoint, we would predict, however, that migration should be density dependent, with a higher fraction of individuals leaving a particular site when the local population density is high. Furthermore, we will argue below that dispersal should also be dependent on other factors at the current site which affect fitness. Our overall objective is to explore the effect of this ‘fitness-driven dispersal’ on population dynamics.
In common with several previous studies, let us consider the case where each generation can be split into two distinct phases: demographic processes and dispersal events. This temporal segregation is biologically plausible for species such as butterflies where most of the lifecycle is spent in relatively immobile immature states, followed by a final highly mobile egg-laying stage. Consider a newly-emerging butterfly which is faced with the choice of remaining in its current habitat or dispersing to another before laying its eggs. To maximize the fitness of its genes, it should remain in its current site if its expectation is that it could obtain more viable offspring there than if it dispersed to another habitat. Say there are M female butterflies in the current site, and each lays λ eggs which hatch into females. Competition for resources is likely to introduce density-dependence in juvenile survival. We follow convention and express this by saying that the probability of each of these offspring reaching maturity is a decreasing function of M (the number of adults which laid eggs at this site). For example, let this juvenile survival probability P(M) be given by
where the parameter a scales the carrying capacity, and b represents the strength and form of competition (the limit b → 1 is considered to represent contest competition and b → ∞ to represent scramble; see Doebeli 1995 for a fuller discussion). Now, if the butterfly stays in the current site, then its expected fitness return will be λP(M). The butterfly should only disperse if it can increase its expected fitness return by doing so. For all but the most trivially simple of systems, this expectation cannot be calculated analytically. However, clearly it will depend on the probability of the individual reaching another population, the number of other adults laying eggs in that population and other aspects of the environment at that location (represented by the values of λ, a and b). In nature, individuals will never have sufficiently accurate information to make a precise estimate of conditions on other sites; hence, dispersal may be well represented by a rule of thumb using only information about local conditions. One reasonable rule to use would be for individuals to disperse if their expected fitness return on the current patch falls below some fraction (Δ) of the maximum (λ): i.e. individuals only disperse if P(M) < Δ. In population terms, this means that there is a critical population density (Mc) which satisfies
If the current population density is lower than this, then there will be no emigration. However, if the population density is higher than Mc, then individuals will leave until the population density falls below the threshold. Thus, from this fitness argument, dispersal will be density-dependent. We can also see that it is related to the constants a and b which represent the environmental conditions on the site. This conclusion stands in contrast to the assumption in previous models of density-dependent dispersal, where the density-dependence in the dispersal function was unrelated to demographic parameters (even though these factors control the effect of competitor density on fitness). For example, the model of Scheuring & Janosi (1996) has a threshold parameter k (directly analogous to our Mc) which they vary whilst keeping the carrying capacity constant. From our argument above, this is equivalent to varying individuals’ criterion for moving (represented by Δ above). However, varying k in a linear fashion causes a strongly non-linear change in Δ, so Scheuring & Janosi's results are difficult to interpret in terms of the effect of changing movement criteria by individuals on population dynamics.
The aim of this paper will be to use a simple mathematical model of a collection of linked populations, to explore how changes in the propensity of individuals to disperse is related to the local density and how these changes affect population dynamics. Unlike any previous models, the propensity of individuals to disperse will also be related (through the fitness argument above), not only to the local population density, but also to other aspects of the quality of the current site (such as its carrying capacity) which relate the local population density to reproductive success. Furthermore, the site which individuals finally settle on will also depend on the same fitness considerations.
Consider an L*L array of distinct spatial positions. Each of these positions supports a local population of a given species which reproduces in distinct generations. At the end of the migration phase of a given generation (t), we denote the population at a given position (i, j) as Ni, j,t. These individuals then each produce a constant number of offspring λ. The probability of these offspring surviving the reproductive phase is given in equation 1, so that the number of individual offspring at each site at the start of the next dispersal phase (Mi, j,t+1) is given by
For simplicity we will assume that a and b take the same value on all sites.
Dispersal works according to the following recipe. We calculate a critical population density Mc from equation 2, i.e.
For every population where the current population density (Mi, j,t+1) is greater than Mc, the difference represents the density of the migrating population from this site. This population is split evenly between the four nearest neighbours. We assume dissipating boundary conditions such that if a neighbour does not exist, then the migrants assigned to it are lost from the system. Migrations occur synchronously on all sites.
After this round of migrations, some sites may have local populations higher than the threshold Mc (because of positive net migration into that site). Further rounds of migration (exactly as described above) occur until the population density on all sites is below (or equal to) the threshold. These populations are then denoted Ni, j,t+1, and another cycle of reproduction and dispersal can occur.
In this model, migration from a natal patch is dependent on the local conditions there (not just the local population density). If an individual migrates, it does so to one of its nearest neighbours. If conditions there are satisfactory, then it will remain and reproduce there; if not, then it will disperse again until either it finds a satisfactory site or is lost from the edge of the lattice. This means that not only is the decision to move fitness dependent, but the final position of the individual depends on decisions about fitness made each time it moves between patches, until it finds a patch which meets its criterion for settling. Other models (such as Hastings & Higgins 1994; Ruxton & Doebeli 1996) have allowed individuals to move further than one nearest-neighbour distance from their neighbouring site, but in these models this was done by applying a fixed dispersal kernel, i.e. the ‘decision’ of individuals as to which patch to settle on is fixed and independent on conditions on that or any other patch. Hence, in these models individuals can settle on patches where their expected reproductive fitness is lower than if they had remained on their natal patch; this cannot happen in our model, where individuals who moved to such a patch would not settle, but rather would disperse again until they found a satisfactory patch. The model definition of an indeterminate number of dispersal rounds can be interpreted as implying that dispersal happens on a much shorter time scale than the length of the dispersal phase of a generation, so that no matter how many dispersal rounds occur, they have all been completed before the next round of reproduction begins.
In this section, we explore the effect of varying the parameter controlling the fitness criterion for migration (Δ) on local and global population dynamics. We also investigate the effect of varying the carrying capacity (a). Both of these parameters affect migration in this model, but would not affect the intrinsic stability of a single population in the absence of migration. Hence, we can explore how this ‘fitness-dependent migration’ affects (and is affected by) population dynamics. We also explore the effect of changing the number of patches in the metapopulation.
As a default, we will consider the case where λ, a, b and Δ are independent of spatial position. If there is no migration, then the dynamics of a single population are governed by equation 1, and have been comprehensively studied (e.g. Bellows 1981; Doebeli 1995). Figure 1(a) shows the behaviour of such a population with parameter values chosen so that the population shows unstable, large-amplitude complex fluctuations. If we take 49 of these populations, start them from different initial values and plot the changes in total population density of the whole ensemble of unlinked patches, then the result would be similar to Fig. 1(b). This plot shows what appears to be small-amplitude complex oscillations about some non-zero value: this reduction in the variance of the signal, compared to a single population, arrives because the oscillations in the different unlinked patches are not synchronized and so will often cancel each other out. This is well known in the study of coupled oscillators (Kaneko 1989).
This situation of 49 unlinked patches can be mimicked in our model by arranging the patches on a 7 × 7 square grid and taking the limit Δ → 0. We now explore the effects on local and ensemble dynamics of increasing Δ, i.e. of decreasing the minimum fitness that individuals are prepared to tolerate without moving. The results are shown in Fig. 2. For extremely low values of Δ, we observe complex behaviour, though typically the ensemble dynamics are much simpler, being either an N-point cycle or a constant equilibrium point. Qualitatively similar results have been observed for a wide range of parameter combinations. Simply, the dynamics of the ensemble have never been observed to be more complicated than that of the constituent populations in isolation. In the sections that follow, we concentrate on an example of the most interesting case, namely that where the intrinsic population dynamics of isolated patches are chaotic.
The equilibrium value (M*) of an isolated population occurs when each individual produces exactly one viable offspring in the next generation, and can be found by solving
We can then find the value of the fitness criterion (Δ) which causes individuals to leave the population only if the current population density is greater than M*. It is easy to show that this critical value of Δ is given by
CASE 1: Δ > ΔC
If we select a higher value of Δ (i.e. Δ > Δc), then the threshold population density Mc is less than the equilibrium M*. In the extreme case, where all the populations start off at the same small value, as we iterate through time, the population in each site will increase until it becomes higher than Mc. Then a bout of migration will occur which will end with all populations having a population density of exactly Mc. Now since Mc < M*, the population in each site will again increase in the next generation, which will trigger another bout of migration until, at the end of the generation, all sites have exactly Mc individuals again. As the system is iterated through time, each site always finishes each generation with exactly Mc individuals. Although more difficult to visualize, this outcome occurs regardless of the initial values given to the populations. Thus, if we pick Δ > Δc then the system will always end up in a spatially homogeneous and temporally constant situation with each patch having exactly Mc individuals at the end of each generation. This explains why the ensemble population density is constant over much of Fig. 2(a). for the parameter values used, the critical value of Δ is 1/7 = 0·14. At this point we can find the value of Mc analytically from equation 4: 1·45; if we multiply this across all 49 sites, we get 71 as the ensemble population density. These values accord well with the extreme left margin of the stable zone in Fig. 2(a).
CASE 2: Δ > ΔC
Now consider what would happen to our model if we set Δ = Δc. If the population density in any patch is greater than M*, then a sequence of migration events will occur until all the patches satisfy Mi,j,t ≤ M*. This mechanism ensures that after a few generations, all patches attain M*, after which there is zero net movement between sites. Thus, irrespective of the initial conditions, Δ = Δc guarantees a spatio-temporal equilibrium point.
CASE 3: Δ > ΔC
In the case where Δ is decreased, so that Mc is slightly greater than M*, then if all populations reach Mc, they will each have the same density after reproduction, which will be below Mc, so no migration will occur. Depending on whether this new population density is above or below M*, the population density in each site will (after the next reproductive bout), then either decrease or increase. Eventually, a value above Mc will be reached, upon which migration will reduce every population down to Mc and the same progression will occur again. In this way, we get a spatially homogeneous population distribution which shows a simple exactly repeating cycle over time. This again is what we see for Δ-values just below Δc, when all the populations undergo a two-cycle between Mc and a lower value (see Fig. 2a). Indeed, from the above argument, it is intuitively clear that whenever Δ < Δc, population cycles will result. The description above suggests that longer period (but still simple) cycles should also be possible, and a four-cycle can indeed be seen for Δ-values around 0·015 (see Fig. 2b).
The simple argument above presupposes synchronization between the sites (in the sense that they simultaneously have the same value). However, synchrony is almost a generic property of the model due to the migration mechanism. Suppose that after a bout of reproduction, sites have different values, some of which are above Mc and some of which are below; after migration the sites which were originally (at different values) above Mc will now all have population density Mc, other sites will have had their population unchanged or increased towards Mc. This process repeating itself will tend to lead to synchronization. Note that synchronization is inevitable if the intrinsic patch dynamics are non-chaotic. However, chaotic intrinsic dynamics (as assumed here) will set up a force which will act to resist synchronization—the precise outcome will be governed by the fitness threshold Δ.
Figure 2(b) shows that complex ensemble behaviour is possible when Δ is extremely low and Mc is high: intraspecific competition always reduces the population density before it reaches Mc—as a result, dispersal never occurs. The populations in each site fluctuate in an asynchronous fashion, producing the low amplitude noisy signal seen in the extreme left of Fig. 2(b). However, complex dynamics can arise following migration (as can be seen for Δ-values around 0·01 in Fig. 2b). Here, the migration level is not large enough to simplify the dynamics and synchronize patches, which perform asynchronous complex fluctuations. However, simple synchronized cycles can be seen for nearby Δ-values. Indeed, in some regions two attractors co-exist, one with simple synchronized cycles and one with complex asynchronous behaviour. Different initial conditions can cause the population to settle on different attractors.
From Fig. 2(b), we can see that for low Δ-values, there appears to be two co-existing attractors, which can be investigated further by comparing Fig. 3(a,b) with Fig. 2(b). Whereas Fig. 2(b) was constructed by picking a new set of random initial conditions for each simulation, Fig. 3 uses the final state of one simulation as the initial conditions for the next. This has the effect of increasing the likelihood that consecutive simulations will settle onto the same attractor. This process was repeated starting the first simulation off with a wide range of different initial conditions; in every case the resulting bifurcation diagram looked like either Fig. 3(a) or b. (The same is true if the process is repeated starting with a high value of Δ and using progressively smaller values.) Both diagrams are the same for Δ-values above Δ≈ 0·012, i.e. for larger values there is only one attractor. However, for low values the attractors can be seen to be characteristically different. Figure 3(a) shows the type of low-amplitude noise similar to Fig. 1(b) and characteristic of the summation of uncorrelated complex dynamics. In contrast, the attractor in Fig. 3(b) mostly shows simple repeating cycles, characteristic of all the populations performing synchronized simple cycles. Even when the ensemble dynamics are more complex (Δ < 0·002), the fact that the ensemble regularly reaches very high and very low values (relative to the equivalent simulations in Fig. 3a) suggests that a high degree of synchrony is still maintained, even for these very low Δ-values when migration is relatively uncommon.
From the arguments given above, we would not expect the number of patches in the metapopulation to have any effect on population dynamics at high Δ-values, numerical simulations (not shown) confirm this. In contrast, it could be that system size affects the population dynamics at low Δ, this can be explored by comparing the 49-patch system of Fig. 2(b), with the nine- and 100-patch systems of Fig. 4(a,b). There is a slight tendency for increasing system size to lead to desynchronized complex dynamics, although this effect is very weak. Generally, system size seems to have very little effect on population dynamics. It does, however, have a strong effect on the number of migration events during each dispersal phase (see Fig. 5). Increasing the system size parameter L increases the mean distance from a patch to the edge and the ratio of interior to edge patches. These effects mean that many more dispersal rounds may be required to reduce all populations below their critical value. Although we have focused so far on the dynamics of the ensemble, one can see that if the habitat fragments are identical, then the mechanism described here will not induce spatial heterogeneity between local populations, and peripheral populations will behave in a generally similar way to those in the centre of the array. Lastly, we find that varying the carrying capacity of patches has no effect on stability. This is not surprising in view of the arguments laid out above, since both the equilibrium value M* and the critical population density Mc have exactly the same (linear) relationship with the carrying capacity (1/a)
Up until now, we have assumed that Δ and, hence, Mc are fixed through time and identical in each patch. Figure 6 shows the consequences of relaxing the assumption that Mc is always constant. In Fig. 6(a) the value of Mc was drawn randomly and independently for each site from a uniform distribution [0·75 Mm, 1·25 Mm], where Mm is the mean value of Mc calculated from equation 4, as previously. Figure 6(b) is the same, except that each site's value of Mc is re-assigned at each generation. Comparison of these figures with Fig. 2(a) shows clearly that the simplification and synchronization effects described previously are robust to the addition of considerable stochastic variation in dispersal thresholds.
In this paper, we have introduced a novel metapopulation model with ‘fitness-dependent dispersal’. The key feature of this model is that an individual's decision to migrate is triggered by its expected future fitness on their current patch. Individuals migrate in a fixed pattern, but (in contrast to previous models) the same fitness criterion is used to assess the suitability of new patches, with individuals continually migrating until they find a patch of suitable quality (which for the simulations described means having a sufficiently low population). Thus, not only is the initial dispersal decision dependent on the local population density, but the choice of site on which an individual settles is also dependent on its local density (and the local density of intervening patches). In this way, our model structure is fundamentally different from the class of models normally studied (e.g. Allen 1975; Hassell et al. 1991; Rohani et al. 1997), because of the way our dispersal rules effectively decouple spatial and temporal scales. Hence, we would expect that this difference in model structure is likely to lead to qualitative differences in predictions. Our conclusions are that fitness-dependent dispersal has three major effects on the population dynamics: it is stabilizing, simplifying and synchronizing. We will now discuss each of these in turn.
It has recently been shown that the local stability of homogenous single-species CMLs like that described here, but with density-independent (or passive) dispersal are dependent only on the intrinsic dynamics of the patches (Hassell et al. 1995; Rohani et al. 1996). If the local populations are stable in isolation, then density-independent dispersal will not destabilize the ensemble; similarly such dispersal cannot stabilize (in a strict sense) an ensemble of intrinsically unstable populations. Similar results have been found in continuous-space models (Neubert, Kot & Lewis 1995). Ruxton 1996a explored the effects of introducing density dependence into dispersal. In that model, there was a single episode of migration per generation and propensity to migrate was density dependent, but the final destination of migrants was not. It was found that adding this sort of density dependence had no effect on local stability, except under very extreme circumstances. Even when the homogeneous equilibrium was destabilized, the system always switched to an alternative equilibrium which was spatially heterogeneous, but temporally constant (a so-called ‘standing wave’).
In the previous modelling framework, where the demographic and dispersal phases are rigidly alternating, the general conclusion was reached that dispersal either has no effect on equilibrium stability or a detrimental one. It was shown that such dispersal could not stabilize a collection of unstable populations (Allen 1975; Hastings 1992; Rohani et al. 1996). In contrast, the fitness-dependent dispersal described here can have a strong stabilizing effect, even when the intrinsic dynamics are highly unstable (see Fig. 2a). Furthermore, extensive numerical investigations suggest that it can never destabilize a system of inherently stable local populations. This form of dispersal is stabilizing whenever individuals demand a high expected fitness return from a patch before being prepared to stay. Specifically, they will only settle on a site if the current conditions mean that they are able to produce at least one offspring which survives to reproduce in the next generation; otherwise they migrate. If their criterion is greater than or equal to this fitness return, then migration will produce a homogeneous stable equilibrium; however, as the fitness criterion increases so the final equilibrium density (which is the one yielding exactly that fitness) decreases. This finding is not surprising since such a high fitness expectation results in individuals undertaking many migration events, which acts to homogenize the density on each patch. In summary, fitness-dependent dispersal has a much stronger stabilizing effect than has been observed in similar models with simpler descriptions of dispersal.
If the fitness criterion is less demanding, then migration does not act to stabilize the dynamics (in a mathematically rigorous sense), but generally acts to simplify them (generally producing a simple cycle, even if the populations are chaotic in isolation. Such simplification is also a feature of the density-independent dispersal of Rohani et al. (1996) and the density-dependent migration of Ruxton (1996a). Furthermore, the simple cycles are spatially synchronized (as will be discussed below). If the fitness criterion for remaining is very undemanding so that migration happens only infrequently, providing the intrinsic dynamics are unstable, then sometimes neither simplification nor synchronization occur. Instead, the noisy equilibrium characteristic of the sum of many populations with uncorrelated complex dynamics can be seen (see Fig. 3a). This behaviour was also observed and extensively described by Scheuring & Janosi (1996) using a model which was functionally very similar to the one described here. However, for parameters where this is observed, there is generally also another co-existing attractor, such that the final behaviour of the system is dependent on initial conditions. The other attractor is characterized generally by simple cycles and synchronization. Even when the dynamics are complex and the synchronization is not complete, there is still strong partial synchrony (see the left edge of Fig. 3b). However, this complicated repertoire of behaviours are confined to a very narrow range of dispersal criteria where migration occurs only when local population density becomes very high.
A considerable number of spatio-temporal population measurements suggest that populations of a given species in distinct regions often appear to cycle in synchrony (see Ranta et al. 1997; Ranta & Kaitala 1997). A parallel theoretical activity has been the search for potential mechanisms which might produce this synchrony (Ranta et al. 1995; Heino et al. 1997; Earn, Rohani & Grenfell 1998). Many candidate mechanisms suggest that spatially distinct regions may be governed by the same external forcing (e.g. by synchronized weather patterns, prey abundance or disease outbreaks), and this spatially correlated forcing induces the population cycles. Ranta et al. (1997) demonstrated that a randomly occurring perturbation reducing reproductive success (by differing amounts between areas) can also have a synchronizing effect, this is the Moran effect (Moran 1953; Royama 1992). Here, we add another potential mechanism to this list: fitness-dependent dispersal. Our simulations suggest that this type of dispersal (characterized by a threshold density) has a very strong synchronizing effect. It can even synchronize dynamics of systems which would show uncorrelated chaotic behaviour in the absence of dispersal. Since chaos is characterized by a divergence in the dynamics of identical systems with very slightly differing initial conditions, this is particularly impressive, as the synchronizing effect of migration has to overcome the desynchronizing effect of the intrinsic local dynamics. Furthermore, we observed a synchronizing effect even for very high threshold values, where migration occurred only occasionally. This has important ecological implications: occasional fitness-dependent dispersal may cause synchronization in an observed system even if (over a short study period) no migration between the different regions is observed. Lastly, and perhaps most interestingly, these results are not dependent on a strong regularity in the action of the dispersal threshold. Even when the threshold values on the sites where distributed in an uncorrelated and highly variable way (Fig. 6a), and varied every generation (Fig. 6b), the synchronizing power of this mechanism is in no way diminished. This provides very suggestive evidence that fitness-dependent dispersal (or other mechanisms generating a dispersal threshold) have a very powerful synchronizing effect on spatial dynamics and that this result is robust against strong heterogeneous stochastic perturbations. We hope that the results presented here will encourage theoreticians to explore the consequences of similar mechanisms in other modelling situations, and empiricists to return to their data with eyes for a fresh potential explanation for observed synchrony.
An interesting recent development in ecology and conservation biology concerns the concept of the Rescue Effect (Brown & Kodric-Brown 1977), where regions of a species’ range that have experienced localized extinctions are recolonized by individuals migrating from surviving populations. At its core, this concept relies on asynchrony between populations. It has been shown theoretically how local dispersal can permit the persistence of populations that would become extinct in isolation, due to the rescue effect (Hassell et al. 1991; Adler 1993; Rohani & Miramontes 1995). Here, we have shown, however, that these conclusions are fundamentally altered when the assumption of passive dispersal is relaxed. In this paper we have shown that fitness-dependent dispersal acts to synchronize.
Making models (especially the strategic type of model described here) requires the making of simplifying assumptions. In view of the interesting behaviour which we report here, it is important to consider how this behaviour might be modified by perturbations to the underlying assumptions of the model. In common with many other researchers, we assume that dispersal is synchronized. This will be realistic for some species where some external trigger acts to synchronize migrants from different populations, but will not hold universally. In fact, relaxation of this assumption makes very little difference to the observed population dynamics, as one would expect from the intuitive arguments given in the results section. We have explored two further alternative rules for the timing of dispersal: one where migration events occurred in purely random order, and one where migration first occurs from the patch with the highest population density (i.e. the lowest per capita fitness). As before, this process continued until all patches had population densities above the threshold. The main effect of this was that the number of migration events per generation decreased considerably compared to the synchronous case (see Fig. 7). However, the population dynamics were not affected (all three dispersal protocols used to produce Fig. 7 produced dynamics visually identical to those of Fig. 2b).
Another assumption which we make is that the density-dependent reproduction phase is completely temporally segregated from the dispersal stage. This will be realistic for some, but not all species. Relaxation of this assumption would require study using an individual-based model, previous work on a simpler system (Ruxton 1996b) suggests that relaxing this assumption would have little qualitative effect on the results described here. Further simulations (not presented) suggest that our results are not dependent on the exact form of the density dependence nor on the use of a square lattice. We have also assumed that dispersal is cost free, in that there are no losses during dispersal except for those individuals lost across the boundary. This assumption will often be a poor representation of real systems (see Ruxton, Gonzalez-Andujar & Perry 1997). It is possible to understand the consequences of adding a cost to dispersal intuitively based on the results we have already obtained. Clearly, such a cost will make dispersal a less attractive option for an individual (i.e. it will effectively reduce Δ). Hence, dispersal will be less in a system where such movements incur a mortality cost. This will mean that the simplifying, synchronizing and stabilizing effects described here will be less strong, although they will not disappear altogether. Indeed, we find that synchronization and simplification occurs even when Δ is very low (Fig. 2b). At present, we assume that the dispersal phase is sufficiently long that individuals can make many moves per generation. Adding a cost to dispersal will reduce the number of migration steps per individual and so will reduce the total number of migratory rounds per generation. Even more importantly, the effects of this perturbation will increase with system size and are likely to be of ecological relevance. Another assumption we have made in the model is that the boundaries of the metapopulation are absorbing. We require this restriction since it provides a mechanism for the loss of individuals from the ensemble population, thus avoiding a situation where the ensemble population is so high that individuals continually disperse and never settle. Adding a mortality risk to dispersal makes infinite dispersal impossible, even with reflecting or repeating boundary conditions. Furthermore, imposing such a mortality risk to all dispersal events will lead to much faster reduction of the ensemble population, since individuals will not have to travel to the boundaries before being eliminated.
Fitness-dependent dispersal, as we have modelled it, has a strong similarity to the method of dispersal used in the individual-based foraging models of Bernstein, Kacelnik & Krebs (1988, 1991). In their model, like ours, individuals migrated from habitats whose quality fell below a threshold. However, unlike our model where the threshold was a fixed property of individuals, in their model the threshold was updated following habitat sampling experience. A similar system of sampling and updating an individual's estimate of inter-patch variation in quality would be an interesting and realistic extension to the model presented here.
Most previous representations of migration in metapopulation models implicitly assume that an individual's propensity to leave a patch and its choice of the patch it settles on are fixed traits, independent of local population density or any other factors which could affect fitness. In many cases, this will lead to individuals lowering their fitness by migrating to a less profitable habitat. The circumstances where we would expect to see such ‘blind’ dispersal in nature will be very limited, as selection will act to produce more adaptive dispersal strategies. At least for our part, we have to admit that our adoption of this dispersal pattern in previous works was based on a desire for simplicity of expression rather than biological realism. Here, at a relatively small cost in mathematical and computational convenience, we have presented an alternative representation of migration; where decisions about when to migrate and where to migrate to are based on seeking to improve subsequent reproductive success. We have shown that (both in terms of stability and synchrony) this type of dispersal has a significantly different effect on population dynamics than conventional blind dispersal. This means that the form of dispersal included in a spatial model can have a very large impact on the predictions of that model. Given that blind dispersal is a poor representation of the actual dispersal mechanism of most natural systems, its adoption on grounds of simplicity may be hard to justify if this selection has a strong influence on model predictions. For this reason, we suggest that modellers need to pay more attention to capturing biological reality when characterizing dispersal, and that further studies like the one presented here are implemented with a view to strengthening our understanding of how the form of dispersal in a given system affects population dynamics. The results presented here give us reason to expect that these effects could be quite dramatic.
P.R. was supported by a NERC post-doctoral research fellowship. G.D.R. acknowledges the support of the Nuffield Foundation and the Royal Society.
Received 13 March 1998;revisionreceived 14 August 1998