TABLE 2.

Selected models of pairwise mutualism

Reference	Eqn	Models for Pairwise Mutualism ($i=1,2$)	Notes
Gause and Witt (1935)	1	$\frac{d{\mathit{N}}_i}{\mathit{dt}}=r_i{\mathit{N}}_i\left(\frac{K_i+{\alpha }_{\mathit{ij}}{\mathit{N}}_j‐{\mathit{N}}_i}{K_i}\right)$	Facultative only
Whittaker (1975)	2 1	$\left\{\begin{array}{c}\frac{d{\mathit{N}}_1}{\mathit{dt}}=r_1{\mathit{N}}_1\left(\frac{K_1+{\alpha }_{12}{\mathit{N}}_2‐{\mathit{N}}_1}{K_1+{\alpha }_{12}{\mathit{N}}_2}\right)\\ \frac{d{\mathit{N}}_2}{\mathit{dt}}=r_2{\mathit{N}}_2\left(\frac{K_2+{\alpha }_{21}{\mathit{N}}_1‐{\mathit{N}}_2}{K_2}\right)\end{array}\right.$	Symbiont (${\mathit{N}}_1$)‐Host (${\mathit{N}}_2$) Obligate ${\mathit{N}}_1$ when $K_1=0$ Parasitism when ${\alpha }_{21}<0$
	2 3	$\left\{\begin{array}{c}\frac{d{\mathit{N}}_1}{\mathit{dt}}=r_1{\mathit{N}}_1\left(\frac{{\alpha }_{12}{\mathit{N}}_2‐{\mathit{N}}_1}{{\alpha }_{12}{\mathit{N}}_2}\right)\\ \frac{d{\mathit{N}}_2}{\mathit{dt}}=\frac{r_2{\mathit{N}}_2}{K_2}\left(K_2+\frac{aD{\mathit{N}}_1}{D+{\mathit{N}}_2}‐{\mathit{N}}_2\right)\end{array}\right.$	Symbiont (${\mathit{N}}_1$)‐Host (${\mathit{N}}_2$) Obligate ${\mathit{N}}_1$
Vandermeer and Boucher (1978)	1	$\frac{d{\mathit{N}}_i}{\mathit{dt}}={\mathit{N}}_i\left(r_i+{\alpha }_{\mathit{ij}}{\mathit{N}}_j‐{\alpha }_{\mathit{ii}}{\mathit{N}}_i\right)$	Legume (${\mathit{N}}_1$)‐Rhizobium (${\mathit{N}}_2$) Obligate when $K_i=r_i/{\alpha }_{\mathit{ii}}\le 0$
Addicott (1981)	4	$\frac{d{\mathit{N}}_i}{\mathit{dt}}=r_i{\mathit{N}}_i\left(\frac{K_i‐{\mathit{N}}_i}{K_i}\right)\left(1+\frac{{\alpha }_{\mathit{ij}}{\mathit{N}}_j}{K_i}\right)$	Aphid (${\mathit{N}}_1$)‐Ant (${\mathit{N}}_2$) Facultative only See Table S1
Wolin and Lawlor (1984)	5	$\frac{d{\mathit{N}}_i}{\mathit{dt}}={\mathit{N}}_i\left(r_i‐\frac{b{\mathit{N}}_i}{1+m{\mathit{N}}_j}‐d{\mathit{N}}_i\right)$	cultative only Reduces intra‐specific limitation in birth ($b$) to at most 0 See Table S1
	6	$\frac{d{\mathit{N}}_i}{\mathit{dt}}={\mathit{N}}_i\left(r_i‐\left(b‐m{\mathit{N}}_j+d\right){\mathit{N}}_i\right)$	Reduces $b$ without limit
Wright (1989)	7	$\frac{d{\mathit{N}}_i}{\mathit{dt}}={\mathit{N}}_i\left(r_i‐c_i{\mathit{N}}_i+b_{\mathit{ij}}\frac{a_{\mathit{ij}}{\mathit{N}}_j}{1+a_{\mathit{ij}}{h}_{\mathit{ij}}{\mathit{N}}_j}\right)$	Pollinators & other forager mutualists See Table S1
Zhang (2003)	8	$\frac{d{\mathit{N}}_i}{\mathit{dt}}=R_i{\mathit{N}}_i\left(c_i‐{\mathit{N}}_i‐a_i{\left({\mathit{N}}_j‐b_i\right)}^2\right)$	teractions between species at the same trophic level $‐\infty <b_i<\infty$
Neuhauser and Fargione (2004)	9 1	$\left\{\begin{array}{c}\frac{d{\mathit{N}}_1}{\mathit{dt}}=r_1{\mathit{N}}_1\left(\frac{K_1+{\gamma }_{12}{\mathit{N}}_2‐{\mathit{N}}_1}{K_1+{\gamma }_{12}{\mathit{N}}_2}‐a{\mathit{N}}_2\right)\\ \frac{d{\mathit{N}}_2}{\mathit{dt}}=r_2{\mathit{N}}_2\left(\frac{K_2+{\alpha }_{21}{\mathit{N}}_1‐{\mathit{N}}_2}{K_2}\right)\end{array}\right.$	Plant (${\mathit{N}}_1$)‐Mycorrhizae (${\mathit{N}}_2$) Facultative only
Graves et al. (2006)	10	$\frac{d{\mathit{N}}_i}{\mathit{dt}}={\mathit{N}}_i\left(r_{i0}+\left(r_{i1}‐r_{i0}\right)\left(1‐e^{‐k_i{\mathit{N}}_j}\right)‐a_i{\mathit{N}}_i\right)$	Lichens Obligate when $r_{i0}〈0,r_{i1}+r_{i0}〉0$ See Table S1
Thompson et al. (2006)	11 12	$\left\{\begin{array}{c}\frac{d{\mathit{N}}_1}{\mathit{dt}}=\left({\rho }_1{b}_1{\mathit{N}}_1+I_1\right)\left(1‐\frac{{\mathit{N}}_1}{S_1}\right)‐\left(d_{1_{\text{min}}}+\frac{d_{1_{\text{max}}}‐d_{1_{\text{min}}}}{1+c_1{\mathit{N}}_2}\right){\mathit{N}}_1\\ \frac{d{\mathit{N}}_2}{\mathit{dt}}=\left({\rho }_2{b}_2{\mathit{N}}_2+I_2\right)\left(1‐\frac{{\mathit{N}}_2}{S_2+{\mathit{N}}_1}\right)‐\left(d_{1_{\text{min}}}+\frac{d_{2_{\text{max}}}‐d_{2_{\text{min}}}}{1+c_2{\mathit{N}}_1}\right){\mathit{N}}_2\end{array}\right.$	Hermit crabs (${\mathit{N}}_1$)‐ Anemones (${\mathit{N}}_2$) Closed system when $I_i=0$, ${\rho }_i=1$ Obligate when ${\rho }_i{b}_i<d_{i_{\mathit{max}}}$ See Table S1
Holland and DeAngelis (2010)	13	$\frac{d{\mathit{N}}_i}{\mathit{dt}}={\mathit{N}}_i\left(r_i+c_i\left(\frac{a_{\mathit{ij}}{\mathit{N}}_j}{h_j+{\mathit{N}}_j}\right)‐q_i\left(\frac{{\beta }_{\mathit{ij}}{\mathit{N}}_j}{e_i+{\mathit{N}}_i}\right)‐s_i{\mathit{N}}_i\right)$	Bidirectional Consumer‐Resource e.g., Plant (${\mathit{N}}_1$)‐Mycorrhizae (${\mathit{N}}_2$) Obligate when $r_i=0$
	13 7	$\left\{\begin{array}{c}\frac{d{\mathit{N}}_1}{\mathit{dt}}={\mathit{N}}_1\left(r_1+c_1\left(\frac{a_{12}{\mathit{N}}_2}{h_2+{\mathit{N}}_2}\right)‐q_1\left(\frac{{\beta }_{12}{\mathit{N}}_2}{e_1+{\mathit{N}}_1}\right)‐s_1{\mathit{N}}_1\right)\\ \frac{d{\mathit{N}}_2}{\mathit{dt}}={\mathit{N}}_2\left(r_2+c_2\left(\frac{a_{21}{\mathit{N}}_1}{h_1+{\mathit{N}}_1}\right)‐s_2{\mathit{N}}_2\right)\end{array}\right.$	Unidirectional e.g., Plant (${\mathit{N}}_1$)‐Pollinator (${\mathit{N}}_2$)
Fishman and Hadany (2010)	14 15	$\left\{\begin{array}{c}\frac{d{\mathit{N}}_1}{\mathit{dt}}={\mathit{N}}_1\left(\frac{\eta \alpha {\mathit{N}}_2}{1+\alpha {\mathit{N}}_1+\alpha \beta {\mathit{N}}_2}‐b‐c{\mathit{N}}_1\right)\\ \frac{d{\mathit{N}}_2}{\mathit{dt}}={\mathit{N}}_2\left(\frac{\mu \alpha {\mathit{N}}_1}{1+\alpha {\mathit{N}}_1+\alpha \beta {\mathit{N}}_2}‐d\right)\end{array}\right.$	Plant (${\mathit{N}}_1$)‐Pollinator (${\mathit{N}}_2$) Obligate only
Kang et al. (2011)	16 1	$\left\{\begin{array}{c}\frac{d{\mathit{N}}_1}{\mathit{dt}}={\mathit{N}}_1\left(r_f\left(\frac{a{\mathit{N}}_2^2}{b+a{\mathit{N}}_2^2}\right)‐r_c{\mathit{N}}_2‐d_1{\mathit{N}}_1\right)\\ \frac{d{\mathit{N}}_2}{\mathit{dt}}={\mathit{N}}_2\left(r_a{\mathit{N}}_1‐d_2{\mathit{N}}_2\right)\end{array}\right.$	Fungal garden (${\mathit{N}}_1$)‐Leaf cutter ant (${\mathit{N}}_2$) Obligate only
Martignoni et al. (2020)	17 18	$\left\{\begin{array}{c}\frac{d{\mathit{N}}_1}{\mathit{dt}}={\mathit{N}}_1\left(r_p+\frac{q_{\mathit{hp}}\alpha {\mathit{N}}_2}{d+{\mathit{N}}_1}‐q_{\mathit{cp}}\beta {\mathit{N}}_2‐{\mu }_p{\mathit{N}}_1\right)\\ \frac{d{\mathit{N}}_2}{\mathit{dt}}={\mathit{N}}_2\left(q_{\mathit{cm}}\beta {\mathit{N}}_1‐\frac{q_{\mathit{hm}}\alpha {\mathit{N}}_1}{d+{\mathit{N}}_1}‐{\mu }_m{\mathit{N}}_2\right)\end{array}\right.$	Plant (${\mathit{N}}_1$)‐Mycorrhizae (${\mathit{N}}_2$) Obligate ${\mathit{N}}_2$
Hale et al. (2021)	19 7	$\left\{\begin{array}{c}\frac{d\mathit{P}}{\mathit{dt}}=P\left[b_P\left(f+\phi \frac{a\mathit{AP}}{1+ah\mathit{P}+a\mathit{AP}}\right)g‐s_P\mathit{P}‐d_P\right]\\ \frac{d\mathit{A}}{\mathit{dt}}=A\left[b_A+\epsilon \frac{a\mathit{P}}{1+ah\mathit{P}}‐s_A\mathit{A}‐d_A\right]\end{array}\right.$	Plant (${\mathit{N}}_1$)‐Pollinator (${\mathit{N}}_2$) Obligate ${\mathit{N}}_1$ when $b_{1}fg‐d_1\le 0$; obligate ${\mathit{N}}_2$ when $b_2‐d_2\le 0$
	20 7	$\left\{\begin{array}{c}\frac{d\mathit{P}}{\mathit{dt}}=P\left[b_{P}fg‐\left(s_P‐\sigma \frac{a\mathit{A}}{1+ah\mathit{P}+a\mathit{A}}\right)\mathit{P}‐d_P\right]\\ \frac{d\mathit{A}}{\mathit{dt}}=A\left[b_A+\epsilon \frac{a\mathit{P}}{1+ah\mathit{P}}‐s_A\mathit{A}‐d_A\right]\end{array}\right.$	Plant (${\mathit{N}}_1$)‐Seed Disperser (${\mathit{N}}_2$) Facultative ${\mathit{N}}_1$ only Obligate ${\mathit{N}}_2$ when $b_2‐d_2\le 0$

A full list of models cited in the main text is included in the supplementary information (Table S1). Equations largely follow the notation from the original citations. All parameters are positive (>0) unless otherwise specified. Models with unique mathematical forms are given unique equation numbers. We encourage the readers to refer to the original references for the model derivations and interpretation of parameters. Notes include inspiring system and obligacy, if specified by authors.