variable parasitoid sex ratios and their effect on host-parasitoid dynamics

Variable Parasitoid Sex Ratios and their Effect on Host-Parasitoid DynamicsAuthor(s): M. P. Hassell, J. K. Waage and R. M. MayReviewed work(s):Source: Journal of Animal Ecology, Vol. 52, No. 3 (Oct., 1983), pp. 889-904Published by: British Ecological SocietyStable URL: http://www.jstor.org/stable/4462 .Accessed: 22/06/2012 16:05

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Journal of Animal Ecology (1983), 52, 889-904

VARIABLE PARASITOID SEX RATIOS AND THEIR EFFECT ON HOST-PARASITOID DYNAMICS

BY M. P. HASSELL, J. K. WAAGE AND R. M. MAY*

Department of Pure and Applied Biology, Imperial College, London S W7 2AZ and *Department of Biology, Princeton University, Princeton, New Jersey 08540, U.S.A.

SUMMARY

(1) Three forms of density-dependent parasitoid sex ratios are discussed: (a) where sex ratio is a function of the size of the adult female population (P), (b) where sex ratio depends on the ratio of females to hosts (P/N) and (c) the case of heteronomous hyperparasitoids, where sex ratio is related to the frequency of encounters with parasitized and unparasitized hosts (yielding male and female progeny, respectively).

(2) The dynamic properties of difference equation models including each of these mechanisms in turn are displayed. Sex ratios as a function of P or P/N are considered within the framework of conventional host-parasitoid models. Heteronomous hyperparasitoids, however, require a new model structure. In each case, models are explored in which encounters with hosts are either assumed to be random (Poisson) or contagiously distributed with the probability of parasitism given by the negative binomial distribution.

(3) With random host encounters, sex ratios as a function of P or P/N can be sufficient to be the sole cause of stability of the interaction, but only with 'fine tuning' of the relevant parameters. These critical parameters values, however, broaden considerably when some additional stability is added via contagion in the distribution of parasitism.

(4) In the special case of randomly acting heteronomous hyperparasitoids, the unusual result occurs of a neutrally stable interaction over a wide range of host rates of increase. The slightest additional stability added to the interaction (e.g. by some contagion in the host attacks) is sufficient to 'tip the balance' and convert the neutral stability to a locally stable equilibrium. We conclude that such interactions should be highly stable in the real world.

INTRODUCTION

Mathematical models of host-parasitoid population dynamics have recently become increasingly realistic by incorporating a variety of functional responses, mutual interference between parasitoids, non-random distributions of parasitism, and additional host and parasitoid mortalities (Hassell 1978). One aspect of parasitoid biology, however, which has been comparatively neglected is the sex ratio of parasitoids. Existing population models assume that parasitoids are either entirely female (thelytoky, a rare phenomenon) or that they exhibit a fixed sex ratio (e.g. Bellows 1979). However, for a very large group of parasitoids, the parasitic Hymenoptera, these assumptions may not be appropriate. Parasitic wasps are known to exhibit considerable variability in sex ratio, a property closely associated with their haplodiploid mode of reproduction in which unfertilized eggs become females. The act of fertilization is thought to be mediated by the ovipositing female in response to varying internal and external stimuli.

0021-8790/83/1000-0889 $02.00 ? 1983 British Ecological Society

'889

A largely anecdotal literature indicates that individual hymenopterous parasitoids will alter the sex ratio of their progeny in response to such factors as host species, host size, host density and parasitoid density (Flanders 1939; Viktorov 1976; Kochetova 1978; Waage 1982a). Only recently has an evolutionary basis been proposed for much of this behaviour. One of the most powerful sex ratio theories is that of 'local mate competition', originally proposed by Hamilton (1976) and later elaborated by a variety of authors (see Charnov (1982) for a review). This theory is appropriate to situations where female

parasitoids exploit discrete patches of hosts, and where mating of their offspring is largely restricted to the patch on which they emerge.

Local mate competition theory then predicts that the optimal sex ratio for a parasitoid wasp will depend upon the number of other female wasps colonizing the patch. When this number is large, such that mating occurs at random between the offspring of many females, a sex ratio near 0.5 (proportion males) is predicted, commensurate with Fisher's (1930) predicted optimum for panmictic populations. As the number of female colonists declines, the optimal sex ratio will decrease until, with only one female per patch, a highly female-biased sex ratio is predicted, where each female produces only sufficient sons to mate all her daughters. At the population level this implies a decrease in sex ratio as parasitoid density increases and/or as the tendency for individuals to aggregate in patches of high host density increases. In nature, conditions appropriate to local mate competition are frequently met in parasitoid populations, particularly in gregarious species which mate on the host at emergence.

Experimental support for the theory of local mate competition has come from studies on sex ratio variation, both within (Werren 1980) and between (Waage 1982a) parasitoid species. Density-dependent shifts in sex ratio have also been experimentally demonstrated in a number of laboratory parasitoid populations. The behavioural mechanisms which generate this variation may involve the perception of chemical traces left by other females on the patch of hosts (Viktorov & Kochetova 1973) or physical contact with other foraging parasitoids (Wylie 1976, 1979). A less well documented but perhaps more ubiquitous cause of variation is the tendency for some parasitoid species to lay male eggs first during an oviposition bout (Waage 1982b). When this occurs, the sex ratio of the

progeny produced from a patch of hosts is likely to increase as more females search, since each will tend to lay fewer eggs, thereby producing a higher proportion of males in their progeny. There is also evidence, particularly from some gregarious parasitoid species that male larvae are competitively superior to females (Chacko 1969). Changes in sex ratio at high parasitoid densities may thus arise without any alteration in parasitoid behaviour, but simply due to increasing levels of larval competition. This would be difficult to distinguish in experimental systems from the changes due to differential sex allocation.

The dynamic implications of such density-dependent sex ratios were first noted by Viktorov (1968), who believed that they evolved as a means of self-regulation in parasitoid populations to prevent the overexploitation of hosts. This hypothesis is less probable than that of local mate competition since it invokes a questionable degree of group selection; but it does at least focus on density-dependent changes in parasitoid sex ratios as a factor contributing to the stability of host-parasitoid interactions. Just how important these changes are to population dynamics has been briefly considered by Waage & Hassell (1982) and is the primary concern of this paper. Equilibrium levels and stability conditions will be displayed for host-parasitoid models in which density-dependent sex ratios are described in three ways. First, we will consider sex ratio shifts dependent entirely on the density of adult females P, as shown in Fig. 1 where host density N is held constant.

890 Parasitoid sex ratios

M. P. HASSELL, J. K. WAAGE AND R. M. MAY

10(a) (b)

0.8 -

0-6 -

t 04 - 0'-4

a)

E 0-2 8 0 1 2 3 4 5 6 0 2 4 6 8 10 12 o

(o (d) 0

o

0'6 a 0-8 -< - \

06 - -

0-4 -

0-2 0 5 10 15 20 25 30 0 10 20 30 40 50

Adult female parasitoids

FIG. 1. Four laboratory examples showing the proportion of female progeny as a decreasing function of the density of adult female parasitoids. In each case, the curve represents eqn (7) fitted by a standard non-linear least squares technique. (a) Dahlbominus fuliginosus (Nees), parasitizing cocooned larvae of Neodiprion lecontei (Fitch) (data from Wilkes 1963); c = 0.89, , = 27-0. (b) Trichogramma evanescens Westwood, parasitizing eggs of Mamestra brassicae L. (Waage & Lane 1984); a= 0.87, P = 6.0. (c) Nasonia vitripennis (Walk.) parasitizing pupae of Musca domestica L. (data from Wylie 1965); a= 0-81, / = 48-0. (d) N. vitripennis parasitizing

pupae of Phaenicia sericata (Meigen) (data from Walker 1967;) a= 0 94, / = 19.0.

Second, since both P and N will vary from generation to generation, we adopt the ratio P/N as a more plausible determinant of sex ratio changes. This, however, is unlikely to be the last word. Close examination of some published studies (Walker 1967; Waage & Lane 1984) reveal that sex ratios are not simply a function of P or P/N alone, but some more complex function of both P and P/N (sex ratio increases with increasing P, even for constant P/N). Finally, we turn to an unusual mechanism of sex allocation found in certain aphelinid wasps which exhibit divergent male ontogenies (Walter 1983) and provide yet a further example of density-dependent sex ratios. In these heteronomous hyperparasitoids females develop as primary parasitoids, while males develop as hyperparasitoids on hosts already containing female larvae. This results in density-dependence since increasing parasitoid densities lead to higher levels of superparasitism and hence greater male production.

MODELS INCLUDING PARASITOID SEX RATIO

We commence with the familiar difference equation model for a host-parasitoid interaction discussed at length by Hassell (1978):

N,+ = ANtf (N, Pt) (la)

Pt+, = cN[ 1 -f(Nt, Pt)].

891

(lb)

Here N and P are the hosts and female parasitoids, respectively, in successive generations t and t + 1; A is the host's net rate of increase;f is a function defining the probability of a host evading parasitism; and c is the average number of adult parasitoids emerging from each host parasitized. This model will be used as a vehicle for examining the dynamic effects of parasitoid sex ratios as a function of either Pt or the ratio Pt/Nt. The example of a divergent male ontogeny, however, will require a rather different model structure.

Two forms for the function f will be adopted. In the first place, we assume random encounters with hosts, an unlimited functional response and no mutual interference, to give the well-known expression of Nicholson (1933) and Nicholson & Bailey (1935) namely:

f= exp (-aPt). (2)

Here the constant a is the per capita searching efficiency of the parasitoids. With constant i and c, eqns (la, b) now inevitably produce expanding oscillations if either population is disturbed from its unstable equilibrium position. While the use of eqn (2) is convenient in highlighting the contribution of density-dependent sex ratios to population stability, it is now clear that a contagious distribution will often provide a better basis for the functionf, particularly if some form of heterogeneity is included in the system (Griffiths & Holling 1969; May 1978; Hassell 1980; M. P. Hassell & R. M. Anderson, unpublished). Appendix I is thus devoted to the case where the probability of parasitism is given by the negative binomial distribution (May 1978); in this case:

aP, \ f= + t) (3)

where k is the parameter of the negative binomial governing the degree of contagion in the distribution (increasing contagion as k decreases). The model (eqn (la, b)) is now stable for all k < 1. The use of such simple expressions with uncomplicated dynamics has the advantage that the qualitiative effects of parasitoid sex ratios in the models are more easily displayed.

Sex ratio as afunction of Pt Let s(Pt) be a function defining the proportion of females (i.e. sex ratio = - s) in the

parasitoid population in generation t. Takingfas in eqn (2), the full model now becomes:

Nt+ = ANt exp (-aP,) (4a)

Pt+ = s(Pt)cNt[1 - exp (-aPt)]. (4b)

The equilibrium populations are simply found by setting Nt+ = Nt = N* and P,+ = Pt =

P*, to give

aP* =InA (5a)

A In A 1 acN* = . (5b)

-1 s(aP*)

Not surprisingly, the equilibria are raised as the male bias in the equilibrium parasitoid population increases (see Fig. 7).

The neighbourhood stability properties of eqns (4a, b) are obtained in the usual way


M. P. HASSELL, J. K. WAAGE AND R. M. MAY 893 4

(a) (b)

3

8 2

012 3410 12 34 5 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10

Host rate of increase (X)

FIG. 2. (a) Generalized local stability diagram in terms of the parameter 0 from eqn (6) and the host rate of increase, L, for the host-parasitoid interaction in eqns (4a, b) where sex ratio is a function of the density of adult female parasitoids, P,. The populations are locally stable within the hatched area whose boundaries are defined in eqn (6), exhibit expanding oscillations in region A, and increase geometrically in region B. Superimposed on the diagram are contours for different values of the parameter combination af/ from eqn (8), where af/ = 0.01, 0.5, 1, 2, respectively, from upper to lower contour. (b) As in (a), but now for the case where the probability of parasitism is given by the negative binomial distribution with k = 1.5 (i.e. eqns (A-1, A-2) with f defined in eqn (3)). The boundary conditions are given in eqn (A-5) and the

a/I-contours are obtained from eqn (A-6).

(e.g. Hassell & May 1973; May 1978). The model proves stable if, and only if, the following criterion is satisfied:

/A+1\ 2 + In A (n-1 A 1

> >> --n (6) 2 In A ~- I In A where

\s d(aP) P=P*

These boundaries are displayed in Fig. 2(a), the hatched area denoting the parameter combinations giving a locally stable equilibrium. Outside this, in region A, the density-dependence is so weak that the interaction is unstable with expanding oscillations, while in Region B the density-dependent constraints on the parasitoid population are such that both populations increase geometrically.

As a specific example of s, we have arbitrarily chosen the expression:

a/I s(P,) = + , (7) fi + Pt

which has been fitted to the four examples in Fig. 1 using a standard non-linear least squares technique. The terms a and /f are both constants whose effects are made clear by the examples in Fig. 3(a, b). Thus, (1 - a) represents the sex ratio when Pt - 0, as shown by the intercepts in Fig. 3(a). In the real world, a will tend to be a species-specific characteristic corresponding roughly to the proportion of females in the progeny of a wasp, achieved when free of any crowding effects. Figure 3(b) shows the outcome of varying /?, which governs the rate at which the sex ratio increases with increasing wasp density. Thus 1// is the initial slope of the curves as they fall away from the intercept at a.

Parasitoid sex ratios

08

E 0-6

o 0

Oo* a-

a-

0 10 20 30 40 50 0 Parasitoid density

10 20 30 40 50

FIG. 3. Numerical examples showing the decline in the proportion of female progeny (s) with increasing parasitoid density predicted from eqn (7). (a) / = 30 and a varying as shown. (b) a=

0.95 and /varying as shown.

Evaluating 0 in eqn (6) using this specific examples of s now gives:

1 0=n + a

In A + a/i (8)

Contours for this expression are superimposed on Fig. 2(a) for various values of the product a/ (a dimensionless term comparable to aP* in eqn (5a)). The interaction is locally stable where the contours lie within the shaded area, and unstable with expanding oscillations where they fall below it. (Values of 0 lying above the shaded area are not possible with s(P,) defined as in eqn (7).)

Figure 2(b) shows the corresponding situation withfdefined by the negative binomial expression in eqn (3) with k = 1 5. The effect of introducing such contagion in parasitoid attacks on hosts is primarily to lower the stability boundaries, which now envelop all the a/ contours drawn. An alternative way of looking at this is shown in Fig. 4. The broken

k = -5

COL 0 0 C) 1-

0 2 3 4 5 6 7


10

FIG. 4. Critical values of aft below which the interaction described for Fig. 2 is locally stable, calculated from the right-hand inequality of eqn (A-5) for different host rates of increase, A. The solid curves are for different values of k as shown and the broken line for the Poisson case

(k -, oo).

894


line corresponds to Fig. 2(a) and shows the critical value of af, below which a Nicholson-Bailey interaction is stable. Thus, stability is impossible for all a/ > 2 and for all AL > 4.92. The solid lines in Fig. 4 show how much greater the critical a/i-values can become when some additional stability is added via contagion in the host attacks.

Stability in these models is thus promoted by strongly density-dependent sex ratios (small /), any factors promoting contagion in the distribution of attacks on hosts (small k) and small net rates of increase of the host population (A). Note that the interaction is always stable if k < 1, with or without any sex ratio effects. Larger values of k may be sufficient for stability, but only if combined with sufficient density-dependence in the parasitoid sex ratios. (Note that ain eqn (7) is only a scaling constant affecting equilibrium size and has no effect on stability.)

Sex ratio as afunction of (Pt/Nt) We now assume that the proportion of female parasitoids in eqn (4a, b) is a function of

the ratio (Pr/Nt) instead of depending on Pt alone. The equilibrium populations remain as given in eqns (5a, b), but with s(P*/N*) replacing the previous term s(aP*) in eqn (5b). The conditions for local stability are, however, different and given by:

( -A_ I A- 1 9- 1 lnJ

, /1 -1

1

2-?nA 1In(9) where

e c A a ad(P*/cN*)) The locally stable area is shown in Fig. 5(a) (hatched) and provides an interesting

contrast with those in Fig. 2(a). The conditions for stability are now more restrictive, with no chance of a stable equilibrium for /A > 2.37 whatever form the density-dependent sex ratios take. The contours drawn on these figures are for the specific analogue of eqn (7), namely:

s(P t/N) = (10) e + (Pt/cNt) giving

A ) (E+ (P*/cN*)] 2

The parameter e replaces fl of eqn (7) from which it differs in being dimensionless. Using this particular example for s, the contours must lie below y = 1, which further constrains the conditions under which a Nicholson-Bailey model can be stabilized.

Figure 5(b) shows the extent to which the stability conditions are broadened when some contagion in the distribution of parasitism (k = 1-5) is included. Finally, Fig. 6 is to be compared with Fig. 4, and shows the critical e-values for different A below which the interaction is stable for the Poisson case (broken line) and for various degrees of non-random parasitism (solid lines).

There are thus two significant differences from the results of the previous section. First, the conditions for stability are appreciably narrower than when s was just a function of the density of adult female parasitoids. Second, stability now depends not only on the parameter e but also upon a (the maximum proportion of females as Pt - 0) with increasing a enhancing stability.

895


5-(z) | / (b)

4-

3-

i 2 3 4 5 1 2 3 4 5


FIG. 5. (a) Generalized local stability diagram in terms of the parameter y from eqn (9) and the

host rate of increase, l, for the interaction in eqns (4a, b) where sex ratio is now a function of the

ratio (Pt/Nt). As in Fig. 2, the populations are stable within the hatched area whose boundaries are here defined in eqn (9). Superimposed on the diagram are contours for different values of e

from eqn (10), where e = 0.01, 0-1, 0-5, 1, respectively, from the upper to lower contour, and

a = 0.8. Values of y greater than unity are not possible from eqn (10). (b) As in (a), but now

adopting the negative binomial expression in eqn (3) with k = 1.5 (cf. Fig. 2(b)). The boundary conditions are given in eqn (A-8) and the e-contours are again obtained from eqn (10) evaluated

at equilibrium.

\k=\5

4-

3

0 I 2 3

Host rate of increase (X) FIG. 6. Critical value of e below which the interaction described for Fig. 5 is locally stable, calculated from the right-hand inequality of eqn (A-5) for different host rates of increase, A. The solid curves are for different values of k as shown and the broken line for the Poisson case

(k - oo) (cf. Fig. 4).

Sex ratios in heteronomous hyperparasitoids In this example of a divergent male ontogeny, females are assumed to emerge from hosts

parasitized only once and males from hosts parasitized more often. Assuming again a Poisson distribution of encounters with hosts, the fraction of female progeny is given by the ratio of the number of hosts encountered only once (q1) to the total fraction encountered (1 - q):

s(Pt)= = . (11) 1 - q, exp (aPt) - 1

The appropriate population model now differs from eqns (la, b) in that the 'parasitoid equation' is based on the first term of the Poisson distribution instead of the one-minus-zero term as before, giving

.. . I


I -

CL 4- , ___ -o

I/ ............

2 4 6 8 10 Host rote of increase (X)

FIG. 7. Equilibrium host and adult parasitoid populations (scaled as acN* and aP*, respectively) in relation to the host rate of increase, A. (. . ?) Parasitoid equilibria from eqn (5a) or (13a). Note that this curve applies to all three models discussed in the text where k - oo. (---) Host equilibria from eqn (5b) for different values of s(aP*) or s(P*/N*) as shown. (?)

Host equilibria from the host-heteronomous hyperparasitoid interaction in eqns (13a, b).

Nt+ = ANt exp (-aPt) (12a)

Pt+ = cNt[aPt exp (-aPt)]. (12b)

From this the equilibrium populations are:

aP* = InA (13a)

acN*= l. (13b)

Just as in the Nicholson-Bailey model, they depend only on the balance of the host's net rate of increase, A, and the parasitoids' searching efficiency, a, but now the host

equilibrium, N*, is much more sensitive to changes in i, as shown in Fig. 7. The stability properties of the model, however, are quite different. Defining

1 ds

\s d(aP) P=p*

as before, we have the same stability conditions (6) as when sex ratio was just a function of female parasitoid density. However, evaluating 0 from eqns (12a, b) now gives

1 0= (14)

A-1 ln ;

which is the same as the right hand condition of eqn (6). We have, therefore, the unusual result, best known from the Lotka-Volterra predator-prey equations, where the interaction is neutrally stable with the amplitude of oscillations dependent on the initial

population sizes as shown in Fig. 8. This is true for all A < e4, when the left hand condition of eqn (6) is no longer satisfied.

The interesting implication of this is that any other factor that adds the slightest stability is sufficient to 'tip the balance' and make the interaction stable. Hence, with a negative binomial distribution of parasitoid attacks (see Appendix II) where:

/( a-k ( -k-5a) ql=aPt 1 + (15a) ( k

897


-t 25- o

I

_ 20 -

O 15-

10-

5-

10 20 30 40 50 Generations

FIG. 8. Two numerical examples of the host-heteronomous hyperparasitoid model in eqns (12a, b), showing the neutrally stable cycles dependent on the initial host (N,) and parasitoid (PI) densities. In both cases a = 0.01, l = 2 and c = 1. ( ) N, = 30, P = 10. (---) N = 20, P1 =8.

1-0=l- 1 + (15b)

'the interaction becomes locally stable for all k < oo provided that

i k-2

Such a divergent male ontogeny thus provides one of the most striking mechanisms for the stability of a host-parasitoid interaction.

DISCUSSION

While experimental evidence of density-dependent sex ratios in parasitoids is as yet limited, the theory of local mate competition predicts that this phenomenon will be widespread in parasitoids which exploit and mate on patchily distributed hosts. Thus, the models in this paper reveal what may prove to be a widespread factor contributing to the stability of host-parasitoid interactions.

The variety of possible mechanisms for density-dependent sex ratios make the modelling of this process difficult. We have chosen two extreme possibilities to explore: (i) where sex ratio is a function of the size of the adult female population, P, and (ii) where it depends on the ratio of females to hosts, P/N. If parasitoids respond to their own population density away from host resources (e.g. at emergence or mating), then P is an appropriate variable. On the other hand, P/N is more realistic if parasitoid density is perceived during the oviposition, since the frequency with which a parasitoid encounters conspecifics, or their traces, will depend upon the number of parasitoids and the number of hosts which they are exploiting. Shifts in sex ratio which result from the selective mortality of female larvae at high levels of superparasitism will also be best described by P/N. (Superparasitism leading to non-selective larval mortality is outside the scope of this paper, but provides yet another density-dependent mechanism acting on the parasitoid population, and may prove to be a further contributor to stability in interactions where parasitoids are egg-limited.) Doubtless, the different factors influencing sex ratio will vary between parasitoid species,

898


but the bulk of available experimental evidence suggests that sex ratio is some combined function of both P and P/N (Walker 1967; Waage & Lane 1984).

The dynamic effects of density-dependent sex ratios are easily summarized in general terms. First, the host equilibrium increases with increasing sex ratio (1 - s). Second, any density-dependent increase in sex ratio will contribute to the stability of the interaction. The extent of this, however, differs according to the mechanism involved. In general, sex ratio shifts dependent on P/N will be less stabilizing than those dependent on P alone. Thus, while both can be sufficient to stabilize a Nicholson-Bailey model, this is impossible for all A > 2.37 when using s(P/N), but only becomes impossible for A > 54.6 with s(P). Even when feasible, however, the stability requires fine tuning of the appropriate parameters governing the form of the density-dependent relationship. These parameters differ in the two cases considered. With s(P), stability hinges on P (or the dimensionless equivalent, aB), which is a measure of the rate at which the proportion of females, s, falls from the maximum value, a, achieved as P - 0 (see Fig. 3(b)). Decreasing f enhances this density-dependent decline and thus adds to stability. With s(P/N), however, stability hinges on both E (the equivalent of a/), and on a, high values of which enhance stability by widening the range of s involved in the density-dependent relationship.

In the real world, one is most unlikely to find interactions that are as unstable as the Nicholson-Bailey model in the absence of density-dependent sex ratios. Various forms of heterogeneity, for example, are often likely to be important contributors to population stability. Density-dependent sex ratios as a function of P or P/N will act in concert with these, adding to the overall stability of the interaction. We have illustrated this in a simple way by showing how much greater becomes the range of sex ratio parameters causing stability when some degree of contagion in parasitoid attacks is introduced (e.g. k = 1 5 in eqn (3), where k < 1 is required for stability if s is constant). In short, these sex ratio shifts can only be the sole cause of stability of a Nicholson-Bailey model if the parameters are carefully chosen within a narrow range of possible values. They thus do not have the same pervasive stabilizing effect as some other documented factors such as heterogeneity and mutual interference.

The eqns (7) and (10) for the proportion of females in the progeny, although arbitrarily chosen, have parameters, a and p or e, that are biologically meaningful. Unfortunately, the available literature is insufficient to determine the most plausible values for these and how they might vary between different kinds of parasitoids. Local mate competition theory constrains sex ratios to a maximum of 0.5, although selective mortality of female larvae may push this even higher (as is probably the case for the Nasonia examples in Fig. 1). Thus, very low values of f or e, which generate rapid shifts to very high sex ratios, are probably unrealistic. The theoretical constraints on a are perhaps clearer. This parameter relates closely to the proportion of female progeny produced by wasps in the absence of competition from conspecifics, i.e. when local mate competition is complete. Parasitoid species which exhibit such high levels of local mate competition, where mating on the patch is often between the progeny of a single female, will generally produce very low sex ratios in the absence of conspecifics (Waage 1982a), and therefore exhibit high values of a. Models with very low values of fl, e or a are therefore unlikely to reflect patterns in the real world.

A completely different mechanism of sex allocation is found in heteronomous hyperparasitoids, which provide a fascinating case of density-dependent sex ratios. In contrast to parasitoids that regulate their sex ratio in response to P or P/N, these heteronomous hyperparasitoids appear to exert very little control over their sex ratio. We have taken the simplest case where the sex ratio is determined by the frequency of

899

900 Parasitoid sex ratios 300- ( 10 (b)

240 - 8 - hosts

180- 6 - 0

60 -

60- 2 A ~~~~~~~60 ~ ~ ~~- 2 4 1 ~~~Het hypers Hyperparasitoids

20 40 20 40 60 80 100 Generations

FIG. 9. (a) Numerical example from eqns (A-13), (A-14) and (A-15) showing an unstable host-parasitoid-hyperparasitoid interaction obtained with a = 0.67, a' = 1, k = 3, k' = 0.7 and A = 2. (b) As in (a), but for the equivalent host-heteronomous hyperparasitoid-hyperparasitoid interaction obtained by substituting eqn (A16) for (A14) and keeping all parameters values the

same.

encounters with parasitized and unparasitized hosts (yielding male and female progeny, respectively). In nature, however, it is likely that ovipositing females will show somewhat greater flexibility, possibly limiting the production of at least some females. Heteronomous hyperparasitoids are found mainly in the aphelinid subfamily, Coccophaginae, which have proved important agents in the biological control of various scale insects and mealybugs (Huffaker & Messenger 1976). The coccophagines have contributed to some of the most stable, and hence most successful, host-parasitoid interactions in biological control, which may bear some relation to the striking effect on stability of this method of sex allocation. The resulting density-dependent sex ratio is sufficient to render the simplest model (eqns (12a, b)) neutrally stable for a wide range of host rates of increase (G < e4) (see Fig. 8). Consequently, it only requires the smallest additional density-dependent effect for these neutral cycles to give way to a stable equilibrium.

This marked stability has an effect on the dynamics of more complex species systems. Thus, an interaction with a host, heteronomous hyperparasitoid and normal hyperparasitoid is significantly more stable than the equivalent host-parasitoid-hyperparasitoid interaction discussed by May & Hassell (1981). This is illustrated by a numerical example in Fig. 9. These results confirm some general speculation that parasitoids exhibiting deviant male ontogenies may stabilize both their own populations and the communities to which they belong (Williams 1977; Viggiani 1981). Interestingly, the natural communities of scale insect and mealybug parasitoids to which heteronomous hyperparasitoids belong are often relatively large (e.g. Kfir & Rosen 1980). These include both primary and hyperparasitoids, and some species with divergent male ontogenies, including heteronomous hyperparasitoids. It is possible that the dynamic stability conferred by these heteronomous hyperparasitoids may be an important factor in the persistence of the communities to which they belong.

ACKNOWLEDGMENTS

This work was supported in part (RMM) by the NSF, under grant DEB81-02783, and in part (JKW) by a grant from the Agricultural Research Council.

M. P. HASSELL, J. K. WAAGE AND R. M. MAY 901

REFERENCES Bellows, T. S. (1979). The modelling of competition and parasitism in laboratory insect populations. Ph.D.

thesis, London. Chacko, M. (1969). The phenomenon of superparasitism in Trichogramma evanescens and T. minutum Riley

(Hymenoptera, Trichogrammatidae). I. Beitrage zur Entomologie, 19, 618-635. Charnov, E. L. (1982). The Theory of Sex Allocation. Princeton University Press, Princeton, New Jersey. Fisher, R. A. (1930). The Genetical Theory of Natural Selection. Oxford University Press, Oxford. Flanders, S. E. (1939). Environmental control of sex in hymenopterous insects. Annals of the Entomological

Society of America, 32, 11-26. Griffiths, K. J. & Holling, C. S. (1969). A competition submodel for parasites and predators. Canadian

Entomologist, 101, 785-818. Hamilton, W. D. (1967). Extraordinary sex ratios. Science, 156,477-488. Hassell, M. P. (1978). The Dynamics of Arthropod Predator-Prey Systems. Princeton University Press,

Princeton. Hassell, M. P. (1980). Foraging strategies, population models and biological control: a case study. Journal of

Animal Ecology, 49,603-628. Hassell, M. P. & May, R. M. (1973). Stability in insect host-parasite models. Journal of Animal Ecology, 42,

695-726. Huffaker, C. B. & Messenger, P. S. (Eds.) (1976). Theory and Practice of Biological Control. Academic Press,

New York. Kflr, R. & Rosen, D. (1980). Parasites of soft scales (Hemiptera: Coccidae) in Israel: an annotated list. Journal

of the Entomological Society of Southern Africa, 43, 113-128. Kochetova, N. I. (1978). Factors determining the sex ratio in some entomophagous Hymenoptera.

Entomological Reviews, 57, 1-4. May, R. M. (1978). Host-parasitoid systems in patch environments: a phenomenological model. Journal of

Animal Ecology, 47, 833-844. May, R. M. & Hassell, M. P. (1981). The dynamics of multiparasitoid-host interactions. American Naturalist,

117,234-261. Nicholson, A. J. (1933). The balance of animal populations. Journal of Animal Ecology, 2, 132-178. Nicholson, A. J. & Bailey, V. A. (1935). The balance of animal populations. Part I. Proceedings of the

Zoological Society of London, 1935, 551-98. Viktorov, G. A. (1968). Influence of density on the sex ratio in Trissolcus grandis Thoms (Hymenoptera,

Scelionidae). Zoologischeskii Zhurnal, 4, 1035-1039. Viktorov, G. A. (1976). Ekologiya Parazitov-Entomofagov. Izdatelstbvo Nauka, Moscow. Viktorov, G. A. & Kochetova, N. I. (1973). The role of trace pheromones in regulating sex ratio in Trissolcus

grandis (Hymenoptera, Scelionidae). Zhurnal Obshchei Biologii, 34, 559-562. Viggiani, G. (1981). Hyperparasitism and sex differentiation on the Aphelinidae. The Role of Hyperparasitism

in Biological Control: A Symposium, pp. 19-26. Agricultural Science Publications, University of California, Berkeley.

Waage, J. K. (1982a). Sib-mating and sex ratio strategies in scelionid wasps. Ecological Entomology, 7, 103-112.

Waage, J. K. (1982b). Sex ratio and population dynamics of natural enemies-some possible interactions. Annals of Applied Biology, 101, 159-164.

Waage, J. K. & Hassell, M. P. (1982). Parasitoids as biological control agents: a fundamental approach. Parasitology, 84, 241-268.

Waage, J. K. & Lane, J. A. (1984). The reproductive strategy of a parasitic wasp. II. Sex allocation and local mate competition in Trichogramma evanescens. Journal of Animal Ecology, 53 (in press).

Walker, I. (1967). Effect of population density on the viability and fecundity of Nasonia vitripennis Walker (Hymenoptera, Pteromalidae). Ecology, 48,294-301.

Walter, G. H. (1983). 'Divergent male ontogenies' in Aphelinidae (Hymenoptera: Chalcidoidea): a simplified classification and a suggested evolutionary sequence. Biological Journal of the Linnean Society, (in press).

Werren, J. H. (1980). Sex ratio adaptations to local mate competition in a parasitic wasp. Science, 208, 1157-1159.

Wilkes, A. (1963). Environmental causes of variation in the sex ratio of an arrhenotokous insect, Dahlbominus fuliginosus (Nees) (Hymenoptera, Eulophidae). Candadian Entomologist, 95, 181-202.

Williams, J. R. (1977). Some features of sex-linked hyperparasitism in Aphelinidae (Hymenoptera). Entomophaga, 22, 345-350.

Wylie, H. G. (1965). Some factors that reduce the reproductive rate of Nasonia vitripennis (Walk.) at high adult population densities. Canadian Entomologist, 97,970-977.

Wylie, H. G. (1976). Interference among females of Nasonia vitripennis (Hymenoptera: Pteromalidae) and its effect on the sex ratio of their progeny. Canadian Entomologist, 108, 655-661.

Wylie, H. G. (1979). Sex ratio variability of Muscidifurax zaraptor (Hymenoptera, Pteromalidae). Canadian Entomologist, 111, 105-107.

(Received 25 October 1982)


APPENDIX I

Stability properties of host-parasitoid models with sex ratio effects This appendix outlines the analysis of the dynamic properties of the host-parasitoid

models defined in the text, for the general case of the negative binomial functional response,f(Pt), defined by eqn (3). What follows is, indeed, the veriest outline, because the basic ideas and techniques have been presented in full in similar contexts on several previous occasions (May 1974; Hassell & May 1973, 1974; May 1978).

First, we consider the case where the sex ratio depends only on Pt:

N + = Ntf (Pt), (A-1)

Pt+ = cNs(Pt)[1 -f(Pt)]. (A-2)

This generalizes eqns (4a) and (4b). The equilibrium parasitoid population, P*, immediately follows from ,Af= 1, or, using

eqn (3) forf,

Y* - aP* = k[2klk -1]. (A-3)

The equilibrium host population, N*, is then given by

X* - caN* =- (A-4) I-i s(Y*)

For a linearized or local stability analysis, we follow the standard routine. Put acNt = X* + xt and aPt = Y* + yt, rewrite eqns (A-1) and (A-2) in terms of xt and Yt (discarding terms of second or higher order to get a pair of linear equations), and then put xt, Yt - (a)t to express the asymptotic dynamics in terms of the eigenvalues a. As the system is a second order one, these eigenvalues will obey a quadratic equation of the form a2 + aa + b = 0, and the eigenvalues will have modulus less than unity (corresponding to a locally stable equilibrium) if and only if the Schur-Cohn condition 2 > 1 + b > lal is satisfied. Turning the crank on the standard machinery, we arrive at the stability criterion

1 ;L-l/k( + 1) l1-l/k 1 + > > . (A-5)

Y* 2(- 1) (At- 1) Y*

Here 0 is as defined following eqn (6). The right-hand inequality comes from the '1 > b' criterion, and growing oscillations (usually leading to limit cycle behaviour) will ensue if the inequality is violated. The left-hand inequality comes from the '1 + b > la I' criterion, and exponential runaway usually follows if it is not satisfied.

Results are quoted in the main text for the Nicholson-Bailey limit, in which parasitoids search independently randomly (corresponding to a Poisson functional response, obtained from eqn (3) as the limit k - oo). For this limit of k - oo, eqn (A-3) gives Y* -+ In L (eqn (5a) of the main test), and thence the general eqn (A-5) reduces to eqn (6).

For the special sex-ratio function defined by eqn (7) in the main text, we have in general that

0= .(A-6) ap + Y*

Here Y* is given by eqn (A-3), and eqn (A-6) reduces to eqn (8) in the Poisson limit k -+ oo. We see that the left-hand inequality in eqn (A-5) is automatically satisfied by this

902


particular expression for 0. The right-hand inequality is illustrated for various values of k in Fig. 4. It can be seen that this right-hand inequality is always satisfied for k < 1; that it can be satisfied for all A for k < 2; and that it requires a/ < 2k/(k - 1) for k > 2. These and other biological implications of the mathematical results are discussed in the main text.

Second, we summarize the analogous calculations for the case where the sex ratio depends not on Pt, but rather on the ratio Pt/Nt. The host-parasitoid dynamics are again described by eqns (A-1) and (A-2), except that now the sex-ratio function s is a function of P/N; for notational simplicity we write s = s(zt), with z, Pt/(cNt).

The equilibrium expression for Y* = aP* is still given by eqn (A-3). The equilibrium host population is then calculated from the relation

X*s(z*) = AY*/( - 1), (A-7)

with X* - acN* and z* = Y*/X*. Small perturbations about this equilibrium may be analysed in the standard way, as

outlined above. The equilibrium is stable against small disturbance provided

2 + y*-rl/k(G + 1)/(A - 1) y*-l/k/( _ 1)- 1

2- y*.-/1k 1 - y*-/k

Here the definition of y follows eqn (9) in the main text, and Y* is given by eqn (A-3). As for eqn (A-5), the left-hand inequality comes from the Schur-Cohn criterion '1 + b > a I', and the right-hand inequality from '1 > b'. The Poisson limit, k - oo (whence Y* -* In A), leads immediately to the relatively simple result of eqn (9).

The particular functional form of eqn (10) for s(z) leads, via eqn (A-8) to the stability boundaries illustrated in Figs 5 and 6. Specifically, it can be shown that the parameter E in eqn (10) (against which the ratio P/cN is to be compared) must satisfy e < 2ak/(k - 1) for stability to be possible with k > 1; this upper limit is illustrated (as A -A 1) in Fig. 6. For k < 1, of course, the system is always stable, no matter how small the sex-ratio effects (May 1978).

The above analysis is purely a linearized one. Numerical studies, and extensive previous experience with other broadly similar host-parasite models (Hassell 1978), suggest that local and global stability criteria are coincident.

APPENDIX II

Stability properties of host-heteronomous hyperparasitoid interactions

Throughout this paper, we have made the phenomenological assumption that the distribution of parasitoid attacks can be described by a negative binomial distribution, with mean aPt and clumping parameter k (Hassell 1978; May 1978). The probability that a given host will escape attack, q0, is thus given by the zero term of this distribution (eqn (3)), and the probability for a host to be found exactly once, q1, is correspondingly given by eqn (15a).

Under the assumptions adumbrated in the main text, female heteronomous hyperparasitoids produce female offspring only from those hosts attacked exactly once. It follows that, for such creatures,

Nt+, = ANtqo(P), (A-9)

Pt+ = cNtql (Pt)

903

(A-10)

Here q0 =f is given by eqn (3) and q, by eqn (15a). Obviously we could analyse the dynamic properties of these equations directly. The presentation can, however, be abbreviated by noting that eqn (A- 10) constitutes a special case of eqn (A-2), with the sex ratio s(Pt) defined as

p q aP(1 + aP/k)1 -k s(P) = . (A-ll) 1I- qo 1-(1 + aP/k)-k

In this special case, the stability-determining quantity 0 (defined following eqn (6)) is

2l-i/k A-l/k

= .- ? . -(A-12) (L- 1) Y*

Here Y* = aP* is again as given by eqn (A-3). In the Poisson or Nicholson-Bailey limit, k -- oo, this reduces to eqn (14).

Substituting eqn (A-12) into the stability criterion (A-5), we see that the right-hand inequality is always satisfied for finite k. In the limit k -+ co, the inequality becomes an equality, corresponding to neutral stability (as discussed in the main text). The left-hand inequality is always satisfied if k < 2; for k > 2, the inequality reduces to the requirement that L < [(k + 2)/(k - 2)1k. The implications of these results are discussed in the text.

Figure 9 presents an example illustrating how an unstable host-parasitoid-hyperparasitoid system can be stabilized by the primary parasitoid being an heteronomous hyperparasitoid. This result extends our earlier analysis (May & Hassell 1981) of host-parasitoid-hyperparasitoid systems with no sex-ratio effects, where the dynamics of host (NA), parasitoid (P,) and hyperparasitoid (Qt) populations obey

Nt+ = N(1 + aPt/k)k, (A-13)

Pt 1 = Nt[1 - (1 + aPt/k)-k](1 + a'Qt/k')-k, (A-14)

Qt+1 =Nt[ -(1 + aPt/k)-k][l- ( + a'Qtk')-k']. (A-15)

For heteronomous hyperparasitoids, eqn (A-14) is replaced by the appropriately modified version of eqn (A-10):

P,+ = NtaPt(1 + aPt/k)--k(l + aQt/k)-k'. (A-16)

The other two equations remain unaltered (assuming hyperparasitoids attack both female and male parasitoids); here we have, for simplicity, put c = 1. A full account of the linear stability properties of this system, contrasting the results with those for the simpler host-parasitoid-hyperparasitoid system discussed by May & Hassell (1981), is available on request. The analysis is cumbersome and technical, and Fig. 9 seems adequate for the broadly qualitative points we seek to make.

REFERENCES

Hassell, M. P. (1978). The Dynamics of Arthropod Predator-Prey Systems. Princeton University Press, Princeton.

Hassell, M. P. & May, R. M. (1973). Stability in insect host-parasite models. Journal of Animal Ecology, 42, 695-726.

Hassell, M. P. & May, R. M. (1974). Aggregation in predators and insect parasites and its effect on stability. Journal of Animal Ecology, 43, 567-594.

May, R. M. (1974). Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton. May, R. M. (1978). Host-parasitoid systems in patch environments: a phenomenological model. Journal of

Animal Ecology, 47, 833-844. May, R. M. & Hassell, M. P. (1981). The dynamics of multiparasitoid-host interactions. American Naturalist,

117,234-261.

Article Contentsp.889p.890p.891p.892p.893p.894p.895p.896p.897p.898p.899p.900p.901p.902p.903p.904Issue Table of ContentsJournal of Animal Ecology, Vol. 52, No. 3 (Oct., 1983), pp. 663-1008Movement Patterns in Sphaeridium: Differences between Species, Sexes, and Feeding and Breeding Individuals [pp.663-680]Breeding and Natal Dispersal of the Goldeneye, Bucephala clangula [pp.681-695]Maximizing Energy Delivery to Dependent Young: A Field Experiment with Red-Backed Shrikes (Lanius collurio) [pp.697-704]Demography and Dynamics of a Stoat Mustela erminea Population in a Diverse Community of Vertebrates [pp.705-726]Factors Affecting the Breeding Success of the Mute Swan Cygnus olor [pp.727-741]Demographic Responses of a Chipmunk (Eutamias townsendii) Population with Supplemental Food [pp.743-755]Regulation of Breeding Density in Microtus pennsylvanicus [pp.757-780]The Activity of Free-Ranging Wood Mice Apodemus sylvaticus [pp.781-794]An Experimental Test of Competition for Space between Blackcaps Sylvia atricapilla and Garden Warblers Sylvia borin in the Breeding Season [pp.795-805]Spacing and Breeding Density of Willow Ptarmigan in Response to an Experimental Alteration of Sex Ratio [pp.807-820]Aggregation and the `Ideal Free' Distribution [pp.821-828]Honey Bee Foraging Ecology: Optimal Diet, Minimal Uncertainty or Individual Constancy? [pp.829-836]Predation, Cover, and Food Manipulations During a Spring Decline of Microtus townsendii [pp.837-848]Density-Dependent Prey Selection in the Water Stick Insect, Ranatra linearis (Heteroptera) [pp.849-866]Patterns of Parasitism by Torymus capite on Hosts Distributed in Small Patches [pp.867-877]Age-Specific Survival, Fecundity and Fertility of the Adult Blowfly, Lucilia cuprina, in Relation to Crowding, Protein Food and Population Cycles [pp.879-887]Variable Parasitoid Sex Ratios and their Effect on Host-Parasitoid Dynamics [pp.889-904]Time and Energy Limits to Brood Size in House Martins (Delichon urbica) [pp.905-925]Territory Size and Population Density in Relation to Food Supply in the Nuthatch Sitta europaea (Aves) [pp.927-935]Patch Time Allocation and Parasitization Efficiency of Asobara tabida, a Larval Parasitoid of Drosophila [pp.937-952]Goshawk Predation on Tetraonids: Availability of Prey and Diet of the Predator in the Breeding Season [pp.953-968]Fertility and Body Weight in Female Red Deer: A Density-Dependent Relationship [pp.969-980]Population Dynamics of the Common Toad (Bufo bufo) at a Lake in Mid-Wales [pp.981-988]Changes in the Population of Pike (Esox lucius) in Windermere from 1944 to 1981 [pp.989-999]Reviewsuntitled [p.1001]untitled [p.1001]untitled [p.1002]untitled [pp.1002-1003]untitled [pp.1003-1004]untitled [p.1004]untitled [pp.1004-1006]untitled [p.1006]Short Notices [pp.1007-1008]Erratum [p.1008]Back Matter

variable parasitoid sex ratios and their effect on host-parasitoid dynamics

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