Rabbits multiply when foxes are scarce, foxes thrive when rabbits are plentiful, and each population drags the other through cycles. The Lotka–Volterra equations make this precise with two products — the encounters between the species — and their phase portrait is a family of closed orbits around a coexistence equilibrium. The linearisation there is a centre, which decides nothing, but the system has a conserved quantity that closes every orbit.

The fur records of the Hudson’s Bay Company, which bought lynx and hare pelts for a century: the numbers of the two animals rose and fell in cycles of about ten years, the lynx peaks following the hare peaks.

$x'=x-xy$, $y'=-y+xy$, prey across and predators up. Every orbit around the coexistence point $(1,1)$ is closed and runs anticlockwise: the prey grow, the predators follow, the prey collapse, the predators starve, and the cycle repeats.

With $x$ the prey and $y$ the predators, $x'=ax-bxy$ and , all four constants positive: prey grow at rate on their own and are eaten at a rate proportional to encounters, ; predators die at rate on their own and grow by eating. The equilibria are , a saddle, and the coexistence point , where the linearisation is a centre. The quantity below is conserved, so every orbit in the positive quadrant is a closed curve: the populations cycle for ever.

• $x'=x-xy$, $y'=-y+xy$: equilibria and ; at the Jacobian is , a centre.

Differentiate V along a solution. The factor multiplying x′ is d − c/x, and x′ is x times a − by, so their product is dx − c times a − by. The same computation for y gives by − a times dx − c. These are equal and opposite, so V never changes: each orbit lies on a level curve of V, and the level curves around the coexistence point are closed.

Differentiate V along a solution.

Since x′ = x(a − by).

Since y′ = y(dx − c).

The two products are equal and opposite.

Predators eat prey at a rate proportional to how often they meet, and that is the product of the two populations.

Great work! You now know how to:

• build the Lotka–Volterra model from encounters
• prove that it conserves a quantity and that its orbits close
• read the averages and the effect of harvesting
• say why a centre of the Jacobian is not enough here