Journal of Animal Ecology 2005 74, 612-618 Predicting the growth of a small introduced muskox population using population prediction intervals EINAR J. ASBJ0RNSEN*, BERNT-ERIK S^THER*, JOHN D. C. LINNELL|, STEINAR ENGENf, REIDAR ANDERSEN*! and TORD BRETTEN§ *Department of Biology, Norwegian University of Science and Technology, N- 7491 Trondheim, Norway; ^Norwegian Institute for Nature Research, Tungasletta 2, N-7485 Trondheim, Norway; ^Department of Mathematical Sciences, Norwegian University of Science and Technology, N-7491 Trondheim, Norway; and §Oppdal Bygdealmenning, N-7240 Oppdal, Norway Summary 1. A key issue in ecology is the prediction of future population fluctuations. Such population predictions are fundamental for population-viability analysis and are essential for assessing the implications of various management actions. Development of reliable population predictions is however, difficult because it requires estimation and modelling of the separate effects of the deterministic components of the population dynamics as well as the stochastic influences on the population fluctuations. Here we model the stochastic dynamics of an introduced population of muskox Ovibos moschatus in the Dovrefjell mountains of central Norway, using a simple model without density regulation. Our aim is to examine quantitatively factors affecting the accuracy of the population projections by applying the concept of Population Prediction Interval (PPI). 2. The long-term growth rate was s - 0-0511, assuming no density dependence. The environmental variance was relatively large (6e2 = 0-0159). This gives a deterministic growth rate of f - 0-0591. However, accounting for losses due to various kinds of human activities resulted in a nearly doubling of s (s - 0-0980). 3. Autumn temperature and late winter snow depth were each able to explain a significant proportion of the annual variation in population growth rates. 4. The impact of environmental stochasticity made the PPI wide after only a few years. Uncertainties in the estimates of the population parameters were quite small and had a minor impact on the PPI. 5. A sensitivity analysis showed that ignoring demographic stochasticity led to an overestimate of the environmental variance al, but that the impact on the width of the PPI was small. 6. This study shows that reliable projections of future population growth, even based on simple population models without density regulation, are dependent on assessment of the accuracy in the population predictions that must be based on estimating and modelling the stochastic influences on the population dynamics. Key-words: environmental stochasticity, muskoxen, Ovibos moschatus, Population Prediction Interval, population viability analysis. Journal of Animal Ecology (2005) 74, 612-618 doi: 10.1111/j. 1365-2656.2005.00946.X © 2005 British Ecological Society Introduction An important challenge for ecologists in the years to come will be to develop predictions of future fluctuations in population size, especially for species of manage- Correspondence: John Linnell, Norwegian Institute for Nature Research, Tungasletta 2, N-7485 Trondheim, Norway. E-mail: john.linnell@nina.no ment concern. Reliable population projections will require information about expected changes of the population, determined by the specific growth rate, carrying capacity and form of the density regulation. In addition, we need to estimate and to model the stochastic influences on the population dynamics. Demographic stochasticity is caused by random differences between individuals in their ability to reproduce or survive (Engen, Bakke & Islam 1998), with 613 Population prediction intervals for muskox © 2005 British Ecological Society. Journal of Animal Ecology, 74. 612-618 strongest effects on the population growth rate at small population sizes (Leigh 1981; Lande 1993, 1998). Environmental stochasticity refers to stochastic variation that affects all individuals in a certain group similarly and is important both in small and large populations (May 1973; Leigh 1981; Lande 1993). Unfortunately, separating the relative contribution of demographic and environmental stochasticity requires a combination of individual-based data on fitness-variation and accurate long-term time-series on population fluctuations (Saether & Engen 2002), which are rarely available. Finally, reliable population projections are also dependent on assessment of how uncertainties in the estimates of the parameters affect the accuracy of the population predictions (Ludwig 1996; Saether etal. 2000; Ellner et al. 2002). Following the introduction of the concept of the population prediction interval (PPI) into population viability analysis (Dennis, Munholland & Scott 1991), it became possible to examine quantitatively how stochasticity and parameter uncertainties affect future population predictions. Following Saether et al. (2000), Engen & Saether (2000) and Saether et al. (2002a), a PPI is defined as the stochastic interval that includes the population size with probability (1 - |3), where |3 is the probability that the variable we want to predict is not contained in the stochastic interval. This means that the confidence of the PPI decreases with increasing |3. When large stochastic effects are present, or population estimates are uncertain, the PPI soon becomes wide (Saether et al. 2000, 2002a; Engen, Saether & Moller 2001) . Furthermore, the PPI will increase as the time elapsed since the last observation increases. The specific growth rate at small population sizes r is an important parameter influencing the expected dynamics of the population. Unfortunately, this parameter is extremely difficult to estimate (Taylor 1995; Saether et al. 2000), because in populations fluctuating around the carrying capacity interpolation over a large range of population sizes is often necessary (Aanes et al. 2002) . Translocation of individuals (IUCN 1987) can provide here an important source of information for obtaining estimates of this parameter because usually a limited number of individuals are introduced into a pristine environment. In general, the success of such translocations increases with habitat quality, the release of wild-caught instead of captive animals and increasing numbers of released animals (Griffith et al. 1989; Wolf et al. 1996; Wolf, Griffith & Garland 1998; Komers & Curman 2000). Also, a strict regulation of human caused mortality is often required (Spalton, Lawrence & Brend 1999). In addition to their use as a tool for preventing species extinction, translocations often provide the possibility to study general ecological processes (Sarrazin & Barbault 1996; Komers & Curman 2000). Here we will use translocation of muskox Ovibos moschatus from eastern Greenland to the mountain range Dovre in Central Norway to parameterize a simple population model without density regulation that can be used to improve our ability to predict future fluctuations of small populations. Population dynamics of ungulates are affected strongly by climate (see review in Saether 1997). Different studies of ungulates have shown an effect from summer climate affecting juvenile as well as adult mortality, and furthermore, variation in summer climate has been reported to influence body weight, inducing variation in fecundity among cohorts (Albon, Clutton-Brock & Guinness 1987; Gaillard et al. 1997; Solbergera/. 1999). The reason for this variation is often linked to the effect summer climate has on the forage quality. Effects from winter climate are often associated with high mortality during harsh winters (Tener 1965; Smith 1984; Saether 1997; Aanes, Saether & 0ritsland 2000; Solberg et al. 2001). However, the influence of variation in climate-induced stochasticity on the population growth rate of ungulates has been quantitatively examined in only a few cases (Coulson et al. 2001; Saether et al. 2002b). The magnitude of such effects is important because they generally reduce the long-term growth rate of the population (Lande, Engen & Saether 2003) and hence increase the vulnerability of small populations. As a result of intensive hunting, the global distribution of the muskox Ovibos moschatus was restricted to certain areas in northern Canada and eastern Greenland at the beginning of the 20th century (Boertmann et al. 1992; Ferguson & Gauthier 1992; Lent 1998). In efforts to decrease the risk of extinction many reintroductions and introductions were made, and the species is now increasing globally. After an unsuccessful attempt prior to the Second World War, in the period 1947-53 muskoxen were taken from eastern Greenland and introduced into a mountain range in central Norway (Myrberget 1987). In this study, we analyse the dynamics of this muskox population. An important focus is to use the theoretical framework presented in Lande et al. (2003) to estimate essential population parameters. We use these estimates to predict future population sizes, illustrating the application of the concept of PPI (Saether et al. 2000). Finally, we relate annual variation in population growth rates to local climate, in order to quantify the effects of this environmental stochasticity on the long-term growth rate of the population. Study area The study was conducted in the Dovre Mountains (DM), located in central Norway at 62°2'N, 9°3'E (see Asbjornsen 2002 for map). The mountain range is a typical alpine tundra environment, having considerable variation in terrain structure. Altitudes vary from 600 m above sea level in valley bottoms to > 2000 m on peaks. This includes open plateaus, U-shaped valleys and rugged terrain. Alpine tundra vegetation dominates, which ranges from snow-bed vegetation to lichen heath and grass heath communities and barren rock. The area is among the driest in Norway, with average precipitation levels less than 500 mm per year. Normally, 614 the area is snow-covered from late October to late May. E. J. Asbjornsen The area is bisected by a major two-lane highway and et al. an intercity railway that both remain open year-round. Most of the range of muskoxen is within a national park and a military training area. During winter, the population feeds on steep slopes and windblown ridges, owing to their preference for low snow cover and low plasticity to snow-rich environments (Alendal 1973; Thing 1984; Klein 1992; Nellemann 1997, 1998). In the period 1947-53, 27 calves and yearlings were introduced in DM. Most of these individuals died, and by the end of August 1953 probably only 10 individuals were left alive (Alendal 1973). Today's population descends from these individuals. E(JV(+11 N,) = rN, and var(Ar,+1 I N,) = a2dN, + aeN;': eqn la eqn lb Here r is the specific population growth rate, ad is the demographic variance and a2 is the environmental variance. The first order approximation of the mean and variance in Xt+i - In N,+i is then: E(Xt+1\X,) = r-1-a2e-1-e-x-a2d and eqn 2a Methods var(Xt+l\Xt) = ade-x' +a2. eqn 2b © 2005 British Ecological Society. Journal of Animal Ecology, 74. 612-618 POPULATION CENSUSES AND REGISTERED MORTALITY Summer censuses were performed from late July until early September. Numbers of muskoxen in DM until 1983 were based on censuses performed and presented by E. Alendal (see Alendal 1973 for presentation of parts of these censuses). After 1983, local rangers performed the summer censuses presented in this study. The method for these censuses involved walking through areas that traditionally contained muskoxen and then counting the animals. Normally around 10 people participated in these censuses (T. Bretten pers. comm.). Given the open landscape, the high visibility and sedentary nature of the species, and the familiarity that the rangers have with muskoxen in the area, it is likely that these minimum counts came close to representing total counts. Recorded age- and place-specific mortality of the muskoxen in DM was collected from rangers, local newspapers and game departments. In total, 265 animals were lost to the population during the study period. Human-caused mortalities dominated (74 were shot, 63 were killed by the train, six died in research immobilization attempts and five were accidentally poisoned), but natural mortalities also occurred (12 died in lightning strikes and 11 died in avalanches or falls). Many animals also dispersed. In total 64 were found more than 10 km from the population's distribution. Most of these animals died or were shot (included in the above numbers) but a small propagule established another population in Sweden (Laikre et al. 1997). POPULATION MODEL No evidence for density regulation was found because the change in population size from one year to another was not related to population size, either on an absolute scale (correlation coefficient = 0-06, n - 30, P > 0-1) or on a logarithmic scale (correlation coefficient = -0-28, n - 30, P > 0-1, see also Asbjornsen 2002). We therefore adopted a population model without density regulation writing N, for the population size at time t so that: Climate effects The model described by eqns 2a and b can be rewritten = E(JT,+11 X,) + UdadNN, + Utat eqn 3 where Ud and Ue are independent variables with zero mean and unit variance. We can use eqn 3 to examine how different climate variables affect fluctuations in population size. We introduce the climate variables yif as random effects, writing: Ueoe = laiyi,+ Ua, eqn 4 where U is another normalized variable and 2 is the component of the environmental variance that cannot be explained by fluctuations in the covariates. This leads to the relation: a] = vard^.^.,) + a2, eqn 5 so that the covariates together explain a fraction: p = var(Ea,j;,i,)/[var(Ea,j;,i,) + 2 years old with 83% of the calves surviving their first year of life, which for an equal sex ratio gives 0-245 female recruits per adult female. To obtain a maximum estimate of the demographic variance (Saether et al. 2004), adult survival rate was assumed equal to the average yearling survival rate (70%), giving $2d = 0-240. Inclusion of demographic variance reduced the width of the PPL After 10 years the width of the PPI was 19-2% smaller with a2, included than without any demographic stochasticity. Discussion This study demonstrates large uncertainties in predicting the future growth of an introduced small population of muskox in Central Norway, owing mainly to the effects of environmental stochasticity (Fig. 1). A significant proportion of this variability was explained by climate during autumn and late spring. This illustrates the importance of including assessments of uncertainties in the population projections when making predictions about future growth of small populations (Fieberg & Ellner 2000; Ellner et al. 2002). The population growth rates estimated in this study were smaller than those recorded in some other introduced populations of this species. After an introduction in Nunivak Island, Alaska, the stochastic growth rate s - 0-0980 + 0-0207 (estimated from a time-series presented in Spencer & Lensink 1970; assuming no census error and demographic stochasticity) was 91 -7% higher than on DM. Similarly, the censuses from muskox introductions in Angujaartorfiup Nunaa, West Greenland (Pedersen & Aastrup 2000) indicated s - 0-21 + 0-10 during the expansion phase 1986-90. Similar high growth rates have also been indicated in several North American muskox populations (Gunn et al. 1991; Reynolds 1998). One reason for the lower population growth rate of the population in DM is that muskoxen in this area are killed frequently by various kinds of human activities. This happens mainly because a railroad that passes through important habitat kills up to several animals per year, and because animals that end up in populated areas are often shot. If the number of animals that are known to be killed by humans are subtracted from the mean of EN, (eqn 1), we obtain almost a doubling of s (s - 0-0980 + 0-0186) in DM. In contrast, the effects on a* was negligible (a2 - 0-0163 + 0-0042). Thus, a major reason for the reduced growth rate of the muskox population in DM compared to other introduced muskox populations (see Henaff & Crete 1989; Gunn et al. 1991; Reynolds 1998) was the killing of animals by humans. However, the growth rate of the DM population was still smaller than the estimates for many introduced ungulate populations (Komers & Curman 2000; Saether et al. 2002b). Several events of emigration from DM to other areas (Asbjornsen 2002) may have contributed to a reduction of the overall growth rate. A significant proportion of the environmental stochasticity was explained by variation in climate during autumn and late winter. These climate effects remained even after accounting for the effects of known killings by humans. In correspondence with our results, survival of muskox calves in the Arctic Wildlife Refuge in north-eastern Alaska was related negatively to the snow depth in May and June (Reynolds 1998). Thus, the long-term growth rates of muskox populations are affected by stochastic climate effects operating during the nonbreeding season, as has been found for many other ungulate populations (Saether 1997; Coulson, Milner-Gulland & Clutton-Brock 2000). The PPI predicted a high probability of future growth of the population (Fig. 1). Several studies have indicated that uncertainties in parameter estimates may strongly influence the accuracy of the predictions (Saether et al. 2000; Saether & Engen 2002). This was not the case in the present case where neglecting uncertainties in the parameter estimates had only a small effect on the width of the PPI (Fig. 1). This illustrates one of the advantages of population studies of introduced populations, because simple models without density dependence and population estimates over a wide range of population sizes make it possible to obtain unbiased estimates of population growth rates at low densities. This is a crucial parameter, for instance in population viability analysis (Lande et al. 2003), that is often extremely difficult to estimate (Taylor 1995). In ungulates, long-term population growth rates at small population sizes between 0-05 and 0-20 seem typical for re-introduced ungulates (Komers & Curman 2000; Saether et al. 2002b). In most of the analyses, we assumed that the demographic variance was small enough to be ignored. However, Komers & Curman (2000) found a strong non-linear effect of propagule size on subsequent population growth that suggests that demographic stochasticity may be important. Our sensitivity analyses showed that neglecting the demographic variance led to an overestimate of a2, as has been noted previously by Engen et al. (2001). Because the contribution of the demographic variance to the variance in the change in population size decreases with population size (eqns lb, b), this results in an increase in the width of PPI owing to too large estimates of a2. However, the effect was small, suggesting that demographic stochasticity is of minor importance for predicting the long-term growth of ungulate populations that have not gone extinct during the first period after introduction. 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