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.