Demographic and environmental stochasticity in predator–prey metapopulation dynamics

Authors

  • MICHAEL B. BONSALL,

    Corresponding author
    1. Department of Biological Sciences, Imperial College London, Silwood Park Campus, Ascot, Berkshire SL5 7PY, UK
      Mike Bonsall, Imperial College London, Silwood Park Campus, Ascot, Berkshire SL5 7PY, UK. Tel: 0207 5942360; Fax: 0207 5942339; E-mail: m.bonsall@imperial.ac.uk
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  • ALAN HASTINGS

    1. Department of Environmental Science and Policy, University of California Davis, Davis, CA 95616, USA
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Mike Bonsall, Imperial College London, Silwood Park Campus, Ascot, Berkshire SL5 7PY, UK. Tel: 0207 5942360; Fax: 0207 5942339; E-mail: m.bonsall@imperial.ac.uk

Summary

  • 1We studied the metapopulation dynamics and persistence of an extinction-prone predator–prey interaction. We show that the dynamics of the system are influenced by both stochastic and deterministic processes.
  • 2Using host–parasitoid metapopulation data, we develop appropriate descriptors of the local within-patch population dynamics. In particular, we show that the local dynamics are well described by a Markov chain. We show that the local dynamics are determined predominately by demographic stochastic processes and that the deterministic signal is relatively weak.
  • 3To test the hypothesis that the persistence of predator–prey metapopulations is affected by habitat size, local population dynamics and different types of (demographic or environmental) stochasticity, we fit population models to the regional metapopulation time-series. Contrary to expectations this demographic noise is, however, undetectable at the regional scale and is masked by an environmental noise process. We show that by linking patches together, the predicted environmental noise is effectively decreased as metapopulation size increases.
  • 4Using a simple spatial stochastic model, we illustrate that the effects of demographic noise are masked rapidly at the regional scale due to the statistical effects of the central limit theorem. We discuss the implications of this for understanding the dynamics and persistence of metapopulations.

Introduction

The metapopulation processes of immigration, emigration and extinction have a prominent role in understanding the population dynamics of single-species (Hanski 1999), pairwise (Ellner et al. 2001) and multispecies interactions (Holt 1997; Bonsall & Hassell 2000). The premise that the persistence of extinction-prone interactions could be enhanced through the inclusion of some spatial processes was acknowledged originally in a number of early studies (e.g. Nicholson & Bailey 1935; Andrewartha & Birch 1954). Generically, increased persistence occurs as habitat subdivisioning and heterogeneity acting at the regional scale effectively reduces the probability of global extinction by allowing rescue of locally extinct patches. However, not until relatively recently has the extent to which space can affect species interactions been more thoroughly explored (Gilpin & Hanski 1991; Hanski & Gilpin 1997; Tilman & Karieva 1997).

For instance, simple predator–prey interactions are known to be locally unstable (Lotka 1925; Volterra 1926; Nicholson & Bailey 1935) and, in certain cases, non-persistent (Nicholson & Bailey 1935). The inclusion of spatial processes was hypothesized initially to increase the persistence of the system (Nicholson & Bailey 1935; Skellam 1951) and demonstrated in a unique series of laboratory microcosm experiments by Huffaker (1968; see also Huffaker, Shea & Herman 1963). The persistence of simple predator–prey interactions was shown to depend on the inclusion of architecturally complex habitats (Huffaker et al. 1963; Pimentel, Nagel & Madden 1963). In fact, Pimentel et al. (1963) argued that laboratory predator–prey microcosms may not be simplistic and the inclusion of habitat complexity (e.g. baffles for restricting dispersal) could clearly affect the persistence of the system. Empirical observations of metapopulation processes in predator–prey interactions have established that dispersal and shifting dynamic patterns are key drivers in predator–prey interactions. Nachman (1991), for example, illustrated that mite predator–prey metapopulations were unstable and non-persistent at the individual plant scale but at the regional, greenhouse-scale systems were maintained through a shifting mosaic of interactions between predator and prey.

More recently, the effects of spatial structure on predator–prey interactions have been extended to assess whether metapopulation processes (local breeding populations, risk of extinction, probability of recolonization, asynchronous dynamics) can influence the persistence of these resource–consumer interactions (Ellner et al. 2001; Bonsall, French & Hassell 2002). For instance, Ellner et al. (2001) have shown that persistence in a mite predator–prey system is influenced more by modification of the predator's search behaviour than the subdivisioning of the habitat into local patches that are accessed only by dispersal. In a host–parasitoid metapopulation, Bonsall et al. (2002) demonstrated that the conditions for metapopulation persistence can be evaluated and showed that the dynamics of this predator–prey interaction were influenced by local demographic processes and habitat structure.

It is well known that populations are influenced by uncertainty or random perturbations. Stochasticity can affect the persistence and dynamics of populations (May 1973; Roughgarden 1975; Lande 1993; Ludwig 1996). In particular, stochastic perturbations acting within a population or across a metapopulation can affect the likelihood of local or global extinction, and understanding how such stochastic processes influences the deterministic population dynamics (Casdagli et al. 1991; Rand & Wilson 1991) remains a focus of contemporary ecological research (Dennis et al. 2001; Grenfell, Bjørnstad & Finkenstadt 2002). Such random perturbations can be manifest into two ways: as demographic or environmental stochasticity.

Demographic stochastic effects depend on the intrinsic uncertainty associated with an individual's reproduction, survival and dispersal, and is most influential in small populations. Essentially, these sampling effects are averaged out as population size increases and Bartlett (1960) showed for a variety of different ecological scenarios how demographic stochasticity can influence the dynamics of single-species, competitive and predator–prey interactions. In a series of developments, these ideas were extended to illustrate how demographic stochasticity (associated with births and deaths only) can affect colonization in island–mainland models of species diversity (MacArthur & Wilson 1967), is likely to be negligible as population size increases (May 1973), has important implications for conservation and species risk assessment (Kokko & Ebenhard 1996) and can lead to Allee effects (Lande 1998). More recently, the role of demographic noise on the dynamics of single-species populations has been thoroughly explored. For example, the dynamics of Tribolium castaneum have been shown to display a range of population dynamic behaviours from stable dynamics through to seemingly chaotic dynamics (Costantino et al. 1995, 1997). Using an age-structured model skeleton, Dennis et al. (2001) illustrated that the dynamics of T. castaneum showed significant departures from the predicted dynamics. These departures could, however, be adequately described by coupling demographic noise and the deterministic age-structured model. Similarly, the effects of demographic noise coupled with non-linear density dependence has been shown to influence the observed dynamics in childhood diseases (e.g. Grenfell et al. 2002). In particular, the predictability of measles dynamics has been shown to depend on the interaction between population size, noise and the underlying deterministic process. Grenfell et al. (2002) suggest that demographic noise can have a major effect on the dynamics of measles even when population sizes remain relatively large.

In contrast, environmental stochasticity refers to randomness imposed on a population or metapopulation by the environment. Principally, these density-independent processes are manifest through fluctuations in the birth rate or in the carrying capacity (Roughgarden 1975; Leirs et al. 1997; Grenfell et al. 1998; Heino 1998). For instance, Grenfell et al. (1998) illustrated how the dynamics of Soay sheep across the St Kilda archipelago are correlated by climatic variabilities and Coulson et al. (2001) extended this analysis to show how age-structure and environmental noise can act concomitantly to lead to variable dynamics in this single-species interaction. More recently, it has been predicted that in relatively simple single-species interactions environmental noise acting on births can lead to sustained oscillations whereas fluctuations in the deterministic system would decay away (Greenman & Benton 2003). Understanding how populations filter this environmental noise (Lundberg et al. 2002; Greenman & Benton 2003) has important implication for realistically interpreting not only single-species but also multispecies interactions. Moreover, the form or degree of correlation in the environmental noise can affect the dynamics of single-species (Dennis et al. 1995), pairwise (Ripa & Ives 2003) and multispecies (Petchey et al. 1999) interactions in counter-intuitive ways by modifying individual species interaction strengths (May 1973).

Stochasticity has, however, many largely unexplored consequences for metapopulation dynamics. It is entirely plausible that the population processes of birth, death, immigration and emigration are all influenced by local and regional stochastic fluctuations, and the effects of stochasticity at the regional scale on resource–consumer interactions could operate in non-intuitive ways. Here, using empirical data from a predator–prey metapopulation, we compare and contrast different forms of noise (demographic, environmental) acting on the dynamics and persistence of a pairwise host–parasitoid interaction. We begin by introducing the empirical system and the statistical methods of model fitting. Here, we emphasize that rather than selecting the most appropriate population model (although this is a important aspect of model fitting) our aim is to evaluate the different forms of stochasticity in a predator–prey metapopulation. In particular, our hypothesis is that demographic noise should be influential within local patches and scale to affect the persistence and dynamics of the regional, predator–prey metapopulation systems. We show that the interaction is influenced heavily by demographic stochasticity; however, this noise is masked at the regional metapopulation scale. The study concludes with a discussion of the effects of environmental and demographic stochasticity on population and metapopulation persistence, dynamics and scaling.

Materials and methods

In this section we outline the data collection methods, the mathematical models and the statistical and computational approaches for fitting the models to the data.

experimental protocol

Laboratory microcosms were used to explore the role of stochasticity and spatial structure on the interaction between Callosobruchus chinensis L. (Coleoptera: Bruchidae) and its specialized parasitic wasp Anisopteromalus calandrae (Howard) (Hymenoptera: Pteromalidae). The detailed experimental design and rationale have been outlined elsewhere (Bonsall et al. 2002) and here we give a brief overview of the methods used to collect the metapopulation time-series data.

Clear plastic boxes were used as the baseline cell for the study. Single cells were linked together (with 2·5 mm holes in the centre of each wall) to create metapopulations. Links to neighbouring boxes (acetate barriers) allowed dispersal to the nearest two, three or four neighbours depending on the position of the cell in the metapopulation. Previous studies have shown that this hole size (25 mm), open for 2 h each week, was sufficient to allow (limited) dispersal of both hosts and parasitoids (French & Travis 2001). Elsewhere (Bonsall et al. 2002) we have shown how important it is to control for the effects of increased resource availability on the persistence of a metapopulation as habitat size increases.

Experimental arenas (four-cell and 49-cell systems) were seeded over a 3-week period by introducing eight pairs of hosts into a single cell in the metapopulation experiments. After hosts had established, four pairs of parasitoids were released over a 2-week period. 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 dead insects each week from every cell from four-cell (12 replicates) and 49-cell (four replicates) systems (Figs 1 and 2).

Figure 1.

Time-series of total host (black line) and wasp (grey line) abundance from the four-cell predator–prey metapopulations. Here, we show four replicates of the predator–prey interaction to illustrate the observed metapopulation dynamics. Total population abundances were determined through weekly counts of dead and alive wasps and beetles in all cells. Mean persistence time of the four-cell systems was 31·25 weeks.

Figure 2.

Time-series of total host (black line) and wasp (grey line) abundance for the 49-cell predator–prey metapopulations. Total population abundances were determined through weekly counts of dead and alive wasps and beetles in all cells. Mean persistence time of the 49-cell systems was 42·25 weeks.

local population dynamics

To test the hypothesis that the local-patch population dynamics (within a metapopulation) were influenced by stochasticity we develop a series of Markov chain models. In particular, we assume that given the state of a patch (empty, host-only, wasp-only or host–wasp) in the metapopulation (X(t)) at time t, its state (X(t + 1)) at time t+ 1 is independent of the states at previous times. That is:

image(eqn 1)

The time evolution of this stochastic process is described completely by a set of one-step transition probabilities that describe how the process will move to state j at time t+ 1 given that it is in state i at time t. As the Markov chain must be in some state at any census point, all row sums of the transition matrix equal 1. We determine the transition matrices for the four-cell and 49-cell metapopulations based on presence/absence data of wasps and hosts and test the assumption that the changes in patch status are Markovian by determining the limiting distribution from each transition matrix. Using these Markov chain models we then determine the expected times (ti) to absorption into state (0, 0) given that the system is in a host-only state, wasp-only state or host–wasp state using well-known properties of finite Markov chains (Norris 1999):

image(eqn 2)

where N is the number of states in the Markov chain, pij are the individual transition probabilities and t0 = 0.

regional population dynamics

Given that the host–parasitoid interaction operates in cells where generations are overlapping (all stages are present simultaneously), the population dynamics are described most appropriately by a continuous-time predator–prey model with logistic prey growth:

image(eqn 3)
image(eqn 4)

where r is the growth rate of host (H), K is the host carrying capacity, f(P(t)) is the functional form for parasitism and dP is the adult parasitoid (P) density-independent death rate. To determine the expected numbers of hosts and parasitoids at the next census point these equations are then solved over a fixed time period (tt+ 7):

image( eqn 5)
image(eqn 6)

noise

The principal aim of this study is to explore how different forms of stochasticity affect the dynamics and persistence of predator–prey metapopulations. In order to evaluate the role of stochasticity on the predator–prey metapopulations we consider two forms of the stochasticity: demographic and environmental. In this section we derive the likelihood functions for estimating parameters for the logistic predator–prey model from the regional metapopulation time-series.

Demographic stochasticity

Demographic stochasticity is the variability in population size due to the probabilistic events of birth, death and dispersal. Under demographic stochasticity we considered that events that change population size are the outcome of a Poisson (Po) distribution. As such a stochastic version of the logistic predator–prey model can be represented as:

image( eqn 7)
image(eqn 8)

This stochastic model with demographic noise gives the conditional one-time step ahead distributions of the species (H and P) as independent, discrete probability distributions. Maximum likelihood estimates for the unknown parameters in the stochastic version of the logistic predator–prey model can be determined by considering an appropriate likelihood function. Quite straightforwardly, the likelihood functions to be maximized are Poisson and of the form:

image(eqn 9)

where M is the set of parameters (predator–prey model) to be estimated from the model (using the simplex optimization algorithm − see below). The overall log-likelihood for the logistic predator–prey model is then of the form:

image(eqn 10)

where Pj and Hj are the observed numbers of predators and prey, respectively, at each census point j, Ej1 and Ej2 are the expected numbers of prey and predators (determined from the predator–prey model) at each census point j.

Environmental stochasticity

Environmental noise is the variability in population size due to fluctuations in abiotic conditions. Under environmental stochasticity, changes in population size occur due to the deterministic processes of birth, death and dispersal and the effects of random noise acting additively (on a log scale − see Dennis et al. 1995) to determine population size at the next census point. The stochastic version of the logistic predator–prey model under environmental noise is of the form:

image(eqn 11)
image(eqn 12)

Here ν = (ν1, ν2) represents a vector that is assumed to have a bivariate normal distribution with mean 0, variances inline image and inline image and covariance ρ. It is assumed that the covariance between successive time-points is small (the autocorrelations are relatively weak) and as such uncorrelated. The appropriate likelihood function for this form of stochasticity is a bivariate Gaussian distribution of the form (Kotz et al. 2000):

image( eqn 13)

where Pj and Hj are the observed numbers of predators and prey, respectively, at each census point j, Ej1 and Ej2 are the expected numbers of prey and predators (determined from the predator–prey model) at each census point j, and σ1, σ2 and ρ are the standard deviations for prey and predators and the correlation between prey and predator numbers, respectively. Maximum likelihood estimates of the parameters (M) are those that minimize the negative log-likelihood. For the bivariate Gaussian distribution, the negative log-likelihood function is of the form:

image( eqn 14)

Model fitting

To minimize the negative log-likelihoods for the different stochastic implementation of the continuous-time predator–prey model (eqns 7–8, 11–12) we use a simplex algorithm (Nedler & Mead 1965; Press et al. 1992) incorporating a numerical integration routine (implemented in C). This optimization method, although slow, is relatively robust at finding the minima and maxima of functions (Press et al. 1992). This method works by using only the likelihood evaluations and after initialization the simplex minimizes the negative log-likelihood through a series of reflections and contractions. Termination criteria are used to identify when the distance moved by the simplex is smaller than a specified tolerance value (10−10). To ensure that the optimization routine reached a global minimum, the algorithm was re-run with one of the vertices initialized with the parameter values of the observed minimum (Press et al. 1992).

Results

local demography and dynamics

A test of the hypothesis that the local dynamics (variability in host abundance) and habitat size affect metapopulation persistence shows that the persistence of the predator–prey interaction is dependent on the interaction between variability in abundance and metapopulation size (metapopulation size : host variability F1,12 = 6·67, P = 0·024). However, the size of the metapopulation had a significant main effect independent of local variability on abundance (metapopulation size F1,12 = 5·113, P = 0·0431) with interactions in larger systems [mean persistence time = 42·25 (7·375 SE)] persisting for, on average, 11 weeks longer than the smaller systems [mean persistence time = 31·25 (1·813 SE)].

To investigate how local demography and habitat size interact the proportion of time that the metapopulations were in one of four different states (empty, host-only, wasp-only, host–wasp) was estimated from presence–absence data. These data indicate that in the four-cell metapopulations patches were empty with probability 0·392, occupied only by hosts with probability 0·2064, occupied only by wasps with probability 0.1968 or occupied by host and wasp with probability 0·2048. Similarly, presence-absence data highlight that patches in the 49-cell metapopulations were empty with probability 0·6568, occupied only by hosts with probability 0·1247, occupied only by wasps with probability 0·0938 or occupied by host and wasp with probability 0·1287. The proportion of patches in each particular state is the observed empirical limiting distribution of probabilities for the metapopulations.

To explore the role of demographic stochasticity on the persistence of the metapopulation model we use a series of stochastic process models. For the four-cell metapopulations the overall presence–absence Markov chain transition matrix for empty (0, 0), host only (H, 0), wasp only (0, W) and host–wasp (H, W) states is:

image(eqn 15)

for empty cells (0, 0), host only (H, 0), wasp only (0, W) and host–wasp (H, W) cells. From this transition matrix it can be seen that there is almost equal probability of an empty cell being colonized by the beetle (0·0995) or the wasp (0·0724). Similarly, once hosts and wasps have established the probability that the cell remains occupied by hosts and wasps is 0·4101. The limiting distribution of this transition matrix is (0·2436, 0·2794, 0·2645, 0·2126). In this limit, it can be seen that there is almost equal probability of observing the four-cell metapopulation in any particular state. Comparing the limiting distribution of this transition matrix and the empirical distribution of probabilities confirms that, at the local patch scale, the beetle and wasp populations can be described by a Markov chain or demographic stochastic processes. The predicted mean times to absorption into state (0, 0) are (H, 0) → (0, 0) = 27·7851 weeks (0, W) → (0, 0) = 23·0766 weeks and (H, W) → (0, 0) = 24·8579 weeks.

Similarly, for the 49-cell metapopulations the overall presence-absence transition matrix for empty (0, 0), host only (H, 0), wasp only (0, W) and host–wasp (H, W) states is:

image(eqn 16)

Again, comparisons of the limiting distribution of this transition matrix (0·6856, 0·1316, 0·0932, 0·0896) and the empirical distribution of state probabilities confirms that the local demographics are well described by a stochastic Markov chain. Following local extinction, the colonization probability of empty patches [transition from (0, 0) to (H, 0)] by hosts is 0·0713. Once hosts and wasps have established on a patch the persistence probability [transition from (H, W) to (H, W)] is 0·3093. The mean absorption time into state (0, 0) are (H, 0) → (0, 0) = 15·9769 weeks (0, W) → (0, 0) = 11·93869 weeks and (H, W) → (0, 0) = 16·6697 weeks. Phase-plots of the dynamics from these larger 49-cell metapopulations (Fig. 3) also illustrate that the predator–prey dynamics are likely to be influenced by low population densities and demographic noise; however, there is a deterministic signal (anticlockwise spiral in the host–parasitoid abundances) at higher density.

Figure 3.

Phase-space plots of total host and wasp abundance (solid points) in the 49-cell metapopulations. The metapopulation dynamics are determined predominantly by demographic processes acting on small populations. However, there is an anticlockwise spiral in changes in abundances characteristic of the deterministic predator–prey interaction. Arrows denote population change through time.

stochasticity and regional dynamics

To confirm that the dynamics of these predator–prey metapopulations are driven principally by stochastic fluctuations rather than the deterministic dynamic associated with predator–prey interactions, we linearized the predator–prey model (eqns 3 and 4) about the steady state (the average host and parasitoid abundances) and fitted this linearized model to the overall abundance data from the longer (49-cell metapopulation) time-series. The linearized dynamics of the logistic predator–prey model take the form of:

image(eqn 17)
image(eqn 18)

where H* and P* are the steady-state values of host and parasitoid abundances, respectively, and h(t) and p(t) are small perturbations in host and parasitoid numbers from the steady-state, respectively. Plots of residuals against the observed values and the predicted vs. the observed values (Fig. 4) indicate that the dynamics within the 49-cell metapopulation show a systematic departure from the dynamics expected in this deterministic predator–prey interaction. This corroborates the stochastic models that noise appears to influence the dynamical fluctuations in these predator–prey metapopulations.

Figure 4.

Linearized model (eqns 17–18) fits to 49-cell metapopulation series: (a) residuals vs. observed abundances and (b) predicted vs. observed abundances for the time-series from each of the larger metapopulation systems. These patterns highlight that the dynamics of the predator–prey interaction show systematic departures from those expected under the deterministic logistic predator–prey model (eqns 3, 4).

To explore the hypothesis that different forms of stochastic noise affect the dynamics of the predator–prey metapopulations, model parameter estimates and goodness-of-fit criterion based on information-theoretic approaches (Burnham & Anderson 2002) under environmental (Gaussian likelihood) and demographic (Poisson likelihood) noise were compared for each metapopulation time-series. Likelihood estimates for each of the four-cell (12 replicates) and 49-cell (four replicates) metapopulations are given in Tables 1 and 2. From comparisons of the likelihoods, it turns out that the predator–prey model incorporating environmental noise (Gaussian likelihood) is a more parsimonious descriptor p (based on Aikake information criterion (AIC)] of total abundance in both the four-cell and 49-cell metapopulations (even if the underlying processes are essentially determined by demographic stochasticity). In fact, differences in AIC values (Δi) suggest that there is little support (Burnham & Anderson 2002) for the role of demographic stochasticity at the regional metapopulation scale in these predator–prey interactions. To check that the error structures under the logistic model were appropriate one-step-ahead predictions, correlations between observed and predicted host and wasp abundances, and the dependence of error on time were assessed for each model for the larger, 49-cell metapopulations (Figs 5 and 6). Under Gaussian errors, the one-step-ahead estimates of abundance are good predictions of the individual host and wasp time-series (Fig. 5a,b), there was an overall positive correlation between observed and expected abundances (Fig. 5c,d) and there was no dependency of the residuals on time (Fig. 5e,f). Similar results were obtained for the logistic model with Poisson errors for the one-step ahead predictions, the relationship between observed and expected abundances and patterns in the residuals (Fig. 6). Finally, to assess whether coupling patches together acts to reduce the environmental variance (and potential facilitate persistence of the interaction), the magnitude of the predicted variances from the regional dynamics in the four-cell and 49-cell systems were compared. The estimated variance per patch for the four-cell and 49-cell metapopulations were 397·085 and 8·024, respectively.

Table 1.  Goodness-of-fit (using AIC = −2*loglik + 2*np) for the Gaussian and Poisson likelihoods for the logistic prey-predator model (eqns 3–4) on the regional time-series from the four-cell metapopulations. Δi is the difference in AIC between the Poisson and Gaussian likelihoods. The fit to all the time-series (Overall) reveals that the Gaussian process is the most parsimonious description of the regional dynamics. Under a linear functional response (f(P) = α · P(t)) the maximum likelihood parameter estimates (with 95% CI) for the Gaussian model are σ1 = 39·854, σ2 = 11·832, ρ = −0·204, r = 2·03 (± 3·077), K = 27·553 (± 15·468), α = 0·802 (± 0·2) and dP = 0·221 (± 0·022)
ReplicateGaussian likelihoodAIC (np = 7)Poisson likelihoodAIC (np = 4)Δi
1 269·627 553·254 430·649  869·298  316·044
2 286·631 587·262 702·467 1412·934  825·672
3 189·497 393·262 330·774  669·584  276·322
4 199·807 413·614 508·926 1025·852  612·238
5 255·120 524·24 432·205  872·41  348·17
6 204·544 423·088 404·851  817·702  394·614
7 247·743 509·486 398·618  805·236  295·750
8 211·885 437·77 398·721  805·442  367·672
9 273·725 561·45 621·030 1250·06  688·61
10 303·161 620·322 709·567 1427·134  806·812
11 368·741 751·4821089·922 2187·844 1436·362
12 361·842 737·684 728·638 1465·276  727·592
Overall3408·3736830·7468489·8416987·6810156·934
Table 2.  Goodness-of-fit (using AIC = −2*loglik + 2*np) for the Gaussian and Poisson likelihoods for the logistic prey-predator model (eqns 3–4) on the regional time-series from the 49-cell metapopulations. Δi is the difference in AIC between the Poisson and Gaussian likelihoods. The fit to all the time-series (Overall) reveals that the Gaussian process is the most parsimonious description of the regional dynamics. Under a linear functional response (f(P) = α · P(t)) the maximum likelihood parameter estimates (with 95% CI) for the Gaussian model are σ1 = 19·828, σ2 = 10·224, ρ = −0·053, r = 0·48 (± 0·069), K = 68·769 (± 13·469), α = 0·051 (± 0·007) and dP = 1·126 (± 0·181)
ReplicateGaussian likelihoodAIC (np = 7) Poisson likelihoodAIC (np = 4)Δi
1 269·627 375·73 279·920 567·84 192·11
2 286·631 745·854 682·5971373·194 627·34
3 432·217 878·434 882·4301772·86 894·426
4 504·2131022·426 843·8861695·772 673·346
Overall1369·1 2752·23082·1746172·3483420·148
Figure 5.

Error structure for the Gaussian model (eqns 11, 12) for the larger, 49-cell metapopulations. The panels are (a) representative observed and one-step-ahead predicted host abundances, (b) representative observed and one-step-ahead predicted wasp abundances, (c) correlation between observed and predicted abundances for hosts from all four 49-cell systems (ρ = 0·35), (d) correlation between observed and predicted wasp abundances from all four 49-cell systems (ρ = 0·35), (e) residual errors vs. time for all host series and (f) residual errors vs. time for all parasitoid series.

Figure 6.

Error structure for the Poisson model (eqns 7–8) for the larger, 49-cell metapopulations. The panels are (a) representative observed and one-step-ahead predicted host abundances, (b) representative observed and one-step-ahead predicted wasp abundances, (c) correlation between observed and predicted abundances for hosts from all four 49-cell systems (ρ = 0·35), (d) correlation between observed and predicted wasp abundances from all four 49-cell systems (ρ = 0·35), (e) residual errors vs. time for all host series and (f) residual errors vs. time for all parasitoid series.

stochastic metapopulation model

In this section we investigate the role of local demographic stochasticity on the dynamics at the regional scale by developing a stochastic metapopulation model. We show that given a fixed connectivity matrix (that is, the fraction of emigrants and immigrants from and to each patch is known) the role of demographic stochasticity on the persistence and dynamics of metapopulations of differing sizes can be evaluated.

Model structure

A stochastic version of the logistic predator–prey model (eqns 3, 4) is derived in which the sizes of the host (H) and the parasitoid (P) populations take integer values and the transitions (births, deaths and parasitism) are stochastic processes (Bartlett 1960). These transitions are summarized in Table 3. Dispersal is also a stochastic process and is assumed to occur once every seven time units (once per week). The number of hosts (Hd) and parasitoids (Pd) leaving a patch are random variables:

Table 3.  Within-patch transitions and probabilities for the stochastic version of the logistic predator–prey model (eqns 3, 4). Transitions within a small interval tt+ δt occur at random across the metapopulation
DescriptionTransitionProbability
Host birth in ith patch HiHi + 1r·Hi·δt
Host death in ith patch{HiHi− 1inline image
Parasitism in ith patchHi →Hi− 1f(Hi; Pi) ·δt
Pi →Pi + 1 
Parasitoid death in ith patch Pi →Pi − 1dP·Pi·δt
image(eqn 19)
image(eqn 20)

Here the number of individuals dispersing during a small time interval, δt, is drawn from a binomial distribution, B(N, p), where N is the population size and p is the probability that an individual disperses. Dispersing individuals have an equal probability of entering any one of their neighbouring patches and the number of neighbouring patches depends on lattice size and patch position.

Persistence and dynamics

We investigate the effects of demographic noise on the persistence and dynamics for five differently sized metapopulations. Replicated regional dynamics from lattice sizes of two, four (2 × 2), eight (2 × 4), 12 (3 × 4) and 16 (4 × 4) patches were generated using the stochastic version of the predator–prey model and analysed using the different likelihood models (eqns 10, 14). Representative dynamics from the five different metapopulations are illustrated in Fig. 7. Mean persistence times range from 76·64 (11·41 SE) generations for the two-patch systems to 707·00 (22·46 SE) generations for the 16-patch system.

Figure 7.

Representative model dynamics for the (a) two-patch, (b) four-patch, (c) eight-patch, (d) 12-patch and (e) 16-patch stochastic predator–prey metapopulations. Transition probabilities are given in Table 3. Dispersal events occur every seven time units and are described by eqns 19, 20. Times between subsequent events are described by draws from a uniform distribution.

Maximum likelihood estimation of parameters for the predator–prey model (eqns 3, 4) confirm that even when the underlying noise processes are known to be due to demographic stochasticity, the most parsimonious description of the regional dynamics is a predator–prey model incorporating environmental noise (Fig. 8). Identification of demographic stochastic effects at the regional scale is confounded by the role of noise operating across multiple patches. This is a simple statistical phenomenon and arises due to the central limit theorem: adding patches in which population experience demographic stochasticity (that are essentially Poisson) eventually approximates a regional population experiencing environmental stochasticity (that is essentially Gaussian).

Figure 8.

AIC values for the Gaussian (grey bar) and Poisson (black bar) likelihood fits to replicated time-series from the stochastic model (Table 3) for different-sized metapopulations. As metapopulation size increases, the role of demographic noise is masked by the increased number of patches experiencing this form of noise. This stochasticity is approximated by a Gaussian distribution.

Discussion

Here, we have shown that the persistence and dynamics of predator–prey metapopulations are influenced by stochastic processes. In particular we have shown, using a series of stochastic models, that the persistence and dynamics of a predator–prey interaction are principally influenced by demographic stochasticity at the patch level but this is masked at the regional metapopulation level. Although it has been illustrated that limited dispersal and demography affect the persistence of this predator–prey interaction (Bonsall et al. 2002), understanding the additional effects of stochasticity has led to a number of additional insights.

For instance, it is well established that noise can act to reduce the strength of species interactions and foster coexistence (May 1973). Persistence of these predator–prey metapopulations is affected by the concomitant effects of limited dispersal and stochasticity both acting to reducing the strength of the interspecific interaction between predators and prey. Metapopulation systems of the C. chinensis–A. calandrae interaction are known to persist longer than the single-cell interactions (Bonsall et al. 2002). In this study, we extend this finding and demonstrate that the effects of noise and habitat size act concomitantly to affect persistence. In particular the smaller, four-cell metapopulations persist, on average, for 11 weeks less than the larger 49-cell lattices and experience 50 times more variability. However, it is possible and probable that the type, nature and the strength of the stochasticity can affect population persistence in a more adverse way. For instance, demographic stochastic processes such as an individual's probability of reproduction, survival or dispersal are known to affect the persistence of populations and are particularly manifest in low populations (Lande 1998; Dennis 2002). This noise, coupled with ecological processes such as interaction strengths, scalings and dynamics, is also likely to influence population and metapopulation persistence (Engen & Saether 1998; Engen, Bakke & Islam 1998; Grenfell et al. 2002).

metapopulation dynamics and persistence

It is relatively well established that the dynamics of species interactions are influenced by deterministic and stochastic processes (Bonsall, Jones & Perry 1998). The dynamics of a variety of vertebrate (e.g. Stacey & Taper 1992; Leirs et al. 1997; Grenfell et al. 1998) and invertebrate (Higgins et al. 1997) populations are known to be influenced by the magnitude (Pimm, Jones & Diamond 1988), the shape (Ludwig 1996) and the temporal autocorrelation (Lundberg et al. 2002) in variability. These effects of noise on the population dynamics of single- and multispecies interactions are principally manifest through environmental processes (e.g. climatic variability). However, demographic noise is equally likely to influence the dynamics of populations, particularly when population sizes are small (Kuussaari et al. 1998) or have specific life-history characteristics (Kokko & Ebenhard 1996). In metapopulation interactions, the role of noise on the observed population dynamics can affect the degree of synchrony between populations (Heino 1998) and/or the observed population abundances (Grenfell et al. 2002). In the present study, we showed that Markov chains describe the (demographic) transition processes that predominate in this predator–prey interaction at the local patch scale where population densities are low. However, there remains a clear deterministic cycle (anticlockwise spiral in host–parasitoid abundances) at higher densities that underpins the interaction between the host and its parasitoid (see Fig. 3).

Empirical and theoretical studies of metapopulation dynamics (Levins 1969, 1970; Hanski 1991) suggest that persistence is a function of area, isolation and the number of patches available for colonization (Hanski, Moilanen & Gyllenberg 1996). In practice, however, the notion that a minimum amount of habitat must be available and is a necessary condition for metapopulation persistence is prone to oversimplification (Hanski et al. 1996). Incorporating stochastic colonization and extinction processes (such that metapopulation extinction is a function of the number of extant populations) can affect estimates of the persistence time of metapopulations by increasing the probability of global extinction (Gurney& Nisbet 1978). A corollary of this is that stochasticity acting through the local-patch processes of births, deaths and parasitism is also likely to impact on the persistence of a metapopulation. Further, it has been suggested that metapopulation persistence is underpinned by four key criteria (Hanski 1999). These are that (1) patches are capable of supporting locally breeding populations, (2) patches are at risk of extinction, (3) asynchronous dynamics operate between patches and (4) recolonization is possible. These processes have been evaluated for the C. chinensis–A. calandrae interaction (Bonsall et al. 2002) and one of the key requisites is that asynchrony in local dynamics allows the rescue of extinct patches. It is predicted that populations in synchrony have a higher risk of extinction (2000Ranta, Kaitala & Lundberg 1998; Earn, Levin & Rohani 2000), and understanding how the effects of demographic noise and deterministic processes scale from the local patch to the metapopulation to influence the regional dynamics and synchrony of resource–consumer interactions remains an open question.

metapopulation scaling and noise

Noise is known to scale with population size and affect the persistence (extinction risk; Hanski 1998) and the dynamics of non-linear ecological systems (Grenfell et al. 2002). We have explored the role of noise at both the local patch scale and the regional metapopulation scale, and demonstrate that demographic noise can be masked at the broader spatial scales. Evidence from the time-series reveal that this result is relatively robust: demographic processes dominate in the predator–prey interaction at the local scale in different metapopulation structures.

Appropriately, we have used linear (stochastic) and linearized (deterministic) models to understand the role of demographic stochasticity on the predator–prey metapopulation. As noted, the rank importance of demographic noise and deterministic processes often scale with population size (Grenfell et al. 2002). However, there remains the likelihood that these stochastic and deterministic factors scale with other ecological factors such as assemblage diversity, habitat geometry or environmentally driven processes. For instance, spatial environmental correlation and environmental stochasticity have been used to explore the persistence of single-species metapopulations (Hanski 1998; Palmqvist &Lundberg 1998). Although extinction risk is a function of local population size and environmentally correlated processes (Ranta et al. 1998), the concomitant role of the strength of species interactions and demographic noise is also likely to influence population and metapopulation persistence.

In the current study we predict that demographic stochastic effects can be masked at larger spatial scales by the statistical phenomenon of the central limit theorem: as sample size increases, the mean of the samples drawn from any distribution will approach a Gaussian distribution. It is also entirely plausible that demographic and environmental processes interact with the deterministic process to affect the persistence and dynamics of this predator–prey interaction. The availability of the time-series and the ability to resolve mechanistic detail about the ecology of the interaction as the spatial resolution increases suggests that it is possible to distinguish between stochastic processes due to demographic or environmental noise from those associated the deterministic dynamics. Disentangling the contributions of environmental and demographic noise processes from the underlying deterministic attractor in resource-consumer metapopulations remains relatively unexplored and will be developed further in a future study.

In conclusion, this paper illustrates that the dynamics and persistence of a predator–prey metapopulation are underpinned by stochastic processes. We have argued that different forms of the stochasticity can have differential effects on the predator–prey interaction: environmental noise is predicted to reduce the impact of interspecific interactions and allow coexistence (May 1973). However, demographic noise is more likely to enhance the likelihood of extinction (Engen, Bakke & Islam 1998). We have shown that the predator–prey metapopulations are influenced by the demographic structure and dynamics operating at the local scale. Separating these stochastic processes from the underlying deterministic dynamics is necessary if metapopulation persistence criteria (Adler & Nuereberger 1994; Hanski & Ovaskainen 2000) are to be applicable and quantifiable.

The stochastic models developed and used in the present study highlight the need to understand the form and strength of stochasticity operating at the local and regional scales as well identifying the deterministic ecological processes as principal determinants of species persistence in metapopulation assemblages.

Acknowledgements

M.B.B. is a Royal Society University Research Fellow. The work was partly funded under NSF grants DEB 0083583 and 0213026 to A.H. We also acknowledge the support of the NERC Centre for Population Biology (CPB) visitor's programme.

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