Competition for one nutrient with internal storage and toxin mortality

This study presents a mathematical model of two species competing in a chemostat for one resource that is stored internally, and who also compete through allelopathy. Each species produces a toxin to that increases mortality rate of its competitor. The two species system and its single species subsystem follow mass conservation constraints characteristic of chemostat models. Persistence of a single species occurs if the nutrient supply of an empty habitat allows it to acquire a threshold of stored nutrient quota, sufficient to overcome loss to outflow after accounting for the cost of toxin production. For the two-species system, a semitrivial equilibrium with one species resident is unstable to invasion by the missing species according to a similar threshold condition. The invader increases if acquires a stored nutrient quota sufficient to overcome loss to outflow and toxin-induced mortality, after accounting for the cost of the invader's own toxin production. If both semitrivial equilibria for the two-species system are invasible then there is at least one coexistence equilibrium. Numerical analyses indicate another possibility: bistability in which both semitrivial equilibria are stable against invasion. In such a case there is competitive exclusion of one species, whose identity depends on initial conditions. When there is a tradeoff between abilities to compete for the nutrient and to compete through toxicity, the more toxic species can dominate only under nutrient-rich conditions. Bistability under such conditions could contribute to the unpredictability of toxic algal blooms.