and present address: T. G. Benton, School of Biological Sciences, Zoology Building, University of Aberdeen, Aberdeen, AB24 2TZ, UK. Tel: 01224 272399; E-mail: firstname.lastname@example.org
1Population dynamics results from an interplay between the environmental state and population density. With many organisms there is structure to the life history, and this structure has important consequences for the population's density dependence and its interaction with environmental noise, and therefore its population dynamics. Perturbing population structure, such as through harvesting, may therefore affect the way that the populations respond to stochastic environmental variation.
2We conducted three experiments on populations of soil mites kept under controlled conditions and harvested a constant proportion of eggs, juveniles or adults. The experiments (a) minimized environmental variability, (b) created environmental variability by randomizing food supplies, and (c) provided excess food and repeatedly subdivided the populations to maintain them below carrying capacity.
3We find that harvesting different stages has marked effects on stage structure, which differ between constant and variable environments. For example, harvesting adults decreases the number of adults, harvesting eggs increases the number of adults and harvesting juveniles has no effect under near constant conditions, whereas in a variable environment harvesting adults and juveniles reduces adult numbers, but harvesting eggs has no effect.
4As well as changing the mean age structure, harvesting can change the variance of the different stages. For example, in a constant environment harvesting juveniles does not change the variability in juvenile numbers or population size, but in a variable environment harvesting juveniles increases the variability in the size of the juvenile class and hence the total population variability.
5For populations that are kept at one-third to one-half carrying capacity (approximately where the maximum sustainable yield should result) harvesting of the different age classes has marked positive effects on the population growth rates. Harvesting different age classes causes changes in the density dependence, which explains the way in which the population parameters respond.
6In conclusion, the population response to harvesting depends on the stage/age structure and the way it changes with harvesting and environmental conditions. Managing economically important populations, subject to harvesting, should consider structured life histories.
In the simplest case (a closed population, no predators, uniform environment, etc.), the number of organisms largely depends on the amount of resources available and the way those resources are partitioned between members of the population. The interplay between the two governs the density dependence of the population: on average when the population density is low, there are more resources available per head, survival and reproduction are high and so the population grows. Conversely, when density is high, there are fewer resources available per head, survival and reproduction are lower and the population declines. In a constant environment, density dependence will often lead to a stable equilibrium in population size. At this equilibrium population numbers will be constant, as will the age or stage structure (the ratio of numbers of individuals in different ages or stages).
This equilibrium system behaviour is unlikely to occur in real populations because real populations will be subject to constant perturbation. This perturbation can come about through variation in the environment, whether stochastic or periodic, or perturbations can arise from differential mortality, notably by pathogens, parasitism or, in an applied context, harvesting. Theory suggests that the response will depend on the current age structure, the way the age structure is perturbed and the way density dependence operates. For example, the effects of a perturbation will differ depending on whether or not density dependence is over-compensatory (May 1974), and whether the density dependence is a function of total population size, the numbers in a specific class, or if it is more complex (Grant & Benton 2003; Hellriegel 2000). In addition, many dynamical models can show multiple stable equilibria, where one of the equilibria is a stable point (Briggs et al. 1999; Greenman & Benton 2004). Perturbations can then move the dynamics between the different attractors (Cushing et al. 2003). For example, a one-off harvest of adults could reduce their numbers, freeing up resources, which allows greater juvenile survival, causing greater competition, reduction in recruitment and a lower, but still stable, number of adults.
A given environmental state (e.g. amount of resources at time t) is likely to cause different population responses, depending on the stage structure. This is because organisms of different ages respond differently. For example, food-deprived juveniles are likely to survive less well than juveniles with food, but food-deprived adults could increase their survival if they decrease reproductive effort in response to food shortage. Hence, the change in population size may differ depending on whether the population is largely made of adults or juveniles. Such stage-structured effects are likely to desynchronize populations, even if they experience similar patterns of environmental noise (Benton et al. 2001). Modelling population dynamics for quantitatively accurate, as well as qualitatively accurate, predictions therefore requires incorporation of (st)age-specific details (Astrom et al. 1996; Jonzen & Lundberg 1999; Hellriegel 2000; Coulson et al. 2001; Kokko 2001). Field studies have indeed shown that (st)age structure is an extremely important determinant of population dynamics, whether it be the population response to the weather, number of competitors, predators or harvesting (Higgins et al. 1997; Stenseth et al. 1999; Hellriegel 2000; Coulson et al. 2001; Clutton-Brock & Coulson 2002; Reid et al. 2003).
The sustainability of exploited populations requires density-dependent renewal: when population density is decreased by harvesting, the remaining individuals compensate by surviving or reproducing better. Given the importance of stage structure on population dynamics, the interplay between harvesting, stage structure and population dynamics has only recently begun to receive attention in the applied literature. The majority of empirical studies (1) compare populations which are harvested or released from harvest, and describe changes in stage structure associated with harvesting (Adams et al. 2000); (2) report analysis of population records and associating changes with changes in harvesting (Fryxell et al. 1999; Solberg et al. 1999; Murphy & Crabtree 2001); (3) build models parameterized with data to predict the effects of harvesting (e.g. Langvatn & Loison 1999; Saether et al. 2001); or (4) build general models which are not specific to any one system (e.g. Kokko 2001; Jonzen et al. 2002; Runge & Johnson 2002). Nicholson (1957), as part of his seminal series of blowfly experiments, reports the differential effects of harvesting adults or larvae. However, such properly controlled experimental studies of harvesting are rare: without a range of empirical studies modelling different systems becomes difficult, because the assumptions that are necessary (about density dependence and how it works across stages, and how density interacts with the environmental variability) may make important differences in the realized dynamics, and without empirical systems to compare the actual dynamics with model dynamics one does not know whether the assumptions made have validity.
Two general questions are of particular importance. First, in a stage- or age-structured population, does harvesting different (st)ages have differential effects on population size? This is of particular importance in targeting the stage to be harvested: should you harvest large and leave small or vice versa? This question has often been tackled using elasticity analysis of a matrix model to see the response of population growth rate (λ) to changes in the survival of the different stages (Benton & Grant 1999), but these models typically do not include density-dependence and so may not predict the changes in stage structure that result from changes in density. Secondly, does harvesting different (st)ages have differential impacts on the way the population responds to environmental variability? Of particular interest would be if harvesting one particular stage caused the impact of environmental variability to be magnified, increasing the overall variability of the population and making it more extinction-prone.
It is with this background we undertook this study. We used replicated populations of a soil mite, Sancassania berlesei (Michael), and applied proportional harvesting regimens to three stages in the life cycle (eggs, juveniles and adults). Comparison between the harvested and non-harvested control populations allows the impact of harvesting on stage structure to be elucidated. We undertook this experiment under three ecological scenarios: ‘near constant’ conditions, variable conditions, and in excess food conditions as would be found in populations below carrying capacity.
We used replicated populations of the soil mite S. berlesei. This is an acarid mite, with a life cycle that consists of egg, three juvenile instars and adults. In this study we subdivide the life cycle into three stages: eggs, juveniles and adults. As discussed elsewhere (Benton, Cameron & Grant 2004) this subdivision, with hindsight, leaves out important detail as any model using the ‘juvenile’ class does not capture the asymmetries between juveniles of different sizes. Similarly, subsequent work (Plaistow et al. 2004) illustrates the plasticity of age-and-size at maturity and that adults may differ considerably in size, fecundity and survival. It would therefore be biologically reasonable to subdivide individuals into more classes than the three we use here (whether instars, sizes or ages or some combination). However, in the applied context of this work, lumping of (st)ages into ‘juvenile’ and ‘reproductive’ classes is often forced due to costs involved in censusing populations (it would, for example, require 2× the work to census each instar rather than the three stages). The harvesting and associated census strategies are based on realistically simple stage-structured harvesting regimes: eggs, juveniles (equivalent to harvesting individuals below a critical size) and adults (equivalent to harvesting individuals above a threshold size).
The mites’ generation time varies with food supply, but under good conditions the minimum egg–egg time is 11 days (Beckerman et al. 2003). Cultures are kept in a constant-temperature (24 °C) incubator, in sealed 20 mm-diameter, flat-bottomed glass tubes, with a plaster of Paris substrate, and fed on dried yeast, on which both adults and juveniles feed. Adults and juveniles differ in size and therefore competitive ability. Populations are censused by counting mites using a Leica MZ8 binocular microscope and a hand counter. In each tube, a sampling grid is scratched onto the plaster base. Replicate cultures were initiated with ∼100 adults from stock culture. There were four treatments: the unmanipulated control and 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 at random; for the adult counts, males and females were counted, and harvested separately to avoid biasing the sex ratio. 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 populations in the ‘excess food’ experiment were given an excess of food (30 + granules of yeast per day). To avoid these populations growing to carrying capacity, on average every second day following the census, the population was split in two by removing half the mites. Half the adults were removed at random. Juveniles and eggs, which are typically spread uniformly across the tube, were removed from two randomly selected quarters of the tube. Following the removal, the remaining mites were re-counted and the appropriate harvesting undertaken. Analysis of age structure used the data from the days on which the population was reduced by half; analysis of population growth rates used the data from the day on which half the population was removed and the next day. Population growth rate, λ, was defined as Nt+1/Nt, where Nt+1 are the preharvest counts on day t+ 1, and the Nt are the post-harvest counts on day t.
The constant experiment was conducted over 86 days in late 2000, and the variable experiment over 101 days in spring 2001. The excess food experiment was conducted in two batches. Part one consisted of six tubes (two control, four adult manipulation) and ran for 54 days in early summer 2001. Part two consisted of 12 tubes (four egg, four juvenile and four control tubes) and ran for 30 days in late summer 2001. Populations were initiated and then allowed to equilibriate (constant: 34 days, variable: 21 days, excess food: used tubes set up for the constant or variable experiments). Following the onset of manipulation, the constant populations showed transient changes prior to settling onto new equilibria (Fig. 1). For the analyses described below for the constant and variable experiments, we used data from two weeks after the onset of manipulation to reduce the impact of transients. This gave 22 counts (taken over 35 days) for the constant experiment and 46 counts (taken over 65 days) for the variable experiment. The excess food experiment gave rise to 16 counts (part 1) or 14 counts (part 2).
Bootstrap resampling was used extensively to estimate confidence limits for the descriptive statistics reported per treatment group (means, variances, stage structure, population growth rates, etc.). Resampling was stratified by culture tube within each treatment group, to ensure that there were no biases due to tube effects, and 1000 resamples were taken. These samples were used to estimate the bias-corrected and adjusted (BCa) 95% confidence intervals (CI). If the statistic's 95% CI for treatment groups do not overlap then, by definition, the statistic differs across groups at P < 0·05 level.
We investigated the relationship between the adult numbers at the next time step as a function of density at the current time. Models were fitted using restricted maximum likelihood (REML) to control for the random effects of tube. In addition, the presence of temporal autocorrelation was investigated by specifying an AR(1) correlation structure within the linear mixed effects model. The model-fitting process started with all interaction terms and simplified until only significant terms remained.
The population growth rate was estimated as r = ln[Nt+1/(Nt − Ht)], where Nt is the population size (eggs + juveniles + adults) at time t, and Ht is the number harvested. The theta logistic model:
was fitted to the data using non-linear regression. This regression equation contains three parameters: the y-intercept, r0, interpreted as the maximum growth rate at low population density, the x-intercept K, interpreted as the carrying capacity and θ which allows the effect of density on population growth rate to be non-linear (Fowler 1981).
the experimental time-series: constant and variable conditions
Under the ‘constant’ feeding regime, the population dynamics show a rapid approach to stage-specific equilibria (Fig. 1). In the control tubes, adult numbers fluctuate little after about 20 days, juveniles are more variable but the time-series is stationary from about 40 days. Egg numbers do not become stationary until day 60. In the manipulated tubes, there are clear treatment effects following the initiation of harvesting at day 35. For example, adult numbers increase over about 15 days in the egg-harvesting treatment, decrease markedly over a few days in the adult harvesting treatment, but fluctuate over a 30–40-day period in the juvenile harvesting treatment. This fluctuation itself corresponds to fluctuations in juvenile numbers: on the onset of harvesting juveniles, the numbers of juveniles decrease. However, this is a transient change: juvenile numbers recover before declining again; adult numbers then decrease as juveniles increase and then increase as juveniles again decrease (Fig. 1). This pattern is suggestive of density-dependent interactions between the stages.
Under the variable feeding regime, the population dynamics are markedly different from the inherently equilibrium dynamics of the populations in constant conditions (Fig. 2). The control tubes indicate that the different stages reflect the environmental variation differently: mean adult counts fluctuate over a magnitude of about 3×, juveniles 6× and eggs by over three orders of magnitude (mean counts across the four replicated tubes vary from 1 to 1230). Again the effects of harvesting are discernible in the time-series graphs. For example, harvesting juveniles appears to increase the variation in juvenile numbers and harvesting adults seems to increase the variability in adult numbers (Fig. 2).
the effects of harvesting on means and variances: constant and variable experiments
After a period of transience for 2 weeks following the onset of manipulations, the mean numbers of the different stages differ depending on both the food regime (constant vs. variable) and the stages harvested. In the constant experiment there are significant differences in the counts, relative to the controls, for each stage class (Fig. 3). Of note is that harvesting adults reduces egg, juvenile and adult counts relative to the control and, as a result, has the biggest effect on overall population size (Benton et al. 2004). Conversely, harvesting eggs increases significantly the numbers of adults. This positive effect of harvesting eggs is also seen in the mean stage counts in the variable food experiment (Fig. 3). Harvesting eggs leads to a significant increase in the number of juveniles in the variable experiment but not the constant experiment, and as a result has an overall positive effect on population size when food is variable (Benton et al. 2004). That harvesting of eggs has a positive effect, but harvesting juveniles does not, suggests that the positive effect comes about not by changing juvenile numbers but by changing juvenile age structure. Harvesting eggs reduces the number of eggs which hatch into the smallest juvenile class. Conversely, harvesting juveniles reduces the number of all juvenile instars. The first instar is the smallest and competitively likely to be the weakest, so reducing numbers at this stage might, like thinning seedlings, lead to a greater survival from the vulnerable stage. This applies particularly in the variable environment, because the environmental variability imposes a strong cohort structure on the population (a food pulse leads to a pulse of eggs hatching together) increasing the competition within the smallest juvenile instar.
In the variable food experiment, the proportion of juveniles is, on average, higher than in the constant experiment (mean proportions in each stage are 0·094 ± SE 0·0014 adults, 0·517 ± 0·0052, juveniles and 0·388 ± 0·0048 eggs under the constant food and 0·182 ± 0·007, 0·747 ± 0·041 and 0·199 ± 0·030, respectively, in the variable food experiment). Harvesting juveniles has a significant affect on both adult numbers and the more numerous juvenile numbers. As a result, harvesting juveniles causes the greatest effect in terms of total population size in the variable environment (Benton et al. 2004).
Stage-structured harvesting leads not only to effects on mean stage structure, but also the variance of each stage (Fig. 4). Harvesting eggs in the constant food experiment (black dots in Fig. 4) leads to significantly greater variance in the numbers of eggs, and also a greater variance in juvenile numbers (on the exclusion of one control tube whose variance in juvenile numbers was significantly greater than the other three; Table 1). All harvesting regimes increased the variance in the egg counts (although egg-harvesting only did so significantly). Only egg-harvesting increased the variance of juvenile numbers; but harvesting juveniles and adults led to significant increases in variances in the males, females or combined adult counts (Fig. 4). Under variable food (Fig. 4, white dots) harvesting eggs tends to decrease the variance, whereas harvesting juveniles increases the variance in juvenile numbers and harvesting adults increases the variance in adult numbers (Fig. 4).
Table 1. Estimated standard deviations and their confidence intervals (CI) for juvenile counts within the four control tubes from the constant experiment
Replicate 4 has a significantly greater variability than the other three replicates.
The mean population size is a summary of the experimental effects but it omits information on the way that density dependence might change. For a single stage, adults, we fitted linear mixed-effects models to estimate the relationship between the current and future densities, taking into account the number of juvenile competitors (Table 2). We found significant effects of treatment on the density dependence of adult numbers (Table 2). In all cases there are significant treatment × density interactions, making the slopes of the At vs. At+1 relationship for each treatment group non-parallel. In particular, in the constant experiment the slopes of all the harvested treatments are lower than that of the control, indicating that the effect of adult density is lower in harvested populations. The point at which the control and treatment curves cross also differs (for example, for the egg harvesting treatment the intercept occurs at about 250 adults, whereas for the adult harvesting it occurs at about 140). Below these densities, harvesting has positive effects relative to the control. Hence, if the adult density was reduced enough by harvesting, a positive effect on adult numbers might also occur (as found in the ‘excess food’ experiment and by Nicholson 1957) or, conversely, if the adult density was greater, the positive benefit of egg harvesting would decrease or disappear. In addition, there is a change in density dependence caused by the environmental variability. Under constant conditions, adult numbers are a function of treatment and previous adult numbers. Under variable conditions, the effect of previous juvenile densities also becomes significant. There is an interaction between juvenile and adult densities: juveniles have a negative impact on adult densities when both adults and juveniles were common in the recent past.
Table 2. Linear mixed effects models estimating the relationship between current adult numbers, treatment and past densities
For both the constant and variable experiments, the initial models were the same, including treatment, past adult and juvenile numbers and all interactions. Models were simplified to leave only significant terms. The ‘statistic’ is the likelihood ratio for the test for autocorrelation and the F statistic for all the fixed effects’ terms. The random effects (tube effects) are not reported here. The treatment group coefficients are contrasts with the control group (E, J and A represent harvesting of eggs, juveniles and adults).
E 78·16530 22·12557
J 26·91554 17·17775
A 33·18191 16·60314
A 0·544056 0·396707
Treatment × ln(At−1+ 1)
E 0·004769 0·089728
J 0·008701 0·093518
ln(At−1+ 1) × ln(Jt−1+ 1)
the effects of harvesting in ‘excess food’ environments
The concept for the ‘excess food’ environment was to allow the population to exist below carrying capacity, and hence for the population to be able to grow. To this end the populations were split every 2 days, and fed some 15× the daily food than the other two populations. To assess the changes in density dependence across the experiments, we fitted theta-logistic models to the population growth rates, pooling data across tubes (Fig. 5, Table 3). The fitted equations indicate that the three experiments differed in the carrying capacities but not in the shape of the density-dependence (Table 3). The predicted carrying capacity for the excess food experiment was 4973 (Table 3) and the median density in the culture tubes was 1840 (n = 258 counts), hence the experimental procedure was successful at reducing the density to about one-third of carrying capacity. In contrast, in the constant experiment the median density was 1907 (n = 687) and the predicted carrying capacity was 1997, and in the variable experiment the median density was 1343 (n = 847) and carrying capacity 1489.
Table 3. Parameters of the theta logistic equation fitted to the data in Fig. 5
The theta logistic equation describes density dependence as a function containing three parameters: r0, the population growth rate at negligible density, K, the carrying capacity and θ, which allows the effect of density on population growth rate, r, to be non-linear. Values of θ < 1 indicate a concave relationship. 1,2,3Different superscripts represent significant differences between the parameters, as estimated using t-tests.
That the slope of density dependence is the same in the three experiments explains the result that harvesting from a stage increases the population growth rate of that stage (Fig. 6a): density dependence is as strong in the excess food conditions as it is at carrying capacity. Reducing the density leads to release from competition and a positive effect on population growth rate. In the terminology of elasticity analysis (Benton & Grant 1999; Caswell 2001) there are negative elasticities of λ. Similar results are obtained (although with smaller effect sizes) if the population growth rate analysis is undertaken for populations at equilibrium (the constant experiment).
As well as altering growth rates, harvesting populations that are below carrying capacity leads to changes in stage structure (Fig. 6b). These changes typically make sense in the light of density dependence. For example, harvesting adults reduces the proportion of eggs (because there are fewer adults laying), but increases the proportion of juveniles (because they have fewer competitors, so more survive), whereas harvesting juveniles increases the proportion of eggs, because the adults obtain more per capita food to invest in laying.
Many previous studies have investigated the relationship between harvesting and age structure (e.g. Langvatn & Loison 1999; Murphy & Crabtree 2001; Strickland et al. 2001; Marboutin et al. 2003). Such studies are based typically on either covariation of harvesting with demographic variation in the field, and hence correlational in nature, or are model-based. As with Nicholson's (1957) study, our approach is experimental, where we perturb the stage structure, as would occur with stage-structured harvesting of commercially important species. Our harvesting is conducted in environments that are close to constant (where harvesting is the main perturbation), in environments that fluctuate in the food supply (so harvesting is an additional perturbation), and in environments where the population is well below carrying capacity (as may be the case in exploited species).
From this experimental approach there are four key results. First, harvesting different stages has differential effects on the other stages − from no effect to negative or positive effects. Secondly, the impact of harvesting also affects the variability of each stage, typically by increasing it. Thirdly, the effects of harvesting depend on the environment experienced by the organism − whether the environment is constant (and density dependence is strong) or fluctuating (with density dependence varying over time). Finally, we show here that populations given excess food and kept considerably below carrying capacity are still subject to density dependence of the same form as populations close to carrying capacity and, as a result, respond in similar ways to populations closer to the equilibrium.
the (st)age structured effect of harvesting
We have shown that selective harvesting alters both the stage structure and the total population size, and is dependent on the stage harvested. This is very noticeable under near constant conditions, where the onset of harvesting leads to a transient change prior to the system settling on a new stable state with altered stage structure. Moving between population equilibria in response to perturbations is a known phenomenon in both theory (Bjørnstad & Grenfell 2001) and from laboratory systems, most notably the Tribolium system in which altering adult mortality can lead to a rich range of dynamics (Cushing et al. 2003).
The changes in both stage structure and population size can be either positive or negative depending on the stage harvested and the stage considered. Negative effects are expected (e.g. harvesting adults reduces adult numbers), but positive effects (e.g. harvesting eggs increases adult numbers) are rarely considered, even though they were first shown and discussed by Nicholson (1957) in his blowfly study. Indirect positive effects have been noted recently in an analysis of the Tribolium population model (Grant & Benton 2003), and empirically in blowflies (Moe et al. 2002). Indirect positive effects come about through changes in density, where reduction of numbers at one stage in the life cycle leads to a lessening of density-dependent effects on other stages in the life cycle. Therefore, to have the possibility of identifying indirect positive effects requires density-dependent stage-structured models. Common techniques for identifying the effects of perturbations on population size or growth rates, such as sensitivity analysis of density independent models, cannot, by definition, predict any positive effects that harvesting may have. The existence of indirect positive effects raises intriguing possibilities for population managers, as it suggests that targeted harvesting (or biocontrol strategies) may be beneficial in terms of total population size. Hence populations that are threatened could, under some circumstances, benefit from continued harvesting rather than a complete cessation of exploitation.
harvesting affects population variability
It has been widely recognized that there is a link between population variance and persistence (Ripa & Lundberg 1996; Halley & Iwasa 1998; Heino 1998; Vucetich et al. 2000; Reed et al. 2003). We have shown that a proportional harvesting strategy can interact with the inherent density dependence and increase the variation in (st)age distributions under both near constant and variable conditions. This supports the assertion that the interaction between stochastic harvesting and density dependence, especially when other density independent factors vary, is an important determinant of population variability (Jonzen et al. 2001, 2002 ). For example, harvesting juveniles tends to amplify variability in juvenile numbers, and therefore the numbers recruiting into adulthood and, in turn, the numbers of eggs laid (Fig. 4). The origin of such effects presumably arises from a 15% reduction in juvenile density affecting competition (within juveniles and between juveniles and adults) in a non-linear way with density.
the effects of harvesting depend on the environment
The way an organism responds to its environment depends on the environment itself (food supply, competitors) and the organism's history (Beckerman et al. 2002). For example, the natal environment can profoundly affect growth, maturation and fecundity later in life (Lindstrom 1999; Metcalfe & Monaghan 2001). If the population density is constant, at equilibrium, individuals will always have experienced a competitive environment where, on average, food is rare. Conversely, if the population fluctuates, individuals may have been born into a low-density environment where food was more freely available, even if the current environment is high density. A perturbation to density caused by harvesting may therefore lead to different responses in the constant, variable and low density environments as individuals respond differently depending on the interaction between the current environment and their history (Plaistow et al. 2004).
populations below k still experience density dependence
Although we maintained populations in the excess food experiment at about one-third carrying capacity, the density dependence remained strong. This is indicated by the positive responses to harvesting (the positive responses are significant within each stage, and insignificant, but consistent, between stages) (Fig. 6). The θ-logistic fits suggest that the population growth rates begin to increase rapidly only when population sizes get to below about 10% of the carrying capacity. For any other density (> K/10) the addition of an individual changes the population growth rate by the same amount.
The value of θ determines the shape of the density-dependent relationship. With θ < 1, changes in density have little effect until the density is low. Conversely, with θ > 1, changes in density have the greatest effect around carrying capacity. Fowler (1981) linked θ to the life history, with longer-lived organisms, such as mammals, having θ > 1, and shorter-lived organisms, such as mites, having θ < 1. Recent analyses of the life history of birds (Saether & Bakke 2000; Saether et al. 2002) show that smaller, short-lived and highly fecund species typically have values of θ < 1 and larger longer-lived species with θ > 1. Empirical examples seem to indicate that θ < 1 is more typical where density dependence in non-linear (11 of 13 examples; Sibly & Hone 2002).
If values of θ < 1 are indeed common, then it implies that density dependence will remain strong considerably below carrying capacity, and weaken only when the population density becomes very low. This implies that conservation models with no density dependence, using the justification that the organisms are below carrying capacity, may be flawed. It also implies that the maximum sustainable yield (MSY) of harvested populations should be at a smaller fraction of carrying capacity than traditionally estimated. If the MSY occurs at low population densities, then harvesting to this population density, coupled with our finding that harvested populations may become more variable, would increase the population's extinction risk owing to both demographic and environmental stochasticity.
The utility of model-based approaches to species management depends crucially on whether the models capture enough of the biology to predict the population response to perturbations. Our experimental approach has illuminated a number of factors that, in this system at least, would be necessary to model properly the population dynamics of this species. First, density dependence is inherent in this system, even when the population is considerably below carrying capacity. Secondly, the life history is structured and demographics depend on competition within and between stages. Hence, perturbing one stage has consequences (positive or negative) for other stages. Density-independent models are able to predict negative responses only to harvesting, and non-structured density-dependent models are unlikely to be able to capture the complexities of the population response. Thirdly, the way a population responds depends not only on the stage harvested, but also on details of its environment and the pattern of perturbations over time. For example, populations at the same density may respond to the same perturbation in different ways depending on the recent history of perturbations. In the short term this is likely to be because of delayed life-history effects. Over the longer term consistent environmental differences (in perturbation history, quality, harvesting, etc.) are likely to lead to evolved differences in responses.
NERC provided the funding (NER/MS/2000/00289), Stirling University temporarily provided a supportive research environment and the Centre for Population Biology at Silwood Park provided facilities during the writing. Andrew Beckerman and Craig Lapsley helped throughout. Rob Smith and two anonymous referees provided constructive feedback on the manuscript.