This model explores how cooperation, exploitation and privatization coexist in a microbial community built around a single shared carbon source, sucrose. Three metabolic strategies compete for it: the public metabolizer secretes invertase, hydrolysing sucrose into glucose that becomes freely available to the whole community; the cheater takes up that glucose without paying the metabolic cost of producing the enzyme; and the private metabolizer imports and digests sucrose intracellularly, avoiding both the cost of sharing and the risk of exploitation.

Population growth follows a generalized Lotka-Volterra model with density-dependent self-limitation for each strategy, an exploitation term through which cheaters benefit at the public metabolizer's expense, and a competition term between public and private metabolizers for the shared sucrose pool. Because the same right-hand sides can be read as per-capita growth rates, the model doubles as a three-strategy evolutionary game: each strategy's fitness is exactly its payoff, and the interior equilibrium where all three coexist corresponds to the point where all three payoffs are equal.

The public metabolizer needs a growth-rate advantage over the private metabolizer to sustain the community: this model's analysis finds stable three-way coexistence only for a narrow window of that ratio (roughly 1.2-5x), with the most robust balance around 1.2-2x. Outside that window the system collapses to a boundary state, private-only dominance if the public metabolizer grows too slowly, or loss of the private metabolizer if it grows too fast relative to the public strategy.

\begin{align*}
  \frac{d \text{Public}}{dt} &= r_P \cdot \text{Public} - \alpha \cdot \text{Public} \cdot \text{Cheater} - \beta \cdot \text{Public} \cdot \text{Private} - \eta \cdot \text{Public} \cdot \text{Public} \\ 
  \frac{d \text{Cheater}}{dt} &= \alpha \cdot \text{Public} \cdot \text{Cheater} - \nu \cdot \text{Cheater} \cdot \text{Cheater} \\ 
  \frac{d \text{Private}}{dt} &= r_M \cdot \text{Private} - \beta \cdot \text{Public} \cdot \text{Private} - \gamma \cdot \text{Private} \cdot \text{Private}
\end{align*}