Single vs. multiple introduction in biological control: the roles of parasitoid efficiency, antagonism and niche overlap

Authors


Brent S. Pedersen, Insect Biology, Wellman Hall, University of California, Berkeley, CA 94720–3112, USA (fax +510 642 7428; e-mail bpederse@nature.berkeley.edu).

Summary

  • 1Theoretical studies have presented conflicting conclusions about the value of introducing multiple parasitoid species to control insect pests, depending on the presence and form of niche separation in the model. We investigated the impacts of multiple species introduction on control using a modelling approach.
  • 2We started with a discrete-time model of a pest, a primary parasitoid and a parasitoid that can interact antagonistically with the primary parasitoid as well as the pest. Implicit niche separation between parasitoids occurs via aggregated encounters with pests. We subsequently modified the simple model to include explicit niche separation and a refuge from parasitism, allowing the assumption of implicit niche separation to be relaxed. Multiple introduction is deemed beneficial if equilibrium pest densities are lower after the release of the interactive parasitoid.
  • 3The outcome of biological control under the simple model depends largely on the search efficiency of the primary parasitoid. The primary parasitoid can co-exist with an antagonistic interactive parasitoid, and multiple introduction is beneficial provided that the level of antagonism by the interactive parasitoid is not too high.
  • 4In the more complex model with explicit niche separation, the ratio of primary search efficiencies of the two parasitoids, niche occupancy and degree of aggregation are shown to be important predictors of the outcome of multiple introduction. We demonstrate the importance of refuge breaking by an additional parasitoid as a compelling reason for multiple introduction. In addition, antagonistic interactions between parasitoids can be mediated by overall efficiency and explicit niche separation.
  • 5For both implicit and explicit niche separation scenarios, the interactive parasitoid was excluded at high levels of niche overlap. Minimum pest densities with a three-species equilibrium were similar between the two scenarios.
  • 6Synthesis and applications. After consideration of explicit niche separation in a two parasitoid–one pest system, we conclude that multiple introduction is often a sound strategy in biological control, despite potential antagonistic interactions from competition, cleptoparasitism or facultative hyperparasitism. In selecting parasitoids for introduction, practitioners should evaluate the potential for niche separation between parasitoids and the overall efficiency of each parasitoid (primary search efficiency and aggregation of search) rather than dismiss a biological control candidate because of moderate antagonistic interactions.

Introduction

Classical biological control, the introduction of a natural enemy to reduce the density of an invasive pest species to a tolerable level, has a long history in cultivated systems (Waage & Mills 1992; Van Driesche & Bellows 1996). Parasitoids have proven to be the most promising natural enemies in attaining suppression of insect pests (Greathead 1986; Greathead & Greathead 1992; Mills 2000). Simple mathematical models, dating back to Lotka (1925), Volterra (1926), Thompson (1924), and Nicholson & Bailey (1935), have been used to conceptualize the trophic interactions occurring among insect pests and natural enemies, thereby aiding in the selection of biological control agents for pest suppression (Hassell 1978; Hochberg & Hawkins 1992; Murdoch & Briggs 1996; Mills 2001).

Biological control practitioners have long debated the merits of single vs. multiple introduction of parasitoid species (Ehler 1990). Those promoting single introduction feel that the superior parasitoid (the one capable of reducing pest densities to the lowest level) may be excluded, or its efficacy compromised, by competition in the event of multiple introductions (Turnbull & Chant 1961; Watt 1965; Turnbull 1967; Ehler 1982, 1985). The supporters of multiple introduction claim that the best parasitoid(s) will persist, maintaining low pest densities despite competitive interactions. Complications fuelling the debate include the extent of temporal and spatial overlap between parasitoids (May & Hassell 1981; Kakehashi, Suzuki & Iwasa 1984; Hochberg 1996), stage structure and life histories (May & Hassell 1981; Godfray & Waage 1991; Briggs 1993; Briggs, Nisbet, & Murdoch 1993), the order of parasitism (May & Hassell 1981), the effects of a refuge from parasitism (Kakehashi, Suzuki & Iwasa 1984; Hochberg 1996), and the potential for competitive and hyperparasitic interactions between parasitoid species (May & Hassell 1981; Hogarth & Diamond 1984; Hochberg 1996; Klopfer & Ives 1997).

An additional concern for target pests that undergo complete metamorphosis is that a greater range of potential interactions occurs among species within a parasitoid assemblage that could influence the outcome of multiple introductions (Mills 1994, 2003). Although obligate hyperparasitism and competition are frequent among parasitoids associated with pests that show incomplete metamorphosis (e.g. Homoptera), facultative hyperparasitism, cleptoparasitism and competitive advantage due to ectoparasitism or developmental strategy are common among parasitoids of pests with complete metamorphosis (e.g. Lepidoptera, Hymenoptera, Diptera and Coleoptera). Here we refer to these interactions as antagonism, and to a parasitoid that exhibits antagonism as an interactive parasitoid, in contrast to a primary parasitoid that develops only on the pest. The disparity in the ability of an interactive parasitoid to locate previously attacked hosts vs. healthy hosts we refer to as ‘preference’. Although it has been suggested that hyperparasitoids and cleptoparasitoids should be excluded from biological control introductions (Ehler 1985; Mills 1990), the role of preference in determining the outcome of multiple introductions has yet to be explored in detail.

In order to provide better guidelines for biological control practitioners with respect to multiple introductions and parasitoid interactions, we addressed these issues through use of a model system. We delineated this system with a host, a primary parasitoid and an interactive parasitoid with varying degrees of niche overlap. Although the biology of each type of interaction is unique (Mills 2003), we are able to encompass the spectrum of antagonistic interactions with a single mathematical formulation modified from Hochberg (1996). With this interactive parasitism model, we show how the spectrum of preference determines the outcome of biological control in a one host–two parasitoid system where aggregated search provides implicit niche separation. We then augment the model to provide explicit niche separation and a stabilizing refuge from parasitism (following Kakehashi, Suzuki & Iwasa 1984), which allows us to compare the effects of implicit and explicit niche separation, as potentially different biological mechanisms, within the same model. We use this model to delineate situations under which multiple introduction is the preferable strategy for biological control as it relates to niche separation and parasitoid antagonism.

Simple model

We employ the discrete-time Nicholson–Bailey (1935) framework representing a host species with non-overlapping generations and two associated parasitoids. The traditional nomenclature, where Ng, Pg and Qg, respectively, denote population densities of the host, primary parasitoid and interactive parasitoid at generation g, is adopted. We introduce the model with the following form:

image(eqn 1a)
image(eqn 1b)
image(eqn 1c)

In each equation, the right-hand side maps the population density at generation g to a new density at g+ 1. The model is a derivation from that of Hassell (1978) and May & Hassell (1981), and has the same form as that of Hochberg (1996) except for the order of occurrence of parasitism and density dependence. With parameters as listed in Table 1, equations 1a–c represent a system where the primary parasitoid (P) attacks an earlier host stage than the interactive parasitoid (Q) at a rate of 1 −fPN. Hosts parasitized by P become inaccessible for direct primary parasitism by the interactive parasitoid Q and available only for the various antagonistic interactions encompassed by the term 1 − fQP. Juvenile hosts (λNg) not attacked by the primary parasitoid are available for primary parasitism by the interactive parasitoid at a rate of 1 − fQN. With the number of female offspring per parasitized host set to cP = cQ =c = 0·5, the parasitoids are solitary, laying a single egg per encounter with an equal allocation to both sexes. Host density dependence in generation g, given by:

Table 1.  Listing of model parameters and their biological interpretation
ParameterBiological interpretation
λNumber of offspring per host (in the absence of parasitism and density-dependent mortality)
KCarrying capacity of hosts
Search efficiency of:
aPN primary parasitoid on host
aQN interactive parasitoid on host
aQP interactive parasitoid on previously attacked hosts
βPMaximum hosts that the primary parasitoid can parasitize
βQMaximum previously attacked and novel hosts that the interactive parasitoid can parasitize
cNumber of female offspring per parasitized host
Negative binomial parameter:
kP aggregation of search by the primary parasitoid
kQ aggregation of search by the interactive parasitoid
Proportion of hosts exposed to parasitism by:
s primary parasitoid
t both parasitoids and to antagonism by the interactive parasitoid
u interactive parasitoid
v neither parasitoid
image(eqn 2)

governs the host population in the absence of parasitism such that host density either converges to or oscillates around the carrying capacity K depending on the value of λ, the number of offspring per host. This form of limitation is derived from Ricker's (1954) work on fish populations and, although widely applied in population models, represents a simplification of the true nature of density dependence in insect populations (Bellows 1981; Getz 1996). As formulated, density-dependent mortality acts on individuals surviving parasitism with a magnitude dependent on the number of hosts present before parasitism (for supporting empirical evidence see Lane & Mills 2003). Although the ordering of density dependence and parasitism has been shown to be important (May & Hassell 1981; Wang & Gutierrez 1981), under successful biological control the hosts are suppressed by parasitism to a level where density-dependent mortality is minimal. This then allows us to concentrate more specifically on the effects of parasitism.

With the negative binomial model used to distribute encounters among hosts, the general form of the escape function, fYX, of a host X from parasitoid Y becomes:

image(eqn 3)

where ɛ is the mean number of encounters per host and k is the aggregation parameter. We use the form derived by Getz & Mills (1996) that is given for each of the three host–parasitoid combinations by:

image(eqn 4a)
image(eqn 4b)
image(eqn 4c)

where a parameter with a single subscript is dependent only on the parasitoid (e.g. βY, the fecundity of parasitoid Y), while a parameter with two subscripts is dependent upon the characteristics of both the parasitoid and its host (e.g. aYX, the search efficiency of parasitoid y for host x). An important assumption of equation 3 is that the encounters of the two parasitoids aggregate independently, leading to separation of the parasitoid niches.

To account for differences in the densities of hosts susceptible to parasitism (a consequence of the order of parasitism), we include δ in the denominator of equations 4a–c to scale the total hosts available for each interaction as follows:

image(eqn 5a)
image(eqn 5b)
image(eqn 5c)

Because the primary parasitoid attacks first, it can potentially encounter the entire population of immature hosts (equation 5a). The interactive parasitoid distributes its eggs (up to a maximum of βQ) between immature hosts that escape the primary parasitoid (equation 5b) and those previously parasitized by the primary parasitoid (equation 5c). Including both sets of potential hosts in the denominators of equations 4b and 4c ensures that the interactive parasitoid is limited by a single complement of eggs. Incorporating equations 5 and 4 into equation 3 gives the proportion of individuals escaping parasitism. Thus, the escape function (equation 3) permits host encounters by the parasitoid to occur as an aggregated (k → 0) or random (k → ∝) process, and for the parasitoid to be limited by fecundity (β → 0) or by search (a → 0).

There is no analytical solution to equation 1 because of the inclusion of host density in the negative binomial escape function and the host density dependence term. Therefore, the system was analysed by numerical simulation, using sequential introductions of the two parasitoids to simplify interpretation, although the final outcome applies equally well to a more typical simultaneous introduction. For all simulations, the model was run for 400 generations after introduction of the host at its carrying capacity and the primary parasitoid at a density of 1. At generation 401, the interactive parasitoid was initialized at a density of 1 and the system was run for an additional 400 generations. This interval was determined to be more than sufficient to remove all transient behaviour from the system. In cases where the system converged to a cycle rather than to a stable-point equilibrium, we used the average density over one period of the cycle as an estimate of population abundance at equilibrium.

Preliminary analysis of the model for a single parasitoid and a host with a relatively low rate of population growth (Hassell & May 1973) revealed that equilibrium host densities were primarily affected by search efficiency and that fecundity had minimal effect on equilibrium host densities past the minimum threshold value that corresponds to the onset of fecundity limitation (Fig. 1). Therefore, we held parasitoid fecundity constant at 35 (but for other examples of parameter sets that show a greater influence of fecundity limitation see Lane, Mills & Getz 1999) and varied only the search efficiency of the interactive parasitoid across the entire spectrum of parasitoid interactions (Fig. 2) from primary parasitism (along the x-axis) to obligate hyperparasitism (along the y-axis). Following Beddington, Free & Lawton (1978) and Hochberg (1996), the merit of introducing a second parasitoid was judged by the ratio of the density of N at the N, P, Q equilibrium (inline image) to the density of N at the N, P equilibrium (inline image). Values of inline image less than 1 (shown in white; Fig. 2) indicate that it is beneficial to introduce Q, while those greater than 1 (shown in grey; Fig. 2) indicate an increase in equilibrium host densities.

Figure 1.

The influence of search efficiency (a) and fecundity (β) on the (log) equilibrium host density (N). λ = 2, K = 10 000, kP = 0·5, c = 0·5.

Figure 2.

Regions of parameter space representing the different forms of antagonistic interaction between parasitoids relative to the combined search efficiency/preference of the interactive parasitoid for unparasitized hosts (aQN) and for hosts previously parasitized by the primary parasitoid (aQP).

Simple model results

When the primary parasitoid has low search efficiency, this simple model suggests that it is beneficial to introduce a second parasitoid irrespective of the form or intensity of its interactions with the primary parasitoid, unless it is close to functioning as an obligate hyperparasitoid (Fig. 3a). The interactive parasitoid is unable to exclude the primary except in cases where its search efficiency for healthy or previously parasitized hosts greatly exceeds that of the primary parasitoid. When the primary parasitoid is excluded, the interactive parasitoid functions solely as a primary parasitoid and its greater search efficiency leads to reduced host densities.

Figure 3.

Outcome of multiple introduction for a primary parasitoid with a low (a) and high (b) search efficiency. Shaded areas indicate an increase in host density with multiple introduction, and contours depict the regions of co-existence. Dotted lines indicate values of aQP used in the bifurcation diagrams below over the same range of aQN used in the figures above. The bifurcation diagram shows 1200 generations of host densities after the introduction of Q with 4800 transient generations discarded. λ = 2, K = 10 000, βP = βQ = 35, kP = kQ = 0·5, c = 0·5.

The interactive parasitoid has greater potential to be detrimental to host suppression when the primary parasitoid is efficient and able to suppress host densities to extremely low levels (Fig. 3b; note the order of magnitude difference between the three-species equilibrium densities in the bifurcation diagrams below Fig. 3a,b). When the interactive and primary parasitoids both function as primary parasitoids (aQP = 0), the two parasitoids co-exist and greater host suppression is achieved (along x-axis). If the search efficiency of the interactive parasitoid is too low, it cannot establish, otherwise a three-species equilibrium exists throughout the parameter range, with greater host suppression still occurring over a broad range of preference for the interactive parasitoid. However, when the interactive parasitoid is largely antagonistic, it reduces the density of the primary parasitoid that would otherwise suppress host densities, and control is lost. In addition, if the interactive parasitoid is highly efficient but shows limited antagonism, the interaction results in large-amplitude cycles and/or chaos (see bifurcation diagram below Fig. 3b). In this case, the average host density over the period of the cycle (or 400 generations of chaotic behaviour) is greater than its density at the stable-point two-species equilibrium for the primary parasitoid alone, making multiple introduction a poor choice under these conditions.

Niche separation model

In this section, the consequences of multiple introduction are explored for a primary and interactive parasitoid with niches that overlap each other and that of the host to varying degrees. An absolute refuge, wherein a fraction of hosts is free from parasitism, adds stability to the system regardless of the distribution of parasitoid encounters. The niche separation model is formulated as a modification of equation 1:

image(eqn 6a)
image(eqn 6b)
image(eqn 6c)

where:

image(eqn 7)

The parameters s, t, u and v delineate regions of overlap between the niches of the host and the two parasitoids (Fig. 4; adapted from Kakehashi, Suzuki & Iwasa 1984). Although we refer to these parameters as regions, they are not spatially isolated areas in a habitat but rather abstractions delineating any form of niche separation that could be expected in nature. A proportion of the host population occupies an absolute refuge from parasitism (v), t is the proportion of hosts in the region of overlap of parasitoid niches, and s(u) is the proportion of hosts uniquely available to parasitoid P(Q). Note that s, t, u and v are fractions of the host population that must always sum to 1. Further, as t approaches unity (s, u, v → 0) there is complete overlap between all three species and the model reduces to equation 1. As v approaches unity, the host is in complete refuge from both parasitoids. As shown in Fig. 4, the escape functions for parasitism in each region are weighted by their respective fraction of available hosts in the population. This yields the overall parasitism function G, the proportion of hosts parasitized. The stability imparted by this niche separation allows us to move away from the traditionally used form of implicit niche separation of equation 3. We do this by taking advantage of the fact that as k → ∝ the parasitoids encounter hosts randomly.

Figure 4.

A schematic representation of the regions of niche overlap between the host (s + t + u + v), primary parasitoid (s + t) and interactive parasitoid (t + u) in the niche separation model of Kakehashi, Suzuki & Iwasa (1984), showing the parasitism occurring in each subpopulation of hosts.

The mean encounter rate ɛXY is structurally identical to that of equation 4 but the egg complements of the two parasitoids are distributed not only across all host species, but also across available hosts in all regions of niche overlap. This requires the modification of equation 5, the functions governing the number of accessible hosts, which after expansion for clarity becomes:

image(eqn 8a)
image(eqn 8b)
image(eqn 8c)

As in equation 5, equation 8 denotes the number of hosts that could potentially be encountered by a parasitoid. The primary parasitoid, P, can attack immature hosts in s and t (equation 8a). The interactive parasitoid Q can encounter a novel set of hosts in region u and hosts escaping parasitism by P in region t (equation 8b) or hosts previously attacked by P in t (equation 8c). Again, as t → 1, equation 8 becomes identical to equation 5 and equation 6 reduces to equation 1.

It is worth noting that before the introduction of Q at generation 401, fQPfQN = 1, which eliminates all interactions in regions t and u, with the exception of (1 − fPN)fQP in t, which becomes indistinguishable from the corresponding interaction (1 − fPN) in region s. This reduces the interactions to a single equation, (s + t)(1 −fPN), which is the total rate of attack by the primary parasitoid in both regions to which it has access. Thus, before generation 401 subpopulations Nu and Nv are in absolute refuge. Upon the introduction of Q, Nu is susceptible to parasitism by Q and Nv is the only subpopulation of hosts in refuge.

Niche separation results

Simulations were performed as for the simple model and are presented in similar fashion with parameter space divided by the ratio inline image = 1. We examine four special cases of niche separation with regard to search by the two parasitoids. In the first case we consider a slight deviation from the simple model of equation 1, the inclusion of an absolute refuge (v > 0) so that a fraction of hosts is unavailable to parasitism, and the stabilizing aggregation of parasitoid encounters with hosts can be relaxed, thereby reducing the level of implicit niche separation. In the second case, we consider the effects of an interactive parasitoid (Q) that is able to break the host refuge and parasitize a host subpopulation that was previously in refuge from the resident parasitoid (u > 0). We show for this scenario how the size of the broken refuge (u) relative to the overall refuge (v) determines the level of host suppression. In the third case, we examine scenarios where there is an absolute refuge (v > 0) and varying degrees of parasitoid niche overlap (t > 0) when the interactive parasitoid does not break the host refuge (u = 0). This enables us to explore the effect of an interactive parasitoid that attacks only the same hosts that are available to the primary parasitoid but differs in search efficiency and antagonism. Finally, we make a direct comparison between explicit niche separation and aggregation of encounters. We show the effectiveness of each in promoting host suppression, stability and co-existence.

the influence of implicit niche separation (aggregated encounters) with complete overlap of accessible hosts and a host refuge

In order to demonstrate the importance of both the level of niche overlap and antagonistic interactions on the outcome of biological control, we introduce either a mild facultative hyperparasitoid (aQP < aQN), an ectoparasitoid (aQN = aQP) or a cleptoparasitoid (aQP > aQN) into a system with a primary parasitoid that has a low (aPN = 0·25), moderate (aPN = 0·50) or high (aPN = 1·00) search efficiency (Fig. 5). The normally destabilizing effect of more random search is offset by an explicit host refuge (v = 0·2), allowing the outcome of the interaction to be examined in relation to the level of aggregation of search by both resident and introduced parasitoids.

Figure 5.

The influence of implicit niche separation (aggregated encounters) with complete overlap of accessible hosts and a host refuge on the outcome of multiple parasitoid introductions for various levels of antagonism by the interactive parasitoid and search efficiency of the primary parasitoid. As in Fig. 3, shaded areas indicate an increase in pest densities upon multiple introduction. λ = 4, K = 10 000, βP = βQ = 35, kP = kQ = 0·5, c= 0·5, s = 0, t = 0·8, u = 0, v = 0·2.

The two most important influences on the outcome of multiple introduction are the ratio of the search efficiencies of primary to interactive parasitoids for unattacked hosts (aPN/aQN) and the level of aggregation of search by the primary parasitoid (kP). When the ratio of search efficiencies is less than unity (Fig. 5a,b,d), the greater search efficiency of the interactive parasitoid leads to greater host suppression even in the face of substantial antagonism, except for high levels of aggregation of search by the interactive parasitoid. As highly aggregated search limits the fraction of hosts that a parasitoid can encounter, it moderates the overall efficiency of a parasitoid in terms of the number of hosts attacked. This makes it more likely that the introduction of an interactive parasitoid with less aggregated search than the primary would be beneficial. This is indicated by the fact that the areas of greater host suppression (white; Fig. 5) occur to the right (less aggregated search by Q) and down (more aggregated search by P) within each part of the figure (Fig. 5). The exact reverse occurs when the ratio of primary search efficiencies for unattacked hosts is greater than unity (Fig. 5f,h,i), such that the interactive parasitoid is only beneficial when the overall efficiency of the primary parasitoid is compromised by highly aggregated search.

The role of antagonism in the outcome of biological control can be seen from the positive diagonals across the sets of figures (Fig. 5d,b, Fig. 5g,e, c and Fig. 5h,f), where the ratio (aPN/aQN) is constant. Along each of these diagonal sequences there is a striking similarity, indicating that the ratio of the primary search efficiencies is a more important factor than the level of antagonism by the introduced parasitoid in determining the outcome of biological control. The exception to the similarity along diagonals (with equal primary search ratios) is Fig. 5g, where it is always beneficial to introduce Q, as opposed to Fig. 5e,c, where it is only beneficial to introduce Q when the search by the interactive parasitoid is less aggregated than that of the primary. This is because the search efficiency for unattacked hosts by both parasitoids is sufficiently high to offset the inefficiency of more aggregated search by the interactive parasitoid.

an interactive parasitoid that is able to break the host refuge from a primary parasitoid

In this case, the primary parasitoid is unable to encounter a fraction of hosts that is available to parasitism by the interactive parasitoid (u = 0·5 − v). The equilibrium density of hosts before the introduction of the antagonistic parasitoid (inline image) remains unchanged at 303·4 (Fig. 6a) and 218·3 (Fig. 6b) because the size of the initial refuge is constant (u + v = 0·5).

Figure 6.

A bifurcation diagram of the equilibrium host densities after the introduction of (a) a facultative hyperparasitoid (aQN = 0·75, aQP= 0·25) that breaks the host refuge from a primary parasitoid with lower search efficiency (aPN= 0·25), and (b) a cleptoparasitoid (aQN= 0·25, aQP= 0·75) that breaks the host refuge from a primary parasitoid with high search efficiency (aPN= 0·75). At each value on the x-axis, 1000 generations are plotted along the y-axis. The shaded line shows the average host density for each point on the x-axis. Along the x-axes, the host refuge (v) varies from 0·5 to 0 while the proportion of hosts available to attack only by the interactive parasitoid (u) varies from 0 to 0·5. There is extinction resulting from a loss of stability in both cases at u ≈ 0·5. λ = 2, K = 10 000, βP = βQ = 35, kP = kQ = 2, c = 0·5, s = 0, t = 0·5, 0 ≤ u ≤ 0·5, 0 ≤ v ≤ 0·5, u + v = 0·5.

An intuitively plausible case for multiple introduction is one in which the search efficiency of a primary parasitoid is low (aPN = 0·25) and a mild facultative hyperparasitoid with greater primary search efficiency (aQN = 0·75, aQP = 0·25) is released (Fig. 6a). However, this plausible case provides a qualitatively similar outcome to that of an intuitively implausible case (Fig. 6b), a primary parasitoid with high search efficiency (aPN = 0·75) and an antagonistic cleptoparasitoid with lower primary search (aQN = 0·25, aQP = 0·75). This similarity demonstrates that the breaking of the initial host refuge, even to a small degree, both overrides any potentially antagonistic effects of an interactive parasitoid and leads to a substantial reduction in host density. Regardless of the efficiency of the primary parasitoid and the antagonism of the interactive parasitoid, multiple introduction results in a beneficial outcome as long as there is sufficient host refuge to stabilize the interaction (v ≈ 0·15 for the set of parameters used in the present example). Once the size of the refuge diminishes below a minimum threshold, the stable-point host equilibrium is lost (Fig. 6a,b) and increasing oscillations result in elevated mean host densities and eventual extinction of the primary parasitoid (u > 0·45).

an interactive parasitoid with a niche contained within that of a primary parasitoid

To determine the effect of an interactive parasitoid that can encounter only a portion of the same hosts that are accessible to a primary parasitoid, we consider the scenario of equal primary search efficiencies (aPN = aQN) for various levels of overlap and antagonism (Fig. 7). Before the release of the interactive parasitoid, subpopulations of hosts in s and t are equally susceptible to parasitism by the primary parasitoid, but following the release the two parasitoids compete in the area of niche overlap (t). In addition, the line aQP = 0·5 along the y-axis separates a mild facultative hyperparasitoid (below) from an increasingly antagonistic cleptoparasitoid (above).

Figure 7.

Outcome of multiple introduction when the interactive parasitoid exploits only subpopulations of hosts available to the primary parasitoid, and has variable preference for healthy and previously parasitized hosts. Along the x-axes the fraction of hosts available to both parasitoids (t) varies from 0 to 0·8, while the subpopulation available exclusively to the primary parasitoid (s) declines from 0·8 to 0 (see Fig. 3 for further explanation). λ = 3, K = 10 000, aPNaQN= 0·5, βP = βQ = 35, kQ = 5, c = 0·5, s = 0·8 − t, u = 0, v = 0·2.

Under these conditions the degree of aggregation of encounters by the primary parasitoid and the level of antagonism by the interactive parasitoid are key determinants of the success of multiple introduction (Fig. 7). When levels of niche overlap and antagonism are low, the interactive parasitoid cannot persist in the system. The region where the interactive parasitoid cannot persist is greater in Fig. 7b than in Fig. 7a because of the greater overall efficiency of a primary parasitoid with less aggregated search. Similarly, at high levels of overlap (t → 1 − v) and antagonism (aQP → 1) the primary parasitoid is excluded. This exclusion can result in reduced host densities when the interactive parasitoid is more efficient due to less aggregated search (Fig. 7a; kQ > kP), but can also result in a loss of control if the interactive parasitoid is less efficient (Fig. 7b; kQ < kP).

The relative efficiencies of the two parasitoids with respect to aggregation of search also affect the likelihood of increased host densities in the region of parameter space supporting their co-existence. Interestingly, when the interactive parasitoid is more efficient than the primary parasitoid (Fig. 7a), the grey area is bounded at aQP ≈ 0·5, the border between milder and stronger antagonism (Fig. 2). In contrast, when the primary parasitoid is more efficient, even mild antagonism results in an increase in host densities such that reduced host suppression occurs through most of the parameter space that supports parasitoid co-existence (Fig. 7b). The primary parasitoid is able to persist under these conditions because it attacks first, but the encounters of previously attacked hosts by the later-attacking interactive parasitoid reduce the abundance of the primary parasitoid and thus host densities increase.

a comparison of explicit and implicit niche separation

Here we delineate two scenarios with the niche separation model to explore the similarity of the effects of explicit niche separation and implicit separation arising from aggregation of parasitoid encounters. For both scenarios there is an absolute refuge (v = 0·2) and no antagonism (aQP= 0). In the first scenario (Fig. 8a), both parasitoids search randomly for hosts (kP =kQ= 10 000) and the outcome of multiple introduction is considered in relation to increasing levels of explicit niche overlap (t). The unique niche of each parasitoid varies according to s=u = (0·8 − t)/2 and declines as the level of overlap (t) increases, with the consequence that, when Q is excluded, u adds to the absolute refuge. The second scenario, that of implicit niche separation (Fig. 8b), has an explicit niche overlap fixed at t= 0·8 and niche separation is attained through increasing aggregation of parasitoid encounters (k → 0·001). Thus, in both scenarios, the parasitoid niches are most distinct at the left side and show greatest overlap at the right side of each figure.

Figure 8.

Direct comparison of explicit (a) and implicit (b) niche separation with no antagonism. This bifurcation diagram is created, as in Fig. 6, with 1000 generations plotted for each point along the x-axis. Explicit niche separation between parasitoids increases across the axis as s=u = (0·8 − t)/2 with kP = kQ = 10 000 (a). For the implicit case, niche separation increases non-linearly across the axis (b). Dotted lines mark the point of extinction of the interactive parasitoid (Q) and thus the loss of the three-species equilibrium. λ = 3, K = 10 000, aPNaQN= 0·25, aQP= 0, βP = βQ = 35, c = 0·5, v = 0·2.

Both scenarios show a similar range of three-species co-existence in relation to level of niche separation, although the scale along the axis for the explicit effect is linear while that for the implicit effect is logarithmic (Fig. 8a,b). At higher levels of niche overlap parasitoid Q is excluded, and the system returns to a two-species equilibrium (inline image) that also becomes unstable at the highest levels of overlap. The topology of the two scenarios is also similar, in that when a three-species equilibrium exists host densities are lower than for the corresponding two-species equilibrium. It is interesting to note that the minimum host densities achieved with a three-species equilibrium are also remarkably similar in both scenarios.

The implicit niche separation scenario allows two identical parasitoids to maintain a stable-point equilibrium even at very low levels of niche overlap (Fig. 8b; at kP = kQ→ 0), in contrast to the explicit niche separation scenario (Fig. 8a; at t→ 0). However, in the explicit scenario host densities are maintained at lower equilibrium densities across the range of niche overlap, permitting co-existence in comparison to the implicit niche overlap scenario. Intermediate levels of aggregation of search (kP = kQ→ 0·1) lead to less host suppression, with host densities minimized at both the highest levels of aggregation (kP = kQ→ 0·001) and at levels occurring just before exclusion of parasitoid Q (shown by the vertical dotted line) for the parameter set used).

Discussion

The aim of classical biological control is to suppress pest densities to economically viable levels via the action of introduced natural enemies. Although a single parasitoid can be sufficient to suppress an invasive pest, multiple introduction, either simultaneous or sequential, has been a frequent practice in biological control (Waage & Mills 1992). However, multiple introduction can result in complex interactions from various forms of antagonism that may or may not be beneficial to the aim of biological control, and for the practitioner the compatibility of such interactions has been unclear. For example, hyperparasitism, although traditionally thought to be detrimental (DeBach 1964; Luck, Messenger & Baberi 1981; Mills 1990), is known to promote stability in model systems (Luck, Messenger & Baberi 1981; Briggs 1993), as has also been shown to be the case for cleptoparasitism (Munster-Swendsen 1985). In addition, as secondary parasitism by facultative hyperparasitoids can involve a fitness cost relative to primary parasitism (Grandgirard et al. 2002), such parasitoids may typically prefer to attack primary hosts and could therefore be more innocuous or even beneficial than generally believed. The ecological significance of other forms of parasitoid antagonism is even less well understood, and thus biological control practitioners have been understandably cautious in considering interactive parasitoids for classical introductions. In view of this we have examined how various forms of antagonism can affect the outcome of multiple introductions and how the antagonisms are, in turn, affected by both explicit and implicit forms of niche separation among the parasitoids.

Previous theoretical studies have promoted either single or multiple parasitoid introduction depending on the assumptions of the model. In most cases, the influencing assumptions are those that generate some form of niche separation. The discrete-time Nicholson & Bailey (1935) model framework with random search contains no explicit niche separation between parasitoids competing for the same host species and, therefore, does not allow for the co-existence of parasitoids (the parasitoid with the greatest attack rate causes the extinction of the less efficient parasitoid). By incorporating an aggregated distribution of parasitoid encounters (implicit niche separation via independent distributions of host encounters) and host density dependence into the Nicholson–Bailey model, May & Hassell (1981) concluded that: (i) an introduced primary parasitoid is always beneficial given that its distribution of host encounters is completely independent from that of the resident parasitoid; and (ii) the introduction of an obligate hyperparasitoid always raises host densities above the equilibrium set by a single resident parasitoid. Hogarth & Diamond (1984) used a modified version of the May & Hassell (1981) model to show further that multiple introduction is a sound practice even in the face of parasitoid competition, as differential survival of the competing parasitoids could facilitate both co-existence and host suppression. Similarly, Hochberg (1996) used a modification of the May & Hassell (1981) model to show that a primary parasitoid with a high search efficiency and adequate fecundity could co-exist with a facultative hyperparasitoid. In general, equilibrium host densities were increased, but when search by the primary parasitoid was more aggregated, the effective refuge created for the pest made it more likely that the facultative hyperparasitoid would lower pest densities.

In the current study (simple model) we extended the scenario examined by Hochberg (1996) to include other forms of antagonism, intermediate to the two extremes of primary parasitism and obligate hyperparasitism, such as ectoparasitism and cleptoparasitism. We show that with implicit niche overlap imposed by aggregated search, the outcome of biological control depends on the search efficiency of the primary parasitoid and the level of antagonism by the interactive parasitoid (Fig. 3). We find that the primary parasitoid is more likely to co-exist with an interactive parasitoid, of any form, if the primary search efficiency of the interactive parasitoid is less than that of the primary parasitoid, and if the level of antagonism is not extreme. In this case, however, with equivalent levels of aggregated search for both parasitoids, the degree of suppression of the pest is primarily determined by the search efficiency of the primary parasitoid.

Kakehashi, Suzuki & Iwasa (1984) contrasted the findings of the May & Hassell (1981) negative binomial model with a negative polynomial model of parasitoid encounters. With complete niche overlap, as defined by the negative polynomial model, single introduction was always the best strategy. However, by using an absolute refuge to promote stability (explicit niche separation model), we show that multiple introduction can often be beneficial under conditions of complete niche overlap when the interactive parasitoid is more efficient than the primary parasitoid, due either to greater search efficiency or less aggregated search (Fig. 5). In fact, we find that the most important determinant for the outcome of biological control with complete niche overlap is the ratio of the primary search efficiency of the primary to that of the interactive parasitoid.

Kakehashi, Suzuki & Iwasa (1984) also developed a model of explicit niche separation, in which they showed that when the niche of each parasitoid was sufficiently distinct, multiple introduction was the preferable strategy. By extending this approach we have been able to examine the importance of refuge breaking (Hochberg & Hawkins 1994), niche sharing, antagonism and forms of niche separation on the outcome of biological control. We show that even a small incursion by an interactive parasitoid into an absolute refuge results in a profound suppression of pest density and overrides the detrimental effects of any antagonism that an interactive parasitoid brings into the system (Fig. 6). This clearly indicates that refuge breaking is a more important consideration than potential antagonism in the selection of parasitoids for multiple introduction. Antagonism does play an important role, however, when an interactive parasitoid is unable to access the subpopulation of pests in refuge from a primary parasitoid. When constrained to share the niche of a primary parasitoid, the effects of antagonism by an interactive parasitoid are determined by the relative efficiencies of the two parasitoids (Figs 5 and 7). When the primary parasitoid has greater search efficiency or has access to a greater fraction of hosts through less aggregated search, even mild forms of antagonism by an interactive parasitoid, such as mild facultative hyperparasitism and ectoparasitism, can frequently be detrimental to pest suppression. However, provided the interactive parasitoid is more efficient than the primary parasitoid in either primary search or access to pests, the outcome of multiple introduction can often be beneficial unless the interactive parasitoid shows a distinct preference for pests previously attacked by the primary parasitoid, as is the case for cleptoparasitism and strong facultative hyperparasitism. Finally, we note that explicit niche separation provides a more consistent suppression of pests by two identical parasitoids in the absence of antagonism, than does the implicit niche separation imposed by an assumption of a negative binomial distribution of parasitoid encounters (Fig. 8). Thus the paradox of biological control (Luck 1990; Murdoch 1990), in which stability is achieved at the expense of pest suppression, can be seen to be an artefact of the use of the negative binomial to provide the stabilizing effect of an implicit pest refuge from parasitism.

One of the greatest challenges in the practice of biological control is to develop a set of practical criteria for the selection of the most effective parasitoid species from the assemblage existing in the region of origin of an invasive pest. Within the framework of the interactive parasitoid model with explicit niche separation that we have presented in this study, there are two key factors that influence the outcome of multiple introduction in biological control and would be of value for practitioners to consider in the selection of parasitoids for use in introductions. The first of these is the overall efficiency of the parasitoids. This was determined in our model by the ability of the parasitoid to locate (primary search efficiency) and gain access to (aggregation of search) the pest. Although not limiting in the current study, fecundity, or maximum number of hosts that can be attacked in the case of gregarious parasitoids, can also affect the overall efficiency (Hochberg 1996; Lane, Mills & Getz 1999). We found that parasitoids with a lower overall efficiency are unlikely to provide additional control and more likely to be detrimental if they have antagonistic properties. However, the second crucial determinant of the outcome of multiple introduction, the extent and form of niche separation, can mediate the antagonistic effects arising from issues of overall efficiency. For instance, the breaking of a proportional refuge from parasitism by an introduced parasitoid is the most compelling reason for multiple introduction, even if it involves use of a hyperparasitoid or cleptoparasitoid, as refuge breaking overrides the effects of even strong antagonism. However, even if an additional parasitoid cannot break a pest refuge, it can still be beneficial to biological control provided that it has higher overall efficiency and does not have a preference for pest individuals previously parasitized by other parasitoids.

Acknowledgements

We thank Cherie Briggs, Michael Hochberg and Sebastian Schreiber for their comments on an earlier version of the manuscript. This work was supported by an EPA-STAR graduate student fellowship to Brent Pedersen.

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