Competitive exclusion in a two-species chemotaxis model

Abstract

We consider a mathematical model for the spatio-temporal evolution of two biological species in a competitive situation. Besides diffusing, both species move toward higher concentrations of a chemical substance which is produced by themselves. The resulting system consists of two parabolic equations with Lotka-Volterra-type kinetic terms and chemotactic cross-diffusion, along with an elliptic equation describing the behavior of the chemical. We study the question in how far the phenomenon of competitive exclusion occurs in such a context. We identify parameter regimes for which indeed one of the species dies out asymptotically, whereas the other reaches its carrying capacity in the large time limit.

Fortune favours the brave: Movement responses shape demographic dynamics in strongly competing populations

“…Our model is based on the Lotka-Volterra competition model (Lotka, 1932;Volterra, 1926) (Painter & Sherratt, 2003;Horstmann, 2011;Stinner et al, 2014), usually in 144 the context of cell biology, and have some resemblance to our model.…”

On a comparison method to reaction-diffusion systems and its applications to chemotaxis

“…In other words, under assumptions (34), we have that all solutions (u(t), v(t)) and (u(t), v(t)) of (29), with both pairs (u(t 0 ), v(t 0 )) and (u(t 0 ), v(t 0 )) positive, ultimately approach the equilibrium solution (33). Therefore, applying (14) and Lemma 3.1 we conclude…”

“…Comparison methods based on upper and lower solutions have been applied to a large number of reaction-diffusion systems as the extensive literature shows (see for instance [10] and reference therein). Comparison methods based on ODE s systems have already used in the last decades, see for instance [7], [15] and [8] for chemotactic systems of two PDE s and extended to parabolic-parabolic-elliptic chemotactic systems in [16], [9] and [14]. We study a reaction-diffusion parabolic system "weakly coupled" (i.e.…”

Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant

“…A more general case is considered in Negreanu and Tello [18] where both species persist in time. "Strong competition assumptions", Le., when the competitive parameters between the species are "large", drive the system to the extinction of the weaker species, see Stinner, Tello and Winkler [22] for details. Parabolic-Parabolic-Elliptic models with external application of chemoattractant have also been considered in Negreanu and Tello [19] where the stability of the solutions is described for a large range of parameters.…”