1 Introduction

1 Introduction

Most practical or real world reaction-diffusion problems involve two interacting components (see [?, ?]). A typical model is the following Lotka-Volterra type competition system for two components

, are population densities at time at point ; is diffusion coefficients, is the coefficients of the specific rate of population growth, and are coefficients of interspecific competition, and are coefficients of intraspecific competition.
In this paper we consider a problem with a free boundary for a system of parabolic equations of the reaction-diffusion type describing a model of population dynamics in biology.
A free boundary problem for a reaction-diffusion system was first introduced by Z.Lin [?], where a prey-predator model was studied, and the existence results were provided. Other studies of reaction-diffusion models with free boundary can be found in [?, ?, ?, ?, ?, ?] and the references cited therein.
Statement of the problem. It is necessary to determine the functions , and satisfying the conditions
(1)
(2)
(3)
(4)
(5)
(6)
where ; is free (moving) boundary, that represents the propagation front is determined together with , functions; , – the initial population densities are in the region .
Throughout this paper, we will put forward the following assumptions:
, , , , , , are positive constants, ;
;
in ,
, , , .
Model (1)-(6) describes the growth of two populations which are both diffusing through and interacting with each other. and represent spatial densities and when is bounded the boundary condition (4) imply that the populations are confined to , i.e. there is no migration across the boundary of .
When , the problem (1)-(6) was investigated in the works [?, ?] and proved a spreading-vanishing dichotomy. In [?, ?], the authors studied the free boundary problem for a reaction-diffusion system with a linear convection term. They obtained a dichotomy result and presented a constant asymptotic spreading speed of the expanding front.
Many other theoretical results for general models with free boundary have been achieved in [?, ?, ?, ?] and references cited therein.
The rest of paper is organized as follows. First we establish two-sided bounds for , and , and then a Hölder norm bounds for , . In sections 3, 4 we prove uniqueness and existence results.