Year 0.0Deer 0 (eq. 0)Wolves 0 (eq. 0)Kills / yr 0Equilibrium -

Top-down map of the 300 m x 300 m arena. Dots: deer (pale = fleeing). Triangles: wolves. Orange ring: chasing. Green ring: feeding (handling time). Dashed circles: a wolf's detection (14 m) and a deer's flee trigger (16 m). Click the map to pick a wolf.

Wolf and Deer: Rosenzweig-MacArthur Cycles, Holling Response and a Live Phase Portrait — 2D Map and Phase Plane

2D companion: the same individual-based model and equations as the 3D version, drawn from above, with the phase plane, the measured Holling curve and the wolves' time budget.

Switch to 3D

Deer (prey)

Wolves (predator)

Start and speed

Live readouts

Phase plane: wolves against deer

Green curve: deer nullcline (deer stop changing). Orange line: wolf nullcline. Small arrows: direction of (dN/dt, dP/dt) from the equations. White: individual wolves and deer; cyan dashed: the equations from the same start.

Holling curve: kills per wolf per year

Curve: F(N) = aN/(1 + a h N), ceiling 1/h dashed. The yellow dot is the live measurement (kills in the last 12 months per wolf) at the current deer count.

Wolf time budget

Deer over time

Wolves over time

Experiments

Each button runs headless simulations with the current settings and tabulates the result.

Independent numerical checks

Controls

Every slider acts at once on the animals and on the equations. Start numbers (deer, wolves) apply on Restart.
Time-lapse: the slider sets model years per real second.
Click or tap the map to pick a wolf: its detection and catch ranges are highlighted.
"Ranges" shows or hides the dashed range circles. "Add a wolf" and "Add 10 deer" rescue a population that has died out.

The science

Equations (Rosenzweig-MacArthur 1963). Deer N and wolves P obey dN/dt = r N (1 − N/K) − F(N) P and dP/dt = e F(N) P − m P, where the Holling type II functional response is F(N) = a N / (1 + a h N) (deer killed per wolf per year), h = handling time (days feeding after a kill, divided by 365), e = wolves born per deer eaten and m = wolf death rate. Setting h = 0 and K → ∞ gives the classic Lotka-Volterra pair with neutral cycles, conserved quantity V = e a N − m ln N + a P − r ln P and small-cycle period T = 2π/√(r m).

Equilibrium and stability. Coexistence needs e > m h; then N* = m / (a (e − m h)) and P* = r (1 − N*/K)(1 + a h N*) / a. The equilibrium is stable when N* > (K − 1/(a h)) / 2; above K_c = 2N* + 1/(a h) a Hopf bifurcation turns the spiral into a limit cycle (the paradox of enrichment).

What the 2D view adds. The map shows the movement rules directly: one step is one day on a 300 m reflecting arena; deer random-walk (about 9 m/day) and flee at 18 m/day from a wolf within 16 m; a wolf wanders at 26 m/day, chases the nearest deer within 14 m at 34 m/day and kills with probability 0.5 per attempt, then feeds for Th days. The time-budget bar shows the share of wolves feeding; in the Holling model the fraction of a wolf's time spent handling is F(N)·h, which is why the kill rate saturates near 1/h. The phase plane draws the vector field (dN/dt, dP/dt) of the equations, so you can see why trajectories circle the equilibrium.

Simplifications. No age, sex, season, terrain, packs or disease; uniform arena; the encounter coefficient a is a result of the movement geometry (default a = 0.30 was measured from it). Rates r, m, e and the sizes are illustrative, tuned for a small arena, not wolf or deer field data. With few animals, demographic noise makes the individual run wander around the equations and sometimes go extinct. Physics module identical to the 3D version.

Spot-check list. Rosenzweig and MacArthur (1963) criterion; Holling (1959) type II response; the numbers above are model constants, not measurements.