Table 1.

Some common nonlinear patterns in ecology and (two possible) response functions for describing them [for a more complete list of nonlinear functions see Miguez et al. (2017)]

Pattern		Function		Ecological context
J‐shaped		Exponential	y = aebx	Population ecology (population growths without resource constrains). Eco‐physiology (temperature responses). Epidemiology (outbreaks).
		Power	y = axb
Saturating		Michaelis–Menten	y = ax/(b + x)	Population ecology (type‐II functional response in predator–prey dynamics). Community ecology (resource competition). Eco‐physiology (photosynthetic curves). Forest ecology (light availability in canopy). Production ecology (fisheries, fruit quality). Epidemiology (infection rates).
		Monomolecular	y = a[1 − e − bx]
S‐shaped (sigmoidal)		Logistic	$y=1/\left[1+e^{-\left(a+bx\right)}\right]$	Life history (individual biological growths). Population ecology (population growths with resource constrains, type‐III functional response in predator–prey dynamic). Forest ecology (stand dynamics).
		Gompertz	$y=a{e}^{-e^{-bx}}$
Hump‐shaped (unimodal)		Ricker	y = axebx	Population ecology (capture rates varying with prey size in predator–prey dynamic). Community ecology (richness species varying with productivity or disturbance gradients). Eco‐physiology (optimums). Fire ecology (fire activity along global productivity gradient).
		Beta	$y=a\left[1+\frac{b-x}{b-c}\right]{\left[\frac{x}{b}\right]}^{\frac{b}{b-c}}$

Terms in equations: y = response variable; x = explanatory variable; e = constant (the base of the natural logarithm); a, b, c = parameters. Parameter values must be in a certain range to match the pattern (e.g., power functions result J‐shaped curves when b > 1, but inverted J‐shaped if b < 0 and decreasing increments with 0 < b < 1). Function names and parameterization vary according to the context in which they are used (see Bolker, 2008).