# Estimating population density from indirect sign: track counts and the Formozov–Malyshev–Pereleshin formula

> How can you turn tracks in the snow into a count of animals? A formula long used in Russia, but barely known elsewhere, does it from three numbers: how many fresh tracks cross a transect, how long the transect is, and how far the animal travels in a day. Using 40 years of deer track counts and trailing in Russia’s Sikhote-Alin reserve, the authors showed the formula is sound, and explained what limits its precision.

Summary by Kersey Lawrence, Original Wisdom. Updated 5 November 2026. Page: https://www.originalwisdom.com/tracking-literature/2006-stephens-zaumyslova-estimating-population-density-indirect/

## Citation and access

- **Authors:** P. A. Stephens, O. Yu. Zaumyslova, D. G. Miquelle, A. I. Myslenkov and G. D. Hayward
- **Year published:** 2006
- **Journal or source:** Animal Conservation
- **Full citation:** Stephens, P. A., Zaumyslova, O. Yu., Miquelle, D. G., Myslenkov, A. I. and Hayward, G. D. 2006. Estimating population density from indirect sign: track counts and the Formozov–Malyshev–Pereleshin formula. Animal Conservation 9(3): 339–348.
- **DOI:** 10.1111/j.1469-1795.2006.00044.x
- **Access status:** Publisher link only
- **Where to find it:** https://doi.org/10.1111/j.1469-1795.2006.00044.x

## Tags

- **Species:** Mammals > Red deer; Mammals > Siberian roe deer; Mammals > Sika deer
- **Region:** Asia > Russia
- **Tracking:** Footprints; Snow tracks; Spoor age estimation; Track counts; Tracking; Tracks and signs identification; Trailing
- **Behaviour and ecology:** Behaviour; Daily travel distance; Movement
- **Methods and technology:** Akaike Information Criterion (AIC); Bootstrapping; Formozov–Malyshev–Pereleshin (FMP) formula; Population estimation; Radio telemetry; Simulation
- **Data type:** Existing data; Modelling or theoretical
- **Observer qualification:** Reported but not validated
- **Discipline:** Population trends; Wildlife tracking
- **Year published:** 2006

## Detailed keywords

- **Species:** Studied: Manchurian red deer (Cervus elaphus xanthopygus), Siberian roe deer (Capreolus pygargus) and sika deer (Cervus nippon). Mentioned: these deer are important prey of the Siberian or Amur tiger (Panthera tigris altaica). Tracks of other ungulates and carnivores were also recorded in the surveys but not analysed. Trees named in the study area: Mongolian oak (Quercus mongolica), Korean pine (Pinus koraiensis), larch (Larix komarovii), birches (Betula costata, Betula lanata and others), basswood (Tilia amurensis), fir (Abies nephrolepis) and spruce (Picea ajanensis).
- **Location and habitat:** Sikhote-Alin State Biosphere Zapovednik (SABZ), north-east Primorye Krai, Russian Far East: 4000 km² of strictly protected ecosystems, with access restricted to scientists and forest guards, in the low Sikhote-Alin Mountains (most peaks below 1200 m) along the Sea of Japan. Habitat zones: oak forest near the coast (770 km²); mixed deciduous and conifer forest with Korean pine (1605 km²); and boreal conifer forest inland (1610 km²). The reserve is made up of 14 drainage basins. Winter snow.
- **Topics:** winter transect counts (the main game-counting method in the Russian Federation); converting track counts from an index into absolute density; the Formozov–Malyshev–Pereleshin (FMP) formula, D = (π/2) × x / (S × M), where x is the number of track crossings, S is the transect length and M is the mean daily travel distance; daily (24-hour) travel distance and what affects it (time of year, snow depth, group size, habitat, mast); track age (tracks from the last 24 hours); precision, survey effort and population density; group size and uneven distribution; stratification; survey bias from non-random routes; track misidentification; independent validation.
- **Statistics or models:** Geometric probability derivation of the FMP formula; simulations (a 2500 km² area; 10,000 replicates per scenario); one- and two-factor models of travel distance compared with the Akaike Information Criterion adjusted for small samples (AICc); nonparametric bias-corrected and accelerated (BCA) bootstrapping (5000 replicates); post-stratification by drainage basin, weighted by transect length and drainage area.
- **Data type:** Existing data: winter track counts in SABZ from 1962 to 2002 (just under 100 km of transects in 1964 to nearly 1800 km in 1985), and winter movement records of 24-hour trails. Modelling: simulations.

## Synopsis

### 1. What they did

In the Russian Federation, many game animals are counted each winter by counting tracks in snow that cross a fixed network of transects. These counts are mostly used as relative indices of abundance. Russian biologists convert them into absolute density with an analytical formula known as the Formozov–Malyshev–Pereleshin (FMP) formula, but it was barely known in English-language science. The authors presented the formula and its derivation, tested it with simulations, and applied it to 40 years of deer track counts from Sikhote-Alin State Biosphere Zapovednik (SABZ).

The formula

- It works out the chance that a transect crosses an animal’s movement path, from the transect’s length and the animal’s daily travel distance.

- So density can be estimated from three things: the number of track crossings counted, the total transect length, and the species’ mean daily (24-hour) travel distance.

- Only tracks of known age, made within the last 24 hours, can be used.

Track counts

- From winter 1962 to winter 2002, forest guards and scientific staff of SABZ counted tracks crossing transect routes, broken into segments of a single habitat type.

- They recorded only tracks made within the last 24 hours, and ignored recrossings that were evidently made by the same animal.

- The three deer species were told apart by the relative size and shape of their tracks.

Daily travel distances, by trailing

- Radio-collared red deer were located on consecutive days at the same time, and their tracks were then followed from beginning to end.

- Or a deer was seen, usually in the late morning, and the next day its tracks were followed from that point until it was seen again after 24 hours.

- Coordinates were recorded along each trail, including every change of direction. Records where the observers had clearly disturbed the animal (for example, it bolted) were excluded.

- They tested whether time of year, snow depth, group size, habitat or mast (acorn or pine nut crop) affected travel distance.

Simulations

- Real movement paths were placed at random in a simulated 2500 km² area. Densities ranged from 0.25 to 10 animals per km², transects from 250 m to 10 km, and total survey effort from 10 to 1000 km.

- Each scenario was repeated 10,000 times to check whether the formula gave the true density and how precise it was.

Field estimates

- Density was estimated for each habitat zone, with confidence intervals from bootstrapping.

### 2. What they found

Daily travel distances

Deer

Records

Mean daily travel distance

Sika deer (all records)

10

2.78 km

Red deer, early winter (December–January)

27

1.52 km

Red deer, late winter (February–March)

63

1.29 km

Roe deer, early winter

11

2.19 km

Roe deer, late winter

51

0.89 km

- Red deer travel distances were surprisingly consistent: 1.35 km on average overall. No model explained much of their variation, but time of year featured in all the top models.

- For roe deer, time of year alone explained over 50% of the variation, and group size added more. Group size could not be used to refine the density estimates, because the track counts did not record it.

- There were no strong signs that habitat type affected travel distance.

Simulations

- In every scenario, the formula’s average estimate was very close to the true density (0.981 to 1.018 of it). So it is theoretically sound and is not biased by the shape of animals’ paths.

- Precision improved with higher population density and more survey effort. Transect length mattered little, beyond transects of about 2 km.

- Below about 2 animals per km², or about 250 km of total survey effort, estimates were quite uncertain. At 2 per km² with 250 km of effort, confidence intervals were about 25–35% of the mean.

- Very low-density populations (below about 0.5 per km²) are unlikely to be surveyed effectively this way.

- These are best cases. Animals travelling in groups reduce precision: precision is that of density divided by mean group size. Red and roe deer travel in groups of about 2, and sika deer in much larger groups. An uneven distribution also reduces precision.

Sikhote-Alin

- The track counts could be used to estimate the abundance of all three deer species, with confidence intervals.

- All three appear to have increased in SABZ over the 40 years, but not consistently across species or habitats.

### 3. What they concluded

- The FMP formula is simple to apply and theoretically sound, and it has great potential where density must be estimated from indirect signs. It could help explain how track-count indices relate to absolute density. With some changes, it might also suit other rare events, such as camera-trap triggers.

- The formula’s use is contentious in Russia, for five reasons:

- Its theoretical basis: the simulations support it more robustly than earlier sketch-based tests.

- Bias from the shape of movement paths: none was found, provided transects are randomly oriented and repeat surveys are spaced out. In SABZ they are separated by at least a month.

- Survey routes on accessible trails or valley bottoms: these may over- or under-sample areas the animals favour. Random transects are the best way to test for this bias.

- Movement varies with conditions: more data are needed. The authors recommend building a library of daily travel distances under different conditions, and checking whether movement changes with population density.

- Misidentified tracks: this “remains an unknown source of error and should be assessed rigorously”. Older tracks would give longer travel distances and better precision, but track quality degrades quickly, which increases misidentification.

- Independent validation is essential, by a method other than track counts. The authors suggest that direct observation with distance sampling is the most promising option.

- For SABZ, they recommend collecting more 24-hour movement data; recording the make-up of travelling groups in track counts; collecting distance data for all ungulate sightings; and recognizing the factors that limit precision in light of the monitoring program’s goals.

### 4. How tracking was or was not used

- Track counts: observers identified deer tracks in snow to species by relative size and shape. They counted only tracks made within the last 24 hours, and ignored evident recrossings by the same animal.

- Trailing: the daily travel distances that the formula depends on came from following individual deer’s 24-hour trails in snow and recording coordinates along the way.

### 5. Tracker or observer certification

- Reported but not validated. Over 40 years, the counts were made by forest guards and scientific staff of SABZ “with extensive field experience, having undergone training in species identification and data collection protocols”. No certification or test of their skill is reported.

- The authors state that track misidentification remains an unknown source of error that should be assessed rigorously.

### 6. Other technologies, skills or innovations

- Radio collars were used to relocate red deer on consecutive days before their trails were followed.

- Universal Transverse Mercator (UTM) coordinates were recorded along each trail.

- Computer simulations tested an analytical formula with far more replication than earlier hand-drawn tests.

- Bias-corrected and accelerated (BCA) bootstrapping was used to estimate confidence intervals.

### 7. Implications for conservation (from the paper)

- Why absolute density matters: indices of abundance are seldom equivalent across habitats or large areas. Many management tasks can be done with more confidence when absolute density is known, such as setting hunting quotas, predicting population viability, and understanding predator–prey relationships. Converting an index into density also forces an explicit look at its error and uncertainty.

- Where the method suits: it is easily applied, and suits places where direct counts are impractical, such as dense forest with poor visibility.

- What to plan for: good survey design, stratification, reliable travel distances and independent corroboration are essential. Limits on precision, especially at low density, need to be weighed against the goals of each monitoring program.

### 8. Implications for tracking (Original Wisdom's observations)

_These are Original Wisdom's observations, not the authors'._

- The formula needs two tracking skills: aging tracks correctly, since only tracks from the last 24 hours count, and trailing an animal for a full 24 hours to measure how far it travels.

- The authors name track misidentification as an unmeasured source of error. Testing observers’ identifications would be one way to measure it.

## Cite this summary

Lawrence, K. 2026. Summary of: Estimating population density from indirect sign: track counts and the Formozov–Malyshev–Pereleshin formula (Stephens and Zaumyslova et al., 2006). Original Wisdom, Tracking Literature. https://www.originalwisdom.com/tracking-literature/2006-stephens-zaumyslova-estimating-population-density-indirect/