### Abstract

- Top of page
- Abstract
- Introduction
- Methods
- Results
- Discussion
- Acknowledgements
- References

**1.** Survival rates and natalities for a population of snowshoe hares in the Yukon were estimated independently of and simultaneously with estimates of population change during the increase phase of a hare cycle.

**2.** Simple demographic models are used to show that even though the estimated survival rates and natalities were high relative to previously published estimates, the observed demographic parameters are unable to explain the extent of population increase, and we conclude that some of these parameters must be underestimates.

**3.** A sensitivity analysis is used to examine the potential influence of changes in these demographic parameters on the population growth rate. During most years of the hare cycle the population growth rate is potentially most sensitive to changes in juvenile postweaning survival. Only during crash years is adult survivorship likely to be a more important determinant of the rate of population change.

**4.** Examination of previously published data sets on two full population cycles suggests that while survival rates are positively correlated with population growth rates, their incorporation into demographic models results in frequent underestimation of the rate of population increase.

### Introduction

- Top of page
- Abstract
- Introduction
- Methods
- Results
- Discussion
- Acknowledgements
- References

Population dynamics has a great advantage over community dynamics in having an arithmetic of numerical change. If changes in numbers in a population are tracked over a time period, estimates of birth, death and migration rates should be consistent with these changes. Surprisingly, very few vertebrate ecologists use this powerful arithmetic to check on the accuracy of their estimated parameters of births and deaths. In this paper population arithmetic is applied to a fluctuating population of snowshoe hares (*Lepus americanus* Erxleben) to see whether it is possible to reconstruct the numerical population changes from independent estimates of the components of births and deaths. This analysis leads to the conclusion that empirically measured survival rates are likely to be underestimated. A sensitivity analysis of snowshoe hare demography was then conducted to understand better the consequences of underestimating different age-specific survival rates for our predictions. In order to evaluate the generality of these findings, this analysis is extended using two previously published data sets to examine the demography of snowshoe hares over their full population cycle.

Snowshoe hares are typical of small mammals in having a high birth rate and a correspondingly high death rate. Individual females may have 3–4 litters during the summer, spaced at approximately 5-week intervals, with litter sizes from 3 to 8, so that as many as 20 offspring may result from one breeding season. The maximum rate of increase of snowshoe hares is thus tenfold per year, assuming an even sex ratio. Observed rates of increase over the 10-year population cycle vary from about 0·2–4·0 per year, with correspondingly high variation in both birth and death rates over the cycle (Krebs *et al*. 1986; Keith 1990).

### Results

- Top of page
- Abstract
- Introduction
- Methods
- Results
- Discussion
- Acknowledgements
- References

The basic demographic parameters are summarized in Table 1 and 2. Figure 1 shows how adult survival varied through the year, and also shows that this survival rate is well modelled by assuming a constant 28 day survivorship.

The comparison of observed hare numbers in the spring of 1996 and the numbers predicted projecting from the spring of 1995 is made in Table 3. The projections all substantially underestimate the observed increase in hare numbers. The observed population growth rate averaged over the five grids was 3·8, the growth rates that result from the use of the demographic parameters underestimate this figure by between 30 and 75%. Evaluation of the significance of these underestimates is complicated by the absence of any knowledge about the probable distribution of the expectations, however, even a conservative evaluation of the difference is significant on the food addition grids, and a Fisher's combined probability test for an effect on all three control grids using the conservative *P*-values from the application of Chebyshev's inequality yields χ2 = 10·94, (6 d.f., *P* = 0·090). A Fisher's combined probability test on the bootstrapped *P*-values indicates a significant discrepancy (χ^{2} = 16·18, 6 d.f., *P* = 0·013) on the three control grids taken together.

Table 3. Observed and predicted hare numbers on each of the five grids, projected from spring 1995 to spring 1996. Parentheses enclose the standard deviation of the observed or predicted value. 95% CI obtained from the program capture (White *et al*. 1982) for observed population estimates and 95% CI obtained by the bootstrap procedure for the predicted population estimate are given below. The prediction is derived from equation 1, the conservative P-values test the null hypotheses that the difference between the observed abundance for spring 1996 and the prediction is equal to zero. The test is performed using Chebyshev's inequality, which is valid for any distribution of the difference. The bootstrap *P*-value uses the overlap between the bootstrapped distribution of expected abundances and a modified exponential gamma distribution fitted to observed spring 1996 census information to test the null hypothesis that the observed spring 1996 abundance is not different to the bootstrapped expectation Grid | Observed Spring 1995 | Observed Spring 1996 | Predicted Spring 1996 | Conservative *P*-value | Bootstrap *P*-value |
---|

Control 1 | 11 (1·98) 9–19 | 36 (1·51) 34–40 | 22 (6·29) 11–44 | 0·217 | 0·104 |

Control 2 | 5 (1·52) 4–11 | 25 (2·13) 24–38 | 10 (3·68) 1–28 | 0·082 | 0·050 |

Control 3 | 17 (1·85) 15–24 | 53 (3·36) 50–65 | 34 (8·55) 22–58 | 0·237 | 0·059 |

Food 1 | 31 (5·18) 25–47 | 130 (11·32) 113–159 | 34 (10·27) | 0·025 | – |

Food 2 | 52 (8·07) 41–74 | 191 (18·07) 163–235 | 56 (16·95) | 0·034 | – |

Closer inspection of the first half of this projection period, spring 1995 to autumn 1995 (Table 4a-b), is suggestive of the origin of the observed discrepancies. The severity of the underestimates of the total population is moderated (as they have less time to compound) but is still very substantial on the food grids (Table 4a). However, if the analysis is conducted exclusive of recruitment, then the projections fall mostly midway within the possible range of observed adult numbers (Table 4b). Inspection of the second half of the projection (autumn 1995 to spring 1996) once again reveals consistent underestimation (Table 4c), significantly so on three out of five occasions. Taken together, these results suggest that while adult survivorship is underestimated, the principal source of error originates in the estimates of recruitment parameters.

Table 4a. Observed and predicted hare numbers on each of the five grids, projected from spring 1995 to autumn 1995. Otherwise details as for Table 3 **Table 4b**. Observed and predicted adult hare numbers on each of the five grids, projected from spring 1995 to autumn 1995. The range of observed adults in the autumn results from assuming all trapped animals in the unclassified cohort are juveniles (lower value) or adults (upper value)Table 4c. Observed and predicted hare numbers on each of the five grids, projected from autumn 1995 to spring 1996. Predictions are based on assuming that autumn-trapped animals in the unclassified cohort are all adult (this assumption gives the largest predicted spring populations). Otherwise details as for Table 3 The sensitivities and elasticities are reported in Table 5. The elasticities suggest growth rate is potentially at least five times more sensitive to changes in adult and postweaning juvenile survival probabilities than to changes in preweaning survival, and at least 15 times more sensitive relative to changes in litter size parameters. Furthermore, analysis of three different hare populations reveals that the observed coefficient of variation in 28-day juvenile survival is consistently about twice that observed in adult survivorship. This suggests that throughout the hare cycle, variation in the 28-day juvenile survival rate is three times as influential in snowshoe hare demography as variation in 28-day adult survivorship and preweaning survivorship, and 30 times as important as variation observed in litter sizes and fourth litter pregnancy rates (Table 5). The analysis predicts that the absence of a fourth litter will decrease the growth rate by ≈ 15% (but note that this is likely to be an overestimate because the low fourth litter survival parameters have been pooled with the higher survival rates of earlier litters). In Table 5 the sensitivities and elasticities are evaluated using the observed demographic parameters for the increase year 1995–96. In Fig. 2(a) the elasticities have been calculated for values of adult and juvenile postweaning survivorship observed for the years 1977–84 (data from Krebs *et al*. 1986). The results suggest that juvenile survival remains the most sensitive demographic parameter over most of the cycle, the important exception being in sharp decline years when adult survival assumes prime influence (Fig. 2a). This result is also observed for the parameters of a cyclic hare population in Alberta (Fig. 2b) using the data from Keith & Windberg (1978). In decline years Übr*LJ*_{28} falls to such low values that small changes to any of the recruitment parameters are unlikely to affect the population dynamics.

Use of pooled data on juvenile survivorship results in a predicted growth rate for the control grids of 2·4, compared with the observed value of 3·8. The influences of the three dominant parameters on this growth rate are shown in graph form in Fig. 3(a–c). The topography of these surfaces close to the observed values is described by the sensitivities reported in the second column of Table 5. The dominant influence of postweaning juvenile survivorship is apparent. The discrepancy between observed and predicted growth rates is not resolved even when survival parameters are inflated to their upper 95% confidence limits. Figure 4(a,b) shows contour plots of the effects of different combinations of Übr*LJ*_{28} and *LA*_{28} on predicted population growth rate. Overlayed on this figure are measured combinations of Übr*LJ*_{28} and *LA*_{28} together with the observed corresponding population growth rates for the Kluane and Rochester populations shown for the full cycle. Observed and predicted growth rates are positively correlated for both populations (for the Kluane population *r* = 0·67; for the Rochester population *r* = 0·91). Note that even though the natalities used in these reconstructions are from a year of high hare density increases (with high rates of preweaning survival likely), the observed growth rates are still frequently underestimated.

### Discussion

- Top of page
- Abstract
- Introduction
- Methods
- Results
- Discussion
- Acknowledgements
- References

This analysis reveals significant discrepancies between estimates of population changes deduced from trapping methods, and those predicted on the basis of observed demographic parameters derived from radio-telemetry studies. While the discrepancies are consistent (there was not a single expectation that exceeded the observed population estimate) and the magnitude in some cases substantial, evaluating their significance is problematic. Despite improvements made to the program capture in the 1991 release, variances from the jack-knife estimator associated with estimated population sizes are sometimes suspected of being negatively biased (Rosenberg, Overton & Anthony 1995; J.D. Boulanger & E. Rexstad, personal communication). For these reasons we have attempted to be conservative in our statistical evaluation of the encountered discrepancies. For a specified standard deviation, Chebyshev's inequality applies to the maximally dispersed distribution and the bootstrapping procedure is conservative in a number of respects (variances in initial population sizes are overestimated by the exponential gamma distribution, by a factor of at least two for the spring to spring predictions; and variances around litters sizes are considerably overestimated by use of a Poisson distribution).

While survival parameters were estimated from telemetry studies and therefore do not confound dispersal with mortality, we recognize that we have omitted the effects of dispersal from the census study. We first ask if the observed underestimation in population growth might be explained by immigration into our study areas. Because the snowshoe hare cycle is synchronous over most of Western Canada and Alaska (Keith 1990), there are unlikely to be any large scale movements into the study area. However, there are certainly small scale movements within the study area. In particular, areas provided with supplemental food in the form of rabbit feed attract immigrants (Boutin 1984). The discrepancies in Food 1 and Food 2 are interpreted as a strong indication of the role of immigration in driving population changes in supplemental food areas. This is shown particularly clearly in Table 4c for the winter period during which numbers on the food grids fell only slightly, in spite of all the winter losses of radio-tagged hares (Table 1).

The control areas also show an underestimation of population increase from 1995 to 1996 and this cannot be explained by movements. We can see no reason to expect immigration into the control grids and we reject this hypothesis to explain the discrepancies. This leaves only four possibilities: (i) that reproductive parameters are underestimated; (ii) that adult survival is underestimated; (iii) that juvenile survival after weaning is underestimated; and (iv) that juvenile survival before weaning is underestimated. We reject the idea that reproductive parameters are biased, because they are at or above similar estimates available in the literature and agree with independent data from road-killed autopsies (C.I. Stefan, unpublished data). Also, because the elasticities of litter sizes are all very low, it is unlikely that any believable change in these reproductive estimates could balance the demographic equation.

Adult survival estimated from mark–recapture models like Jolly–Seber have, in the past, underestimated adult survival estimated from radio-telemetry by as much as 15% (Boutin & Krebs 1986). We therefore predict that using trapping data to estimate survival as opposed to telemetry would result in an even greater underestimate of growth rate. W. Hochachka (unpublished information) compared adult hare survival of individuals that were radio-collared and those not collared in the same population and showed that hare survival was decreased by radio-collaring. W. Hochachka (unpublished information) speculated that the trauma of live-trapping was the major cause of this additional mortality, because hares that were captured more often suffered a greater reduction in survival. Hare mortality rates might also be increased in the live-trapping areas because predators were attracted to these areas in winter as a result of ease of access on packed snowshoe trails (Murray & Boutin 1991). We would thus tentatively suggest that the adult survival rates presented in Table 1 might be underestimated by no more than 5%. If we increase adult hare survival by 5% (from 0·923 per 28 days to 0·969), we still cannot balance the demographic equation (Fig. 3a,b). It is concluded that a negative bias in adult survival is part of the reason for the observed discrepancy on the control grids, but it is unlikely to be the whole explanation.

It is not clear which of the two components of juvenile survival could be biased. We can only observe that, in general, the postweaning survival of juveniles is nearly as high as adult survival, and because on biological grounds it seems likely that being a juvenile is at least slightly more risky than being an experienced adult, we would guess that the observed postweaning survival estimate is accurate to within 1–2%. We suspect that the preweaning survival estimate is also negatively biased. We do not know this for certain, but we anticipate that the disturbance of caging pregnant females and moving the marked juveniles back into the field may increase losses at least slightly in this life stage. If adult survival rates were increased by 4%, and juvenile postweaning survival by 2%, then preweaning survival would require ≈ 13% increase to explain the observed discrepancies. However, the growth rate is extremely sensitive to juvenile postweaning survival, a 3% increase in this parameter would require only a 5% increase in preweaning survival rates.

In conclusion, the demographic equation does not balance for snowshoe hares. This imbalance is most credibly corrected by making small changes (< 5%) to all survival parameters. We think that hare survival estimated from radio-telemetry is likely to be negatively biased, and that the effects of caging pregnant females prior to parturition may have a negative impact on estimation of preweaning survival. We suggest that additional work needs to be undertaken to determine exactly why radio-collared hares suffer added mortality and to reduce any impact of live-trapping on individuals. The use of alternative methods to estimate recruitment would also be useful to increase the accuracy of our estimates of preweaning mortality.

The sensitivity analysis is fully consistent with long-standing conclusions that juvenile survivorship is of prime importance in determining changes in snowshoe hare populations (Green & Evans 1940; Keith & Windberg 1978; Krebs *et al*. 1986). It should be noted that the derivatives in Table 5 are the slopes of curves evaluated at particular parameter values, and are likely to be misleading if used for very large parameter perturbations. The model makes few questionable assumptions. For the year from March 1995 to March 1996 we are satisfied that adult survivorship is well modelled as a constant value independent of seasonal modification (see Fig. 1). However, this may not always be true, and past studies have found autumn and early winter survivorships to correlate more highly with growth rate than survival rates over other seasons (Krebs *et al*. 1986). The model assumes that all demographic parameters apply equally and independently to members of the population, i.e. that no covariances exist among demographic parameters as they apply to individuals at any particular time. The existence of such covariances could conceivably introduce systematic bias into the predictions of the model. Cyclic snowshoe hare populations are highly dynamic and it is perhaps not surprising that their growth rates are most sensitive to juvenile survivorship. Only in crash years, when juvenile survivorship is so low that small changes make no difference, does λ become more sensitive to adult survival rates.

Previous studies have found correlations between measures of survival and population growth rate; for example Krebs *et al*. (1986) observed correlations between adult autumn survival and population growth rate (*r* = 0·49), and juvenile autumn survival with population growth rate (*r* = 0·60). Keith & Windberg (1978) observed analogous correlations of 0·65 and 0·8. In this study correlations were noted between observed and expected growth rates (where expected growth rates are generated using equation 6) of 0·67 and 0·91 for these two data sets, respectively. That it is possible to achieve these correlation values without accounting for annual variation in natality is indicative of the insensitivity of growth rate to these parameters. The possibility that these populations are sensitive to different demographic parameters at different stages of the cycle (see Fig. 2) may help to explain why correlation coefficients between growth rates and different survival rates are not higher. High sensitivities are a prerequisite for tight correlations.

We encourage other population ecologists to utilize simple methods to check on the accuracy of their estimates of demographic components. In particular, the data presented here suggest considerable caution should be used in inferring missing demographic components from the demographic equation. Using changes in numbers and estimated death rates to infer birth rates can lead to misleading interpretations of demographic change if the death rate estimates are strongly biased.

### Acknowledgements

- Top of page
- Abstract
- Introduction
- Methods
- Results
- Discussion
- Acknowledgements
- References

The empirical data used in these analyses were gathered by many technicians and graduate students over a long period of time. In particular, we would like to thank V. Nams, M. O’Donoghue, J. McDowell, C. Doyle and F. Doyle for their efforts in the field. Base facilities were maintained by the Arctic Institute of North America and managed by A. Williams. The field work was funded by Natural Sciences and Engineering Research Counsel grants to C.J. Krebs and the Kluane Collaborative Special Project and Arctic Alpine Grants. E. Gillis was supported by a Natural Sciences and Engineering Research Counsel PGS-A award. This publication is No. 124 of the Kluane Boreal Forest Ecosystem Project. We are grateful to Dennis Chitty, Karen Hodges, Harald Steen and two anonymous referees for helpful comments on this manuscript. We would like to thank John Boulanger, Jim Nichols and Eric Rexstad for very helpful discussions regarding the jack-knife estimator and its confidence intervals.