Analysis of an intra- and interspecific interference model with allelopathic competition

Abstract

In this paper, we propose an original model of two-microbial species competing for a single nutrient in the chemostat including general intra- and interspecific density-dependent growth rates with allelopathic interactions. Each species produces a toxin that affects the growth of other species as well as its own growth. The removal rates are distinct and include the specific death rate and the autotoxicity of each species. We establish an in-depth mathematical analysis by determining the multiplicity of all steady states of the three-dimensional system and their necessary and sufficient conditions of existence and local stability according to the operating parameters, which are the dilution rate and the inflowing concentration of the substrate. To describe the asymptotic behavior of the process according to these control parameters, we first determine theoretically the operating diagram. Using MATCONT software, these theoretical results are validated numerically but it reveals the cusp bifurcation that occurs by varying two parameters. The one-parameter bifurcation diagram in the dilution rate shows that there can be either transcritical or saddle-node bifurcations. Finally, we apply our results to a particular model in the literature without intra- and interspecific interference but with only allelopathic effects of the second species on the first species. We show that one of the coexistence steady states is locally exponentially stable when it exists, whereas in the literature they have not been able to demonstrate that the stability condition of this steady state is always fulfilled.