Stabilization in a two-species chemotaxis system with a logistic source

2013

Nonlinearity

In this paper, we study a system of partial differential equations describing the evolution of a population under chemotactic effects with non-local reaction terms. We consider an external application of chemoattractant in the system and study the cases of one and two populations in competition. By introducing global competitive/cooperative factors in terms of the total mass of the populations, we obtain, for a range of parameters, that any solution with positive and bounded initial data converges to a spatially homogeneous state with positive components. The proofs rely on the maximum principle for spatially homogeneous sub-and super-solutions.

Section: A Priori Estimates and Comparison Methodsmentioning

confidence: 61%

“…Similar comparison arguments have been used in the context of chemotaxis in other papers, see for instance [26,28] or [23]. In [23], the method is applied to a case of non-constant chemotaxis coefficient, defined by x(w) = xo(N -u), which becomes negative for a large concentration of u.…”

Section: F{u) = U + Fmentioning

confidence: 99%

On a competitive system under chemotactic effects with non-local terms

2013

Nonlinearity

“…The Parabolic-Parabolic-Elliptic cases with competitive terms, Le., when there exists an explicit interaction between the species, have also been studied in the last years by different authors. In Tello and Winkler [20] the Parabolic-Parabolic-Elliptic problem is analyzed under "weak" competitive assumptions, which drive it to positive homogeneous steady states. A more general case is considered in Negreanu and Tello [18] where both species persist in time.…”

Section: Du Du Dwmentioning

confidence: 99%

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

Journal of Differential Equations

2015

We study the behavior of two biological populations "w" 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 satisfles an ODE, more precisely, , t >0, w¡ = h(u, v, w), x e Q, f > 0, under appropriate boundary and initial conditions in an «-dimensional open and bounded domain Q. We consider the cases of positive chemo-sensitivities, not necessarily constant elements. The chemical production function h increases as the concentration of the species "w" and ' V increases. We flrst 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, Xi an d the size of fj,¡. Finally, some examples of the theoretical results are presented for particular functions h and xi •

“…We now follow the proof in [10], and thanks to (2.13) We replace s by <f>(s) in the integral terms of the system and once we prove the lemma, we may eliminate <f>. We consider the following functions…”

Section: Mathematical Analysismentioning

confidence: 99%

Asymptotic stability of a mathematical model of cell population

Journal of Mathematical Analysis and Applications

2014

We consider a simplified system of a growing colony of cells described as a free boundary problem. The system consists of two hyperbolic equations of first order coupled to an ODE to describe the behavior of the boundary. The system for cell populations includes non-local terms of integral type in the coefficients. By introducing a comparison with solutions of an ODE's system, we show that there exists a unique homogeneous steady state which is globally asymptotically stable for a range of parameters under the assumption of radially symmetric initial data.