Article
A CompetitionDiffusion System Approximation to the Classical TwoPhase Stefan Problem
Japan Journal of Industrial and Applied Mathematics (Impact Factor: 0.32). 06/2001; 18(2):161180. DOI: 10.1007/BF03168569
ABSTRACT
A new type of competitiondiffusion system with a small parameter is proposed. By singular limit analysis, it is shown that
any solution of this system converges to the weak solution of the twophase Stefan problem with reaction terms. This result
exhibits the relation between an ecological population model and waterice solidification problems.
any solution of this system converges to the weak solution of the twophase Stefan problem with reaction terms. This result
exhibits the relation between an ecological population model and waterice solidification problems.

 "The study of Stefanlike problems arising in ecology over bounded domains can be traced back to the work of [16], who studied the population segregation patterns. See also [11]. In [15], the author considered a predator–prey model with the Stefantype condition over a bounded domain. "
Article: The minimal habitat size for spreading in a weak competition system with two free boundaries
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ABSTRACT: In this paper, we focus on the dynamics for a Lotka–Volterra type weak competition system with two free boundaries, where free boundaries which may intersect each other as time evolves are used to describe the spreading of two competing species, respectively. In the weak competition case, the dynamics of this model can be classified into four cases, which forms a spreading–vanishing quartering. The notion of the minimal habitat size for spreading is introduced to determine if species can always spread. Some sufficient conditions for spreading and vanishing are established. Also, when spreading occurs, some rough estimates for spreading speed and the longtime behavior of solutions are established. 
 "We assume that the free boundaries move according to onephase Stefan condition, which is a kind of free boundary conditions widely used in the study of melting of ice [38], wound healing [9], and population dynamics [7] [12] [13]. The derivation of onephase or twophase Stefan conditions in population models as singular limits of competitiondiffusion systems can be found in [26] [27] etc. When β = 0 (i.e., there is no advection in the environment), the qualitative properties of the problem (P ) was studied by Du and Lin [12] for logistic nonlinearity f (u) = u(1 − u). "
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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. 
 "Among others, they obtained the global existence, uniqueness, regularity and asymptotic behavior of solutions for the problem. Later [4] [5] [8] [12] studied similar strong competitive models. Recently Du and Lin [6] and Du and Lou [7] studied a free boundary problem, which is essentially the problem (1.6) in case v ≡ 0. They constructed some semiwaves to characterize the spreading of u which represents the density of a new species. "
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ABSTRACT: We study two systems of reaction diffusion equations with monostable or bistable type of nonlinearity and with free boundaries. These systems are used as multispecies competitive model. For twospecies models, we prove the existence of a traveling wave solution which consists of two semiwaves intersecting at the free boundary. For threespecies models, we also prove the existence of a traveling wave solution which, however, consists of two semiwaves and one compactly supported wave in between, each intersecting with its neighbor at the free boundary.
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