Global Asymptotic Behavior of a Chemostat Model with Discrete Delays

Abstract

This paper studies the global asymptotic behavior of an exploitative competition model between n species in a chemostat. The model incorporates discrete time delays to describe the delay in the conversion of nutrient consumed to viable biomass and hence includes delays simultaneously in variables of nutrient and species concentrations. In the case where only two species are engaged in competition, it is shown that competitive exclusion holds for any monotone growth response functions. Sufficient conditions are also obtained for the model to exhibit competitive exclusion in the n-species case. In regard to the delay effects on the qualitative outcome of competition, it is demonstrated that when the delays are relatively small, the predictions of the model are identical with the predictions given by corresponding models without time delays. However, including large delays in the model may alter the predicted outcome of competition. The techniques used also work when different removal rates are permitted, and in this case there are even new results in the no-delay case.

Competitive Exclusion for Chemostat Equations with Variable Yields

“…The reader is referred to [24], for complements and details on the CEP, and to [16] for recent results and a discussion on competitive exclusion. Most of the results on the CEP for (4) and (7) have been based on Lyapunov functions [3,9,14,21,22,26,27] . For a survey of constructing Lyapunov functions in the chemostat, the reader is referred to [10].…”

“…Let us prove that (27), (27) and (27) hold. Since f 1 (S) = q 1 (S) − D 1 is increasing, the function f 1 (S) changes sign only at S = λ 1 and hence, (27) is satisfied. Since…”

“…It was often considered in the literature [4,14,26,27]. For this class of systems the main result in [21] is Corollary 3.1 (Theorem 2.1 in [21]).…”

“…Using (27) and (28), the first term of the above sum is non-positive for 0 < S < 1 and equals 0 if and only if S = λ 1 . Using (29), the second term is non-positive for 0 < S < 1 and equals 0 if and only if x i = 0 for i = 2 · · · N .…”

“…Notice that, as in Lemma 1.3, the condition (29) implies that λ 1 < λ i for all i ≥ 2. Indeed, if f i (S) > 0 for some S ≤ λ 1 , then f 1 (S) ≤ 0, so that the inequality (29) Proof Assume that (27), (28) and (29) hold. Notice that (27) is the same as (14) and (28) is the same as (15).…”