*Presidential address to the British Ecological Society, University of Leicester, 5 January 1999.

# Host–parasitoid population dynamics*

Article first published online: 25 DEC 2001

DOI: 10.1046/j.1365-2656.2000.00445.x

Additional Information

#### How to Cite

Hassell, M. P. (2000), Host–parasitoid population dynamics. Journal of Animal Ecology, 69: 543–566. doi: 10.1046/j.1365-2656.2000.00445.x

#### Publication History

- Issue published online: 25 DEC 2001
- Article first published online: 25 DEC 2001

- Abstract
- Article
- References
- Cited By

### Introduction

- Top of page
- Introduction
- A basic framework
- Heterogeneity in host–parasitoid interactions
- Multispecies systems
- Metapopulation dynamics
- Conclusion
- References

This address deals with just one kind of natural enemy, insect parasitoids, and illustrates some ways in which our understanding of their dynamical interaction with hosts has advanced over the past 25 years or so. Parasitoids comprise some 10% or more of all metazoan species, and largely belong to two families, the Diptera (two-winged flies) and the Hymenoptera (sawflies, bees, wasps and ants). Excellent introductions to the biology of parasitoids can be found in Clausen (1940), Askew (1971) and Godfray (1994). Adult female parasitoids lay one or more of their eggs on, in or close to the body of their host, usually an immature stage of another insect, which is then consumed over a period of days or weeks by the feeding parasitoid larva or larvae. As in most true parasites, all the food necessary to complete development comes from a single host, but like true predators this almost always leads to the death of the host, albeit with a delay until the parasitoid larva is fully developed.

Parasitoids have long been popular subjects for ecological study for several reasons. First, they are important for biological pest control and this has stimulated much empirical and theoretical work on the attributes that make parasitoids effective pest control agents. Secondly, parasitoids are ideal subjects for developing relatively simple population models. This is mainly because it is only the adult females that search for hosts, and because the act of finding a host is normally followed by oviposition. The success in finding and attacking hosts therefore closely defines parasitoid reproduction, which means that (i) host–parasitoid models can have a much simpler structure than corresponding predator–prey models in which all predator stages may attack prey with different effectiveness, and (ii) reproduction is less closely defined by prey consumption. Finally, many species of parasitoids and their hosts can readily be cultured in laboratory microcosms, and this has greatly increased the amount of empirical information on host–parasitoid interactions under controlled conditions. This experimental approach to population dynamics has rightly been extolled by Kareiva *et al*. (1989).

Much of the work on the dynamics of host–parasitoid interactions has taken a mechanistic approach. Components of the interaction are investigated using simple experiments, and their dynamical effects examined in a step-wise way within population models. The overall objective is to have a detailed understanding of how the important processes operating in the host and parasitoid life cycles affect the dynamics of the populations. These components may be the fundamental demographic parameters, features of the life histories, effects of other interacting species or other features of the habitat such as patchiness of resources and variability of resource quality. In each case, empirical information is needed to describe the components and define their relationship with key variables such as population density. From this, a description is obtained of each component in an appropriate model framework. Finally, analysis of the parameterised model indicates the dynamical effects of that component. Such a mechanistic approach to population dynamics is demanding of data and requires a particularly close interaction between data and model development.

This address illustrates how the dynamics of host–parasitoid interactions may be influenced by three major ecological processes: by spatial patchiness, by interactions with other species, and by metapopulation structure. A common underlying theme is that parasitism does not occur at random and that spatial and other processes lead to aggregated distributions of parasitism amongst the host population. The realisation of how important spatial processes are to population dynamics in general has led to a revolution in the subject (Wiens 1989), and as a result many of the earlier, simple host–parasitoid models are now viewed as rather special limiting cases. As is usually the case, developing the mathematical models has sometimes proved less of a challenge than designing and executing appropriate studies to collect the data. But it is encouraging that there are now several examples where empirical field studies and models have been drawn fairly close together (e.g. Hassell 1980; Jones *et al*. 1993; Reeve *et al*. 1994; Murdoch *et al*. 1996). But problems of spatial scale still pervade ecology (Levin 1992, 1994). The interactions discussed in this paper involve either a single host and parasitoid species interacting together or simple webs, and are discussed at two very different spatial scales. On the one hand, there are *local populations* characterised by more-or-less complete mixing of individuals at some point(s) during the generation period. On the other hand, there are *metapopulations* formed by collections of local populations linked by some degree of dispersal each generation between the individual local populations. But first a basic framework is outlined upon which many of the developments in modelling host–parasitoid interactions have been built.

### A basic framework

- Top of page
- Introduction
- A basic framework
- Heterogeneity in host–parasitoid interactions
- Multispecies systems
- Metapopulation dynamics
- Conclusion
- References

A modelling tradition, initiated mainly by entomologists with insect hosts and their parasitoids in mind (Thompson 1924; Nicholson 1933; Varley 1947), assumes that populations have discrete and synchronized generations. This is in contrast to Lotka–Volterra models (Lotka 1925; Volterra 1926), which start with the assumption that the generations of the interacting populations overlap completely and that birth and death processes are continuous. Discrete generations inevitably introduce a one-generation time lag between the act of parasitism and the resulting change in host populations, and it is the presence of these time lags that represent the fundamental difference between the two kinds of model. Although the discrete and continuous frameworks reflect fundamentally different kinds of life cycle, both classes of model have been used to demonstrate how a wide range of comparable features of host–parasitoid interactions influence population dynamics

The usual framework for discrete-generation host–parasitoid models is given by

*N*_{t + 1}= λ*N*_{t}*f*(*N*_{t},*P*_{t})

*P*_{t + 1}=*cN*_{t}[1 -*f*(*N*_{t},*P*_{t})]eqn 1

where *P* and *N* are the population sizes of the searching adult female parasitoids and the susceptible host stage, respectively, in successive generations *t* and *t* + 1. In the host equation, the parameter λ is the *net* finite rate of increase of hosts in the absence of the parasitoids, which may be density-dependent or assumed to be a constant. It depends on the hosts' fecundity, sex ratio, any immigration and emigration, and all host mortalities other than parasitism itself. The function *f*(*N*_{t}*,P*_{t}) defines the fraction of the *N*_{t} hosts escaping parasitism; one minus this term (within the square brackets in the parasitoid equation) therefore gives the fraction of hosts parasitized. All assumptions about the efficiency of parasitoids at finding and parasitizing hosts are thus contained within this term. Finally, *c* is the average number of adult female parasitoids emerging from each host parasitized (often assumed to be one, that corresponds to parasitoids with solitary larvae)*.* It depends upon the sex ratio of the parasitoid progeny, any mortality suffered within hosts and any mortalities of the subsequent adult female parasitoids prior to searching for hosts in the next generation. Clearly, these simple equations subsume a huge amount of host and parasitoid biology, and the apparent simplicity of the model is deceptive: to be parameterized for a particular host–parasitoid system requires detailed life table information on both populations (e.g. Hassell 1980; Jones *et al*. 1993).

The best known example of model 1 is that of Nicholson (1933) and Nicholson & Bailey (1935) who explored in depth a model in which the following assumptions about parasitism were made. First, the parasitoids are never egg-limited and encounter hosts in direct proportion to host abundance. The total number of encounters with hosts is therefore given by *N*_{enc}=*aN*_{t}*P*_{t}, where *a* is the per capita searching efficiency which Nicholson called the ‘area of discovery’. Secondly, these *N*_{enc} encounters are distributed randomly amongst the population of equally susceptible hosts. There is thus either no avoidance of superparasitism or, if the parasitoids *can* avoid superparasitism, they do so instantaneously without affecting subsequent performance in any way. These two assumptions are at the core of Nicholson's so-called ‘competition curve’ in which the proportion of hosts escaping parasitism is given by the zero term of the Poisson distribution, exp(– *aP*_{t}), where *aP*_{t} are the mean encounters per host, N_{enc}/*N*_{t} *= aP*_{t}. Thus, one minus this zero term is the probability of a host being attacked. Substituting into model 1 gives:

*N*_{t + 1}= λ*N*_{t}exp( -*aP*_{t})

*P*_{t + 1}=*N*_{a}*= cN*_{t}[1 - exp ( -*aP*_{t})]eqn 2

where *N*_{a} is the number of hosts that are parasitized irrespective of the number of times they have been encountered.

The dynamical properties of the Nicholson–Bailey model are well known. A host–parasitoid equilibrium always exists depending on the values of *a, c and*λ, and this is always locally *unstable* with the slightest perturbation leading to oscillations of rapidly increasing amplitude (Fig. 1a). This instability, compared to the neutrally stable Lotka–Volterra model, arises from the one-generation time lags between cause and effect that are inherent in these difference equation models (May 1973) and which enhance the degree that parasitism acts as a delayed density-dependent, or second-order feedback process (Varley 1947; Berryman & Turchin 1997).

Although unstable oscillations have been observed in a few simple laboratory host–parasitoid and predator–prey experiments (e.g. Burnett 1958; Huffaker 1958; Hassell & May 1988), such instability is hard to reconcile with the results from other laboratory systems in which the interactions are much more stable (e.g. Utida 1957; Huffaker 1958; Huffaker *et al*. 1963; Fujii 1983; Bonsall & Hassell 1997, 1998; Shimada 1999) (and see Fig. 2 below) and, more generally, with the long-term persistence of natural systems. Nicholson (1947), anticipating the current vogue for metapopulations, suggested one means by which oscillatorily unstable local populations may persist. Assuming that the interaction occurs in distinct and separated areas, the ‘cycle of increase in numbers, followed by … extermination, proceeds independently in different parts of the occupied country; so at all times some groups are increasing and some decreasing in numbers…. Consequently when one considers a large tract of country, the abundance [of both host and parasitoid].. remains more or less constant; whereas in any small area of the same country the fluctuation in numbers ….may be violent.’ Such metapopulation persistence is considered in more detail below.

There are, however, several other important ways in which the Nicholson–Bailey model can be modified that both add realism and allow the populations to persist (Hassell 2000). Many of these involve elaborating the parasitism function *f*(·). For example, May (1978) started from the premise that the assumption that hosts are parasitized at random is an unlikely proposition in the real world where host individuals are bound to vary in their spatial location, phenotype, and stage of development. It is much more likely, therefore, that the risk of being parasitized will vary within the host population, leading to an overall distribution of parasitoid attacks that is more aggregated than random. His model, instead of being based on the Poisson distribution, makes the specific assumption that survival from parasitism is described by the zero term of the negative binomial distribution so that *f* in eqn 1 is given by:

where *k* is the index defining the degree to which the distribution of parasitism amongst the host population is aggregated (most aggregated as *k* 0, becoming random (i.e. Poisson) as *k*∞). The properties of this model are importantly different from those of the Nicholson − Bailey model. The equilibria are locally stable provided that the distribution of parasitism is sufficiently aggregated, or, more specifically, if and only if *k* < 1 (Fig. 1b). The stabilizing effect of small *k* stems from the way that parasitism is heterogeneously distributed amongst the host population. This is further explored in the next section.

### Heterogeneity in host–parasitoid interactions

- Top of page
- Introduction
- A basic framework
- Heterogeneity in host–parasitoid interactions
- Multispecies systems
- Metapopulation dynamics
- Conclusion
- References

No study has had a greater influence in publicising how heterogeneity can affect population dynamics than C.B. Huffaker's classic experiments with predatory and prey mites feeding on oranges (Huffaker 1958; Huffaker *et al*. 1963). While these experiments tell us little about the detailed mechanisms by which spatial patchiness promotes persistence, they have been the inspiration for the development of many models for spatially structured predator–prey systems (e.g. Hilborn 1975; Caswell 1978; Hastings 1978; Crowley 1979; Nisbet & Gurney 1982). An example from a host–parasitoid experiment is shown in Fig. 2; the interaction is unstable in a more-or-less homogeneous environment, but persists in a relatively stable interaction when the beans, which are the host resource, are confined within small patches. As with Huffaker's mites, increased partitioning of the environment into discrete patches reduced the chances of extinction and allowed the populations to persist at levels well below the host or prey's carrying capacity.

The term ‘heterogeneity’ will be used here in a specific way. Following Chesson & Murdoch (1986), it is defined in terms of the variation in risk of parasitism between different individuals in the host population. For example, the Nicholson–Bailey model is recovered if the relative risk of parasitism is uniformly distributed per patch, while the negative binomial model of May (1978) is obtained when the risk is gamma distributed. In this terminology, therefore, a habitat is only ‘heterogeneous’ in so far as it leads to ‘aggregation of risk’ of parasitism between host individuals. The importance of this measure lies in the way that it can be used to quantify the stabilizing effect of an aggregated distribution of parasitism amongst host individuals in a population.

One of the most obvious ways in which heterogeneity of risk of parasitism can arise is in a patchy environment where the level of parasitism varies between patches. But it can also arise in quite different ways. For example, host and parasitoid life cycles may not be properly synchronized so that some hosts are less at risk from parasitism, or escape completely, due to the phenological mismatch of life cycles that do not coincide. Or, there may be phenotypic variation between host individuals such that some hosts are able to reduce their risk of parasitism by virtue of their physiology or behaviour. Heterogeneity of risk is thus a pervasive feature of natural interactions. A very clear exposition of how this heterogeneity helps to stabilize host–parasitoid interactions of the form of model is given by Taylor (1993).

#### Spatial patterns of parasitism

Let us consider an environment made up of discrete patches of food plants upon which an insect herbivore species with discrete generations feeds. The herbivore is attacked by a specialist parasitoid species whose adults coincide temporally with the susceptible host stage. On emergence, the adult hosts and female parasitoids disperse from their natal patches and move amongst the patches ovipositing. We have, therefore, total populations of *N*_{t} hosts and *P*_{t} adult female parasitoids distributed amongst the *n* patches such that within any one patch there are *P*_{i} parasitoids searching for *N*_{i} hosts. Suppose first that the parasitoids divide themselves evenly amongst the patches, and that within any patch they have a linear functional response, a constant searching efficiency and encounter hosts at random. Percentage parasitism is now the same in each patch in any one generation, and the Nicholson–Bailey model for the whole population is recovered exactly. All host individuals therefore suffer the same risk of parasitism.

Such a uniform risk of parasitism across patches is countered by a wealth of empirical evidence. The two laboratory examples in Fig. 3 are clear cases where the searching parasitoids tend to congregate in the patches of high host density. In one case, the resulting pattern of parasitism is positively density-dependent, while in the other, it is just the opposite – parasitism is inversely density-dependent. This difference stems from the different functional responses of the two species in the following way. *Trybliographa rapae* has a short handling time giving a relatively high maximum attack rate per parasitoid (i.e. high upper asymptotes of their functional responses). The pattern of parasitism thus reflects the distribution of parasitoids, and hence the density-dependent patterns in Fig. 3b. In contrast, the egg parasitoid, *Trichogramma evanescens*, has a much longer handling time and hence a lower maximum attack rate per female. Individual parasitoids are therefore restricted in their ability to exploit high host density patches and this leads to the inverse density dependence in Fig. 3d. Examples illustrating this mechanism are shown in Fig. 4.

While data combining information on the distribution of searching parasitoids and the resulting patterns of parasitism are difficult to collect from the field (but see Waage 1983; Driessen & Hemerik 1991), it is relatively easy to quantify patterns of parasitism without bothering about the distribution of searching adults: hosts can be sampled from a range of patches, taken back to the laboratory and the incidence of parasitism then determined, usually by rearing out the next generation or by dissection. Of the 201 different examples listed in the reviews of Lessells (1985), Stiling (1987), Walde & Murdoch (1988) and Hassell & Pacala (1990), 59 show direct density-dependent patterns of parasitism, 53 show inverse patterns and 89 show parasitism uncorrelated with host density per patch. Some examples are shown in Fig. 5. Using the terminology of Pacala *et al*. (1990) and Hassell *et al*. (1991b) the direct and inverse density-dependent patterns arise from ‘*host-density-dependent heterogeneity*’ (HDD), and the density-independent patterns arise from ‘*host-density-independent heterogeneity*’ (HDI).

Much of the interest in these different patterns has focused on their potential importance to population dynamics. Early theoretical work with discrete generation interactions tended to emphasize how density-dependent patterns could promote stability (e.g. Hassell & May 1973, 1974; Murdoch & Oaten 1975; Beddington *et al*. 1978). Others have pointed out that both the inverse patterns and the density-independent ones can be equally important for stability (Hassell 1984; Chesson & Murdoch 1986; Hassell & May 1988; Walde & Murdoch 1988; Pacala *et al*. 1990; Hassell *et al*. 1991b). Simple models shed light on this complex picture.

#### A model with hdd parasitism

The simple model framework can be applied to interactions in an explicitly patchy environment as long as *f*(*N*_{t},*P*_{t}) represents the *average*, across all patches, of the fraction of hosts escaping parasitism. The function *f*(*N*_{t},*P*_{t}) therefore depends on both survival from parasitism *within* patches as well as the spatial distributions of hosts and parasitoids *between* the *n* patches. Let us consider the simple scenario of a habitat with *n* patches as above, within each of which parasitism is random and determined by a type II functional response. The distribution of hosts and parasitoids from patch to patch is defined by α_{i} and β_{i}, the fractions of total hosts and total searching parasitoids, respectively, in the *i*th patch, so that *f* is given by:

where *a* is the searching efficiency per patch and *T*_{h} is the handling time (Hassell & May 1973). More specifically, we will assume that the parasitoids aggregate in patches of high host density following the simple power relationship:

where µ is an index of parasitoid aggregation and *w* is a normalization constant such that the β_{i} values sum to unity (Hassell & May 1973). The index, µ, can describe a wide range of parasitoid distribution patterns: they will be evenly distributed across patches if µ = 0, and increasingly tend to aggregate in patches of high host density as µ rises until, in the limit µ∞, they will all congregate in the single patch of highest host density leaving the remainder as complete refuges. Lastly, if µ < 0 the parasitoid distribution is reversed, with their local abundance now inversely correlated with host density per patch. The way that these different patterns can promote stability (assuming, for simplicity, a linear functional response within patches) is shown by the following example in which the hosts are aggregated according to a negative binomial distribution:

Here *p*(*j*) is the probability of having *j* hosts in a patch from the negative binomial distribution and *P*(*j*) are the number of parasitoids in a patch with *j* hosts (defined by eqn 5). Analysis of this model shows that sufficiently strong direct *or* inverse density-dependent distributions of parasitoids can strongly stabilize the interactions (Fig. 6a), while insufficient parasitoid aggregation leads to local instability (if the host rate of increase is above some threshold level, a range of interesting global dynamics can occur (Rohani *et al*. 1994)). Different degrees of parasitoid aggregation can also have a marked effect on equilibrium levels (Fig. 6b). Parasitoids have the greatest effect in reducing host equilibria when their distribution most closely tracks that of the hosts (i.e. µ = 1). Increasing aggregation (positive or negative) in this model leads to higher host abundances, simply because the parasitoids, once initially distributed, are confined to their respective patches, irrespective of the degree of host exploitation (see below).

#### A model with hdi parasitism

There has been much more emphasis on exploring the effects of HDD than HDI heterogeneity, partly because of the pivotal role it has been assumed to have in promoting population stability, and partly because behavioural ecologists have wished to make the link between foraging behaviour and population dynamics (Bernstein *et al*. 1988; Bernstein *et al*. 1991; Godfray 1994). It is now clear, however, that HDI heterogeneity can be at least as important a process in promoting population persistence.

Let us consider a specific example where all the heterogeneity in parasitism is independent of the spatial distribution of hosts (HDI), as in Fig. 5d, and assume that this is achieved by the distribution of searching parasitoids being independently *aggregated* from patch to patch following a gamma distribution. We also assume that within any one patch the parasitoids exploit hosts at random and have a constant per capita searching efficiency (i.e. a linear functional response). The fraction of hosts escaping parasitism is now given by:

where *g*(ε) is the gamma probability density function for parasitoids per patch with unit mean and variance 1/α (α is a positive constant governing the shape of the density function) and *a* is the usual per capita searching efficiency of the parasitoid. In each patch, therefore, host survival by *P*_{t}ε randomly searching parasitoids is given by the zero term of a Poisson distribution with mean *aP*_{t}ε (Murdoch *et al*. 1984a; Chesson & Murdoch 1986). Very much the same derivation would apply were the parasitoids uniformly distributed across patches but their searching efficiency, *a*, varied according to a gamma distribution (Bailey *et al*. 1962). The stability properties of the model are straightforward: the populations will be stabilized by the HDI heterogeneity as long as there is sufficient variance in the distribution of parasitoids or, more specifically, if α < 1.

Interestingly, eqn 7 also reduces exactly to May's (1978) negative binomial model. The phenomenological index of aggregation, *k*, is now explicitly related to the degree of density-independent aggregation of the searching parasitoids (α) (Chesson & Murdoch 1986; Pacala *et al*. 1990; Hassell *et al*. 1991b). The stability criterion, *k* < 1 from the negative binomial model, therefore corresponds exactly in this spatially explicit model to α < 1. Similar stabilizing properties have also been demonstrated from related models with density-independent parasitism by Reeve *et al*. (1989), further emphasizing that density-independent patterns of parasitism are potentially important as a stabilizing mechanism.

#### Optimal foraging

The parasitoid aggregation described above involves an inflexible aggregation strategy in which individuals are distributed amongst the patches at the start of each generation, where they then remain confined irrespective of how heavily the hosts become exploited and whether or not there are more profitable patches elsewhere. The parasitoids in this section are much more adaptive in being able continually to seek out the best patches currently available. The large body of literature on optimal patch use stems from the classic paper by Charnov (1976) in which a forager leaves a patch when the instantaneous rate of gaining fitness (via food, reproductive sites, etc.) is reduced to the maximum rate achievable in the environment as a whole (e.g. Stephens & Krebs 1986; Krebs & Kacelnik 1991; Krivan 1996, 1997). Applied to parasitoids, this means an optimal distribution of parasitoid searching effort that maximizes the rate of parasitism for the individual searching parasitoids, depending on the spatial distribution of hosts and any behavioural and life history constraints. Cook & Hubbard (1977), Hubbard & Cook (1978) and Comins & Hassell (1979) developed models for such optimally searching parasitoids. The Comins–Hassell model involves individual parasitoids moving amongst patches, perfectly allocating their searching time in relation to the details of the host distribution and the value of the parasitoid searching efficiency and handling time (see also Royama 1971; Lessells 1985). At the start of the searching period, each parasitoid is therefore found in the patch with the highest host density. As soon as the rate of encountering healthy hosts in this patch is reduced to the level of encounter that could be achieved in the second most profitable patch, the parasitoids divide their search between patches 1 and 2 (but not equally if the handling time is greater than zero since more time will be ‘wasted’ in patch 1 in handling already parasitized hosts). So the process continues, with the set of exploited patches broadening until, if there were sufficient parasitoids with enough time, all patches would be reduced to the same level of profitability. The relative distribution of parasitoids therefore changes during the searching season, being most aggregated at the start and becoming less so as time progresses.

The dynamical effects of this continuous redistribution and evening-out of the numbers of parasitoids per patch is shown in Fig. 7 in terms of the overall searching efficiency in relation to the numbers of parasitoids searching. The horizontal, broken, line is for randomly searching parasitoids; their probability of finding a host (and therefore their searching efficiency) is the same in all patches and throughout the searching season, irrespective of the total parasitoid searching effort, *PT*. Curve B is for parasitoids with a fixed aggregation strategy (e.g. see eqn 5). At low to intermediate values of *PT* the parasitoids are more efficient than their random counterparts because of their aggregation in the patches of high host density (assumed µ = 1). But at higher values of *PT* their inability to move away from these patches once well exploited greatly reduces their efficiency to the point that they fare even worse than the random searchers. In contrast, optimal foragers (Curve A) are the most efficient at low *PT* when they all congregate in the most profitable patches. But as *PT* increases and they tend to reduce all patches to the same profitability, their efficiency comes closer and closer to that of the random parasitoids. The contribution of all these patterns to stability depends on the negative slope of these lines evaluated at the *PT* equilibrium (Hassell 1978; Comins & Hassell 1979). Thus, if *PT* is very large the optimally foraging parasitoids will contribute very little to stability, much as if they were searching randomly. But at lower values of *PT* optimal foraging leads to aggregated distributions of parasitoids and HDD parasitism, which in turn can be a powerful mechanism for stability. In short, the stabilizing effects of optimal foraging depend on how much the aggregation of risk early in the searching season has been reduced by the end of the season (Hassell & Pacala 1990).

More broadly, we can conclude that the contribution to stability by ‘HDD parasitoids’ that redistribute within generations will tend to decrease as parasitoid density rises and as the duration of the searching period increases. However, most of the empirical HDD patterns of parasitism referred to above (e.g. Fig. 5) have come from host samples taken *after* the period of exposure to parasitoids; redistribution during the searching season has therefore not tended to equalize parasitism across patches. It seems unlikely, therefore, that parasitoids are generally sufficiently abundant or have enough searching time to reduce all patches to similar levels of profitability.

#### A more unified approach

We have seen that both HDD and HDI patterns of parasitism can under certain conditions contribute markedly to persistence. This Section outlines a general criterion for the contribution of HDD and HDI parasitism to stability, and then applies this to data that are readily available in the field.

Let us first turn back to the model for HDI aggregation above, in which the parasitoids aggregate independently of local host density according to a gamma distribution with a unit mean density per patch and variance of 1/α = 1/*k* (stable as long as *k* < 1) where α is a positive constant determining the shape of the density function. An alternative way of expressing this stability criterion makes use of the square of the coefficient of variation of searching parasitoids per patch (*CV*^{2}= variance/mean^{2}), which in this case is simply defined by 1/*k*. The stability condition *k* < 1 is therefore identical to the condition *CV*^{2} > 1 (May (1978); Chesson & Murdoch 1986; Hassell & May 1988; Pacala *et al*. 1990); in other words, the interaction will be stable if the distribution of searching parasitoids per patch, measured as the square of the coefficient of variation (*CV*^{2}), is greater than one (see Hassell *et al*. (1991b) for full details on the estimation of *CV*^{2}). Interestingly, this specific criterion approximates quite well to the requirements for stability across a broad spectrum of discrete-generation host–parasitoid models (Pacala *et al*. 1990; Hassell *et al*. 1991b).

A useful feature of the *CV*^{2} > 1 rule is that it is readily decomposed into its constituent parts of HDD and HDI heterogeneity, both of which can be quantified directly from empirical data collected in the field. The basis of this is the assumption that the distribution of searching parasitoids is given by the same power function as in eqn 5, to which is added the gamma distributed residual, ε:

where *n* and *p* are the local host and parasitoid densities, respectively. Thus, the magnitude of any HDD aggregation is given by the value of the aggregation index, µ, and HDI aggregation is given by the magnitude of the variance of ε. Full details of the analysis are given in Pacala & Hassell (1991). Their key conclusion is that the *CV*^{2} > 1 rule may be approximated as:

*CV*^{2}≈*C*_{I}*C*_{D}- 1eqn 9

where *C*_{I} is the HDI component given by *C*_{I} = 1 + σ^{2} in which σ^{2} is the variance of ε, and *C*_{D} is the HDD component given by *C*_{D} = 1 + *V*^{2}µ^{2} in which *V* is the weighted coefficient of variation of the host density per patch. The *CV*^{2} > 1 rule therefore applies if *C*_{I}*C*_{D} > 2, which can arise from HDI alone if *C*_{I} > 2, HDD alone if *C*_{D} > 2, or some combination of the two.

Applying the *CV*^{2} > 1 rule to field data is relatively straightforward in the few cases where there are data on the actual distribution of searching parasitoids in relation to host density per patch (Driessen & Hemerik 1991; Reeve *et al*. 1994). But this suffers from two major drawbacks. First, the data are hard to collect in the field and second, as the examples above show, there are clearly other factors than the distribution of adults determining heterogeneity in the risk of parasitism. It is possible, however, to estimate *CV*^{2} directly from the distribution of parasitism rather than from the distribution of adult parasitoids. This has two main advantages: (i) the data are far easier to collect and are widely available in the literature; and (ii) the estimates will encompass other factors that contribute to aggregation of risk. The procedure for doing this is based on *inferring* the distribution of searching parasitoids from the pattern of parasitism, assuming the two are linked by a linear functional response (Pacala & Hassell 1991). In other words, it works back to what the *CV*^{2} of adult parasitoids per patch would have had to have been if their distribution were the sole determinant of the observed levels of parasitism and if their functional responses were linear. In practice, it involves estimating the key parameters in eqn 9 from data on levels of parasitism and host density per patch: µ and σ^{2} from maximum likelihood estimates, and *V*^{2} directly from the data on the distribution of hosts. *C*_{I} and *C*_{D} are thus obtained and thence the value of *CV*^{2} (see Pacala & Hassell 1991).

The values of *C*_{I} and *C*_{D}, and thence *CV*^{2}, have been estimated for the five examples in Fig. 5. In Fig. 5a there is a clear direct density-dependent pattern of parasitism. The estimated *CV*^{2} is 1·16, which indicates a level of heterogeneity in parasitism that, were it typical from generation to generation, should be just sufficient to stabilize an interaction of the form of model with a linear functional response without the need for any additional stabilizing mechanisms. The relatively large value for *C*_{D} (= 2·06) and the small value for *C*_{I} (= 1·05) indicate that virtually all of this stabilizing heterogeneity comes from the density-dependent parasitism (HDD), which in turn depends upon *both* the aggregated distribution of parasitism and the highly aggregated spatial distribution of hosts (*V*^{2} = 0·66). The importance of having sufficient host clumping is illustrated by the example in Fig. 5b. In this case there is pronounced inverse spatial density dependence, but the relatively weak host clumping (*V*^{2} = 0·25) leads to low values of *C*_{D} (= 1·29) which, together with the small amount of HDI heterogeneity, leads to very little stabilizing heterogeneity (*CV*^{2} = 0·37). The remaining two data sets (Fig. 5c, d) show no evidence of HDD parasitism. In Fig. 5c there is also no appreciable effect of HDI parasitism. But in Fig. 5d the HDI variation is very pronounced (*C*_{I} = 8·25), producing marked stabilizing heterogeneity (*CV*^{2} = 7·33). Thus, although appearing erratic, the data in Fig. 5d actually contain more evidence of factors that could stabilize dynamics than do any of the previous examples.

In an analysis of 65 such data sets, *CV*^{2} < 1 in 47 of them indicating rather weak levels of stabilizing heterogeneity. In the remaining 18 cases, *CV*^{2} > 1 indicating that heterogeneity if consistently at that level ought to be sufficient to stabilize the populations (Hassell & Pacala 1990; Pacala *et al*. 1990). Interestingly, in the great majority of these, HDD was insignificant and HDI heterogeneity all important. In short, and contrary to what is still a popular view, this suggests that while both density-dependent and density-independent patterns of parasitism can contribute to the stability of host–parasitoid interactions, it is the density-independent ones that are the more important.

Apart from providing a rough indication of the amount of stabilizing heterogeneity in discrete-generation, coupled host–parasitoid interactions, the *CV*^{2} > 1 rule also has the merit of being readily divided into HDD and HDI heterogeneity, both of which can be estimated from available field data on the distribution of parasitism. But there are several factors which will affect the applicability and validity of the rule. For example, most of the available information comes from single-site studies carried out within a single generation; the crucially important information on the *typical* levels of heterogeneity over a period of time is thus usually lacking. In the few cases where *CV*^{2} has been estimated over a period of time, it has been found to be quite variable (Redfern *et al*. 1992; Jones *et al*. 1993). Values of C*V*^{2} > 1 should thus really be interpreted as levels of heterogeneity of risk which, if typical over a period of time, would be sufficient to stabilize the host–parasitoid interaction without the need for any other stabilizing mechanisms.

As well as temporal variability in spatial heterogeneity, the C*V*^{2} > 1 rule will also be confounded by the presence of other important density-dependent processes acting upon the host population, due to resource limitation, generalist predators, pathogens and so on. In general, and not surprisingly, less heterogeneity of risk will be needed if these other processes contribute to the stability of the host population. A further problem is that estimates of *CV*^{2} from field data on parasitism involve extrapolating from parasitism per patch to the density of searching adult parasitoids assuming a linear functional response and hence a close correspondence between the distribution of searching parasitoids and parasitism. Ives (1992), however, has emphasized that the relationship between searching parasitoids and parasitism can be much weaker if the parasitoids have a strong type II functional response. More heterogeneity will then be needed to overcome the destabilizing effect of the type II functional response. An excellent review of these and other caveats is given by Taylor (1993).

### Multispecies systems

- Top of page
- Introduction
- A basic framework
- Heterogeneity in host–parasitoid interactions
- Multispecies systems
- Metapopulation dynamics
- Conclusion
- References

Our current understanding of host–parasitoid population dynamics comes largely from models of two-species interactions. While these simple systems correspond quite well to some natural examples in the field in which hosts are effectively attacked mainly by one parasitoid species (e.g. Askew & Shaw 1986; Hawkins & Lawton 1987; Hawkins 1988), many natural systems involve intricate webs of interacting species (e.g. Memmott *et al*. Askew 1961; Hawkins & Goeden 1984; Askew & Shaw 1986; Memmott & Godfray 1993; Memmott *et al*. 1993, 1994; Rott *et al*. 1998; Müller *et al*. 1999). Because they contain so many host species variously linked by different kinds of natural enemies, a complete description of the population dynamics of each species in the system is not a feasible way of trying to understand the processes structuring and maintaining the community as a whole.

Recent work on quantifying host–parasitoid food webs is beginning to provide important information needed to relate population dynamic theory to questions of community structure (e.g. Memmott *et al*. 1994; Schonrogge *et al*. 1996; Müller *et al*. 1999). In particular, *quantitative webs* in which the abundance of all host and parasitoid species are recorded (Memmott *et al*. 1994; Müller *et al*. 1999; Schonrogge *et al*. 1999) enable one to gauge to what extent webs may be broken down into functional compartments, recognized as semi-discrete units connected to the community as a whole by relatively few and weak connections. Such units are therefore dynamically compartmentalized to some extent (Pimm & Lawton 1980), and are the most amenable to being modelled as separate entities.

The more that webs can be broken down into such small functional units, the more relevant it becomes to examine how such communities are influenced by the population dynamics of their constituent species (Begon *et al*. 1997; Holt 1997a). Most work in this area has built on the basic two-species Lotka–Volterra or Nicholson–Bailey frameworks by adding one extra species to give systems of two natural enemies attacking one host or prey (May & Hassell 1981; Hogarth & Diamond 1984; Kakehashi *et al*. 1984; Hassell & May 1986; Godfray & Waage 1991; Briggs *et al*. 1993), two hosts or prey attacked by a common natural enemy (Roughgarden & Feldman 1975; Comins & Hassell 1976; Holt 1977; Comins & Hassell 1987; Holt & Lawton 1994), or a tritrophic community composed of a host, parasitoid and hyperparasitoid (Beddington & Hammond 1977; Hassell 1978, 1979; May & Hassell 1981; Wilson *et al*. 1997). This section, by way of an example, examines just one of these systems and considers how the basic host–parasitoid dynamics can be considerably influenced by an additional host species.

#### One parasitoid attacking two hosts

Much of the early theoretical work on the dynamics of two prey species sharing a common natural enemy species was inspired by Paine's (1966; 1974) classic study on the role of the starfish, *Pisaster*, in promoting the coexistence of its competing prey. The central question addressed by this work was to deduce how a shared natural enemy may influence the coexistence of prey species competing both intra- and interspecifically for the same resource (e.g. Cramer & May 1972; May 1974; van Valen 1974; Murdoch & Oaten 1975; Roughgarden & Feldman 1975; Comins & Hassell 1976; Fujii 1977; Hassell 1979). More recently, the emphasis has changed towards examining the effects of a natural enemy shared between *non-competing* prey, and how this may give rise to ‘apparent competition’ (Holt 1977, 1984; Jeffries & Lawton 1984; Holt & Lawton 1993; Holt *et al*. 1994). In this section we consider both of these approaches in turn.

##### Directly competing hosts

Consider the following scenario. Two competing insect herbivore species (*N*_{1} and *N*_{2}) are attacked by a common parasitoid species (*P*) to give:

where

In the absence of parasitism, we have a simple, discrete-generation, two-species competition model with host intrinsic rates of increase, *r*_{i}, carrying capacities, *K*_{i}, and the usual competition coefficients, α_{i}. The zero growth isoclines for the two host species are linear, as in the comparable Lotka − Volterra model, and coexistence is thus feasible only if there is some niche separation (α_{1}α_{2} < 1). Coexistence is impossible if there is complete niche overlap (α_{1}α_{2} ≥ 1) because the destabilizing effects of interspecific competition always outweigh the intraspecific density dependence.

Let us now introduce parasitism defined by the negative binomial model, where *a*_{i} is the per capita searching efficiency and *k*_{i} is the degree of aggregation of attacks amongst hosts (see eqn 3). In this way non-random parasitism is subsumed within the model. First, we note that if the parasitoids treat both host species identically (i.e. *a*_{1} = *a*_{2} and *k*_{1} = *k*_{2}) and if the hosts have equal rates of increase (*r*_{1} = *r*_{2}), then the natural enemy has no effect on the conditions for competitive coexistence (van Valen 1974; May 1977a), although it can alter the local stability properties of the coexisting hosts (Comins & Hassell 1976). There are, however, two main ways in which the introduction of the parasitoids can alter the conditions for host coexistence. (i) If the parasitoids are more effective against one host than the other (i.e. because *a*_{1}≠*a*_{2} and/or *k*_{1}≠*k*_{2} and/or *r*_{1}≠*r*_{2}), then the host isoclines will be lowered by differential amounts. This makes it possible for the parasitoid to maintain a three-species equilibrium under conditions where the two host species alone could not coexist, provided that the interspecific competition is not too strong (α_{1}α_{2} < 1). (ii) If there is ‘switching’ which allows the proportion of a particular host attacked to change from less than expected to greater than expected as the relative abundance of that host increases (Ivlev 1961; Murdoch 1969; Lawton *et al*. 1974; Cornell 1976; Cock 1978), the most abundant host type at any time will suffer the greatest percentage mortality. This can have a profound effect on the dynamics of the competing host species (Elton 1927; Murdoch & Oaten 1975), creating a potentially stable, three-species equilibrium where none existed before, even if the host species show complete niche overlap (α_{1}α_{2} ≥ 1) (Roughgarden & Feldman 1975; Comins & Hassell 1976; May 1977b; Teramoto *et al*. 1979).

Switching, if a widespread property of natural systems, could clearly be an important process shaping community structure. But after considerable interest in the 1970s in demonstrating switching experimentally and in exploring its dynamical impact, there has been relatively little recent interest in the process. This is largely because it has been very difficult to demonstrate unambiguously in the field (e.g. Murdoch *et al*. 1984b) and also because interest has moved to other behavioural responses that can result in density-dependent changes in the frequency of attack (Sutherland 1983, 1996).

##### Apparent competition

Let us now turn to a simpler three-species system of two *noncompeting* host species (*N*_{1} and *N*_{2}) linked only by their shared parasitoid species *P*:

*N*_{i, t + 1}= λ_{i}*N*_{i,t}*f*_{i}(*P*_{t}),*i*= 1,2

*P*_{t+ 1}=*N*_{1,t}[1 –*f*_{1}(*P*_{t})]+*N*_{2,t}[1 –*f*_{2}(*P*_{t})] eqn 12

The interesting feature of such systems is that, although the hosts do not compete directly for resources, the parasitoids produce the same kind of reciprocal negative effects on each host's growth rate that one would expect to observe in cases of true interspecific competition. This arises in the following way. Suppose the abundance of one host species increases. Parasitoid numbers as a result will also increase, and this in turn will lead to greater levels of parasitism on the other host species, whose population levels will therefore be depressed. In other words, just the result one would expect from a classical manipulation experiment to detect interspecific competition were one unaware of the shared natural enemy. Because of these parallels with interspecific competition, Holt (1977, 1984) christened this type of indirect interaction as ‘apparent competition’. Subsequently, Holt & Kotler (1987) also distinguished between ‘short-term apparent competition’ involving the immediate effects of behavioural responses of the natural enemy to an increase in one of its hosts or prey, and ‘long-term apparent competition’ involving the population effects over several generations.

The potential importance of apparent competition in structuring insect communities has been widely discussed (e.g. Lawton & Strong 1981; Freeland 1983; Jeffries & Lawton 1984; Lawton 1986; Holt & Lawton 1994; Berdegue *et al*. 1996; Godfray & Müller 1998; Müller & Godfray 1999). As shown by Holt & Lawton (1993), the effects of apparent competition within the general model are inevitably to lead to one of the host species going extinct, irrespective of the form of *f*(*P*_{t}) (as long as it is independent of *N*_{1} and *N*_{2}) (except in the unlikely situation of the parasitoid having exactly the same equilibrium density when supported on either host species alone). This is an interesting result and harks back to Nicholson (1933) who suggested that for a host–parasitoid equilibrium to exist there must be at least as many parasitoid species as host species, and to MacArthur's (1968) general assertion that there cannot be more species than there are niches. In short, one species of host that never encounters another species which feeds on a completely different resource, can still be responsible for its exclusion because of indirect interactions mediated via the shared natural enemy.

The empirical evidence for the importance of apparent competition in structuring insect communities is still sparse and largely based on field studies showing raised levels of mortality on one host species attributed to the presence of another species (e.g. Settle & Wilson 1990b; Settle & Wilson 1990a; Evans & England 1996; Schonrogge *et al*. 1996). A more concrete example of long-term apparent competition in operation comes from a laboratory study of two stored product moths (*Plodia interpunctella and Ephestia kuehniella*) attacked by a shared parasitoid, *Venturia canescens* (Bonsall & Hassell 1997, 1998). In replicated time series the separate pairwise interactions between the parasitoid and one of the hosts were always stable (Fig. 8a,b). Three-species interactions were then carried out within cages that completely separated the two host species but allowed the parasitoids to roam freely attacking both hosts. These three-species systems never persisted, with *E. kuehniella* always going extinct (Fig. 8c).

This instability is strikingly at odds with the frequent observation in the field of coexisting hosts sharing common parasitoid species, and prompts the question of what realistic biological processes within the framework of model could override the effects of apparent competition. Holt & Lawton (1994) considered several candidate mechanisms; for instance, that (i) hosts are resource limited, (ii) a constant *number* of hosts escape parasitism in refuges (constant *proportion* refuges, however, always fail to allow coexistence), or (iii) switching may occur between the host species. But they conclude that none of these mechanisms are sufficiently general or widespread to have a major impact. Other mechanisms that could sometimes be important for coexistence at certain spatial scales are (i) that random environmental variation affects both host species in different ways so that the two host species achieve higher growth rates under different sets of environmental conditions (Chesson & Huntly 1989; Godfray & Müller 1998), or (ii) that the interaction occurs at a metapopulation scale (see below).

Of all these mechanisms, however, the one that may prove to be the most important falls under the general heading of switching, simply because of the ease with which it may arise from the aggregative behaviour of parasitoids in a patchy environment (Royama 1970; Comins & Hassell 1987; Bonsall & Hassell 1999). Suppose, for example, that parasitoid searching patterns are modified in response to host density (e.g. Lewis & Tumlinson 1988; see Godfray 1994 for a review; Papaj & Vet 1990; Turlings *et al*. 1990) leading to aggregation on whichever host species is the most abundant at the time. This is a mechanism that will produce a net switching effect, which can be strongly stabilizing in a three-species interaction (Bonsall & Hassell 1999). In short, with this kind of switching arising so readily from a straightforward process of parasitoid or predator aggregation in a patchy environment, it may well prove to be a major factor in the coexistence of hosts or prey, irrespective of whether or not they are directly competing with each other.

### Metapopulation dynamics

- Top of page
- Introduction
- A basic framework
- Heterogeneity in host–parasitoid interactions
- Multispecies systems
- Metapopulation dynamics
- Conclusion
- References

The spatial interactions between hosts and parasitoids discussed above assume complete mixing of the dispersing individuals of both populations at least once during each generation period. A very different picture can emerge, however, if the habitat is on a much larger scale relative to the characteristic dispersal rates of the organisms. These larger areas now contain entire local populations that are only partially linked by dispersal, which in turn means that some degree of autonomous population dynamics can easily occur over a period of generations. Such an ensemble of partially independent local populations forms a ‘metapopulation’, a term originally coined by Levins (1968, 1969, 1970).

Metapopulation models have developed in quite different ways (see reviews by Taylor 1988, 1991; Kareiva 1990; Hanski & Simberloff 1997; Nee *et al*. 1997; Hanski 1999). In the original formulation of Levins (1968) space was implicit; the habitat was effectively divided into an infinite number of patches of equal size and quality, all of which are equally in contact with each other and are either empty, or occupied by the species in question. The models record the rate of change in the proportion of occupied patches as a function of colonization and extinction parameters without any explicit consideration of local dynamics. There have been many developments based on this framework; for example, allowing stochastic colonizations and extinctions (e.g. Gurney & Nisbet 1978; Hanski 1991, 1997a), spatially correlated environmental stochasticity (e.g. Hanski 1996) and varying patch quality, population growth, and extinction and colonization rates (see reviews by Hanski 1997a; Hanski 1999). There have also been extensions to describe two-species interactions (e.g. May 1994; Holt 1997b; Nee *et al*. 1997) and multispecies communities (Tilman 1994; Wennergren *et al*. 1995), looking at how trade-offs between extinction and colonization can facilitate coexistence.

Another class of metapopulation models has taken a spatially explicit approach in defining habitats or patches within a grid-like arrangement so that each patch has specific neighbours. Some of these ‘cellular’ models also record only different states of habitat occupancy (e.g. Maynard Smith 1974; Wolfram 1984; Crawley & May 1987; Wilson *et al*. 1993; Durrett & Levin 1994; Nowak *et al*. 1994; Rand & Wilson 1995; Rand *et al*. 1995), while in others local populations are represented by continuous variables and have explicit dynamics (e.g. Reeve 1988; Hastings 1990; Taylor 1990; Comins *et al*. 1992; Solé & Valls 1992; Rohani & Miramontes 1995; Swinton & Anderson 1995; Wilson *et al*. 1997, 1998).

There have been many recent attempts to detect metapopulation dynamics under natural conditions (see Harrison 1991; Schoener 1991; Harrison & Taylor 1997; for reviews). Clear examples amongst arthropods come from Lepidoptera populations (e.g. Thomas & Hanski 1977; Harrison *et al*. 1988; Hanski *et al*. 1995; Gyllenberg & Hanski 1997; Hanski 1997b; Harrison & Taylor 1997; Lei & Hanski 1997; Maton & Harrison 1997; Nieminen & Hanski 1998; Saccheri *et al*. 1998; Hanski 1999); mites (e.g. Sabelis *et al*. 1991; Walde 1991, 1994, 1995), waterfleas (Hanski & Ranta 1983) and spiders (Schoener & Spiller 1987a, 1987b; Spiller & Schoener 1990). There has also been an elegant laboratory demonstration of metapopulation dynamics by Holyoak & Lawler (1996) working with a protist predator–prey interaction in a system of interconnected patches.

This section examines explicitly spatial metapopulations applied to host–parasitoid systems, looking in particular at how population persistence is affected by metapopulation structure and restricted dispersal rates. For the main part, the total environment is assumed to be subdivided into local habitats (= patches) that are arranged in a grid. We shall assume that the hosts and parasitoids have discrete generations each divided into two distinct phases: (i) *a between-patch dispersal phase* in which a proportion of adult hosts and parasitoids leave their natal patch to colonize other patches according to some dispersal rule that determines the degree of mixing within the metapopulation as a whole, and (ii) *a within-patch parasitism and reproduction phase* in which the local host population is parasitized and the surviving healthy hosts go on to reproduce. Such models, with discrete time and space but continuous population state, have been dubbed ‘Coupled Map Lattices’ (Kaneko 1992; Solé & Valls 1992; Solé*et al*. 1992; Kaneko 1993).

#### Local dispersal

There are many different scenarios within any given metapopulation structure. Let us consider, for example, the situation where in each generation a certain fraction of adult hosts, µ_{H}, and adult female parasitoids, µ_{P}, leave the patch from which they emerged, while the remainder stay behind to complete their life cycle in their original patch. The dispersing individuals spread outwards and, for convenience, we will assume that they colonize equally the eight nearest neighbours of the patch from which they emerged. Longer-range dispersal can therefore only occur through repetition of these single-patch movements over a number of generations (more complex dispersal rules are discussed in Rohani & Miramontes (1995) and Wilson *et al*. (1997)).

The model is therefore defined in two parts. First, the equations for the dispersal stage in each patch are given by:

*Ν′*_{ι,t}= (1 - μ_{N})*N*_{i,t}*+*μ_{N}*N*_{i,t}

*P′*_{ι,t}= (1 - μ_{P})*P*_{i,t}*+*μ_{P}*P*_{i,t}eqn 13

where *N*_{i,t} and *P*_{i,t} are the predispersal host and parasitoid population densities in patch *i* at time *t*, *Ν′*_{ι,t} and *P′*_{ι,t} are the densities after dispersal, and *N*_{i,t} and *P*_{i,t} are the host and parasitoid populations averaged over the eight nearest neighbouring patches (Hassell *et al*. 1991a; Comins *et al*. 1992). Slightly different definitions are needed for patches along the boundary of the arena, depending on whether one assumes *cyclic*, *absorbing* or *reflective* boundary conditions; but as long as the arenas are sufficiently large, the type of boundary condition generally makes little difference, except that simulations with cyclic boundaries tend, not surprisingly, to produce more symmetrical spatial patterns of abundance.

The second part of the model defines the dynamics within a patch:

*N*_{i,t + 1}= λ*N′*_{i,t}*f*(*P*′_{ι,t})

*P*_{i,t + 1}= c*N′*_{i,t}[1 -*f*(*P*′_{ι,t})]eqn 14

where λ is the constant host rate of increase, and survival from parasitism is defined either by the negative binomial or Nicholson–Bailey models.

This introduction of explicit space with nearest neighbour movement has profound effects on dynamics (Hassell *et al*. 1991a; Comins *et al*. 1992; Hassell *et al*. 1994; Comins & Hassell 1996). These are outlined here under the five headings below.

##### Stable local populations

As found by Reeve (1988), the system as a whole will be stable if the within-patch, local populations are also stable (i.e. *k* < 1). In other words, stable and persisting local populations prevent the ‘turnover’ of habitat patches that is a characteristic feature of many model metapopulations. The same applies to comparable systems with single, or competing, species interactions (Hassell *et al*. 1995; Rohani *et al*. 1996). A number of factors can confound this simple conclusion. Most obviously, varying habitat ‘quality’ affecting the demographic rates of the local populations is bound to introduce different dynamics depending on the nature of the heterogeneity that is imposed.

##### Spatial patterns of abundance

The dynamics become markedly different once local population dynamics are assumed to be unstable (e.g. Nicholson–Bailey). Any metapopulation persistence that now occurs will result from the spatially distributed nature of the interaction, rather than from within-patch stability. The combination of restricted dispersal and ‘boom and bust’ local patch dynamics leads to a tendency for travelling waves of host and parasitoid abundance within the metapopulation arena. Because of the within-patch instability, the waves continually leave a wake of more-or-less empty patches behind them. The state of different patches thus becomes highly asynchronous and the metapopulation as a whole now persists much more readily than its constituent parts could on their own.

The deterministic spatial patterns of dynamics that can be obtained by this process are striking and varied, and fall into three categories labelled ‘spatial chaos’, ‘spiral waves’ and ‘crystal lattices’ (Hassell *et al. *1991) (Fig. 9). The spatial chaos is characterized by the populations fluctuating from patch to patch with no long-term spatial organization; each randomly orientated wave front persists only briefly. The spiral waves differ in that the waves of local population densities tend to rotate in either direction around relatively immobile focal points. A time series taken from a particular local population would be characterized by fairly regular cycles produced by the regular ‘passage’ of the arm of the spiral through the habitat, except at the focus of the spiral where the population density would remain relatively constant. Local populations ‘swept’ by the trailing arms of the spiral fluctuate with increasing amplitude the further they are away from the focus. The characteristic wavelength of these spirals can vary greatly depending on the parameters of the model. Finally, the so-called ‘crystal lattices’ appear as more-or-less regular spacing of relatively high and low density patches within the grid. These three different patterns depend very much on the relative dispersal rates of the hosts and parasitoids, as shown in Fig. 10. The region of clearly defined spirals is the largest, while the crystal lattice region is very small and is only observed for very low host rates of increase.

Theory is far ahead of experiment or observation in this area. Direct observations of these kinds of spatial patterns of dynamics in the field present enormous logistical problems. It may be possible, however, to determine properties of the population density distributions which are diagnostic of spirals or spatial chaos (for example, particular patterns of delayed covariance). This could then lead to indirect tests of the existence of these self-generated patterns in nature, that will also distinguish them from patterns of population abundance driven mainly by environmental randomness.

##### Metapopulation persistence

One of the most striking features of metapopulations is the ease with which they tend to persist, even if the local populations are unstable, as long as there is sufficient asynchrony in the state of the different patches. The spatial patterns in Fig. 9 all promote such asynchrony, and are associated with characteristic types of total population fluctuation: large-amplitude irregular cycles from chaotic spatial patterns, smaller-amplitude irregular cycles from spiral waves and stable populations from crystal lattices.

Although this persistence appears robust, there is always the possibility of extinction, which becomes increasingly likely as the size of the total habitat decreases (Fig. 11). In the limit of very small arena widths (*n* ≤ 3) and all individuals dispersing from their respective patches (µ_{N} = µ_{P}= 1), local and metapopulation models become the same. As the arena size increases, the probability of extinction rapidly decreases. But even with very large arenas, the metapopulation will not persist indefinitely. Increasing the arena size decreases this probability of extinction but never eliminates it. For a given size of arena, the host and parasitoid dispersal rates also influence the probability of extinction. As dispersal increases, adjacent patches generally become more synchronous and the host and parasitoid distributions become more autocorrelated in space. The characteristic spatial scale of the dynamics thus increases, and it becomes more difficult to fit the self-maintaining pattern into the available space. In much the same way, fragmentation of the whole environment runs the risk of disrupting the metapopulation dynamics, either by reducing the number of local populations below some level required for the combined metapopulation to persist, or by interfering with the dispersal required to link the locally unstable local populations (often leading to population outbreaks as the ability of the parasitoid to regulate the host is disrupted). The extent of this depends largely on the characteristic spatial scale of the dynamics (Hassell *et al*. 1993).

The striking spatial patterns and persistence properties of metapopulations can be observed from completely uniform and regular environments with no spatial or temporal variation in any of the parameters. But how robust are these properties to various forms of random variation; in particular, to environmental and demographic stochasticity? Environmental stochasticity is most easily represented by random variation in the demographic parameters from patch to patch (Reeve 1988; Comins *et al*. 1992; Hassell *et al*. 1993; Ruxton & Rohani 1996). In this case there is little disruption of the spatial patterns observed from the corresponding deterministic cases, even with quite appreciable levels of random variation, primarily because the variation is on a much smaller spatial scale than the spatial patterns themselves (except for the crystal lattices that are easily disrupted). Only if the random noise is spatially correlated, with a scale of variation comparable to the characteristic scale of the spatial dynamics, does it markedly affect the outcome by disrupting the spatial patterns.

Demographic stochasticity can also have a marked effect on the dynamics of metapopulations. One feature of host–parasitoid metapopulations that have oscillatorily unstable local populations is the extremely low population sizes that can occur within patches, often involving fractions of a host or parasitoid individual. This raises the possibility that some of the properties described above are artefacts of allowing such fractional individuals to exist. Wilson *et al*. (1997) and Wilson & Hassell (1997) explored this using a stochastic individual-based model (SIB), which shows increased extinction rates and larger population fluctuations whenever average population densities get very low.

#### Multispecies metapopulations

The basic two-species metapopulations above may be easily extended to a range of three-species host–parasitoid systems in which either two host species share a single parasitoid species (H-P-H), a single host species is attacked by two parasitoid species (P-H-P) or there is an interaction between a host, parasitoid and hyperparasitoid (H-P-Q) (Hassell *et al*. 1994; Comins & Hassell 1996; Wilson & Hassell 1997). In particular, it will be interesting to determine if ‘diffusive’ dispersal in these more complicated metapopulations has the same profound effects of promoting persistence and, if so, how this changes the conditions for the coexistence of competing species. The three types of interaction can be modelled in much the same way as for the two-species host–parasitoid systems and full details are given in Comins & Hassell (1996).

All three models give similar results to the extent that a third species (be it another host, another parasitoid or a hyperparasitoid) can coexist for long periods within the spatial dynamics (spiral waves or chaos) generated by an existing two-species interaction, although the conditions for doing so are considerably restricted compared to the corresponding two-species system and depend upon some kind of fugitive coexistence (Hutchinson 1951; Horn & MacArthur 1971; Levins & Culver 1971; Hanski & Ranta 1983; Nee & May 1992; Hanski & Zhang 1993). For, example, in the two parasitoid–one host system, coexistence occurs most easily when the two parasitoid species have very different dispersal rate, provided that the low dispersal rate is matched by high within-patch searching efficiency, and vice versa. Similarly, in the apparent competition case of two hosts and one shared parasitoid (M. Bonsall & M. Hassell, unpublished), coexistence occurs most readily when the two host species have very different dispersal rates and the relatively immobile species has either the higher rate of increase or is less susceptible to parasitism. And, as in the two-species models, introducing demographic stochasticity decreases the probability of persistence (Wilson *et al*. 1998).

Another interesting property of these models is that coexistence tends to be associated with some degree of self-organizing niche separation between the competing species (Hassell *et al*. 1994; Comins & Hassell 1996; M. Bonsall & M. Hassell, unpublished). This is best seen when the spatial dynamics show clear spirals. For example, in the case of two competing parasitoids with very different dispersal rates, the relatively immobile species tends to be confined to the central foci of the spirals where it is the most abundant species, and the highly dispersive species occupies the remainder of the ‘trailing arm’ of the spirals, as shown in Fig. 12. Since the foci of spirals are relatively static in these models, the less mobile species appears to occur only in isolated, small ‘islands’ within the habitat, much as if these were pockets of favourable habitat. As the dispersal rates become less divergent between the species, the niche of the less dispersive species spreads further into the arm of the spirals. These intriguing properties, however, appear to be quite sensitive to model conditions. For example, the segregation in Fig. 12 tends to disappear if there are threshold populations below which host and parasitoid cannot fall, and is not found at all in the SIB model above (Wilson & Hassell 1997); the ‘focus-living’ species gradually becomes eliminated from the foci, without being able to reinvade the vacant foci. Introduction of a small fraction of individuals capable of long-range dispersal, however, permits reinvasion of the vacant spiral foci and the phenomenon is once again observed (Comins & Hassell 1996).

The possibility that coexistence is promoted by trade-offs between competitive and dispersal abilities is well recognized. What is different here is that the spatial metapopulation structure within which these trade-offs occur, may itself be self-generated by virtue of the dynamics of the interacting species, leading to competitors separated in space in a self-organized way. An important challenge is to develop tests that will enable one to distinguish between species segregated in this way due to their internal dynamics from those whose distribution depends on a non-uniform environment.

### Conclusion

- Top of page
- Introduction
- A basic framework
- Heterogeneity in host–parasitoid interactions
- Multispecies systems
- Metapopulation dynamics
- Conclusion
- References

Although the theory of host–parasitoid interactions is relatively well advanced and there is a large amount of empirical information on host–parasitoid systems in the literature, there are several major areas where much work still needs to be done. For example, while many of the demographic components of host–parasitoid systems have been separately quantified by field measurement and laboratory experiment, there are still relatively few detailed long-term studies of insect populations and their natural enemies in which the essential life-table parameters have been measured and appropriate mechanistic models of the system developed. The great value of such long-term data lies in the relative ease with which one can then develop parameterized models of the system.

The role of environmental and demographic stochasticity on host–parasitoid interactions has only recently been explored in a rather general way. Yet, these effects will almost certainly be very important since they can interact with non-linear regulatory processes in complicated ways. Environmental noise, for example, can help or hinder the persistence of host–parasitoid associations, depending on the frequency spectrum of the noise and how it interacts with density-dependent mechanisms. Chesson (1986), Chesson & Case (1986) and colleagues have made a start in codifying these effects in general, but much remains to be done. This is an area where laboratory microcosms should have an important role to play. Stochasticity can be manipulated precisely and the way that it affects the key demographic components can then be evaluated.

Another major development in which much crucial information is lacking is in the development of a population-based community ecology of parasitoids. Understanding parasitoid communities requires an understanding of the dynamics of the basic host–parasitoid linkages, on which much work has been done. But it is now clear how much more complex are the dynamics of systems with three or more species. Unravelling these will be difficult, but the simple and often strong trophic link between parasitoids and their hosts suggests that this has a better chance of succeeding than many equivalent programmes revolving around other types of animal trophic relationship.

Finally, many people have speculated that heterogeneities are crucial to the persistence of host–parasitoid systems. At the local scale this involves heterogeneity in the host risk of parasitism, which can arise in many different ways. Of these, density-dependent aggregation by parasitoids in patches of high host density may not as important as has been thought, but heterogeneity in host defences and aggregation irrespective of host density may be more important. Comprehending these processes in an explicitly spatial setting has been a preoccupation of theorists for a long time. But, despite much quantification of the heterogeneities themselves, there is still little *direct* empirical evidence, apart from in a few laboratory studies, pinpointing how these heterogeneities influence population dynamics in the field. Replicated, long-term, manipulation experiments in the field and laboratory will be invaluable in resolving this long-standing issue.

The integration of theory and observation is also much needed in studying heterogeneities at the metapopulation scale. Key questions revolve around the measurement (and types of analysis) needed to diagnose population density distributions indicative of spirals or spatial chaos, and methods of distinguishing self-generated spatial patterns from those arising due to environmental randomness. Direct observations at the metapopulation scale in the field present enormous logistical problems, but Hanski's (1999) field study of the Glanville fritillary metapopulation in SW Finland is showing the way.

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- A basic framework
- Heterogeneity in host–parasitoid interactions
- Multispecies systems
- Metapopulation dynamics
- Conclusion
- References

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Correspondence: Prof. M.P. Hassell, FRS, Department of Biology, Imperial College at Silwood Park, Ascot, Berks SL5 7PY. Tel: 01344 294207. Fax: 01344 874957. E-mail: m.hassell@ic.ac.uk