A resource ratio theory of cooperation

R* model platform

Consider a resource competition model for two species (A and B) competing for two limiting resources (R₁ and R₂) such as nitrogen and phosphorus. The following equations describe resource supply, uptake and leaching (Tilman 1982; DeAngelis 1992; Daufresne & Loreau 2001) in a two species competition model. The change in the amount of resource i, R_i, where i is the resource index 1 or 2 is:

(1)

where S is the species index A or B, I_i is input, l_i is the leaching or loss rate, and g_Si is the amount of nutrient i taken up by species S, a function of the biomass of species S, N_S, and the availability of the two resources R₁ and R₂. All model variables and parameters are found in Table S1.

We assume that the two resources are essential and non-substitutable, and there is a fixed stoichiometry of individuals of either species with ratio requirement α_S,

We follow population dynamics in terms of nutrient flows and pools, using the amount of resource 1 in biomass. The change of resource 1 in biomass of species S, N_S, is:

(2)

where g_S1 is the amount of resource 1 taken up by species S per unit of time, and m_S is the rate of nutrient lost from the biomass either through metabolism (e.g. through tissue mortality rate) or through death of the organism.

In the absence of trading, the two resources are taken up in proportion to the species’ ratio requirement α_S:

(3)

Resource uptake is dependent upon biomass (N_S) and resource availability (R_i), with a minimum function that describes growth following Liebig’s law of the minimum:

(4)

where f_Si is the maximal resource i uptake rate function for species S. Here we will consider specifically linear and Monod type uptake functions (Appendix S1), as well as a generic function f_Si that can be any monotonically increasing function. Note that the uptake kinetics for a nutrient is more efficient when it is limiting (uptake is then at the maximum value) than when it is non-limiting (where uptake is limited by the other, limiting nutrient), consistent with empirical observations (Turpin 1988). Also note that we do not deal with plastic or potentially evolving allocation towards the uptake of resource 1 vs. 2. We assume that the allocation to the uptake machineries for both resources are fixed, and that maximum uptake of both resources is feasible – it is realized when the two resources are co-limiting.

The two species differ in their ability to deplete a resource when limiting ( for species S, resource i) and resource ratio requirements (α_S). Resource depletion (, see Appendix S1) is the minimum level of the resource pool that allows the plant to maintain a positive growth rate and defines a zero net growth isocline (ZNGI; Fig. 1a). From this basic modelling structure, Tilman (1982) demonstrated that we can assess resource depletion exerted by each species assuming some initial resource supply point, to understand how various patterns in affect competitive outcomes and species coexistence (Fig. 1a). In particular, stable coexistence is predicted when each species requires more of the resource that is most limiting to that species (Fig. 1b; Appendix S1; Tilman 1982). Extreme resource supply regions, of either resource, result in competitive exclusion by one species or the other (Fig. 1b). In this manuscript we investigate the effect and evolution of trade in situations that can potentially lead to coexistence as in Fig. 1b. In Appendix S1, we generalize our conclusions to situations with potential founder control (Appendix S1).

Allowing trading in the model

For any resource supply point within the region of potential coexistence, uptake of one resource is constrained because the other resource becomes limiting (Fig. 1a). This excess of potentially accessible resource is left unused in the traditional resource ratio model because it cannot be utilized owing to a rate-limiting constraint imposed by the other resource. When two species coexist, each species has potential access to the very resource that limits the other species (Fig. 1b). This creates the opportunity for each species to over-consume non-limiting resources and exchange these resource for the other, rate limiting, resource by relaxing the assumption that species do not acquire resources unless they can immediately apply them to growth.

In our trading model, we adopt the assumption that the non-limiting resource can potentially be acquired at the same rate as when it’s limiting, thus creating an excess E_Si of the non-limiting resource i for the individual of species S:

We then allow that a fraction of this excess resource, t_S may be transferred to a trading partner of the other species, thus increasing the resources potentially available for the growth of the partner (Appendix S1). This resource transfer may, or may not, be reciprocated. The resulting growth rate for an individual of species S with trading strategy t_S in a partnership with trading partner P with strategy t_P is given by:

Note that we assume no cost for uptake of excess resource nor transfer to or from partner, thus giving no incentive for cheating.

Trading strategy optimization

Imagine that competing individuals of the two species have the option to function as a single unit exchanging resources. We can calculate the properties of a two species unit that maximizes ecosystem resource utilization and resource depletion (Appendix S1). In this optimally trading pair, individuals take up both resources at their physiological maximum rate and those resources are used to produce biomass of both species. As demonstrated for example in Tilman (1982), in a pure resource competition framework, such as the one presented here, the competitive strategy that wins in competition is the one that can deplete resources below a level of positive growth for its competitors. As we show in Appendix S1, an optimally trading unit as defined above does indeed do that. The region of resource depletion points where optimally trading units can drive resource depletion below that of species acting on their own is graphically depicted by the orange shaded region in Fig. 1c. Every combination of resource supply within this shaded region carries the potential of optimal resource depletion by a trading unit. In this case, an optimal pair is capable of depleting resources down to the orange line, below the level achieved by two species competitive coexistence (equilibrium point C* in Fig. 1b). Thus, we can delimit the environmental conditions where optimal trading is potentially the best competitive strategy, and the result is an increased potential for coexistence between species (Fig. 1c) relative to traditional resource competition (Fig. 1b). Competitive simulation runs (Appendix S1) show that optimal trading units are the only trading pairs that cannot be invaded by any other trading pair or non-trading individual: they are evolutionary stable strategies in the sense of Maynard Smith (1982).

The conditions that support potential trading are fairly general. First, the two resources must be non-substitutable and essential, i.e. growth is a minimum function and the ZNGI are L shaped. Second, one species must be relatively better at depleting one of the resources, i.e. and . Economists refer to this condition as comparative advantage (Schwartz & Hoeksema 1998). This model, however, makes simplifying assumptions that may be unrealistic. We assume that the cost of excess resource acquisition and resource transfer between species is negligible. As discussed later, this means that there is no benefit to cheating. These assumptions are likely to be violated in many natural systems. Increasing the cost of maintaining resource exchange would decrease the capacity for trading pairs to outperform competing individuals.

Effect of trade on population sizes

An interesting outcome of this simple model, however, is that while cooperation maximizes system-wide resource depletion and biomass production, it does not necessarily increase biomass production for both of the trading partners. To assess the population consequences of trading, we compare optimal trading outcomes with those of both single species resource consumption and two species competition. These situations encompass the three possible combinations of presence/absence of trading and presence/absence of competition (Table 1).

There are three possible outcomes of in terms of species biomass relative to trading or not trading. Biomass achieved per species in competition (COMP) is always lower than species biomass in monoculture (MONO). However, trading (TRAD) can result in the maximum biomass (TRAD > MONO > COMP), intermediate biomass (MONO > TRAD > COMP) or the least biomass (MONO > COMP > TRAD) for each species (Fig. 2). Resource supply is critical in determining the outcome biomass for each species. Trade maximizes potential biomass for each species when there is a low supply of the resource for which a species is a superior competitor (Fig. 2). Thus, both species increase potential biomass when resource supplies are low for both resources. Biomass outcomes are intermediate between their performance in isolation and their performance competing with the other species when resource supplies are high and not lacking in the resource for which the focal species is a good competitor (Fig. 2). Finally, biomass with trade is lower than in competition when resource supply is high and close to the region where competitive exclusion would result in the elimination of the other species in a competitive relationship. Although the exact parameter space configuration is system specific, this array of outcomes suggests that trading interactions span a competition – antagonism – mutualism continuum when assessing interactions with their effect on biomass or population size.

Natural selection and the evolution of trade

We alter the framework to an individual-based model to assess whether trade is able to evolve from a competitive precursor. Each individual in this model has a strategy that is described by two traits: (a) the fraction of resources that it gives to its trading partner t_S and (b) its propensity to associate with a partner p_S. Free individuals may form pairs at random from the pool of available individuals at a rate increasing with their propensity to associate with partner [rate of association is (p_A+ p_B)/2 per unit time]. A pair can break down as a consequence of either individual defecting from the partnership, at rate 1 − (p_A + p_B)/2 per unit time, or as a consequence of the death of one of the individuals. Resources are lost from individuals continuously through a metabolic rate. Death of individuals occurs for two reasons. First, if nutrient supply does not meet demand, an individual loses biomass and does not survive below a threshold size. In addition, background mortality is captured by a constant low rate of individuals randomly selected to die and be removed from the population at each time step. Population size is constant and similar for both populations, enabling 1 : 1 pairing between individuals and matching with theoretical predictions for optimal trading. Replacement of dead individuals come from asexual reproduction. Probability of reproduction is proportional to biomass. Offspring inherit their parents strategies for both trading and partnership fidelity, with some probability of mutation. In other respects (i.e. resource use) this model is analogous to the previous deterministic model.

In the absence of trade this model behaves exactly as we expect. The system quickly moves to an equilibrium abundance for each species (Fig. 3a), and nutrients are depleted to the two species equilibrium (Fig. 3b). When trade and continued association in partnerships are allowed to evolve (Fig. 3c–f), then the dynamics go to a new, and different, equilibrium abundance of the two species (Fig. 3c). This new equilibrium depletes resources to a lower equilibrium point (Fig. 3d). This lower depletion is achieved by the evolution of trade (Fig. 3e) and long-term relationship with partner (Fig. 3f). Under these conditions, the natural selection drives evolution towards forming long associations and the optimal trading strategy predicted by our theoretical calculations. This combination of traits is more efficient at depleting nutrients and can outcompete any other strategy. This result is robust to whether nutrient levels are set such that both species benefit or whether nutrient levels are close to the competitive exclusion barrier, where trading results in a seemingly parasitic relationship between species. However, it does require that the system starts with coexisting competitors, and that mortality be sufficiently low so that long-term partner association between individuals is possible.

This natural selection example illustrates the paradox highlighted in the previous section. At each time step individuals that join in trading partnerships enjoy a selection advantage over those that do not trade or do not remain in trading relationships, yet the end result may be that one species ends up with a lower net biomass than is found under a non-trading condition (Fig. 3c). Total nutrient uptake in the system is increased as a consequence of trade, but the advantage may accrue to just one of the two species. Individuals of the ‘losing’ species, however, must evolve towards trade or else they are eliminated from the environment by their trading conspecifics that have higher relative fitness. Those results were reasonably robust to adding a moderate cost to trading. We ran simulations where trading strategies had a lower nutrient uptake, and the reduction was proportional to the level of trading. Trading took much longer time to evolve and evolved to lower levels, and the biomass of both species were lower than without a cost (results not shown).

Resource trade and coexistence

The model presented here predicts that conditions allowing coexistence with competition could actually favour natural selection towards establishing a cooperative trading relationship. Many biological attributes may prevent resource exchange, but the model suggests two important observations. First, it should be possible for cooperation to evolve from a competitive precursor. This observation is complementary to Roughgarden (1975), who predicted mutualisms evolving from parasitic precursors. Second, this model shows that natural selection can favour the evolution of trading relationships despite a ‘parasitic’ effect of that relationship on biomass or population size.

A number of biological attributes may preclude competitors from exchanging nutrients. Nevertheless, the observation that cooperation broadens the domain of coexistence (Fig. 1c) is an important observation in terms of considering the maintenance of diversity. The only conditions for there to exist a trading strategy that outperforms a competitive strategy are low mortality to enable long-term partnerships between individuals of the two species, and that each organism is less efficient than the other organism at depleting one of the two resources. This closely corresponds to conditions highlighted by comparative advantage biological market models where the poorer competitor only need to experience a differential cost in acquiring the two resources (Schwartz & Hoeksema 1998; Hoeksema & Schwartz 2003). Trading thus widens an existing coexistence region, and may create a coexistence region in the founder-control situation (Appendix S1). However, coevolution towards an optimal trading pair requires long-term contact between individuals of the two species, therefore coexistence. Outside regions of coexistence, an optimal trading pair will displace non-trading individuals, but it needs to have evolved from a different environment, or for other reasons.

Evolution of trading

We showed that trading should evolve when there is coexistence. This comes about because when an individual feeds its partner, its partner increases in abundance, and if its partner also happens to trade, that in turn helps the target individual. This positive feedback brings the species to an efficient level of trading.

What about cheaters? Cheaters are individuals that take the resources from their partners without providing goods or services in return (Bronstein 2001; Yu 2001). Because we assume a strict resource competition, and no cost to trading, there is an alignment of interest of the two partners to deplete the resources optimally: optimal trading is the only way they can become together a super-competitor unit that survives at such low resource levels and exclude all other competing strategies. To demonstrate this alignment of interest, we evolved the trading traits the other way around: individual strategies determined how much they received from their partner rather than how much they gave to their partner. They evolved to the same optimal trading level, and evolution towards trading occurred faster (results not shown). In our model, partner fidelity (Sachs et al. 2004) evolved to enable stable partnerships and the evolution of trading. Cooperation was not costly to individuals, and it was beneficial to the individual. Theory does not predict the evolution of cheating in such case. Cheaters might evolve if the alignment of interest was broken, for example if not all individuals could enter a trading relationship, or if parasites were allowed to evolve in the system. There is no gain for cheating if there are no costs to cooperate, it is just not an option for competitors if it prevents them from surviving at low resource levels.

The mutualism – antagonism – competition continuum

One of the most intriguing result we obtained was that trade could evolve even though it would decrease the total population biomass of one of the partner species. It is well known that fitness and population sizes are often unrelated, so that populations might even evolve to extinction (Webb 2003). In our model, fitness is determined by the ability to grow and survive and reproduce at low resource availability. Trading enables an optimal ‘trading unit’ to optimize resource use and deplete resources to lower levels than any other strategies can tolerate (Fig. 3). At high and unbalanced resource levels (Fig. 2), the optimal balance shifts resources away from the competitively superior species, as resource use is optimized when some of the resources previously consumed by the target species are shifted to its partner species.

It is important to realize that depending on the criterion used to define mutualism, we arrive at qualitatively very different conclusions. An effect-based definition, separating pairwise interactions along the classical (+, 0, −) gradient in interaction effects can classify interactions depending on their net effect on population sizes, population growth rates or individual fitness (Abrams 1987). One possible criterion is based on population size response to partner removal: whether population size achieved by either species in optimal trading is larger or smaller than population size achieved in the absence of the partner (Abrams 1987). With this definition in mind, we would define resource exchange relationships as spanning a continuum from mutualism (+,+) at low resource supplies, through antagonism (+, −) at intermediate resource supplies and to competition (−, −) at high resource supplies.

However, we suggest that mutualism and parasitism should be defined not by population-level effects but by fitness effects, as seems to be the consensus in mutualism studies (Abrams 1987). In a resource competition setting, individual fitness is based on its ability to grow and survive at low resource density. It is clear that optimal trading enables survival and growth at lower levels of resources, and therefore provides a positive effect on individual fitness is all the region of resource supply points where an optimal trading pair is the best competitive strategy. A fitness-based definition would thus find that the interaction is mutualistic wherever it occurs, even though it presents a context dependent outcome on population size. As we argue below, an appropriate test to contrast the effect of trading on fitness vs. total biomass would be to perform new competition experiments.

Finally, there is yet another definition of mutualism, based on the evolutionary dynamics of an interaction (i.e. mutualistic evolution as defined in de Mazancourt et al. 2005). If both partners actively invest to put an interaction in place through the process of natural selection, then we would call resource exchange a mutualism no matter what the resource supply and biomass outcome. Our evolutionary simulation results were robust to trading strategies bearing a moderate cost, although evolution of trading took much longer (results not shown). Therefore the interaction would be a mutualism according to that criterion too, even if population biomass decreased.

Our model demonstrates that, across a large spectrum of resource supply, there is selection to pair up and establish trading relationships. The key factor that explains this apparent paradox is the fact that population size, or the biomass achieved by any individual, is not a measure of fitness. Fitness is determined by the ability of an individual to survive at low resource abundance, thus depleting the resources until all competitors are driven to extinction. An optimal trading pair is a super-competitor at all resource supply levels. In that sense, there is a strong benefit of the interaction for both species engaging in the interaction at all resources supplies in the area where an optimal trading pair is the best competitive strategy, whatever the population consequences (Fig. 2).

Generally, the model we present here supports the empirical observations of fungal – plant interactions moving along the continuum from mutualism to saprophytes (e.g. Koide et al. 2008) and parasitism (e.g. Egger 2006). In addition, this model framework may explain the evolution of mycorrhizae from saprophytic precursors (e.g. Hibbett et al. 2000). Assume species A is a mycorrhizal fungus, species B is a plant; resource 1 is phosphorus, and resource 2 is light (Fig. 4a). The fungus cannot have positive growth on its own, as it cannot use light to photosynthesize. Fungus A can however assist the plant to deplete light and convert it into the carbon source the fungus requires. We therefore focus on the region of resource space that is below the mycorrhizae’s ZNGI (Fig. 4a). We find that the model makes predictions that match existing experimental results (Johnson 1993; Hibbett et al. 2000; Rice & Currah 2002; Egger 2006; Koide et al. 2008). We can follow the effect of the trading partnership on the biomass of both partners along gradients of phosphorus at high and low light levels (Fig. 4). When phosphorus supply is too low, the plant cannot grow either with or without the mycorrhizae (Fig. 4d,e). As phosphorus supply increases, the plant cannot grow independently but can grow in association with the mycorrhizae. At higher levels of phosphorus supplies, the plant grows better with the mycorrhizae than without. Further increasing the level of phosphorus supply, the plant grows better without the mycorrhizae (Fig. 4d). Finally, at yet higher levels of phosphorus supply, the plant competitively excludes the mycorrhizae (Fig. 4d,e). Johnson (1993) showed that mycorrhizae from higher P soils appear less mutualistic. Our theory predicts that even though mycorrhizae might appear less mutualistic or even parasitic at some phosphorus levels, it might still be in plant’s best interest to enter the association and thus become a super-competitor. We thus question whether the interaction is really parasitic when the plant appears to grow less well in the presence of a mycorrhizae, and suggest that there could be an increased fitness for the plant due to the interaction. This could be tested with a direct competition experiment between a plant associating with its mycorrhizae and an otherwise identical plant genetically modified to inhibit mycorrhizal association.

Resource exchange models and mutualism theory

Mutualism theory is proceeding along multiple lines (Hoeksema & Bruna 2000). One body of work relies principally on game theory to address the issue of cheating within a cooperative relationship (Doebeli & Knowlton 1998; Doebeli & Hauert 2005). Progress in this field has focused on conditions that would result in stable cooperation in the face of selection for self-interested defection from a cooperative relationship (Doebeli & Hauert 2005). Market theory has developed as an alternative approach to understanding mutualism (Noe & Hammerstein 1994; Schwartz & Hoeksema 1998; Hoeksema & Schwartz 2003; Akcay & Roughgarden 2007).

Our model contributes to this expanding literature by placing biological markets within the context of intratrophic resource competition and population dynamics. We show that trading should evolve under strict global resource competition, whenever conditions are sufficient to support stable coexistence, and long-term partnership is possible. This is an important observation because virtually all other models of mutualistic behaviour emerge from parasitic precursors (e.g. Boucher 1985; de Mazancourt et al. 2005), or simply do not address the evolution of the relationship (Hoeksema & Schwartz 2003; Doebeli & Hauert 2005).

Ecological theory associated with mutualism is expanding and diversifying as we better understand the breadth and variety of interspecific relationships that may be defined as mutualism. We have used a simple, and realistic, variation from a classic resource exchange model to expand ecological theory in two realms. First, we use this platform to broaden our understanding of the possible outcomes of resource competition and how we might understand resource competition to influence the maintenance of species diversity. Second, we use this model to expand our understanding of mutualisms and show how a mutualism may evolve from a competition precursor. Finally, by examining predicted biomass outcomes derived from different regions of resource supply, we show that selection may paradoxically favour partnering and engaging in resource exchange despite the fact that the resulting biomass production or population size may decrease from that in a strictly competitive environment.

Our models provide an important first step to understand the evolution of resource trade from a competitive precursor. Further development should probably address the question of resources made available to a range of individuals rather than just one partner. It could potentially be adapted to study the evolution of microbial social interactions (West et al. 2006) or nutrient exchange between plants through mycorrhizal networks (Simard et al. 1997). It could include policing mechanisms or evolution of specialization once in a trading relationship. Nevertheless, our model represents a broad new landscape in which to consider positive interspecific interactions.