Results

In the model, parameters e and d are the main drivers for coexistence. In the case of e , coexistence in the producer-cheater system is a product of the balance of two opposing forces; invading producer species that drive down enzyme production, and resource strain from cheaters that drives up enzyme production. These forces are at equilibrium when enzyme production investment is at the critical threshold (e* ). As the investment in enzyme production increases in a producer-cheater mixture, the trade-offs between producer biomass and enzyme production reach a point where enzyme production requires too much energy and becomes unsustainable, causing the producer population to crash (indicated in Figure 2 with solid red vertical lines; e  = 0.2067). However, another threshold for enzyme production investment (where e  ≠ 0) exists (indicated in Figure 2 with dashed vertical lines). In monoculture, selection favours producer mutants that invest less in enzyme production (i.e., with lower evalue), because these mutants can always successfully invade the producer population at equilibrium due to their higher per-capita growth rates (Figure 3A). Over time, this process reduces the production of the enzyme, which reduces the available resource, in turn reducing the population abundance of the producer. Eventually, producer abundance slowly drifts towards a critical production threshold,e *(e *m; Figure 2A; e  = 0.0008). As investments in enzyme production drift lower thane *m, the total population size reaches zero abundance, going extinct and causing system collapse (Figure 3C).
In contrast, given the biologically realistic, literature-driven, parameter values we chose for the model, we observed an interesting dynamic when a cheater is present. In a producer-cheater mixture, the cheater creates a resource strain that is strong enough to prevent selection from driving producer enzyme production down to its critical limit (e *c; Figure 3B; e  = 0.0009). This strain on resources creates a discontinuous shift in equilibrium abundance, such that below a different critical threshold,e *c, equilibrium abundance suddenly drops from positive to zero (as opposed to the slow continuous drift towards zero that occurs in producer monocultures). Moreover, in the mixed culture case where the sudden shift happens, residual resources in the system allow for the possibility of “evolutionary rescue”. That is, if a new producer mutant should arise with an enzyme production rate that falls above e *c, it will be able to successfully invade the system, and will ultimately increase enzyme abundance sufficiently to stabilize the system (Figure 3D). In other words, the presence of a cheater allows for the possibility of long-term persistence of both strains, whereas a pure producer monoculture is doomed to relatively rapid extinction.
While resource diffusion does not exhibit the dual-threshold nature of enzyme production investment, it does control producer abundance in a producer-cheater system. As diffusion approaches higher values, producer access to its resource is impeded by the cheater, causing a population extinction and therefore a system collapse (Figure 4,d  = 0.152373). Similarly, if diffusion is too slow (Figure 4,d  = 0.05), the cheater’s access to the resource is restricted, and the cheater population goes extinct, eventually causing the system to crash following the producer-only evolutionary dynamics described above. Importantly, for the range of biologically realistic model parameters that we consider here, cheaters cannot overgrow the producers.