Warming at the population level: Effects on age structure, density, and generation cycles

Abstract The impact of climate change on strongly age‐structured populations is poorly understood, despite the central role of temperature in determining developmental rates in ectotherms. Here we examine the effect of warming and its interactions with resource availability on the population dynamics of the pyralid moth Plodia interpunctella, populations of which normally show generation cycles, a consequence of strong and asymmetric age‐related competition. Warming by 3°C above the standard culture temperature led to substantial changes in population density, age structure, and population dynamics. Adult populations were some 50% larger in warmed populations, probably because the reduced fecundity associated with warming leads to reduced larval competition, allowing more larvae to develop to adulthood. Warming also interacted with resource availability to alter population dynamics, with the generation cycles typical of this species breaking down in the 30° populations when standard laboratory diet was provided but not when a reduced nutrient poor diet was used. Warming by 6° led to either rapid extinction or the persistence of populations at low densities for the duration of the experiment. We conclude that even moderate warming can have considerable effects on population structure and dynamics, potentially leading to complete changes in dynamics in some cases. These results are particularly relevant given the large number of economically important species that exhibit generation cycling, in many cases arising from similar mechanisms to those operating in P. interpunctella.

with strong age-structured effects in a population, namely generation cycles (Knell, 1998).
Generation cycles can also be generated by interactions with a natural enemy such as a parasitoid, especially when the reproductive phase of the host's life cycle is relatively short and the generation time of the natural enemy is roughly one-half or one-and-a-half
For the present study, we used replicated, multigeneration laboratory mesocosms to test the effects of warming on populations of the pyralid lepidopteran Plodia interpunctella. P. interpunctella is a model insect system which has been extensively studied by population ecologists to help us understand phenomena such as the effect of age structure and asymmetric competition on population dynamics (Bjørnstad et al., 1998;Briggs et al., 2000;Gurney et al., 1983) and the long-term dynamics and evolution of populations with natural enemies present (Begon, Sait, & Thompson, 1996;Boots, Childs, Reuman, & Mealor, 2009;Boots & Mealor, 2007;Knell, Begon, & Thompson, 1998a;Sait, Begon, & Thompson, 1994;Sait, Liu, Thompson, Godfray, & Begon, 2000). Laboratory populations of P. interpunctella typically exhibit generation cycles (Sait et al., 1994) which are believed to be a consequence of the strong asymmetric competition between larval stages: Larger 4th and 5th instar larvae are aggressive and will not only compete for food with the younger larvae but also cannibalize them along with eggs and sometimes pupae (Cameron, Wearing, Rohani, & Sait, 2007;Reed, 1998).
This strong asymmetric interaction means that the age structure of populations of P. interpunctella becomes synchronized, causing generation-length cycles (Briggs et al., 2000;Sait et al., 1994). Both statistical modeling of population time series (Bjørnstad et al., 1998) and mathematical population modeling (Briggs et al., 2000) support this and emphasize the importance of the asymmetry in competition between older and younger larvae in driving these generation cycles (Bjørnstad et al., 1998).
Predicting the effects of increased temperature on populations exhibiting generation cycles is not straightforward. Grover, McKee, Young, Godfray, and Turchin (2000) found no consistent effect of warming on cycling in Daphnia, but the authors were not able to tell whether the cycles in their populations were generation cycles or multigeneration cycles. In the case of P. interpunctella, there has been a considerable amount of research on the life-history effects of temperature, summarized in Table 1. Work on the strain used in these experiments found that increasing temperature from 27°C (the standard laboratory rearing temperature) to 30°C reduces the egg-to-eclosion period by 2-3 days and adult lifespan by 2.5 days, with 50% of unmated females surviving for 12.5 days at 27°C and 10 days at 30°C (Laughton, O'Connor, & Knell, 2017), suggesting that generation cycles should become shorter with increasing temperature. This assumes that other temperature effects on life-history characters in P. interpunctella do not influence the behavior of the populations, however, and it is possible that changes in factors like recruitment could override this: Lifetime fecundity of females is reduced by 20%-25% by an increase from 27 to 30°C (Laughton et al., 2017). Increasing temperature has also been found to have a pronounced effect on eggto-adult survival in P. interpunctella, with the proportion of eggs leading to eclosing adults changing from roughly 60% at 27°C to between 20% and 30% at 33°C (Parrett & Knell, 2018). These reductions in fecundity and survival could potentially weaken intraspecific competition, thereby reducing the tendency of the populations to cycle.
A further potential influence of global change that is likely to interact with warming is resource availability. As the climate changes, so will the availability and quality of resources such as food (Tylianakis, Didham, Bascompte, & Wardle, 2008), and so populations experiencing warming will also experience multiple other changes to their environment. Food quality is known to affect population dynamics in P. interpunctella. Some studies used a very rich diet based on commercial baby food fortified with yeast, glycerol, and honey, and these produce clear and well-defined cycles (Boots et al., 2009;Sait et al., 1994). Other work has used a less rich diet based on bran, and P. interpunctella populations reared on this show less well-defined cycles or no cycles at all, probably because of a reduction in population density in the latter populations (Knell et al., 1998a;McVean, Sait, Thompson, & Begon, 2002a). Because of this importance of resource quality in P. interpunctella, here we manipulate resource availability as well as temperature, with half of our populations being raised on standard laboratory diet and half on a low nutritional content diet.
Given the previously described effects of poor diet on P. interpunctella population dynamics, we predict that generation cycles will be less likely in populations given poorer quality diets. How poor quality food might interact with temperature to affect P. interpunctella populations is not clear, but given the importance of resourcebased competition in this system and the effects of temperature on fecundity and survival, it is possible that these factors will interact with food quality to alter population dynamics.
An outbred stock culture of P. interpunctella

| Incubators
In order to produce a properly replicated study, separate incubators were built, one for each population. These consisted of an expanded polystyrene box (40 × 30 cm base and 22 cm height) containing a vivarium heat mat connected to a thermostat. Each heat mat was covered with a layer of vermiculite to ensure convective rather than radiative heat transfer, and each incubator was fitted with a standard PC fan to ensure good circulation. Population cages were propped on supports approximately 5 cm above the heat mats, and a window was cut in the front of each incubator and double glazed with acrylic sheet to allow light in. All incubators were kept in a controlled temperature (CT) room at 26°C and were repositioned within the CT room every few weeks to ensure no effects of shelf location on population dynamics. iButton Thermochron data loggers (Maxim Integrated, San Jose, California www.maximintegrated.com) were used to monitor temperature in each incubator.

| Experimental design
Eighteen populations were established, each using 100 mixed sex (assumed to be 50:50 based on preliminary surveys) larvae from the outbred line. Populations were kept in plastic containers (25 × 25 cm base and 15 cm height) in individual incubators at 27°C, 30°C, or 33°C ± 0.5°C, and on one of two diets: either a standard laboratory diet (as above, 10:1:1 ratio of organic wheat bran, brewer's yeast, and glycerol), or a nutritionally stressful "poor" diet (replacing 50% of the wheat bran with methylcellulose, an indigestible bulking agent).
Treatment combinations followed a fully factored design, giving three replicate populations for each temperature × diet combination. All populations were exposed to the same 12L:12D light cycle.
The base of each population container was divided into a grid of six large and three small sections. After an initial undisturbed 2-week establishment period, each week on rotation one large and one small section of diet was removed and replaced with the appropriate fresh diet. The extracted sections contained both live larvae and dead adults. Each small section of diet (3 weeks old) was sorted, all larvae were removed, and the number, instar (3rd, 4th, or 5th), and sex (testes become visible at 4th and 5th instar) recorded. 1st and 2nd instar larvae were not counted because their extremely small size made it impossible to obtain accurate counts. After sorting, larvae were returned to their original populations. Dead adults were collected from the entire population container, including those from both the small and large sections, and all were sexed and counted before being discarded. The populations were allowed to run continuously, with larval data collected for 62 weeks, and dead adult data collected for 82 weeks, approximately 14 generations.

| Data analysis
All analyses of population data were carried out using a truncated time series, with the first 10 weeks removed so as to avoid bias from transients associated with the initial setup of the populations.
Quantitative population-level measures such as mean population sizes were compared using linear models, with nonsignificant terms removed to produce a minimal adequate model (Zuur, Ieno, & Walker, 2009), whereas qualitative measures such as whether generation cycles were present or not were compared using Fisher's exact test.
The dynamics of each population were explored by examining spectral densities (Shumway & Stoffer, 2017), complemented by wavelet analysis (Cazelles et al., 2008). Spectral analysis is a technique which decomposes a time series of data into its component frequencies, and the relative importance, or density, of these frequencies is then displayed on a periodogram, with peaks in the spectral density indicating periodicities in the data. The population data were square-root transformed prior to this analysis in order to normalize the errors, and approximate statistical significance thresholds for these analyses were calculated as the 95th quantile for spectral densities of 10,000 replicates of white noise drawn from a negative binomial distribution with the mean and shape parameter equal to those for the time series in question.
Wavelet analysis also looks for periodicities in the data but does not assume "stationarity" in the time series, meaning that if the dynamics change over the course of a time series this can be visualized, so if, for example, there are cycles in a time series that change period this will be detected and visualized. Because the long-period cycles in these populations have wavelengths which are greater than 25% of the total length of the time series, these were removed from the data by smoothing with a loess function of span 0.4 prior to analysis.
Wavelet transformations were carried out using the WaveletComp package (Roesch & Schmidbauer, 2018) using a Morlet wavelet and plotted with 5% significance contours calculated using the default (white noise) options.
All analyses were carried out in R v. 3.5.0 (R Development Core Team, 2013). Full analysis details and the corresponding R code are available in Supporting information Appendix S1. Figure 1 shows the time series for each population. All populations at 27 and 30°C persisted for the entire 82 weeks. All of the 33°C populations from the standard diet treatment went extinct in a few generations, but two of the 33°C populations from the poor diet treatment persisted for the full period. The populations that went extinct all did so in less than 6 months, so the high temperature populations either went extinct rapidly or persisted for a long time. Figure 2a shows the mean population sizes (as weekly counts of dead adults) for each population. There was a highly significant effect of temperature on population size (F 2,9 = 43.6, p = 0.00002), with the 30°C treatments having the highest numbers of adult moths and the two 33°C populations which persisted having the lowest. There was no significant effect of diet quality or the interaction between diet quality and temperature. By contrast with the data for adults, there were fewer larvae in the 30°C treatment than in the 27°C one (Figure 2b), and the two 33°C treatments which persisted had very low larval numbers. As with the adult population data, there was no effect of diet quality on larval population sizes, but a highly significant effect of temperature (F 2,11 = 48.4, p < 0.00001).

| RE SULTS
Visual inspection of Figure 1 also suggests that the 30°C populations are more variable than the 27°C ones, but this is an artifact of their larger population sizes. If the coefficients of variance (CVs) are calculated for each population (see Figure 6 in was a significant interaction between temperature and diet on the proportion of 5th instar larvae such that overall the age structure of the population became more even (F 1,7 = 6.91, p = 0.034). Either increasing the temperature from 27 to 30°C or changing from standard to poor diet quality reduced the proportion of 5th instar larvae by 6%-8%, but when the temperature was 30°C there was little effect of changing from standard to poor diet on age structure. Both of these treatments therefore lead to a more even distribution of larval age classes. Interestingly, this trend did not appear to continue for the two 33°C populations which persisted. Despite having the highest temperature and poor diet, the age structure of both larval populations was intermediate, with proportionally more 5th instar larvae than any of the 30°C and poor diet populations, but proportionally fewer than the 27°C and standard diet populations.
The spectral density plots ( Figure 4) show that there are two important periodicities in the data: first a long-period cycle of between 25 and 35 weeks, as indicated by the peak on the left hand side of the graphs, and second, in some populations, short-period generation cycles of 6 weeks overlaid onto the longer term cycles. This is shown by the peak that in most cases aligns with the vertical dotted line, which indicates a frequency of 1/6 on each graph. These longer period cycles varied between populations, with some being approximately 36 weeks and some shorter (approximately 24 weeks). Note that because of the length of these cycles the spectral analysis will only fit peaks corresponding to periods of 72, 36, 24, 18, etc., so the precise lengths of the long-period cycles are hard to estimate.
Regarding the shorter period cycles, the spectral densities reveal strong and statistically significant peaks corresponding to a period

2008) which finds strong evidence for 6-week cycles in the 27°C
F I G U R E 3 Warming to 30°C and reducing the quality of the available diet both make the larval age structure more even, with proportionally fewer 5th instar larvae and proportionally more younger larvae. 3rd, 4th, and 5th instar larvae were recorded, but 4th instars are not shown for clarity. 1st and 2nd instar larvae are too small to count reliably. The effect seems to be somewhat reversed in the two populations at 33°C which persisted populations and the 30°C populations reared on poor diet ( Figure 5)there is a clear a "ridge" of highly significant wavelet power corresponding to a period of roughly 6 weeks extending across most of the time series. This was clearer in some (e.g., 27°, poor diet, replicate 2) than in others (e.g., 27°, standard diet, replicate 1) and in none of the pop-

| D ISCUSS I ON
Most attempts to understand how warming will affect life on Earth have focussed on species persistence and extinction, and how this will relate to community and ecosystem functioning. Here we take an alternative approach and look in detail at the population biology of a single species system. We find that even when a population persists and is apparently healthy, its ecology can be significantly altered by even a moderate degree of warming. An increase of 3°C from the standard laboratory rearing temperature for Plodia interpunctella leads to substantial changes in the moth's population biology. Adult populations are larger but larval populations are smaller, and the age structure of the larval populations changes, with a more even distribution of age classes in the warmer populations. Temperature interacts with resource availability to determine the population dynamics of the system, with the generation cycles which are typical of P. interpunctella populations seen in all cases at 27°C and in 30°C populations with poor diet, but not in the 30°C populations given standard diet. Overall, the 30°C populations were less variable than the 27°C ones.
Many of these changes with warming are likely to be related to changes in the strong competition between larvae that is thought to be the driver for most of the patterns seen in P. interpunctella population data (Bjørnstad et al., 1998;Sait et al., 1994). When resources are scarce the larger 4th and 5th instar larvae not only outcompete smaller larvae, but will also cannibalize eggs, smaller larvae, each F I G U R E 4 Spectral densities for each of the time series from populations which persisted for the full experiment. Peaks indicate periodicity in the time series, and frequency (the x-axis) is equal to 1/the wavelength in weeks. Peaks, which cross the dashed line, are statistically significant. All of the populations at 27°C and the "poor diet" populations at 30°C show strong peaks at a wavelength of 6 weeks (indicated by the dotted vertical lines), indicating regular cycles of one generation in length-although one of these, that for replicate 3 of the 30°C poor diet populations is only slightly over the significance threshold. The 30°C populations given standard diet, by contrast, all show multiple, marginally significant peaks with little consistency. One 33°C population shows generation cycles, but the other does not. All time series also show peaks at frequencies close to zero, which arise from the multigeneration cycles which are apparent in the time series in addition to the generation cycles other and pupae (Cameron et al., 2007;Reed, 1998). At 30°C, P. interpunctella egg-to-adult survival is reduced and overall fecundity is lower (Laughton et al., 2017;Parrett & Knell, 2018). This reduction in recruitment means that competition between larvae is likely to be reduced, which is consistent with the more even age distributions of larvae found at 30°C. Reduced larval competition is also consistent with the dramatic and, at first sight, counterintuitive increase in adult populations under temperature stress at 30°C. A series of studies have shown that if competition is over-compensating, then an increase in mortality or a decrease in recruitment can lead to an apparently paradoxical increase in population size if it occurs at a life stage prior to competition acting (reviewed in (Schröder, Leeuwen, & Cameron, 2014). In the present case, it appears that one consequence of this is that a greater number of larvae reach adulthood.
Our predictions about the effects of temperature and diet quality on the generation cycling typical of P. interpunctella populations are not consistent with the patterns we found. In particular, poor diet here seems to be associated with a greater likelihood of generation cycling, and there is no indication of a reduction in cycle period with higher temperatures. Strikingly, generation cycling was lost in the 30°C, standard diet treatments but not in the poor diet populations at the same temperature. Modeling suggests that more asymmetric competition and higher fecundity are both associated with a greater tendency to display generation cycles (Briggs et al., 2000). Reduced fecundity and increased mortality at 30°C will reduce the degree of competition between larvae and make generation cycling less likely.
We suggest that the increased resources available in the standard diet treatment reinforce this effect by further decreasing competition, leading to the lack of generation cycling seen in these populations. The more even distributions of larval ages seen in the 30°C populations is consistent with this, although the distribution for larvae in the poor diet treatment, which did show generation cycling, F I G U R E 5 Wavelet power spectra of population time series. The x-axis gives time in weeks, and the y-axis indicates the period. The power is indicated by color, with blue indicating low power, whereas green, yellow, and red show increasing power, and the paler regions at the left and right hand sides of the plots show the regions where effects associated with the beginning and end of the time series reduce certainty. The time series identified by spectral analysis (Figure 4) as showing 6-week generation cycles, including all of the 27°C populations, the 30°C poor diet populations, and the second replicate of the 33°C poor diet treatment all show areas of high power at a period of 6 weeks, with those with particularly strong 6-week peaks in the spectral analysis showing "ridges" of power across the time series, indicating that generation cycling is occurring for most of the period that the populations were maintained. The 30°C populations maintained on standard diet, by contrast, show brief periods of regularity at a variety of periods, and only the second one shows any peak of power near the 6-week periodicity found in the other time series. The 33°C population which does not show generation cycles (the first one) has a very brief period of cycling at the start of the time series after which there is no real periodicity. Populations which became extinct during the experiment are not shown is similar to that for the standard diet treatment at this temperature, suggesting a possible further effect of food quality which we do not currently fully understand. One possible way for food quality to affect dynamics arises from the effects of egg cannibalism on population behavior. This was found to be necessary for generation length cycling in the models of Briggs et al. (2000) and in the absence of egg cannibalism age-structured models of P. interpunctella dynamics predict half-generation length cycles. Lower recruitment and abundant resources in the 30°C, standard diet populations could cause a reduction in egg cannibalism, reducing the likelihood of generation cycles and possibly accounting for the transient shorter period cycles seen in these cases.
The maintenance of 6-week cycles in the 30°C, poor diet populations and also in the one 33°C population which cycled is intriguing, and given the reduction in development time and adult lifespan associated with higher temperatures (Laughton et al., 2017), it seems counterintuitive. A partial solution may lie in the fecundity schedules of the adult moths, which are similar at 27 and 30°C, with most females laying the majority of their eggs in the first few days of adulthood (Laughton et al., 2017). This means that the effective generation time might be more similar at these temperatures than would otherwise be thought. Alternatively, it is possible that the poor diet associated with generation cycles at both higher temperatures could lead to extended fecundity periods and longer adult lives. Reports are mixed regarding dietary effects on development time in P. interpunctella (Littlefair, Nunn, & Knell, 2016;McVean, Sait, Thompson, & Begon, 2002b) synthetic diet (i.e., a less nutritious diet) were associated with faster development. However, more cellulose was also strongly associated with longer adult lives in P. interpunctella (Littlefair et al., 2016). If this extended adult lifespan also leads to a longer period during which eggs are laid, this could compensate for shorter development associated with temperature and maintain cycle length at 6 weeks.
Long-period population cycles have been reported from P. interpunctella populations before, but only with natural enemies present: The presence of both a granulosis virus and the parasitoid Venturia canescens leads to multigenerational cycles of 3-4 generations (Begon et al., 1996). This behavior was ascribed by the authors to the reduction in asymmetrical larval competition caused by these natural enemies, which seems to dampen generation-length cycles and lead to more classical predator-prey type cycles. More recently, Boots et al. (2009)  is likely that a moderate amount of warming will also cause some of these to display similarly altered population dynamics. At a minimum, such changes would alter harvest, management, and control requirements, and they could, potentially, cause severe impacts on agriculture and fisheries. More generally, age structure is a crucially important factor in the population ecology of the majority of animal species (Begon, Townsend, & Harper, 2005), and these data demonstrate that moderate warming can alter age structures considerably, with important consequences for population structures and behavior. Understanding the effects of climate change at a population level is crucial for conservationists, pest control practitioners, and natural resource managers and this study makes clear the necessity of in depth, early research in order to predict what these effects will be.

ACK N OWLED G M ENTS
We thank Nina Wedell for providing some of the moths used to form our outbred stock line and Cian O'Connor and James Westmancoat for technical assistance. We are also grateful to two anonymous reviewers for their helpful suggestions regarding a previous version of the manuscript.

CO N FLI C T O F I NTE R E S T S
The authors declare that they have no competing interests.

AUTH O R S ' CO NTR I B UTI O N S
RK and AL designed the study. AL was responsible for the practical work and data collection. RK analyzed the data. RK wrote the first draft of the manuscript, and AL and RK revised it and produced the final version.

DATA ACCE SS I B I LIT Y
The datasets used here are available from the Dryad repository at https://doi.org/10.5061/dryad.4fg24s2, along with a file containing the R code for all of the analyses and data visualizations presented here. Figure 2a (main text) shows the mean adult population sizes for each population. We can fit a model to these data to assess effect sizes and the statistical significance of the effects of temperature and food quality. Because most of the high temperature populations did not persist, we will restrict the analysis to the 27 and 30° treatments only.

M I N I M A L M O D EL
Model selection by removal of nonsignificant terms leaves us with only temperature as an explanatory variable.
Since food quality has no significant effect we have a few more df to play with so we can include the two persistent populations at 33° as well Overall, temperature has a substantial effect on the mean population sizes. Increasing the temperature from 27 to 30° increases mean population size roughly 1.5 times, but going from 27° to 33° means that those populations manage to persist at the higher temperature have a mean size which is roughly a third of the 27° populations and a fifth of the 30° ones. Figure 6 shows the coefficients of variability for the adult population trajectories for each population. 30° populations were slightly less variable than 27° populations. The 33° populations that persisted had much higher CVs, but this is possibly more of an effect of their low mean population size than anything else.

VA R I A B I LIT Y O F A D U LT P O PU L ATI O N S
Can we discern any statistically significant patterns in the variance of the adult populations?
No significant interaction term and no effect of food. We can include the 33° populations in a final model.

F I G U R E 6 Coefficients of variation for each population
CVs are significantly lower in the 30° populations and then significantly raised in the 33° populations. There is no discernable effect of food quality. Figure 2b (main text) shows the mean counts per week for the total number of larvae per food section. By contrast with the data for adults, there were fewer larvae in the 30° treatment than in the 27° one. The two 33° treatments which persisted had very low numbers.

L A RVA E
Analysis of these data is similar to that for the adult populations. An initial model fitted to the 27 and 30° data only is reduced to one with only temperature as an explanatory variable and we can add in the two persisting 33° populations to that to give a final minimal adequate model with temperature as an explanatory variable but not food quality. By contrast with the adult populations, larval populations are smaller in the 30° treatment than in the 27° one, but as with the adult populations the two 33° populations which persisted have very low larval counts.

AG E S TRU C TU R E O F TH E L A RVA L P O PU L ATI O N
Time-averaged proportions of 3rd, 4th, and 5th instar larvae are shown in Figure 3 (main text). There are clear differences between treatments, with the highest proportions of 5th instar larvae in the 27 degree good food populations and the lowest proportions being in the 30° populations.
These differences in the proportions are statistically significant. They can be analyzed using a linear model with the mean number in one stadium (either 5th or 3rd instar larvae, it makes little difference) as a response variable and with the total number of larvae as a covariate.

LAUGHTON ANd KNELL
Alternatively, an analysis using the proportion of 5th instar larvae as the response: The age structures of the larval populations are therefore significantly different between treatments, with a significant interaction between temperature and food quality. The 27°, good food populations have the most 5th instar larvae, and either increasing the temperature or changing to poor food decreases the relative abundance of 5th instar larvae in favor of younger animals. Food quality has little effect on age structure for the 30° populations, however, all of which have roughly similar age structures.

TI M E S E R I E S A N A LYS I S LO N G -PE R I O D C YCLE S
The smoothed spectral density plots show that in fact there are two important periodicities in the data-firstly a long-period cycle of between 35 and 25 weeks, as indicated by the peak on the left hand side of the graphs, and secondly short-period generation cycles of 6 weeks overlaid onto the longer term cycles. This is shown by the peak that in most cases aligns with the vertical dotted line, which indicates a frequency of 1/6 on each graph.
These long-period cycles are visualized in Figure 7. Some populations show longer period cycles (approx. 35 weeks) and some shorter (approx 25 weeks). Note that because of the length of these cycles the spectral analysis will only fit peaks corresponding to periods of 72,36,24,18, etc., so the precise lengths of the long-period cycles are hard to estimate.
There is no apparent association between temperature and cycle length, but there is between food quality and cycle length. If we consider the 27 and 30° populations, all the poor food populations show short cycles (six out of six), whereas most of the good food populations show longer cycles (4 out of 6). This is not quite statistically significant: F I G U R E 7 The 18 population time series, square root transformed, with a fitted Loess smoother which makes the long-period cycles clear