TABLE 1.

The historical development of theory of mutualism

	Linear benefits	Saturating benefits (intraspecific)	Saturating benefits (interspecific)	Cost‐benefit models & shifting net effects	Consumer‐resource approach
Representative work	Gause and Witt (1935) proposed the first mutualism model as a modification of the Lotka–Volterra equations	Whittaker (1975) proposed that benefits to a host population from a symbiont should saturate per host individual due to extrinsic factors	Wright (1989) proposed that benefits should saturate with interspecific density, due to constraints on handling time	Hernandez (1998) proposed that benefits increase at low partner density, but interaction becomes negative at high partner density	Holland and DeAngelis (2010) proposed that resource supply and consumption processes directly affect per‐capita growth rate
Mechanisms included	Benefit increases per‐capita growth rate (low‐density effect), equilibrium density (high‐density effect), or both	Per‐capita benefit accrual decreases as: Resources or space become limiting*, Substrates to receive or attract benefits become limiting, Competition for benefits increases. * “extrinsic” factors; all other listed limitations are “intrinsic” to the mutualism	Rate of benefit accrual decreases as (effective) partner density becomes limiting, or due to satiation, search time, or handling time. Benefits may also be subject to intraspecific limitations	Partners have nonlinear effects, with positive effects (net benefits) at low recipient or partner densities and negative effects (net costs) at high densities. Benefits accrue due to facilitation at low density. Costs accrue due to exploitation or competition at high density	Benefits accrue due to consumption of resources (or services) supplied by a partner. Costs accrue by supplying resources to a partner or having resources consumed
Characteristic assumptions	Benefit is a linear function of partner density	Benefit increases per‐capita growth rate and equilibrium density, but saturates with increasing recipient density	Benefit increases per‐capita growth rate and equilibrium density, but saturates with increasing partner density. Recipient experiences additional self‐limitation	Net effects are represented directly as a non‐monotonic interspecific function or emerge from the balance between interspecific benefit and cost functions	Consumption is an interspecific process. Services are approximated as function of partner density or consumption rate. Costs accrue in demographic or foraging parameters (“fixed costs”), or are functions of partner consumption rate (“variable costs”)
Characteristic predictions	Unbounded growth between facultative partners with strong interactions. Stable coexistence between facultative partners with weak interactions. Extinction of obligate partners below a certain density threshold or unbounded growth above such threshold with strong interactions. Extinction of obligate partners with weak interactions	Stable coexistence in feasible interactions, regardless of interaction strength or obligacy. Threshold between extinction of obligate partners and stable coexistence when at least one partner is obligate. Coexistence is non‐oscillatory (stable node)	Same predictions as in intraspecific saturating models	Diverse dynamics, depending on the model and its parameterization: Predictions of saturating models, but coexistence may be oscillatory (stable spiral). Mutualistic coexistence, competitive coexistence, or competitive exclusion. Mutualistic coexistence, parasitic coexistence, or extinctions	Fixed costs: same predictions as in saturating models. Variable, linear costs: same predictions as saturating models, but coexistence may be oscillatory. Variable, nonlinear costs: mutualistic coexistence or overexploitation by consumers leading to collapse; coexistence may be oscillatory
Citations	Gause and Witt (1935), Whittaker (1975), Vandermeer and Boucher (1978), Goh (1979), Addicott (1981), Gilpin et al. (1982)	Whittaker (1975), May (1976), Soberón and Martinez del Rio (1981), Dean (1983), Wolin and Lawlor (1984), Parker (2001)	Wells (1983), Pierce and Young (1986), Wright (1989), Graves et al. (2006), Thompson et al. (2006), Fishman and Hadany (2010), Johnson and Amarasekare (2013), García‐Algarra et al. (2014)	Tonkyn (1986), Hernandez (1998), Holland et al. (2002), Neuhauser and Fargione (2004), Wu et al. (2019)	Holland and DeAngelis (2010), Kang et al. (2011), Revilla (2015), Martignoni et al. (2020), Hale et al. (2021)