Weather-mediated regulation of olive scale by two parasitoids

Authors

  • Jacques Rochat,

    1. INRA, Unité Santé des Plantes et Environnement, 37 bd. du Cap, 06600 Antibes, France; and
    2. Ecosystem Science, 151 Hilgard Hall, University of California, Berkeley, CA 94720, USA
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  • Andrew Paul Gutierrez

    Corresponding author
    1. Ecosystem Science, 151 Hilgard Hall, University of California, Berkeley, CA 94720, USA
      Dr A.P. Gutierrez, University of California, Ecosystem Science, 151 Hilgards Hall, Berkeley, CA 94720, USA. Fax: 1 (510) 643–5098.Carpediem@Nature.Berkeley.Edu
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Dr A.P. Gutierrez, University of California, Ecosystem Science, 151 Hilgards Hall, Berkeley, CA 94720, USA. Fax: 1 (510) 643–5098.Carpediem@Nature.Berkeley.Edu

Summary

  • 1 The effects of temperature and interspecific competition among two parasitoids (Aphytis maculicornis (=paramaculicornis) (Masi)–Coccophagoides utilis Doutt) of the olive scale (Parlatoria oleae (Colvée)) were examined, and the reason for the successful biological control of olive scale explained.
  • 2 An age-structure distributed maturation time model of this system was developed that simplifies many of the details of prior physiologically based models.
  • 3 Temperature-dependent physiological indices were used to scale fecundity and survivorship rates from their maximum values.
  • 4 The distributed maturation time population dynamics model captured the variance in temperature-related development times required to simulate the dynamics of the system.
  • 5 A type III ratio-dependent functional response was used to estimate parasitism rate on olive scale and also served to stabilize our single patch model.
  • 6 The model confirms the Huffaker & Kennett (1966) conclusions concerning the role of weather and the relative contribution of the two parasitoids in the regulation of olive scale in California olive.

Introduction

The successful regulation of a species by natural enemies depends on many biotic and abiotic factors and their interactions (Huffaker, Messenger & DeBach 1971). The need to evaluate such complex systems is increasing as new invasive species are introduced, and concerns grow about the effects of global warming on the composition and dynamics of extant food webs (e.g. Kareiva, Kingsolver & Huey 1993). However, data sufficient to evaluate such problems are often sketchy because the data were gathered with other objectives in mind and gathering the requisite data sets is often prohibitively expensive (Gilbert et al. 1976). In addition, the question of how to study complex consumer-resource dynamics remains contentious with several competing paradigms available (see Hawkins & Cornell 1999). An approach that has been gaining attention is the use of physiologically based models that include resource availability as well as the effects of weather (Gutierrez & Wang 1977; Gutierrez & Baumgärtner 1984; Gurney et al. 1996). Physiologically based supply–demand models have been applied successfully to several field tritrophic systems (see Gutierrez 1996). In these studies, the processes of resource acquisition and allocation (Petrusewicz & MacFayden 1970) were modelled, and weather was used to drive the model that simulated the field data. This experience provided sufficient insights for determining the simplification that could be made in such studies and provides guidelines as to the amount of information required to analyse biologically complex trophic interaction.

Here we use physiologically based indices outlined by Gutierrez et al. (1994) to model the population dynamics of the economically important olive scale (Parlatoria oleae (Colvée)) and two of its introduced parasitoids (Aphytis maculicornis (=paramaculicornis) (Masi) and Coccophagoides utilis Doutt in central California. The indices characterize the effects of temperature on developmental rates, fecundity and survivorship on the field dynamics of the three species. Published bionomics and field data as well as observations and intuition of Huffaker & Kennett (1966) were used to formulate the model.

The biology of the olive scale system

Olive scale is a typical armoured scale (Diaspididae) producing both male and female progeny (Pinhassi, Nestel & Rosen 1996). Approximately two-thirds of the 100 progeny (Garcia 1971) produced are males, but the sex ratio may vary among feeding sites within olive (Arambourg 1986). The short-lived males develop more rapidly than females (37 days vs. 46 days, Broodryke & Doutt 1966). Three species of aphelinid parasitoids were introduced to combat olive scale (OS) in California, but only A. maculicornis (A) and C. utilis (C) established (Huffaker & Kennett 1966) and are currently providing good control. The basic interactions of the species are depicted in the lower right hand inset in Fig. 1, while the complex stage specific relationships are illustrated in the larger figure.

Figure 1.

The stage-specific interactions of olive scale (OS), Aphytis maculicornis (A) and Coccophagoides utilis (C) in California olive.

The biology of C. utilis is based on Broodryke & Doutt (1966) and Kennett, Huffaker & Finney (1966). Mated C. utilis females attack endoparasitically olive scale male and female larvae and only unmated females are produced. Second stage scales (both sexes) are preferred and adult scale are rarely attacked. Male progeny are produced by arrhenotoky when unmated females ectoparasitize sibling female pupae (adelpho-hyperparasitism). Colonies of C. utilis may be initiated using only mated females because the developmental times of immature females are variable (38–50 days) and adult longevity is relatively long (10 days), assuring temporal overlap of males and unmated females. Males require 32–35 days to complete development and also live 10 days. Auto- and hetero-hyperparasitism in aphelinids was examined by Mills & Gutierrez (1996).

The parasitoid A. maculicornis is a thelytokous ectoparasitic species (all progeny are females) that attacks parasitized and unparasitized olive scale (Hafez & Doutt 1954; Kennett et al. 1966). The parasitoid prefers large hosts but does not attack adult scale. Little information is available on its developmental times and reproductive rates, hence the rates for the related well-studied thelytokous Aphytis chilensis Howard on oleander scale (Aspidiotus nerii Bouché) were used (Pizzamiglio 1985). In cases of multiple parasitism with C. utilis, the ectoparasitic A. maculicornis is always the victor (Broodryke & Doutt 1966).

The phenological data on the three species in central California reported by Huffaker & Kennett (1966) provided critical information on their thermal limits. Average temperatures in the Fresno area vary throughout the year from 10 to 35 °C (Anonymous 1983). All species are active all year long but at greatly reduced levels during winter. Aphytis is active primarily in spring and fall and Coccophagoides is most active during the hot summer. These observations and the details of the biology described above form the basis for the development of our population dynamics model. We first present a general model and then show how the biology of the three species is incorporated.

The dynamics model

Six linked age-structured functional populations are used to model the system. The functional populations are: female olive scale (population index n = 1), males olive scale (n = 2), A. maculicornis (n = 3), males of C. utilis (n = 4), virgin females of C. utilis (n = 5) and mated females of C. utilis (n = 6). Each population has different biological characteristics, average developmental times and variance, and may respond differently to temperature (Table 1). The temperature-related developmental biology is captured using a distributed maturation time model (equation 1, see Manetsch 1976; Vansickle 1977). The time-varying dynamics of each population (Nn(t)) is described by a separate set of ordinary differential equations (equation 1) that include the time-varying effects of temperature and biotic interactions. Plant & Wilson (1986) showed the relationship of the distributed maturation time model to the deterministic age-structured McKendrick (1926)von Förster (1959) model, and DiCola, Gilioli & Baumgärtner (1999) review field and theoretical applications of the approach. The relationship of this model to the delayed differential equation model used by Gurney, Nisbet & Lawton (1983) is shown in Appendix 1.

Table 1.  Parameter values for the olive scale system*
 FunctionParameterUnitOlive scaleA. maculicornis MaleFemaleC. utilis Mated f.
FemaleMale
  • *

    Initial conditions: 10 individuals per age-class, except for mated females of C. utilis (n = 6): (0,0,0,0,0,0,10,10) (special cases: 100 individuals instead of 10 in the three populations of C. utilis (n = 4, 5, 6)).

  • a

    a In parentheses, duration of immature stages.

Population index n     1  2   3     4  5  6 
Life span  Δ(Topt)days  76 (46)a40 (37)a 28 (21)a   31 (24)a31 (24)a31 (24·5)a 
Number of age-classes k    8  8   8     8  8  8 
Developmental rateR(T) =1/Δ(t)a    0·860  0·852   0·845     0·860  0·860  0·860 
  b    0·731  0·719   0·709     0·400  0·400  0·400 
  cday−1    0·014  0·027   0·037     0·029  0·029  0·029 
  Tm °C  25·525·5 23·4   303030 
Survivorship to temperature φ (υ)(T) θ min °C   −5 −5  −5    −5 −5 −5 
   θ opt °C  2424 22   272727 
   θ max °C  4545 40   454545 
  b  1212 12   121212 
Temperature scaling on functional response φ (f)(T) θ min °C    5  5   5   101010 
   θ opt °C  2424 22   282828 
   θ max °C  3737 32   393939 
  b    1·5  1·5   1     1  1  1 
Functional responsef(N, H) α (0)   –    0·25     0·9  0·25  0·25 
  aday−1   –    0·5   10  0·5  0·5 
Age-specific fecundity β (x)amindays  46 21   242424 
  aoptdays  49 23·1   25·124·724·7 
  amaxdays  76 28   313131 
  b100   0·5 100  2  2 
   β maxday−1    3 10     61010 
Total fecundity β (x)dx   84 28 2323 
      on (1)on (2)on (5)on (5)on (5)on (1)on (2)
Preference ξ(x)amindays   – 1212  0  24·5141212
   α optdays   – 42·625·5  0–14  24·5–3114–2418.815
   α maxdays   – 462714  31244627
  b   –  1  1 + ∞ + ∞ + ∞  5  5
   ξ max   –  1  0·45  0·25    1  1  1  0.45

Assume a functional population n (e.g. males, mated and virgin females) composed of kn age classes Nn ,i(t), i = 1, 2, … , kn. Ignoring the index n for the functional population, the population dynamics of the ith age class may be modelled as:

image( eqn 1)

Ageing at time t occurs via flow rates inline image

from age class Ni−1 to Ni. Births (r0 = X(t)) from all reproductive females to this population enter the first age class (see below), and the eldest exit at maximum age

via the last age class inline image. The variable

Δ is the mean developmental life span of individuals in the population, hence dividing Δ by k gives the age width (Δ/k) of each age class. The observed variance of developmental times (σ2)may be estimated from data as k = Δ2/σ2 and incorporated into the model by an appropriate value of k (Manetsch 1976; Severini et al. 1990). Theoretically, the distribution of developmental times of cohorts follows a gamma distribution. Large variance in developmental times requires a small value for k and a small variance require large value usually < 50. A value of k = 8 was selected for all populations to capture the variance of observed developmental times for all populations (Table 1).

Developmental time of poikilotherms in days varies with temperature T, hence Δ(T(t)) = Δ(t). Nutrition and other environmental variables also affects developmental times (Nisbet & Gurney 1983; Stone & Gutierrez 1986), but here only the effects of temperature were considered. At any time t, the total density of individuals in the population across all age classes equals.

image( eqn 2)

The time-varying age-specific mortality rate (µi(t)) is the sum of all death rates due to all causes including net migration (see below) and may take values −∞ < µi (t) < +∞. The transfer of individuals from one functional population to another (e.g. unmated to mated females) is included for convenience as a separate term (ɛi(t) ≥ 0).

The total number of equations in our system is

inline image where kn is the number of age classes

for the nth population (i.e. six functional populations each with eight age classes). The system of ns equations was solved using the NDSolve function in Mathematica® (Wolfram 1999). Equilibrium of the system was assumed when the rate of change in the density of female olive scale eggs plus crawler (n = 1, i = 1) over three years met the criterion inline image. Convergence to this criterion occurred rapidly. Of course, the model requires initial conditions to ensure the uniqueness of the solution.

Incorporating the biology into the model

Temperature-dependent development

Data on the developmental rate of olive scale are from Pinhassi et al. (1996), but only partial developmental rate data are available for A. maculicornis (Hafez & Doutt 1954) and C. utilis (Broodryke & Doutt 1966). The three species are active all year round, but each species has a very different thermal response to temperature requiring different time scales in the model. None of the species has a period of dormancy and all survive the short periods of non-freezing temperatures common during winter and the high temperatures of midsummer.

The lowest average monthly temperature for growth and reproduction of the olive tree is 10 °C (Marchyllie 1981). Young stages of olive scale suffer mortality during periods of high summer temperatures. A. maculicornis is active mostly during autumn and spring, while C. utilis is active mainly during the hot summer. The lower thermal limits for all species is assumed near 0 °C, but the phenology of the species in the field suggest that the temperatures for their maximal development have the following order: A. maculicornis < olive scale < C. utilis. Above this peak, developmental rates slow and approach zero at high temperatures.

The developmental rates functions for the three species are depicted in Fig. 2a. There is ample evidence in the literature for this kind of relationship. Homopteran examples are blue alfalfa aphid (Acyrthosiphon kondoii Shinji) (Summers, Coviello & Gutierrez 1984), oleander scale (Aspidiotus nerii Bouchè) (Pizzamiglio 1985) and spotted alfalfa aphid Therioaphis maculata (Buckton) and its three parasitoids (Messenger 1964) (see Gutierrez 1996). The shapes of the developmental rate functions on temperature are right skewed convex (humped) having specific thermal limits. We used a modified form of Janisch’s (1925) catenary model (equation 3, Gutierrez 1996) to characterize the developmental rates R(T(t)) of the species as a function of temperature (T(t)) at time t (Fig. 2a).

Figure 2.

Temperature-dependent developmental rates (1/developmental time, (a)), fecundity (progeny per day, (b)) and survivorship (per day, (c)) of the species in the olive scale system (olive scale (OS) and its two parasitoids Aphytis maculicornis (A) and Coccophagoides utilis (C)), and the rates through time computed using a 10–35 °C variation in the annual temperature pattern (a′, b′, c′, respectively).

image( eqn 3)

Tm is roughly the inflexion point at high temperatures and a, b and c are constants. In the Janisch model, a = b but in our modification a > b captures the correct shape of the relationship. Other developmental rate functions could be used just as well.

The dynamics of all populations are driven by average daily temperature computed using a cosine function of time t.

image( eqn 4)

The parameter is the yearly average temperature, is the annual amplitude, Tmax and Tmin are the maximum and minimum temperatures, and d is the Julian date when T(t) is at maximum: T(d) = Tmax. The parameter d is fixed at day 198·25 so that the maximum temperature occurs in mid-July. The developmental rates of all species over the year for the temperature pattern Tmin = 10 °C and Tmax = 35 °C are shown in Fig. 2a′. Observed temperatures could also be used but the computation time is increased inordinately due to the integration method used. Other temperature regimes are also evaluated to show their effects on system dynamics.

Temperature effects on fecundity and survivorship

The physiological details of resource allocation to growth and reproduction were outlined by Petrusewicz & MacFayden (1970). More generally, the birth and death rates for each species are formulated as convex index functions of temperature and level of resource acquired (Gutierrez 1992). Data on the effects of temperature on resource assimilation to growth and reproduction, and survivorship are widely available, and yield convex relationships on temperature (see Gutierrez 1996). The convex shape of the assimilation function may be explained as the net of the acquisition (a type III function) and respiration rates (i.e. the Q10 rule, van’t Hoff (1884)) corrected for assimilation efficiency. The normalized assimilation rate has a maximum at the optimum temperature θopt. Survivorship is also affected by the assimilation rate, and this suggests that a common function with species specific parameters may be used to characterize temperature effects on both fecundity and survivorship (Fig. 2b,c) (Gutierrez 1992). We use (equation 5) to capture the shape of this relationship.

image( eqn 5)

Ignoring population indices, the general parameters are: A = θopt − θmin, B = θmax − θopt, θmin and θmax are the minimum and the maximum temperature thresholds, respectively, and b is a positive constant that determines the shape of the function (i.e. b→ 0: spike, b→ ∞: uniform). The function φψ(T(t)) is used as a scalar of maximum fecundity (ψ = f) and survivorship (ψ =T) (see below). The species-specific parameters are given in Table 1 and the functions are illustrated in Fig. 2b,c.

Fecundity

Detailed age-specific fecundity data are available for olive scale (Garcia 1971) and C. utilis (Broodryke & Doutt 1966), and several Aphytis species (Rosen & DeBach 1979; Viggiani 1984), but only rough estimates could be found for A. maculicornis (Hafez & Doutt 1954; Rosen & DeBach 1979). The patterns of maximum age (x)-specific fecundity β(x) occurring under optimal temperature for each population over its reproductive age window [amin, amax] are shown in Fig. 3.

Figure 3.

Age-specific fecundity patterns for the different species in the olive scale system (olive scale (OS) and its two parasitoids Aphytis maculicornis (A) and Coccophagoides utilis (C)). The dot indicates the average life span for each species.

image( eqn 6)

The parameter βmax is the fecundity at age aopt, A = aopt − amin, B = −amax −aopt, and b a positive constant (see Table 1).

Fecundity is also influenced by searching success of the parasitoids (see below). The resource acquisition rate of organisms (their functional response; Lawton, Hassell & Beddington 1975; Beddington, Hassell & Lawton 1975) depends on resource availability, consumer demand and search capacity as well as temperature. If consumer population (Np) attacks its resource populations (Hp, see its definition below), then its searching success is characterized by the Gutierrez-Baumgärtner model (Gutierrez 1996) (equation 7). This per capita model is a special case of Watt’s model (1959).

image( eqn 7)

The function Dp( ) = Dp(Np(t)) is the maximum demand for resources by population Np(t), and αp( ) = αp (Hp(t)) is the proportion of the resource population Hp(t) not in a refuge (i.e. availability). For the olive scale (p = 1,2), its resource species (the olive tree) is not modelled, hence its functional response reduces to fp(Np(t)) = Dp(Np(t)). The functions Dp( ), αp( ) and Hp(t) are defined below.

The total demand for resources by consumer p is

image( eqn 8)

where βp(x) is the maximum age-specific fecundity rate of individuals Np of age x (equation 6). The availability rate of the resource populations Hp(t) for the consumer p is:

image( eqn 9)

where α(0)p and ap are constants with ap chosen so that the model fp( ) (equation 7) is weakly type III. Note that fp( ) tends to a type II model as αp → +∞.

Some populations may be consumers under some circumstances and resource populations under others (e.g. parasitoids), hence we have a possible ns × ns interaction matrix of potential consumer and resource populations. Resource density across all resource populations for consumer p is:

image( eqn 10)

where 0 ≤ ξp,n(x) ≤ 1 is the preference function of the pth consumer population for the nth resource population of age x (see Fig. 4). ξp,n,i (discretized ξp,n(x) per age-class i) may be zero for some populations or age-classes, effectively removing them from the computation. In some cases, a resource population may include itself (p = n, e.g. auto-parasitism in virgin females of C. utilis). The functional form of ξp,n (x) is the same as the function β(x) (equation 6), with the maximum preference defined by the parameter ξmax (Table 1).

Figure 4.

Age-specific preference functions of consumer species for resource species (olive scale (OS) and its two parasitoids (Aphytis maculicornis (A) and Coccophagoides utilis (C)).

Mating in C. utilis is also viewed as a resource acquisition process (e.g. Gutierrez et al. 1994) and (equation 7) is also used to estimate it. The mating of unmated females may be viewed as a mortality rate µ(P)n = 5,i(t) (equation 11, see next section) that transfers individuals to the same age class (i) in the mated population via ɛn = 6,i(t) in equation 1:

image( eqn 11)

The birth rate of population Np(t) (Beddington et al. 1976) accrues from predation across all of its resources (Hp(t)) (Lawton et al. 1975) as (see equations 1, 5, 7):

image( eqn 12)

The function Xp(t) may be corrected for sex ratio as required and includes the effect of age structure of both consumer and resource populations (fp(Np(t),Hp(t))), as well as the effects of temperature on fecundity (equation 5, i.e. φ(f)p(T(t)), see Gutierrez et al. 1994).

The net mortality rate µ

Mortality may be due to predation (parasitism), temperature, or it may be due to shortfalls in resource acquisition rates.

Predation. Age-specific predation on a resource species (µ(P)n,i(t)) may accrue via the action of several consumers and hence is the sum of the mortality caused of all consumer species. Thus, mortality in the ith age class of the nth population is computed as:

image( eqn 13)

where ξp,n,i is the consumer p-specific preference for the resource population n and φ(f)p(T(t)) is the effects of temperature on the functional response fp( ) of the consumer.

Temperature-dependent mortality. The effects of temperature (T) on age-specific survivorship (φ(T)n(T(t))) are also of the general form (equation 5). The effects are assumed the same for all kn age-classes of the population. For simplicity, age specific mortality µ(T)n,i(t) is:

image( eqn 14)

This function sets the minimum and maximum temperature above or below which individuals of the population cannot survive.

Mortality due to resource shortfalls. Olive scale and immature parasitoid stages both feed on olive and hence they die at a rate proportional to resource (R) availability (Gutierrez 1992). Olive scale dynamics are assumed logistic because the dynamics of the olive tree were not modelled, and hence a carrying capacity of K = 106 is assumed. Both unparasitized and parasitized olive scales older than the crawler stage are included in computing the effective scale density. This assumes that the crawler stage has little impact on the collective resource and that the rate of resource utilization is the same across the other age-classes of parasitized and unparasitized scale. The mortality rate due to resource shortfalls (µ(R)n,i(t)) is computed as:

image( eqn 15)

where the parameter ωn,i defines whether population age-class Nn,i feeds on olive. In terms of the Lotka–Voterra competition model, the first ratio in (15) is the per capita reproductive rate of the nth population and the second ratio is the proportion of the carrying capacity consumed. A smaller carrying capacity could have been chosen without qualitatively affecting the results.

Total mortality. The total mortality is the sum of the rates due to all factors and is incorporated in equation 1 as:

image( eqn 16)

If a different integration scheme is used, another method must be used to incorporate the mortality in the model.

Simulation results

The range of average temperatures in the Fresno area of the great Central Valley of California varies roughly between 10 °C in mid-winter and 35 °C during mid-summer (Anonymous 1983). This is the standard temperature pattern used in our study (equation 4). The full range of temperatures at 5-degree increments for both minimum and maximum (0–45 °C) are used to examine the effects of temperature regimes on the dynamics of the olive scale–A. maculicornis–C. utilis system and to estimate the limits of each species.

Temperature-mediated interaction of olive scale– A. maculicornis–C. utilis system

Using the pattern of temperatures in the Fresno area, without parasitism, olive scale increases to the carrying capacity K = 106 under the standard regime (not shown), but when the parasitoids are included, the populations settle to a low equilibrium with bounded fluctuations (Fig. 5) (cf. Murdoch 1994). The 3-year equilibrium dynamics are those of the system after it has reached equilibrium from arbitrary initial conditions (Table 1). The type III properties of the functional response model produce the stability observed in the model. A type II model would lead to the extinction of the interacting populations.

Figure 5.

The bounded equilibrium patterns of total log densities of olive scale (OS) and its two parasitoids (Aphytis maculicornis (A) and Coccophagoides utilis (C)) using the Central California 10–35 °C variation in annual temperatures (OS,A,C). bsl00086 , Olive scale; bsl00088, Aphytis maculicornis;bsl00090, Coccophagoides utilis.

The model predicts that the parasitoid Aphytis has a sharp period of activity in spring, declines in summer and has a longer period during late summer and autumn. Coccophagoides is predicted to have strong activity during late summer. The phenology of the three species and the level of control of olive scale are similar to those observed in the field (Huffaker & Kennett 1966).

Olive is grown across a variety of temperate climates, hence a cline of five temperature patterns with fixed 15 °C annual amplitude was examined. The range of the patterns is from a cool 5–20 °C to a hot 25–40 °C (Fig. 6). This set of temperature patterns serves to illustrate the range of system dynamics, but another cline of temperature would yield a similar variation of dynamics. In cool regimes Coccophagoides populations go extinct and while Aphytis populations go extinct at high temperatures. At intermediate temperatures favourable for all three species, both parasitoids persist and regulate olive scale with minimal fluctuation.

Figure 6.

The bounded equilibrium patterns (total log densities) of olive scale and its two parasitoids using five annual temperature patterns with a 15 °C annual amplitude (OS, A, C). bsl00086, Olive scale; bsl00088, Aphytis maculicornis;bsl00090, Coccophagoides utilis.

The relative control of olive scale by the parasitoids, singly or in combination, is mapped across all temperature regimes (Fig. 7). The minimum temperatures are on the abscissa and the maximum temperature on the ordinate, with average temperatures falling along the diagonals. Below average temperatures of 10 °C, all species become extinct because it is below the threshold for olive scale reproduction. Similarly, all species become extinct above average temperatures of 35 °C and in temperature regimes that fluctuate above 45 °C. Within the favourable range, olive scale populations explode to the carrying capacity in the absence of the parasitoids (Fig. 7a,a′). When only the parasitoid A. maculicornis is in the system, regulation of olive scale is greatest at low temperatures (up to 20 °C average) and decreases with increasing temperatures until A. maculicornis becomes extinct above an average of 30 °C (Fig. 7b,b′). If only C. utilis is in the system, regulation of olive scale is best above 25 °C and decreases as temperature decline breaking-down completely at low temperatures where C. utilis goes extinct (Fig. 7c,c′). This case could not be observed in the field as A. maculicornis was introduced in California before C. utilis (Huffaker & Kennett 1966). If both parasitoids are introduced, excellent regulation of olive scale is predicted across the full range of temperatures. This occurs despite the fact that Aphytis continues to become extinct at high temperatures and C. utilis becomes extinct at low temperatures (Fig. 7d,d′). In the intermediate range of temperature, both parasitoids coexist and provide better control than either parasitoid alone. As an aside, increases in temperature fluctuations increased the variance of olive scale population fluctuations, but these results are not shown.

Figure 7.

Mapping of the response of different possible combinations of olive scale and its two parasitoids to temperatures: presence absence (a, b, c, d) and the maximum log density of total olive scale (a′, b′, c′, d′).

Some additional novel interactions merit mention. In temperature regimes 10–30, 15–20, 15–25 and 20–20 °C, establishment of Coccophagoides depends on its initial population levels. At 10–25 and 15–15 °C, the build-up of olive scale is so fast that mortality due to logistic competition between parasitized and unparasitized olive scale excludes C. utilis from the system. Aphytis is also displaced at 20–40 °C due to this same competition. Aside from these novel findings, the model verifies Huffaker & Kennett (1966) intuition that weather (e.g. temperature in this case) affects the interactions of olive scale and its two parasitoids and the level of regulation of olive scale itself.

Discussion

The generality that weather sets the limits for trophic interactions among poikilotherms and influences the level of control natural enemies exert is well accepted in biological control (Huffaker et al. 1971). Discussion of field examples may be found in van den Bosch, Messenger & Gutierrez (1981) and theoretical analyses are found in Schreiber & Gutierrez (1998). For example, there are elements of weather effects in the classic case of the biological control of the cottony cushion scale (Icerya perchasi Maskell) by the vedalia beetle (Rodolia cardinalis Muls.) and Crytochaetum iceryae (Will.) in California (Quezada & DeBach 1973). In the California red scale (Aonidella aurantii (Maskell)) system, the parasitoid Aphytis lingnanensis Compere is thought to have been competitively displaced by A. melinus DeBach (DeBach & Sundby 1963) because the latter has size-dependent advantage in sex allocation (Murdoch, Briggs & Nisbet 1996). No temperature effects were included in the Murdoch et al. model, yet it is known that in hot environments, or when plants are stressed, the proportion of small individuals may increase, aiding the displacement when size selection is a factor. Such an effect caused the displacement of the encyrtid cassava mealybug parasitoid Apanogyrus diversicornis Howard by A. lopezi DeSantis in the field and was explained by a physiologically based model (Gutierrez et al. 1988, 1993). The idea that abiotic factors limit the distribution and abundance of species is an old one and was first proposed by von Liebig (1840) and later Shelford (1931). Indices of favourability to temperature and other factors were used by Fritzpatrick & Nix (1970) to describe the limits of pasture plants in Australia and by Gutierrez et al. (1974) to summarize the phenology of cowpea aphid (Aphis craccivora Koch) in South-east Australia. Sutherst, Maywald & Bottomly (1991) incorporated some of these ideas into a GIS program widely used for assessing the bioclimatic limits of pests. The indices in these studies are similar to the temperature and resource-dependent physiological indices used here to factor fecundity and survivorship from the maximum in the olive scale–Aphytis maculicornis–Coccophagoides utilis system. The use of such indices in population dynamics models was proposed by Gutierrez et al. (1988, 1994), and successfully implemented here in simplified form. The parameters for our indices are rough estimates based on partial data and field observation, and yet proved sufficient to explain the dynamics of the system. More exact parameter values may be estimated from laboratory age-specific life-table experiments conducted at different temperatures (e.g. Messenger 1964), and the effects easily included in the model. We would not, however, expect gross changes in the observed dynamics of our olive scale system.

However, how much detail is required in a population dynamics model to capture the underlying causes for the observed dynamics? The trade-offs between increased biological realism and mathematical tractability in the analysis of stability and other properties of ecological model are well appreciated (see Wang & Gutierrez 1981; Godfray & Waage 1991). The recent population ecology literature shows increasing attention to physiologically based approaches (Gurney et al. 1996; Berryman & Gutierrez 1999; Hodkinson 1999), as well as aggregation, metapopulation dynamics (Gutierrez et al. 1999), host size selection and other factors (Briggs et al. 1993, 1995; Gutierrez et al. 1993). Increases in biological detail in models must occur if we are to answer more subtle questions about specific interactions in the field, and to make site-specific predictions. Briggs et al. 1993; Murdoch, Chesson & Chesson (1985) compared the olive scale system to the red scale system and argued that equilibrium in the olive scale system should be viewed in a metapopulation context (see Gutierrez et al. 1999 for counter arguments). This was not required here as the important aspects of the olive scale system proved to be the effects of temperature on development, survivorship and reproduction of the interacting species. The use of a distributed maturation time dynamics model allowed the observed variance of developmental time to arise naturally in the model assuring the requisite overlap of the different species and their substages, and the weakly type III modification of the Gutierrez–Baumgärtner functional response provided stability in the model. The use of physiological indices to capture abiotic effects rather than including the physiological details facilitated realistic programming of the complex biology and its evaluation using the software package Mathematica® (Wolfram 1999). We note that similar results would have accrued developing the model using standard computer languages.

Model predictions

The thermal limits of olive scale are wider than that of its two parasitoids. In the field, Aphytis is active during cooler periods (spring and autumn) and declines during the hot summer. In contrast, Coccophagoides is most active at hotter periods of mid- to late summer and declines during the winter. Similar thermal relationships were demonstrated between the spotted alfalfa aphid and its three parasitoids (Messenger 1964). Control of olive scale by either parasitoid alone under conditions observed in Central Valley of California were predicted to be poor, while control by both parasitoids was considerably better. These results are qualitatively similar to what was observed by Huffaker & Kennett (1966) and accrued despite the fact that the parameters of the model were rough approximations. The model allowed investigation of the effects of various temperature patterns on the control of olive scale and, not surprisingly, the simulation results suggest that control is best in the central portion of the thermal range of the three species and poorer at the extremes (Huffaker et al. 1971). Fully parameterized versions of the model would have application to the regional analyses of the dynamics of this system because site-specific weather (and edaphic factors) determines the presence and the extent of the biotic interactions that may occur. The model could be used to determine the possible impact of climate change on the degree of pest regulation (e.g. Hodkinson 1999; Gutierrez 1996, 2000) and food web composition (Kareiva et al. 1993; Schreiber & Gutierrez 1998) as well as the risk from new pest introductions.

Acknowledgements

We wish thank S.J. Schreiber for his useful comments on the appendix and N.J. Mills, H.C.J. Godfray and an anonymous referee for excellent comments on an earlier draft of the paper. This work was supported by the International Relations Directorate of the Institut National de la Recherche Agronomique (France) and the Division of Ecosystem Sciences at the University of California at Berkeley.

Received 17 April 2000; revision received 16 January 2001

Appendix

Appendix 1

Origin of the age-structured distributed-delay model and its link with delay-differential equations

Origin of the age-structured distributed-delay model

Let N(t,x)dx be the population density at chronological time t in the physiological age range x to x + dx. Let β(t,x) and µ(t,x) be the birth and death rates which are function of age x. In a small increment of time dt, the number of the population of age x that dies is µ(t,x) N(t,x)dt. The birth rate only contributes to N(t,0). The conservation law for the population says that

image( eqn A.1)

Dividing equation A.1 by dt gives

image( eqn A.2)

where dx/dt = v(t,x) is the developmental rate of individual organisms. If dx/dt = 1, equation A.2 simplifies to the McKendrick (1926)von Förster (1959) model.

The approximation of the partial derivative inline image

with the backward difference inline image is com

monly called up-wind scheme (McKendrick 1926). The upwind numerical scheme for equation A.2 is described by the system of ordinary differential equations:

image( eqn A.3)

It provides the ground for development of models that have been called boxcars (e.g. de Wit & Goudriaan 1978), sojourn time model (DiCola et al. 1999), and time-distributed models (e.g. Manetsch 1976).

The distributed-delay model used in this paper is characterized by the assumption:

image( eqn A.4)

The ageing rate v(t) of individual organisms is thus assumed to be independent of the age over their maturation time Δ(t). Then, equation A.3 (with boundary conditions) takes the form:

image( eqn A.5)

with inline image is the maturation flux

from the ith to the (i + 1)th age-class, inline image the recruitment into the system, and Y(t) =

rk(t) the maturation flux out of the system. In the absence of mortality, the distribution of maturation times has a gamma distribution.

Links with delay-differential equations

Let there be a Gamma distribution of parameters (k, inline image) i.e. the function f(t) defined for t ≥ 0 by:

image( eqn A.6)

with k and Δ being positive constants.

f(t) is maximum for t = tM = Δ(1 − 1/k), which tends to Δ as k → ∞, and

inline image

Since inline image when k is large,

inline image

Moreover, whatever ɛ > 0,

inline image

Since x ≥ log(1 + x), inline imagef(Δ + ɛ) = 0.

In the same way, whatever ɛ, 0 < ɛ < Δ, inline imagef(Δ − ɛ) = 0.

Hence, when k → ∞, inline image

Recall that inline image, therefore:

image( eqn A.7)

Taking the Laplace transform L{g(t)}(s) = L{g(t)} of equation A.5 gives

image( eqn A.8)

with the sum of the k equation A.8 being equal to:

image( eqn A.9)

Replacing recursively from L{N1(t)} in equation A.8 gives the expression of L{Y(t)} as a function of L{X(t)}:

image( eqn A.10)

Since inline image, using convergence

(equation A.7) when k → ∞, and using equation A.10 for L{Y(t)}, equation A.9 becom.

image( eqn A.11)

where inline image.

Then, taking the inverse Laplace transform of equation A.11, the term inline image determines the solution for 0 ≤ t < Δ

image( eqn A.12·1)

until the survivors of the initial population N(0) leave the population at maturity when t = Δ, and the convolution inline image gives the solution for t > Δ

image( eqn A.12·2)

This delay differential equation is equivalent to the Gurney et al. (1983) formulation of lumped age-classes models, for one age-class:

image( eqn A.13)

where R(t) is the recruitment into the population N at time t, M(t) the maturation flux out of that population, δ the mortality rate, and τthe maturation time.

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