Lotka-Volterra Calculator

The formula

dPrey = (a*N - b*N*P) dt ; dPred = (d*N*P - g*P) dt

Cycles, not equilibrium

The Lotka-Volterra equations are famous for what they do not do: settle. Prey recover, predators multiply on the abundance, predators overshoot, prey crash, predators starve, and the whole thing goes round again indefinitely. The system orbits its equilibrium point without ever arriving.

The counter-intuitive result

The equilibrium prey population depends only on the PREDATOR's parameters, and the equilibrium predator population only on the PREY's. Improving conditions for prey does not raise the prey population at equilibrium — it raises the predator population. This genuinely surprises people and it falls straight out of setting both derivatives to zero.

What it leaves out

There is no carrying capacity for the prey, no handling time for the predator, no age structure, no space and no other species. Real predator-prey systems have all of these, and the lynx and hare data everyone cites turn out to be driven substantially by vegetation cycles the model never sees. It is a beautiful starting point rather than a description.

How much to trust a footprint

Life-cycle figures for the same product routinely differ by a factor of two between studies, because deciding what to count — the farm, the packaging, the shop, the drive home, the disposal — is a methodological choice rather than a measurement. The numbers here are central estimates from published meta-analyses. They are reliable for comparing one option against another, which is what people actually use them for, and not reliable to the second significant figure.

The other thing worth keeping in view is scale. Individual choices matter and they are not where the emissions are: the difference between the best and worst quartile of beef producers is larger than most people's entire dietary footprint, and that is a question about agriculture policy rather than about shopping.