Content uploaded by Fengquan Li
Author content
All content in this area was uploaded by Fengquan Li on Dec 11, 2014
Content may be subject to copyright.
A preview of the PDF is not available
The diffusive competition problem with a free boundary in strong heterogeneous environment and weak heterogeneous environment
Abstract
In this paper, we consider the diffusive competition problem consisting of an
invasive species with density u and a native species with density v. We
assume that v undergoes diffusion and growth in , and u exists
initially in , but invades into the environment with spreading front
. To understand the effect of the dispersal rate , the initial
occupying habitat , the initial density of invasive species
(u), and the parameter (the ratio of the invasion speed of the free
boundary and the invasive species gradient at the expanding front) on the
dynamics of this free boundary problem, we divide the heterogeneous environment
into two cases: strong heterogeneous environment and weak heterogeneous
environment. A spreading-vanishing dichotomy is obtained and some sufficient
conditions for the invasive species spreading and vanishing is provided both in
the strong heterogenous environment and weak heterogenous environment.
Moreover, when spreading of u happens, some rough estimates of the spreading
speed are also given.
ResearchGate has not been able to resolve any citations for this publication.
We discuss a free boundary problem for a general reaction-diffusion equation in a one-dimensional interval with Dirichlet conditions on both fixed and free boundaries. The problem models the spreading of an invasive or new species in which the free boundary represents a spreading front of the species and is described by Stefan-like condition. The main purpose of this paper is to study the asymptotic behavior of solutions as t→∞. This helps us to understand whether the spreading is successful or not. It will be shown that the asymptotic properties are closely related to positive solutions for the corresponding elliptic problems. Moreover, we will apply general results to the free boundary problems for diffusion equations with logistic or bistable nonlinearity.
In this paper we investigate two free boundary problems for a Lotka-Volterra
type competition model in one space dimension. The main objective is to
understand the asymptotic behavior of the two competing species spreading via a
free boundary. We prove a spreading-vanishing dichotomy, namely the two species
either successfully spread to the entire space as time t goes to infinity and
survive in the new environment, or they fail to establish and die out in the
long run. The long time behavior of the solutions and criteria for spreading
and vanishing are also obtained. This paper is an improvement and extension of
J. Guo and C. Wu.
In fact, we can only prove this estimate under the further requirement . The original proof of (2.3) is not strict because the coefficient function tends to zero if . The last part of Theorem 2.1 of [1] should be revised and so this theorem should be rewritten as:
This paper concerns a diffusive logistic equation with a free boundary and seasonal succession, which is formulated to investigate the spreading of a new or invasive species, where the free boundary represents the expanding front and the time periodicity accounts for the effect of the bad and good seasons. The condition to determine whether the species spatially spreads to infinity or vanishes at a finite space interval is derived, and when the spreading happens, the asymptotic spreading speed of the species is also given. The obtained results reveal the effect of seasonal succession on the dynamical behavior of the spreading of the single species.
We study the diffusive logistic equation with a free boundary in time-periodic environment. Such a model may be used to describe the spreading of a new or invasive species, with the free boundary representing the expanding front. For time independent environment, in the cases of one space dimension, and higher space dimensions with radial symmetry, this free boundary problem has been studied in [Y.-H. Du and Z.-G. Lin, SIAM J. Math. Anal. 42, No. 1, 377–405 (2010; Zbl 1219.35373); erratum ibid. 45, No. 3, 1995–1996 (2013; Zbl 1275.35156)], [Y.-H. Du and Z.-M. Guo, J. Differ. Equations 250, No. 12, 4336–4366 (2011; Zbl 1222.35096)]. In both cases, a spreading-vanishing dichotomy was established, and when spreading occurs, the asymptotic spreading speed was determined. In this paper, we show that the spreading-vanishing dichotomy is retained in time-periodic environment, and we also determine the spreading speed. The former is achieved by further developing the earlier techniques, and the latter is proved by introducing new ideas and methods.
We study nonlinear diffusion problems of the form ut = uxx + f(u) with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundaries representing the expanding fronts. For monostable, bistable, and combustion types of nonlinearities, Du and Lou ["Spreading and vanishing in nonlinear diffusion problems with free boundaries," J. Eur. Math. Soc. (JEMS), to appear] obtained a rather complete description of the long-time dynamical behavior of the problem and revealed sharp transition phenomena between spreading (limt→∞u(t, x) = 1) and vanishing (limt→∞ u(t, x) = 0). They also determined the asymptotic spreading speed of the fronts by making use of semiwaves when spreading happens. In this paper, we give a much sharper estimate for the spreading speed of the fronts than that in the above-mentioned work of Du and Lou, and we describe how the solution approaches the semiwave when spreading happens.
This short paper concerns a diffusive logistic equation with a free boundary and sign-changing coefficient, which is formulated to study the spread of an invasive species, where the free boundary represents the expanding front. A spreading–vanishing dichotomy is derived, namely the species either successfully spreads to the right-half-space as time and survives (persists) in the new environment, or it fails to establish itself and will extinct in the long run. The sharp criteria for spreading and vanishing are also obtained. When spreading happens, we estimate the asymptotic spreading speed of the free boundary.
We investigate the spreading speed and spreading-vanishing dichotomy
determined by a free boundary model introduced in Du and Lin.
This model gives an alternative approach to the study of
spreading of invasive species, which is usually studied through the
traveling wave solution approach. We firstly correct some mistakes
regarding the range of spreading speed in Du and Lin, which reveals
an interesting relationship between the spreading speeds determined
by the two different approaches. Then we use numerical analysis to
determine the parameter ranges for which a simple approximate
formula for the spreading speed holds. Finally we examine the
threshold parameter range for the spreading-vanishing dichotomy and
the pattern of the spreading radius curve with different initial
data and in various heterogeneous environment.