Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. I. Asymptotic dynamics in fixed unbounded domain

Abstract

The current series of research papers is to investigate the asymptotic dynamics in logistic type chemotaxis models in one space dimension with a free boundary or an unbounded boundary. Such a model with a free boundary describes the spreading of a new or invasive species subject to the influence of some chemical substances in an environment with a free boundary representing the spreading front. In this first part of the series, we investigate the dynamical behaviors of logistic type chemotaxis models on the half line R + , which are formally corresponding limit systems of the free boundary problems. In the second of the series, we will establish the spreading-vanishing dichotomy in chemoattraction-repulsion systems with a free boundary as well as with double free boundaries.

Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary

“…where ν > 0 in (1) is a positive constant, a, b, χ i , λ i , and µ i (i = 1, 2) are nonnegative constants, and a(t, x) and b(t, x) satisfy the following assumption, Biological backgrounds of the chemoattraction-repulsion system are discussed in the paper ( [25]) and the free boundary backgrounds of ( 1) are discussed in the paper ( [1]). The free boundary condition in ( 1) is also derived in [1] based on the consideration of "population loss" at the front which assumes that the expansion of the spreading front is evolved in a way that the average population density loss near the front is kept at a certain preferred level of the species, and for each given species in a given homogeneous environment, this preferred density level is a positive constant determined by their specific social and biological needs, and the environment.…”

“…Especially, [34] first studied the global solvability, boundedness, blow-up, existence of non-trivial stationary solutions and asymptotic behaviors of the chemoattraction-repulsion system without the logistic term. [12] also have some important works related to the boundedness, blowup of system (1) with Neumann boundary conditions in a bounded domain. Among the fundamental problems in studying chemotaxis models are the existence of nonnegative solutions which are globally defined in time or blow up at a finite time and the asymptotic behavior of time global solutions.…”

Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. II. Spreading-vanishing dichotomy in a domain with a free boundary

“…where ν > 0 in (1.1) is a positive constant, and in both (1.1) and (1.2), χ i , λ i , and µ i (i = 1, 2) are nonnegative constants, and a(t, x) and b(t, x) satisfy the following assumption, Biological backgrounds of (1.1) and (1.2) are discussed in the first part of the series ( [2]). The free boundary condition in (1.1) is also derived in [2] based on the assumption that, as the expanding front propagates, the population suffers a loss of constant units per unit volume at the front, and that, near the propagating front, the population density is close to zero. Formally, (1.2) can be viewed as the limit system of (1.1) as h(t) → ∞.…”

“…The objective of this series is to investigate the asymptotic dynamics of (1.2) and the spreading and vanishing scenario in (1.1). In the first part of the series ( [2]), we studied the asymptotic dynamics of (1.2). In this second part of the series, we study the spreading and vanishing scenario in (1.1).…”

“…In this second part of the series, we study the spreading and vanishing scenario in (1.1). To state the main results of the current paper, we first recall some results proved in [2]. Let…”