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 d = 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.