Carrying Capacity Calculator
Growth peaks at half the carrying capacity, which is why fisheries are managed there.
The formula
dN/dt = rN(1 - N/K), and growth peaks at N = K/2
Growth is fastest in the middle
A population grows most rapidly not when it is smallest but at exactly half its carrying capacity, where there are plenty of individuals reproducing and still plenty of room. Below that there are too few parents; above it, too little food. The curve is symmetric about that point.
Which is why fisheries are managed at half
Maximum sustainable yield is taken at half the unfished stock, because that is where the population replaces itself fastest. Fishing a stock down to a quarter does not give a quarter of the yield — it gives less than at half, and the stock is closer to collapse besides. Most historical fishery failures came from treating the peak as a floor rather than a ceiling.
Carrying capacity is not a fixed number
It depends on food, water, disease, predation and weather, all of which move. A population sitting at what looked like equilibrium can crash when a dry year lowers K beneath it. Treating K as a constant is the model's largest simplification and the one that fails first.
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.