SECR

Overview

Animals move within a two-dimensional arena and are sampled by a fixed detector array across repeated occasions. Density is estimated from who was caught, at which detectors, and on which occasions, with no assumption of a fixed observable strip.

Detection is spatial and imperfect:

each individual has a latent activity centre that is never directly observed;
detectors sample probabilistically around those centres over multiple occasions;
the estimator infers density from capture histories and detector geometry alone.

Glossary

Controls

Control	Description
Speed	Playback speed: Slow / Normal / Fast.
Seed	RNG seed for animal and detector placement. Reset = same layout. 🎲 New = new population and layout.
Detector type	Named preset that adjusts \(g_0\) and \(\sigma\) to approximate a sensor type: Default, Camera trap, Live trap, Acoustic detector.
\(g_0\)	Baseline detection probability at zero distance from an animal’s activity centre (0.05–1.0). Updates live. Default: 0.20.
\(\sigma\)	Home range / detection scale (km). Controls how steeply detection probability drops with distance from the activity centre. Updates live. Default: 0.15 km.
N	Number of animals in the arena. Takes effect on Reset. Default: 10.
K	Number of sampling occasions per run. Takes effect on Reset. Default: 10.
Movement	Named movement preset: Typical resident, Sedentary, Wide-ranging, or Nomad. Sets \(\tau\) and \(f\).
τ (mobility)	Home range crossing time in occasions. Low \(\tau\) = restless / fast-moving; high \(\tau\) = sedentary. Takes effect immediately. Default: 5 occ.
f (fidelity)	Site fidelity multiplier: tightens the effective home range around the activity centre. Higher = tighter. Default: 1.0×.
Detectors	Detector layout: Grid (regular array), Random (random placement), or Click to place (interactive). Takes effect on Reset.
n_grid	Grid dimension (\(N \times N\)). Visible when Detectors = Grid (3–7, default 4). Takes effect on Reset.
Show detection surface	Overlays the spatial detection probability surface \(p(x,y)\) on the simulation canvas. Updates live when \(g_0\), \(\sigma\), or detectors change.
Show activity centres	Shows true activity centres (•) and estimated centres (×, mean of capturing detector positions) for each individual.

Estimates readout

Symbol	Meaning
Occasion (\(k\))	Current occasion index.
Individuals (\(M\))	Number of distinct animals detected at least once so far.
Captures	Total capture events across all individuals and occasions.
ESA	Effective sampling area (km²): \(\int\!\int p(x,y)\,dx\,dy\) over the arena, given current \(g_0\), \(\sigma\), and detector layout.
\(\hat{D}\)	Estimated density: \(M / \mathrm{ESA}\).
True \(D\)	True density = \(N / \text{arena area}\).
Home range (\(\pi\sigma_{\rm eff}^2\))	Area of a circle with radius \(\sigma_{\rm eff} = \sigma / \sqrt{f}\): approximate home range size (km²).
Recapture rate	Total capture events divided by (\(M \times k\)): average captures per individual per occasion.
Coverage (\(M/N\))	Proportion of the true population detected at least once.

Key symbols

Symbol	Meaning
\(g_0\)	Baseline detection probability at zero distance (dimensionless, 0–1).
\(\sigma\)	Detection / home range scale (km).
\(\tau\)	Mobility parameter: home range crossing time (occasions).
\(f\)	Site fidelity multiplier (dimensionless).
\(K\)	Number of occasions.
\(N\)	True number of animals.
\(M\)	Number of detected individuals.
\(\mathrm{ESA}\)	Effective sampling area (km²).
\(\hat{D}\)	Estimated density (animals per km²).
\(D\)	True density (animals per km²).
Activity centre	Latent centre of an individual’s home range; never directly observed.
Capture history	Record of which individual was detected at which detector on which occasion.

Arena geometry

The arena is a square with side length ARENA_KM = 2.0 km. It is divided into two zones:

Zone	Variable	Value
Inner study area	`INNER_KM`	1.2 km × 1.2 km
Buffer zone (each side)	`BUFFER_KM`	0.4 km

The buffer is ≈ 2.7σ at the default σ = 0.15 km. This is a standard SECR convention: animals whose activity centres lie within ~2σ of the detector array contribute meaningfully to detections. Animals further out are effectively invisible.

The canvas is a square. PX_PER_KM = canvasSize / ARENA_KM. All world coordinates are in km; pixel coordinates are computed for display only using worldToPx(wx, wy).

Inner study area corners: (innerX0, innerY0) = (BUFFER_KM, BUFFER_KM) = (0.4, 0.4).

Animal placement

placeAnimals(N, ARENA_KM, ARENA_KM, placementRng) in src/secr-engine.js.

Each of the \(N\) animals is placed by drawing activity-centre coordinates uniformly:

cx = rng() * arenaW
cy = rng() * arenaH

The animal starts exactly at its activity centre: x = cx, y = cy. Velocity is initialised to zero: vx = 0, vy = 0.

Activity centres span the full arena (inner + buffer), not just the inner study area. This is correct: real populations extend beyond any detector array, and some of those individuals can still be detected.

The true density is trueD = N / (ARENA_KM²) = N / 4.0 animals/km².

Detector placement

placeDetectors(layout, innerW, innerH, innerX0, innerY0, rng, gridN, nRandom) in src/secr-engine.js.

All detectors are placed within the inner study area only.

Grid layout

gridN × gridN detectors on a regular grid. Cell size dx = innerW / gridN, dy = innerH / gridN. Each detector at:

x = innerX0 + (col + 0.5) * dx
y = innerY0 + (row + 0.5) * dy

Default gridN = 4 → 16 detectors. Range: 3–7, giving 9–49 detectors.

Random layout

16 detectors placed by independent uniform draws within the inner study area:

x = innerX0 + rng() * innerW
y = innerY0 + rng() * innerH

Uses the seeded placement RNG.

Custom layout

Detectors are managed directly by sketch.js. Clicking within the inner study area places a new detector; clicking an existing detector removes it. Pressing “Remove all” calls clearDetectorsFn() which empties the array and notifies analytics. Custom detectors are preserved through Reset (the positions are re-indexed but not regenerated).

Seed and RNG architecture

Three separate RNGs are used:

RNG	Seeding	Purpose
`placementRng = createRng(seed)`	Seeded (Alea PRNG)	Animal centres, grid/random detector positions
`movementRng = new Math.seedrandom()`	Fresh unseeded each `initSim()`	Per-frame OUV movement steps
`detectionRng = new Math.seedrandom()`	Fresh unseeded each `initSim()`	Bernoulli detection draws each occasion

Same seed → same animal and detector layout. Movement paths and detection outcomes vary on every replay, even with the same seed.

Seed range: 0–99999. Each browser session starts with a random seed. Entering and committing a seed value in the input field immediately calls initSim().

Movement model (OUV)

Each frame during a run, every animal takes one stepAnimal() call from src/secr-engine.js. The model is Ornstein-Uhlenbeck in velocity space (OUV), also called an Ornstein-Uhlenbeck process in position via second-order Langevin dynamics.

Frame-level update

u1 = max(rng(), 1e-10),   u2 = rng()
r  = sqrt(-2 * log(u1)),  θ = 2π * u2
nx = r * cos(θ) * noiseScale          // Box-Muller N(0, noiseScale²)
ny = r * sin(θ) * noiseScale

vx_new = vx * (1 - drag) + springStr * (cx - x) + nx
vy_new = vy * (1 - drag) + springStr * (cy - y) + ny
x_new  = x + vx_new
y_new  = y + vy_new

where drag = 0.08 is a fixed constant.

Wall reflection

If an animal would exit the arena, its position is reflected and the outward velocity component is negated:

if x < 0:       x = -x,           vx = -vx
if x > arenaW:  x = 2*arenaW - x, vx = -vx
(same for y)

Parameter derivation

springStr and noiseScale are computed each frame from the slider values:

const fpk = FRAMES_PER_OCC[speed]; // frames per occasion
const drag = 0.08;
const springStr = drag / (tau * fpk);
const noiseScale = sigma * drag * sqrt(2 / (fidelity * tau * fpk));

The derivation comes from the overdamped continuous-time Langevin equation:

Position autocorrelation time \(\tau_x = \mathrm{drag} / K_s\), so \(K_s = \mathrm{drag}/(\tau \cdot \mathrm{fpk})\).
Equilibrium variance: \(\mathrm{Var}(x) = \mathrm{noiseScale}^2 / (2 K_s \cdot \mathrm{drag}) = \sigma^2 / \mathrm{fidelity}\).

This gives:

\(\tau\) (occasions): controls position autocorrelation time (how long the animal takes to cross its home range).
fidelity: controls noise amplitude and range tightness. Higher fidelity = smaller effective home range radius = \(\sigma / \sqrt{\mathrm{fidelity}}\).

Speed and frames per occasion

Label	Frames per occasion
Slow	120
Normal	60
Fast	15

At 60 fps and Normal speed, each occasion takes 1 second of real time.

Display smoothing

The physics simulation runs at true positions (x, y). A separate display position (dx, dy) is lerped toward the true position each frame:

dx_new = dx + 0.15 * (x - dx)
dy_new = dy + 0.15 * (y - dy)

The DISPLAY_LERP = 0.15 factor creates smooth animation without affecting the detection computation. Detection uses (cx, cy), not (dx, dy).

Movement presets

Presets set \(\tau\) and fidelity simultaneously.

Preset	\(\tau\) (occasions)	Fidelity	Effect
Resident	5.0	1.0	Default. Moderate range size, moderate return.
Sedentary	6.0	3.0	Tighter home range (radius ÷ √3). Slower crossing time.
Wide-ranging	2.0	0.4	Larger effective range (radius × √2.5). Faster crossing.
Nomad	1.0	0.2	Very fast, large range; barely constrained to a centre.

Detection function

One occasion = one call to tryDetectsOnOccasion(animals, detectors, g0, sigmaEff, rng).

For every (animal \(i\), detector \(j\)) pair:

Distance from the animal’s activity centre to the detector:

d_ij = hypot(animal.cx - detector.x, animal.cy - detector.y)

Detection probability:

g_ij = g0 * halfNormal(d_ij, sigmaEff)
     = g0 * exp(-d_ij² / (2 * sigmaEff²))

Bernoulli draw: detected = detectionRng() < g_ij.
If detected: record { animalId, detectorId }.

Each animal can be detected by multiple detectors on the same occasion. Each (animal, detector, occasion) triple produces one capture event. The capture history therefore records the detector index too, not just presence/absence.

Detection is based on (cx, cy) (the latent activity centre), not on the display position (dx, dy) or the physics position (x, y). This is consistent with the SECR model: the detection function \(g(d)\) is a marginalised home-range detection function, integrating over all locations within the home range that a centre at \((c_x, c_y)\) implies.

Effective detection σ

sigmaEff = sigma / sqrt(fidelity)

Higher fidelity compresses the home range. sigmaEff is passed to tryDetectsOnOccasion and to all ESA computations. The UI slider reads sigma; the underlying detection range is automatically adjusted.

Detector type presets

These set g0 and sigma simultaneously.

Preset	\(g_0\)	\(\sigma\) (km)	Interpretation
Default	0.20	0.15	Balanced starting point
Camera trap	0.25	0.08	High detection within narrow field
Live trap	0.40	0.05	Very localised, high if animal passes
Acoustic	0.12	0.40	Wide detection radius, moderate probability

Effective sample area (ESA)

Single-occasion \(p_1(x, y)\)

The probability that an animal with activity centre at \((c_x, c_y)\) is detected by at least one detector on a single occasion:

\[ p_1(c_x, c_y) = 1 - \prod_{j=1}^{J} \bigl(1 - g_0 \cdot g(d_j)\bigr) \]

where \(d_j = \|(c_x, c_y) - \text{det}_j\|\) and \(g(d) = \exp(-d^2 / 2\sigma_{\rm eff}^2)\).

This is computed as a log-sum for numerical stability:

logNonDetect += log(1 - g0 * halfNormal(d, sigma) + 1e-15);
p1 = 1 - exp(logNonDetect);

\(K\)-occasion ESA

The probability of being caught at least once across \(k\) independent occasions:

\[ p_k(c_x, c_y) = 1 - (1 - p_1(c_x, c_y))^k \]

ESA is then the numerical integral over the full arena:

\[ \mathrm{ESA}(k) = \sum_{\rm cells} \bigl[1 - (1 - p_1)^k\bigr] \cdot \Delta A \]

The arena is discretised into 60 × 60 = 3600 cells for ESA computation (both in the analytics D̂ historical series and in the estimates strip). cellArea = arenaW * arenaH / (60 * 60).

For the D̂ convergence chart, the detection surface (p_1 for each cell) is computed once and reused for all \(k = 1, 2, \ldots, k_{\rm current}\), by applying the \(k\)-exponent formula cell-by-cell. This avoids recomputing the surface \(k\) times.

The estimates strip uses computeESA_K from src/secr-engine.js, which is equivalent: a grid reduction over the surface.

Density estimator

\[ \hat{D}(k) = \frac{M(k)}{\mathrm{ESA}(k)} \]

where:

\(M(k)\) = number of unique animal IDs in captures with occasion \(\leq k\).
\(\mathrm{ESA}(k)\) = \(K\)-occasion ESA using sigmaEff and the current detector layout.

\(\hat{D}\) is plotted at each completed occasion as a convergence trace. True density is N / ARENA_KM².

This is an educational estimator only: it uses the true \(g_0\) and \(\sigma\) parameters and does not fit a likelihood model to the capture histories. A full SECR MLE would estimate \(g_0\), \(\sigma\), and \(D\) from the data.

Analytics panels

Four panels rendered by secr/analytics.js using D3. They subscribe to the shared state and re-render on every notification (detector change, parameter change, or occasion completion). Position updates do not trigger re-renders.

1. Detection surface heatmap

Shows the single-occasion detection probability \(p_1(c_x, c_y)\) over the full arena as a 50 × 50 colour grid using the YlOrRd D3 colour scale. A cell is transparent if \(p < 0.001\).

The heatmap redraws whenever g0, sigma/sigmaEff, or the detector layout changes. It is skipped entirely when the “Show detection surface” overlay is not visible; this avoids expensive grid computation on every occasion tick.

The inner study area boundary is overlaid as a dashed blue rectangle. Detector positions are overlaid as small blue dots.

2. Capture history matrix

One row per captured animal (in order of first capture), one column per completed occasion. Cells show a coloured background and the detector IDs (comma-separated) for all detectors that caught the animal on that occasion. Empty occasions show a grey cell.

Row labels are coloured boxes A–J matching the per-animal colours used in the simulation display.

3. D̂ convergence chart

Green line: \(\hat{D}(k)\) for \(k = 1, \ldots, k_{\rm current}\). Red dashed horizontal line: true \(D\). X-axis: occasion index. Y-axis has expand-only tracking as in the DS module.

4. Running estimates strip

Field	Value
\(k\)	Current / target occasions
\(M\)	Unique individuals caught
Total captures	Count of all (animal, detector, occasion) events
ESA	\(k\)-occasion ESA (km²) at current \(k\)
\(\hat{D}\)	\(M / \mathrm{ESA}(k)\)
True \(D\)	\(N / \mathrm{ARENA\_KM}^2\)

Additional derived quantities shown below the table:

Estimated home-range area: \(\pi \sigma_{\rm eff}^2\) (km²).
Mean recapture rate: total captures / (\(M \times k\)), computed only when \(M > 0\) and \(k > 0\).

Phase states

The simulation has four phases, tracked in phase:

State	Label	Transition
`idle`	▶ Run	After `initSim()`, before first Run press
`running`	⏸ Pause	While occasions are ticking
`paused`	▶ Resume	After Pause press
`complete`	↺ Run again	After `currentK >= targetK`

“Run again” (pressed when complete) adds another K occasions to targetK and continues on the same population; it does not reset the animals or detectors.

Control taxonomy

Live update (no reset required)

Control	Effect
\(g_0\) slider	Updates detection probability and surface immediately
\(\sigma\) slider	Updates `sigmaEff`, detection surface, and ESA immediately
Fidelity \(f\) slider	Updates `sigmaEff` (via `sigma / √f`) and surface
\(\tau\) slider	Updates spring strength and noise scale for movement immediately
Detector type preset	Sets \(g_0\) and \(\sigma\); same as moving those two sliders
Show detection surface checkbox	Toggles heatmap visibility; triggers redraw
Show activity centres checkbox	Toggles centroid overlay

Reset required

Control	Effect
\(N\) slider	Number of animals (placement regenerated)
\(K\) slider	Occasions per run
Distribution layout	Grid / random / custom
Grid size \(n_{\rm grid}\)	Only applicable in grid mode
Seed	Entirely new placement

Add/remove detectors in custom mode is the one exception: detector changes take effect immediately without a full reset, because updateDetectors() notifies the analytics and the surface redraws.

Assumptions and limitations

The density estimator uses known true \(g_0\) and \(\sigma\). A real SECR analysis would estimate these from the capture histories.
\(g(d)\) is half-normal only. No other detection function is implemented.
The movement model is the same for all individuals; there is no individual variation in \(\tau\) or fidelity.
The arena has reflecting boundaries. In reality, populations extend beyond any study area.
Multiple detections of the same individual by the same detector on different occasions are tracked independently. The capture history records every (animal, detector, occasion) event, not just first capture.
Habitat is uniform; habitatFn = () => 1 throughout.