Mathematical Models Involving Multiple Resource Limitation

Abstract

<p>A model of the chemostat involving two populations of microorganisms competing for two perfectly substitutable resources is developed and analyzed. A general class of functions is used to describe nutrient uptake, one which allows for the effect that the concentration of each resource has on the amount of the other resource consumed. The model significantly generalizes those previously studied. The dynamics of this model are then compared with the dynamics of the classical growth and two-species competition models, as well as models involving two perfectly complementary resources.</p> <p>It is not surprising that the above competition for two resources can result in the coexistence of the two competitor populations. However, an example is also given in which the extinction of one population is averted by the introduction of its competitor. Thus, exploitation of common resources promotes diversity in some circumstances. This situation is investigated further and a more general description is given.</p> <p>A model of single-species growth on two resources is then presented. For a given dilution rate, the medium in the growth vessel is enriched by increasing the input concentration of one of the resources. Enrichment is considered beneficial if the carrying capacity of the environment is increased. Analytic methods are used to determine the effects of enrichment on the asymptotic behaviour of the model for different dilution rates. The existence of a threshold value for the dilution rate is established. For dilution rates below the threshold, enrichment is beneficial, regardless of which resource is used to enrich the environment. When the dilution rate is increased beyond the threshold, it becomes important to consider which resource is used to enrich the environment. For one of the resources it is shown that, while moderate enrichment can be beneficial, sufficient enrichment leads to the extinction of the population. For the other resource, enrichment leads from washout or initial condition dependent outcomes to survival, and is thus beneficial.</p> <p>The growth model is then extended to include a single predator population. Using the threshold value for the dilution rate established in the growth model, the effects of enrichment on the asymptotic behaviour of the resultant predator-prey model are investigated. Here, enrichment is considered beneficial if it can lead from washout for some positive initial conditions to survival of both species for any positive initial conditions. For dilution rates below the threshold, enrichment is beneficial, regardless of which resource is used for enrichment. As in the growth model, it becomes important to consider which resource is used to enrich the environment when the dilution rate is above the threshold. For one of the resources, moderate enrichment can be beneficial, while sufficient enrichment leads to a regime in which washout is possible. For the other resource, sufficient enrichment is beneficial.</p>