Stability of the chemostat system including a linear coupling between species

Abstract

<p style='text-indent:20px;'>In this paper, we consider a resource-consumer model taking into account a linear coupling between species (with constant rate). The corresponding operator is proportional to a discretization of the Laplacian in such a way that the resulting dynamical system can be viewed as a regular perturbation of the classical chemostat system. We prove the existence of a unique locally asymptotically stable steady-state for every value of the transfer-rate and every value of the dilution rate not exceeding a critical value. In addition, we give an expansion of the steady-state in terms of the transfer-rate and we prove a uniform persistence property of the dynamics related to each species. Finally, we show that this equilibrium is globally asymptotically stable for every value of the transfer-rate provided that the dilution rate is with small enough values.</p>

Stabilization of the chemostat system with mutations and application to microbial production

“…where D s ∈ R n×n is the diagonal matrix D s ∶= diag(𝜇 1 (s), … , 𝜇 n (s)) and u is the Perron root of the quasi-positive irreducible matrix § A 𝜀,s ∶= D s + 𝜀M ∈ R n×n , Note that the first equality in (7) defines the vector x as the Perron vector of A 𝜀,𝜎 up to a positive multiplicative constant, that is, x = 𝜅a 𝜀,𝜎 for some 𝜅 > 0 where a 𝜀,𝜎 is the unitary Perron vector of A 𝜀,𝜎 . 17 The second equality in (7) allows to uniquely define the eigenvector vector x as x = 1 𝜅 a 𝜀,𝜎 where 𝜅 ∶= a 𝜀,𝜎 ⋅ 1, 1 ∶= (1, … , 1) ∈ R n , and ⋅ denotes the standard inner product of R n .…”

“…• In Section 3.1, we apply the PMP on (17) which allows us to derive properties on optimal controls.…”

“…[12][13][14][15][16] The main objective of this article is to study the stabilization of the chemostat system whenever mutations for microorganisms are incorporated. [17][18][19] This means that, in addition to the classical equations of the chemostat, each species is able to convert into neighbor ones. It is relevant whenever the number of species is high in order to take into account evolution.…”

“…It is relevant whenever the number of species is high in order to take into account evolution. 17,20,21 Such a phenomenon leading to a coupling term between species may be related to transposable elements, gene transfer, 22,23 or mutations (among other possible factors). In the sequel, we employ the wording "mutations" to describe the coupling between species that is considered throughout this work.…”

“…We end-up this section by relating the desired steady-state (defined thanks to a given nominal value of the substrate) to the coexistence equilibrium when considering a constant dilution rate. 17 We also discuss global stability properties when Haldane's kinetics are considered in place of Monod functions. In Section 3, we introduce the optimal control problem related to the maximization of species and we first derive properties on optimal controls using the PMP.…”