Population responses to perturbations: predictions and responses from laboratory mite populations

Authors

  • T. G. BENTON,

    Corresponding author
    1. School of Biological and Environmental Sciences, University of Stirling, Stirling FK9 4LA UK
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  • T. C. CAMERON,

    1. School of Biological and Environmental Sciences, University of Stirling, Stirling FK9 4LA UK
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    • Present address: T. C. Cameron, Earth & Biosphere Institute, School of Biology, University of Leeds, Leeds LS2 9JT.

  • A. GRANT

    1. Centre for Ecology, Evolution & Conservation, School of Environmental Sciences, University of East Anglia, Norwich NR4 7TJ, UK
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Correspondence and present address: T. G. Benton, School of Biological Sciences, Zoology Building, University of Aberdeen, Aberdeen, AB24 2TZ, UK (e-mail t.benton@abdn.ac.uk; tel. +44 1224 272399).

Summary

  • 1Mathematical models are frequently used to make predictions of the response of a population to management interventions or environmental perturbations, but it is rarely possible to make controlled or replicated tests of the accuracy of these predictions.
  • 2We report results from replicated laboratory experiments on populations of a soil mite, Sancassania berlesei, living in ‘constant’ or ‘variable’ environments. We experimentally perturbed vital rates, via selective harvesting, and examined the population-level responses. The response depends on the stage manipulated and whether there is environmental variability. Increased mortality usually decreased population size and increased population variability. However, egg mortality in a variable environment increased total population size.
  • 3We used time-series analysis to construct a stage-based population model of this system, incorporating the responses to both density and variation in food supply.
  • 4The time-series model qualitatively captures the population dynamics, but does not predict well the way the populations will respond to the change in mortality. Elasticity analysis, conducted on the model's output, therefore did not lead to accurate predictions.
  • 5The presence of indirect positive population effects of a negative perturbation, but only in a variable environment, suggests that predicting the population response will require the incorporation of density dependence and environmental stochasticity. That the considerable biological complexity of our time-series model did not allow accurate predictions suggests that accurate prediction requires modelling processes within a stage class rather than trying to make do with simple functions of total density.

Introduction

Determining the likely way that a population will respond to a change in one or more of its demographic parameters is a critical problem both in ‘pure’ ecology and for applied questions such as population management and prediction of the consequences of environmental change. Typically, demographic and or behavioural information is collected and used to parameterize a model. The parameters are then perturbed in turn, and their affect on the model's population growth rate or population size predicted.

Constructing density-independent matrix models is straightforward, as is their perturbation analysis, so they are widely used in conservation biology (for reviews see Benton & Grant 1999a; De Kroon, van Groenendael & Ehrlen 2000; Caswell 2001; Caswell 2000; Fieberg & Ellner 2001). However, there is an increasing recognition that population dynamics are a complex interplay between stage or age structure, environmental variability and density dependence (Benton & Beckerman, in press; Bjørnstad et al. 1999; Higgins et al. 1997; Leirs et al. 1997; Ellner et al. 1998; Grenfell et al. 1998; Grenfell et al. 2000; Bjørnstad & Grenfell 2001; Coulson et al. 2001; Fromentin et al. 2001; Clutton-Brock & Coulson 2002). The corollary of this is that populations are often unpredictable, and that ‘the devil is in the detail’ (Clutton-Brock & Coulson 2002) when it comes to understanding the way that they behave. It has been claimed that a density-independent analysis will often be a very close approximation to the results of a density-dependent and stochastic analysis (Caswell 2001), but this is certainly not invariably true (Grant & Benton 2003).

If the ‘devil is in the detail’, purely model-based approaches to predicting population dynamics in response to perturbations may leave out biologically important factors, and therefore not properly estimate effects. Conversely, it is often difficult to observe the direct effect of perturbations in the field, where replication and confounding factors are difficult to manage. Laboratory model organisms, on the other hand, may provide a suitable system with which to investigate the population response to perturbation. Our first aim was to perturb the vital rates of our study organism in a controlled and replicated way and observe the population response under both constant and stochastic environmental conditions. Our second aim was to attempt to reconstruct the biology of the model organism in a matrix model using time-series analysis of data on stage abundances in these same experiments. For field populations, we often only have a qualitative understanding of the biology of a species and time-series data of this type. From this start point, can we construct a model to capture the details of the population responses to perturbations?

We have used two sorts of models: simple matrix models based on mean values of the parameters and a simulation model, incorporating information on the response to food and density, parameterized from the time series. We conducted perturbation analyses differently on these models. First, the mean projection matrices, estimated from the time series and from the simulation model, were subjected to typical linear perturbation analysis, which estimates the sensitivity of population growth rate, λ, to independent changes in each of the vital rates. Such sensitivity (or proportional sensitivity = elasticity) analysis has repeatedly been used to estimate the vital rates that would benefit most from management, or the vital rates that are under the strongest selection pressure (Benton & Grant 1999a; De Kroon et al. 2000; Caswell 2001). This use is based on two assumptions: (i) that when the population is free to grow (e.g. at low density) the elasticity of λ predicts the change in population growth rate for a given manipulation, and (ii) when λ is constrained by density dependence that a change in λ predicts a long-term change in population size.

Our second approach was to simulate the time-series model and conduct two further types of elasticity analysis. Evolutionary ‘fitness’ is the invasibility of a new type into a resident population (Metz, Nisbet & Geritz 1992; Rand, Wilson & McGlade 1994), rather than the population growth rate per se. In a linear situation, strategies with the largest λ will invade residents with smaller λ. In a time-varying situation, this is no longer the case, and fitness is better estimated by the invasion exponent, ϕ, than λ (Benton & Grant 2000; Brommer 2000). Elasticities of λ are only an approximation of the elasticities of ϑ, although the approximation may be quite good (Grant & Benton 2000; Caswell 2001). In addition to the elasticities of ϑ, we used the model output to calculate the way the population size changed with changes in the vital rates. The elasticities of λ, ϑ and population size could then be compared with the elasticities estimated empirically from the population experiments.

Methods

experimental methods

We used replicated populations of the soil mite Sancassania berlesei (Michael). This has a life cycle that consists of egg, three juvenile instars and adults. The generation time varies with food supply, but under good conditions the minimum egg–egg time is 11 days (Beckerman et al. 2003). Cultures were kept in a constant-temperature incubator (24 °C), in sealed, 20-mm diameter, flat-bottomed glass tubes, with a Plaster of Paris substrate, and fed on dried yeast. Populations were censused by counting mites using a Leica MZ8 binocular microscope and a hand counter. In each tube, a sampling grid was scratched on the plaster base. Replicate cultures were initiated with c. 100 adults from stock culture. There were four treatment groups, with four replicate populations in each: the unmanipulated control and three treatments reducing the survival of eggs, juveniles or adults. Each day the numbers of individuals in the manipulated stage were counted and 15% were removed – giving a 15% reduction in daily survival for the target groups. Counts of the non-target groups in each population were made on average 5 days per week. All counts and manipulations were undertaken by the same individual (T.C.C.).

Each of the three experiments was conducted on four replicate populations per treatment (16 populations in total). The populations in the ‘constant’ experiment were maintained at equilibrium by the supply of limiting food (0·0030 g of powdered yeast per culture per day). Thus, density dependence was strong and environmental variability was low. The populations in the ‘variable’ experiment experienced large variability in their food supply (28 × 0·0015 g granules of yeast were supplied in random amounts over random days throughout each 2-week period). The constant experiment was conducted over 86 days in late 2000, and the variable experiment over 101 days in spring 2001. Populations were initiated and then allowed to equilibrate (constant: 34 days; variable: 21 days). Following the onset of manipulation, the constant populations showed transient changes prior to settling onto new equilibria. For the analyses described below, we used data from two weeks after the onset of manipulation to allow transients to disappear. This gave 22 counts (taken over 35 days) for the constant experiment, and 46 counts (taken over 65 days) for the variable one.

analysis

Bootstrapping was used to estimate the confidence limits for the difference between mean population sizes for the treatments. Stratified resampling (by tube within treatment group) was conducted 1000 times for each treatment group, the difference between the means of these resamples and the means of the control group gave the bootstrap distribution, from which the 95% Bias Corrected and Adjusted percentiles were calculated. The difference between the treatment and control means was then converted to an elasticity (see below).

developing the time-varying matrix model

The matrix model is of the form:

image(eqn 1)

where E, J and A refer to egg, juvenile and adult numbers, respectively. Survival terms are α, for egg survival, β, for juvenile survival, and γ for adult survival. F is fecundity (eggs per adult), H is the rate of egg hatching and M is maturation to adulthood. Subscripts give time (in days). If the matrix is labelled B, with its elements being bij, and the vector of population sizes N, then equation 1 can be written BtNt = Nt+1.

The system of equations given by the matrix (equation 1), defines terms to allow the estimation of the parameters from the experimental time series:

image(eqn 2)
image(eqn 3)
image(eqn 4)

where MtJt = Rt+1, the number of recruits at time t + 1.

estimatingα, h and f

Equation 2 indicates that the number of eggs laid (AtFt) is Et−1 − αEt, from this it follows that αt is survival of the eggs (δ) that do not hatch (i.e. αt = δ(1 – Ht)). We have no information on daily egg survival, but do have information on the proportion of eggs hatching on each day (over the 6-day hatching period), and therefore the total proportion surviving to hatch. If we assume that egg survival is constant and that all eggs left alive on day 6 hatch, then we can estimate the daily egg survival as the root of the polynomial:

δ6 − δ5bP1 − δ4bP2 − δ3bP3 − δ2bP4 − δbP5 − bP6 = 0

where b is the overall proportion of the brood to hatch, and P is the proportion hatching on days 1–6 following laying. Thus, knowing b and P1 … P6 allows estimation of δ. Assuming these quantities are constant over time, then if the number of eggs laid in the first cohort at t = 1 is known, the number hatching, and the number left alive, on each subsequent day can be estimated. For the rest of the time series of egg counts, it then becomes a book-keeping exercise: for each of the cohorts of eggs laid over the previous 6 days, estimate the numbers of eggs alive and not yet hatched, sum them and subtract from the current egg count to estimate the eggs laid in the last interval. This book-keeping exercise therefore estimates the time series of the eggs laid (and thus the fecundity), and also time series of the number of eggs hatching, from which Ht can be estimated as the number hatching divided by the number of eggs at the previous count.

time-series analysis

To estimate the parameters of the matrix model, time-series analysis was conducted on the data from the four control tubes. All analyses were conducted in Splus (Insightful Corp., Seattle, WA, USA). A model was fitted by starting with a minimal model and increasing its complexity (stepwise forwards); and also by starting with a full model (up to three-way interactions) and simplifying it (stepwise backwards). The final model was therefore approached iteratively from different directions, ensuring its robustness. This was necessary as the models incorporated a large number of potential explanatory variables, all of which were of known biological importance. Inclusion of a factor in the model was based on the effect that the inclusion of the term had on the model's AIC (Akaike Information Criterion) statistic, as inferential statistics could not be used owing to non-independence of the time-series data. Analysis was conducted on the time series of females; adult numbers in the model were estimated as twice the predicted numbers of females as the sex ratio is 1 : 1.

Fecundity

Owing to the nonlinear relationship between fecundity and feeding, fecundity was used as a dependent variable in a nonlinear least-squares regression. The model was fitted forwards and backwards, with terms added and deleted manually, until all remaining terms in the model had 95% profile confidence intervals which excluded 0, and removing any term increased the AIC.

Juvenile survival

GLMs (Generalized Linear Models) were fitted of the form:

(Jt − Ht−1Et−1) ∼ f(Jt−1, ♀t−1, days-since-feeding … ).

The dependent variable was rounded to the nearest integer, and models were fitted with quasi-likelihood GLMs (with log link and variance equal to the mean). Quasi-likelihood accounted for the overdispersion in the model. The minimal model was the model from which removal of any terms increased the AIC, and addition of no further terms reduced it. Substituting the fitted values into equation 3 allows estimation of βt during the time series.

Adult survival and recruitment

GLMs were fitted of the form:

(♀t + 1) ∼f(♀t, Jt, days-since-feeding … ).

The fitted values of ♀t+1 were used to estimate both the number of recruits, Rt+1, and γt, as (from equation 4), and expressing in terms of females: ♀ = A/2): ♀t+1= Rt+1/2 + γt/. From experiments reported elsewhere (e.g. Beckerman et al. 2003 and unpublished), we know that female survival is strongly related to the food supplied over her lifetime (primarily through the negative trade-off between fecundity and survival). We also know that females and juveniles are unequal competitors, so female per capita food is affected much more by the numbers of females than the numbers of juveniles. We thus subsetted the variables in the model into ones most likely to be related to female survival (variables with females in, for example, ♀t−1, ♀-at-last-feeding; and variables related to past patterns of food supply). The fitted values from this subset of variables were deemed to represent the component γtt, from which γt = fit/♀t, and the difference between the fits for the whole model and the partial model were deemed to represent the number of new females recruiting into adulthood at t+ 1. Existing data (e.g. Beckerman et al. 2003) indicated the likely shapes of the functions relating density to survival and recruitment, so selection of variables for the model included an extra criterion (as well as AIC): that of biological pragmatism. If exclusion of a variable on statistical grounds caused the resulting fits to change from the a priori known pattern to something not biologically reasonable, its inclusion was reconsidered.

estimating the mean matrix using quadratic programming

First, we estimated a mean projection matrix from the population time series using Wood's quadratic programming methods (see Caswell 2001, section 6·2·2). We used the average matrices estimated from (i) three long-term time series for mites kept under low levels of variability in food (seven balls of yeast supplied at random throughout a given week: details in Benton, Lapsley & Beckerman (2001)), and (ii) from five replicate time series supplied with food at the same level of variability used in the second experiment (14 balls over 2 weeks: details in Benton, Lapsley & Beckerman (2002)). The values of the elements of these matrices are influenced by density dependence, but are constant.

elasticities ofϑ, λ and population size from the matrix model

The matrix-simulation model was coded in FORTRAN77. It generates a feeding schedule (either constant food, or a random food supply, using the same algorithm used to generate the laboratory feeding regimes), then uses the fitted relationships from the time-series analysis to simulate the parameters of the matrix model, and thus the numbers of individuals at subsequent time steps.

For each simulation, the ‘resident’ population was started and allowed to reach equilibrium over 7500 time steps. An ‘invader’ population was then started, whose life history was identical in all respects with the residents’, except that one vital rate (i.e. matrix element) was reduced by 0·001% (constant food) or 5% (variable food). The larger perturbation for the variable model is necessary to increase the power of the comparison between the ‘baseline’ and ‘perturbed’ life histories. The dynamics of the invader population were followed for 1500 time steps, after which least-squares regression was used to estimate the invader's rate of population change (i.e. the slope of the regression of the natural logarithm of invader population size against time). This process of invasion was replicated 200 times. The average slope of the regressions gives the invasion exponent, ϑ. The elasticity of ϑ to changes in the vital rates (the individual terms of the projection matrix, bij) is given by:

image(eqn 5)

Over the 300 000 time steps of each simulation, the vital rates of the resident (i.e. the time-specific matrices) were averaged to give a mean matrix for linear matrix analysis, where the elasticities are calculated using:

image(eqn 6)

The mean population size over the 300 000 time steps allowed the calculation of population-size elasticities, using:

image(eqn 7a)
image(eqn 7b)

The E refers to mathematical expectation, estimated by the mean value. We refer to these as the elasticities of the mean log population size (equation 7a) and the log mean population size (equation 7b). Which is more appropriate to use may depend on the question under investigation (Grant & Benton 2000). The mean log population size elasticity is going to be robust to outliers, but will weight population troughs more than population peaks. The converse will be the case with the log mean population size elasticity.

Results

population effects of manipulations: the data

Populations fed food at a constant rate show damped oscillations leading to an equilibrium (Benton et al. 2001; Fig. 1); those fed food in a variable regime exhibited much more variable dynamics (Fig. 1). Manipulating mortality has dramatic effects on the average population size, which depend on the stage harvested (Figs 2 and 3). Under constant food, reducing adult survival by 15% has the biggest effect on population size, reducing it by 18·4%. Under the variable food regime, reducing juvenile survival has the biggest effect, reducing mean population size by 21·4%, and reducing egg survival actually increased population size by 9·7%. Changing survival rates can also change the population variance: populations undergoing a reduction in juvenile survival in the variable-food experiment are significantly more variable than the other three treatments (Fig. 2b, Bartlett's test, P < 0·0005), indicating that processes within the juvenile stage play an important role in buffering environmental stochasticity. Variability in the food supply reduces the mean population size, even though the populations receiving variable food received it at the same mean rate as the near constant food regime (see Table 4).

Figure 1.

The mean time series from the four unmanipulated control populations in the two experiments: Constant food (top row) and Variable food (bottom row). The panels are (l to r) egg, juvenile and adult counts. Each point is the average of the counts from the four populations, error bars are SE.

Figure 2.

The population sizes (E + J + A) of the treatment groups plotted against the corresponding population sizes for the controls for each day following the onset of harvesting (day 34 in the constant experiment (a), and day 21 for the variable food experiment (b)). The symbols reflect mean population size (n = 4 replicate tubes), with SE for the four groups: control and 15% daily reduction in eggs, juveniles and adults. The dashed line gives the 1 : 1 relationship.

Figure 3.

Treatment effects on the mean of the ln-transformed counts of total population size (E + J + A) for constant and variable food experiments. Treatment groups are E = egg survival decreased, J = juvenile survival decreased, A = adult survival decreased, C = control. Points are means (of the four population sizes per treatment groups). Bootstrapping was used to estimate the 95% CI.

Table 4.  Comparison of population sizes from the control tubes and the output of the time series model
 Data from experimentOutput from model
AdultsJuvsEggsTotalAdultsJuvsEggsTotal
  1. The outputs from the model come from a representative run (500 time steps). The descriptive statistics for the ‘constant’ controls were calculated from day 27 onwards. The ‘variable food’ regime was based on 28 balls of yeast, randomly supplied over each 14-day period. The ‘constant’ regime was based on 2 balls equivalent weight of powdered yeast per day.

Variable food
 Mean206·6 952·1316·11477224·2 980·3329·01532
 SD 84·0 422·4343·1 476 55·5 419·2388·0 584
 Range21–47556–25120–1600524–271861–417100–26230–1909357–3483
 n225 228229 225500 500500 500
Constant food
 Mean202·11048·5853·92104·5260·91162·6475·91899·4
 SD 25·2 283·1234·9 390·5    
 n160 160160 160    

elasticity analyses: empirical elasticities

The difference between mean population sizes for the different treatments can be converted to elasticities (eqn 7), giving the proportional change in population size for a change in egg, juvenile or adult survival (Fig. 4). Two points arise from Fig. 4. First, the pattern of elasticities is different between the constant and the variable food regimes. This is especially clear with the change in the mean log population size (Fig. 4a), where the rank order of elasticities changes. The difference in pattern is significant for the mean log population size elasticity, but not for the log mean population size elasticities (where the confidence limits are wider owing to the measure's sensitivity to outliers). Second, the elasticity for egg survival in a variable environment is significantly less than zero: harvesting eggs increases the total population size. The data suggest that fewer eggs leads to more juveniles and more adults (see Cameron & Benton 2004). This increase does not occur with harvesting juveniles, which suggests that harvesting eggs or juveniles has different effects owing to the effect on age structure rather than juvenile numbers per se. Young juveniles are the most numerous, and naturally suffer the greatest mortality. By harvesting eggs, there is a reduction in competition amongst this class, compared to harvesting juveniles of random ages. This allows the fewer hatchlings to get a better ‘head start’ in growth, leading to greater overall recruitment out of the most sensitive stage. In a variable environment, the food pulses lead to pulses of eggs being laid, which, when they hatch, lead to enhanced competition between the smallest juveniles. In the constant environment, cohort structure decreases, as it is not ‘reset’ by changes in the competition caused by environmental variability. The changes in age structure due to harvesting are discussed more fully elsewhere (Cameron & Benton 2004).

Figure 4.

Empirical elasticities of population size, for constant and variable experiments. (a) empirical elasticities of the mean log population size (b) elasticities of log mean population size. Error bars are 95% CI, estimated by bootstrapping. Treatment groups are E = egg survival decreased, J = juvenile survival decreased, A = adult survival decreased.

parameterizing the matrix model

The data from the control tubes in the variable food experiment (see Fig. 1) were used to derive the equations that relate the environment (food supply, density) to the vital rates used in the matrix model.

Egg hatching and egg survival

Eggs laid at the same time may hatch up to 6 days after laying (T.G. Benton, unpublished data). For those eggs that do hatch, the proportions hatching on days 1–6 (P1… P6) are 1%, 11%, 32%, 30·0%, 25% and 1%, respectively. The proportion of eggs that eventually hatch (b) is related to adult density the day before the eggs are laid. The highest proportion hatches at intermediate densities; hatching rates at low and high densities are lower. This relationship was fitted with a quadratic regression (b = 0·457 + 0·272 lnA − 0·034 (lnA)2, F2,2 = 158, P = 0·006). To simplify the estimation of the time series of eggs laid, we assume that b is fixed. The median density of adults in the control cultures (Fig. 1) is 198, giving an estimate of the average proportion hatching of 95%. This figure, plus the daily hatching proportions, allows an estimate of the average daily egg survival, δ, of 0·9862.

Determinants of fecundity

There is a clear nonlinear relationship between fecundity and the days since food was last supplied to the culture (Fig. 5), so a nonlinear regression curve was fitted to the data. The important parameters in the model were found to be related to the food supply over previous days, as well as adult and juvenile densities (see Table 1). The model explained 67% of the variance in the reconstructed fecundity time series.

Figure 5.

Examples of the relationships estimated for the matrix model from the time-series data from the control populations on variable food. (a) and (b) show the model in Table 3 used to estimate recruitment and adult survival. (a) shows estimated recruitment (line is 6 d.f. spline) (b) shows the model's predicted values vs. observed. The line is a regression fitted to the points (y = 10·4 + 0·90x), indicating that the fitted values slightly underestimate the data (c). Fecundity was estimated from the difference in egg counts from day to day, taking into account estimates of hatching and egg survival (these are shown plotted against the time since last feeding, circles). A nonlinear regression was then fitted, the fits from which are the crosses, offset between days. See Table 1 for details of the fecundity model.

Table 1.  Coefficients of a nonlinear regression relating estimated fecundity to the environment
CoefficientDependent variableValueSEt-valueΔAIC
  1. Determinants of fecundity, where Ft = (eggs laidt+1/♀t). The significant parameters in the model were: days since last food (days), the amount of food at the last feeding (lastfood), the interval between previous two meals (int), the total food received over the last 4 days (foodt−1·t−4), the food received on the day of laying (foodt), the numbers of females on the day of laying(♀t), and the number of juveniles on the previous day (Jt−1). The model fitted was:

  2. ln(Ft+1) = (c1 · days + c2 · int · foodt−1·t−4 + c3 · foodt + c4 · days · foodt) · exp(c5 · days + c6 · ♀t + c7 · int + c8 · lastfood + c9 · days · int · lastfoodc10 · Jt−1).

  3. The AIC of the full model is 36·7, and the change in AIC with the deletion of each term in the model is given in the column ΔAIC. The null variance is 106 on 403 d.f., the residual sum of squares is 34·9 on 394 d.f., so the model accounts for 67% of the variance.

C1Days   6·624110·74760      8·8636·7
C2Int · foodt−1·t−4−0·014530·00216  −6·72 2·5
C3Foodt−2·264220·47797  −4·73 4·7
C4Days · foodt   2·194450·46679      4·70 4·5
C5Days−0·930080·03500−26·5741·6
C6t−0·005890·00064   −9·19 7·4
C7Int−0·049270·00896  −5·50 3·4
C8Lastfood   0·021840·00291      7·50 4·4
C9Days · int · lastfood   0·000920·00011      8·16 4·6
C10Jt−1−0·000440·00007  −6·37 3·4

Determinants of juvenile survival

Juvenile survival was modelled in a GLM, with (Jt– Ht−1 · Et−1) as the dependent variable. The minimal adequate model contained terms describing the food supply, adult and juvenile densities on the previous day and adult densities on the day when food was last supplied (see Table 2). This minimal model explained 81·6% of the variance in the dependent variable. The fitted values from this model were substituted into equation 3, allowing estimation of βt.

Table 2.  Coefficients of the GLM relating juvenile survival to the environment
CoefficientValueSET-valueΔAIC
  1. Juvenile survival was modelled as a quasi-likelihood GLM (with log link and variance equal to the mean), with (Jt– Ht−1· Et−1) as the dependent variable. The minimal adequate model contained the following terms: ln(t−1), ln(Jt−1), days since last meal (days), the interval between the previous meals (int), the number of females on the day of last feeding (ln(♀lastfd)). The null deviance is 76971 on 379 d.f., the residual deviance is 14176 on 372 d.f., so the model accounts for about 82% of the deviance. The AIC of the full model is 15018, and the change in AIC with the deletion of each possible term in the model, without breaking hierarchy rules, is given in the column ΔAIC.

Intercept   8·060190·34377  23·4 
ln(♀t−1)−1·763190·07254−24·3 
ln(Jt−1)−0·191250·04854  −4·0 
Days   0·005950·00041  14·3127
Int   0·027010·00220  12·2 
ln(♀lastfd)   0·085080·00654  13·0 
ln(♀t−1) · ln(Jt−1)   0·244450·01031  23·7485
ln(♀lastfd) · int−0·005850·00048−12·1 69

Determinants of adult numbers: survival and recruitment

The matrix model (equation 1) indicates that At+1 =Rt+1 + γ.At, so we modelled ♀t+1 as the dependent variable in a GLM with the predictor variables being female and juvenile densities and various food-supply variables (see Table 3, Fig. 5). The resulting model is complex (with 15 coefficients), and it explains approximately 92% of the variance of the data. One coefficient in the model is not important on statistical grounds (removing it slightly decreases the AIC, and its t-value << 2), however, removing it makes the estimated relationships between recruitment, adult survival and density a poor fit to existing data (Beckerman et al. 2003). Hence the term (ln(Jlastfd)) was retained.

Table 3.  Coefficients of the GLM relating female numbers at t and t+ 1, used to estimate recruitment and female survival
CoefficientValueSEt-valueΔAIC
  1. Female numbers were modelled as a quasi-likelihood GLM (with log link and variance equal to the mean), with (♀t+1) as the dependent variable. See previous legends for explanations of the terms. A subset of the coefficients (marked with * in table) were used to estimate the females at time t + 1 as a function of females and food. The fits from these coefficients were used to estimate female survival, and the difference between those fits and the fits of the full model were used to estimate recruitment. The null deviance is 7643 on 379 d.f., the residual deviance is 640 on 365 d.f., so the model accounts for about 92% of the deviance. The AIC of the full model is 702, and the change in AIC with the deletion of each possible term in the model, without breaking hierarchy rules, is given in the column ΔAIC.

Intercept* 1·796570·44267 4·1 
ln(♀t)* 0·913380·11143 8·2 
ln(♀t–1)*−0·324050·05339−6·1 
ln(♀lastfd)*−0·333080·08569−3·9 
ln(Jlastfd) 0·011390·00828 1·4−2·6
Days*−0·045550·01614−2·8 
Int* 0·057790·01512 3·8 
Lastfood 0·001870·00065 4·1       4·8
foodt−1·t−4 0·001370·00059 2·3 
foodt−0·001250·00089 1·4 
foodt−1·t−4 foodt 0·000970·00031 3·2 6·4
ln(♀t) · ln(♀lastfd)* 0·072730·01845 3·9 12·0
ln(♀t) · days* 0·007770·00351 2·2 1·3
ln(♀t) · int*−0·027450·00536−5·1 22·7
ln(♀t−1) · int* 0·015620·00436 3·6 9·3

the matrix-model simulation

To assess the model's fit, we generated ‘one step ahead’ predictions by using the model to predict the population at time t + 1 given the data up to time t. Using the data from which the model was derived gives predictions close to the observed (median prediction/observed = 0·996, IQ range = 0·89–1·10, 95% CI = 0·66–1·38). A laboratory time series, from an unrelated experiment in 1997/8, but run with similar levels of food variability, gave one-step ahead predictions which were less consistent with the model, but still, on average, respectable (median prediction/observed = 1·03, IQ range = 0·84–1·26, 95% CI = 0·58–1·85). Since 1997/8 the mites are likely to have evolved within culture (T. G. Benton, unpublished data), and aspects of the culturing have also changed.

The stochastic variable in the matrix model is the food supply, which is selected using the same algorithm that is used to generate the laboratory feeding regime (28 balls of yeast per 14 days, given at random amounts and random times). The model captures much of the dynamics of the laboratory populations (compare Figs 1 and 6, Table 4). The model's simulations produces time series that give rise to periodograms with similar frequency spectra, which are typically dominated by low frequency fluctuations, i.e. they are noticeably ‘reddened’.

Figure 6.

Example of 250 days of output from the matrix model. The vital rates are functions of food and density, and food is supplied at random, using the same algorithm as in the experiment.

The functions used in the model were derived from the ‘variable food’ regime, and the model simulates the population dynamics reasonably well under variable food. When the model runs on constant food the fit is not so good. The predicted population sizes for the different life stages are different from those observed by varying amounts (adults +29%, juveniles +11% and eggs −44%, total population size −9·7%).

elasticity analyses: model-based elasticities

The mean matrices estimated directly from the data and from the simulation model (Table 5), were used to estimate the elasticities of λ using standard techniques (Caswell 2001). The density dependent, stochastic, analogues of the elasticities of λ, are the elasticities of ϑ, the invasion exponent. The elasticities of ϑ should sum to 1·0, and so the sum can be used as a check of numerical precision: the constant elasticities of ϑ sum to 0·998 and the variable to 0·994. The elasticities of population size were also estimated by numerical differentiation. All these elasticities are shown in Fig. 7.

Table 5.  The mean matrices estimated from the temporally invariant matrix model (QP) and from the density dependent, stochastic matrix model
ElementQP (low variability)QP (high variability)Constant MeanVariable MeanVariable SDCV
  1. The elements of the temporally invariant matrix were estimated by quadratic programming (QP) from time series. The simulation model was run for 300 000 time steps and means and SDs estimated. λ for QP (low variability) 0·99188, QP (high variability) 0·94969, constant matrix is 1·0000, λ for variable matrix is 1·003817. The coefficient of variation (CV) is the ratio of the SD to the mean.

α0·7650·3760·725180·645130·191420·29672
F0·4640·7830·501320·402160·507601·26218
H0·1550·4180·261070·341120·191400·56109
β0·8740·7530·893120·924370·028750·03110
M0·0100·0170·230220·021370·011200·52410
γ0·9650·8950·897410·900940·060100·06671
Figure 7.

Elasticities from the matrix models. The top two panels show the elasticities estimated for the constant and variable conditions (▵ lambda = elasticities of λ, □ invasion = elasticities of ϑ, ○ log mean = elasticities of log mean population size, • mean log = elasticites of mean log population size, * quadratic = elasticites of λ from the mean matrix estimated using quadratic programming). The bottom two panels show the different elasticities in proportion to the elasticities of λ from the mean matrix of the simulation model (Table 5). The horizontal line indicates proportionality. The elasticities of population size were scaled to the interval (0,1) to allow direct comparison with the other elasticities (the sum of the log mean population elasticities is 7·8711 and 4·3842, constant and variable, respectively; and mean log is 7·8711 and 4·1108). Error bars are parametric 95% CI. Jitter has been added to the lower x axis, to aid clarity.

The elasticities of λ from the matrices estimated from the data, using quadratic programming, and from the simulation model, differ and also depend on the environmental variability (Fig. 7). The elasticities of λ estimated from the simulation model matrix suggest that adult survival has the biggest elasticity in constant conditions and juvenile survival under variable conditions. This difference occurs because the average life history is different in the two environments. Conversely, the elasticites of λ estimated by the quadratic programming matrix do not vary so much between the constant and variable conditions, and suggest the population response to perturbing adult survival should be 4·7 times that of perturbing juvenile survival (whereas the data suggest it should be 2·3 times in constant and 1·1 or 0·7 times in variable environments, Fig. 4).

Concentrating on the results of the simulation model, there are significant differences between the elasticities of λ, ϑ and population size, in both the constant and variable regimes, indicating that the elasticities of λ are only an approximation to the other elasticities (Fig. 7, bottom figures). Notably, the egg survival elasticity of population size under constant food is only 20% of the elasticity of λ, and the adult survival elasticity is 47% larger. Under constant food regimes, the elasticities of λ and ϑ are identical, but differ from the elasticities of population size, as density dependence is not a simple function of population numbers (Grant & Benton 2003). Under variable food, there are significant differences between the elasticities of λ, ϑ, log mean population size and mean log population size (Fig. 7). The mean difference between the λ elasticity and the corresponding elasticities of ϑ, log mean or mean log population size is 27% (median 13%, interquartile range 8–48%, maximum 78%).

In the constant environment the observed ratios of the population size elasticities (E : J : A, 0·08 : 0·28 : 0·64, Fig. 4) most closely match the corresponding ratios for the matrix model's population size elasticities (0·03 : 0·34 : 0·63) rather than the elasticities of λ or ϑ (0·13 : 0·42 : 0·44), and certainly differ from the mean data matrix's elasticity of λ (0·07 : 0·16 : 0·77). Under variable food, the two ways of estimating empirical elasticities differ: the elasticity of juvenile survival is bigger when the elasticity is estimated with the mean log population size. This pattern most closely corresponds with the population size elasticities predicted by the matrix model (whether log mean or mean log), suggesting the data produce more extreme points than the model, and thus affecting the log mean population size estimates more strongly. The matrix model also does not predict negative elasticities of egg survival in the variable environment (Figs 4b and 7). The absolute sizes of the egg survival elasticities are also under-estimated by all the elasticities from the time-varying matrix model. In the variable environment, the observed negative elasticity means that there is no close fit to any of the predicted elasticities, although the rank order of the empirical mean log population size elasticities is correctly predicted (by all types of elasticity estimated from the matrix model). This is not the case for the empirical log mean population size elasticities, where the adult population size elasticity remains the largest elasticity under variable conditions; this situation does not correspond to any predictions from the model.

Discussion

Populations living in ‘constant’ and ‘variable’ environments exhibit markedly different population dynamics. In addition to environmental variability affecting the population dynamics (mean population size and variance) it also affects the way that the population responds to perturbations, in this case manipulation of survival, via selective culling of eggs, juveniles or adults. Three empirical results are of note. First, the biggest effect on population size occurs through manipulating adult survival in the constant conditions and juvenile survival in variable conditions. Secondly, under variable conditions, manipulating juvenile survival increases the population variance as well as reducing the mean population size. Thirdly, there is an indirect positive effect: reducing egg survival increases the overall population size.

Environmental variability is undoubtedly important in determining both how organisms’ life histories are expressed, through phenotypically plastic responses (Fox & Savalli 1998; Stearns 2000; Beckerman et al. 2003), how life histories evolve (Benton & Grant 1999b; Coulson et al. 1998; Rose, Rees & Grubb 2002) and how the population responds to the environment in terms of its size and variance (Bjørnstad & Grenfell 2001; Benton et al. 2002; Clutton-Brock & Coulson 2002). It is therefore not surprising, although rarely if ever demonstrated, that the same perturbation can give rise to different population responses depending on the environmental variability. Hence, empirically, there is a context-dependence of population response. In the results described here, for example, culling juveniles creates a relatively small population size effect when the environment is close to constant, but reduces population size markedly, and increases population variance, when the population is variable.

The indirect positive effect of harvesting eggs illustrates the importance of density dependence, as by reducing numbers of eggs (and therefore hatchlings), it allows greater survival of juveniles past the critical larval instar, much as thinning seedlings leads to greater overall survival in plants. This positive population effect is only detectable in the variable environment, indicating that there is an interesting interaction between the environmental variability and density dependence. This interaction is likely to be due to the way that the fluctuating environment can create cohorts of individuals that hatch together following a pulse of food. Cohort structure is less evident in a population at equilibrium. In the variable environment, a pulse of food creates a large pulse in eggs and therefore the larvae when they hatch (Cameron & Benton 2004, Fig. 2). Removing eggs reduces the size of the peak and therefore the strength of competition in the weakest competitive class. With exploitation competition, a reduction in the peak cohort size may cause a nonlinear increase in juvenile survival. Harvesting juveniles at random does not affect the cohort structure in the same way, thereby retaining excessive competition in the hatchlings. The importance of the stage structure within juveniles suggests that modelling this explicitly may improve the model fit.

Such indirect effects have recently been noted in the population response of blowfly to cadmium toxicity (Moe, Stenseth & Smith 2002), in an analysis of the Tribolium larva pupa adult (LPA) model (Grant & Benton 2003), as well as in responses to tropic interactions (Schmitz, Hamback & Beckerman 2000; Vonesh & Osenberg 2003). These effects are most likely to arise if the system is over-compensatory and therefore potentially close to instability, such as will occur if there are marked asymmetries in competitive ability between different (st)ages, or if there is cannibalism of the smaller/younger organisms by older/larger ones. These mechanisms also have in common that they create cohorts of individuals that compete strongly – either through environmental variability or due to intrinsic or extrinsic mechanisms creating periods when competition is excessive and periods when competition is weak.

We developed from the extensive time-series data a density-dependent stochastic matrix model. This model, by incorporating the responses to density of competitors (adults and juveniles) and the stochastic forcing variable, food, is more biologically complex than would usually be possible in management situations. Using this model, we estimated the elasticities of λ, ϑ and population size. Additionally, using time-series data, we estimated directly a mean matrix, from which we calculated the elasticites of λ. We have therefore been able to compare our empirical elasticities with those predicted by the models, and therefore ask ‘Does elasticity analysis provide reliable information on the population responses to change in the vital rates?’. In the constant environment, the rank order of the elasticities from the models is correct, although the magnitude of the difference between the adult and juvenile survival elasticities is not reflected by the data. Scaling the elasticities (so that the sum of the α, β and γ elasticities = 1·0) allows direct comparison between the different methods. The 95% CI of the egg survival empirical elasticity includes all the elasticities from the matrix models. However, the 95% CI of the juvenile and adult survival elasticity exclude the elasticity of λ (from both mean matrices) and ϑ. Thus, in a ‘constant environment’, the density dependence and uncontrolled stochasticity ensure that λ elasticities do not predict the observed population response as well as the population size elasticities.

In the variable environment, the empirical elasticities of log mean and mean log population size differ considerably, and differ from the elasticities predicted by the matrix models. Notably, the empirical elasticity of egg size is negative, whereas the models’ elasticities are all positive. The matrix model, although containing an unusually large amount of biological information from the time series, does not predict negative elasticities for egg survival. Additionally, the absolute size of the empirical elasticities of population size are not accurately predicted by the model. The data suggest that perturbing juvenile or adult survival has largely the same effect on log mean population size (the ratio of elasticities is 0·9), but perturbing juvenile survival has about 1·5 times the effect of adult survival on the mean log population size (ratio 1·47). The model predicts that this difference should be 1·11 and 1·88 times, respectively.

That the population size and variance (and therefore the elasticities) change differentially with changes in the vital rates between the constant and variable environments suggests that any attempts to predict population responses to changes in their environment should incorporate stochasticity into their modelling framework. How this is done will itself have large consequences, so where possible should be based on underlying biological understanding. If this does not exist, through lack of data, ignoring its potential effects may be dangerous, and different scenarios should be modelled to assess the system's sensitivity to environmental variability. Our results chime with other recent studies specifically examining elasticities (Wisdom, Mills & Doak 2000; Cross & Beissinger 2001; Fieberg & Ellner 2001; Orzack & Tuljapurkar 2001; Stephens et al. 2002; Tuljapurkar, Horvitz & Pascarella 2003). Likewise, the existence of a positive population response to harvesting indicates that populations that experience density dependence should incorporate it in analyses (Vonesh & De la Cruz 2002; Matos, Freckleton & Watkinson 1999; Grant & Benton 2003). As this positive population response is found only in variable environments, it further suggests that both density dependence and environmental variability should be incorporated.

Any model will always be an approximation to reality, and although the model may never lie, how good an approximation to reality it is requires experimental verification. The time-series model developed here uses both a large amount of data (including replicated time series), a large number of parameters, and manages to capture quantitatively the population size and variance in the variable environment (Table 4). Yet the model fails to capture some of the details of the population behaviour (such as the positive effect of egg harvesting). This is likely to be because the model does not provide for size-structured variation among juveniles, where the hatchlings are very vulnerable to competition and mechanical damage by trampling from older juveniles. This deficiency of matrix models has recently been identified in a theoretical study by Pfister & Stevens (2003). Given that we find that ‘the devil is in the detail’, very simple approaches, such as the elasticity analysis of λ from a temporally invariant matrix model may be unlikely to offer much chance of quantitatively accurate population-level prediction, especially in density-dependent situations where the elasticities of λ are constrained to be positive. Our recommendation for managers would be to estimate a mean matrix and do elasticity analysis of λ, but to check for the presence of both strong density dependence and environmental variation and examine their potential consequences using models. Elsewhere (Cameron & Benton 2004) we show that density dependence is as strong at 33–50% of carrying capacity as it is at carrying capacity, indicating that relative rarity is not sufficient to allow the assumption of a lack of density dependence in a population.

It is possible to model systems with simple models (the LPA model is elegantly simple and works well). However, such models/systems may be rare in a temporally varying world. The mites’ performance depends on both current food supplies (which depend on density, age structure and food supply) and previous food supplies, both within their lifetime and across generations (S.J. Plaistow et al., unpublished data). Under constant conditions, such as are the usual conditions in the laboratory, this temporal-condition-dependence is hidden. Models developed for temporally varying conditions, such as field data (e.g. Zeng et al. 1998; Coulson et al. 2001; Clutton-Brock & Coulson 2002), suggest complexity may be necessary for many systems.

Acknowledgements

NERC provided the funding (NER/MS/2000/00289), the Centre for Population Biology provided facilities during the writing. Andrew Beckerman and Craig Lapsley helped, in various ways, throughout the project.

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