MICHAEL B. BONSALL, DAVID R. FRENCH and MICHAEL P. HASSELL

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1 Ecology , Metapopulation structures affect persistence of predator Blackwell Science, Ltd prey interactions MICHAEL B. BONSALL, DAVID R. FRENCH and MICHAEL P. HASSELL Department of Biological Sciences and NERC Centre for Population Biology, Imperial College at Silwood Park, Ascot, Berkshire SL5 7PY, UK Summary 1. We investigate the metapopulation dynamics of an extinction-prone host parasitoid interaction. 2. Single cell host parasitoid interactions show diverging oscillations resulting in rapid extinction of both host and parasitoid. By linking cells together into metapopulations and controlling for increased availability of resource, persistence time of the trophic interaction is enhanced. 3. Metapopulation persistence is influenced by spatial structure and local demographics. Using nonlinear mixed models, we show that the persistence time of these host parasitoid metapopulations is determined by habitat size and local within-patch population dynamics. 4. We use a metapopulation model to show how the role of local demographic population structure and habitat size act concomitantly to affect persistence. Key-words: Anisopteromalus, Callosobruchus, mixed models, space, time series. Ecology (2002) 71, Ecological Society Introduction The role of space is central in determining the patterns of species distribution and abundance. Although the explicit inclusion of spatial structure into population ecology is a relatively recent development (Gilpin & Hanski 1991; Hanski & Gilpin 1997), a number of early studies proposed that some spatial mechanism might increase the persistence of ecological systems (Nicholson & Bailey 1935; Wright 1940; Andrewartha & Birch 1954). For example, the role of space was postulated as a principal factor promoting persistence of predator prey interactions (Nicholson & Bailey 1935; Huffaker 1958; Huffaker, Shea & Herman 1963; Pimentel, Nagel & Madden 1963). Nicholson and Bailey illustrated theoretically that predator prey interactions, and in particular host parasitoid interactions, are unstable due to the inherent time-lag in the numerical response of the predator to changes in prey abundance (Nicholson & Bailey 1935). Given the ubiquity and diversity of host parasitoid interactions (Askew 1971; Hassell 1978, 2000), the mechanisms leading to the persistence and stability of Correspondence: Mike Bonsall, Department of Biological Sciences, Imperial College at Silwood Park, Ascot, Berkshire, SL5 7PY. Tel.: ; Fax: m.bonsall@ic.ac.uk these enemy victim interactions have been the focus of considerable research interest (e.g. Hassell 1978, 2000; Hochberg & Ives 2000). One predominant mechanism promoting the persistence and stability of this trophic interaction is spatial heterogeneity (Bailey, Nicholson & Williams 1962; Hassell & May 1974; Hassell et al. 1991a): some individuals by virtue of their spatial distribution are more likely to be parasitized than others. More recently, spatially explicit mechanisms acting at the scale of the metapopulation (populations linked by limited dispersal) have been proposed as factors leading to the persistence of host parasitoid interactions. Limited dispersal and asynchronous dynamics between subpopulations can allow increased persistence of otherwise non-persistent interactions (Levins 1969, 1970; Gilpin & Hanski 1991; Hanski & Gilpin 1997). The role of spatial structure has been postulated as a principal mechanism increasing persistence time of single host single parasitoid interactions (Allen 1975; Hassell, Comins & May 1991b), competing parasitoid interactions (Hassell, Comins & May 1994; Comins & Hassell 1996) and shared enemy interactions (Hassell et al. 1994; Bonsall & Hassell 2000). In general, four central conditions for metapopulation level persistence have been identified (Hanski 1999). These are that (i) habitat patches can support locally breeding populations, (ii) all patches are at risk

2 1076 M. B. Bonsall, D. R. French & M. P. Hassell Fig. 1. Schematic diagram of the experimental four-cell metapopulation. Cells were linked and dispersal was restricted using acetate gates. of extinction, (iii) recolonization must be able to occur, and (iv) the dynamics between patches are asynchronous. Here, we evaluate these conditions for a trophic interaction and test the hypothesis that the persistence of an extinction-prone host parasitoid interaction is enhanced by metapopulation processes. We show that persistence is not only affected by metapopulation size but that it is also influenced by the local demographic and population dynamic processes operating within the metapopulations. We begin by describing the host parasitoid system used and the empirical results obtained. A metapopulation model is then presented to investigate our findings and these results are discussed with reference to recent developments in metapopulation theory. Materials and methods Laboratory microcosms were used to test the role of spatial structure on the persistence of the interaction between the bruchid beetle, Callosobruchus chinensis (L.) (Coleoptera: Bruchidae), and its specialist pteromalid parasitoid, Anisopteromalus calandrae (Howard) (Hymenoptera: Pteromalidae). Clear plastic boxes ( mm) were used as the baseline cell for the study. Single cell C. chinensis populations (replicated four times) and host parasitoid populations (replicated 12 times) were established to assess the local dynamics of this predator prey interaction. Metapopulation systems were created by joining cells together with 2 5 mm holes placed in the centre of each wall to permit dispersal to the nearest two, three or four neighbours depending on the position of the cell in the metapopulation. Preliminary experiments showed that this hole size (2 5 mm), open for 2 h each week was sufficient to allow (limited) dispersal of both hosts and parasitoids (French & Travis 2001). Simple acetate barriers between cells were used to restrict dispersal (see Fig. 1). Four different multi-patch systems were investigated: square lattices comprising four-cell systems linked by limited (replicated 12 times) or unlimited (replicated four times) dispersal and 49-cell systems linked by limited (replicated four times) or unlimited (replicated four times) dispersal. These treatments were used to test the effects of limited dispersal. Controls (unlimited dispersal treatments) were used to ensure that any effects on persistence were due to limited dispersal and not simply due to the increase in resource availability. Experiments were seeded over a 3 week period by introducing eight pairs of hosts in the single cell (population experiments) or cell 1 (metapopulation experiments: Fig. 1). After hosts had established, four pairs of parasitoids were released over a 2-week period. Parasitoid release into the single cell experiments followed two strategies: early (1 and 2 weeks after the start of the experiment) or late (10 and 11 weeks after the start of the experiment). In the metapopulation experiments, the parasitoid was released 8 and 9 weeks after the start of the dispersal regime. In the unlimited dispersal treatments, the parasitoid was released 8 and 9 weeks and weeks into the 49 and four-cell arenas, respectively. Black-eyed beans (Vigna unguiculata (L.) Wapl. (Leguminosae)) were replaced following a 4-week resource renewal regime with three beans being replaced each week. Removed beans were stored in ventilated vials for a further 4 weeks and any newly emerging animals were released back into the appropriate cell. Time series were obtained by counting both alive and removed dead insects each week from every cell. In the multi-patch systems, insects were counted before and after the dispersal period (2 h). This established protocol allowed the net number of individuals

3 1077 Host parasitoid metapopulation dynamics Fig. 2. Representative time series (of total square-root abundance) for (a) C. chinensis in single cells, (b) C. chinensis A. calandrae in single cells, (c) four-cell host-parasitoid metapopulation with unlimited dispersal, (d) four-cell host-parasitoid metapopulation with restricted dispersal, (e) 49-cell host-parasitoid metapopulation with unlimited dispersal, and (f ) 49-cell host-parasitoid metapopulation with restricted dispersal. Solid black line is C. chinensis and dashed line is A. calandrae. emerging in each cell to be recorded. All experiments were undertaken under controlled environmental conditions (30 C, 70% r.h., 16 : 8 light : dark cycle) and all statistical analyses were completed in S-Plus 2000 (Mathsoft 2000). Results DYNAMICS OF SINGLE-CELL HOST ALONE AND HOST PARASITOID INTERACTIONS Representative dynamics of the single-cell bruchid and bruchid parasitoid interaction, the four-cell metapopulations and the 49-cell metapopulations are illustrated in Fig. 2. The population time series of the single-cell C. chinensis populations can be described by linear stochastic processes. Time series analysis of the single species population dynamics reveals no evidence for nonlinear processes. We fit linear autoregressivemoving average (AR-MA) time series statistical models (up to AR order 10 and MA order 6 to the detrended time series with the likelihood and AIC conditioned on the same number of observations) to the single species (C. chinensis) populations. Statistical model simplification and criticism show that both density-dependent and density-independent processes underpin the structure of the observed population dynamics (Table 1). In particular, the observed dynamics (Fig. 2a) are composed of both density-dependent (autoregressive) and density-independent (moving average) processes. Although there is a mathematical equivalence between these two stochastic processes (Box & Jenkins 1970), associating autoregressive and moving average processes with density dependence and independence is an appropriate first approximation to understanding the dynamics of ecological time series. Finite order combinations of both these stochastic processes are often necessary to provide the most parsimonious descriptions of the process structures underpinning ecological time series. Analysis reveals that first or second order densitydependent processes are the best descriptors for the autoregressive component of the dynamics while the most parsimonious density-independent structures are either first or second order (Table 1). Given that the populations interact in a continuous manner with overlapping generations and sampling occurred at frequencies greater than one generation, the statistical density-dependent structures identified (Table 1) suggest the dynamics of C. chinensis are underpinned by direct density-dependent processes. As the populations were maintained in controlled environments, the identified statistical density-independent components arise through the differences in demographic or age-structure operating within and between the populations.

4 1078 M. B. Bonsall, D. R. French & M. P. Hassell Table 1. Summary of the time series statistical models, order of the density-dependent (AR) and density-independent (MA) components, AIC values and parameters values (± SE) for the single-species, C. chinensis time series Series AR MA AIC Model parameters (± SE) I AR: (0 064) MA: (0 056) II AR: (0 038) MA: (0 015) III AR: (0 046) (0 04) MA: (0 049) (0 046) IV AR: (0 055) MA: (0 049) In the absence of parasitoids, all the host populations persisted for over 75 weeks, representing approximately 20 generations. However, the introduction of A. calandrae significantly decreased the probability of persistence (χ 2 = 13 8, d.f. = 1, P < 0 001) with an average time to extinction of 5 38 (± 1 43 SE) weeks. Survival analysis showed that there was no significant difference of different parasitoid release times on the persistence of the interaction (χ 2 = 1 0, d.f. = 1, P = 0 324). Mean time to extinction for the early (week 1 2) release protocol was 6 93 (± 2 73 SE) weeks and 3 75 (± 0 55 SE) weeks for the late (week 10 11) release protocol. PERSISTENCE OF HOST PARASITOID METAPOPULATIONS To test the hypothesis that the extinction-prone host parasitoid interaction can persist as a result of metapopulation dynamics, the times to extinction of single cell, four-cell and 49-cell systems were considered. Survival analysis showed that the size of the metapopulation was a significant factor (χ 2 = 11 4, d.f. = 2, P < 0 001) with interactions in larger systems persisting longer (Fig. 3). In order to corroborate that these differences were due to the effects of restricted dispersal and not an effect of increased resource availability, persistence times between limited and unlimited dispersal systems were compared. Significant differences were observed between limited and unlimited dispersal systems with host parasitoid interactions in the limited dispersal systems persisting longer (χ 2 = 6 2 d.f. = 1, P = 0 013). To explore synchrony in local host densities and assess whether patch dynamics were correlated, we examined the correlation coefficients of beetle and wasp densities between each of the patches and all other patches in each of the metapopulations. Local host densities were weakly positively correlated (ρ = 0 071, t = 2 476, P = 0 013) between patches in the four-cell metapopulations but showed no correlation between patches in the 49-cell (ρ = 0 001, t = 0 065, P = 0 948) metapopulations. Whereas, local wasp densities showed no correlation between patches in the fourcell (ρ = 0 025, t = 0 801, P = 0 423) or in the 49-cell (ρ = 0 019, t = 0 985, P = 0 325) metapopulations. DEMOGRAPHIC STRUCTURE AND METAPOPULATION PERSISTENCE Given that age-structure can affect the within-patch populations dynamics (see above), we postulate that the mechanisms responsible for the differences in persistence are influenced by the local patch demographic processes operating within the metapopulations. To explore the mechanistic basis for the observed differences in persistence, we fitted nonlinear mixed models (Pinheiro & Bates 2000) to the replicated single-cell and four-cell host parasitoid systems. From the analysis of persistence of the metapopulations (see above), we have assumed that persistence declines exponentially with time. However, the withinpatch dynamics and habitat size may also affect the rate of decline and time to extinction. To explore this hypothesis and the mechanistic basis for the observed differences in the persistence of host parasitoid interaction in different sized metapopulations, we use the following baseline statistical nonlinear mixed model: n ij = φ 1i exp( φ 2i t ij ) + ε ij eqn 1 where n ij is the population abundance in the i th arena (i = 1 24) at the j th time point ( j = 1 30), φ β b i φi = and β (fixed effects) represent φ = β b 2 2 2i the average value of the individual parameters (φ i ) in the sample of arenas and b i (random effects) represent the deviations of φ i from the population average. Random effects are assumed to be independent for different arenas and the within-group (patch) errors (ε ij ) are also assumed to be independent for each different arena and time point, and also independent of the random effects. Wasp abundance is the adopted surrogate for assessing the persistence of the trophic interaction. Arena-specific (dashed line) and population average (solid line) predicted rates of decline in wasp abundance (open circles) for the nonlinear mixed model where the within-group (patch) errors are autocorrelated and metapopulation size is included as a covariate are shown in Fig. 4. Although the nonlinear model is appropriate for describing the decline in wasp abundance, the incorporation of population demographic structure and metapopulation size increase the explanatory

5 1079 Host parasitoid metapopulation dynamics Fig. 3. Mean (± SE) persistence times for the host parasitoid interaction under the five different treatments. Survival analyses show significant differences in persistence under different metapopulation sizes and dispersal treatments. Fig. 4. Persistence of the trophic interaction (wasp abundance) for (a) the 12 single cell populations and (b) the 12 4-cell metapopulations. Each panel shows the overall (solid line) and the individual arena fit (dashed line) of the nonlinear model (eqn 1) and wasp abundance (open circles). power of the statistical model. Time to extinction is influenced by the demographic effects operating within the patches: by modelling the within-group (patch) errors as a set of autocorrelated (AR(1)) errors, a likelihood ratio test showed that local demographic effects have a significant influence on the persistence of the host parasitoid interaction (Table 2). Further, incorporating metapopulation size as an explanatory variable is also a significant determinant of persistence of the host parasitoid interaction (Table 2). A metapopulation model In this section we investigate the role of demographic structure and population dynamics by developing a stochastic metapopulation model. Although the model is relatively generic, we use it to corroborate our empirical observations that demography can affect the persistence time of a metapopulation. Given a fixed connectivity matrix (that is the fraction of emigrants and immigrants from each patch is known)

6 1080 M. B. Bonsall, D. R. French & M. P. Hassell Table 2. Summary of the nonlinear mixed model. Likelihood ratio tests show that within-arena demographics and habitat size are significant determinants of the persistence of the host parasitoid interaction Model d.f. AIC BIC Log likelihood L.R.T. P-value Independent errors AR(1) errors AR(1) errors AR(1) errors + size we evaluate the role of different demographic processes on the persistence of different metapopulation structures. MODEL STRUCTURE The model is structured as a metapopulation model in that the within-patch dynamics are governed by stochastic processes principally described by mixed autoregressive (AR: density-dependent) and movingaverage (MA: density-independent) processes. For instance, dynamical changes on a patch driven by a second order density-dependent processes (N i,t, N i,t 1 ) and a second order density-independent processes (Z i,t, Z i,t 1 ) are described by: N i,t+1 = f(n i,t, N i,t 1, Z i,t, Z i,t 1, ε t ) eqn 2 of dispersal conditions. Here, it is assumed that individuals only disperse to the nearest four neighbours with periodic boundaries. For a regularly spaced four patch (2 2) metapopulation (Fig. 1), the connectivity matrix would take the form: 0 di λ = 2 di 2 0 di di di d 2 2 i 0 d i 2 d i 2 0 eqn 4 where d i is the fraction of individuals dispersing from each patch. where f( ) is the stochastic process underpinning the local dynamics and ε t is an identically, independently distributed random variable. Using this time series based approach for modelling the within-patch dynamics allows different demographic population structures to be analysed without specifying the precise mechanistic basis underpinning the local dynamics. We explore six different stochastic processes that have direct biological interpretation: AR(1), AR(1)-MA(1), AR(1)-MA(2), AR(2), AR(2)-MA(1) and AR(2)-MA(2). An AR(1) process is analogous to a direct density-dependent effect that leads, in the main, to equilibrium dynamics while an AR(2) process is a delayed density-dependent effect giving rise to oscillatory or equilibrium dynamics ( Royama 1992). The moving average effects account for correlated errors reflecting changes due to demographic stochastic effects operating within the populations. Net change of individuals in a patch due to dispersal is described by: N, + = ( 1 d ) N + λ N it d i i ij j j= 1 p eqn 3 where N i and N j are the numbers of individuals in the i th and j th patch, λ is the connectivity matrix: λ ij determines the fraction of individuals leaving the j th patch and entering the i th patch, and d i is the density-independent dispersal rate (d i is simply the sum of fraction dispersing to each patch from the i th patch: d i = λ ij ). The j= 1 connectivity matrix (λ) can be used to describe any set p EFFECT OF DEMOGRAPHIC STRUCTURE ON PERSISTENCE TIMES We begin by investigating the effects of demographic structure on persistence time for the six different models for a 12-patch metapopulation (Table 3). Demographic structure affects the median and mean time to extinction. The median time to extinction decreases as correlated effects become more prevalent at the local patch scale. The mean time to extinction is always greater than the median time to extinction and the distribution of extinction times is right-skewed: most metapopulations are extinct after a small number of generations while a few persist for many generations. However, coefficients of variation show little difference in the variability in times to extinction under AR(1) or AR(2) processes. This suggests that other factors such as metapopulation size may have a concomitant role in determining the persistence of these multi-patch systems. EFFECT OF METAPOPULATION SIZE AND DEMOGRAPHIC STRUCTURE ON PERSISTENCE TIME We explore the effects of habitat size and different demographic structures on metapopulation persistence time for the six different models (Fig. 5). Under an AR(1) process (Fig. 5a) persistence time is a function of local demographic processes and habitat size. Persistence decreases as the correlated moving average effects

7 1081 Host parasitoid metapopulation dynamics Table 3. Summary statistics (mean, median, standard deviation, coefficient of variation) for persistence times (in generations) for different demographic models in a 12 patch (3 4) metapopulation over 100 generations Demographic model Summary Statistics AR MA Mean Median SD CV , , , , , , , , , , , , , , , , , Fig. 5. Model predictions for the effects of demographic structure on metapopulation persistence under (a) an AR(1) process and (b) a AR(2) process. Solid black line is the pure autoregressive process, dashed line is when the demographic processes incorporate an 1st order moving average process and the dot-dash line is when the demographic processes include a 2nd order moving average process. become more influential on the dynamics at the local scale. Similarly, under AR(2) processes (Fig. 5b), persistence time declines as the correlated moving-average effects become prevalent at the local scale. For example, in an eight patch system, persistence times vary from 22 generations for an AR(2)-MA(2) process up to 62 generations when the moving-average effects are negligible (Fig. 5b). As habitat size increases persistence time asymptotes and is unaffected by differences in local patch dynamics. This effect of habitat size is consistent under both AR(1) and AR(2) processes: in large multipatch systems, persistence times often exceed 100 generations. Discussion Here we have demonstrated that an extinction prone trophic interaction can persist as a result of metapopulation processes. Such studies remain relatively rare (Harrison & Taylor 1997) despite the wealth of research on metapopulation dynamics (Gilpin & Hanski 1991; Hanski & Gilpin 1997). We have shown that the four conditions for metapopulation persistence (Hanski 1999) can be evaluated for this host parasitoid interaction: (i) patches support local breeding populations of both host and parasitoid, (ii) all populations have a high risk of extinction, (iii) recolonization is possible

8 1082 M. B. Bonsall, D. R. French & M. P. Hassell through limited dispersal, and (iv) as habitat size increases, the dynamics between patches are not synchronous. Moreover, we have also demonstrated that the within-patch population dynamics also have a significant influence on metapopulation persistence. In the single species interaction, the identified stochastic processes from time series analysis are entirely consistent with other studies on bruchid population dynamics. It is known that C. chinensis competes for resources through contest competition (the autoregressive processes) (Utida 1959; Fuji 1969; Bellows 1982a,b). Coupled with age-structure effects (the moving-average processes), this density dependence can give rise to the observed fluctuating dynamics (Utida 1941a,b; May et al. 1974; Bellows & Hassell 1988; Kuno, Kozai & Kubotsu 1995). The introduction of the parasitoid leads to divergent dynamics characteristic of unregulated host parasitoid interactions. Anisopteromalus calandrae overexploits its host and due to time lags in development this leads to the eventual extinction of both host and parasitoid (Nicholson & Bailey 1935; Hassell 1978, 2000). In comparison to other studies on predator-prey dynamics in metapopulations, we show that the spatial structure has a primary influence on the persistence of the trophic interactions (Huffaker 1958; Allen 1975; Hassell et al. 1991b; Nachman 1991; Holyoak & Lawler 1996). Recently, Ellner et al. (2001) have shown that habitat structure per se has a limited role in the persistence of mite predator prey metapopulations: persistence arises through reduced attack by predatory mites on spatially subdivided prey. By controlling for resource availability and linking populations together through limited dispersal, we have shown in this study that the spatial subdivision promoted persistence of the extinction-prone host parasitoid interaction. Further, by appropriately modelling the correlation structure of the within-group (patch) errors (Diggle, Liang & Zeger 1994; Pinheiro & Bates 2000) we partitioned out the effects on persistence due to local demography and metapopulation structure. Metapopulation persistence is a nonlinear function of within-patch dynamics and habitat size. Although a number of studies have assessed the role of patch size and/or area (Pulliam 1988; Day & Possingham 1995), patch quality (Hanski & Singer 2001) and dispersal (Kuno 1981), the role of local patch demographics on metapopulation dynamics has only recently been considered (Hastings 1991; Gyllenberg & Hanski 1992; Gyllenberg, Hanski & Hastings 1997). Here, we have shown empirically and theoretically that there is an effect due to the legacy of individuals. Individuals leaving a resident patch and dispersing to a new patch may still have an influence on the persistence of the resident patch through delayed density-dependent effects and also an effect on the new patch through direct density-dependent processes. Dispersal allows individuals to affect the demographic structure and dynamics of the local population as well as the overall metapopulation. By developing a series of stochastic metapopulation models where the number of patches and within-patch dynamics vary, we showed that the effects of local demographic structure can have a significant impact on the persistence of species interactions. While it is known that density dependence operating at the local scale is necessary for metapopulation persistence (Hanski, Foley & Hassell 1996), a corollary of this is that persistence is also influenced by the structure of demographic processes operating within the patches. In general, it appears that correlated stochastic effects decrease the median time to extinction. The effect of stochasticity disrupts the regulation of the local within-patch population dynamics and can increase the likelihood of metapopulation extinction. Recently, a number of metapopulation characteristics have been shown to depend on particular demographic parameters (Hanski 1998; Lande, Engen & Sæther 1998; Nachman 2000). Although we assumed that the correlated stochastic effects influence the demographic structure within patches, it is clear that the effects of both environmental and demographic stochasticity will act concomitantly on local populations to affect persistence. Recently Nachman (2000) has shown how such stochastic events may push populations into regions of parameter space where long-term persistence is impossible. Populations with narrow ranges of demographic parameters will be more susceptible to extinction events. Further, density-dependent processes such as migration (Sæther, Engen & Lande 1999; Nachman 2000) and population carrying capacity (Lande 1993; Hanski 1998) may also affect local population dynamics and, consequently, determine extinction events in metapopulations. Several studies have illustrated that simple indices and measures may be used to determine the persistence of metapopulations (Mangel & Tier 1993; Adler & Nuereberger 1994; Hanski & Ovaskainen 2000). Here we have shown that persistence of the same metapopulation structure may be variable depending on the underlying demographic processes. The stochastic models used in the present study indicate the need to evaluate the role of within-patch population dynamics, as well as patch area, isolation and quality, as a determinant in the measure of persistence of interactions in fragmented landscapes. Acknowledgements MBB is supported by a Royal Society University Research Fellowship. We also acknowledge the support of the Natural Environment Research Council. References Adler, F.R. & Nuereberger, B. (1994) Persistence in patchy irregular landscapes. Theoretical Population Biology, 45,

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