Pick a scenario; every slider stays editable.
Territories: each lynx's 95 % utilisation area (strong tint) and overlap zones (pale tint). The flags are dens. The map is drawn at 1 unit = 1 km; animals, trees and hares are enlarged so you can see them.
Each sweep runs six 160-day simulations headlessly (about 10 s) and plots the measured result against the theory.
Running...
The science
Mechanistic home-range model. Each lynx is a correlated random walker: dv = (-γ v + F(x)) dt + √(2γ) sv dW, dx = v dt, with force F = -ω2(x - xden) + sv2[β ∇ln R - b ∇ln(1 + Mforeign)]. The first term pulls the animal back to its den, R is the (smoothed) prey density (habitat selection with exponent β) and Mforeign is other animals' scent-mark density. This follows the idea of Moorcroft, Lewis and Crabtree (1999) of territory formation by scent avoidance, in a simple form.
Exact reference case. With β = b = 0 the stationary distribution is Gaussian with per-axis variance σ2 = sv2/ω2, so the 95 % utilisation area is A95 = 2π σ2 ln 20. The page checks the simulation against this closed form. With the default sv = 4 km/d and ω = 1.8/d, σ = 2.2 km and A95 is about 93 km², the order of magnitude of a female Eurasian lynx range (typically tens to a few hundred km², strongly dependent on prey; the numbers here are illustrative).
Prey and hunting. Hares regrow logistically, dP/dt = r P (1 - P/K) with r = 0.04/d, and a kill removes one hare spread over about 1 km. The intake rate is Holling type II, a P/(1 + a h P) with a = 0.5 km²/d and handling time h = 0.4 d. Energy is simple bookkeeping: +1 kg of meat per kill, -0.9 kg/d of need (illustrative values, not field measurements); hungry animals roam up to 35 % faster.
Scent marks. Marks are deposited at the lynx position and decay with life time τM. Avoiding foreign marks makes boundaries emerge without any rule about territories. The overlap shown is the share of an animal's 95 % cells that also lie in a neighbour's 95 % set.
Honest simplifications. Females only, fixed den sites, no kittens, no males, no seasons in the dynamics (the Winter button changes only colours), no terrain effect on movement (the relief is decoration), and one prey species with a static carrying-capacity map. A fixed 160-day horizon shows a quasi-steady state, not a long-term population cycle. Real lynx use trails, snow depth and a changing prey base. Do not use these numbers for management.
Scale of the picture. The map is 24 km by 24 km; animals, trees and hares are drawn hundreds of times larger than life, and the relief is exaggerated.