On a parabolic‐elliptic chemotaxis system with periodic asymptotic behavior

Abstract

We study a parabolic‐elliptic chemotactic PDEs system, which describes the evolution of a biological population “u” and a chemical substance “v” in a bounded domain normalΩ⊂Rn. We consider a growth term of logistic type in the equation of “u” in the form μu(1 − u + f(t,x)). The function “f,” describing the resources of the systems, presents a periodic asymptotic behavior in the sense limt→∞supx∈Ω|f(x,t)−f∗(t)|=0, where f ∗ is independent of x and periodic in time. We study the global existence of solutions and its asymptotic behavior. Under suitable assumptions on the initial data and f ∗, if the constant chemotactic sensitivity χ satisfies χ<μ/2, we obtain that the solution of the system converges to a homogeneous in space and periodic in time function.

On the numerical solution to a parabolic-elliptic system with chemotactic and periodic terms using Generalized Finite Differences

“…where "σ(N (x, t) − U )" describes the growth rate of "U " and N (x, t) = 1 + f (x, t) is the carrying capacity of the system. Function "f " considered in [17] and in the present paper, describing the resources of the systems, presents a periodic asymptotic behavior in the sense…”

“…It is natural that these resources depend on time and present some kind of periodicity caused, for instance, by the seasons of the year and consequently the population presents some seasonal behavior. The motivation of including such term can be found in [17] where it was proved the global existence of solutions and its asymptotic behavior produced by the logistic growth which counteracts the blow-up tendency produced by chemotaxis. Under suitable assumptions on the initial data and g, if the constant chemotactic sensitivity η satisfies η < σ/2 the authors obtained that the solution of the system converges to a homogeneous in space and periodic in time function.…”

“…Under suitable assumptions on the initial data and g, if the constant chemotactic sensitivity η satisfies η < σ/2 the authors obtained that the solution of the system converges to a homogeneous in space and periodic in time function. Under certain assumptions,( see Hypotheses (1.5)- (1.11) in [17]), it is expected that the solution of the system converges to a periodic function function depending on σ and g(t) which can be explicitly computed.…”

“…For instance, in [20] the authors proved that all solutions of the non-stationary system approach the steady state (1, 1) for large times. Finally, we consider a function f (x, t) fulfilling the assumptions made in [17] with the periodic asymptotic behavior stated in (2). We present numerical examples of all three cases in Section 3, for both regular and irregular domains.…”

Solving a reaction–diffusion system with chemotaxis and non-local terms using Generalized Finite Difference Method. Study of the convergence

“…In the present paper we shall use matrix arguments. The differences between the parabolic-elliptic system and the parabolic-parabolic one are also significant in the analytical proof of the continuous models, as can be seen in [7] and [8]. Furthermore, the novelty of this paper is the introduction and discretization of non-local terms, which is a challenging task and a problem of growing interest.…”