Model description

Our model includes two species – an enzyme producer,NE , and a “cheater”, NC , growing in in a common environment (Figure 1). The producer species releases exoenzymes that degrade the substrate into resources that are directly used for growth and production of exoenzymes. The cheater species does not produce the exoenzyme, thereby allowing itself to allocate more resources for growth. The cheater, however, depends on the diffusion of resources from the immediate area around the producers. Within the context of our model, we wanted to test which parameters control coexistence and how similar parameter conditions influence population abundances of producer monocultures vs producer-cheater cocultures.
Other than differences in enzyme production, we assume that species share the same vital rates and requirements and that species occupy two spatially proximal, but separate patches. Specifically, population dynamics for the enzyme producer and cheater follow
\(\frac{dN_{E}}{\text{dt}}=\left(rR_{E}-\left(e+m\right)\right)N_{E}\)Eq. (1a)
\(\frac{dN_{C}}{\text{dt}}=\left(rR_{C}-m\right)N_{C}\) Eq. (1b)
where r is the intrinsic growth rate, m is the mortality rate, RE and RC are, respectively, resource concentration in the vicinity of the enzyme producer vs. the cheater, and e describes the cost of enzyme production. Thus, the enzyme producer always has a lower per-capita growth rate at any given level of resource availability.
Enzyme production is controlled by the abundance of the enzyme producer, and follows
\(\frac{\text{dE}}{\text{dt}}=eq_{z}N_{E}-m_{z}E\) Eq. (2)
where qz converts between the energetic cost of enzyme production to the producer vs. the rate of enzyme production, andmz describes the rate at which the enzyme breaks down over time. Note that parameter e therefore jointly relates to the cost of enzyme production and the rate at which new enzyme is produced. Furthermore, in the absence of enzyme producers, all enzymes eventually break down and the concentration reduces to zero.
Resource dynamics in the model are controlled by the concentration of available substrate S , the concentration of enzymes that break down the substrate into usable resources, E , and the rates at which resources are taken up by species and diffuse between regions. We assume an open system, in which new substrate enters at a rate proportional to the current concentration, and in which resources associated with dead biomass are flushed from the system. Resource and substrate dynamics follow
\(\frac{dR_{E}}{\text{dt}}=\text{gES}-qE_{E}rN_{E}-dR_{E}+dR_{C}\)Eq. (3a)
\(\frac{dR_{C}}{\text{dt}}=-dR_{C}+dR_{E}-\text{qr}R_{C}N_{C}\)Eq. (3b)
\(\frac{\text{dS}}{\text{dt}}=a\left(S_{0}-S\right)-\text{gES}\)Eq. (3c)
where g describes the rate at which the enzyme breaks down the substrate, q describes the quantity of resource needed to produce a biomass unit, d is the diffusion rate governing movement of resource from the region around the enzyme producer to the region around the cheater, a describes the rate at which new substrate enters the system, and S 0 is the maximal substrate concentration in the absence of enzyme producers.
At equilibrium, substrate concentration and resource concentrations in the vicinity of the enzyme producer and the cheater can be calculated as
\(R_{E}^{*}=\frac{m\ +\ e}{r}r\text{\ \ \ \ \ \ \ }\text{if}\text{\ \ \ \ \ \ }N_{E}^{*}>0,\ \ \ \ \ \ \text{else}\text{\ \ \ \ \ \ }R_{E}^{*}=0\)Eq. (4a)
\(R_{C}^{*}=\frac{m}{r}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{if}\text{\ \ \ \ \ \ }N_{C}^{*}>0,\ \ \ \ \ \ \text{else}\text{\ \ \ \ \ \ }R_{C}^{*}=R_{E}^{*}\)Eq. (4b)
\(S^{*}=\frac{aS_{0}}{mz}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ }\text{if}\text{\ \ \ \ \ \ }N_{E}^{*}>0,\ \ \ \ \ \ \text{else}\text{\ \ \ \ \ \ }S^{*}=S_{0}\)Eq. (4c)
Because of the energetic cost of enzyme production, equilibrial resource concentrations are always higher in the vicinity of the enzyme producer. Note that these concentrations simultaneously describe the equilibrial resource concentration, and the minimum resource requirement needed for positive equilibrial abundance (Tilman, 1982), meaning that the cheater is effectively a “better” competitor than the enzyme producer, and, in the absence of the spatial gradient, would be able to drive it to extinction. Thus, if the diffusion rate is too high, the enzyme producer will ultimately be driven extinct, and the system will collapse.
Because enzyme and resource concentrations are primarily controlled by the enzyme producer, both the equilibrial enzyme concentration and the equilibrial abundance of the cheater can be described with relatively simple functions, based on the demographic rates and equilibrial concentration of the enzyme producer
\(N_{C}^{*}=\frac{\text{de}}{\text{mqr}}\) Eq. (5a)
\(E^{*}=\frac{eq_{z}N_{E}^{*}}{m_{z}}\) Eq. (5b)
In contrast, the equilibrial abundance of the enzyme producer, although analytically tractable, follows a more complex set of functions, which include a monoculture equilibrium value (i.e., whenNC*  = 0), a stable coexistence point where bothNE* andNC*  > 0, and an Allee point. If the population size of the enzyme producer falls below this Allee point and is held there long enough for resource and enzyme concentrations to equilibrate, the enzyme producer will ultimately be drawn towards extinction, because it is unable to counteract the diffusion gradient of resources being drawn towards the cheater. The full formulas for these three equilibria are available in the supplement (Table S1).
For the purpose of examining the model under realistic empirical conditions, we use parameter values reported in empirical studies. We use growth and mortality rates of Escherichia coli , a model organism in experimental microbiology and biotechnology applications. As a substrate, we identified cellulose as an appropriate carbon source. Cellulose requires specialised extracellular enzymes, cellulases, to be broken down into monosaccharides like glucose, which can be used by most bacterial species (see Table 1 for parameter values and relevant literature). For parameter qz , a conversion term describing the relative energy cost of enzyme production vs biomass production, we chose units for e such that the costs to growth are equal to enzyme production rate (i.e.,qz  = 1). Note that this choice does not change the generality of our results (other than complicating the interpretation of the units in which e is measured), but rather, facilitates model testing by reducing the dimensionality of our parameter space.
Parameters e and d are the primary variables of interest and therefore we allow them to vary in our model, in order to find ESSs that allow for coexistence. When diffusion rate = 0, all resources are privatised by the producer, causing complete exclusion of the cheater. As d approaches infinity it is assumed that diffusion increases (e.g., producers and cheaters are mixed in a very dense culture). Similarly, as we will show, varying e , the cost of enzyme production, in our model reveals maximum and minimum (non-zero) coexistence thresholds that influence both the producer monoculture and the producer-cheater mixture.