How predictable are aphid population responses to elevated CO2?

Authors


J. A. Newman, St. Peter's College, New Inn Hall Street, Oxford, OX1 2 DL, UK (fax +44 1865278855, e-mail: jonathan.newman@zoo.ox.ac.uk).

Summary

  • 1Experiments investigating the population responses of aphids to CO2 enrichment have yielded results suggesting that aphid populations will be both larger under elevated CO2 and that they will be smaller under elevated CO2. Most studies have failed to reject the null hypothesis of no difference in population sizes due to atmospheric CO2 concentration. This diversity of results has led some investigators to conclude that aphid responses are not general, and that every aphid–plant interaction may be unique and unpredictable a priori. We use a single, general, mathematical model to consider the population responses of cereal aphids to grass grown under different CO2 concentrations.
  • 2The model shows that it is possible to explain any of the three observed results: larger populations, smaller populations, or no difference, and that which of these three outcomes arises may depend critically on the interaction between aphid nitrogen requirements and the nitrogen fertility of the soil. The model also shows that the qualitative results will depend on how sensitive the aphid species is to increases in its own density. Past studies have shown that aphids increase their production of winged offspring in response to increasing aphid density. The model predicts that, in general, aphid species that have lower nitrogen requirements and that are less sensitive to their own density will be more likely to have larger populations in elevated CO2 compared to ambient CO2.
  • 3Differences between aphid species (and clones) in their nitrogen requirements and the strength of their density-dependent response have not been widely reported in the literature. Also, the nitrogen fertility of the soil has rarely been manipulated in experiments on aphid responses to rising CO2 levels. The model suggests that the diversity of population responses of aphids may be both understandable and predictable in the context of such an interaction.

Introduction

Global change ecology has progressed from asking questions about plant responses to elevated atmospheric CO2 concentrations (denoted [CO2]e) to asking questions about the effects of CO2 concentration (denoted [CO2]) on the wider community and at more than one trophic level. Studies of the effects of [CO2], as moderated by the plant, on herbivores have begun to accumulate. The most thorough review of this literature (Coviella & Trumble 1999) combines the results from approximately 80 studies of species from 22 families spread over 8 orders (see also Bezemer & Jones 1998; Whittaker 1999, 2001). The review concludes that the effect of [CO2]e will not only be highly species-specific, but also specific to each insect–plant system. This conclusion is sometimes summarized by saying that insect–plant interactions may be ‘idiosyncratic’.

Following the same theme as Coviella & Trumble, the title of Bezemer et al.'s (1999) paper asked ‘How general are aphid responses to elevated atmospheric CO2?’ They conducted a study on the impact of [CO2]e on two aphid species feeding on a series of host plants. Among the conclusions they reached were that long-term population responses to [CO2]e cannot be reliably predicted from detailed measurements on individual aphids and that aphids show speciesspecific responses to [CO2]; the same suggestion was recently echoed by Hughes & Bazzaz (2001).

Before we reach the conclusion that aphid–plant interactions are idiosyncratic, we need to reconsider what sort of generality we might seek in the results of such studies. At one end of the continuum would be the generality that all aphid species have similar, e.g. all larger (or smaller), populations under [CO2]e. At the other end of this spectrum would be the conclusion that every single aphid species by plant species interaction will be different, i.e. there are no generalities. Somewhere in between, however, is the conclusion that aphid responses to [CO2]e might be widely different, but understandable in terms of a single general mechanism of response. We would have simply to find the mechanism(s) by which to recognize the generality. An example of just such a case was presented by Thornley & Cannell (2001) in the context of soil carbon responses to a global rise in temperature.

In this paper we use a mathematical model to explore the question of predictability and generality in the response of aphids to [CO2]e. We reach the conclusion that aphid population responses may be predictable in the context of a general interaction between soil N fertility and the aphids’ N requirement and density-dependent response in winged morph production.

In order to set the model in context, we first review the literature on aphid responses to [CO2]e. Are all aphid responses to [CO2]e in the same direction (i.e. always larger or always smaller)? We agree with Bezemer et al.'s (1999) conclusion that the simple answer to this question is ‘no’. We have reviewed 10 refereed research papers that look at 12 species of aphid feeding on 18 species of plants under four different levels (including ambient) of [CO2]. The results of these studies are summarized in Table 1. Whittaker (1999) conducted a similar review of 10 aphid species concluding that ‘two have shown a significant increase, two a non-significant increase, two faster reproduction or development times, and four show population decrease or no change (Whittaker 2001).’ Of the 25 results in our review, in which there were no statistically significant differences, undoubtedly some will have failed to reject the null hypothesis because the null hypothesis was true, and some may have failed to reject the null hypothesis because they lacked sufficient statistical power to do so, even though the null hypothesis was false (i.e. a type II error). But without formal analyses of the statistical power of the tests, it is difficult to interpret these results. Of the 14 results that clearly indicate a response, only seven come from freely growing populations. The remaining seven results investigated some aspect(s), or parameter(s) of population growth. One possible reason behind Bezemer et al.'s (1999) conclusion that ‘that long-term population responses to elevated CO2 can not be reliably predicted from detailed measurements on individual aphids’ is that not all the studies will have looked at sufficient parameters to predict long-term population growth. For example, knowing that the daily rate of nymph production is higher under [CO2]e, as Awmack, Harrington & Leather (1997) found, cannot by itself be expected to predict the total aphid response. For that we would need both mortality and fecundity rates (and potentially density-dependent responses such as differences in morph production). Furthermore, of the 14 cases in which the aphids showed a statistically significant response (positive or negative) only six of these also demonstrated a statistically significant response by the host plant. In the other cases, either the plant response was not investigated, or the study lacked sufficient statistical power to detect plant performance differences (or, possibly, the null hypothesis was true).

Table 1.  Published effects of [CO2]e on aphid populations. The results are divided by response variable and cross classified by the statistical significance of the aphid response, and by the statistical significance of the plant response (the suggestion being that we might have different expectations of the aphid response in cases where there was not a significant plant response). The table shows that there are nine results that, all other things being equal, suggest larger aphid populations under [CO2]e (sum of column two), five results that suggest smaller aphid populations under [CO2]e (sum of column three) and 25 results that showed no statistically significant effect on the aphid populations (sum of column four). Note that these 39 results are not independent as they come from only 10 papers (Thompson, Brown & Woodward 1993; Salt, Fenwick & Whittaker 1996; Awmack et al. 1997; Docherty et al. 1997; Bezemer, Jones & Knight 1998; Diaz et al. 1998; Bezemer et al. 1999; Newman et al. 1999; Whittaker 1999; Hughes & Bazzaz 2001)
Plant response to [CO2]eAphid population response to [CO2]e
Significant increaseSignificant decreaseNo significant change
 Fecundity
Significant response011
No significant change215
 RGR or development rate
Significant response004
No significant change202
 Intrinsic rate of increase (rm)
Significant response000
No significant change101
 Population size
Significant response238
No significant change204

We pursue these issues, and explore the kind of information needed to consider the generality of response of aphid populations to [CO2], using a model.

The model

The model is a system of coupled differential equations based upon that of Johnson & Thornley (1984, 1985). This is a mechanistic and stochiometrically complete model of carbon (henceforth denoted C) and N growth physiology for a temperate grass. It comprises 15 state variables (differential equations): four age classes of root mass, shoot mass and leaf area index (LAI), C substrate, N substrate and soil N. The model has five driving variables: temperature, day length, light intensity; N input rate, and [CO2]. This model and its more detailed successors (collectively known as the Hurley Pasture Model) have been widely explored and the responses of the plants to these driving variables are well documented (Thornley 1998; references therein).

The plant model is coupled, here, to an aphid population dynamics model (see Appendix) to explore the response of aphids to [CO2] (as moderated through the plant, see also Thornley, Fowler & Cannell 1991; Thornley & Cannell 1997). The aphid submodel comprises 10 coupled differential equations. It is a stage based model of the two different aphid morphologies (alate and apterous), each with four juvenile instars and one adult stage.

The joint model thus comprises two dynamically linked submodels: the plant submodel and the aphid submodel. These submodels are dynamically linked such that the aphid population growth depends upon the amount of C and N in the plant substrate pools and aphids remove mass from these pools through feeding. Plant growth and resource allocation also depends upon the mass of C and N in these substrate pools.

Both submodels, and thus the joint model, lack entirely any spatial structure (see Parsons, Schwinning & Carrére 2001 for discussion of related spatially explicit models). Parameters and results are represented on a m−2 basis. The model should be thought of as characterizing local grassland populations of aphids rather than more regional responses. Note also that the model is presented without aphid predators or parasites. This matches experiments in which the aphids are conventionally caged and are intentionally not subjected to these higher trophic influences. See Appendix (model realism and model sensitivity sections) for further discussion of the assumptions and limitations of the model.

Illustrations of model behaviour

Illustrations represented in this section use variable temperature, day length and light intensity based on seasonal means for central and southern Britain (latitude 51·54° N, elevation 50 m); see Thornley (1998) chapter 7.

effects of the environment: n × [co2]interaction

Figure 1 shows the predicted change in population size through a spring and summer, starting from 5 initial aphids, plotted against Julian date (day of the year, equivalent to 31 May to 28 September). Under ambient CO2 concentrations (henceforth [CO2]a) we see that as the soil fertility rises relative to the N requirement of the aphid species, aphids become less limited by N and more limited by other processes (mainly density, which both reduces plant quality and increases alate production). Under [CO2]e the effect is the same, but a comparison of the [CO2]a and [CO2]e results suggests that populations will be smaller under [CO2]e than under [CO2]a, and that the degree of reduction depends on the N requirement of the aphid and the N input into the system.

Figure 1.

Model results showing the effects of varying N requirements and [CO2]. N requirements are depicted relative to that estimated for R. padi. In each figure, from top to bottom, the lines represent N requirements from 10% below to 10% above the R. padi estimate (i.e. 0·9–1·1 in steps of 0·05; 0·9 and 1·1 are depicted in bold lines for clarity). Results are for (a) [CO2]a (360 p.p.m.) with no additional N fertiliser; (b) [CO2]a but with the addition of 20 kg N ha−1 per month of fertiliser; (c) [CO2]e (700 p.p.m.) and no additional N; and (d) [CO2]e and 20 kg N ha−1 month−1 fertiliser.

In Fig. 2, for a fixed N requirement (cf. a given aphid species) we plot the ratio (relative to [CO2]a = 360 p.p.m.) of final population size in relation to [CO2] and N input. The isocline of 1 indicates combinations of [CO2] and N input that result in final populations (at day 270) that are the same size as if they were grown in [CO2]a under the same N input. Isoclines < 1 indicate combinations in which the final population sizes are smaller under [CO2]e and isoclines > 1 indicate combinations where the final population sizes are larger. The figure suggests that for low N input, increasing the [CO2] decreases aphid population size but for higher N input, increasing the [CO2] increases the population size.

Figure 2.

Population responses relative to [CO2]a = 360 p.p.m. The contours represent the ratio of final population size (day 270) under the xy combination to the final population size under (x, 360 p.p.m.). The x-axis is expressed in kg N m−2. N applied every 15 days; aphid N requirement is 1.

differences among aphid species

In the model, aphid morph differentiation is affected by three parameters: temperature, plant quality and the aphid species’ response to crowding (aphid density). Because temperature is a driving variable and plant quality is determined dynamically in the model, we consider only the effects of the density-dependent response inline image, see Appendix, eqn 1c). This parameter represents the maximum tolerated aphid density (m−2) by which all offspring produced are alate (winged) morphs. Assuming a tiller density of 2000 tillers m−2, we can easily convert this parameter to the more familiar aphids tiller−1 and this latter quantity is displayed in Fig. 3. inline image is likely to be highly species-specific, and may even vary substantially among clones (see also Müller, Williams & Hardie 2001). Figure 3 shows that, in general, aphid species that have lower N requirements and weaker density-dependent responses in morph production (i.e. are more tolerant of crowding) are more likely to have larger populations under [CO2]e than under [CO2]a. Also, the more N we add to the system, the more likely aphid populations will be larger under [CO2]e regardless of the species’ N requirement or density-dependent response.

Figure 3.

Effects of the aphid density-dependent response. The x-axis is the maximum aphid density that the species can tolerate before all offspring produced are alates (winged). In the model, this is represented by inline image. The x-axis can be translated into inline image by multiplying by 2000 tillers m−2 (assuming that as an average figure) to yield aphids m−2. The isoclines show combinations of aphid N requirement and aphid density-dependent response for which population sizes are the same in [CO2]e (700 p.p.m.) and [CO2]a (360 p.p.m.). The isoclines represent different N inputs (AN, kg m−2 per 15 days). In general, the weaker the aphids’ density-dependent response and the lower the aphids’ N requirement, the more likely it is that populations will be larger under [CO2]e. Regardless of the aphids’ N requirement and strength of its density-dependent response, the more N fertiliser that is added to the soil, the more likely that the populations will be larger under [CO2]e.

Discussion

The model predicts that any of the three outcomes in response to [CO2] enrichment are possible: larger, smaller, or similar sized populations. Which outcome occurs depends critically on the interaction between the rate of N input to the soil, and the N demand and density-dependent response of the aphid species. We suggest that such an interaction may explain the diversity of results in the literature on aphid responses to [CO2]e. This result extends the work of others, for example by Cannell & Thornley (1998), who have demonstrated that grassland ecosystem responses to [CO2]e depend critically on the N status of the soil.

revisiting the literature

Different aphid species are represented in this model by changing the N requirements, and the aphids’ density-dependent response in alate production, inline image (Appendix, eqn 1c). Both of these parameters have the potential to alter the qualitative predictions of the model (e.g. whether populations will be larger or smaller under [CO2]e). Because both will probabably vary among aphid species and clones, and are rarely measured, it is perhaps not surprising that the phenomena reported in the literature suggest no single (simple) response of aphids to [CO2]e. Rather, the model suggests that comparisons of aphid population sizes in elevated compared to [CO2]a may result in any possible conclusion. It would be nice to use the literature to test these qualitative predictions of the model, but this is problematic because the two predicted mechanisms, N requirement and density-dependent responses, are not reported (and indeed are probably unknown) in the published studies of aphid responses to [CO2].

In place of this deficiency of information, our model is based on some basic assumptions about nutrient demand and aphid population dynamics. To test the qualitative predictions of any such model, it is essential to have estimates of the N requirements of specific aphid species as well as N input to the soil/plant system, and yet specific aphid N requirements are poorly understood or reported in the literature. The parameter estimate on which we based the N requirement in the model (and then varied it to explore its importance), was derived from studies of the developmentally early nutritional requirements of the pea aphid (Prosser, Simpson & Douglas 1992; Abisgold, Simpson & Douglas 1994; Simpson, Abisgold & Douglas 1995), for we could find little information in the literature on any other aphid. Perhaps more surprisingly and importantly, experiments on aphid population growth in [CO2]e rarely state the N input used, and in any case do not vary these inputs (of the studies summarized in Table 1, only Diaz et al. (1998) and Docherty et al. (1997) experimentally vary these inputs); embarrassingly, this is also true of our own experimental study (Newman et al. 1999).

Data on density dependence in aphid morph production is even more elusive than aphid N requirements. Several researchers have experimentally demonstrated such density-dependent responses in morph production, as depicted here and in particular for R. padi (e.g. Johnson 1965; De Barro 1992). Müller et al. (2001) have thoroughly reviewed this literature and suggest that aphid responses to density can be quite variable (and sometimes quite weak), and yet we predict that the qualitative results of population growth studies may critically depend on this parameter.

testing the model

One complaint or criticism of this model will be that it is not very useful for experimental situations as aphid N requirements and density-dependent morph production responses will rarely if ever be known, and are difficult to estimate. The model suggests that, despite the difficulty involved in their estimation, these parameters are absolutely critical to our understanding of the interaction between aphids and plants in [CO2]e. That is, our a priori prediction of population responses will be hampered as long as we fail to gain a strong grasp on these two parameters. In fact, the model implies that it will be fruitless to study this interaction without at least experimentally varying the N input, in each and every experiment, regardless of our specific scientific interest in this hypothesis.

So how can we estimate aphid N requirement for use in a model such as this? Perhaps the best approach, if the most challenging, is something along the lines of Abisgold et al. (1994) in which artificial diets were constructed and the response surface of aphid growth to sugar and amino acids was estimated. However, this technique is likely to be too time consuming to be generally useful. We suggest that it will be possible to estimate both the aphids’ N requirement and the shape of the function relating performance to plant quality by generating data on aphid performance (e.g. estimates of their intrinsic rate of increase, rm and plotting these as a function of plant N substrate concentrations (i.e. non-structural N).

The density dependence of morph production in aphids is not conceptually difficult to estimate. Throughout, we model the aphid response to ‘local’ density (e.g. density per tiller, as in the experiments of Johnson (1965) and De Barro (1992)) but express that response on the scale of m−2. As we point out in the Appendix, it would be incorrect to try and parameterize this model by measuring the aphid density response on a m−2 basis, unless one assumes that aphids are homogeneously distributed in space – an assumption that we do not make. Aphid density responses could be measured following previous experimental protocols. What is called for now is for the same protocol to be applied to a number of aphid species (and/or clones) under the same controlled experimental conditions, to assess the degree to which this parameter does vary among species (or clones). Additionally, from a modelling perspective, the emphasis of such experiments must move away from testing the null hypothesis of no effect and toward the estimation of the precise functional response.

general conclusions

Our approach has been to abstract what we see as the fundamental mechanisms of the aphid–plant interaction, and to consider the dynamic consequences of these mechanisms. In essence, we have generated a testable hypothesis, and in testing this hypothesis we may find out that it is wrong. However, by proposing an hypothesis based on some known mechanisms of aphid nutrition and population growth, we are at least able to highlight what studies might be necessary to begin to explain, and so to understand, the diversity of aphid responses to [CO2]e. Identifying a mechanism by which to see aphid responses as ‘general’ (as opposed to ‘idiosyncratic’) is surely a valuable goal, as a general explanation can greatly streamline our progress in identifying [CO2] impacts, i.e. we would not have to study every aphid–plant system! General explanations, such as the one proposed here, are what society demands in order to assess mitigation policy options and necessities.

Acknowledgements

We thank Steve Simpson, Julie Abisgold, Caroline Awmack and Richard Harrington for extremely useful discussions of aphid nutrition and growth. Joy Bergelson also provided valuable discussion. We appreciate the comments of Dave Raffaelli and two anonymous referees.

Appendix

aphid submodel

Others have used simulation modelling techniques to investigate aphid population growth. The most commonly used approach seems to be that of Carter and collaborators (summarized in Carter 1985; references therein). We chose to write our own aphid submodel for three main reasons. First, Carter's aphid population model is based on physiological time rather than chronological time which did not fit well with our choice of plant growth submodel. We model the same phenomenon by making the aphid vital rates functions of both time and temperature (as discussed below). Second, additional to the problems associated with the use of physiological time, Carter's model is designed to work with a phenomenological rather than mechanistic plant growth model. Lastly, in Carter's model there is no dynamic linkage between plant quality and aphid population growth. While Carter's model has been shown to produce a good fit to some data sets, its use of statistically based functions rather than mechanistic functions make it better suited to problems of pest forecasting on crops and not as well suited to hypothesis generation or theoretical exploration of mechanistic explanations. We wish to stress that the aphid model that we generated is not particularly novel, it is simply better suited to the research goals than other available models. We placed less emphasis on accurate numerical prediction (as is necessary with pest forecasting) and more emphasis on producing a general model of aphid population dynamics that is qualitatively accurate and that would be suitable for making cross species comparisons and generating mechanistic hypotheses. Below we describe the submodel.

The aphid submodel is a stage-based model of aphid population dynamics. The submodel is linked dynamically to the plant submodel via the substrate pools of C and N as these pools are assumed to represent the quality of plant forage available to the aphid. Aphid feeding reduces the quantity of both C and N substrate, which is then unavailable to the plant for growth and maintenance.

The aphid model was motivated by our previous experimental work with Rhopalosiphum padi (L.) (the bird cherry-oat aphid Newman et al. 1999), and wherever possible parameter estimates are based on this species. However, we propose that the model is simple and fundamental, and so by varying some or all of the aphid parameters we can represent aphid species generally. Here we consider population growth only during the spring and summer, and thus consider only asexual reproduction.

Fecundity rate

De Barro (1992) has shown that fecundity depends mainly on temperature, and only weakly, to not at all, on plant quality and aphid density. We follow Thornley (1998,§3·11) in modelling the temperature response as a sigmoidal cubic function (which takes it's name from the value of q in eqn 1a, where by default q= 2, which produces a cubic function in T) which is mathematically transparent and of the correct general shape. So, the total number of nymphs produced per adult aphid per day is given by

image(eqn 1a)

where mβΤ = 3·2, qβΤ = 1·75, T0,β = −2, Tref,β = 20, Tmax,β = 35 °C (parameters derived from Dean (1974), table 3).

Alate production

Some of the nymphs will develop into apterous (wingless) adults, others into alate (winged) adults. This proportion is a function of plant quality, aphid density and temperature (Johnson 1965, 1966a,b; De Barro 1992; see also; Müller et al. 2001). We first consider each of these factors separately and then we consider their combined effect.

Temperature.  The proportion of alate offspring as a function of temperature is modelled by a ‘switch-on sigmoid’ which again is mathematically transparent and of the correct general shape. Thus,

image(eqn 1b)

where Th = 0·3435 is the value of T/inline image at which alate production is half-saturated (approximately 15·5 degrees), inline image = 45 °C set to match the plant model (Thornley 1998), and nβ,T controls the sharpness of the response. These parameter estimates were derived from the data in table 4 of De Barro (1992).

Aphid density.  The proportion of alate offspring as a function of aphid density is also modelled as a ‘switch-on sigmoid’:

image(eqn1c)

where Aden,h= ½ is the value of Aden/inline image at which alate production is half-saturated, Aden is the aphid density (m−2) and is given by eqn 4e, inline image is the maximum tolerated aphid density (m−2, this is the density at which all offspring will be alate morphs) and nρ, Aden = 6 controls the sharpness of the response. Experiments demonstrating the effects of density on alate production are conducted on the scale of aphids per tiller. As the number of aphids per tiller increases, so does the proportion of alate offspring. As mentioned previously, this model has no explicit spatial scale, parameter estimates used in the model are on a m−2 basis. Although we express the term inline image on a m−2 basis, this parameter is not expected to bear any resemblance to a field measurement of the number of aphids m−2; the only way that it would is if aphids were homogeneously distributed in space – an assumption that we definitely are not making. inline image is simply a ‘local density’ expressed in different (larger) units. Using this approach, we estimate that inline image ≤ 105, which we obtained from measures of tiller density (Eberson & West 1996) and aphids per tiller (Skirvin, Perry & Harrington 1997). In the examples used in this paper we have set as a default inline image ≤ 5 × 105 and discuss the consequences of this assumption later.

Plant quality.  The proportion of alate offspring as a function of plant quality is modelled using a ‘switch off sigmoid’:

image(eqn 1d)

where inline image and Qpl is a dimensionless index of plant quality considered below in the section on plant quality. The parameters represent best guesses. While it is very clear that plant quality affects alate production, previous experiments are not very useful for modelling this function.

The combined effects of temperature, aphid density and plant quality.  Putting these three factors together, we assume that the proportion of alate offspring is given by

image(eqn 1e)

if max {ρal,T, ρal,A&#x030c;den, ρal,Qpl} < 1, otherwise inline imageal = 1. Equation 1e represents a weighted average of the three factors, giving more weight to the factor(s) that indicate the highest proportion of alate offspring.

Development rates

Effects of temperature.  We use the development times reported by Elliot & Kieckhefer (1989). We again follow Thornley (1998) in modelling the temperature response as a sigmoidal cubic function. Thus, the development rates (day−1) are given by

image( eqn 2a)

if inline image, otherwise δi,j = 0. In eqn 2ai = ap,al to indicate apterous or alate, and j = 1, … , 4 to denote the instar. All parameter estimates are derived from Elliot & Kieckhefer (1989; p. 135): inline image(day−1). The scaling parameters, inline image denote the differences in development rates between the instars.

Effects of plant quality.  Lower plant quality results in slower development rates. We use a ‘switch-off’ sigmoid to model the effects of plant quality.

image(eqn 2b)

where mfδ = 0·1, nfδ = 4, Qρl,fδ,h = 0·5. The parameter values represent a best guess as no information is available from the literature. Equation 2b changes sigmoidally from mfδ at low values of the plant quality index, to 1 for high values of the plant quality index. By varying mfδ we can explore the strength of the plant quality response.

Combining eqns 2a and 2b we get

image(eqn 2c)

Equation 2c implies that, at the worst levels of plant quality, development rates are 10% of their maximum temperature-dependent rate.

Mortality rates

Effects of temperature. Dean (1974) contains data on survival rates at different temperatures. The survival rates at a given temperature can be reasonably approximated by a linear function. For each temperature we replaced survival with mortality and calculated the slope of the mortality by time function. These slopes were then plotted against temperature and used to derive a general function of mortality rate (day−1) as a function of temperature. This function seems to be adequately described by

image(eqn 3a)
image

Effects of plant quality.  We assume that mortality rate is affected by plant quality, although the available data is not suitable for parameter estimation. We make the modest assumption that, at the lowest levels of plant quality, mortality rates increase by 10%. Here we use a switch-off sigmoidal function

image(eqn 3b)

where inline image. Equation 3b takes the value of 0·006 (i.e. inline image, which is 10% of µmax) when Qpl = 0 and goes to 0 as Qpl goes to 1. Combining eqns 3a and 3b we obtain

image(eqn 3c)

State variable dynamics

In eqns 4a–d below, Jm,n denotes the population size (m−2) of each juvenile instar where m = ap,al denotes the wing morphology (apterous or alate) and n = 1, ... , 4 denotes the particular instar. Aap and Aal denote the population size (m−2) of apterous and alate adult aphids, respectively. The parameters inline imageal, β, inline image and inline imagem,n (subscripts same as J) where defined in eqns 1e, 1a, 3c and 2c, respectively.

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

We assume that alate adults disperse and that immigration from outside the population is negligible and can be ignored. This seems a reasonable assumption as dispersal is presumably the purpose of producing winged offspring, and as we are modelling local aphid populations rather than regional populations, immigration seems unlikely to be an important factor in the local population dynamics. Because we assume that alate aphids disperse upon reaching the adult stage, total local aphid density is given by

image(eqn 4e)

Plant quality

For the purposes of intake, we divide aphids into two groups based on body size: instars 1–3; and instar 4 and adults.

The total amounts of N and C substrate that are available (kg) to the aphids are

WN,avail = WN − WN,unavail, WC,avail = WC − WC,unavail(eqn 5a)

Together, these two assumptions simply recognize the fact that aphids cannot remove all of either the N or C substrate from a plant. The parameters for the unavailable substrates are approximate values, set equal to the initial values for these state variables WN, unavail = 0·0037 and WC,unavail = 0·015 (kg).

So total available substrate (kg) and N and C fractions are given by

image(eqn 5b)

The population sizes of instars 1–3 and 4 – adult are,

image(eqn 5c)

Next, we consider the individual intake. The total quantity of substrate that one aphid must ingest, subject to its maximum intake rate, to meet its daily N requirement is

image(eqn 5d)

These parameters are expressed in kg (to match the units of the plant submodel) and were estimated from published and unpublished data on body size and intake (Prosser et al. 1992; Douglas 1993; Abisgold et al. 1994; Simpson et al. 1995; also R. Harrington unpublished data, S. Simpson unpublished data).

Nreq,1→3 = 3·33 × 10−9, Nreq,4→A = 7·65 × 10−9,
Ωmax,1→3 = 4·2 × 10−8, Ωmax,4→A = 1·35 × 10−7

In equations (5e and 5f), the subscript j denotes the two groups: instars 1–3; and instar 4 and adults.

The fractional demands on the total available substrate pool are

image(eqn 5e)

The total individual intake subject to the constraint of substrate availability is

image(eqn 5f)

Next we consider the population intake. Daily intake of N and C substrate is given by

image(eqn 5g)

Now, we use the results above to derive a plant quality index. Plant quality increases as the total N intake increases to meet the required intake

image(eqn 5h)

where h = 0·5, inline image, parameter estimates represent best guesses which produce reasonable relationships. Equation 5h varies from 0 to 1 as aphids go from completely failing to obtain N to completely satisfying their N requirements.

Model sensitivity

It is essential to ask how our assumptions influence the behaviour of the model. To answer this we need to make clear the distinction between qualitative behaviour and quantitative behaviour of the model. The qualitative behaviour of the model refers to results such as the general dynamics of population growth, e.g. the shape of the time series trajectories in Fig. 1 or the existence of the contours in Fig. 3. The quantitative behaviour of the model refers to issues like precisely what is the population size under a given set of conditions, or exactly where are the contours in Fig. 3 (see Thornley & Johnson 2000; for more discussion on model philosophy and validation). From our sensitivity analysis, it was clear that the quantitative behaviour of the model is affected by our parameter estimates, but that the qualitative behaviour is not. That is, we would reach the same general conclusions for any reasonable guess for the unknown parameter values.

We should be clear that without further, purposeful, parameter estimation (that is, research designed specifically to estimate the parameters) the quantitative predictions of the model are unlikely to be very accurate. However, as our purpose in constructing this model is not pest forecasting but hypothesis generation, the model as it stands seems sufficiently robust. We consider this issue in the discussion.

Model realism

The aphid population and plant growth submodels are coupled via an abstraction of plant quality. Aphid nutritional requirements are considerably more complex than the amount of substrate N in the plant (e.g. Weibull 1988). So there is certainly a good argument for more sophisticated (i.e. less abstract) models of plant quality. However, the problem with this is that plant growth models capable of responding to [CO2]e, light and temperature contain insufficient depiction of the detailed biochemistry to model dynamically the production in the plant of the relevant amino acids. Aphid nutrition may also depend on sugar concentration per se (Simon, Dedryver & Pierre 1991; Abisgold et al. 1994), and this complication will be explored in subsequent work. Lastly, aphid nutritional performance depends on symbiotic bacteria in the aphid gut (Douglas 1998) that synthesize essential amino acids and so help to buffer the aphid against changes in plant quality. This aspect is also not represented in the model. The role of such symbionts may have to wait for progress on a quantitative biochemical model of amino acid production. Perhaps the best justification for using the particular abstraction that we have chosen, is that N input to the host plant has been shown to affect aphid population performance, even if the exact mechanism is not faithfully reproduced (or known). Certainly this model is a crude representation of the way in which aphid performance depends on nutrition, and further models and experiments will be necessary to test the usefulness of the initial hypotheses raised by this model of the biological interaction it seeks to explain.

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