How can inter-specific interferences explain coexistence or confirm the competitive exclusion principle in a chemostat?

Abstract

In this paper, I consider two species feeding on limiting substrate in a chemostat taking into account some possible effects of each species on the other one. System of differential equations is proposed as model of these effects with general inter-specific density-dependent growth rates. Three cases were considered. The first one for a mutual inhibitory relationship where it is proved that at most one species can survive which confirms the competitive exclusion principle. Initial concentrations of species have great importance in determination of which species is the winner. The second one for a food web relationship where it is proved that under general assumptions on the dilution rate, both species persist for any initial conditions. Finally, a third case dealing with an obligate mutualistic relationship was discussed. It is proved that initial condition has a great importance in determination of persistence or extinction of both species.

Analysis of a Fractional-order “SVEIR” Epidemic Mo del with a General Nonlinear Saturated Incidence Rate in a Continuous Reactor

“…R 5 + , the closed non-negative cone in R 5 , is positively invariant by the system (2.6) [22,23,24,25,26,27,28,29,30,31,32,33]. More precisely, let m = min(mS, mV , mE, mI , mR), then I get Proposition 1.…”

Mathematical Analysis of a Fractional-order "SIR" Epidemic Model with a General Nonlinear Saturated Incidence Rate in a Chemostat

“…+ , the closed non-negative cone in R 3 , is positively invariant [22,23,24,25,4,26,27,28,29,30,31,32,33] for the system (2.6). More precisely, Proposition 1.…”

“…Now, if R = 1, then D α V2 = 0 if and only if S = Sin and the largest compact invariant set in {(S, I) ∈ Ω : D α V2 = 0} is the singleton {Ē}. Therefore, by the LaSalle's invariance principle (see, for instance, [34]), {Ē} is globally asymptotically stable (for other applications, see [26,27,28,31,33]).…”

Semi-analytical solution for a system of competition with production a toxin in a chemostat

“…The parameter k represents the fraction of potential growth devoted to producing the toxin. For more information on the mathematical modeling and analysis of subject, we can see for example the works of El Hajji et al [5][6][7]. Using a procedure of normalization including an operation of reduction of variables of problem (1.2) can lead to the reduced form [11] ds…”