Present address: Department of Biological Sciences, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK.
Predicting the impact of stage-specific harvesting on population dynamics
Article first published online: 8 JUN 2009
© 2009 The Authors. Journal compilation © 2009 British Ecological Society
Journal of Animal Ecology
Volume 78, Issue 5, pages 1076–1085, September 2009
How to Cite
Carslake, D., Townley, S. and Hodgson, D. J. (2009), Predicting the impact of stage-specific harvesting on population dynamics. Journal of Animal Ecology, 78: 1076–1085. doi: 10.1111/j.1365-2656.2009.01569.x
- Issue published online: 29 JUL 2009
- Article first published online: 8 JUN 2009
- Received 16 October 2008; accepted 22 April 2009 Handling Editor: Tim Benton
- Daphnia magna;
- population projection matrix;
- transfer function analysis
- Top of page
- Materials and methods
- Supporting Information
1. Perturbation analyses of population projection matrices predict the response of a population’s growth rate to changes in lifestage-specific vital rates. Such predictions have been widely used in population management but their reliability remains hotly debated.
2. We grew replicate populations of the water flea Daphnia magna in controlled, density-independent conditions and subjected treatment populations to harvesting of the largest lifestage. We predicted the growth rate of treatment populations using sensitivity analysis (a linear approximation), and transfer function analysis (TFA; which captures nonlinear responses) applied to projection matrix models parameterized from the control populations.
3. When perturbation analyses considered only the direct effect of harvesting on adult survival, the growth rate of harvested populations (averaging 0·051) was significantly overestimated (average of 0·112) by TFA and non-significantly underestimated (average of 0·012) by sensitivity.
4. When the indirect effects of harvesting on other vital rates were accounted for in a structured perturbation, TFA gave accurate predictions (average growth rate of 0·068), while sensitivity gave significant underestimates (average of −0·043).
5. Our results demonstrate two crucial sources of error that may influence predictions of the impacts of demographic perturbations on population dynamics. First, impacts of stage-specific harvesting are inherently nonlinear, hence predictions based on sensitivity must be treated with caution. Second, stage-specific perturbations can change non-target demographic rates, even in the absence of adaptation.
6. Population managers should consider both nonlinear and indirect effects of perturbations when designing management interventions. We encourage the development of methods to assess the robustness of predictions to unforeseen perturbation structures and indirect harvesting impacts.
- Top of page
- Materials and methods
- Supporting Information
A key question in population ecology is how a population’s growth rate will respond to changes in vital rates such as recruitment and mortality. The answer to this question has both an ecological and an evolutionary component. Recent interest in adaptive responses of phenotypes to harvesting of old or large individuals (Conover & Munch 2002), for example reduced age at first reproduction in exploited fish stocks (Walsh et al. 2006), has outpaced fundamental understanding of the link between harvesting strategies and population dynamics. This link defines the ecological impact of harvesting on population growth and sustainability, and helps to quantify the selection pressures imposed on exploited populations.
Predicting the relationship between harvesting strategy and population growth is generally achieved by perturbation analysis of life cycle models. Where a population is structured into stages, stage-specific vital rates can be compiled as a population projection matrix (PPM), of which the dominant eigenvalue, λmax, measures the asymptotic growth rate of the population. Partial derivatives of λmax with respect to changes in PPM elements are known as sensitivities (Demetrius 1969; Caswell 2001) or, on a log–log scale, elasticities (de Kroon et al. 1986; these estimate proportional changes in λmax in response to proportional changes in vital rates). PPM elements with the highest sensitivities or elasticities are predicted to give the greatest change in population growth rate (PGR) when perturbed by a fixed amount, and thus represent promising targets for population conservation actions (e.g. Crouse, Crowder & Caswell 1987; Crooks, Sanjayan & Doak 1998; Shea et al. 2005). Those with the lowest sensitivities or elasticities may indicate the stage-specific harvesting strategies to which the population is least vulnerable (Nault & Gagnon 1993; Olmsted & Alvarez-Buylla 1995). Sensitivity or elasticity is often extrapolated to estimate a population’s asymptotic growth rate following perturbation (Caswell 2001).
Any use of sensitivity or elasticity makes the important approximation that λmax has a linear (or log–log linear) response to changes in vital rates (de Kroon, van Groenendael & Ehrlén 2000). Alternatively, nonlinearity can be accounted for by simulating the perturbed PPM and recalculating λmax (Caswell 2000) or analytically, by applying transfer function analysis (TFA; Hodgson & Townley 2004). All of these predictive perturbation analyses assume that unperturbed and perturbed populations are close to their respective stable stage structures. They also assume that the PPM is subject to no changes in vital rates beyond those of the original perturbation (including no density dependence of the vital rates; Grant & Benton 2003). These limitations have led to calls for caution in the predictive use of sensitivity and elasticity analyses (Ehrlén & van Groenendael 1998; Mills, Doak & Wisdom 1999; Caswell 2000; Menges 2000). However, perturbation analyses must involve the implicit prediction of growth rates following substantial perturbations, if they are to be useful.
Bierzychudek (1999) provided an early test of PPM predictions, concluding that the dynamics of a population of Arisaema triphyllum since 1982 bore little resemblance to the predictions of the PPM constructed at that time (Bierzychudek 1982). Lindborg & Ehrlén (2002) also found PPMs to give poor predictions of population growth, in Primula farinosa, but Kuhn et al. (2001) found that an age-structured PPM gave good predictions of the PGR of Americamysis bahia exposed to various toxin concentrations. These examples only tested the accuracy of projecting the unperturbed population into the future. Fewer studies of PPMs have tested the predictions of perturbation analyses, and these have typically been for complex models including the effects of density dependence and/or stochasticity. Benton, Cameron & Grant (2004) constructed PPMs for laboratory mite populations subject to perturbation (reduced survival rates) and density dependence, with and without environmental stochasticity. They found that elasticities derived from mean PPMs gave inaccurate predictions of the effects of perturbations. Benton et al. (2004) considered within-stage processes and environmental stochasticity to have been responsible. A still more complex model for flour beetles (Dennis et al. 1995), including stochasticity as well as density dependence, made good predictions of the effect of population disturbances (one-off addition or removal of individuals). It seems that perturbation analyses applied to simple models, ignoring such effects as stochasticity and density dependence, may not give satisfactory predictions when the real populations are subject to these processes. The construction of more complex models has had mixed results in compensating for this. We take a different approach. Rather than making our models more complicated, we simplify our populations by minimizing any density dependence, transient effects, environmental stochasticity, differential cues for plastic responses and adaptation. We then investigate whether basic methods of perturbation analysis can predict the PGR reliably when such simplified and highly controlled populations are harvested.
The freshwater crustacean Daphnia magna Straus is an excellent model organism for stage-structured demographic analysis. Under favourable conditions, females reproduce parthenogenetically, producing large broods of female offspring at intervals of 3–4 days (Cox, Naylor & Calow 1992). Offspring grow without reproducing, until they reach adult size at around 8 days old (Cox et al. 1992), leading to fecundity rates which are strongly size dependent. In this study, we grew unperturbed populations of D. magna and populations in which harvesting reduced a single element of the PPM (representing the survival rate of the largest size class) to zero. The conformation of these populations to simple PPM models was maximized by eliminating, as far as possible, density dependence, transient effects and environmental stochasticity. Under these conditions, we hypothesized that predictions of the growth rate of the harvested populations made using linear perturbation analysis (sensitivity) would be less accurate than those of a nonlinear analysis (TFA). Harvesting caused indirect effects on PPM elements other than the one directly targeted. It is extremely unlikely that these indirect effects were caused by phenotypic plasticity or adaptation because we used isogenic, parthenogenetic stock, the timeframe of the experiment was short and the control and harvested populations were subject to standardized treatment. The indirect effects of harvesting allowed us further to hypothesize that structured analyses, accounting for indirect effects, would outperform simple, unstructured perturbation analyses.
Materials and methods
- Top of page
- Materials and methods
- Supporting Information
Population establishment and maintenance
Twelve populations of D. magna were established in November 2007, using 23–25 parthenogenetic descendents of a single individual (Blades Biological Ltd, Edenbridge, UK), per population (Table 1). Each population was maintained in 800 mL of hard water COMBO medium (Kilham et al. 1998) in a 1·0-L polypropylene beaker. A 100-mL sub-compartment made of polypropylene and 150 μm nylon mesh was suspended in each beaker, and held a sub-population (see below) apart from the main population. The mesh size of the sub-compartment prevented the passing of Daphnia, but allowed free movement of suspended algae. Daily feeding and replenishment of medium removed 150 mL of medium from each population (after mixing) and replaced it with 150 mL of fresh medium containing 3·64 × 105 cells mL−1 of thawed Chlamydomonas reinhardtii from frozen stocks. Beakers were also cleaned daily to remove algal biofilms. Algal food was grown before the experiment in hard water COMBO medium from inoculum CCAP 11/32A supplied by CCAP, Dunstaffnage Marine Laboratory, Oban, UK.
|Size class||Minimum (μm)||Maximum (μm)||Initial frequency|
The stage structure and population size of each main and sub-population were determined daily using a horizontal filtration channel. The main population was reduced to 400 mL without removing any Daphnia, and poured into the upstream end of the filtration channel. This contained a series of nylon mesh screens (John Stanier & Co., Manchester, UK) of decreasing aperture size, allowing the separation of Daphnia into six size classes (Table 1). The channel was 25 mm wide and filled to a depth of 38 mm. A peristaltic pump recycled medium to the upstream end at a rate of 750 mL min−1. The resulting flow speed of c. 20 mm s−1 was sufficient to ensure that Daphnia did not remain upstream of screens which they could easily pass through, but weak enough to prevent visible damage by compression against finer screens. After 2 min, during which Daphnia caught in eddies were freed manually, Daphnia were removed from each section of the channel for manual counting, harvesting (see below) and thinning. The sub-population, in 25 mL of medium, was then added to the channel and quantified in a similar manner.
Experimental treatment (harvesting) and thinning
The experimental treatment required the daily separation of size six Daphnia (Table 1) from the others of their population. This was achieved by releasing them into the sub-compartment of their population’s beaker after each day’s quantification. Individuals of other size classes, whether captured from the main or sub-population, were released into the main beaker. In treatment populations, any size six individuals captured in the sub-pot were discarded. Their offspring from the previous 24 h, however, were released into the main population, preventing any direct effect of harvesting on fecundity. This experimental treatment reduced the probability with which the largest size class survived and remained in their size class (element a6,6 of the PPM A) to zero. Because the animals to be harvested were left in the population until census, no other element of the PPM was directly affected by the treatment.
To prevent density dependence, a population was thinned whenever its total size before harvesting exceeded 60. Thinning without changing the population’s stage structure was achieved by discarding half of the Daphnia in each size class after stage structures had been quantified and harvesting treatments applied. Odd numbers were rounded up or down with equal probability.
Construction of population projection matrices
Wood’s method (Wood 1994; Caswell 2001) allows the parameterization of a PPM from a time series of population structure; a so-called ‘inverse method’. Quadratic programming selects PPM parameters, subject to specified constraints, that minimize the sum of squared deviations between observed and fitted abundances for all stages and time intervals together. We modified Caswell’s (2001) Matlab program for the parameterization of PPMs by Wood’s method, such that the population structure at the beginning and end of each projection interval were defined independently. We also translated it for Scilab (INRIA-ENPC 2006; script available from the authors). For our matrix parameterization, the population structure at the start of each projection interval was that of the released main or sub-population, and the population structure at the end of each projection interval was that of the main or sub-population captured a day later, before any harvesting or thinning. Because PPM elements were not calculated from underlying vital rates, our parameterization method makes no assumptions regarding the timing of reproduction relative to census. However, for the purposes of interpretation, they may be considered birth-flow populations (Caswell 2001), in which fecundity values depend on the raw survival rates of both parents and offspring.
Time series of PGR highlighted an initial period of transient fluctuation, from which data were discarded. PPMs were subject to the constraints shown in Fig. 1. Reproductive activity was restricted to sizes five (producing young which could grow up to size three by first census) and six (producing young of up to size four). Individuals could grow by up to three stages in a day, remain in the same stage or regress by one stage. While the chitinous carapace of Daphnia makes substantial reductions in size impossible, individuals may regress a stage between one day and the next either from chance acting on individuals of borderline size, or perhaps because of small changes in their external dimensions caused by flexion in the carapace. The division of each population into main and sub-populations gave additional information for matrix parameterization (e.g. size one individuals in the sub-population could only be the result of reproduction by a size six parent). Main and sub-populations therefore made independent contributions to the parameterization of each population’s PPM.
Comparison of treatment and control projection matrices
A population perturbation may have both direct and indirect effects on the vital rates comprising a PPM. We define direct effects as those which may be predicted using the PPM, and the classification of its census as post-breeding, pre-breeding or continuous breeding (Caswell 2001). For example, harvesting adults clearly has a direct effect on adult survival in the PPM, but if it takes place before reproduction in a projection interval is complete, it will also have a calculable direct effect on fecundity. By contrast, indirect effects of perturbation on the vital rates of a PPM act through harvesting-induced changes in properties of the population or environment which are not accounted for in the PPM and thus cannot be estimated without further knowledge of the system.
Because our PPMs were parameterized using data which excluded the direct effect of harvesting, we expected those from control and treatment populations to be indistinguishable. However, if harvesting had indirect effects on the vital rates comprising the PPM elements, we would expect differences between treatment and control populations. We used a randomization method to test for significant differences between PPMs parameterized for control and treatment populations. For each of the 36 elements of the PPM, we calculated mean values for control and treatment populations. The square of the difference between these two means was our test statistic for each PPM element, and the sum of squared differences gave a single test statistic to compare control and treatment PPMs. Data were then randomized 10 000 times by randomly re-labelling each population as control or treatment (924 distinct permutations possible) and recalculating the test statistics. Post-hoc comparisons of each PPM element between treatment and control populations were not adjusted for multiple testing, as the aim was to identify which elements were responsible for the overall difference between treatment and control populations. All randomization tests were carried out using the Poptools add-in (Hood 2006) in Excel Professional Edition 2003 (Microsoft Corporation, Seattle, Washington, USA).
Prediction of harvested population growth rates
The harvesting imposed on treatment populations had the direct effect of reducing PPM element a6,6 to zero. Other PPM elements were not directly affected; culled individuals were left alive until the end of the interval, allowing them to produce offspring or regress to a smaller size class. However, as indirect effects of harvesting could have occurred, we present methods for analysing harvesting as a structured perturbation (where several PPM elements may be directly or indirectly affected by a single perturbation), of which the simple perturbation (affecting only element a6,6) is a special case. A perturbed PPM, Apert, may be described as:
where A is the unperturbed (control population) PPM, p is the magnitude of the perturbation, and B is a structure matrix of the same dimension as the PPM. We chose to scale B such that p is equal to the change in element a6,6. b6,6 is thus equal to one in every case, and for the simple perturbation, all other elements of B equal zero. Where the comparison of control and treatment PPMs indicated a significant difference in a PPM element, however, the corresponding element of B for a structured perturbation was set as the effect of perturbation on that element, relative to the effect on element a6,6. For example, if a perturbation reduced element a6,6 by 0·2 and element a5,5 by 0·1, b5,5 would equal −0·1/−0·2 = 0·5.
Structured perturbation analysis by sensitivity uses the chain rule. The sensitivity of λmax to p is the product of the sensitivity of λmax to element aij and the sensitivity of element aij to p, summed over all ij (Caswell 2001):
where n is the dimension of the PPM and S is the n × n sensitivity matrix whose elements sij are the partial derivatives of the PPM’s dominant eigenvalue λmax with respect to each element aij of the PPM. Multiplication of the sensitivity ∂λmax/∂p by p extrapolates sensitivity to give the predicted change in the dominant eigenvalue λmax. In the simple case where a single PPM element aij is perturbed, the sensitivity ∂λmax/∂p is equal to sij, and the predicted change in λmax is equal to the product of sij and p.
Sensitivity and TFA are both tools for prospective perturbation analysis (Caswell 2000), but whereas sensitivity estimates lambda as a function of perturbation magnitude, TFA calculates perturbation magnitude as a function of any specified eigenvalue λpert (Hodgson & Townley 2004). TFA requires that the structure matrix B be transformed into one or more (q in total) pairs of column (d) and row (e) vectors such that:
The transfer function for λpert, G(λpert) is calculated as:
where I is an identity matrix of the same dimension as the PPM A. The eigenvalues of G(λpert)−1 are p; the perturbation magnitudes that achieve the desired eigenvalue λpert (Hodgson, Townley and McCarthy 2006). For simple perturbations, a single pair of vectors d and e defines the perturbation. G(λpert) is thus a scalar, e(λpertI−A)−1d, whose reciprocal is p.
Transfer function analysis and sensitivity were each applied for both simple and structured perturbations, giving four predictions of λpert from each control population. Predicted PGR were calculated as ln(λpert). Scilab (INRIA-ENPC 2006) was used for the calculation of all predictions.
Comparison of predicted and observed population growth rates
Observed PGR in each harvested population were calculated at each time interval as:
where Nt−1 is the total size of the population released (after any thinning or harvesting) at time t−1, while Nt is the size of the population captured at time t before thinning, but after any harvesting. PGR therefore excludes the effect of thinning, but includes the effect of harvesting. Division by the interval in days between times t−1 and t (It−1,t) adjusts PGR for minor variation around the standard projection interval of one day. PGR was then averaged over time (excluding any initial period of transient dynamics) to give a single value for each treatment population.
We used a randomization method to compare observed values of PGR with those predicted by each of the four methods of perturbation analysis. The test statistic was the square of the difference between the mean observed PGR and the mean predicted PGR (averages of six treatment or control populations respectively). Data were then randomized 10 000 times by randomly re-labelling each PGR value as observed or predicted (924 distinct permutations possible) and recalculating the test statistics.
- Top of page
- Materials and methods
- Supporting Information
Establishment of asymptotic growth
Because of a shortage in the stock populations, experimental populations were set up with fewer small individuals than were present in the stable stage structure suggested by previous experiments, and experienced some initial transient effects (Table 1; Fig. 2). Also, several females produced resting eggs at the start of the experiment, presumably because of high densities before the experiment began.
Populations experiencing transient effects are valid for matrix parameterization (indeed some transient effects are essential; Gross, Craig & Hutchinson 2002) but not for the measurement of asymptotic PGR. Initial fluctuations in PGR appeared to have subsided by mid-November (Fig. 2); we therefore used data from 19 November 2007 onwards for all further analyses. Almost all individuals after this date reproduced parthenogenetically; in the rare circumstance that females were found producing resting eggs, these individuals were preferentially removed during thinning.
The initial transient effects on PGR left treatment populations rather depleted. Any population whose total size fell below 15 was completely replaced with the individuals routinely thinned from a control population (i.e. half of the pre-thinning population; between 31 and 47 individuals). Three treatment populations were thus replaced before the start of data collection on 19 November 2007, and the other three were replaced shortly afterwards. The first release–recapture interval following repopulation was discarded from analyses, to avoid transient effects before the first harvesting of the new population.
Construction and comparison of population projection matrices
Our modified version of Wood’s method gave PPMs for each population (Fig. S1 in the Supporting Information) which were biologically plausible, and showed consistency within treatment and control populations (overall matrices in Fig. 3). However, there were significant differences between PPMs for treatment and control populations (randomization test; P = 0·0036). In treatment populations, elements representing reproduction by large adults (a1,6 and a2,6), large adult survival without regression (a6,6) and accelerated growth of stage 3 (a5,3) were significantly lower than in control populations (randomization tests; two-tailed P = 0·0418, 0·0016, 0·0141 and 0·0492, respectively). Note that differences in element a6,6 are in addition to the direct effect of harvesting; the rate at which adults survived and remained in size six was lower in treatment populations even before they were experimentally harvested. Element a5,6 (representing regression from large to small adults) was significantly higher (P = 0·0204) in treatment than in control populations. The overall large adult survival rate (regardless of any changes in size class) was thus reduced in treatment populations because of the lower value of a6,6, but increased due to the higher value of a5,6. These two effects cancelled out, such that total large adult survival before harvesting did not differ between treatment and control populations (randomization test for the sum of a5,6 and a6,6 among the 12 populations; two-tailed P = 0·957).
Because control and treatment PPMs differed significantly beyond the direct effect of perturbation, perturbation analyses were carried out both for direct perturbation to element a6,6 and for the structured perturbation represented by matrix B or vectors di and ei (Fig. 3). For TFA, two pairs of vectors d and e were necessary (q = 2); the first representing changes to PPM elements in the sixth column and the second representing changes to a5,3 (Fig. 3). The structured perturbation included a strong, negative, indirect effect on fecundity. Structured perturbation analyses thus predicted lower values of λmax for any given negative perturbation than did the equivalent simple perturbation analysis (eigenvalue–perturbation curve for combined control populations in Fig. 4; log-transformed predictions for the specific harvesting regime imposed on the treatment populations in Fig. 5).
For unstructured perturbations, TFA significantly overestimated PGR in the harvested populations (P = 0·0023; Fig. 5), while the predictions of sensitivity, although low, were not significantly different from observed values (P =0·0673; Fig. 5). In contrast, when structured perturbations were applied, sensitivity gave significant underestimates of PGR in the harvested populations (P = 0·0049; Fig. 5) while TFA gave predictions which were not significantly different from observed values (P = 0·3534; Fig. 5).
- Top of page
- Materials and methods
- Supporting Information
The widespread use of perturbation analyses demands a closer examination of their limitations. We set out to test the importance of nonlinear responses to perturbation, by closely controlling our experimental populations to maximize conformity with density independence and isolation of harvesting effects. Despite our best efforts another source of error, indirect effects of perturbation, also affected the quality of predictions. This allowed us to demonstrate techniques that account for both nonlinearity and indirect effects, and to highlight the possible confounding of ecological and evolutionary responses to harvesting.
Nonlinearity in the eigenvalue–perturbation curve
The direct effect of harvesting was to reduce PPM element a6,6 to zero. It is known (Cohen 1978, 1981) that the relationship between the dominant eigenvalue λmax of a PPM and perturbations to an element of its main diagonal is necessarily convex. When such an element is perturbed, the extrapolation of linear sensitivity must therefore underestimate λmax of the perturbed PPM to some extent. As the PGR of a population at stable stage distribution is the logarithm of λmax, PGR would also be underestimated. The application of sensitivity to control PPMs did indeed underestimate the PGR of harvested populations (Figs 4 and 5) although this underestimation was not significant when only the simple perturbation, corresponding to the direct effects of harvesting, was considered. The degree of nonlinearity was substantial; linear (sensitivity) and nonlinear (TFA) perturbation analyses gave average predictions of PGR which differed by about 0·1 for both simple and structured perturbations (Figs 4 and 5). According to the PPM element being perturbed and the structure of the PPM, λmax may be a consistently concave, convex or linear function of the perturbation magnitude, or its convexity may depend on the parameter values in the PPM (Carslake, Townley & Hodgson 2008). Sensitivity analysis may therefore produce overestimates, underestimates, or accurate predictions of PGR in perturbed populations in a pattern that is only partially predictable. Many perturbation analyses in population management use elasticity (de Kroon et al. 1986) rather than sensitivity analysis. We could not apply elasticity analysis, because the reduction of a vital rate to zero is an infinite change on the logarithmic scale used in elasticity. On a log–log scale, asymptotic growth rates are always an accelerating (convex) function of perturbation, so the extrapolation of elasticity will always underestimate the growth rate of a perturbed population. The magnitude of error when sensitivity or elasticity is subject to linear extrapolation will vary between PPMs, but can increase dramatically as the size of the perturbation increases. The conclusions of studies based on sensitivity or elasticity should always be backed up by a nonlinear analysis, particularly when the perturbations concerned are large.
Indirect effects of harvesting
If harvesting had changed only the intended element a6,6, then simple perturbation analysis using TFA should have made accurate predictions of PGR in treatment populations, and sensitivity given substantial underestimates. Instead, TFA gave significant overestimates, and sensitivity gave better predictions (Fig. 5). This, supported by the randomization tests, indicated substantial indirect effects of harvesting. We conclude that harvesting changed the age and/or size structure within stage six, resulting in an indirect effect on other PPM elements. Individuals in treatment populations were removed if recorded as size six for successive days, so the remaining size sixes were probably younger and smaller than those in control populations. A smaller average size of these individuals in harvested populations would explain their greater tendency to regress to size five, if there was either measurement error in their division into stages, or a genuine ability to decline slightly in size. Daphnia magna also produce smaller broods earlier in their adult life (Cox et al. 1992), which explains the lower fecundity of the size six individuals in treatment populations. The significant difference between control and treatment PPMs in element a5,3 might result from an effect of a mother’s age or size on the growth of her offspring. Younger or smaller mothers are likely to have accumulated fewer resources, which may result in smaller broods (Cox et al. 1992) and/or poorer quality offspring. Most of the non-significant differences between control and treatment PPMs for the growth of smaller stages (first three columns of Fig. 3a,b) also indicated that small Daphnia in treatment populations were growing more slowly. However, the marginal significance (P = 0·0492) of the result for a5,3 and the very small effect size (transition probabilities of 0·032 and 0·000 in control and treatment PPMs, respectively) means that we cannot rule out this being a chance result, especially given the number of tests conducted.
Population projection matrix analyses assume that individuals within a stage have identical vital rates; violation of which may have important consequences for the reliability of PPM analyses (Pfister & Stevens 2003; Benton et al. 2004). When we applied unstructured perturbation analyses to control populations, differences between control and treatment populations in the age or stage structure (and thus vital rates) within size six appear to have resulted in considerable overestimation of the PGR of treatment populations. This overestimation largely cancelled out the underestimation because of the application of sensitivity analysis, resulting in simple sensitivity predictions which were fortuitously accurate. There is no reason to assume that this would happen in every case; indirect effects of perturbation are difficult to predict, and nonlinearity may lead to overestimation or underestimation by sensitivity, depending on the PPM element being perturbed (Cohen 1978; Kirkland and Neumann 1994; Caswell 1996; Carslake et al. 2008). There must be few natural populations in which vital rates are truly independent of the time an individual has spent in a stage. Hence indirect effects of harvesting are likely to be the rule, rather than the exception. The division of the population into stages carefully defined according to the biology of the population, or into a greater number of arbitrarily defined stages, will reduce these effects although the latter will cause a loss of accuracy when defining the greater number of stage-specific vital rates. Integral projection models (Ellner & Rees 2007), in which vital rates are continuous functions of one or more age or stage variables, may solve these problems, although fewer perturbation tools are currently available for these more complex approaches. Their application in this study was not possible, because size was measured on a discontinuous scale.
We have assumed in this study that the Daphnia did not adapt in any way to the harvesting regime. Our use of asexually reproducing clonal populations makes selection an unlikely candidate cause of indirect changes due to harvesting. We consider phenotypically plastic responses to have been unlikely, as there were no cues in the animals’ environment to suggest higher mortality in treatment populations, other than their own population structure. Most recorded cases of plasticity in Daphnia require some form of environmental stimulus (e.g. Hammill, Rogers & Beckerman 2008). Many studies have identified evolutionary or plastic responses of populations to harvesting (see, for example, Conover & Munch 2002; Walsh et al. 2006). The changes in stage-specific life history traits that we identify, in the presumed absence of evolution and plasticity, illustrate that care must be taken before assuming that demographic changes in non-target age- or size-classes are the result of adaptation. Such indirect effects of harvesting could simply be due to unidentified within-class variation in demographic rates. On the other hand, a longer repetition of our experiment with genetically mixed populations might find adaptation of demographic rates in response to harvesting. Under our harvesting regime (removal of the largest individuals), breeding at a smaller size would clearly be advantageous. Such a response would add to the indirect effects of the perturbation, and thus to the potential inaccuracy of predictions made using simple models.
Improving the accuracy of perturbation analysis
Despite our efforts to make the experimental populations conform to the assumptions of PPM models, simple perturbation analysis overestimated PGR in harvested populations, and sensitivity underestimated it (Fig. 5). Structured perturbation analysis by TFA gave predictions of PGR which were not significantly different from those observed in treatment populations. However, structured perturbation analysis requires data on the structure of the perturbation. Where a perturbation is known to have direct effects on more than one PPM element, this structure may be anticipated a priori. For example, in real harvesting situations, increases in stage-specific mortality are usually imposed continuously (rather than immediately before census, as we did). This has an effect on the fecundity of the targeted stage that is calculable, given knowledge of any seasonality in reproduction. Where effects on other PPM elements are indirect, the perturbation structure may be inferred from a small perturbation and extrapolated, assuming that relative changes in PPM elements are independent of the magnitude of perturbation. We, however, had no a priori expectation of the perturbation structure, and applied only one magnitude of perturbation. This obliged us to infer the perturbation structure from the same perturbed populations whose growth rate was being predicted. Error in the definition of the structure matrix B was limited to that due to the minor variation among populations. In reality, population managers applying a structured perturbation analysis would not have such accurate data on the perturbation’s indirect effects. We considered the indirect effects of our experimental treatment to be due to violation of the PPM assumption that stages encompass individuals with uniform vital rates. If the necessary data are available, careful matching of the chosen model structure to the demographic structure of the population will reduce, but probably never eliminate, this problem. The choice of model structure (e.g. a Leslie vs. a Lefkovitch matrix) may also constrain some PPM elements to have a greater sensitivity or elasticity than others, inevitably favouring some population management strategies over others (Carslake, Townley and Hodgson in press).
It might be asked why the predictions from TFA applied to structured perturbations were not even better, since data from the same populations whose PGR was being predicted were included in that prediction, and both indirect effects of harvesting, and nonlinearity of the eigenvalue–perturbation curve, were accounted for. Perturbation analyses predict the asymptotic PGR. Our treatment populations were in an approximately asymptotic state (Fig. 2), but some fluctuations were inevitable (and indeed essential for PPM parameterization; Gross et al. 2002). The long-term growth rate of a population can be influenced by the standard deviation, as well as the mean value, of vital rates in the PPM (Tuljapurkar, Horvitz & Pascarella 2003). However, we consider our populations to have been at least as close to an asymptotic state as any natural populations for which perturbation analyses might be employed. Further inaccuracies in PGR estimated from nonlinear perturbation analysis may have arisen from minor variation in the projection interval (standard deviation of 30 min around a mean of 1·00 days), which was adjusted for in the direct calculation of PGR, but not in the matrix parameterization process. Finally, we used an inverse method to parameterize PPMs from time series of population structure, in the absence of more precise individual or cohort data. This will have caused some noise in the PPM parameter estimates. Although PPMs parameterized from control and treatment populations were generally distinct, there was considerable variation within each (Fig. S1).
In conclusion, we applied both linear and nonlinear single-element perturbation analyses to deterministic PPMs parameterized from unperturbed populations, but failed to make accurate predictions of the growth rate of harvested populations despite the carefully controlled conditions. This result adds to the calls already made (Silvertown, Franco & Menges 1996; Ehrlén & van Groenendael 1998; de Matos & Silva Matos 1998; Mills et al. 1999) for caution to be applied in the use of PPMs. We consider unintended, indirect effects of the experimental perturbation to have been responsible for much of the prediction inaccuracy which we observed and would encourage the development of methods to assess the robustness of predictions to uncertainty in perturbation structure. The use of further data from experimentally harvested populations allowed reliable predictions to be made. This information is not available in most practical applications, but lower quality information on perturbation structure may be available from a priori reasoning, or the extrapolation of lesser perturbations. We urge those making predictive use of perturbation analysis to choose a model that reflects the complexity of the population’s dynamics and to consider the full structure of the perturbation, including any indirect effects. We encourage the use of the precise, nonlinear techniques now available for such analyses.
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Julia Reger made an important contribution to the collection of experimental data. Tom Richardson constructed the filtration channel. Comments by Pieter Zuidema, Selina Heppell and two anonymous reviewers improved earlier versions of this paper. Work was funded by NERC, the European Social Fund and the University of Exeter.
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Fig. S1. Population projection matrices for each experimental population.
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