The impact of increasing predators on avian prey populations

Transcription

1 1 The impact of increasing predators on avian prey populations Christopher Paul Bell SongBird Survival PO Box 311 Diss Norfolk IP22 1WW 8 9 Predator impact on prey species Key-words: bird census data, population decline, raptors, sequential habitat occupation, songbirds, spurious correlation cp_bell@btinternet.com

2 2 15 Summary Capsule: The possibility that predation has played a role in bird population declines has been largely dismissed following a study by Newson et al. (2010), which concluded that there is little evidence of any association between predator and prey populations. The study methodology involved regression of annual change in prey populations on corresponding change in predator populations, but a reanalysis of the results shows that both positive and negative associations are markedly more frequent than would be expected by chance. This reflects the fact that the method is liable to generate spurious associations whenever there are contemporaneous trends among prey and species populations. Rejection of the significance of predation on the basis of Newson et al. s results has been premature, and a better understanding of its role in bird population declines may be required before these can be reversed.

3 3 27 Introduction The collapse of populations among many species of lowland terrestrial birds has been one of the most striking aspects of biodiversity decline in Britain and Europe over the past half-century. Between the mid-1970s and the mid-1990s, breeding populations of common farmland birds in Britain declined by around 50% and woodland birds by 20% (DEFRA 2015), with similar trends evident more widely in Europe (Vorisek et al. 2010). This period of rapid decline followed a phase of agricultural intensification that peaked at the start of the 1970s (Chamberlain et al. 2000) and coincided with a period of population increase among a range of predators (Gibbons et al. 2007), leading researchers to initially focus on these two factors as potential causal agents (e.g. Fuller et al. 1995) Studies on the effect of predation have included analyses of national census data aimed at testing the credibility of a link between increasing predator populations and declines of potential prey species. Particularly influential has been an analysis by Newson et al. (2010), which showed relatively few negative associations between predators and prey, leading to the conclusion that there is little evidence for predator impacts at the national scale. An unexpected prevalence of positive associations was also cited as evidence that largely exonerates the predators concerned as causal agents in prey population declines Meanwhile evidence has mounted that the principal driver of bird population declines is changes in agricultural practice (Newton 2004), and this has informed the widespread implementation of agrienvironment schemes in the UK, designed to mitigate the consequences of such change. The results of such schemes have been disappointing, however, since bird populations have continued to decline (DEFRA 2015). This lack of success may be due to poor implementation (Davey et al. 2010), however it may also be that the role of predators has been dismissed prematurely following the findings of Newson et al. and similar studies such as those of Thomson et al. (1998) and Chamberlain, Glue & Toms (2009). Here I reassess the methods and findings of Newson et al. and

4 argue that they offer no support for the proposition that bird population declines are unrelated to increasing populations among their predators. 54 Preliminary evaluation of Newson et al. s methods Newson et al. analysed data from the Common Birds Census (CBC, Marchant 1983), covering the period , and from the Breeding Birds Survey (BBS, Harris et al. 2015), covering the years They used a log-linear regression model based on that of Thomson et al. (1998), who used binary measures of predator presence and absence to predict annual change in prey abundance at individual census sites. The method partitions out potentially spurious correlation caused by long-term trends by including separate intercepts for each calendar year, so that the predator variable only emerges as significant if prey population change within calendar years is more negative on average among sites where the predator is present Thomson el al. s model estimated prey abundance in year t using abundance in t-1 as an offset, but Newson et al. achieved the same result by cumulating the effect of annual intercepts and predator indices for years 1 to t-1 at each site (Freeman & Newson 2008). They also used a measure of change in predator abundance from t-1 to t in place of the binary measure of predator presence a, as well as including a range of environmental variables. The analysis assessed the overall effect of predation on each species with a multiple deletion test encompassing all the predator variables, and also performed individual tests in which each predator variable was deleted from the model while retaining all the others For prey species in which the multiple deletion test was non-significant, it was inferred that they are not affected by predation, and no further consideration was given to deletion tests for individual predator species. Such inference is unjustified, however, since predator variables that account for negligible deviance in the model contribute redundant degrees of freedom to the likelihood ratio test. Absence of predator effects can only be safely inferred where no significant Δdeviance emerges

5 during progressive backward elimination of the least significant predator variables in the model, particularly since there are many implausibly related predator/prey combinations (e.g. Great Spotted Woodpecker v Lapwing) b. In any event, the multiple deletion test is a poor indication of whether predation is causing prey populations to decline, because much of the Δdeviance is attributable to positive associations with individual predator species. The two-tailed significance levels reported by Newson et al. for individual predator effects are similarly tangential, and exclude some negative interactions that would be significant under a more appropriate one-tailed test, while highlighting irrelevant positive associations. The large number of positive associations that emerged nevertheless provides a clue as to the underlying processes at work, which can be further investigated by looking at overall patterns in the results. 86 Re-presentation and evaluation of results in Newson et al Newson et al. present regression coefficients with standard errors, from which z-scores and their one-sided probabilities can be calculated. Under a null hypothesis of no effect, the expected distribution of such probabilities is uniform, since by definition there is an equal probability that the corresponding z-scores will appear in any percentile of the asymptotically normal error distribution. However in nine of 17 cases the distribution differs significantly from uniform, and there is a marked tendency for a u-shaped distribution throughout, especially for CBC, indicating that marginally positive or negative relationships are more frequent than expected (Table 1, Fig. 1). This includes the case of Collared Dove, included as a dummy predator to expose any underlying tendency for spurious correlations to emerge, and which shows a markedly u-shaped distribution for CBC data and a prevalence of positive correlations for BBS Some insight into the significance of such relationships can be obtained by comparing the direction of correlation with general population trends in the prey species involved (Table 2). In the case of CBC, out of 91 marginally positive or negative regressions the majority (58) are positive, even though most (36) of these are with generally declining prey species. For BBS a majority of marginal

6 probabilities (31/56) are again positive, and most (19) of these are with generally increasing species. These results demonstrate that although population change among predator and prey species is associated more often than would be expected by chance, many of these associations are unlikely to be a result of direct interactions between the species concerned (see Appendix S2 for additional discussion). Consideration therefore needs to be given as to why the methodology might be generating spurious relationships. 107 Origin of spurious associations Newson et al. performed regression of annual change in prey numbers against annual change in predator numbers on the assumption that prey will decline during a period of predator increase, but return to stability when predator populations stabilize, leading to a negative relationship between the two (Fig. 2a, b). The inclusion of separate intercepts for each calendar year means that temporal covariance within individual sites will not contribute to the overall covariance between predator change and prey change in the model, but it will still determine that which occurs at a given point in time among a sample of sites that vary in the timing of predator increase. However, this also applies where predator increase and prey decline do not generally coincide, which can result in the emergence of positive change-change associations Because there is a tendency for the change-change relationship to cycle, especially where the predator is increasing from zero or a low level (Figure 2b), the emergence of a negative relationship is heavily dependent on the coincidence of rapid population change near the inflexion points of the respective population trends. If these are displaced temporally, the negative change-change relationship snaps over to a positive one (Figure 2 c-f). Change-change analyses may also have counter-intuitive results with other combinations of population trends, such as when predator and prey are both increasing. This will result in a positive association if the increasing trends are coincidental, but again this can change to a negative association if the trends are displaced relative to each other (Figure 2g).

7 Associations between changes in species populations can also arise from variation in the rate of change from place to place, and from habitat to habitat. A species may reach saturation level in its preferred habitat before it begins to colonise habitats or locations that are less preferred (Brown 1969, Fretwell & Lucas 1970). Consequently, although a species may be increasing or decreasing overall, actual changes in population level may only be occurring in more marginal habitats, while populations in preferred habitats remain stable. This will generate associations between species in the rate of population change at any instant in time, which vary according to the relative habitat preferences of the species concerned, even when the underlying population trends are completely independent and unrelated For example, where two species with similar habitat preferences are undergoing population change, they may be essentially stable at census sites that represent their core habitat, and either increasing or decreasing at sites representing more marginal habitat. In either case there will be a positive association between the two species rates of population change at any point in time (Fig. 3a, b). Where the habitat preferences of the two species differ, such that the core habitat of one is marginal for the other, and vice versa, parallel population changes may result in a negative association (Fig. 3c, d). Where one species is increasing while the other is decreasing, the association will be negative if habitat preferences are similar (Fig. 3e), or positive if habitat preferences differ (Fig. 3f). 144 Re-evaluation of Newson et al. s study The analysis presented by Newson et al. has many features that mitigate against the emergence of evidence for predator effects, including use of two-tailed tests and the inclusion of redundant variables/degrees of freedom in their linear models. The analysis also fails to account for density dependence of change in prey numbers (cf. Thomson et al. 1998, Swallow et al. 2015), which are likely to decrease when high and increase when low, regardless of the impact of predation. These

8 issues alone undermine their conclusion of a lack of evidence for impacts of predators at the national scale However the methodology used by Newson et al. suffers from deficiencies that are much more fundamental. It rests first and foremost on the assumption that if a species is not being affected by the predator then no association will emerge. However this is an unsafe assumption for a variety of reasons, and the results presented in Newson et al. suggest that it is a false one. Instead there is a marked tendency for both positive and negative associations to emerge more often than would be expected by chance, and in many cases these are highly unlikely to be a result of any direct interaction between the species involved. The existence of this underlying tendency for spurious associations also means that effects arising from genuine interactions may either be indistinguishable from such associations, or cancelled out by them if they happen to occur in the opposite direction. Newson et al. s assertion that positive associations between predators and potential prey species exonerate the predators concerned is therefore unjustified, and their interpretation of the high frequency of positive correlations as either a behavioural response of birds to an increased predation threat, or a result of predators responding to increased prey availability in improving habitat, is naive Interpretation is made even more difficult by the fact that non-coincident trends occurring among species within individual sites (Figure 2 c-h) can potentially arise where prey decline really is being caused by increased predator numbers. Population increase among predators, including those that involve an expansion in range and colonisation of new territory, is not necessarily captured accurately by bird census data. This is particularly so for the Common Birds Census data used by Newson et al., which is primarily designed to monitor local populations of songbirds by mapping their territories in plots of generally ha (Marchant 1983), and is much less efficient at monitoring populations of larger birds, including many predators (Nicol & Norris 2010). Newson et al. attempted to address this problem through use of predator indices derived from smoothed

9 frequency surfaces calculated for each calendar year. However, the index so derived for any discrete location will differ from the actual abundance of the predator by the amount of its residual from the model estimate, leading to displacement of the predator up-curve relative to the prey downcurve Even if CBC data accurately reflect the situation on the ground it may be misleading, since it only refers to presence during the breeding season, and so fails to capture predator activity occurring at a locality at other times of year. For example Newton (1986) reported that during the recovery of Sparrowhawks from their mid-20 th century population crash in Britain, initial sightings of birds in newly recolonised areas generally occurred outside the breeding season, usually in the autumn. Such visitors could have been impacting local bird populations prior to becoming established as breeders and therefore being recorded by CBC, and this may explain a greater prevalence of negative associations between Sparrowhawk presence and prey decline in October compared to December or March/April (Chamberlain, Glue & Toms 2009) It is equally possible that decline in prey populations may only begin some time after predator presence has begun to increase. More predators inevitably result in mortality that would not otherwise occur, but this may be compensated either by a density-dependent reduction in other forms of mortality, or a density dependent increase in breeding success. An increase in predator presence may, therefore, have no net effect on prey populations or else may only begin to have an effect once it has crossed a threshold (Boyce, Sinclair & White 1999). In the latter case, decline in prey populations may only occur well after predator populations have begun to increase. 195 Conclusion Overall, therefore, it is reasonable to assume that for species where there are underlying population trends over a wide area, there will be a tendency for associations to emerge even where there is no direct interaction between the species, and also that where a predator is affecting prey populations,

10 the resulting association may not necessarily be negative. This can explain the high frequency of marginally positive and negative effects that emerged from Newson et al. s analysis, as well as the higher frequency of such effects in the analysis of CBC data, since CBC covers a period during which many more prey populations were undergoing marked declines than in the period covered by BBS Such are the problems with interpretation of regression analyses of annual change in prey numbers against predator indices, that the approach may have limited utility, and the results presented by Newson et al. cannot support the inference that there is no evidence for an effect of predators on prey populations. This also enjoins caution regarding a recent analysis by Swallow et al. (2015), which indicates a powerful effect of Sparrowhawk abundance on population change in House Sparrows. In the latter case however, the result is similar to that of a previous analysis of the same data by Bell et al. (2010) using a radically different approach involving estimation of average population trends in relation to the timing of predator increase. Use of the two approaches in tandem may, therefore, be the most constructive approach to the use of census data to investigate the problem of predation and bird population decline.

11 Endnotes a. In some cases a lagged effect was modelled using change from t-2 to t-1. b. For scientific names see Appendix 1.

12 References Baillie, S.R., Marchant, J.H., Leech, D.I., Massimino, D., Sullivan, M.J.P., Eglington, S.M., Barimore, C., Dadam, D., Downie, I.S., Harris, S.J., Kew, A.J., Newson, S.E., Noble, D.G., Risely, K. & Robinson, R.A BirdTrends 2014: trends in numbers, breeding success and survival for UK breeding birds. Research Report 662. BTO, Thetford Bell, C.P., Baker, S.W., Parkes, N.G., Brooke, M de L. & Chamberlain, D.E The role of the Eurasian Sparrowhawk (Accipiter nisus) in the decline of the House Sparrow (Passer domesticus) in Britain. Auk 127: Brown, J.L The buffer effect and productivity in tit populations. Am. Nat. 103: Boyce, M.S., Sinclair, A.R.E. & G.C. White Seasonal compensation of predation and harvesting. Oikos 87: Chamberlain, D.E., Fuller, R.J., Bunce, R.G.H., Duckworth, J.C. & M. Shrubb Changes in the abundance of farmland birds in relation to the timing of agricultural intensification in England and Wales. J. Appl. Ecol. 37: Chamberlain, D. E., Glue, D. E. & Toms, M. P Sparrowhawk Accipiter nisus presence and winter bird abundance. J. Ornithol. 150: Davey, C.M., Vickery, J.A., Boatman, N.D., Chamberlain, D.E., Parry, H.R. & Siriwardena, G.M Assessing the impact of Entry Level Stewardship on lowland farmland birds in England. Ibis 152: DEFRA Wild bird populations in the UK, 1970 to Annual statistical release. Department of Environment, Food & Rural Affairs, London. populations-in-the-uk

13 Freeman, S.N. & Newson, S.E On a log-linear approach to detecting ecological interactions in monitored populations. Ibis 150: Fretwell, S.D. & Lucas, H.J On territorial behaviour and other factors influencing habitat distribution in birds. Acta Biotheor. 19: Fuller, R.J., Gregory, R.D., Gibbons, D.W., Marchant, J.H., Wilson, J.D., Baillie, S.R. & Carter, N Population declines and range contractions among lowland farmland birds in Britain. Conserv. Biol. 9: Gibbons, D.W., Amar, A., Anderson G.Q.M, Bolton, B., Bradbury, R., Eaton, M.A., Evans, A.D., Grant, M.C., Gregory, R.D., Hilton, G.M., Hirons G.J.M., Hughes, J., Johnstone, I., Newbery, P., Peach, W.J., Ratcliffe, N.; Smith, K.W., Summers R.W, Walton, P. & Wilson J.D The predation of wild birds in the UK: a review of its conservation impact and management. Research Report no 23, RSPB, Sandy Harris, S.J., Massimino, D., Newson, S.E., Eaton, M.A., Balmer, D.E., Noble, D.G., Musgrove, A.J., Gillings, S., Procter, D. & Pearce-Higgins, J.W The Breeding Bird Survey BTO Research Report 673. British Trust for Ornithology, Thetford. 252 Marchant, J.H BTO Common Birds Census Instructions. British Trust for Ornithology, Tring Newson, S.E., Rextad, E.A., Baillie, S.R., Buckland, S.T. & Aebischer, N.J Population change of avian predators and grey squirrels in England: is there any evidence for an impact on avian prey populations? J. Appl. Ecol. 47: Newton, I The Sparrowhawk. Poyser, London Newton, I The recent declines of farmland bird populations in Britain: an appraisal of causal factors and conservation actions. Ibis 146: Nicoll, M. & Norris, K Detecting an impact of predation on bird populations depends on the methods used to assess the predators. Methods in Ecology and Evolution 1:

14 Swallow, B., Buckland, S.T., King, R. & Toms, M.P Bayesian hierarchical modelling of continuous non-negative longitudinal data with a spike at zero: An application to a study of birds visiting gardens in winter. Biometrical J. 58: Thomson, D. L., Green, R. E., Gregory, R. D. & Baillie, S. R The widespread declines of songbirds in rural Britain do not correlate with the spread of their avian predators. P. R. Soc. B 265: Vorisek, P., Jiguet, F., van Strien, A., Skorpilova, J., Klvanova, A. & Gregory, R.D Trends in abundance and biomass of widespread European farmland birds: how much have we lost? BOU Proceedings Lowland Farmland Birds III.

15 Appendix 1 Scientific names of species mentioned in the text and supporting information. Bird nomenclature follows the British Ornithologists Union British List published 25 February Grey Squirrel Sparrowhawk Buzzard Lapwing Collared Dove Green Woodpecker Great Spotted Woodpecker Kestrel Magpie Jay Carrion Crow Blue Tit Great Tit Coal Tit Skylark Chiffchaff Willow Warbler Garden Warbler Nuthatch Wren Starling Blackbird Sciurus carolinensis Accipiter nisus Buteo buteo Vanellus vanellus Streptopelia decaocto Picus viridis Dendrocopos major Falco tinnunculus Pica pica Garrulus glandarius Corvus corone Cyanistes caeruleus Parus major Periparus ater Alauda arvensis Phylloscopus collybita Phylloscopus trochilus Sylvia borin Sitta europaea Troglodytes troglodytes Sturnus vulgaris Turdus merula

16 16 Song Thrush Mistle Thrush Spotted Flycatcher Robin Dunnock House Sparrow Tree Sparrow Yellow Wagtail Meadow Pipit Chaffinch Bullfinch Greenfinch Linnet Goldfinch Yellowhammer Reed Bunting Turdus philomelus Turdus viscivorus Muscicapa striata Erithacus rubecula Prunella modularis Passer domesticus Passer montanus Motacilla flava Anthus pratensis Fringilla coelebs Pyrrhula pyrrhula Chloris chloris Linaria cannabina Carduelis carduelis Emberiza citrinella Emberiza schoeniclus 273

17 Appendix 2 Discussion of one-sided regression probability distributions that significantly differ from uniform in Figure 1 and Table 1. Buzzard CBC data for Buzzard show a marked bias towards positive correlations, but the prey species concerned tend to be declining. However such correlations are readily explicable in terms of the timing of the increase in Buzzard populations, which took place in the 1990s, whereas declines in prey species generally occurred rather earlier. If Buzzards colonised a significant proportion of CBC sites after the main period of decline among the other species present, the overall result could be a positive relationship of the type illustrated in Figure 2c. It could also be explicable in terms of the scenario illustrated in Figure 3f, whereby the predator moves into habitats or areas where prey are relatively stable at the same time that declining prey species withdraw from the core Buzzard habitats. Buzzard continues to increase rapidly through the BBS period, where there is a preponderance of negative correlations, though none of the species concerned could reasonably have been impacted by Buzzard predation Sparrowhawk CBC data for this species provide the only instance where a significant difference from a uniform distribution is caused by a positive skew of regression probabilities (Table 1, Figure 1), and all of the species with significant negative correlations could plausibly be affected by Sparrowhawk predation. Again, however, caution is required. One of the strongest correlations is with another rapidly increasing species, the Nuthatch, and might be attributable to temporally disjunct increases within locations (Figure 2 g,h), and negative correlations with declining species might be examples of the process in Figure 3e. Sparrowhawk populations were generally stable during the BBS period, so it might be expected that no correlations would occur with prey populations, which is borne out by the almost perfectly uniform distribution of regression probabilities (Figure 1).

18 Corvids All three corvid species show similar patterns with negative skew of regression probabilities during the CBC period (Table 1, Figure 1), the most significant of which are generally with declining species, suggesting either temporal disjunction of population change within sites (Figure 2 c-f) or else the scenario of Figure 3f. Regression probabilities are not significantly different from uniform during the BBS period for Magpie and Jay, and only differ for Carrion Crow because of clustering of z-scores ultimately attributable to low mean counts of the species on BBS sites Collared Dove The general increase in Collared Doves throughout the CBC period results in both positive and negative correlations with declining species, the positive correlations suggesting either disjunct population movements within sites (Fig 2 c-f) or the scenario of Fig 3f, with negative correlations suggesting that of Fig 3e. There is overwhelmingly negative skew of regression probabilities during the BBS period, and most of the positive correlations indicated are with increasing species suggesting parallel spatio-temporal spread of species with similar habitat requirements (e.g. Fig 2a).

19 Appendix 3 Species breakdown of the frequencies in table 2 Species Prey Population Trends Buzzard +ve Robin Blue Tit CBC BBS +ve -ve +ve -ve Wren Goldfinch ± stable Coal Tit Greenfinch Goldfinch Blue Tit Reed Bunting -ve Song Thrush Willow Warbler House Sparrow Linnet Yellowhammer Lapwing Dunnock Skylark Sparrow -hawk +ve Robin Nuthatch Goldfinch Blackbird ± stable Greenfinch Goldfinch Blue Tit -ve Yellow Wagtail Willow Warbler Starling Lapwing Tree Sparrow Bullfinch Reed Bunting Kestrel +ve Wren Nuthatch Robin ± stable Goldfinch Mistle Thrush Tree Sparrow House Sparrow -ve Mistle Thrush Yellowhammer Reed Bunting Lapwing Starling House Sparrow Tree Sparrow Garden Warbler Blackbird Great Spotted Woodpecker +ve Nuthatch Great Tit Dunnock Greenfinch Goldfinch Wren ± stable Coal Tit Chaffinch -ve Willow Warbler Spotted Flycatcher Reed Bunting Yellow Wagtail Starling

20 20 Species Prey Population Trends CBC BBS +ve -ve +ve -ve Magpie +ve Green Woodpecker Blue Tit Great Tit Dunnock Greenfinch Goldfinch ± stable Greenfinch Goldfinch Chaffinch -ve Lapwing Skylark Mistle Thrush Starling House Sparrow Tree Sparrow Bullfinch Yellow Wagtail Blackbird Willow Warbler Spotted Flycatcher Starling Jay +ve Green Woodpecker Nuthatch Great Tit Green Woodpecker Chiffchaff ± stable Garden Warbler Chiffchaff Greenfinch Goldfinch Greenfinch Lapwing Blue Tit -ve Dunnock Blackbird Song Thrush Spotted Flycatcher House Sparrow Linnet Bullfinch Reed Bunting Yellow Wagtail Tree Sparrow Yellow Wagtail Carrion Crow +ve Green Woodpecker Robin Chaffinch ± stable Garden Warbler Linnet Tree Sparrow Yellowhammer -ve Lapwing Meadow Pipit Willow Warbler Starling Starling

21 21 Species Collared Dove Prey Population Trends +ve CBC +ve -ve +ve -ve Tree Sparrow Linnet Reed Bunting Blue Tit Chaffinch Robin Great Tit Dunnock Chiffchaff Greenfinch Goldfinch BBS Blackbird ± stable Garden Warbler Chiffchaff Blue Tit Yellowhammer House Sparrow -ve Spotted Flycatcher Tree Sparrow Lapwing Dunnock Song Thrush Starling Grey Squirrel +ve Wren Blackbird Greenfinch ± stable Blue Tit Coal Tit Chaffinch -ve 317

22 Table 1. Kolmogorov-Smirnov One-Sample tests for a uniform distribution of one-tailed probabilities for regression coefficients of predator variables in Newson et al. (2010) Species CBC BBS D P D P Buzzard *** * Sparrowhawk * Kestrel Great Spotted Woodpecker Magpie ** Jay *** Carrion Crow *** * Collared Dove * *** Grey Squirrel All *** ** 320

23 Table 2. Frequency of regressions of prey population change on predator population change with one-tailed probabilities <0.05 (i.e. marginally negative) or >0.95 (marginally positive) in relation to general prey population trends over the period of the relevant survey (Baillie et al. 2014). Prey Population Trends CBC BBS P>0.95 P<0.05 P>0.95 P<0.05 +ve ± stable ve All See Appendix 3 for a breakdown by predator and prey species.

24 Figure 1. Frequency of one-tailed probabilities for regression coefficients of predator variables in Newson et al. (2010) Figure 2. (a) Predator (solid circles) population change from year t to t+1 modelled as N t+1 =N t +N t *(1- N t /10), and prey (open circles) population change modelled as N t+1 =N t +0.7*N t *(1-N t /30)-0.03*N t *P t, where P t = predator abundance. (b) Prey change in a) expressed as loge(n t+1 /N t ) plotted against predator change in the same year expressed as loge(n t+1 +1/N t +1)(cf. Newson et al. 2010). (c) As in a), but prey population modelled as N t+1 =N t +0.7*N t *(1-N t /30)-0.03*N t *P t+7. (d) Prey change in c) plotted as in b). (e) As in a) but prey population modelled as N t+1 =N t +0.7*N t *(1-N t /30)-0.03*N t *P t-5. (f) Prey change in e) plotted as in b). (g) As in a) but prey population modelled as N t+1 =N t +N t *(1-N t /30). (h) Prey change in g) plotted as in b). Dashed line in b), d) and f) represents maximum likelihood regression estimate Figure 3. Spurious relationships resulting from simultaneous changes in species populations. Boxes represent habitats or census locations. 0, + and represent change in population during a one year period for two species. (a) Species share preferred habitat/core area and expand into marginal habitats/areas. (b) Species share the same preferred habitat/core area and contract from marginal habitats/areas. (c) Species differ in preferred habitat/core area and expand into marginal habitats/areas. (d) Species differ in preferred habitat/core area and contract from marginal habitat/areas. (e) Species share the same preferred habitat/core area and one expands into while another contracts from marginal habitat/areas. (f) Species differ in preferred habitat/core area and one expands while another contracts from marginal habitat/areas. Correlation plot represents relationships that emerge from a change-change analysis of the process illustrated. 346

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