The science
The story
When wolves were released into Yellowstone in 1995–96 (roughly 30–40 animals from Canada; sources quote 31 for the first two winters, up to 41 counting later releases), the northern elk herd was about 17–19 thousand animals. Counts later fell to roughly 5–8 thousand (approximate; wolves were only one cause, alongside hunting, bears, cougars, drought and weather). Willow and aspen browsed to stubs for decades began to grow again in some places, and beaver colonies increased. This is the textbook trophic cascade: effects that run down the food chain wolves → elk → plants → beavers → streams.
What is debated
A trophic cascade can be density-mediated (predators kill herbivores, so there are fewer of them) or behaviourally mediated (herbivores change where and how they feed because of the “landscape of fear”; Brown, Laundré & Gurung 1999; Laundré, Hernández & Altendorf 2001). Ripple & Beschta (2004, 2012) argued for strong behavioural effects on willow and aspen. Others found weaker or patchy effects: Kauffman et al. (2010) saw little recruitment of tall aspen outside protected spots; Creel & Christianson (2008) found vigilance costs but weak effects on elk reproduction; Marshall et al. (2013) tied willow recovery to hydrology and beaver activity as much as to wolves. Evidence that rivers changed shape because of wolves is the most contested link; here it is a qualitative illustrative index only. Use the “Wolves remove elk / Fear only” switch to see both pathways in the model.
The model (what runs on screen)
- Landscape. 30×30 grid, 300 m valley, a meandering river, wooded gullies and edges (cover c∈[0,1]) and open meadows. Grass capacity Gmax and willow capacity Wmax are larger near the river.
- Risk field. R = (0.3 + 0.7c)·[0.55·(1−e−K) + 0.45·(1−e−V/1.1)], with K = Σwolves exp(−d²/2σ²) (σ = 22 m, current wolf positions) and V the wolf visit memory (decay time 1.5 y; kills and chases add to it).
- Elk foraging–vigilance trade-off. Vigilance v(R) = v₀ + (vmax−v₀)·R/(R+R½); intake I = Imax(1−v)·F/(F+h). Habitat choice maximises payoff u = I/(1+γn) − λ·0.8·R − move cost (an ideal-free-distribution-style rule, Fretwell & Lucas 1969/70; n = animals already there).
- Energy and demography. dE/dt = kin·I − kout·E; births ∝ max(0, E−0.45); starvation hazard when E<0.36; background mortality 0.06 y−1.
- Vegetation. dB/dt = rB(1−B/Bmax) − browsing, advanced with the exact logistic solution B(t+Δt) = K/(1+((K−B)/B)e−rΔt); elk prefer willow (weight 2.2). A tiny seed floor lets browsed stands recover.
- Wolf packs. Each pack has a home-range centre and radius (territory slider) that drifts toward prey. When hungry (satiation < 0.6) it makes attempts at a rate ∝ n/(n+40) (search-limited). Stalk: abort hazard 0.05+0.25(1−cnear) per second over 3 s; strike distance ds = 22 − 13cnear (cover lets wolves get closer). Capture probability p = pmax·e−ds/16·(0.4+0.6·c̄)·(1.25−0.6Etarget)·(1+0.08(m−1)). Open ground, healthy prey and long distances mean low p.
- Beavers. Eight river reaches. A colony settles with rate 0.55·smoothstep(0.22,0.5,willow) and dies out when willow < ~0.2; ponds and dams grow with time; bank stability = 0.65·willow + 0.35·pond, erosion index = 1 − stability (illustrative only).
- Comparison ODE. dE/dt = rE(1−E/K) − W·kmaxE/(E+Eh) (Holling 1959 type II; logistic prey; Lotka 1925 / Volterra 1926 lineage). The dashed line shows what a pure “kill” model predicts for the same number of wolves, so you can see how much the spatial and behavioural model departs from it.
Honest simplifications
- Time is compressed: 1 s ≈ 1 year at warp 1, so a pack makes dozens of attempts per year. All attempts follow the same rules; a sample of them is animated (stalk, chase, outcome), the rest are resolved instantly. A successful capture is shown only as the elk leaving the herd count with a soft fade and a “−1 elk” note.
- One elk marker stands for ~50 elk; wolves are individuals. Animals and trees are drawn enlarged. Real numbers quoted above are approximate; the model is calibrated qualitatively, not fitted to Yellowstone data.
- Other predators (bears, cougars), hunting by people, snow, drought, disease and the many other factors in the real park are not modelled. Beaver and river effects are illustrative.
References (plain text)
Lotka 1925; Volterra 1926; Holling 1959 (functional response); Fretwell & Lucas 1969/70 (ideal free distribution); Brown, Laundré & Gurung 1999 and Laundré et al. 2001 (ecology/landscape of fear); Ripple & Beschta 2004, 2012; Creel & Christianson 2008; Kauffman et al. 2010; Marshall et al. 2013; Beschta & Ripple 2016 (riparian recovery after wolf reintroduction). The page is educational and not affiliated with any agency.