Space and noise

Populations are rarely structured simply by births and deaths: the role of space and spatial structure are central to the patterns of distribution and abundance of predators and prey. Although the direct, explicit inclusion of spatial structure is a relatively recent development (Giplin and Hanski, 1991; Hanski and Giplin, 1997; Hanski and Gag-giotti, 2004), a number of early ecological studies argued that space might affect the persistence of different ecological interactions (e.g. Nicholson and Bailey, 1935, Andrewartha and Birch, 1954; Hassell and May, 1973).

Spatial scale can have profound influence on the dynamics, distribution, and abundance of predators and prey. By aggregating patches of predators and prey into increasingly larger units of habitat (e.g. leaf, twig, branch, tree, forest), different ecological processes and mechanisms become important to the dynamics of the interaction. For instance, the statistical patterns associated with the distribution of parasitism are clearly dependent on the scale of the observations (Heads and Lawton, 1983) and what constitutes a patch (Waage, 1979). However, as spatial scale increases, the size of the samples at each scale decline but the range of densities increases. This makes predicting the outcome at different spatial scales difficult, which is clearly highlighted in a comprehensive study of the mortality factors affecting the cynipid gall-former Andricus quercuscalicis (Hails and Crawley, 1992). Mortality of this gall-former can be attributed to one of nine different causes, five of which were due to predation. Patterns associated with predator-induced mortalities were shown to vary across spatial scales and between years. For instance, bird predation on galls was positively density dependent across all scales in one year and highly variable (positive and negative) the following year. Similarly, parasitism by Mesopolobus fuscipies varied from tree to tree, with some trees showing positive and others showing negative density dependence as spatial scale changed. It is important to note that the persistence of predator-prey interactions does not necessarily require density-dependent processes to operate at all times, in all places, or at all scales (e.g. Taylor, 1988): it is entirely plausible that the persistence of these interactions is masked by population redistributions and stochasticity.

The dynamical implications of mixing of predators and prey populations have been widely explored (Allen, 1975; Reeve, 1990; Hassell et al, 1991b). Assuming implicit space, Hassell and May (1988) show that incomplete mixing of hosts and parasitoids can allow the persistence of an otherwise unstable predator-prey interaction. This is particularly marked if the host completely mixes but the parasitoid is sedentary. Under this extreme scenario, patches of low host density are often unable to support a parasitoid population and are effectively refuges from parasitism leading to a stable host-parasitoid interaction (Hassell and May, 1988). Other spatially implicit representations of predator-prey interactions have explored the stability properties of the overall regional predator-prey interaction compared with the localized interaction (Reeve, 1990; Wilson et al., 1998). Reeve (1990) concluded from his study that the dynamical stability properties at the regional scale were essentially the same as those observed in the localized interaction. If the local population dynamics were unstable then extinction of the predator-prey interaction was expected at the regional scale (Reeve, 1990). However, details about the mechanistic interaction between predator and prey can disrupt this general finding. Rohani et al. (1996) showed for a range of ecological models that the broad stability effects did not differ between local and regional scales. However, under high predator (or prey) overdispersion (k << 1), then the regional-scale dynamics might be unstable even though the local dynamics are stabilized (by the highly nonlinear effects of parasitism; May, 1978a). Similarly, the coupled effects of predation and environmental noise can lead to disparity between the regional and local dynamics (Reeve, 1990). Stochastic variation in host fecundity and parasitoid attack rate between patches can destabilize the local dynamics but prevent extinction of the interaction at the regional scale (due to patch turnover and rescue). Other forms of sto-chasticity, such as demographic stochasticity (that associated with the inherent fluctuations of birth, death, and dispersal) have similar effects (Wilson et al., 1998). Coupled with restricted dispersal, demographic stochasticity introduces heterogeneity among predator-prey patches, leading to persistent interactions at the regional scale but with local extinction of patches.

More recently, it has been shown that predator-prey metapopulations can be influenced by both stochastic and deterministic processes (Bonsall and Hastings, 2004). Exploring the local and regional dynamics of the interaction between the bruchid beetle C. chinensis and its parasitoid, Anisopter-omalus calandrae, Bonsall and Hastings (2004) showed how demographic stochastic processes dominate at the local scale yet this noise is unde-tectable at the regional scale. By fitting different population models to the regional predator-prey time series, it was shown that identifying such demographic stochasticity is confounded by noise operating differently in different patches. This leads to noise being mis-identified as environmental rather than demographic stochastic perturbations, and is a consequence of the simple statistical phenomenon of the central limit theorem. By aggregating predator-prey patches which are experiencing demographic stochasticity (often described by a Poisson distribution), we can create the illusion that the regional predator-prey interaction is experiencing 'environmental' stochasticity (often described by a normal distribution). Consequently, the type of noise and its effects operating in predator-prey interactions at different spatial scales can be easily mis-interpreted (Bonsall and Hastings, 2004).

One fundamental way in which space can affect predator-prey interactions is through the processes of limited dispersal linking otherwise local, independent populations. This is the metapopulation paradigm (Levins, 1969, 1970) and it is the central theme in understanding how the dynamics of ecological interactions scale from local to regional levels (e.g. Hanski and Gaggiotti, 2004). Theoretical models of spatially explicit predator-prey interactions reveal that even if the local dynamics are unstable, the regional interaction can persist (Hassell et al., 1991b; Comins et al., 1992; Wilson et al., 1993). This occurs because the local populations tend to fluctuate out of phase, enabling extinct patches to be rescued (through the immigration of prey) and allowing the whole metapopulation to persist. A number of different spatial patterns are associated with such regional dynamics that are generated principally through the process of limited dispersal (Figure 5.9). Under low host and high parasitoid dispersal, crystal lattice patterns may emerge (Figure 5.9c). As host dispersal increases, indeterminate patterns (spatial chaos) are observed (Figure 5.9b), but the predominant type of spatial pattern takes the form of predator-prey spirals (Figure 5.9a). These spirals are characterized by the local population densities forming spiral waves which rotate around relatively fixed focal points. The regional dynamics, however, are relatively complex limit cycles influenced by the position and number of the focal points (which vary through time in a non-repeating way; Hassell et al., 1991b; Comins et al., 1992).

Once the assumption of discrete space or patches is relaxed, we find that the dispersal of predator and prey can lead to a range of dynamical outcomes (Kot, 1992; Neubert et al., 1995). Greater dispersal of the predator (relative to the prey) can lead to a range of dispersal-driven, period-doubling bifurcations resulting in unstable interactions between predators and prey (Kot, 1992). Extending this idea, White et al. (1998) show how wolf-pack territoriality and the spatial interaction between wolves and deer can be described by simple rules

■ill" ,""ni l" I1"»" I . Ill 111 II I.

Figure 5.9 Maps showing the spatial distributions of host and parasitoids from a spatially explicit version of the Nicholson-Bailey model (Hassell et al., 1991b, Comins et al., 1992) with parameters l = 1.3 and a = 0.01. In each case the lattices have absorbing boundaries and interactions are initialized by seeding a single patch with a small number of hosts and parasitoids. The patterns are (a) spiral waves, obtained with host and parasitoid dispersal fractions of 0.5, (b) chaotic (indiscernable) patterns, obtained with host and parasitoid dispersal fractions of 0.2, and (c) crystal lattice patterns, obtained with host dispersal of 0.04 and parasitoid dispersal of 0.9. Lattice sizes were 35 x 35.

of movement behaviour. For example, movement of wolf packs towards regions of higher prey density leads to spatial segregation of predator packs, reduced competition between predators, and allows the establishment of prey gradients between wolf packs (White et al., 1998). These behaviours give rise to spatial patterns in the distribution and abundance of predators and prey (Gueron and Levin, 1993; Gueron et al., 1996).

Territorial structure, and consequently habitat size, has an influential effect on spatially explicit predator-prey interactions. Ecological interactions in small localized places are often prone to extinction and there is a positive relationship between metapopulation persistence and habitat size (Hanski, 1999; Bonsall et al., 2002). Theoretically, lattices of a small, finite size restrict the possibility of asynchrony in the local dynamics (all patches are in phase) and as such patches can not be rescued from extinction (Hassell et al., 1991b; Comins et al., 1992). As habitat size increases, the possibility of asynchronous dynamics increases and the probability of persistence is greater. However, only rarely has this effect of increased persistence due to spatial processes and habitat structure been observed (Huffaker, 1958; Holyoak and Lawler, 1996; Hanski, 1999; Ellner et al., 2001; Bonsall et al., 2002).

As mentioned above, in an original set of experiments, Huffaker and colleagues (Huffaker and Kennett, 1956; Huffaker, 1958; Huffaker et al, 1963) investigated the effects of spatial structure in a mite predator-prey system. First, Huffaker (1958) showed that in the absence of dispersal, the predator-prey interaction was prone to extinction. Second, by manipulating mite movement he showed that the persistence of the interaction could be increased. Finally, in extending the system to more patches and greater complexity, the long-term persistence of the predator-prey interaction could be attributed to the effects of habitat size and dispersal (Huffaker et al., 1963). More recently, the role of spatial structure on the persistence of predator-prey interactions has been thoroughly explored (Holyoak and Lawler, 1996; Ellner et al., 2001; Bonsall et al., 2002). The overriding consensus from this range of studies on different experimental organisms is that persistence of predator-prey interactions is critically dependent on the process of dispersal. For example, recent work on the metapopulation dynamics of C. chinensis and A. calandrae has shown that the persistence of this extinction-prone host-parasitoid interaction is enhanced by metapopulation processes (Bonsall et al., 2002). By controlling for the effects of resource availability affecting the predator-prey interaction, metapopulation persistence was shown to be driven principally by coupling patches through limited dispersal, with larger systems persisting for longer. In a comparable study, Ellner et al. (2001) have shown that habitat structure per se has a relatively weak role in the persistence of predator-prey metapopulations and it is the reduced probability of attack by the predator at the patch scale that allows the system to persist. The role of habitat structure and spatial scale has a central role on the dynamics of predator-prey interactions, and these effects have now been observed to operate in more complicated, multispecies predator-prey assemblages (Hassell et al., 1994; Comins and Hassell, 1996; Bonsall and Hassell, 2000; Bonsall et al, 2005). The special case of interactions between microparasitic diseases and their hosts provide some very striking illustrations of the importance of spatial scale and spatial structure. Consider, in particular, the dynamics of measles in England and Wales (Grenfell et al., 1994, 2001, 2002; Grenfell and Bolker, 1998; Bjornstad et al, 2002). Aggregated data from urban and rural regions show the seasonal, biennial dynamical pattern with epidemics in urban places coming ahead of rural ones (Grenfell and Bolker, 1998). This difference in the timing of the epidemics is due to the strength of local coupling between urban and rural places. Patterns of spatial synchrony are related to population size: the number of cases in larger cities (large population size) tend to be negatively correlated, whereas there is no correlation in case reports between rural places with small population size (Grenfell and Bolker, 1998). This regional heterogeneity leads to hierarchical epidemic patterns. In the small rural places, the infection fades out in epidemic troughs, although this is clearly dependent on the degree of coupling to larger urban places (Finkenstadt and Grenfell, 1998).

These processes give rise to a range of spatial dynamical patterns such that larger cities show regular biennial cycles whereas in small towns disease dynamics are strongly influenced by sto-chasticity (Grenfell et al., 2001). Most recently, the dynamics of measles epidemics have been shown to depend on the balance between nonlinear epidemic forces, demographic noise, environmental forcing, and how these processes scale with host population size (Grenfell et al., 2002).

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