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WEATHER AND SYNCHRONY IN 10-YEAR POPULATION CYCLES OF ROCK PTARMIGAN AND RED GROUSE IN SCOTLAND.

A. WATSON [1,3]

R. Moss [1]

P. ROTHERY [2]

Abstract. Rock Ptarmigan (Lagopus mutus on two adjacent submassifs, and Red Grouse (Lagopus lagopus scoticus) on lower ground between them, showed largely synchronous [sim]10-yr cycles during a [sim]50-yr study on the infertile Cairngorms massif of Scotland. Adult birds of both these Lagopus species were counted along transect walks. Both species showed the very low mid-1940s trough previously recorded for tetraonids in much of northwest Europe. Each of five subsequent peaks in all three populations fell within a year of one another, and 1-2 yr after cyclic high June temperatures at a nearby village. Troughs were less synchronous. A model with lagged June temperatures and fourth-order delayed density dependence, with no input from observed bird numbers after the first 4 yr, gave a good postdiction of Rock Ptarmigan numbers on the bigger submassif for 49 yr, suggesting a weather cycle entraining a Rock Ptarmigan cycle. However, June temperatures had little explanatory value for Rock Ptarmigan numbers on the s maller submassif. Indirect evidence suggested that synchrony between the two Rock Ptarmigan trajectories may have been due partly to emigration from the bigger to the smaller submassif. The population trajectory of Red Grouse resembled that of Rock Ptarmigan on the smaller submassif more closely than the two Rock Ptarmigan trajectories resembled one other. Hence synchrony depended more on local circumstances than on species.

Key words: cycle entrainment; density dependence, delayed; dispersal; interspecific synchrony; intraspecific synchrony; Lagopus; population cycles; postdiction of cycles; Red Grouse; Rock Ptarmigan; Scotland; weather cycles.

INTRODUCTION

Populations of sympatric species often fluctuate in partial synchrony. For example, Moran (1952) showed that Scottish shooting-bag records of Rock Ptarmigan (Lagopus mutus), Capercaillie (Tetrao urogallus), and Black Grouse (Tetrao tetrix) were correlated with those of Red Grouse (Lagopus lagopus scoticus). Cycles in numbers of many northern birds and mammals often involve several species fluctuating together. Thus, Capercaillie, Black Grouse, and Hazel Grouse (Bonasa bonasius) in Finland fluctuated together with [sim]6-yr periodicity (Lindstrom et al. 1995). Similarly, Sharptailed Grouse (Tympanuchus phasianellus), Ruffed Grouse (Bonasa umbellus), and Prairie Grouse (Tympanuchus cupido) in North America showed synchronous [sim]10-yr cycles (Keith 1963). Cycle period, however, is not specific to a species. For example, the circumpolar species Lagopus lagopus (called Willow Ptarmigan, Willow Grouse, and Red Grouse) shows cycles varying in period from 3-4 to [greater than or equal to] 10 yr. and cycles have no t been found in some populations (Watson and Moss 1979). Hence hypotheses about population cycles must explain (1) what drives cycles, (2) intraspecific variations in cycle period, and (3) interspecific synchrony.

Cyclic weather patterns might explain those three aspects, but no such pattern sufficient to cause all three has yet been identified (Moss and Watson, in press). However, a synchronizing effect of extrinsic perturbing factors such as bad weather (Moran 1953), often called the "Moran effect" (Royama 1992, Haydon and Steen 1997, Ranta et al. 1997, Grenfell et al. 1998) remains likely. Randomly imposed low reproductive rates produced synchrony in model populations of tetraonids and red squirrels (Sciurus vulgaris) in Finland (Ranta et al. 1995, Ranta et al. 1997). Sinclair and Gosline (1997) suggested that cyclic weather patterns, initiated by the solar sunspot cycle, might entrain [sim]10-yr cycles of snowshoe hares (Lepus americanus) in the Yukon (i.e., the animal cycles occur irrespective of sunspots, but sunspots influence the timing). Watson et al. (1998) showed that peaks in the [sim]10-yr cycles of Rock Ptarmigan in the Cairngorms massif of Scotland occurred just after high values in a 10-yr cycle of Jun e temperature. Here, we present further data on the relationship between weather and ptarmigan numbers.

Shooting bags of Red Grouse in northeast Scotland between 1849 and 1938 (Mackenzie 1952) had a main period of 6-7 yr (Williams 1985), but reported periods range from 4-5 yr in northern England (Potts et al. 1984) to [greater than or equal to] 10 yr on the Scottish east coast (Moss et al. 1996). For Scottish Rock Ptarmigan, Moran (1952) detected no oscillations in Mackenzie's (1952) data on shooting bags, but Ginzburg and Inchausti (1997) found a 9-10 yr periodicity. In both Red Grouse and Rock Ptarmigan, the same local population can show [sim] 10-yr and [sim]6-yr fluctuations at different times (Moss et al. 1996, Watson et al. 1998). It is therefore worth comparing fluctuations in adjacent Red Grouse and Rock Ptarmigan populations. In the present paper we do so for the Cairngorms massif, where Rock Ptarmigan numbers have shown clear [sim]10-yr periodicity since 1943 (Watson et al. 1998). The three main questions are: do Red Grouse in this inland mountainous area show 10-yr cycles like the adjacent Rock Ptar migan, do the two species fluctuate in synchrony, and can weather explain their population trajectories?

STUDY AREAS

Massifs and submassifs

The landscape comprised hill massifs separated by valleys, with side valleys separating submassifs. Rock Ptarmigan lived on submassifs, mostly above 760 m, and Red Grouse mostly below. Rock Ptarmigan numbers fluctuated in close synchrony within submassifs, and in partial synchrony among submassifs (Watson et al. 1998), Transect "walk data" (Methods) that covered much (65 [km.sup.2]) of submassif DS (170 [km.sup.2]) were highly correlated with total counts of Rock Ptarmigan density on "count area" D within DS (Fig. 1, Table 1, Appendix A). This applied also to walk data from BS (which covered 50 [km.sup.2] of the 70-[km.sup.2] BS submassif) and total densities from count area B. Hence the submassif was an appropriate unit for studying synchrony in Rock Ptarmigan. For Red Grouse, we had walk data from the 54-[km.sup.2] GR that lay mostly between submassifs, and total counts on the 100-ha G (Table 1).

Habitat and weather

Watson (1965) and Watson et al. (1998) described Rock Ptarmigan habitats, count areas, and walk-data areas in more detail. Rock Ptarmigan bred on short arctic-alpine vegetation with a high proportion of heaths (Ericaceae), and used boulders as their main cover. Most Red Grouse in the district bred at altitudes well below the Red Grouse walk-data area GR, on moorland dominated by tall heather (Calluna vulgaris), which formed their main food and cover. GR itself comprised mostly subalpine ground dominated by tallish vegetation, chiefly heather with some blaeberry (Vaccinium myrtillus) and crowberry (Empetrum nigrum). The average boundary between breeding habitats of Rock Ptarmigan and Red Grouse was at 760 m, varying with vegetation height and boulder abundance. Tallish heather with Red Grouse occurred in sheltered hollows [less than or equal to]900 m, and short arctic-alpine vegetation with boulders and Rock Ptarmigan on exposed sites [greater than or equal to]600 m.

The Red Grouse count area G lay at 760-900 m, close to the upper altitudinal limit for Red Grouse. Adjacent to the Rock Ptarmigan count area D, it had no Rock Ptarmigan territories in any year. It comprised mostly subalpine ground with tallish heather, blaeberry, and crowberry, with some patches of short heath. Peaty gleyed podzol soils dominated hollows and lower slopes with tallish heather, and subalpine podzols dominated the freely drained ground with short heath.

Burning is common in moor management, but no fires were burned on Rock Ptarmigan areas DS and BS, or on Red Grouse count area G, except for tiny patches where fires lit lower down had petered out (Watson [1965] and later unpublished observations). Burning was done on GR, the walk-data area for Red Grouse. During the study years it involved not the narrow-strip fires of managed grouse moors, but the wider and less controlled fires typical of management for red deer (Cervus elaphus), which resulted in low densities of Red Grouse (Phillips and Watson 1995).

Data on monthly mean temperature and rainfall (UK Meteorological Office) came from Braemar village, which lay at 340 m in a valley 7 km east of Red Grouse area GR and 7 km south of Rock Ptarmigan area BS. With the prevailing southwest wind from the Atlantic Ocean, the Scottish climate is drier farther east (Birse 1971), and, as is typical worldwide, high hills receive more precipitation and clouds (Green 1974, Watson 1992). It was often noticed during fieldwork that the more westerly DS was duller, foggier, and rainier than BS and the lower-altitude GR.

Grazing mammals, and predators

No domestic sheep were run on the areas since at least 1900, and grazing by red deer was light (Watson 1965), mainly on grass rather than on the heaths eaten by Rock Ptarmigan and Red Grouse. Deer grazed most heavily in grassy valley bottoms, which we excluded from our areas. In principle, mountain hares (Lepus timidus) might compete with Red Grouse and Rock Ptarmigan for heath food, and are a potential alternative food for predators, but were at very low densities on all the areas, [sim]1 adult/[km.sup.2] (Watson and Hewson 1973, Watson et al. 1989). There was also some evidence that mountain hare numbers fluctuated with 10-yr periodicity (A. Watson and R. Moss, unpublished data).

Descriptions of Rock Ptarmigan predators in the Cairngorms area (Watson 1965, Watson et al. 1998) apply also to Red Grouse. All predators were generalists, and their abundance varied little over the years. The main one was red fox (Vulpes vulpes), with Golden Eagle (Aquila chrysaetos) next. Crows (Corvus corone) were scarce (Watson 1996), none being seen in most years on DS and BS, and few on GR.

METHODS

Numbers of adult Rock Ptarmigan and Red Grouse seen per 10 km of transect walks ("walk data") within the walk-data areas were noted (Watson 1965, Watson et al. 1998) in May-September. On the count areas, which were smaller, intensive counts showed total numbers of adult cocks and hens (Jenkins et al. 1963, Watson 1965, Watson and Miller 1976) in spring (April- early May). Rock Ptarmigan numbers below are taken from Watson et al. (1998).

Autocorrelograms and spectral analyses were done as in Watson et al. (1998). Most statistical analyses used SAS procedures (SAS Institute 1996). Density dependence in population trajectories was characterized by autoregressive models (Royama 1992) that relate present to past numbers

[log.sub.e][N.sub.t] = [k.sub.0] + [k.sub.1] [log.sub.e][N.sub.t-1] + [k.sub.2] [log.sub.e][N.sub.1-2]... + [k.sub.n] [log.sub.e][N.sub.t-n] + [[varepsilon].sub.t]

where N are numbers, k constants, t - n specifies the year, and s is error.

For some analyses, we added terms in meteorological variables such as June temperature ([c.sub.n][J.sub.t-n]) where c are constants and I are monthly mean June temperatures) to the right-hand side. We used only meteorological variables that we already knew to be related to ptarmigan numbers, so reducing the likelihood of finding spurious associations. April and May monthly mean temperatures affected the date of blaeberry growth, which was correlated with Rock Ptarmigan breeding success, as were June temperature and rainfall (Watson et al. 1998).

Evidence for delayed density dependence of order 2... n was obtained by inspecting the significance of the [k.sub.2...n] terms from the autoregressions. Because any effect of numbers in year t - n on numbers in year must operate through the intervening years, evidence for delayed density dependence of order n must come from autoregressions that include all terms up to [k.sub.n]. Alternatively, if a regular event that causes cycles occurs in year t - n, its effects should be strongest at lag n - 1. In the latter case, it seems appropriate to seek the [k.sub.n] that accounted best for variation in N. We did this by regressions (SAS PROC REG) using backward elimination (step-down regressions), and Akaike's information criterion (for brevity called AIC below), which is a measure of goodness of fit that incorporates parsimony. In the presence of delayed density dependence, the null hypothesis for direct (first-order) density dependence is [k.sub.1] = 1.0.

We use the term "density dependence," but density itself is unlikely to limit numbers. For example, if resource density differs between areas, the same bird density (number/unit area) will result in different amounts of resource per bird. Hence for analyses we use absolute numbers, except when comparing bird density between count areas, and when comparing bird density on count areas with walk data.

RESULTS

Average densities

Mean Rock Ptarmigan densities on count areas D and B (Appendix A, Fig. Al) were less than half of those on the more fertile soils of the nearby Mounth massif (Watson et al. 1998). Similarly, count area G (Appendix A) held lower densities of Red Grouse (Appendix B) than [sim]20 other widely spread areas where we have studied Red Grouse in northeast Scotland, all on managed moors at lower altitudes (Jenkins et al. 1967, Moss et al. 1975, Watson et al. 1984).

Population trajectories

Walk data and count data showed similar trajectories. Here we present results based on walk data, for which there were longer runs (Table 1). All three trajectories (Fig. 2), for Rock Ptarmigan on DS and BS, and for Red Grouse on GR, showed cycles with an [sim]10-yr period (Fig. 3). Step-down autoregressions (Table 2) for all three trajectories indicated delayed density dependence, with the strongest relationship at year t - 4 (lag 3). When we forced all four terms ([log.sub.e][N.sub.t-1] ... 4) into the autoregression, the relationship at lag 3 remained significant only for Rock Ptarmigan at BS.

Cross correlations were stronger between GR Red Grouse and BS Rock Ptarmigan numbers than between Rock Ptarmigan numbers on BS and DS (Table 3). Hence synchrony depended more on local circumstances than on species.

All three trajectories showed a deep trough in 1945 (DS Rock Ptarmigan) or 1946 (BS Rock Ptarmigan and GR Red Grouse), and fluctuated together until the joint 1961-1962 peak. Thereafter, troughs were sometimes out of phase, but all peaks for the two species came within 1 yr of one another. This involved some remarkable concurrences. For example, in 1980 the Red Grouse were at a trough and the DS Rock Ptarmigan at a peak, but in 1981 the Red Grouse showed a single-year peak, so maintaining the synchrony of the peaks. Similarly, in 1983-1988 the Red Grouse and the BS Rock Ptarmigan were increasing towards their 1989 peaks, but DS Rock Ptarmigan were in decline. Even so, the DS Rock Ptarmigan peaked in 1990. Such concurrences suggested that some common factor synchronized the peaks of all three trajectories.

June temperature

Perhaps this inferred common factor was a 10-yr weather cycle. Watson et al. (1998) showed that high mean June air temperatures at Braemar village (Fig. 2) tended to recur every 10 yr in 1912-1995 (autocorrelation coefficient [r.sub.lag10] = 0.28, P [less than] 0.05). High temperatures occurred as isolated single-year spikes, not as high points of a smooth curve. Watson et al. (1998) also noted that peaks in Rock Ptarmigan walk data at DS in 1943-1995 came 1-2 yr after five of the seven years when mean June temperature at Braemar surpassed 12.5[degrees]C (1950, 1960, 1970, 1976, 1979, 1988, and 1992). They showed that this was unlikely (exact P = 0.0015) on the null hypothesis that June temperatures were in random order. A similar relationship with June temperature occurred for BS in 1945-1995 (P = 0.0018). The data for Red Grouse at GR in 1943-1990 show the same pattern (P = 0.008). Hence June temperatures did not vary at random with respect to bird numbers. In principle, this association could have occurre d simply because temperatures and bird numbers both showed causally unconnected [sim]10-yr cycles, but the following evidence makes this seem unlikely.

The breeding success of Rock Ptarmigan on count area D was related to monthly mean June temperatures at Braemar, and was correlated with population change (Watson et al. 1998). When we added four terms in June temperature ([c.sub.1][J.sub.t-1] to [c.sub.4][J.sub.t-4]) to the full four-term autoregression for DS Rock Ptarmigan (Table 2), three of them ([c.sub.1][J.sub.t-1], [c.sub.2][J.sub.t-2], and [c.sub.4][J.sub.t-4]) were significant (P [less than] 0.05). A step-down regression ([R.sup.2] = 0.72, inclusion level P = 0.1) retained the autoregressive terms in [k.sub.1] and [k.sub.4] (as in Table 2), and also the [c.sub.1][J.sub.t-1], [c.sub.2][J.sub.t-2], and [c.sub.4][J.sub.t-4] terms for June temperature

[log.sub.e][N.sub.t] = -2.19 + 0.74 [log.sub.e][N.sub.t-1] - 0.22 [log.sub.e][N.sub.t-4] + 0.094[J.sub.t-1] + 0.132[J.sub.t-2] + 0.091[J.sub.t-4] (1)

(SE values were 1.08, 0.08, 0.07, 0.048, 0.047, and 0.048 for these six terms, respectively). June temperature terms with longer lags were not significant. A model with [J.sub.t-3] forced into it had a positive but non-significant (P [greater than] 0.1) value for [c.sub.3]. The AIC confirmed Model 1. Neither April nor May monthly mean temperatures, nor monthly mean rainfall in June, explained significantly more variation when added to the above regressions.

We next considered how well June temperatures could explain changes in Rock Ptarmigan numbers at DS without delayed density dependence. We repeated the step-down regression above, but excluded [log.sub.e][N.sub.t-2...4]. This gave ([R.sup.2] = 0.66)

[log.sub.e][N.sub.t] = -3.01 + 0.73 [log.sub.e][N.sub.t-1] + 0.11[J.sub.t-1] + 0.13[J.sub.t-2] + 0.10[J.sub.t-4] (2)

(SE values were 1.16, 0.09, 0.05, and 0.05, respectively). The AIC led to the same model.

We used deterministic versions of Models 1 and 2 to postdict population trajectories (Fig. 4). Stepwise postdictions (Fig. 4a) used observed June temperatures and observed [N.sub.t-1] or [N.sub.t-4] to predict each [N.sub.t]. Omnibus postdictions (Fig. 4b) also used observed June temperatures, but began with Rock Ptarmigan counts from 1943 and 1946 (Model 1) or 1946 only (Model 2) at the start of each run, with no further input from observed bird counts. Both stepwise postdictions gave a good account of the observed trajectory at DS. The main difference was that the last three postdicted peaks occurred a year later than observed ones.

Both the omnibus postdictions (Fig. 4b) revealed most of the main features of the trajectory at DS, but fluctuated less widely than it did. Postdictions from Model 1 fluctuated more widely than those from Model 2, so giving a better fit (correlations between observed and postdicted values of [log.sub.e][N.sub.t] from 1947 to 1995 were [R.sup.2] = 0.53 and [R.sup.2] - 0.29 for the two models, respectively), except that Model 2 made a better postdiction of the unusually high 1971 peak. June temperature, however, had no significant explanatory value for the BS Rock Ptarmigan trajectory, or for the GR Red Grouse one.

A stochastic version of omnibus Model 1, in which random variations were added each year, did not produce a consistently better fit than the deterministic version. Further stochastic simulations of omnibus Model 1, in which temperature varied randomly, showed Rock Ptarmigan numbers continuing to cycle with [sim]12-yr periodicity. From this, one would expect Rock Ptarmigan cycles to continue if the 10-yr weather cycle were to stop. This fits the observation of a Rock Ptarmigan peak in the Cairngorms in 1923 despite no June temperature [greater than] 12[degrees]C in 1912-1932 (Watson et al. 1998). Also, it is consistent with general ideas about how weather cycles might entrain animal cycles (Sinclair and Gosline 1997).

Interactions between DS and BS

We checked whether numbers of Rock Ptarmigan at BS could help to explain numbers at DS, and vice versa. Separate step-down regressions for DS and BS were each started with the four autoregressive terms from DS, the four autoregressive terms from BS, and the four lagged June temperature terms. The DS result, confirmed by AIC, was identical with Model 1 and included no terms in BS numbers.

The BS result, from step-down regression ([R.sup.2] = 0.64, inclusion level P = 0.1), included a term in DS numbers but none in June temperature

[B.sub.t] = 1.50 + 0.8l[B.sub.t-1] - 0.26[B.sub.t-2] - 0.32[B.sub.t-4] + 0.29[D.sub.t-4]

(SE values were 0.48, 0.14, 0.14, 0.11, and 0.13), where [B.sub.t] is [log.sub.e][N.sub.t] for BS and [D.sub.t] is [log.sub.e][N.sub.t] for DS. The AIC indicated a model ([R.sup.2] = 0.67) that also included [D.sub.t-1] and [D.sub.t-2]

[B.sub.t] = 1.37 + 0.79[B.sub.t-1] - 0.24[B.sub.t-2] - 0.32[B.sub.t-4] + 0.32[D.sub.t-1] - 0.42[D.sub.t-2] + 0.43[D.sub.t-4]

(SE values 0.56, 0.14, 0.14, 0.11, 0.20, 0.22, and 0.15).

The correlations between observed values and omnibus postdictions of [B.sub.t] from 1949 to 1995 were [R.sup.2] = 0.05 for the stepwise and [R.sup.2] = 0.14 for the AIC model. Postdictions from the step-down model showed peaks in 1950, 1957, 1967, 1976, and 1986, and troughs in 1955, 1962, 1972, 1981, and 1991, so bearing little resemblance to the observed trajectory at BS. The AIC model did somewhat better, showing peaks in 1951, 1962, 1972, 1982, and 1992, and troughs in 1956, 1965, 1977, and 1990. Although neither model gave good omnibus postdictions, these results indicate that numbers at DS, or some factor associated with them, significantly influenced numbers at BS, but not vice versa.

Shooting

Hardly any Rock Ptarmigan were shot on DS, but many were shot on BS in 1971-1994 (Watson et al. 1998). No Red Grouse were shot on the Red Grouse count area G, but some were shot on part of the Red Grouse walk-data area GR (Appendix B). Very few were shot there before 1971, but more in 1971-1994. Only 31 [km.sup.2] of the 54-[km.sup.2] GR were shot over, and none within 2 km of the count area. When we added log, number of Rock Ptarmigan (or Red Grouse) shot in year t - 1 as a potential explanatory variable to the autoregressions for the BS Rock Ptarmigan (or to the GR Red Grouse), this increased the variation accounted for by [less than]1% (NS). Hence the number of birds shot in fall t was not consistently related to population change between years t and t + 1. This is consistent with the suggestion (Watson et al. 1998) of compensation in winter loss.

DISCUSSION

Cycles and synchrony

Rock Ptarmigan on arctic-alpine habitat and Red Grouse on adjacent subalpine habitat showed [sim]10-yr cycles in the Cairngorms. Processes that drive cycles and those that affect their timing may differ. The evidence in this paper is mostly about timing.

The unusually deep trough in numbers of both species in the mid 1940s (Fig. 2) was widespread, occurring in Red Grouse, Rock Ptarmigan, and other tetraonids in much of Scotland, and in tetraonids in Finland and other parts of northwest Europe (Mackenzie 1952, Siivonen 1948, 1952, Watson et al. 1998). It may have been due to a widespread extrinsic factor, such as bad weather, that brought many populations into synchrony.

Up to the mid 1970s, all three trajectories fluctuated more or less together. After this, the BS and GR trajectories drifted out of phase with the DS trajectory between peaks, but came back into synchrony at the peaks. Correlations (Table 3) suggested that whatever caused the three trajectories to drift out of phase between peaks was more similar on GR and BS than on BS and DS. The close association between peaks in all three trajectories and cyclic peak June temperatures (Fig. 2) suggested that June temperatures played a part in the synchrony among peaks.

June temperature

Models with June temperature made good postdictions of the trajectory at DS (Fig. 4). Models 1 and 2 included temperature terms back to year t - 4 and hence explain how a single year with high temperature can affect population growth rates for [greater than]1 yr. Neither model included a temperature term in year t - 3, a quirk perhaps of no biological significance. The delayed density-dependent term ([k.sub.4][N.sub.t-4]) in Model 1 caused its postdictions to fluctuate more widely, and to fit the DS trajectory better, than Model 2's postdictions. Even Model 1 fluctuated less widely than the observed DS trajectory, suggesting that the instability in the real population exceeded that modeled by [k.sub.4][N.sub.t-4].

Model 1 largely accounts for the association between cyclic peak June temperatures and cyclic peak bird numbers at DS. However, the same synchrony between peak June temperatures and peak bird numbers occurred also at BS and GR, where trajectories were not well postdicted by June temperature. Possible reasons for trajectories drifting out of phase between peaks include differences in weather, soils, and vegetation, and their interactions. Also, shooting may have affected social or genetic structure at BS and GR after 1971, and this may have affected numbers, a possibility not excluded by the finding that the numbers shot had little explanatory value in postdictive models. None of this, however, explains the synchrony in peaks.

Dispersal

Detailed studies on D (Watson et al. 1998), the DS count area (Fig. 1), showed that neither breeding success nor the rate of finding dead adults exhibited direct density dependence. Hence, as peak densities approached, many DS young may have failed to establish themselves locally. Emigration contributed significantly to cyclic declines at D, and some DS emigrants may have dispersed into the more heavily shot population on the smaller BS. The finding that numbers at BS were related to past numbers at DS, but not vice versa, is consistent with net movement from DS to BS, So, the synchrony in peaks between DS and BS could have been due to emigrants from DS. A similar speculation for the Red Grouse at GR requires, a hypothetical population of Red Grouse, equivalent in its dynamics to the Rock Ptarmigan at DS. Most Red Grouse in the district were on moorland habitat at lower altitude than the subalpine GR, but we have no data or their dynamics. Indeed we have no direct evidence or movement for either species. Eve n so, emigration seems a likely explanation of the observed concurrences between peaks, particularly after 1971. The evidence suggests that synchrony among Rock Ptarmigan trajectories was due to the combined effects of June temperature at DS and dispersal from DS to BS.

Mechanisms

Density dependence.--Autoregressions with all four terms in N (Table 2) and step-down autoregression gave somewhat different results about the lags in the delayed density dependence of all three trajectories Other evidence (Watson et al. 1998) shows that Scottish Rock Ptarmigan trajectories on granite hills are characterized by fourth-order (lag 3 yr) density dependence. We conclude that all three trajectories showed delayed density dependence, strongest at a lag of 3 yr. Lags of this length seem characteristic of DS Rock Ptarmigan population dynamics. They occurred between clutch size and past density, between breeding success and past density (Watson et al. 1998), between present and past density (Table 2), and between present density and past June temperatures (Models 1 and 2).

Presumably, past densities and temperatures both affected ptarmigan numbers through continuing effects on the birds' resources, enemies, or population structure. This could not have been due to recruitment from floating subadults, since Rock Ptarmigan breed in their first spring (Rae 1994). Moreover, Watson et al. (1998) gave evidence that known natural enemies, including predators and internal macroparasites, do not drive Scottish Rock Ptarmigan cycles. They suggested that these are driven by intrinsic positive feedback between recruitment in one year and the next. This hypothesis can explain the entrainment of cycles by weather, as well as the mechanism driving cycles.

Hudson et al. (1999) claimed to have prevented population cycles in Red Grouse by administering anthelmintic drugs, and inferred that the cycles were caused by a nematode parasite. In fact, their treated and control populations both showed similar fluctuations and the claimed effect of anthelmintics rested on an apparent difference in amplitude between treated and control fluctuations. However, they used shooting bags to indicate grouse numbers, and the apparent deep troughs in shooting bags on their control sites were because no shooting was done in trough years. Hence the claimed difference in amplitude was not substantiated by the data presented (Lambin et al. 1999, Moss and Watson, in press). In the Cairngorms, cycles in Rock Ptarmigan numbers (Fig. 2) are probably not due to internal macroparasites (previous paragraph), and the close similarity between Rock Ptarmigan and Red Grouse population trajectories suggests that they have much in common.

Entrainment.--Increases in the breeding population might be caused by higher breeding success. A separate effect, suggested for Red Grouse (Watson et al. 1994, Moss et al. 1996) and Rock Ptarmigan (Watson et al. 1998), is that a younger breeding population in year might tend to generate an increase in numbers in year t + 1 because a greater proportion of their offspring is recruited. Thus, other things being equal, higher breeding success due to a warm June in summer t - 4 should result in greater recruitment of young birds into the breeding population of spring t - 3. This entails a younger t - 3 spring population, which should cause more recruitment, resulting in an increase in numbers by spring t - 2, and so on until spring t.

Drive.--The above explanation involves positive feedback between recruitment in years t - 3, t - 2, t - 1, and t, even when breeding success does not vary. When combined with directly density-dependent recruitment, such positive feedback can drive model population cycles in Red Grouse (Mountford et al. 1990, Hendry et al. 1997, Mathiopoulos et al. 1998). Hence, it might explain the instability approximated by the delayed density-dependent term [k.sub.4][N.sub.t-4] in Model 1, as well as the continuing effects of past temperatures.

In the above Red Grouse models, the effects of age structure on recruitment act through kin clusters of philopatric territorial cocks that facilitate the recruitment of their collective male offspring. The crucial prediction, that bigger kin clusters result in greater recruitment, has been confirmed for Red Grouse (MacColl et al., in press). The models predict that the average size of kin clusters is bigger in younger populations, so that cocks hatched into a younger breeding population should have a greater probability of recruitment.

Entrainment of population dynamics to weather patterns

To mimic effects of weather variations on the net growth rate of a population, models of the Moran effect have typically used random perturbations that are independent from year to year (Moran 1953, Royama 1992, Ranta et al. 1995). There is, however, a growing recognition that weather in different years may be autocorrelated (e.g., Hurrell and van Loon 1997). One effect of such autocorrelations on species with intrinsically unstable population dynamics may be to alter time lags (Reddingius 1990) and hence cycle period. Another may be to entrain population cycles by cyclic weather patterns (Sinclair and Gosline 1997), and Rock Ptarmigan and Red Grouse in the Cairngorms may be an example. The geographical scale and rhythms of such postulated multi-species entrainments are likely to depend on the scale and rhythms of the weather pattern. For example, the [sim] 10-yr cycle in Red Grouse at Rickarton moor in 1946-1986 (Moss et al. 1996), near the east coast of Scotland, was not synchronized with the inland subalp ine cycles reported here. Certainly, weather and climate in northeast Scotland can differ markedly on a scale of a few kilometers (Birse 1971), let alone the 90 km between the coastal and subalpine areas mentioned here. In addition, weather may interact with other factors. Watson et al. (1998), for example, showed that the association between warm Junes and Rock Ptarrnigan breeding success on the infertile soils of the Cairngorms did not occur on the more fertile soils of the nearby Mounth massif.

A weakness with this and other hypotheses involving weather is that they are difficult to disprove because large-scale experiments with weather are impracticable. Even so, entrainment to cyclic weather patterns might occur through direct effects of weather on population dynamics. For example, regular variations in weather might cause regular variations in breeding and subsequent density. Or, entrainment might involve animals adjusting the timing of their reproductive effort, such that they optimize fitness. Thus, a 10-yr cycle in reproductive investment might optimize fitness if greater investment coincided with a 10-yr cycle in better weather. In either case, variations in cycle period should reflect spatial and temporal variations in weather.

ACKNOWLEDGEMENTS

We thank S. Gumming for notes on numbers shot, D. Elston for help with statistics, and S. Redpath for comments on the manuscript.

(1.) c/o Institute of Terrestrial Ecology, Banchory, Kincardineshire AB31 4BY, Scotland

(2.) Institute of Terrestrial Ecology, Abbots Ripton, Huntingdon, Cambridgeshire PE17 2LS, UK

(3.) E-mail: awat@ceh.ac.uk

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                  Population data by species, study area,
                 and dates in the Cairngorms of Scotland.
           Walk data [+]                                  Count data [++]
Species      Area        Area ([km.sup.2])   Years           Area
Ptarmigan     DS          65               1943-1995          D
              BS          50               1945-1995 [ss]     B
Red Grouse    GR          54               1943-1990          G
Species    Area (ha)   Years
Ptarmigan     131    1951-1978
               50    1951-1973 [II]
Red Grouse    100    1951-1978 [n]
(+.)Wide-ranging counts from a large part
of each submassif. The total area of Rock
Ptarmigan habitat on submassif DS was 170
[km.sup.2], and on BS the total area
of habitat was 70 [km.sup.2].
(++.)Total counts on defined study areas (Fig. 1).
(ss.)No data from 1958-1961.
(II.)No data from 1953 or 1958-1969.
(n.)No data from 1953.
                Coefficients from autoregressions for Rock
                         Ptarmigan and Red Grouse.
                                          Autoregression coefficients
Trajectory                                         [k.sub.1]
Full model
 Ptarmigan (DS)                               0.86 [+ or -] 0.15
 Ptarmigan (BS)                               0.90 [+ or -] 0.15
 Red Grouse (GR)                              0.62 [+ or -] 0.25
Step-down regression (inclusion criterion
 P [less than] 0.10)
 Ptarmigan (DS)                               0.77 [+ or -] 0.09
 Ptarmigan (BS)                               0.84 [+ or -] 0.14
 Red Grouse (GR)                              0.74 [+ or -] 0.10
Trajectory                                     [k.sub.2]
Full model
 Ptarmigan (DS)                           -0.09 [+ or -] 0.19
 Ptarmigan (BS)                           -0.49 [+ or -] 0.20
 Red Grouse (GR)                           0.23 [+ or -] 0.18
Step-down regression (inclusion criterion
 P [less than] 0.10)
 Ptarmigan (DS)                                   ...
 Ptarmigan (BS)                           -0.27 [+ or -] 0.14
 Red Grouse (GR)                                  ...
Trajectory                                     [k.sub.3]
Full model
 Ptarmigan (DS)                           -0.06 [+ or -] 0.17
 Ptarmigan (BS)                            0.32 [+ or -] 0.21
 Red Grouse (GR)                          -0.08 [+ or -] 0.18
Step-down regression (inclusion criterion
 P [less than] 0.10)
 Ptarmigan (DS)                                   ...
 Ptarmigan (BS)                                   ...
 Red Grouse (GR)                                  ...
Trajectory                                     [k.sub.4]      [R.sup.2]
Full model
 Ptarmigan (DS)                           -0.16 [+ or -] 0.12   0.65
 Ptarmigan (BS)                           -0.36 [+ or -] 0.14   0.61
 Red Grouse (GR)                          -0.25 [+ or -] 0.15   0.60
Step-down regression (inclusion criterion
 P [less than] 0.10)
 Ptarmigan (DS)                           -0.23 [+ or -] 0.07   0.59
 Ptarmigan (BS)                           -0.22 [+ or -] 0.10   0.60
 Red Grouse (GR)                          -0.22 [+ or -] 0.09   0.59
Notes: The biggest AIC values were given by
models in [k.sub.1] and [k.sub.4] for study
areas DS and GR, and by the full model for
area BS. The missing BS data for 1958-1961
(Fig. 2) were estimated by linear
interpolation. Sample size is number of
years, from Table 1.
               Cross and partial cross correlations between
                 Rock Ptarmigan and Red Grouse walk data.
                Cross-correlation
                  coefficients
                                               Red
Population          Ptarmigan     Ptarmigan   Grouse
trajectory            (DS)          (BS)       (GR)
Partial
Ptarmigan (DS)                      0.25       0.48 [*]
Ptarmigan (BS)        -0.02                    0.73 [**]
Red Grouse (GR)        0.31 [*]     0.70 [**]
Notes: Degrees of freedom used to estimate
significance levels were calculated using
Moran's (1952) correction to allow for the
autocorrelation structure in each series. This
reduced the effective series length by a factor
of about 3. Sample size is number of years,
from Table 1.
(*.)P [less than] 0.05,
(**.)P [less than] 0.01.


APPENDIX A

COUNTS COMPARED WITH WALK DATA

An analysis of covariance (SAS GLM procedure, Type III sums of squares) with adult density as the dependent variable [R.sup.2] = 0.89), explained by bird counts (adults/10 kin; [F.sub.1,59] = 173.4, P [less than or equal to] 0.0001), study area ([F.sub.2,59] = 4.7, P = 0.013), and bird count X area interaction ([F.sub.2,59] = 5.34, P = 0.007), showed significant differences between areas in intercept and slope (Fig. Al). These differences were due to the intercept for DS (8.38 [+ or -] 2.21 adults/[km.sup.2] [mean [+ or -] 1 SE of estimate], P = 0.0008) being greater than that for BS or GR (both Ns), and to the slope for BS (0.61 [+ or -] 0.10, P = 0.0003) being less than that for DS (0.89 [+ or -] 0.08, P [less than or equal to] 0.0001) and GR (0.95 [+ or -] 0.04, P [less than or equal to] 0.0001). Hence the differences were among areas, not between species.

These small differences occurred because densities on the different count areas bore slightly different relationships to numbers on the much larger walk-data areas (the latter comprised patchworks of different quality habitats supporting different bird densities). The count area on DS, for example, was chosen for its relatively high density (Watson et al. 1998). Even so, the main conclusion is that densities on count areas were linearly related to walk data. We infer that annual variations in walk data provided a useful index of annual changes in spring density.
                     Number of Red Grouse shot on the
                     54-[km.sup.2] GR walk data area.
Year Number
1971   31
1972   92
1973  280
1974   15
1975   50
1976  146
1977   69
1978  108
1979   98
1980   98
1981  121
1982   38
1983  107
1984   13
1985    0
1986   16
1987    0
1988   59
1989   26
1990  281
1991  130
1992   15
1993   20
1994   35
1995    0
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Author:WATSON, A.; Moss, R.; ROTHERY, P.
Publication:Ecology
Article Type:Statistical Data Included
Geographic Code:4EUUS
Date:Aug 1, 2000
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