Host–parasitoid population dynamics*

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 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 = λNtf(Nt, Pt) P t + 1 = cNt[1 - f(Nt, Pt)]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(Nt,Pt) defines the fraction of the Nt 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 Nenc=aNtPt, where a is the per capita searching efficiency which Nicholson called the ‘area of discovery’. Secondly, these Nenc 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(– aPt), where aPt are the mean encounters per host, Nenc/Nt = aPt. Thus, one minus this zero term is the probability of a host being attacked. Substituting into model 1 gives: N t + 1 = λNt exp( - aPt) P t + 1 = Na = cNt[1 - exp ( - aPt)]eqn 2 where Na 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 andλ, 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). Numerical simulations showing host (○) and parasitoid (● population oscillations from: (a) the Nicholson–Bailey model with parasitoid searching efficiency, a = 0·068 and the host rate of increase, λ = 2; (b) the negative binomial model. The parameters are the same as in (a), except that parasitism is no longer random (k = 0·6 instead of k→∞). 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. Population dynamics of the bruchid beetle, Callosobruchus chinensis (●) feeding on black eyed beans and its pteromalid parasitoid, Anisopteromalus calandrae (○) in a laboratory system (Hassell & May 1988). (a) A non-patchy system with 50 beans uniformly distributed on the arena floor. The parasitoids are introduced in week 19 and become extinct in week 32, allowing the hosts to increase until checked by resources. (b) A patchy system with 50 beans each in an individual container with restricted access to both hosts and parasitoids. (From Hassell 2000.) 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. 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). 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 Nt hosts and Pt adult female parasitoids distributed amongst the n patches such that within any one patch there are Pi parasitoids searching for Ni 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. Average patterns of time allocation (a, c) and parasitism (b, d) per patch in relation to host density per patch from two laboratory systems. (a, b) Ten females of the cynipid parasitoid, Trybliographa rapae, parasitizing larvae of the cabbage at different within of & Hassell 1988). d) females of the parasitoid, Trichogramma parasitizing eggs of the on at different In contrast to (a, b) parasitism is inversely density-dependent, the for the parasitoids to on the with host (Hassell 1982). Numerical examples showing how the of parasitoids and different kinds of functional can lead to both density-dependent and inverse density-dependent spatial patterns of parasitism. (a) An aggregated distribution of searching parasitoids. (b) kinds of functional defining the attack rate per parasitoid within a from a linear functional with a = and = from a with a = and = The resulting overall parasitism per patch with and corresponding to those in that the more than for the of the parasitoids to cause inverse density (From Hassell 2000.) While data information on the distribution of searching parasitoids and the resulting patterns of parasitism are to collect from the field see 1983; & it is relatively to quantify patterns of parasitism without about the distribution of searching hosts can be from a range of patches, taken to the laboratory and the of parasitism then usually by the next generation or by the different examples in the of & Murdoch and Hassell & direct density-dependent patterns of parasitism, inverse patterns and parasitism with host density per examples are shown in Fig. the of et al. and Hassell et al. the direct and inverse density-dependent patterns arise from and the patterns arise from Examples of field studies showing different spatial patterns of parasitism. (a) parasitism of the larvae of cabbage by the cynipid parasitoid, Trybliographa rapae & Hassell 1988). (b) density-dependent parasitism of eggs by the parasitoid, & parasitism of the scale by the parasitoid, et al. parasitism of the by the parasitoid, for a of the (From & Hassell Much of the in these different patterns has on their importance to population theoretical work with discrete generation interactions to how density-dependent patterns (e.g. Hassell & May Murdoch & 1975; et al. have that both the inverse patterns and the can be equally important for (Hassell Chesson & Murdoch Hassell & May & Murdoch et al. Hassell et al. models on this The simple model framework can be to interactions in an patchy environment as long as f(Nt,Pt) the across all patches, of the fraction of hosts escaping parasitism. The function f(Nt,Pt) 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 of a habitat with n patches as within each of which parasitism is random and by a functional The distribution of hosts and parasitoids from patch to patch is defined by and the of total hosts and total searching parasitoids, respectively, in the patch, so that f is given by: where a is the searching efficiency per patch and is the handling time (Hassell & May specifically, will that the parasitoids in patches of high host density following the simple where is an index of parasitoid and is a constant such that the values to (Hassell & May The can describe a wide range of parasitoid distribution they will be evenly distributed across patches if = 0, and tend to in patches of high host density as in the they will all congregate in the single patch of host density the as complete if < the parasitoid distribution is with their local abundance now inversely with host density per The way that these different patterns can for a linear functional within is shown by the following example in which the hosts are aggregated to a negative binomial is the probability of hosts in a patch from the negative binomial distribution and are the number of parasitoids in a patch with hosts by eqn of this model that sufficiently direct or inverse density-dependent distributions of parasitoids can stabilize the interactions (Fig. while parasitoid leads to local instability the host rate of increase is some a range of dynamics can occur et al. of parasitoid can also have a effect on equilibrium levels (Fig. Parasitoids have the effect in host equilibria when their distribution most closely that of the hosts (i.e. = or in this model leads to host because the parasitoids, are confined to their patches, irrespective of the degree of host The effects of parasitoid on and equilibrium (a) between the degree of of parasitoids, in eqn and the amount of host k a negative binomial distribution of hosts per from model with survival from parasitism, given by eqn total number of patches n = searching efficiency a = 1 and host rate of increase λ = (b) Examples of the host equilibrium level with the degree of parasitoid for three different numbers of patches from model with given by and a = λ = 2 and n = n = and n = Hassell in which are There has been much more on the effects of than because of the it has been assumed to have in population and because have to make the between and population dynamics et al. et al. Godfray 1994). It is now however, that heterogeneity can be at as important a process in population Let us consider a specific example where all the heterogeneity in parasitism is of the spatial distribution of hosts as in Fig. and that this is by the distribution of searching parasitoids being independently aggregated from patch to patch following a gamma We also that within any one patch the parasitoids exploit hosts at random and have a constant per capita searching efficiency (i.e. a linear functional The fraction of hosts escaping parasitism is now given by: where is the gamma probability density function for parasitoids per patch with mean and is a constant the of the density and a is the usual per capita searching efficiency of the In each patch, therefore, host survival by randomly searching parasitoids is given by the zero term of a Poisson distribution with mean et al. Chesson & Murdoch much the same were the parasitoids uniformly distributed across patches but their searching efficiency, a, to a gamma distribution et al. The properties of the model are the populations will be by the heterogeneity as long as there is in the distribution of parasitoids or, more specifically, if < 1. eqn also to (1978) negative binomial model. The index of is now to the degree of of the searching parasitoids & Murdoch et al. Hassell et al. The k < 1 from the negative binomial model, therefore corresponds in this spatially model to < 1. stabilizing properties have also been from models with parasitism by Reeve et al. further that patterns of parasitism are important as a stabilizing The parasitoid described an in which individuals are distributed amongst the patches at the start of each generation, where they then confined irrespective of how the hosts become and or not there are more patches The parasitoids in this are much more in being able to the best patches The large body of on patch stems from the classic paper by in which a a patch when the rate of is reduced to the maximum rate in the environment as a whole (e.g. & & 1997). to parasitoids, this means an distribution of parasitoid searching that the rate of parasitism for the individual searching parasitoids, depending on the spatial distribution of hosts and any and life & & (1978) and & Hassell models for such searching parasitoids. The model individual parasitoids amongst patches, their searching time in relation to the of the host distribution and the of the parasitoid searching efficiency and handling time also the start