Table A2.

Competing model formulations

Model	Normalized force:	Normalized recruitment term:	Normalized distortion term:	Second order terms:	No. of parameters:
	φ(t) =	$\frac{d\epsilon \left(t\right)}{dt}=$	$\frac{d\zeta \left(t\right)}{dt}=$
I	ε(t)ζ(t)	$-b\epsilon \left(t\right)+b\left[\frac{\lambda \left(t\right)-{\lambda }_d}{1-{\lambda }_d}\right]$	$-c\left[\zeta \left(t\right)-1\right]+\frac{1}{{\upsilon }_0}\frac{d\lambda \left(t\right)}{dt}$	—	Three: b and c (with units of s−1) and υ0 (unit-less)
II	ε(t)ζ(t)	$-\left(b+\gamma \hspace{0.17em}f\left(x\right)\right)\epsilon \left(t\right)+b\left[\frac{\lambda \left(t\right)-{\lambda }_d}{1-{\lambda }_d}\right]$	$-c\left[\zeta \left(t\right)-1\right]+\frac{1}{{\upsilon }_0}\frac{d\lambda \left(t\right)}{dt}$	—	Four: b, γ, and c (with units of s−1) and υ0 (unit-less)
III	ε(t)ζ(t)	$-b\left(1+\gamma \hspace{0.17em}f\left(x\right)\right)\epsilon \left(t\right)+b\left[\frac{\lambda \left(t\right)-{\lambda }_d}{1-{\lambda }_d}\right]$	$-c\left(1+\gamma \hspace{0.17em}f\left(x\right)\right)\left[\zeta \left(t\right)-1\right]+\frac{1}{{\upsilon }_0}\frac{dl\left(t\right)}{dt}$	—	Four: b and c (with units of s−1) and γ and υ0 (unit-less)
IV	ε(t)ζ(t)	$-\left(b+{\gamma }_{\eta }\hspace{0.17em}f\left(x\right)\right)\epsilon \left(t\right)+b\left[\frac{\lambda \left(t\right)-{\lambda }_d}{1-{\lambda }_d}\right]$	$-\left(c+{\gamma }_x\hspace{0.17em}f\left(x\right)\right)\left[\zeta \left(t\right)-1\right]+\frac{1}{{\upsilon }_0}\frac{dl\left(t\right)}{dt}$	—	Five: b and c (with units of s−1) and γn, γx, and υ0 (unit-less)
V	ε(t)[ζ(t) + κζ1(t)]	$-\left(b+\gamma \hspace{0.17em}f\left(x\right)\right)\epsilon \left(t\right)+b\left[\frac{\lambda \left(t\right)-{\lambda }_d}{1-{\lambda }_d}\right]$	$-c\left[\zeta \left(t\right)-1\right]+\frac{1}{{\upsilon }_0}\frac{d\lambda \left(t\right)}{dt}$	$\frac{d{\zeta }_1\left(t\right)}{dt}=-d{\zeta }_1\left(t\right)+\frac{1}{{\upsilon }_0}\frac{d\lambda \left(t\right)}{dt}$	Six: b, γ, c, and d (with units of s-−1) and κ and υ0 (unit-less)
VI	ε(t)ζ(t)	$-\left(b+\gamma \hspace{0.17em}f\left(x\right)\right)\epsilon \left(t\right)+b\left[{\epsilon }_1\left(t\right)\right]$	$-c\left[\zeta \left(t\right)-1\right]+\frac{1}{{\upsilon }_0}\frac{d\lambda \left(t\right)}{dt}$	$\frac{d{\epsilon }_1\left(t\right)}{dt}=-a\left({\epsilon }_1\left(t\right)-\frac{\lambda \left(t\right)-{\lambda }_d}{1-{\lambda }_d}-\theta \frac{d\lambda \left(t\right)}{dt}\right)$	Six: b, γ, c, and a (with units of s−1) and θ and υ0 (unit-less)

Model legend: I, simple RD model; II, RD model with nonlinear interaction term (NLRD model); III, NLRD model with nonlinear distortion effect on distortion (same nonlinear effect of distortion on both recruitment and distortion); IV, NLRD model with nonlinear distortion effect on distortion (same nonlinear effect of distortion on both recruitment and distortion); V, NLRD model with second-order distortion term; VI, NLRD model with second-order recruitment term.