## 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.

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 (*N _{n}*(

*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.

Function | Parameter | Unit | Olive scale | A. maculicornis | Male | Female | C. utilis Mated f. | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Female | Male | |||||||||||

- *
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(T_{opt}) | days | 76 (46)^{a} | 40 (37)^{a} | 28 (21)^{a} | 31 (24)^{a} | 31 (24)^{a} | 31 (24·5)^{a} | ||||

Number of age-classes | k | – | 8 | 8 | 8 | 8 | 8 | 8 | ||||

Developmental rate | R(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 | |||||

c | day^{−1} | 0·014 | 0·027 | 0·037 | 0·029 | 0·029 | 0·029 | |||||

T_{m} | °C | 25·5 | 25·5 | 23·4 | 30 | 30 | 30 | |||||

Survivorship to temperature | φ _{(υ)}(T) | θ _{min} | °C | −5 | −5 | −5 | −5 | −5 | −5 | |||

θ _{opt} | °C | 24 | 24 | 22 | 27 | 27 | 27 | |||||

θ _{max} | °C | 45 | 45 | 40 | 45 | 45 | 45 | |||||

b | – | 12 | 12 | 12 | 12 | 12 | 12 | |||||

Temperature scaling on functional response | φ _{(f)}(T) | θ _{min} | °C | 5 | 5 | 5 | 10 | 10 | 10 | |||

θ _{opt} | °C | 24 | 24 | 22 | 28 | 28 | 28 | |||||

θ _{max} | °C | 37 | 37 | 32 | 39 | 39 | 39 | |||||

b | – | 1·5 | 1·5 | 1 | 1 | 1 | 1 | |||||

Functional response | f(N, H) | α _{(0)} | – | – | – | 0·25 | 0·9 | 0·25 | 0·25 | |||

a | day^{−1} | – | – | 0·5 | 10 | 0·5 | 0·5 | |||||

Age-specific fecundity | β (x) | a_{min} | days | 46 | – | 21 | 24 | 24 | 24 | |||

a_{opt} | days | 49 | – | 23·1 | 25·1 | 24·7 | 24·7 | |||||

a_{max} | days | 76 | – | 28 | 31 | 31 | 31 | |||||

b | – | 100 | – | 0·5 | 100 | 2 | 2 | |||||

β _{max} | day^{−1} | 3 | – | 10 | 6 | 10 | 10 | |||||

Total fecundity | ∫ β (x)dx | – | 84 | – | 28 | – | 23 | 23 | ||||

on (1) | on (2) | on (5) | on (5) | on (5) | on (1) | on (2) | ||||||

Preference | ξ(x) | a_{min} | days | – | – | 12 | 12 | 0 | 24·5 | 14 | 12 | 12 |

α _{opt} | days | – | – | 42·6 | 25·5 | 0–14 | 24·5–31 | 14–24 | 18.8 | 15 | ||

α _{max} | days | – | – | 46 | 27 | 14 | 31 | 24 | 46 | 27 | ||

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 *k _{n}* age classes

*N*(

_{n ,i}*t*),

*i*= 1, 2, … ,

*k*. Ignoring the index

_{n}*n*for the functional population, the population dynamics of the

*i*th age class may be modelled as:

Ageing at time *t* occurs via flow rates

from age class *N*_{i−1} to *N _{i}*. Births (

*r*

_{0}=

*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 . 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 (*σ*^{2})may be estimated from data as *k* = Δ^{2}/*σ*^{2} 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.

The time-varying age-specific mortality rate (*µ** _{i}*(

*t*)) is the sum of all death rates due to all causes including net migration (see below) and may take values −∞ <

*µ*

*(*

_{i}*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 (

*ɛ*

*(*

_{i}*t*) ≥ 0).

The total number of equations in our system is

where *k _{n}* is the number of age classes

for the *n*th 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 . 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).

*T _{m}* 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.*

The parameter*T̄* is the yearly average temperature,*T̃* is the annual amplitude, *T*_{max} and *T*_{min} are the maximum and minimum temperatures, and *d* is the Julian date when *T*(*t*) is at maximum: *T*(*d*) = *T*_{max}. 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 *T*_{min} = 10 °C and *T*_{max} = 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 *Q*_{10} rule, van’t Hoff (1884)) corrected for assimilation efficiency. The normalized assimilation rate has a maximum at the optimum temperature *θ*_{opt}. 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.

Ignoring population indices, the general parameters are: *A* = *θ*_{opt} − *θ*_{min}, *B* = *θ*_{max} − *θ*_{opt}, *θ*_{min} and *θ*_{max} 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 *φ _{ψ}*(

*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 [*a*_{min}, *a*_{max}] are shown in Fig. 3.

The parameter *β*_{max} is the fecundity at age *a*_{opt}, *A* = *a*_{opt} − *a*_{min}, *B* = −*a*_{max} −*a*_{opt}, 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 (*N _{p}*) attacks its resource populations (

*H*, 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).

_{p}The function *D _{p}*( ) =

*D*(

_{p}*N*(

_{p}*t*)) is the maximum demand for resources by population

*N*(

_{p}*t*), and

*α*

*( ) =*

_{p}*α*

*(*

_{p}*H*(

_{p}*t*)) is the proportion of the resource population

*H*(

_{p}*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

*f*(

_{p}*N*(

_{p}*t*)) =

*D*(

_{p}*N*(

_{p}*t*)). The functions

*D*( ),

_{p}*α*

*( ) and*

_{p}*H*(

_{p}*t*) are defined below.

The total demand for resources by consumer *p* is

where *β** _{p}*(

*x*) is the maximum age-specific fecundity rate of individuals

*N*of age

_{p}*x*(equation 6). The availability rate of the resource populations

*H*(

_{p}*t*) for the consumer

*p*is:

where *α*_{(0)p} and *a _{p}* are constants with

*a*chosen so that the model

_{p}*f*( ) (equation 7) is weakly type III. Note that

_{p}*f*( ) tends to a type II model as

_{p}*α*

*→ +∞.*

_{p}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:

where 0 ≤ *ξ*_{p,n}(*x*) ≤ 1 is the preference function of the *p*th consumer population for the *n*th resource population of age *x* (see Fig. 4). *ξ*_{p,n,i} (discretized *ξ*_{p,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 *ξ*_{p,n} (*x*) is the same as the function *β*(*x*) (equation 6), with the maximum preference defined by the parameter *ξ*_{max} (Table 1).

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 *ɛ*_{n = 6,i}(*t*) in equation 1:

The birth rate of population *N _{p}*(

*t*) (Beddington

*et al*. 1976) accrues from predation across all of its resources (

*H*(

_{p}*t*)) (Lawton

*et al*. 1975) as (see equations 1, 5, 7):

The function *X _{p}*(

*t*) may be corrected for sex ratio as required and includes the effect of age structure of both consumer and resource populations (

*f*(

_{p}*N*(

_{p}*t*),

*H*(

_{p}*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 *i*th age class of the *n*th population is computed as:

where *ξ*_{p,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 *f _{p}*( ) 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 *k _{n}* age-classes of the population. For simplicity, age specific mortality

*µ*

_{(T)n,i}(

*t*) is:

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* = 10^{6} 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:

where the parameter *ω*_{n,i} defines whether population age-class *N*_{n,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 *n*th 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:

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* = 10^{6} 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.

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.

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.

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.