Narrow Results

In this paper, the asymptotical behavior of a chemostat model for E. coli and the virulent phage T4 is analyzed. The basic reproduction number R₀ is proved to be a threshold which determines the outcome of the virulent phage T4. If R₀ < 1, the virus dies out; if R₀ > 1, the virus persists. Sufficient conditions for the Hopf bifurcation are also established. The theoretical results show that increasing the input of nutrient will result in an increase in the equilibrium population density of the virulent bacteriophage T4, but will have no effect on the equilibrium population density of E. coli. The results also show that increasing the input of nutrient or increasing the average lytic time for the infected E. coli can destabilize the interaction between E. coli and T4.

We consider a chemostat model of phytoplankton competing for nitrogen taking into account effects of both intra- and interspecific crowding and the light limitation. We consider crowding as an additive density-dependent mortality rate. Crowding effects may be classified into intra- and interspecific crowding depending on whether the additional mortality is caused by the same or alternate species.

We analyze the existence and local and global stability of single species and coexistence equilibria using the linearization and stability method of Lyapunov. We present a numerical example illustrating the fact that the crowding effects may lead to the bistable coexistence of two phytoplankton species. We demonstrate that the crowding effects and the light limitation affect the outcome of exploitative competition for a single resource and promote coexistence. We also show that while the crowding has a stabilizing effect on phytoplankton community, the light limitation may destabilize the system and produce sustained oscillations.

A mathematical model of competition between plasmid-bearing and plasmid-free organisms for a single limiting resource in a chemostat with distinct removal rates and in the presence of an external inhibitor is analyzed. This model was previously introduced in the special case where the growth rate functions and the absorption rate of the inhibitor follow the Monod kinetics and the removal rates are the same as the dilution rate. Here, we consider the general case of monotonic growth and absorption functions, and distinct removal rates. Through the three operating parameters of the model, represented by the dilution rate, the input concentrations of the substrate and the inhibitor, we give necessary and sufficient conditions for existence and stability of all equilibria. To better understand the richness of the model’s behavior with respect to those operating parameters, we determine the operating diagram theoretically and numerically. This diagram is very useful to understand the model from both the mathematical and biological points of view.