THE LOTKA-VOLTERRA predators and prey, forever chasing
The oldest equations of ecology: prey multiply, predators eat prey and multiply, predators starve without prey. ẋ = αx − βxy, ẏ = δxy − γy. The solution never settles — it oscillates forever, predator peaks lagging prey peaks, circling a fixed point neither reaches. Volterra derived it in 1926 to explain why WWI’s pause in fishing had, paradoxically, raised the share of predator fish in the Adriatic.
THE TECHNIQUE ẋ = αx − βxy ; ẏ = δxy − γy
The demo finds the fixed point (x*=γ/δ, y*=α/β) where both rates vanish — the eye of the oscillation: live demo
HISTORY & CREDIT Lotka 1925 · Volterra 1926
“Predator and prey settle to a balance.” — the basic model never settles; it orbits the fixed point in closed cycles determined by the starting counts. cited
the chase · prey feed predators; predators thin prey; each lags the other. the eye · both rates vanish at x*=γ/δ, y*=α/β — a centre the orbits circle. 1926 · Vito Volterra (and Alfred Lotka 1925) — the Adriatic fishery puzzle.
A balance that is never reached, only circled. dynamics
RECOMMEND FOR I-13 the fixed point, on the compiler
On i-13, with α=4 β=2 δ=1 γ=3 the fixed point x*=3, y*=2 makes both rates exactly zero:
$ i13 run lf_lotka-volterra.i13
RUN OK · 46 step(s)
x_star = 3 y_star = 2
prey_dot = 0 pred_dot = 0 fixed_point = 1
Recommend as a NULL — a fixed point (B43). The centre is a fixed point of the flow, pinned by the parameters; the oscillation is dynamics. Fixed-points auto-NULL. NULL — predators and prey, forever chasing.