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

By Mihaela Negreanu, Ignacio Tello. 2015

100 citations recorded by the source corpus.

Abstract We study the behavior of two biological populations “u” and “v” attracted by the same chemical substance whose behavior is described in terms of second order parabolic equations. The model considers a logistic growth of the species and the interactions between them are relegated to the chemoattractant production. The system is completed with a third equation modeling the evolution of chemical. We assume that the chemical “w” is a non-diffusive substance and satisfies an ODE, more precisely, { u t = Δ u − ∇ ⋅ ( u χ 1 ( w ) ∇ w ) + μ 1 u ( 1 − u ) , x ∈ Ω , t > 0 , v t = Δ v − ∇ ⋅ ( v χ 2 ( w ) ∇ w ) + μ 2 v ( 1 − v ) , x ∈ Ω , t > 0 , w t = h ( u , v , w ) , x ∈ Ω , t > 0 , under appropriate boundary and initial conditions in an n-dimensional open and bounded domain Ω. We consider the cases of positive chemo-sensitivities, not necessarily constant elements. The chemical production function h increases as the concentration of the species “u” and “v” increases. We first study the global existence and uniform boundedness of the solutions by using an iterative approach. The asymptotic stability of the homogeneous steady state is a consequence of the growth of h, χ i and the size of μ i . Finally, some examples of the theoretical results are presented for particular functions h and χ i .

Subfields: Cell Biology, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/341f6642656026f7be59d222486a65d9be1a444a

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Citations in the corpus

On the weakly competitive case in a two-species chemotaxis model (2016, background): “Furthermore, in spatially one-dimensional domains more insight into qualitative behaviour of the system has been obtained in [45] and [46], where global existence of solutions was shown as well as existence of nonconstant steady states by bifurcation analysis.”
On a parabolic–elliptic system with gradient dependent chemotactic coefficient (2018, background): “The system has been also studied for several biological species, see for instance Tang and Tao [33], Tello and Winkler [36], Stinner, Tello and Winkler [32], Negreanu and Tello [27] and [28], Wang and Wu [37] among others.”