Law of Tolerance & Limiting Factors

Environmental Factors

Temperature 20 °C

Water availability 60%

Soil nutrient level 55%

Max growth rate r_max 0.35

Response Curve

Temperature response—

Population State

Limiting factor (Liebig's Law): Temperature

Population N

Carrying capacity K

Growth rate dN/dt

Fitness (min response)

How it works

Shelford's Law of Tolerance: every organism has an optimum value for each environmental factor (temperature, water, nutrients…), with performance falling off toward both a lower and an upper limit — a bell-shaped tolerance curve, not a straight line. Too little water hurts a plant, but so does too much.

response(x) = exp( -((x - optimum)² / (2·breadth²)) )   ∈ [0, 1]

Liebig's Law of the Minimum: population growth is capped not by the average of all factors but by whichever single factor is scarcest relative to what the organism needs — a chain is only as strong as its weakest link.

fitness = min( response(temp), response(water), response(nutrient) )
K = K_max · fitness
dN/dt = r_max · fitness · N · (1 − N / K)

Sliders — move temperature, water and nutrient level; each has its own tolerance curve plotted on the right.
Limiting factor — whichever factor currently has the lowest response value; that is the one actually constraining growth, per Liebig.
Population — follows logistic growth toward a carrying capacity K that shrinks as fitness drops; push any one factor far enough from its optimum and the population declines even if the other two are perfect.

Real-world relevance: this pairing of laws underlies agronomy (identifying which nutrient to add first), conservation biology (predicting range limits under climate change) and aquaculture (setting safe temperature/oxygen bands).

Thriving organism Stressed organism Bare ground

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