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Sensitivity Analysis of Deterministic and Stochastic Simulation Models of Populations of the Sheep Blowfly,Lucilia sericata
J. theor. Biol. (1997) 184, 139–148
Sensitivity Analysis of Deterministic and Stochastic Simulation Models of Populations of the Sheep Blowfly, Lucilia sericata A F, R W N F† School of Biological Sciences, The University of Bristol, Bristol BS8 1UG and †Leahurst Veterinary Field Station, The University of Liverpool, Neston, L64 7TE, U.K. (Received on 11 January 1996, Accepted in revised form 1 August 1996)
Using empirically derived relationships between temperature and development rate, deterministic and stochastic simulation models were constructed to predict the seasonal pattern of abundance of the sheep blowfly, Lucilia sericata. The number of day degrees accumulated each day by a cohort was calculated from the daily temperature pattern and the base temperature threshold for that stage. The diurnal temperature pattern was described by a simple sine curve relationship, based on the maximum and minimum temperatures. The stochastic model uses a Monte-Carlo simulation technique to assign random development rates to each life cycle stage generated from a Weibull distribution fitted to observed variation in development rate for each stage. The deterministic model typically predicts four discrete waves of emergence of L. sericata adults each season, with the fifth generation limited by the inset of dispause. The stochastic model resulted in a similar predicted population but with less discrete generations than the deterministic model and it more closely resembled the pattern of blowfly abundance observed in the field. Sensitivity analysis showed that both a reduction in the base temperatures and the number of day degrees required for life cycle development increased the daily rate of development at a given temperature and resulted in higher mean numbers of flies present each generation. However, the pattern of blowfly abundance observed was more sensitive to variations in base temperatures than day degree requirements. 7 1997 Academic Press Limited
Introduction Models that realistically simulate the behaviour of insect pest populations may be of considerable value, since they may allow patterns of pest abundance to be explained and predicted and new control strategies to be identified and implemented. Such models are typically deterministic, describing the behaviour of a population of identical individuals using mean parameter values (Anderson et al., 1982; Moon 1983; Highley et al., 1986; Lysyk & Axtell, 1987; Leathwick et al., 1992; Wall et al., 1993a). However, in reality, populations are made up of individuals showing inherent variation in physiological responses and resultant behaviour (Wagner et al., 1984a; Dennis et al., 1986). These variations can affect both the patterns of abundance and the stability of the system 0022–5193/97/020139 + 10 $25.00/0/jt960255
to perturbations. Hence, insect population models that ignore this stochasticity may loose much of their value as pest management tools (Phelps et al., 1993). This may have particular importance where the timing of control needs to be linked closely to patterns of first seasonal emergence or of stage-specific events in the pest life cycle. Furthermore, deterministic models may misrepresent small populations where chance factors due to this demographic stochasticity would have greater effect (Hardman, 1976; Fox, 1993; Burkey, 1995). The aim of the work described in this paper was to construct and compare the sensitivity of deterministic and stochastic simulation models of the seasonal patterns of abundance of the sheep blowfly Lucilia sericata Meigen (Diptera: Calliphoridae). This species is a significant pest throughout northern Europe; in 7 1997 Academic Press Limited
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England and Wales in 1988 and 1989 blowfly strike affected over 80% of sheep farms where, on average, over 500000 sheep were struck each year (French et al., 1992). Blowfly Life Cycle Adult female L. sericata lay eggs in the wool of sheep close to the skin surface, selecting areas of high humidity, such as fleece soiled by faeces or urine (Davies, 1948; Cragg, 1955). After hatching, the larvae pass through three stages, feeding on the epidermal tissues and skin secretions (Evans, 1936). This feeding activity causes extensive tissue damage to the host and the lesions produced attract further oviposition and, if untreated, may rapidly lead to death of the infested sheep (Guerrini, 1988). When they have completed feeding the third stage larvae drop to the ground where they undergo a period of dispersal as wandering larvae. The larvae then burrow into the soil before pupariating. Newly emerged adults mate and, after their ovaries have fully matured, females seek out a suitable oviposition site on a host sheep. At each oviposition each female deposits about 200 eggs in a single egg batch (Wall, 1992). Cycles of oviposition and emergence can continue throughout the spring and summer but, as the temperature and photoperiod decline in late August, females produce larvae that cease development mid-way through the wandering phase and burrow into the ground where they overwinter as diapausing larvae (Davies, 1934; Saunders et al., 1985). Blowfly Development Rates The development rates of insects are determined primarily by the temperature to which they are exposed (Beck, 1983; Wagner et al., 1984a; Higley et al., 1986; Leathwick et al., 1992). As a result the
majority of insect population models attempt quantitatively to extrapolate the population growth rate from available temperature data (e.g. Wang, 1960; Wagner et al., 1984b; Phelps et al., 1993; Atzeni et al., 1994). One of the most commonly used methods involves the calculation of day degreees. This technique assumes that development rate is directly proportional to temperature above a minimum threshold and that below this threshold no development occurs (Clements, 1963; Moon, 1983; Preuss, 1983; Phelps et al., 1993). The exact nature of the temperature-development rate relationship differs between species and between life cycle stages within a species. The relationship is usually determined by rearing individuals of different stages under a range of constant temperautres. The rate of development at any temperature is then calculated as the reciprocal of the time taken for 50% of the individuals to complete development at that temperature. A plot of these development rates against temperature shows an approximately sigmoidal relationship, with a linear portion through the middle temperature range. By linear regression, this portion can be extrapolated to where it intercepts the x-axis, giving an estimate of the base threshold temperature for that stage (Higley et al., 1986; Petitt et al., 1991; Bernal & Gonzalez, 1993). The base threshold temperatures for development of pre-adult stages and ovary maturation by adult females of L. sericata have been calculated previously to be around 9°C and 11°C, respectively (Table 1). The difference between the base temperature and the ambient temperature multiplied by the time taken to complete any life cycle stage at that temperature, gives the number of day degrees required for complete development of that stage. The number of day degrees required for each stage of the L. sericata life cycle have been calculated previously and are shown in Table 1.
T 1 Day degree requirements (2S.E.) base threshold temperatures (2S.E.), and the a and b Weibull parameters, fitted to the distribution of the number of day degrees required for completion of pre-adult life-cycle stages and adult egg batch maturation at 25°C Day degree requirement (2S.E.) Post-diapuse larvae Larval wandering stage Pupation First egg hatch Subsequent egg batches
29.7 45.6 126.1 62.0 27.7
(0.3) (5.1) (3.7) (2.3) (0.9)
Base temperature in °C (2S.E.) 9.2 9.5 8.8 11.2 11.1
(0.1) (0.6) (0.7) (1.3) (0.7)
Weibull parameters a
b
30.11 2.64 6.6 9.42 6.59
30.78 49.92 144.62 63.25 30.98
In most day degree models of insect populations the fraction of development that occurs on any one day is calculated as the difference between the base temperature and the mean ambient temperature for that day, divided by the number of day degrees required for that stage. These day degrees are then accumulated each day to predict the duration of each stage (e.g. Wall et al., 1992, 1993a). However, this calculation ignores day degree fractions that will be accumulated when the maximum temperature is above the base threshold but the mean daily temperature lies below it. This condition may apply for several weeks in Spring and Autumn, resulting in an underestimate of the day degrees accumulated by conventional models. Hence, for greater precision, in the present work the minimum and maximum temperatures for each day were used. The diurnal temperature pattern was assumed to resemble a sine curve, passing from a minimum T1 at the start of the day (time t = 0), to a maximum Tmax at the middle of the day (t = 1/2), and returning to a second minimum T2 at the end of the day, when time t = 1 (Gettinby & Gardiner, 1980). This pattern can be represented by a sinusoidal curve from the first minimum to the maximum, and then a second curve from the maximum to the second minimum. For the first half of the day, the relationship is: temperature = B1 sin (2Pt − P/2) + A1 ,
(1)
where B1 = (Tmax − T1 )/2 and A1 = (Tmax + T1 )/2. For the second half of the day, the relationship is: temperature = B2 sin (2Pt − P/2) + A2 ,
(2)
where B2 = (Tmax − T2 )/2 and A2 = (Tmax + T2 )/2. For each life cycle stage, the number of day degrees that is accumulated on any one day is given as the sum of the areas below the temperature curves and above the base threshold for that stage. The time t = t1 represents the point where the temperature curve crosses the base threshold for a given stage in the first half of the day, and t = t2 is where they cross in the second half of the day. If the base threshold lies below T1 , then t1 is set to 0, and full development is assumed for that half day. If the base threshold lies below T2 then t2 = 1 and again, full development occurs for the second half day. Therefore, the number of day degrees accumulated on a single day is the sum of the integral of eqn (1), from t = t1 to t = 1/2 and the integral of eqn (2) from t = 1/2 to t = t2 . The calculated number of day degrees divided by the total number of day degrees required for completion of that stage gives the fraction of overall development undergone on that day. For the next day, T2 is used as the first minimum
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temperature and the new Tmax and T2 are read in. By this method, day degrees are accumulated even if the daily mean temperature lies below the base threshold, as long as the maximum temperature is above. Model Structure The models developed simulate the passage of a single blowfly population throughout a season by daily accumulations of day degrees, based on daily maximum and minimum temperature data. In the models, a cohort is defined as the group of individuals arising from a single oviposition. All individuals in a cohort are assumed to be identical, having undergone the same degree of development. The models iterate on a daily basis, so that each cohort accumulates the calculated development fraction each day, according to its life cycle stage. Egg hatch and the larval feeding stages that occur on the sheep are allowed a constant 3 days for completion, since in the protected microhabitat of the fleece, the rate of development of these stages occurs independently of temperature (Wall et al., 1992). When the accumulated day degree fractions for the off-sheep stages exceed one, the cohort is assumed to have completed the development necessary for that stage (Gettinby & Gardiner, 1980) and the cohort passes on to the next stage of the life cycle. At this point the development fraction for the cohort is set to 0 and accumulation of day degrees resumes for the next stage with the new base threshold and day degree requirements. When a cohort of adults accumulates sufficient day degrees to complete egg maturation, 50% of the individuals in the cohort oviposit, laying 200 eggs each. The models start on day 1, nominally 1st February, with an initial cohort of 100 overwintering larvae in a state of diapause. Daily maximum and minimum temperatures for the year 1990 are used throughout this analysis. As the temperatures increase during early spring, the cohort accumulates day degree fractions until it completes post-diapause development and pupariates. When pupal development is complete, this first cohort emerges as adults. In the model, to simulate the effect of induced diapause at the end of the season, the larvae of females ovipositing after day 210 (30th August) are assumed to enter diapause (Saunders et al., 1985). Adult cohorts remain viable unless the temperature drops below 10°C, when they are assumed to die. During each daily loop of the model, the number of individuals in each cohort is altered by an imposed mortality schedule. The rate of adult mortality imposed is temperature dependent, based on L. sericata population age distributions observed
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in the field (Wall, 1992). This mortality rate allows approximately 15% of females to reach a first oviposition, 7% to reach a second, and 2.5% to reach a third, given the base temperature for maturation of the ovaries and the day degree requirements presented in Table 1. Pre-adult mortality is set at a constant value of 50% per generation, which approximates to that observed in field populations (R. Wall, unpublished results). The maximum population density is limited by the number of oviposition sites available to the blowfly population. For simplicity, in the models presented, it is assumed there is an arbitrary maximum of 1500 suitable oviposition sites available to the blowfly population. Each day 30 new sites become available, up to the maximum number. Oviposition site deprivation does not affect the number of eggs per batch oviposited by females of the blowfly L. cuprina (Barton Browne et al., 1990). Hence, in the models presented here, if the number of flies attempting to oviposit exceed the number of sites available on any one day, females that do not oviposit remain fully gravid until the next day, although they remain subject to the normal rate of adult mortality, which is assumed not to be associated with reproduction.
therefore, uses information relating to both the mean rate of development and the variation in these rates (Wagner et al., 1984a). Each run of the stochastic model produces a different pattern of abundance due to the random assignment of day degree requirements. To allow model outputs to be compared the stochastic model was run a set number times and the predicted population size on each day averaged across runs. In order to determine the optimum number of runs needed to produce consistent average predictions, the model initially was run 5, 10, 15, 20, 25, 30, 35, 40, 45 and 50 times with a standard set of parameters. By repeating each set of runs 20 times, the consistency of the predictions for each number of runs could be assessed. The standard error of these 20 average predictions for each set of runs on days 100, 150, 200 and 250 (Fig. 1) shows that the standard error declines with the increasing number of runs but increases with the number of days over which the model has iterated. As a result of this analysis, 40 runs of the stochastic model was adopted as standard for all the following trials since this gave a relatively consistent average output. 800
Deterministic and Stochastic Models 600
Standard error
In the deterministic model all the individuals are assumed to be identical in terms of their day degree requirements. However, in reality, L. sericata of any one life cycle stage show a positively skewed range of day degree requirements that can be described by a Weibull function (Table 1). Similar distributions in insect populations have been observed by other authors (Sharpe et al., 1977; Phelps et al., 1993). To incorporate the variability observed within the population into the stochastic model, each cohort is assigned a random day degree requirement for each of its life cycle stages, generated from the appropriate Weibull curve (Table 1). This value is retained throughout the duration of that life cycle stage, so that slow developing cohorts remain slow, and fast developing cohorts remain fast (Phelps et al., 1993). The day degree requirements for each stage of development are assigned independently of the requirements of the previous stages and, furthermore, the day degree requirements of each offspring cohort are assigned independently of the development rates of the parent cohort. This process of assigning day degree requirements means that there is random variation between cohorts based on empirically determined functions and the stochastic model,
400
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0 5
15
25 35 Number of runs
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F. 1. The standard error of the mean number of individuals predicted to be present at days 100 (—Q—), 150 (— + —), 200 (—(—) and 250 (—q—) (where day 1 = 1st February) by a stochastic model of the seasonal abundance of the blowfly Lucilia sericata. For each point, the model was run a number of times with a standard set of parameters. Each set of runs was repeated 20 times and the standard error of these 20 average predictions plotted against the number of runs in each set.
Using these models, a series of simulations were carried out, varying the base temperatures and day degree values, to determine the relative importance of each on the observed patterns of blowfly abundance. The effect on the number of adult flies to emerge in a particular generation was assessed by averaging the number of flies of that generation present on each day. To quantify the effect on the timing of a generation, each day number on which adults of a particular generation was present was weighted by the number of individuals present on that day. These were then averaged over all days on which adults of the generation were present, to produce the mean time to the mid-point of each generation.
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The output of the stochastic model (Fig. 2) also predicts the emergence of four generations but these are less distinct with individuals of each generation present both earlier and later than the predictions of the deterministic model. This difference is particularly marked early in the season when temperatures are relatively low. The small number of adults present at about day 75 was due to a cohort emerging particularly early by the chance allocation of low day degree requirements. However, this cohort emerged so early that the temperatures were too cold to support adult individuals (less than 10°C), so the cohort died before it had the chance to oviposit. Sensitivity Analysis
Results The output from the deterministic model predicts that the adults emerge in discrete waves, with fly numbers increasing exponentially with each wave (Fig. 2). The peaks in fly abundance occur at about days 100, 150, 180 and 210. A small number of individuals form a fifth generation, but the majority of offspring from the fourth generation undergo diapause. This pattern of output is similar to that of a previous deterministic model developed for L. sericata (Wall et al., 1992), which has been shown to be a good description of the changes in abundance of wild populations of L. sericata in sheep pastures (Wall et al., 1993a).
The day degree requirements of all life cycle stages were varied simultaneously by up to 240% of the value calculated from laboratory data (Table 1). For clarity, only the effect on the third and fourth generations is shown because these contain the greatest number of individuals. Regressing the mean time to mid-point of each generation against the percentage change in the day degree requirement, shows that variation in the day degree requirement has a significant influence on the timing of each generation (Fig. 3). The mean time of emergence of the third and fourth generations increases linearly with the number of day degrees required to complete
10000
Fly number + 1
1000
100
10
1 0
50
100
150 Day number
200
250
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F. 2. The number of individuals (+1) of the blowfly Lucilia sericata predicted to be present each day (where day 1 = 1st February) by deterministic (dashed line) and stochastic (continuous line) models.
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number of individuals present in each generation (P Q 0.01). However, if the day degree requirements are reduced by over 20%, the numbers do not increase to such high levels as the deterministic model predicts. This is because an increasing proportion of the 40 runs of the stochastic model become extinct by the second generation due to individuals emerging early in the season, before the ambient temperatures are high enough to support reproducing adults. As a result, as the day degree requirements are reduced, the mean number of adults present in the fourth generation is progressively lower than that predicted by the deterministic model (Fig. 4).
240
Mean time (days)
220
200
180
–40
–30 –20 –10 0 10 Day degree adjustment (%)
20
F. 3. The mean time (days) to the mid-point of the third (filled squares) and fourth (open squares) generations of the blowfly Lucilia sericata following adjustment of the day degree requirements in deterministic (dashed lines) and stochastic (continuous lines) simulation models.
the life cycle. The predictions of the deterministic and stochastic models are similar, both of which show that each 10% increase in the day degree requirement of all life cycle stages increases the mean time to mid-point of the third generation by approximately 5.5 days (P Q 0.001), and the mean time to mid-point of the fourth generation by approximately 8 days (P Q 0.001). Both models show that varying the day degree requirements has a significant effect on the number of flies present in each generation. As the number of day degrees required to complete the life cycle is reduced, so the mean number of individuals present in both third and fourth generations increase (Fig. 4). This is because with lower day degree requirements a greater number of individuals survive to oviposit a greater number of times. In the deterministic model, this relationship is highly significant for both the third and fourth generations (P Q 0.001), up to a 20% reduction in day degree requirements. As the day degree requirements are reduced by over 20%, the population in the fourth generation reaches an upper limit of approximately 17000 individuals, a maximum imposed by the limiting effect of the number of oviposition sites available. The stochastic model predicts a weaker, but still significant increase in the
18 000
15 000
Mean fly number
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Both deterministic and stochastic models were run with the base temperatures for each stage reduced by up to 25% of the value calculated from laboratory data (Table 1). The results of the simulations show that as the base temperatures for each stage are decreased, development can take place at lower temperatures and the mean time to mid-point of the third and fourth generations is reduced (Fig. 5). Once again the two models are in close agreement, both predicting that for each 10% decrease in the base temperature of all life cycle stages there is
12 000
9000
6000
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0 –40
0 –30 –20 –10 10 Day degree adjustment (%)
20
F. 4. The mean number of the blowfly Lucilia sericata present at the mid-point of the third (filled squares) and fourth (open squares) generations following adjustment of the day degree requirements in deterministic (dashed lines) and stochastic (continuous lines) simulation models.
increase in numbers with reduction in base temperature is significant (P Q 0.05).
220
210
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160 75
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85 90 Base adjustment (%)
95
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F. 5. The mean time (days) to the mid-point of the third (filled squares) and fourth (open squares) generations of the blowfly Lucilia sericata following adjustment of the base temperature (°C) in deterministic (dashed lines) and stochastic (continuous lines) simulation models.
an approximately 11.5 day decrease in the time to mid-point of the fourth generation (P Q 0.001), and an 8 day decrease in the time to mid-point of the third generation (P Q 0.01). Reduction in the life cycle base temperatures has the effect of increasing the mean number of individuals present in both third and fourth generations (Fig. 6). This is because lower base thresholds allow development to occur at lower ambient temperatures and, as a result a greater number of individuals reach oviposition and survive to oviposit a greater number of times. In the deterministic model, this increase in numbers continues up to a 20% decrease in base temperatures (P Q 0.01). If the base temperatures are reduced by over 20%, the population in the fourth generation reaches its maximum size due to the limiting effect of the number of oviposition sites available. In contrast, as the base temperature is reduced, there is an increased likelihood of each run of the stochastic model going to extinction, due to early emergence of the adults. Once again, therefore, with decreasing base temperatures the mean number of individuals in the third and fourth generations predicted by the stochastic model are progressively smaller than those predicted by the deterministic model, although the
For the stochastic model, the effects of changing the permissible range of variation in day degree requirements for each life cycle stage was examined. The mean day degree requirement for each stage was maintained at its calculated value (Table 1) and the a parameter of the Weibull distribution was altered so that the variation in day degree requirements could be changed by up to 260%. The effect on the mean number of adults in each generation and the timing of each generation were noted, as was the standard error of these means. Changing the variation in day degree requirements between cohorts had no effect on the mean number of adults in each generation, or on the mean time to mid-point of each generation, or on the standard error of the mean number of flies per generation. However, regression shows there is a slight but significant effect on the standard error of the mean time to mid-point of the third generation (P Q 0.001) and fourth generation (P Q 0.01) (Fig. 7). As the variation in day degree requirements becomes greater,
16 000
12 000 Mean fly number
Mean time (days)
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8000
4000
0 75
80
85 90 Base adjustment (%)
95
100
F. 6. The mean number of the blowfly Lucilia sericata present at the mid-point of the third (filled squares) and fourth (open squares) generations following adjustment of the base temperature (°C) in deterministic (dashed lines) and stochastic (continuous lines) simulation models.
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Standard error
1.25
1.05
0.85
0.65
0.45 –60
0 40 –40 –20 20 Change of normal variation (%)
60
F. 7. The standard error of the mean time to the mid-point of the third (filled squares) and fourth (open squares) generations of the blowfly Lucilia sericata following adjustment of the variance in day degree requirements in a stochastic simulation model.
the more the stochastic model will differ from the deterministic model, and so the standard errors of the mean timing of both generations 3 and 4 increase. Discussion Using empirically derived relationships between temperature and development rate, a deterministic simulation model has been developed and used to predict the abundance of the sheep blowfly, L. sericata, throughout a season, from an initial overwintering cohort. A similar model has been shown to explain a significant proportion of the variance in seasonal abundance in the field (Wall et al., 1992; Wall et al., 1993a). A stochastic version of the model, which uses a Monte-Carlo simulation technique to assign development rates at random to each cohort in the simulation, was also developed. These random rates are generated from a Weibull distribution fitted to the observed variation in development rates for each life cycle stage in constant temperature experiments. Each run of the stochastic model produces a slightly different prediction, due to the random element of the model which, when averaged across a number of runs, provide a prediction of mean abundance. The deterministic model typically predicts four discrete waves of emergence of L. sericata adults,
with the fifth generation limited by the onset of diapause. The stochastic model generates a similar predicted pattern but with less discrete generations that more closely resembles the pattern of blowfly abundance observed in the field (Wall et al., 1993a, b). The stochastic model has the further advantage of predicting a range of days on which emergence occurs, with the first adult emerging several days before that of the deterministic model. If control was to be aimed at the first day of emergence, some flies would be missed if prediction was based on the deterministic model alone (Wall et al., 1995). In sensitivity analysis both models suggest that both the base threshold temperatures and the day degree requirements significantly affect the timing of generation peaks. Reducing the base temperature of each life cycle stage allows development to occur at lower temperatures and greater numbers of day degrees to be accumulated each day. Therefore each cohort reaches the next life cycle stage earlier and overall each generation would occur at an earlier date. Similarly, lowering the number of day degrees that need to be accumulated by any stage decreases the time spent in that stage, so that the next generation of adults occurs earlier in the season. Both models show approximately the same rate of change of generation time with varying base temperatures and day degree requirements. Both models also show that the base threshold temperatures and the day degree requirements significantly affect the number of flies present in each generation. By accelerating the rate of development by one of these methods, adult females spend less time maturing their ovaries and so are more likely to oviposit before they die. This results in each generation consisting of the product of more ovipositions than if a slower development rate was assumed. As a result, the number of adults present in each generation increases as the development rate increases. However, reducing the day degree requirements of the base threshold temperature in the deterministic model allows the population to reach its carrying capacity by the fourth generation. This is not the case in the stochastic population because of the increasing number of cohorts which, through the random allocation of low base temperatures or low day degree requirements, emerge and die in Spring before seasonal temperatures are high enough to sustain adult development. The sensitivity analysis suggests that variations in base temperatures and/or day degree requirements may have very different consequences for the observed pattern of blowfly abundance. Both models show that the mean time to mid-point of the third
and fourth generations is highly sensitive to the precise base temperature used. A decrease in base temperature of 1°C would result in an approximately 11 day decrease in the time to mid-point of the fourth generation, and an 8 day decrease in the time to mid-point of the third generation. Such errors are in the range of those observed around the means calculated from the experimental data (Table 1). This observation is of particular importance because the base thresholds for each stage used in the present study, calculated from the observation of development rates in constant temperature experiments, may be inaccurate due to the non-linearity of the temperature–development rate relationship at low temperatures. Hence, it is possible that the base thresholds used in the models are actually over-estimations of the true base temperatures. Such errors may have critical effects when precise timing of pest control is required. In constrast, the results presented here suggest the sheep blowfly system may be far less sensitive to variations in day degree requirements than to variation in base temperature. The pre-adult larval stages that occur off the host require a total of 172 day degrees over a base temperature of 9°C. Both the deterministic and stochastic models show that deviations of up to 20 day degrees are required to increase the mean time to mid-point of the third and fourth generations by approximately 5 and 8 days, respectively. Such errors are considerably greater than the range of variation observed in the experimental data (Table 1). It has been suggested by various authors that development rates measured in constant temperature incubators do not correspond to development rates under the fluctuating temperatures in the field (Clements, 1963; Higley et al., 1986; Hagstrum & Milliken, 1991). In contrast, however, Dallwitz (1984) showed that L. cuprina had similar development rates under constant and fluctuating temperatures, up to 30°C. Such high temperatures would rarely be experienced by L. sericata in the field in northern Europe. This, together with the apparent lack of sensitivity of the abundance pattern to minor variations in day degree accumulation, indicates that the use of day degree requirements estimated in constant temperature incubators in the simulation models may not be an unwarranted oversimplification. Understanding the driving forces behind the population dynamics of L. sericata is only the first step towards managing the sheep strike system as a whole. There are a large range of factors that determine the impact of the fly population on farms
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around Britain, such as other climatic variables, geographical factors and farm management practices. However, the models presented here represent an essential stage in this process allowing the effects of stochasticity and the sensitivity of the blowfly system to variation in base temperatures and day degree requirements to be considered. The support of a Royal Society University Research Fellowship to R. Wall and a LINK grant, supported by Bayer Ltd and the Ministry of Agriculture, Fisheries and Food of the U.K. government are gratefully acknowledged. REFERENCES A, T. E., K, G. G. & S, R. E. (1982). Temperature-dependent model for the postdiapause development and Spring emergence of the European Corn Borer, Ostrinia nubilalis (Hu bner) (Lepidoptera: Pyralidae), in North Carolina. Environ. Entomol. 11, 1307–1311. A, M. G., M, D. G., S, J. P., A, K. A. & B, D. G. (1994). Comparison of the predicted impact of a screwworm fly outbreak in Australia using a growth index model and a life-cycle model. Med. Vet. Ent. 8, 281–291. B B, L., G, A. C. M. & V, W. G. (1990). Readiness to lay in the blowfly Lucilia cuprina is unaffected by oviposition site-deprivation or egg-load. Entomol. Exp. Appl. 55, 33–40. B, S. D. (1983). Insect thermoperiodism. Annu. Rev. Entomol. 28, 91–108. B, J. & G, D. (1993). Experimental assessment of a degree-day model for predicting the development of parasites in the field. J. Appl. Ent. 116, 459–466. B, T. V. (1995). Extinction rates in archipelagos: Implications for populations in fragmented habitats. Conserv. Biol. 9, 527–554. C, A. N. (1963). The physiology of mosquitoes. Int. Monogr. Pure Appl. Biol. 17, 72–79. C, J. B. (1955). The natural history of sheep blowflies. Ann. Appl. Biol. 42, 197–207. D, R. (1984). The influence of constant and fluctuating temperatures on development rate and survival of pupae of the Australian sheep blowfly Lucilia cuprina. Entomol. Exp. Appl. 36, 89–95. D, L. (1948). Observations on the development of Lucilia sericata (Mg.) eggs in sheep fleeces. J. Exp. Biol. 25, 86–102. D, W. M. (1934). The sheep blowfly problem in North Wales. Ann. Appl. Biol. 21, 267–282. D, B., K, W. P. & B, R. C. (1986). Stochastic model of insect phenology: Estimation and testing. Environ. Entomol. 15, 540–546. E, A. V. (1936). The physiology of the sheep blow-fly Lucilia sericata Meig. (Diptera). Trans. Roy. Ent. Soc. 85, 363–378. F, G. A. (1993). Life history evolution and demographic stochasticity. Evol. Ecol. 7, 1–14. F, N. P., W, R., C, K. L. & M, K. L. (1992). Prevalence, regional distribution and control of blowfly strike in England and Wales. Vet. Rec. 131, 337–342. G, G. & G, W. P. (1980). Disease incidence forecasts by means of climatic data. Biometeorology 7, 87–103. G, V. H. (1988). Ammonia toxicity and alkalosis in sheep infested by Lucilia cuprina larvae. Int. J. Parasit. 18, 79–81. H, D. W. & M, G. A. (1991). Modeling differences in insect developmental times between constant and fluctuating temperatures. Ann. Ent. Soc. Am. 84, 369–379. H, J. M. (1976). Deterministic and stochastic models simulating the growth of insect populations over a range of temperatures under Malthusian conditions. Can. Ent. 108, 907–924.
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H, L. G., P, L. P. & O, H. R. (1986). DEGDAY: A program for calculating degree-days and assumptions behind the degree-day approach. Environ. Entomol. 15, 999–1016. L, D. M., B, N. D. & V, A. (1992). A model for Nematodiasis in New Zealand Lambs. Int. J. Parasit. 22, 789–799. L, T. J. & A, R. C. (1987). A simulation model of house fly (Diptera: Muscidae) development in poultry manure. Can. Ent. 119, 427–437. M, R. D. (1983). Simulating developmental time of preadult face flies (Diptera: Muscidae) from air temperature records. Environ. Entomol. 12, 943–948. P, F. L., A, J. C. & B, C. S. (1991). Degree day model for vegetable leafminer (Diptera: Agromyzidae) phenology. Environ. Entomol. 20, 1134–1140. P, K., C, R. H., R, R. J. & F, S. (1993). Monte Carlo simulation method for forecasting the timing of pest insect attacks. Crop Prot. 12, 335–342. P, K. P. (1983). Day-degree methods for pest management. Environ. Entomol. 12, 613–619. S, D. S., MP, J. N. & C, K. D. (1985). Maternal and larval effects of photoperiod on the induction of larval diapause in two species of fly, Calliphora vicina and Lucilia sericata. Exp. Biol. 46, 51–58. S, P. J., C, G. L., DM, D. W. & C, C. L. (1977). Distribution model of organism development times. J. theor. Biol. 66, 21–38.
W, T. L., W, H., S, P. J., S, R. M. & C, R. N. (1984a). Modeling insect development rates: A literature review and application of a biophysical model. Ann. Ent. Soc. Am. 77, 208–225. W, T. L., W, H., S, P. J. & C, R. N. (1984b). Modeling distributions of insect development time: A literature review and application of the Weibull function. Ann. Ent. Soc. Am. 77, 475–487. W, R. (1992). The reproductive output of the blowfly Lucilia sericata. J. Insect Physiol. 39, 743–750. W, R., F, N. & M, K. L. (1992). Effects of temperature on the development and abundance of the sheep blowfly Lucilia sericata (Diptera: Calliphoridae). Bull. Ent. Res. 82, 125–131. W, R., F, N. P. & M, K. L. (1993a). Predicting the abundance of the blowfly Lucilia sericata (Diptera: Calliphoridae). Bull. Ent. Res. 83, 431–436. W, R., F, N. P. & M, K. L. (1993b). Sheep blowfly population control: Development of a simulation model and analysis of management strategies. J. Appl. Ecol. 30, 743–751. W, R., F, N. P. & M K. L. (1995). Population suppression for control of the blowfly Lucilia sericata and sheep blowfly strike. Ecol. Ent. 20, 91–97. W, J. Y. (1960). A critique of the heat unit approach to plant response studies. Ecology 41, 785–790.
Sensitivity Analysis of Deterministic and Stochastic Simulation Models of Populations of the Sheep Blowfly, Lucilia sericata A F, R W N F† School of Biological Sciences, The University of Bristol, Bristol BS8 1UG and †Leahurst Veterinary Field Station, The University of Liverpool, Neston, L64 7TE, U.K. (Received on 11 January 1996, Accepted in revised form 1 August 1996)
Using empirically derived relationships between temperature and development rate, deterministic and stochastic simulation models were constructed to predict the seasonal pattern of abundance of the sheep blowfly, Lucilia sericata. The number of day degrees accumulated each day by a cohort was calculated from the daily temperature pattern and the base temperature threshold for that stage. The diurnal temperature pattern was described by a simple sine curve relationship, based on the maximum and minimum temperatures. The stochastic model uses a Monte-Carlo simulation technique to assign random development rates to each life cycle stage generated from a Weibull distribution fitted to observed variation in development rate for each stage. The deterministic model typically predicts four discrete waves of emergence of L. sericata adults each season, with the fifth generation limited by the inset of dispause. The stochastic model resulted in a similar predicted population but with less discrete generations than the deterministic model and it more closely resembled the pattern of blowfly abundance observed in the field. Sensitivity analysis showed that both a reduction in the base temperatures and the number of day degrees required for life cycle development increased the daily rate of development at a given temperature and resulted in higher mean numbers of flies present each generation. However, the pattern of blowfly abundance observed was more sensitive to variations in base temperatures than day degree requirements. 7 1997 Academic Press Limited
Introduction Models that realistically simulate the behaviour of insect pest populations may be of considerable value, since they may allow patterns of pest abundance to be explained and predicted and new control strategies to be identified and implemented. Such models are typically deterministic, describing the behaviour of a population of identical individuals using mean parameter values (Anderson et al., 1982; Moon 1983; Highley et al., 1986; Lysyk & Axtell, 1987; Leathwick et al., 1992; Wall et al., 1993a). However, in reality, populations are made up of individuals showing inherent variation in physiological responses and resultant behaviour (Wagner et al., 1984a; Dennis et al., 1986). These variations can affect both the patterns of abundance and the stability of the system 0022–5193/97/020139 + 10 $25.00/0/jt960255
to perturbations. Hence, insect population models that ignore this stochasticity may loose much of their value as pest management tools (Phelps et al., 1993). This may have particular importance where the timing of control needs to be linked closely to patterns of first seasonal emergence or of stage-specific events in the pest life cycle. Furthermore, deterministic models may misrepresent small populations where chance factors due to this demographic stochasticity would have greater effect (Hardman, 1976; Fox, 1993; Burkey, 1995). The aim of the work described in this paper was to construct and compare the sensitivity of deterministic and stochastic simulation models of the seasonal patterns of abundance of the sheep blowfly Lucilia sericata Meigen (Diptera: Calliphoridae). This species is a significant pest throughout northern Europe; in 7 1997 Academic Press Limited
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England and Wales in 1988 and 1989 blowfly strike affected over 80% of sheep farms where, on average, over 500000 sheep were struck each year (French et al., 1992). Blowfly Life Cycle Adult female L. sericata lay eggs in the wool of sheep close to the skin surface, selecting areas of high humidity, such as fleece soiled by faeces or urine (Davies, 1948; Cragg, 1955). After hatching, the larvae pass through three stages, feeding on the epidermal tissues and skin secretions (Evans, 1936). This feeding activity causes extensive tissue damage to the host and the lesions produced attract further oviposition and, if untreated, may rapidly lead to death of the infested sheep (Guerrini, 1988). When they have completed feeding the third stage larvae drop to the ground where they undergo a period of dispersal as wandering larvae. The larvae then burrow into the soil before pupariating. Newly emerged adults mate and, after their ovaries have fully matured, females seek out a suitable oviposition site on a host sheep. At each oviposition each female deposits about 200 eggs in a single egg batch (Wall, 1992). Cycles of oviposition and emergence can continue throughout the spring and summer but, as the temperature and photoperiod decline in late August, females produce larvae that cease development mid-way through the wandering phase and burrow into the ground where they overwinter as diapausing larvae (Davies, 1934; Saunders et al., 1985). Blowfly Development Rates The development rates of insects are determined primarily by the temperature to which they are exposed (Beck, 1983; Wagner et al., 1984a; Higley et al., 1986; Leathwick et al., 1992). As a result the
majority of insect population models attempt quantitatively to extrapolate the population growth rate from available temperature data (e.g. Wang, 1960; Wagner et al., 1984b; Phelps et al., 1993; Atzeni et al., 1994). One of the most commonly used methods involves the calculation of day degreees. This technique assumes that development rate is directly proportional to temperature above a minimum threshold and that below this threshold no development occurs (Clements, 1963; Moon, 1983; Preuss, 1983; Phelps et al., 1993). The exact nature of the temperature-development rate relationship differs between species and between life cycle stages within a species. The relationship is usually determined by rearing individuals of different stages under a range of constant temperautres. The rate of development at any temperature is then calculated as the reciprocal of the time taken for 50% of the individuals to complete development at that temperature. A plot of these development rates against temperature shows an approximately sigmoidal relationship, with a linear portion through the middle temperature range. By linear regression, this portion can be extrapolated to where it intercepts the x-axis, giving an estimate of the base threshold temperature for that stage (Higley et al., 1986; Petitt et al., 1991; Bernal & Gonzalez, 1993). The base threshold temperatures for development of pre-adult stages and ovary maturation by adult females of L. sericata have been calculated previously to be around 9°C and 11°C, respectively (Table 1). The difference between the base temperature and the ambient temperature multiplied by the time taken to complete any life cycle stage at that temperature, gives the number of day degrees required for complete development of that stage. The number of day degrees required for each stage of the L. sericata life cycle have been calculated previously and are shown in Table 1.
T 1 Day degree requirements (2S.E.) base threshold temperatures (2S.E.), and the a and b Weibull parameters, fitted to the distribution of the number of day degrees required for completion of pre-adult life-cycle stages and adult egg batch maturation at 25°C Day degree requirement (2S.E.) Post-diapuse larvae Larval wandering stage Pupation First egg hatch Subsequent egg batches
29.7 45.6 126.1 62.0 27.7
(0.3) (5.1) (3.7) (2.3) (0.9)
Base temperature in °C (2S.E.) 9.2 9.5 8.8 11.2 11.1
(0.1) (0.6) (0.7) (1.3) (0.7)
Weibull parameters a
b
30.11 2.64 6.6 9.42 6.59
30.78 49.92 144.62 63.25 30.98
In most day degree models of insect populations the fraction of development that occurs on any one day is calculated as the difference between the base temperature and the mean ambient temperature for that day, divided by the number of day degrees required for that stage. These day degrees are then accumulated each day to predict the duration of each stage (e.g. Wall et al., 1992, 1993a). However, this calculation ignores day degree fractions that will be accumulated when the maximum temperature is above the base threshold but the mean daily temperature lies below it. This condition may apply for several weeks in Spring and Autumn, resulting in an underestimate of the day degrees accumulated by conventional models. Hence, for greater precision, in the present work the minimum and maximum temperatures for each day were used. The diurnal temperature pattern was assumed to resemble a sine curve, passing from a minimum T1 at the start of the day (time t = 0), to a maximum Tmax at the middle of the day (t = 1/2), and returning to a second minimum T2 at the end of the day, when time t = 1 (Gettinby & Gardiner, 1980). This pattern can be represented by a sinusoidal curve from the first minimum to the maximum, and then a second curve from the maximum to the second minimum. For the first half of the day, the relationship is: temperature = B1 sin (2Pt − P/2) + A1 ,
(1)
where B1 = (Tmax − T1 )/2 and A1 = (Tmax + T1 )/2. For the second half of the day, the relationship is: temperature = B2 sin (2Pt − P/2) + A2 ,
(2)
where B2 = (Tmax − T2 )/2 and A2 = (Tmax + T2 )/2. For each life cycle stage, the number of day degrees that is accumulated on any one day is given as the sum of the areas below the temperature curves and above the base threshold for that stage. The time t = t1 represents the point where the temperature curve crosses the base threshold for a given stage in the first half of the day, and t = t2 is where they cross in the second half of the day. If the base threshold lies below T1 , then t1 is set to 0, and full development is assumed for that half day. If the base threshold lies below T2 then t2 = 1 and again, full development occurs for the second half day. Therefore, the number of day degrees accumulated on a single day is the sum of the integral of eqn (1), from t = t1 to t = 1/2 and the integral of eqn (2) from t = 1/2 to t = t2 . The calculated number of day degrees divided by the total number of day degrees required for completion of that stage gives the fraction of overall development undergone on that day. For the next day, T2 is used as the first minimum
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temperature and the new Tmax and T2 are read in. By this method, day degrees are accumulated even if the daily mean temperature lies below the base threshold, as long as the maximum temperature is above. Model Structure The models developed simulate the passage of a single blowfly population throughout a season by daily accumulations of day degrees, based on daily maximum and minimum temperature data. In the models, a cohort is defined as the group of individuals arising from a single oviposition. All individuals in a cohort are assumed to be identical, having undergone the same degree of development. The models iterate on a daily basis, so that each cohort accumulates the calculated development fraction each day, according to its life cycle stage. Egg hatch and the larval feeding stages that occur on the sheep are allowed a constant 3 days for completion, since in the protected microhabitat of the fleece, the rate of development of these stages occurs independently of temperature (Wall et al., 1992). When the accumulated day degree fractions for the off-sheep stages exceed one, the cohort is assumed to have completed the development necessary for that stage (Gettinby & Gardiner, 1980) and the cohort passes on to the next stage of the life cycle. At this point the development fraction for the cohort is set to 0 and accumulation of day degrees resumes for the next stage with the new base threshold and day degree requirements. When a cohort of adults accumulates sufficient day degrees to complete egg maturation, 50% of the individuals in the cohort oviposit, laying 200 eggs each. The models start on day 1, nominally 1st February, with an initial cohort of 100 overwintering larvae in a state of diapause. Daily maximum and minimum temperatures for the year 1990 are used throughout this analysis. As the temperatures increase during early spring, the cohort accumulates day degree fractions until it completes post-diapause development and pupariates. When pupal development is complete, this first cohort emerges as adults. In the model, to simulate the effect of induced diapause at the end of the season, the larvae of females ovipositing after day 210 (30th August) are assumed to enter diapause (Saunders et al., 1985). Adult cohorts remain viable unless the temperature drops below 10°C, when they are assumed to die. During each daily loop of the model, the number of individuals in each cohort is altered by an imposed mortality schedule. The rate of adult mortality imposed is temperature dependent, based on L. sericata population age distributions observed
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in the field (Wall, 1992). This mortality rate allows approximately 15% of females to reach a first oviposition, 7% to reach a second, and 2.5% to reach a third, given the base temperature for maturation of the ovaries and the day degree requirements presented in Table 1. Pre-adult mortality is set at a constant value of 50% per generation, which approximates to that observed in field populations (R. Wall, unpublished results). The maximum population density is limited by the number of oviposition sites available to the blowfly population. For simplicity, in the models presented, it is assumed there is an arbitrary maximum of 1500 suitable oviposition sites available to the blowfly population. Each day 30 new sites become available, up to the maximum number. Oviposition site deprivation does not affect the number of eggs per batch oviposited by females of the blowfly L. cuprina (Barton Browne et al., 1990). Hence, in the models presented here, if the number of flies attempting to oviposit exceed the number of sites available on any one day, females that do not oviposit remain fully gravid until the next day, although they remain subject to the normal rate of adult mortality, which is assumed not to be associated with reproduction.
therefore, uses information relating to both the mean rate of development and the variation in these rates (Wagner et al., 1984a). Each run of the stochastic model produces a different pattern of abundance due to the random assignment of day degree requirements. To allow model outputs to be compared the stochastic model was run a set number times and the predicted population size on each day averaged across runs. In order to determine the optimum number of runs needed to produce consistent average predictions, the model initially was run 5, 10, 15, 20, 25, 30, 35, 40, 45 and 50 times with a standard set of parameters. By repeating each set of runs 20 times, the consistency of the predictions for each number of runs could be assessed. The standard error of these 20 average predictions for each set of runs on days 100, 150, 200 and 250 (Fig. 1) shows that the standard error declines with the increasing number of runs but increases with the number of days over which the model has iterated. As a result of this analysis, 40 runs of the stochastic model was adopted as standard for all the following trials since this gave a relatively consistent average output. 800
Deterministic and Stochastic Models 600
Standard error
In the deterministic model all the individuals are assumed to be identical in terms of their day degree requirements. However, in reality, L. sericata of any one life cycle stage show a positively skewed range of day degree requirements that can be described by a Weibull function (Table 1). Similar distributions in insect populations have been observed by other authors (Sharpe et al., 1977; Phelps et al., 1993). To incorporate the variability observed within the population into the stochastic model, each cohort is assigned a random day degree requirement for each of its life cycle stages, generated from the appropriate Weibull curve (Table 1). This value is retained throughout the duration of that life cycle stage, so that slow developing cohorts remain slow, and fast developing cohorts remain fast (Phelps et al., 1993). The day degree requirements for each stage of development are assigned independently of the requirements of the previous stages and, furthermore, the day degree requirements of each offspring cohort are assigned independently of the development rates of the parent cohort. This process of assigning day degree requirements means that there is random variation between cohorts based on empirically determined functions and the stochastic model,
400
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25 35 Number of runs
45
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F. 1. The standard error of the mean number of individuals predicted to be present at days 100 (—Q—), 150 (— + —), 200 (—(—) and 250 (—q—) (where day 1 = 1st February) by a stochastic model of the seasonal abundance of the blowfly Lucilia sericata. For each point, the model was run a number of times with a standard set of parameters. Each set of runs was repeated 20 times and the standard error of these 20 average predictions plotted against the number of runs in each set.
Using these models, a series of simulations were carried out, varying the base temperatures and day degree values, to determine the relative importance of each on the observed patterns of blowfly abundance. The effect on the number of adult flies to emerge in a particular generation was assessed by averaging the number of flies of that generation present on each day. To quantify the effect on the timing of a generation, each day number on which adults of a particular generation was present was weighted by the number of individuals present on that day. These were then averaged over all days on which adults of the generation were present, to produce the mean time to the mid-point of each generation.
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The output of the stochastic model (Fig. 2) also predicts the emergence of four generations but these are less distinct with individuals of each generation present both earlier and later than the predictions of the deterministic model. This difference is particularly marked early in the season when temperatures are relatively low. The small number of adults present at about day 75 was due to a cohort emerging particularly early by the chance allocation of low day degree requirements. However, this cohort emerged so early that the temperatures were too cold to support adult individuals (less than 10°C), so the cohort died before it had the chance to oviposit. Sensitivity Analysis
Results The output from the deterministic model predicts that the adults emerge in discrete waves, with fly numbers increasing exponentially with each wave (Fig. 2). The peaks in fly abundance occur at about days 100, 150, 180 and 210. A small number of individuals form a fifth generation, but the majority of offspring from the fourth generation undergo diapause. This pattern of output is similar to that of a previous deterministic model developed for L. sericata (Wall et al., 1992), which has been shown to be a good description of the changes in abundance of wild populations of L. sericata in sheep pastures (Wall et al., 1993a).
The day degree requirements of all life cycle stages were varied simultaneously by up to 240% of the value calculated from laboratory data (Table 1). For clarity, only the effect on the third and fourth generations is shown because these contain the greatest number of individuals. Regressing the mean time to mid-point of each generation against the percentage change in the day degree requirement, shows that variation in the day degree requirement has a significant influence on the timing of each generation (Fig. 3). The mean time of emergence of the third and fourth generations increases linearly with the number of day degrees required to complete
10000
Fly number + 1
1000
100
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1 0
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150 Day number
200
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F. 2. The number of individuals (+1) of the blowfly Lucilia sericata predicted to be present each day (where day 1 = 1st February) by deterministic (dashed line) and stochastic (continuous line) models.
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number of individuals present in each generation (P Q 0.01). However, if the day degree requirements are reduced by over 20%, the numbers do not increase to such high levels as the deterministic model predicts. This is because an increasing proportion of the 40 runs of the stochastic model become extinct by the second generation due to individuals emerging early in the season, before the ambient temperatures are high enough to support reproducing adults. As a result, as the day degree requirements are reduced, the mean number of adults present in the fourth generation is progressively lower than that predicted by the deterministic model (Fig. 4).
240
Mean time (days)
220
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–40
–30 –20 –10 0 10 Day degree adjustment (%)
20
F. 3. The mean time (days) to the mid-point of the third (filled squares) and fourth (open squares) generations of the blowfly Lucilia sericata following adjustment of the day degree requirements in deterministic (dashed lines) and stochastic (continuous lines) simulation models.
the life cycle. The predictions of the deterministic and stochastic models are similar, both of which show that each 10% increase in the day degree requirement of all life cycle stages increases the mean time to mid-point of the third generation by approximately 5.5 days (P Q 0.001), and the mean time to mid-point of the fourth generation by approximately 8 days (P Q 0.001). Both models show that varying the day degree requirements has a significant effect on the number of flies present in each generation. As the number of day degrees required to complete the life cycle is reduced, so the mean number of individuals present in both third and fourth generations increase (Fig. 4). This is because with lower day degree requirements a greater number of individuals survive to oviposit a greater number of times. In the deterministic model, this relationship is highly significant for both the third and fourth generations (P Q 0.001), up to a 20% reduction in day degree requirements. As the day degree requirements are reduced by over 20%, the population in the fourth generation reaches an upper limit of approximately 17000 individuals, a maximum imposed by the limiting effect of the number of oviposition sites available. The stochastic model predicts a weaker, but still significant increase in the
18 000
15 000
Mean fly number
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Both deterministic and stochastic models were run with the base temperatures for each stage reduced by up to 25% of the value calculated from laboratory data (Table 1). The results of the simulations show that as the base temperatures for each stage are decreased, development can take place at lower temperatures and the mean time to mid-point of the third and fourth generations is reduced (Fig. 5). Once again the two models are in close agreement, both predicting that for each 10% decrease in the base temperature of all life cycle stages there is
12 000
9000
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0 –30 –20 –10 10 Day degree adjustment (%)
20
F. 4. The mean number of the blowfly Lucilia sericata present at the mid-point of the third (filled squares) and fourth (open squares) generations following adjustment of the day degree requirements in deterministic (dashed lines) and stochastic (continuous lines) simulation models.
increase in numbers with reduction in base temperature is significant (P Q 0.05).
220
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160 75
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F. 5. The mean time (days) to the mid-point of the third (filled squares) and fourth (open squares) generations of the blowfly Lucilia sericata following adjustment of the base temperature (°C) in deterministic (dashed lines) and stochastic (continuous lines) simulation models.
an approximately 11.5 day decrease in the time to mid-point of the fourth generation (P Q 0.001), and an 8 day decrease in the time to mid-point of the third generation (P Q 0.01). Reduction in the life cycle base temperatures has the effect of increasing the mean number of individuals present in both third and fourth generations (Fig. 6). This is because lower base thresholds allow development to occur at lower ambient temperatures and, as a result a greater number of individuals reach oviposition and survive to oviposit a greater number of times. In the deterministic model, this increase in numbers continues up to a 20% decrease in base temperatures (P Q 0.01). If the base temperatures are reduced by over 20%, the population in the fourth generation reaches its maximum size due to the limiting effect of the number of oviposition sites available. In contrast, as the base temperature is reduced, there is an increased likelihood of each run of the stochastic model going to extinction, due to early emergence of the adults. Once again, therefore, with decreasing base temperatures the mean number of individuals in the third and fourth generations predicted by the stochastic model are progressively smaller than those predicted by the deterministic model, although the
For the stochastic model, the effects of changing the permissible range of variation in day degree requirements for each life cycle stage was examined. The mean day degree requirement for each stage was maintained at its calculated value (Table 1) and the a parameter of the Weibull distribution was altered so that the variation in day degree requirements could be changed by up to 260%. The effect on the mean number of adults in each generation and the timing of each generation were noted, as was the standard error of these means. Changing the variation in day degree requirements between cohorts had no effect on the mean number of adults in each generation, or on the mean time to mid-point of each generation, or on the standard error of the mean number of flies per generation. However, regression shows there is a slight but significant effect on the standard error of the mean time to mid-point of the third generation (P Q 0.001) and fourth generation (P Q 0.01) (Fig. 7). As the variation in day degree requirements becomes greater,
16 000
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Mean time (days)
145
8000
4000
0 75
80
85 90 Base adjustment (%)
95
100
F. 6. The mean number of the blowfly Lucilia sericata present at the mid-point of the third (filled squares) and fourth (open squares) generations following adjustment of the base temperature (°C) in deterministic (dashed lines) and stochastic (continuous lines) simulation models.
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Standard error
1.25
1.05
0.85
0.65
0.45 –60
0 40 –40 –20 20 Change of normal variation (%)
60
F. 7. The standard error of the mean time to the mid-point of the third (filled squares) and fourth (open squares) generations of the blowfly Lucilia sericata following adjustment of the variance in day degree requirements in a stochastic simulation model.
the more the stochastic model will differ from the deterministic model, and so the standard errors of the mean timing of both generations 3 and 4 increase. Discussion Using empirically derived relationships between temperature and development rate, a deterministic simulation model has been developed and used to predict the abundance of the sheep blowfly, L. sericata, throughout a season, from an initial overwintering cohort. A similar model has been shown to explain a significant proportion of the variance in seasonal abundance in the field (Wall et al., 1992; Wall et al., 1993a). A stochastic version of the model, which uses a Monte-Carlo simulation technique to assign development rates at random to each cohort in the simulation, was also developed. These random rates are generated from a Weibull distribution fitted to the observed variation in development rates for each life cycle stage in constant temperature experiments. Each run of the stochastic model produces a slightly different prediction, due to the random element of the model which, when averaged across a number of runs, provide a prediction of mean abundance. The deterministic model typically predicts four discrete waves of emergence of L. sericata adults,
with the fifth generation limited by the onset of diapause. The stochastic model generates a similar predicted pattern but with less discrete generations that more closely resembles the pattern of blowfly abundance observed in the field (Wall et al., 1993a, b). The stochastic model has the further advantage of predicting a range of days on which emergence occurs, with the first adult emerging several days before that of the deterministic model. If control was to be aimed at the first day of emergence, some flies would be missed if prediction was based on the deterministic model alone (Wall et al., 1995). In sensitivity analysis both models suggest that both the base threshold temperatures and the day degree requirements significantly affect the timing of generation peaks. Reducing the base temperature of each life cycle stage allows development to occur at lower temperatures and greater numbers of day degrees to be accumulated each day. Therefore each cohort reaches the next life cycle stage earlier and overall each generation would occur at an earlier date. Similarly, lowering the number of day degrees that need to be accumulated by any stage decreases the time spent in that stage, so that the next generation of adults occurs earlier in the season. Both models show approximately the same rate of change of generation time with varying base temperatures and day degree requirements. Both models also show that the base threshold temperatures and the day degree requirements significantly affect the number of flies present in each generation. By accelerating the rate of development by one of these methods, adult females spend less time maturing their ovaries and so are more likely to oviposit before they die. This results in each generation consisting of the product of more ovipositions than if a slower development rate was assumed. As a result, the number of adults present in each generation increases as the development rate increases. However, reducing the day degree requirements of the base threshold temperature in the deterministic model allows the population to reach its carrying capacity by the fourth generation. This is not the case in the stochastic population because of the increasing number of cohorts which, through the random allocation of low base temperatures or low day degree requirements, emerge and die in Spring before seasonal temperatures are high enough to sustain adult development. The sensitivity analysis suggests that variations in base temperatures and/or day degree requirements may have very different consequences for the observed pattern of blowfly abundance. Both models show that the mean time to mid-point of the third
and fourth generations is highly sensitive to the precise base temperature used. A decrease in base temperature of 1°C would result in an approximately 11 day decrease in the time to mid-point of the fourth generation, and an 8 day decrease in the time to mid-point of the third generation. Such errors are in the range of those observed around the means calculated from the experimental data (Table 1). This observation is of particular importance because the base thresholds for each stage used in the present study, calculated from the observation of development rates in constant temperature experiments, may be inaccurate due to the non-linearity of the temperature–development rate relationship at low temperatures. Hence, it is possible that the base thresholds used in the models are actually over-estimations of the true base temperatures. Such errors may have critical effects when precise timing of pest control is required. In constrast, the results presented here suggest the sheep blowfly system may be far less sensitive to variations in day degree requirements than to variation in base temperature. The pre-adult larval stages that occur off the host require a total of 172 day degrees over a base temperature of 9°C. Both the deterministic and stochastic models show that deviations of up to 20 day degrees are required to increase the mean time to mid-point of the third and fourth generations by approximately 5 and 8 days, respectively. Such errors are considerably greater than the range of variation observed in the experimental data (Table 1). It has been suggested by various authors that development rates measured in constant temperature incubators do not correspond to development rates under the fluctuating temperatures in the field (Clements, 1963; Higley et al., 1986; Hagstrum & Milliken, 1991). In contrast, however, Dallwitz (1984) showed that L. cuprina had similar development rates under constant and fluctuating temperatures, up to 30°C. Such high temperatures would rarely be experienced by L. sericata in the field in northern Europe. This, together with the apparent lack of sensitivity of the abundance pattern to minor variations in day degree accumulation, indicates that the use of day degree requirements estimated in constant temperature incubators in the simulation models may not be an unwarranted oversimplification. Understanding the driving forces behind the population dynamics of L. sericata is only the first step towards managing the sheep strike system as a whole. There are a large range of factors that determine the impact of the fly population on farms
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around Britain, such as other climatic variables, geographical factors and farm management practices. However, the models presented here represent an essential stage in this process allowing the effects of stochasticity and the sensitivity of the blowfly system to variation in base temperatures and day degree requirements to be considered. The support of a Royal Society University Research Fellowship to R. Wall and a LINK grant, supported by Bayer Ltd and the Ministry of Agriculture, Fisheries and Food of the U.K. government are gratefully acknowledged. REFERENCES A, T. E., K, G. G. & S, R. E. (1982). Temperature-dependent model for the postdiapause development and Spring emergence of the European Corn Borer, Ostrinia nubilalis (Hu bner) (Lepidoptera: Pyralidae), in North Carolina. Environ. Entomol. 11, 1307–1311. A, M. G., M, D. G., S, J. P., A, K. A. & B, D. G. (1994). Comparison of the predicted impact of a screwworm fly outbreak in Australia using a growth index model and a life-cycle model. Med. Vet. Ent. 8, 281–291. B B, L., G, A. C. M. & V, W. G. (1990). Readiness to lay in the blowfly Lucilia cuprina is unaffected by oviposition site-deprivation or egg-load. Entomol. Exp. Appl. 55, 33–40. B, S. D. (1983). Insect thermoperiodism. Annu. Rev. Entomol. 28, 91–108. B, J. & G, D. (1993). Experimental assessment of a degree-day model for predicting the development of parasites in the field. J. Appl. Ent. 116, 459–466. B, T. V. (1995). Extinction rates in archipelagos: Implications for populations in fragmented habitats. Conserv. Biol. 9, 527–554. C, A. N. (1963). The physiology of mosquitoes. Int. Monogr. Pure Appl. Biol. 17, 72–79. C, J. B. (1955). The natural history of sheep blowflies. Ann. Appl. Biol. 42, 197–207. D, R. (1984). The influence of constant and fluctuating temperatures on development rate and survival of pupae of the Australian sheep blowfly Lucilia cuprina. Entomol. Exp. Appl. 36, 89–95. D, L. (1948). Observations on the development of Lucilia sericata (Mg.) eggs in sheep fleeces. J. Exp. Biol. 25, 86–102. D, W. M. (1934). The sheep blowfly problem in North Wales. Ann. Appl. Biol. 21, 267–282. D, B., K, W. P. & B, R. C. (1986). Stochastic model of insect phenology: Estimation and testing. Environ. Entomol. 15, 540–546. E, A. V. (1936). The physiology of the sheep blow-fly Lucilia sericata Meig. (Diptera). Trans. Roy. Ent. Soc. 85, 363–378. F, G. A. (1993). Life history evolution and demographic stochasticity. Evol. Ecol. 7, 1–14. F, N. P., W, R., C, K. L. & M, K. L. (1992). Prevalence, regional distribution and control of blowfly strike in England and Wales. Vet. Rec. 131, 337–342. G, G. & G, W. P. (1980). Disease incidence forecasts by means of climatic data. Biometeorology 7, 87–103. G, V. H. (1988). Ammonia toxicity and alkalosis in sheep infested by Lucilia cuprina larvae. Int. J. Parasit. 18, 79–81. H, D. W. & M, G. A. (1991). Modeling differences in insect developmental times between constant and fluctuating temperatures. Ann. Ent. Soc. Am. 84, 369–379. H, J. M. (1976). Deterministic and stochastic models simulating the growth of insect populations over a range of temperatures under Malthusian conditions. Can. Ent. 108, 907–924.
148
. E T A L .
H, L. G., P, L. P. & O, H. R. (1986). DEGDAY: A program for calculating degree-days and assumptions behind the degree-day approach. Environ. Entomol. 15, 999–1016. L, D. M., B, N. D. & V, A. (1992). A model for Nematodiasis in New Zealand Lambs. Int. J. Parasit. 22, 789–799. L, T. J. & A, R. C. (1987). A simulation model of house fly (Diptera: Muscidae) development in poultry manure. Can. Ent. 119, 427–437. M, R. D. (1983). Simulating developmental time of preadult face flies (Diptera: Muscidae) from air temperature records. Environ. Entomol. 12, 943–948. P, F. L., A, J. C. & B, C. S. (1991). Degree day model for vegetable leafminer (Diptera: Agromyzidae) phenology. Environ. Entomol. 20, 1134–1140. P, K., C, R. H., R, R. J. & F, S. (1993). Monte Carlo simulation method for forecasting the timing of pest insect attacks. Crop Prot. 12, 335–342. P, K. P. (1983). Day-degree methods for pest management. Environ. Entomol. 12, 613–619. S, D. S., MP, J. N. & C, K. D. (1985). Maternal and larval effects of photoperiod on the induction of larval diapause in two species of fly, Calliphora vicina and Lucilia sericata. Exp. Biol. 46, 51–58. S, P. J., C, G. L., DM, D. W. & C, C. L. (1977). Distribution model of organism development times. J. theor. Biol. 66, 21–38.
W, T. L., W, H., S, P. J., S, R. M. & C, R. N. (1984a). Modeling insect development rates: A literature review and application of a biophysical model. Ann. Ent. Soc. Am. 77, 208–225. W, T. L., W, H., S, P. J. & C, R. N. (1984b). Modeling distributions of insect development time: A literature review and application of the Weibull function. Ann. Ent. Soc. Am. 77, 475–487. W, R. (1992). The reproductive output of the blowfly Lucilia sericata. J. Insect Physiol. 39, 743–750. W, R., F, N. & M, K. L. (1992). Effects of temperature on the development and abundance of the sheep blowfly Lucilia sericata (Diptera: Calliphoridae). Bull. Ent. Res. 82, 125–131. W, R., F, N. P. & M, K. L. (1993a). Predicting the abundance of the blowfly Lucilia sericata (Diptera: Calliphoridae). Bull. Ent. Res. 83, 431–436. W, R., F, N. P. & M, K. L. (1993b). Sheep blowfly population control: Development of a simulation model and analysis of management strategies. J. Appl. Ecol. 30, 743–751. W, R., F, N. P. & M K. L. (1995). Population suppression for control of the blowfly Lucilia sericata and sheep blowfly strike. Ecol. Ent. 20, 91–97. W, J. Y. (1960). A critique of the heat unit approach to plant response studies. Ecology 41, 785–790.