Figure Legends
Figure 1
Schematic of a chemostat model with producer and cheater populations and
a single complex substrate resource. Variables:NE = enzyme producer population;NC = cheater population,S = substrate; E = enzyme;RE = enzyme producer resource;NC = cheater resource. Parameters:a = substrate inflow rate; g = rate of
substrate degradation by enzymes; e = enzyme production
investment by the producer; d = resource diffusion rate;q = quantity of resource required for the production of species
biomass; r = species growth rate;m = species mortality rate.
Figure 2
Effect of parameter e on species abundances and coexistence. The
dash-dotted red vertical line marked e* indicates a criticale value creating an EES for the producer, in the context of
enzyme production investment (e* m = 0.0009).
Solid red vertical lines marked e° show the maximum e value
before the enzyme producer population collapses due to the increased
investment in enzyme production (e = 0.2067). In (A) we show the
effect of the e * on the abundance of the enzyme producer as a
monoculture (blue) and in a mixture (gold). Due to the invasibility of
the producer monoculture by producers with lower e , producer
abundance can eventually drift to zero. On the other hand, the presence
of the cheater in the same situation creates a discontinuous shift from
a positive to a negative equilibrium
(e *c = 0.0008), preventing any further invasion
of lower e producers. Since cheater abundance (B), resource
release (C, D) and enzyme (E) and substrate (F) concentrations are
tightly linked to producer abundance, they follow similar dynamics in
producer-cheater mixtures. In the absence of a producer, substrate
concentration returns to baseline, indicated by the dotted black
horizontal line in (F) Due to the plotting scalee *m and e *c are
overlapping and are both shown with the single e * dash-dotted red
vertical line.
Figure 3
(A) Here we show the invasion rate of producers with variations in their
investment in enzyme production. Producers who invest less in enzyme
production have higher invasion rates and success. At the e for
which the invasion success intersects with zero invasion rate, an ESS of
the producer population emerges (e * = 0.0009).
(B) Effects of model parameter e on the abundance of the enzyme
producer. The solid red vertical line signifies the maximum e before
negative growth occurs (max e = 0.2067) due to allocating too
many resources into enzyme production and not growth. The dash-dotted
red vertical line indicates the lowest possible e , a critical
enzyme production investment threshold, allowed in the model before
negative growth occurs for the enzyme producer, due to lack of resource
release from the substrate (e * = 0.0009). This enzyme production
investment threshold creates an uninvadable ESS — producers with lower
enzyme production investment can no longer invade the producer
population due to the presence of the cheater.
(C-D) Show the effect of lowering the cost of enzyme production belowe *. This change causes a discontinuous shift in equilibrial
abundance for the enzyme producer, driving it towards extinction because
it is no longer able to produce enough resources to overcome the
diffusion gradient towards the cheater. If the cost of enzyme production
remains below the threshold value, the enzyme producer is ultimately
driven extinct (C). If the cost of enzyme production is increased back
above the threshold value before resources are depleted, the enzyme
producer is able to recover (D). This is possible due to residual
resources in the system, allowing the producer population to recover. In
our model, the producer cannot be rescued if its abundance crosses below
the shaded red area. Parameters: a = 0.01; g = 72.64;d = 0.10; q = 0.65; mz = 1.05;r = 2.08; m = 0.11.
Figure 4
Similar to Figure 3, the effect of resource diffusion rate, parameterd , on species abundances and coexistence is shown (A). Increasing
diffusion rate in the producer-cheater mixture (golden line in (A))
reduces resource availability to the producer, leading to extinction and
system collapse. Since cheater abundance (B), resource release (C, D)
and enzyme (E) and substrate (F) concentrations are tightly linked to
producer abundance, the follow the trajectory of producer abundance, in
producer-cheater mixtures. In the absence of a producer, substrate
concentration returns to baseline, indicated by the dotted black
horizontal line in (F).