Article
A competitiondiffusion system approximation to the classical twophase Stefan problem
Japan Journal of Industrial and Applied Mathematics
(Impact Factor: 0.27).
06/2001;
18(2):161180.
DOI: 10.1007/BF03168569

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ABSTRACT: In this paper we study two kinds of free boundary problems for the diffusive prey–predator model over a one dimensional habitat, in which the free boundary represents the spreading front and is only caused by the prey. In these two problems, it is assumed that the species can only invade further into the new environment from the right end of the habitat, and the spreading front expands at a speed that is proportional to the prey’s population gradient at the front. Asymptotic behaviors of solution as are discussed firstly. The acquired results can help us to comprehend whether the expanding is successful or not. Then we establish the criteria for spreading and vanishing.Nonlinear Analysis Real World Applications 08/2015; 24:7382. DOI:10.1016/j.nonrwa.2015.01.004 · 2.34 Impact Factor 
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ABSTRACT: We consider FisherKPP equation with advection: $u_t=u_{xx}\beta u_x+f(u)$ for $x\in (g(t),h(t))$, where $g(t)$ and $h(t)$ are two free boundaries satisfying Stefan conditions. This equation is used to describe the population dynamics in advective environments. We study the influence of the advection coefficient $\beta$ on the long time behavior of the solutions. We find two parameters $c_0$ and $\beta^*$ with $\beta^*>c_0>0$ which play key roles in the dynamics, here $c_0$ is the minimal speed of the traveling waves of FisherKPP equation. More precisely, by studying a family of the initial data $\{ \sigma \phi \}_{\sigma >0}$ (where $\phi$ is some compactly supported positive function), we show that, (1) in case $\beta\in (0,c_0)$, there exists $\sigma^*\geqslant0$ such that spreading happens when $\sigma > \sigma^*$ and vanishing happens when $\sigma \in (0,\sigma^*]$; (2) in case $\beta\in (c_0,\beta^*)$, there exists $\sigma^*>0$ such that virtual spreading happens when $\sigma>\sigma^*$ (i.e., $u(t,\cdot;\sigma \phi)\to 0$ locally uniformly in $[g(t),\infty)$ and $u(t,\cdot + ct;\sigma \phi )\to 1$ locally uniformly in $\R$ for some $c>\beta c_0$), vanishing happens when $\sigma\in (0,\sigma^*)$, and in the transition case $\sigma=\sigma^*$, $u(t, \cdot+o(t);\sigma \phi)\to V^*(\cdot(\betac_0)t )$ uniformly, the latter is a traveling wave with a "big head" near the free boundary $x=(\betac_0)t$ and with an infinite long "tail" on the left; (3) in case $\beta = c_0$, there exists $\sigma^*>0$ such that virtual spreading happens when $\sigma > \sigma^*$ and $u(t,\cdot;\sigma \phi)\to 0$ uniformly in $[g(t),h(t)]$ when $\sigma \in (0,\sigma^*]$; (4) in case $\beta\geqslant \beta^*$, vanishing happens for any solution. 
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ABSTRACT: This paper deals with the diffusive LotkaVolterra type preypredator model with a free boundary over a one dimensional habitat. This problem may be used to describe the interaction between indigenous species and invasive species and the spreading of such two species, with the free boundary representing the expanding front. Our main purpose is to study the spreading and vanishing phenomena and long time behaviors of prey and predator.Communications in Nonlinear Science and Numerical Simulation 06/2015; 23(13):311327. DOI:10.1016/j.cnsns.2014.11.016 · 2.57 Impact Factor
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