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, =  substrate; =  enzyme;RE  =  enzyme producer resource;NC  =  cheater resource. Parameters: =  substrate inflow rate; =  rate of substrate degradation by enzymes; =  enzyme production investment by the producer; =  resource diffusion rate; = quantity of resource required for the production of species biomass; =  species growth rate; =  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 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).