Static Distance Rule

The static distance rule is developed to separate accumulated observations that are more than a home-range length (longest axis) apart, i.e. if observations are made further than a home-range length apart, they are likely to come from different individuals. The rule is based around the maximal home-range length or the diameter of a circle of equivalent size, dependent on the shape of the home range. An index of home-range shape was obtained by dividing maximum home-range length by the maximum width perpendicular to the maximum length axis. We averaged annual values for females followed over > 1 winter. An assumption of this approach is that females display territoriality and have limited home-range overlap. This assumption was tested by calculating percentage home-range overlap (100% minimum convex polygon method; MCP) between neighbours (not possible for Østfold/Akershus area at time of analysis due to small sample size).

Dynamic Distance Rule

This alternative to the static distance rule is developed to separate observations made within one week of each other. We calculated distances moved in a single night from all the linear distances moved by lynx per night (from day bed to day bed). Distances travelled in periods longer than a single night are expressed as the linear distance between the starting point (day 1) and the day time location on each of the following days (i.e. day 1 to day 2, day 1 to day 3, day 1 to day 4). All possible values were extracted from the data sequences, such that, for example, two locations taken four days apart were included in the day 1 to day 4 distance even if the intervening distances (day 1 to day 2 or day 1 to day 3) were not available. For each individual lynx we took the maximum value recorded in each time interval, and then averaged these individual maximum values to get a study area mean. We only included individuals with ≥ 10 movements.

Static Distance Rules

Although there was a high degree of overlap between a few pairs of adult female lynx (> 50% in a single case from Sarek), the overall picture was one with clear intra-sexual territoriality, as average percentage overlap between neighbours was < 10% for all study areas (Table 2). This result conforms to that found for other studies of lynx in Europe (Breitenmoser et al. 1993, Schmidt et al. 1997), and indicates that in the majority of cases the assumption of non-overlap between neighbours which is required for the use of a static distance rule will be fulfilled. This includes our Hedmark study area where lynx have the largest home ranges ever recorded for the species (Linnell et al. 2001).

Table 2

Average overlap between neighbouring winter home ranges of adult female lynx in different study areas calculated using 100% minimum convex polygon.

Because of the very large home ranges of the Scandinavian lynx (Table 3) which are much larger than those reported from other European study areas (Breitenmoser et al. 1993, Schmidt et al. 1997, Stahl et al. 2001), the resulting distance rules are very large (see Table 3). For example, in the low roe deer area, the longest female's home range was 83 km. These maximal lengths were largely due to the fact that many home ranges were long and thin, as indicated by the high length/width ratios (see Table 3). This was due to the fact that many of the home ranges were aligned parallel with steep-sided valley systems in the study areas with greatest topographic relief. We therefore suggest that in areas with deep valley systems, the actual maximal length may be more appropriate, whereas in generally flat areas, the diameter of a circle of equivalent size to the home range may be appropriate.

Table 3

Average sizes, maximal and circular lengths and length/width ratios of winter (1 November - 30 April) home ranges for adult female lynx in different study areas, calculated using 100% minimum convex polygons. Data from Poland are excluded from statistical tests due to small sample sizes. Numbers in parentheses represent the number of radio-collared lynx for which data were available.

Home-range sizes varied dramatically between the different eco-regions of Scandinavia (see Table 3), being larger in areas with apparently low prey density (postulated in Linnell et al. 2001 and tested in Herfindal et al. 2005). Therefore, there is a clear need to adjust distance rules to local conditions. The comparison between the Scandinavian and Polish data shows the extreme sensitivity of this distance rule approach to local variation in home-range size. Some of the Scandinavian home ranges were almost an order of magnitude larger than the Polish, while the resulting static distance rules were five times larger.

Although the use of the mean (of all the individual lynx) home-range lengths may be less conservative than the absolute maximal value (of the individual with longest home range), the lengths averaged here are the maximal lengths of the MCP home ranges, which reflect the extreme borders of home ranges and are rarely used in practice. We therefore believe that, if anything, these rules are very conservative. Although our telemetry data are mainly day-bed data, and lynx may use areas by night that lie beyond the range estimated from day data, this is likely to only add a few kilometres to the estimates at most. Being conservative is an acceptable bias in harvest management where conservation is the only objective. However, in the Swedish reindeer herding areas, compensation for depredation is linked to the documented numbers of family groups in the area. In this context, conservative estimates will have economic consequences for reindeer herders and will lead to continued controversy over lynx population size. In such circumstances, it is recommended that static distance rules be supported by the dynamic distance rules.

Dynamic Distance Rules

Lynx have very irregular movement patterns, tending to be relatively sedentary during the days when they consume a kill, and then moving further while searching for prey (Pedersen et al. 1999, Jędrzejewski et al. 2002). This leads to much variation in daily movement distances, ranging within 0-55 km. Therefore, we have focused on the mean of the maximum individual straight line movement distances for each temporal interval (numbers of days between observations). This value will make quite conservative family group counts as it is based on maximum movement distances of the individuals. As lynx tend to make a kill every 5-10 days (Breitenmoser & Haller 1993, Okarma et al. 1997, Jobin et al. 2000) cumulative maximum distances moved from a starting point tend to rapidly reach an asymptote. In fact, after only two nights, lynx had moved distances of about 60% of the total distance reached after seven nights (Table 4). As for the static distance rules, the dynamic distance rules varied greatly between regions, with furthest movements in the areas with lowest prey density. Distances moved in the Polish study area were up to one fifth of those from the other areas for the intervals where comparative data existed.

Table 4

Mean and range of the maximum distances moved by individual adult female lynx from a start point to seven consecutive day beds. Only females with > 10 movement events per period class are included.

In order to illustrate how the application of the dynamic distance rule works in practice, we calculated the distance between neighbouring females when they were radio-located on the same day in each of the three Scandinavian regions (Table 5). The results clearly show that 1) the average distance between neighbours is greater than the region specific dynamic distance rule, and 2) in > 80% of cases, the neighbours were further apart than the distance rule. This indicates that the application of the dynamic distance rule is likely to achieve accurate separation in the majority of cases. In this test we included all neighbouring females, irrespective of whether they had kittens or not. In real life applications not all neighbours will have kittens, and therefore the risk of error will be smaller (i.e. there is a lower density of family groups than of adult female lynx).

Table 5

Average distances (± SD) between radio-locations of neighbouring female lynx located on the same day, and percentage of observations further apart than the regionally specific dynamic distance rule (DDR), in the three Scandinavian regions.

Practical Considerations and Potential Biases

One important aspect with the application of this count method is to ensure that observations are made during the times of the year when least errors can be made. Lynx reproduction is highly seasonal throughout Europe, with kittens generally being born in May and June (Arnemo et al. 1999, Reinhardt & Halle 1999, Schmidt 1998). All kittens follow their mothers during the first autumn, and do not usually separate until after late February at the age of ≥ 9 months (Schmidt 1998; J.D.C. Linnell & J. Odden, unpubl. data). No kittens that we have studied maintained regular contact with their mother after one year of age. Furthermore, males begin associating with adult females in the build-up to the mating season from late February. Therefore, observations collected from October until late February are least likely to contain false positives (two adult lynx in association being recorded as a family group) or false negatives (juveniles having dispersed or having reduced maternal association).

However, even with these limitations on season, there are occasions when independent lynx (adult males and yearlings of both sexes) can travel with adult females for periods of several days (e.g. Pedersen et al. 1999) even outside the mating season. As the tracks of kittens and adults are almost impossible to distinguish based on size, these events could lead to false positives. It is therefore recommended to show caution towards ‘family groups’ that are detected from only a single observation. As kittens can follow in the exact footsteps of each other and their mother, it is vital that a set of tracks should be snow-tracked for stretches of at least 1 km to determine if it really is from one or more lynx. Determining the number of kittens can require several kilometres of snow-tracking and should therefore only be used with caution as a criteria to separate between different groups. In Norway, we require that tracks of potential family groups be followed for distances of 2 km in all cases, and 3 km in cases where only two lynx are believed to be present (there is a greater chance of a false positive if only two lynx are together, i.e. it is exceptionally rare for three lynx which are not a mother and kittens to be together outside the mating season). Some geographic features such as large ice-free fjords or lakes and high mountain ranges may be useful in separating observations that are closer than the indicated separation distances (Knight et al. 1995). However, care needs to be taken as lynx are good at crossing rivers, frozen fjords or lakes and infrastructure such as roads and railways.

One limitation of these static rules is that they will fail to detect any increases in population size beyond the point where adult female home ranges form a continuous mosaic across the entire area sampled. Beyond this point, it is unclear if home ranges will decrease in size in a density dependent manner, or if females will become less territorial and tolerate greater home-range overlap. Because of heavy hunting pressure in Norway and the Swedish reindeer husbandry area which effectively limits population growth at present (Andersen et al. 2003), this situation is only likely to occur in central and southern Sweden, but may also be an issue in other parts of Europe where lynx are not hunted. Combining static and dynamic distance rules should overcome this weakness to some extent. Also, in practice, not all adult female lynx will reproduce each year; in fact in some populations and in some years almost 50% will either not reproduce or not be followed by surviving kittens in the winter (Andrén et al. 2002). This will make it easier to separate between neighbouring family groups, especially in saturated territorial mosaics. Under these situations it is also important to consider what these rules are actually estimating. Rather than effectively mapping out the distinct groups (which our analysis indicates that they are effective at when lynx occur at low to medium density), at very high density the rules will produce a conservative estimate of the minimum number of lynx that are likely to have been responsible for the observed tracks, even if the occasional track is assigned to the wrong family. This type of problem is inherent in the nature of minimum counts.

However, if there is an impression that the territorial mosaic is super-saturated and that animals are overlapping, the only alternative is to conduct intensive censuses where animals are separated based on back-tracking in the snow (Liberg & Glöersen 2000). Even this method can have errors associated with it as there is a risk that some animals are not located at all, or that through poor-tracking or bad snow conditions, field personnel will fail to connect track sequences that are in fact continuous. Another alternative is to combine the rule-based minimum counts with an index (e.g. track counts; Linnell et al. 2007) which should not suffer from these problems at high density, and has the advantage that it reflects the density of all age and sex classes of lynx, not just the reproductive females. Another approach at very high density would be to select a random sample of areas and perform sampling based estimates, using either tracks (Becker et al. 1998), or faecal DNA methods (Flagstad et al. 2004). In effect, there are no perfect techniques, especially those that are cost-effective, and the limitations of each must be borne in mind when making management decisions based on the results.

When applying these rules, a decision will need to be made as to which rule to use. For example, the most conservative would be to apply the absolute maximum values, although a more realistic approach could be to use the mean of the individual maximums. In Norway, the static rules are applied to produce two estimates, a conservative rule (the mean of the maximum values) and a less conservative rule (the mean of the mean maximum length and the mean circular diameter; Brøseth et al. 2003). In Norway, we begin to separate groups where possible using the static distance rules, resorting to the dynamic rules only when observations are spatially close to each other. The choice depends on the degree to which a conservative result is to be prioritised. What is crucial is that the choice of rules is made using home range and movement data from a population that has a similar ecology and prey density, as the results from Poland effectively illustrate how lynx movements can vary widely between study areas. What we do not yet know enough about is the stability of home-range sizes and movement parameters over time, especially in the face of changing lynx density, changing prey density and social disruption. Ongoing research will hopefully shed light on this important source of potential variation.

This method was developed in Scandinavia where tracks in snow are the most common ‘observation’. Tracks can be easily validated by trained personnel, and in Norway all accepted observations are checked by wardens from the State Nature Inspectorate. Other types of observation such as photographs of females with kittens, and kittens shot or killed in traffic collisions can also be used as spatially explicit evidence for reproduction. However, for all observations it is important that data are validated. From our experience many visual observations of ‘lynx’ are of other species, and we have recorded many incidences of hunters mistaking tracks of red foxes Vulpes vulpes or wolverine Gulo gulo for those of lynx. The application of this methodology for low density species like lynx requires the input of observations from the public. This implies that checking observations is especially important, and that creating a network of experienced personnel to carry out this task is vital (Stahl & Vandel 1998). One important aspect is that, while the public may be motivated to report observations during the first years of such a programme, their interest often decreases with time. It is therefore desirable to maintain some form of structure in the search for observations (only really suitable in areas where good tracking snow exists), such as organising hunters to take part in large-scale, one-day censuses (e.g. Liberg & Glöersen 2000, Odden et al. 2001), or ensuring that a minimum search effort is made each year (Linnell et al. 2007) by having a network of transects that are examined at least once a year.

In summary, we do not view this as a method with universal application. It requires many observations, a structured reporting system, a species with territorial behaviour and local data on movement patterns. It also has the weakness of not being based on a formal statistical sampling procedure. However, it can potentially work for a wide range of species, and is the only method that we were able to devise to produce acceptable results for an area as large as Scandinavia on an annual basis.