2 · Foxes feastFood is plentiful, so foxes multiply — but young foxes take time to grow.
3 · Rabbits crashToo many foxes now. Rabbits fall while foxes are still at their peak.
4 · Foxes starvePrey is scarce, so foxes decline. Few foxes lets rabbits recover.
Foxes grow whenever rabbits are above the blue dashed break-even level. Rabbits peak when foxes become numerous enough to cancel their breeding — but rabbits are still above break-even then, so foxes keep growing. The fox peak comes only after rabbits have fallen back to that level.
Idealized ecosystem: what this leaves out
Only two species. No other prey, predators, disease or people.
Smooth averages. Populations are continuous numbers; real animals are individuals, so tiny populations are in fact fragile and can vanish.
Fixed rules. The hunting, breeding and starvation rates never change with weather, seasons, cover, age or learning.
Even mixing. Foxes meet rabbits in proportion to how many of each exist — no territories, refuges or travel time.
Pictures are illustrative. Each animal figure stands for a group; their walking is decoration, not part of the calculation.
Numerics. Rabbits: growth r·R(1−R/K) minus catches a·R·F. Foxes: gain 0.25·a·R·F minus 0.5·F. Solved with 4th-order Runge–Kutta in small fixed steps (0.02), and populations are clamped so they never go below zero. Results are in model units and model time, not real months or animals.